ASHRAE HANDBOOK design and installation, Traducciones de Tecnología Industrial

¡Descarga ASHRAE HANDBOOK design and installation y más Traducciones en PDF de Tecnología Industrial solo en Docsity! E F10 e E2] + E22 * E2Z3 . F24 e F25 + E26 a + F28 *E29 1997 Fundamentals Contributors and Preface Thermodynamics and Refrigeration Cycles Fluid Flow Heat Transfer Two-Phase Flow Mass Transter Psychrometrics Sound and Vibration Thermal Comfort Indoor Environmental Health Environ. Control for Animals and Plants Sorbents and Desiccants Insulated Assemblies-Fundamentals Insulated Assemblies-Applications Thermal and Water Vapor Trans. Data Ventilation and Infiltration Climatic Design Information Residential Load Calculations Nonresidential Load Calculations Fenestration 1 Factors in Drying and Storing Farm Crops 2 Alr Contaminants 3 Odors 4 Measurement and Instruments F19 Airflow Around Buildings 6 Energy Resources 7 Combustion and Fuels 9 9 Retfrigerants Thermophysical Properties of Refrigerant F20 Secondary Coolants (Brines) F30 Energy Estimating and Modeling Methods F31 Space Air Diffusion F32 Duct Design F33 Pipe Sizing F34 Abbreviations and Symbols F39 Units and Conversions F36 Physical Properties of Materials FS/ Fundamentals of Control F38 Codes and Standards 1997 ASHRAE Handbook Preface The Fundamentals Handbook covers basic principles and includes data for the entire technology of the HVAC&R industry. Although design data and information change little over time, research sponsored by ASHRAE and others continues to generate new information for the ASHRAE Handbooks. In addition, the technical committees that prepare the chapters strive not only to provide new information, but also to clarify existing information, delete obsolete material, and reorganize chapters to make the infor- mation more understandable and easier to use. In this 1997 ASHRAE Handbook the following changes and additions are worth noting. • Chapter 1, Thermodynamics and Refrigeration Cycles, has an improved style of analysis and method of calculating thermody- namic properties. The chapter also discusses zeotropic refrigerant mixtures and includes numerical examples to show how the sec- ond law of thermodynamics can be applied to actual refrigeration cycles. The information on absorption refrigeration cycles has been clarified, and example analyses of various absorption cycles are included. • Chapter 6, Psychrometrics, now includes equations for calculat- ing standard pressure when elevation or temperature is known. • Chapter 8, Thermal Comfort, includes more information on ther- moregulation to help in understanding the physiology underlying comfort and its relation to the thermal environment. New material about the effects of clothing insulation and the perception of draft discomfort is also included. • Chapter 9, Indoor Environmental Health, has been substantially rewritten. It now includes new information that describes the var- ious health sciences, summarizes diseases associated with the indoor environment, compares pertinent indoor air quality stan- dards, and introduces the principles of industrial hygiene. • Chapter 14, Measurement and Instruments, includes new sections on carbon dioxide measurement and data logging devices. • Chapter 15, Airflow Around Buildings, presents simplified meth- ods for estimating the effect of changes in terrain on wind speed profiles. A field-validated model has led to a new procedure for calculating rooftop exhaust stack height. The new model pro- duces stack heights that are about two-thirds as high as the 1993 ASHRAE Handbook requirement. • Chapter 16, Energy Resources, has been moved here from the 1995 ASHRAE Handbook—Applications. Basic energy data have been updated. • Chapter 17, Combustion and Fuels, now includes information on NOx emissions from uncontrolled fuel-burning equipment. Meth- ods for reducing NOx are also discussed. • Chapter 19, Thermophysical Properties of Refrigerants, now includes data for the zeotropic blends R-404A, R-404C, and R-410A, and the azeotropic blend R-507A. Most of the CFC refrigerants have been retained to assist in making comparisons. Revised formulations have been used for most of the hydrocarbon refrigerants and the cryogenic fluids. • Chapters 22 and 23, Thermal and Moisture Control in Insulated Assemblies, contain more information on moisture transport and control. The effects of moisture on the building and its occupants are discussed in more detail. New recommendations and con- struction details for moisture control in three types of climates and in attics, roofs, and crawl spaces are included. • Chapter 25, Ventilation and Infiltration, now has information on nonresidential ventilation, infiltration degree-days, air change effectiveness, and age of air. • Chapter 26, Climatic Design Information, has been substantially expanded. The chapter includes new heating, cooling, dehumidi- fication, and wind design conditions for 1442 locations. • Chapter 29, Fenestration, includes new models for calculating heat transfer in glazing cavities. The solar heat gain section has been rewritten. New sections on condensation resistance, com- plex shading systems, annual energy performance, and durability have been added. • Chapter 30, Energy Estimating and Modeling Methods, has been substantially rewritten. The chapter provides an overview of the various methods available for estimating energy use. A sample heat balance calculation is included. • Chapter 32, Duct Design, has additional information on the ther- mal gravity (stack) effect and on duct system leakage. • Chapter 33, Pipe Sizing, includes a new section on steam conden- sate systems. • Chapter 37, Fundamentals of Control, has been taken from Chapter 42 in the 1995 ASHRAE Handbook—Applications. This move divides the information on controls into two topic areas— one covers fundamentals and the other covers the applications of controls. • Chapter 39, Building Envelopes, is a new chapter that will be moved to the 1999 ASHRAE Handbook—Applications. The technical committee completed the chapter in 1996, and the Handbook Committee decided to place it temporarily in this volume. • Old Chapter 29, Cooling and Freezing Times of Foods, and Chap- ter 30, Thermal Properties of Foods, are not included in this Handbook. They will be included in the 1998 ASHRAE Hand- book—Refrigeration. Each Handbook is published in two editions. One edition con- tains inch-pound (I-P) units of measurement, and the other contains the International System of Units (SI). Look for corrections to the 1994, 1995, and 1996 volumes of the Handbook series that have been noted since March 1995 on the Internet at http://www.ashrae.org. Any changes to this volume will be reported in the 1998 ASHRAE Handbook and on the Internet. If you have suggestions on improving a chapter or you would like more information on how you can help revise a chapter, e-mail bparsons@ashrae.org; write to Handbook Editor, ASHRAE, 1791 Tullie Circle, Atlanta, GA 30329; or fax (404) 321-5478. Robert A. Parsons ASHRAE Handbook Editor Thermodynamics and Refrigeration Cycles 1.3 (11) Equation (6) can be used to replace the heat transfer quantity. Note that the absolute temperature of the surroundings with which the system is exchanging heat is used in the last term. If the temper- ature of the surroundings is equal to the temperature of the system, the heat is transferred reversibly and Equation (11) becomes equal to zero. Equation (11) is commonly applied to a system with one mass flow in, the same mass flow out, no work, and negligible kinetic or potential energy flows. Combining Equations (6) and (11) yields (12) In a cycle, the reduction of work produced by a power cycle or the increase in work required by a refrigeration cycle is equal to the absolute ambient temperature multiplied by the sum of the irrevers- ibilities in all the processes in the cycle. Thus the difference in the reversible work and the actual work for any refrigeration cycle, the- oretical or real, operating under the same conditions becomes (13) THERMODYNAMIC ANALYSIS OF REFRIGERATION CYCLES Refrigeration cycles transfer thermal energy from a region of low temperature TR to one of higher temperature. Usually the higher temperature heat sink is the ambient air or cooling water. This tem- perature is designated as T0, the temperature of the surroundings. The first and second laws of thermodynamics can be applied to individual components to determine mass and energy balances and the irreversibility of the components. This procedure is illustrated in later sections in this chapter. Performance of a refrigeration cycle is usually described by a coefficient of performance. COP is defined as the benefit of the cycle (amount of heat removed) divided by the required energy input to operate the cycle, or (14) For a mechanical vapor compression system, the net energy sup- plied is usually in the form of work, mechanical or electrical, and may include work to the compressor and fans or pumps. Thus (15) In an absorption refrigeration cycle, the net energy supplied is usually in the form of heat into the generator and work into the pumps and fans, or (16) In many cases the work supplied to an absorption system is very small compared to the amount of heat supplied to the generator so the work term is often neglected. Application of the second law to an entire refrigeration cycle shows that a completely reversible cycle operating under the same conditions has the maximum possible Coefficient of Performance. A measure of the departure of the actual cycle from an ideal revers- ible cycle is given by the refrigerating efficiency: (17) The Carnot cycle usually serves as the ideal reversible refriger- ation cycle. For multistage cycles, each stage is described by a reversible cycle. EQUATIONS OF STATE The equation of state of a pure substance is a mathematical rela- tion between pressure, specific volume, and temperature. When the system is in thermodynamic equilibrium (18) The principles of statistical mechanics are used to (1) explore the fundamental properties of matter, (2) predict an equation of state based on the statistical nature of a particulate system, or (3) propose a functional form for an equation of state with unknown parameters that are determined by measuring thermodynamic properties of a substance. A fundamental equation with this basis is the virial equation. The virial equation is expressed as an expansion in pres- sure p or in reciprocal values of volume per unit mass v as (19) (20) where coefficients B’, C’, D’, etc., and B, C, D, etc., are the virial coefficients. B’ and B are second virial coefficients; C’ and C are third virial coefficients, etc. The virial coefficients are functions of temperature only, and values of the respective coefficients in Equa- tions (19) and (20) are related. For example, B’ = B/RT and C’ = (C – B2)/(RT)2. The ideal gas constant R is defined as (21) where (pv)T is the product of the pressure and the volume along an isotherm, and Ttp is the defined temperature of the triple point of water, which is 273.16 K. The current best value of R is 8314.41 J/(kg mole·K). The quantity pv/RT is also called the compressibility factor, i.e. Z = pv/RT or (22) An advantage of the virial form is that statistical mechanics can be used to predict the lower order coefficients and provide physical significance to the virial coefficients. For example, in Equation (22), the term B/v is a function of interactions between two mole- cules, C/v2 between three molecules, etc. Since the lower order interactions are common, the contributions of the higher order terms are successively less. Thermodynamicists use the partition or distri- bution function to determine virial coefficients; however, experi- mental values of the second and third coefficients are preferred. For dense fluids, many higher order terms are necessary that can neither be satisfactorily predicted from theory nor determined from exper- imental measurements. In general, a truncated virial expansion of four terms is valid for densities of less than one-half the value at the I · m· s( )out∑ m· s( )in∑– Q · Tsurr -----------∫–= I · m· sout sin–( ) hout hin– Tsurr ----------------------–= W · actual W · reversible T0 I · ∑+= COP Useful refrigerating effect Net energy supplied from external sources -----------------------------------------------------------------------------------------------------≡ COP Qi Wnet ----------= COP Qi Qgen Wnet+ -----------------------------= ηR COP COP( )rev ----------------------= f p v T,( , ) 0= pv RT ------ 1 B′p C′p2 D′p3 …+ + + += pv RT ------ 1 B v⁄( ) C v 2⁄( ) D v 3⁄( ) …+ + + += R pv( )T Ttp ------------- p 0→ lim= Z 1 B v⁄( ) C v 2⁄( ) D v 3⁄( ) …+ + + += 1.4 1997 ASHRAE Fundamentals Handbook (SI) critical point. For higher densities, additional terms can be used and determined empirically. Digital computers allow the use of very complex equations of state in calculating p-v-T values, even to high densities. The Bene- dict-Webb-Rubin (B-W-R) equation of state (Benedict et al. 1940) and the Martin-Hou equation (1955) have had considerable use, but should generally be limited to densities less than the critical value. Strobridge (1962) suggested a modified Benedict-Webb-Rubin relation that gives excellent results at higher densities and can be used for a p-v-T surface that extends into the liquid phase. The B-W-R equation has been used extensively for hydrocar- bons (Cooper and Goldfrank 1967): (23) where the constant coefficients are Ao, Bo, Co, a, b, c, α, γ. The Martin-Hou equation, developed for fluorinated hydro- carbon properties, has been used to calculate the thermodynamic property tables in Chapter 19 and in ASHRAE Thermodynamic Properties of Refrigerants (Stewart et al. 1986). The Martin-Hou equation is as follows: (24) where the constant coefficients are Ai, Bi, Ci, k, b, and α. Strobridge (1962) suggested an equation of state that was devel- oped for nitrogen properties and used for most cryogenic fluids. This equation combines the B-W-R equation of state with an equa- tion for high density nitrogen suggested by Benedict (1937). These equations have been used successfully for liquid and vapor phases, extending in the liquid phase to the triple-point temperature and the freezing line, and in the vapor phase from 10 to 1000 K, with pres- sures to 1 GPa. The equation suggested by Strobridge is accurate within the uncertainty of the measured p-v-T data. This equation, as originally reported by Strobridge, is (25) The 15 coefficients of this equation’s linear terms are determined by a least-square fit to experimental data. Hust and Stewart (1966) and Hust and McCarty (1967) give further information on methods and techniques for determining equations of state. In the absence of experimental data, Van der Waals’ principle of corresponding states can predict fluid properties. This principle relates properties of similar substances by suitable reducing factors, i.e., the p-v-T surfaces of similar fluids in a given region are assumed to be of similar shape. The critical point can be used to define reducing parameters to scale the surface of one fluid to the dimensions of another. Modifications of this principle, as suggested by Kamerlingh Onnes, a Dutch cryogenic researcher, have been used to improve correspondence at low pressures. The principle of corresponding states provides useful approximations, and numer- ous modifications have been reported. More complex treatments for predicting property values, which recognize similarity of fluid prop- erties, are by generalized equations of state. These equations ordi- narily allow for adjustment of the p-v-T surface by introduction of parameters. One example (Hirschfelder et al. 1958) allows for departures from the principle of corresponding states by adding two correlating parameters. CALCULATING THERMODYNAMIC PROPERTIES While equations of state provide p-v-T relations, a thermody- namic analysis usually requires values for internal energy, enthalpy, and entropy. These properties have been tabulated for many sub- stances, including refrigerants (See Chapters 6, 19, and 36) and can be extracted from such tables by interpolating manually or with a suitable computer program. This approach is appropriate for hand calculations and for relatively simple computer models; however, for many computer simulations, the overhead in memory or input and output required to use tabulated data can make this approach unacceptable. For large thermal system simulations or complex analyses, it may be more efficient to determine internal energy, enthalpy, and entropy using fundamental thermodynamic relations or curves fit to experimental data. Some of these relations are dis- cussed in the following sections. Also, the thermodynamic relations discussed in those sections are the basis for constructing tables of thermodynamic property data. Further information on the topic may be found in references covering system modeling and thermo- dynamics (Stoecker 1989, Howell and Buckius 1992). At least two intensive properties must be known to determine the remaining properties. If two known properties are either p, v, or T (these are relatively easy to measure and are commonly used in sim- ulations), the third can be determined throughout the range of inter- est using an equation of state. Furthermore, if the specific heats at zero pressure are known, specific heat can be accurately determined from spectroscopic measurements using statistical mechanics (NASA 1971). Entropy may be considered a function of T and p, and from calculus an infinitesimal change in entropy can be written as follows: (26) Likewise, a change in enthalpy can be written as (27) Using the relation Tds = dh − vdp and the definition of specific heat at constant pressure, cp ≡ (∂h/∂T)p, Equation (27) can be rear- ranged to yield (28) P RT v⁄( ) BoRT Ao– Co– T 2⁄( ) v 2⁄ bRT a–( ) v 3⁄+ += aα( ) v 6⁄ c 1 γ v 2⁄+( )e γ– v2⁄( )[ ] v 3 T 2⁄+ + p RT v b– ---------- A2 B2T C2e kT Tc⁄–( ) + + v b–( )2 --------------------------------------------------------+= A3 B3T C3e kT Tc⁄–( ) + + v b–( )3 -------------------------------------------------------- A4 B4T+ v b–( )4 ---------------------+ + A5 B5T C5e kT Tc⁄–( ) + + v b–( )5 -------------------------------------------------------- A6 B6T+( )e av + + p RTρ Rn1T n2 n3 T ---- n4 T 2 ----- n5 T 4 -----+ + + + ρ2 += Rn6T n7+( )ρ3 n8Tρ4 + + ρ3 n9 T 2 ----- n10 T3 ------- n11 T4 -------+ + n16– ρ2( )exp+ ρ5 n12 T 2 ------- n13 T3 ------- n14 T4 -------+ + n16– ρ2( )exp n15ρ6 + + ds ∂s ∂T -----    p dT ∂s ∂p -----    T dp+= dh ∂h ∂T -----    p dT ∂h ∂p -----    T dp+= ds cp T ---- dT p∂ ∂h     T v– dp T -----+= Thermodynamics and Refrigeration Cycles 1.5 Equations (26) and (28) combine to yield (∂s/∂T)p = cp/T. Then, using the Maxwell relation (∂s/∂p)T = −(∂v/∂T)p, Equation (26) may be rewritten as (29) This is an expression for an exact derivative, so it follows that (30) Integrating this expression at a fixed temperature yields (31) where cp0 is the known zero pressure specific heat, and dpT is used to indicate that the integration is performed at a fixed temperature. The second partial derivative of specific volume with respect to temperature can be determined from the equation of state. Thus, Equation (31) can be used to determine the specific heat at any pres- sure. Using Tds = dh − vdp, Equation (29) can be written as (32) Equations (28) and (32) may be integrated at constant pressure to obtain (33) and (34) Integrating the Maxwell relation (∂s/∂p)T = −(∂v/∂T)p gives an equation for entropy changes at a constant temperature as (35) Likewise, integrating Equation (32) along an isotherm yields the following equation for enthalpy changes at a constant temperature (36) Internal energy can be calculated from u = h − pv. Combinations (or variations) of Equations (33) through (36) can be incorporated directly into computer subroutines to calculate properties with improved accuracy and efficiency. However, these equations are restricted to situations where the equation of state is valid and the properties vary continuously. These restrictions are violated by a change of phase such as evaporation and condensa- tion, which are essential processes in air-conditioning and refriger- ating devices. Therefore, the Clapeyron equation is of particular value; for evaporation or condensation it gives (37) where sfg = entropy of vaporization hfg = enthalpy of vaporization vfg = specific volume difference between vapor and liquid phases If vapor pressure and liquid (or vapor) density data are known at saturation, and these are relatively easy measurements to obtain, then changes in enthalpy and entropy can be calculated using Equa- tion (37). Phase Equilibria for Multicomponent Systems To understand phase equilibria, consider a container full of a liq- uid made of two components; the more volatile component is des- ignated i and the less volatile component j (Figure 2A). This mixture is all liquid because the temperature is low—but not so low that a solid appears. Heat added at a constant pressure raises the tempera- ture of the mixture, and a sufficient increase causes vapor to form, as shown in Figure 2B. If heat at constant pressure continues to be added, eventually the temperature will become so high that only vapor remains in the container (Figure 2C). A temperature-concen- tration (T-x) diagram is useful for exploring details of this situation. Figure 3 is a typical T-x diagram valid at a fixed pressure. The case shown in Figure 2A, a container full of liquid mixture with mole fraction xi,0 at temperature T0 , is point 0 on the T-x diagram. When heat is added, the temperature of the mixture increases. The point at which vapor begins to form is the bubble point. Starting at point 0, the first bubble will form at temperature T1, designated by point 1 on the diagram. The locus of bubble points is the bubble point curve, which provides bubble points for various liquid mole fractions xi. When the first bubble begins to form, the vapor in the bubble may not have the i mole fraction found in the liquid mixture. Rather, the mole fraction of the more volatile species is higher in the vapor than in the liquid. Boiling prefers the more volatile spe- cies, and the T-x diagram shows this behavior. At Tl, the vapor forming bubbles have an i mole fraction of yi,l. If heat continues to be added, this preferential boiling will deplete the liquid of species i and the temperature required to continue the process will increase. Again, the T-x diagram reflects this fact; at point 2 the i mole frac- tion in the liquid is reduced to xi,2 and the vapor has a mole fraction of yi,2. The temperature required to boil the mixture is increased to ds cp T ----dT T∂ ∂v     p dp–= p∂ ∂cp     T T T 2 2 ∂ ∂ v       p –= cp cpo T T 2 2 ∂ ∂ v       pTd 0 p ∫–= dh cpdT v T T∂ ∂v     p – dp+= s T1 p0,( ) s T0 p0,( ) cp T ---- Tpd T0 T1 ∫+= h T1 p0,( ) h T0 p0,( ) cp Td T0 T1 ∫+= s T0 p1,( ) s T0 p0,( ) T∂ ∂v     p pTd p0 p1 ∫–= h T0 p1,( ) h T0 p0,( ) v T T∂ ∂v     p – pd p0 p1 ∫+= Td dp     sat sfg vfg ------ hfg Tvfg ---------= = Fig. 2 Mixture of i and j Components in Constant Pressure Container 1.8 1997 ASHRAE Fundamentals Handbook (SI) THEORETICAL SINGLE-STAGE CYCLE USING A PURE REFRIGERANT OR AZEOTROPIC MIXTURE A system designed to approach the ideal model shown in Figure 7 is desirable. A pure refrigerant or an azeotropic mixture can be used to maintain constant temperature during the phase changes by maintaining a constant pressure. Because of such concerns as high initial cost and increased maintenance requirements a practical machine has one compressor instead of two and the expander (engine or turbine) is replaced by a simple expansion valve. The valve throttles the refrigerant from high pressure to low pressure. Figure 8 shows the theoretical single-stage cycle used as a model for actual systems. Applying the energy equation for a mass of refrigerant m yields (39) The constant enthalpy throttling process assumes no heat transfer or change in potential or kinetic energy through the expansion valve. The coefficient of performance is (40) The theoretical compressor displacement CD (at 100% volumet- ric efficiency), is (41) which is a measure of the physical size or speed of the compressor required to handle the prescribed refrigeration load. Example 2. A theoretical single-stage cycle using R134a as the refrigerant operates with a condensing temperature of 30°C and an evaporating temperature of −20°C. The system produces 50 kW of refrigeration. Determine (a) the thermodynamic property values at the four main state points of the cycle, (b) the coefficient of performance of the cycle, (c) the cycle refrigerating efficiency, and (d) rate of refrigerant flow. Solution: (a) Figure 9 shows a schematic p-h diagram for the problem with numerical property data. Saturated vapor and saturated liquid proper- ties for states 1 and 3 are obtained from the saturation table for R134a in Chapter 19. Properties for superheated vapor at state 2 are obtained by linear interpolation of the superheat tables for R134a in Chapter 19. Specific volume and specific entropy values for state 4 are obtained by determining the quality of the liquid-vapor mixture from the enthalpy. Fig. 8 Theoretical Single-Stage Vapor Compression Refrigeration Cycle Q4 1 m h1 h4–( )= W1 2 m h2 h1–( )= Q2 3 m h2 h3–( )= h3 h4= COP Q4 1 W1 2 -------- h1 h4– h2 h1– ----------------= = CD m· v3= Fig. 9 Schematic p-h Diagram for Example 2 x4 h4 hf– hg hf– --------------- 241.65 173.82– 386.66 173.82– -------------------------------------- 0.3187= = = v4 vf x4 vg vf–( )+ 0.0007374 0.3187 0.14744 0.0007374–( )+= = 0.04749 m 3 /kg= Thermodynamics and Refrigeration Cycles 1.9 The property data are tabulated in Table 1. (b) By Equation (40) (c) By Equation (17) (d) The mass flow of refrigerant is obtained from an energy balance on the evaporator. Thus The saturation temperatures of the single-stage cycle have a strong influence on the magnitude of the coefficient of performance. This influence may be readily appreciated by an area analysis on a temperature-entropy (T-s) diagram. The area under a process line on a T-s diagram is directly proportional to the thermal energy added or removed from the working fluid. This observation follows directly from the definition of entropy. In Figure 10 the area representing Qo is the total area under the constant pressure curve between states 2 and 3. The area represent- ing the refrigerating capacity Qi is the area under the constant pres- sure line connecting states 4 and 1. The net work required Wnet equals the difference (Qo − Qi), which is represented by the shaded area shown on Figure 10. Because COP = Qi /Wnet, the effect on the COP of changes in evaporating temperature and condensing temperature may be observed. For example, a decrease in evaporating temperature TE significantly increases Wnet and slightly decreases Qi. An increase in condensing temperature TC produces the same results but with less effect on Wnet. Therefore, for maximum coefficient of perfor- mance, the cycle should operate at the lowest possible condensing temperature and at the maximum possible evaporating temperature. LORENZ REFRIGERATION CYCLE The Carnot refrigeration cycle includes two assumptions which make it impractical. The heat transfer capacity of the two external fluids are assumed to be infinitely large so the external fluid tem- peratures remain fixed at T0 and TR (they become infinitely large thermal reservoirs). The Carnot cycle also has no thermal resistance between the working refrigerant and the external fluids in the two heat exchange processes. As a result, the refrigerant must remain fixed at T0 in the condenser and at TR in the evaporator. The Lorenz cycle eliminates the first restriction in the Carnot cycle and allows the temperature of the two external fluids to vary during the heat exchange. The second assumption of negligible thermal resistance between the working refrigerant and the two external fluids remains. Therefore the refrigerant temperature must change during the two heat exchange processes to equal the chang- ing temperature of the external fluids. This cycle is completely reversible when operating between two fluids, each of which has a finite but constant heat capacity. Figure 11 is a schematic of a Lorenz cycle. Note that this cycle does not operate between two fixed temperature limits. Heat is added to the refrigerant from state 4 to state 1. This process is assumed to be linear on T-s coordinates, which represents a fluid with constant heat capacity. The temperature of the refrigerant is increased in an isentropic compression process from state 1 to state 2. Process 2-3 is a heat rejection process in which the refrig- erant temperature decreases linearly with heat transfer. The cycle is concluded with an isentropic expansion process between states 3 and 4. The heat addition and heat rejection processes are parallel so the entire cycle is drawn as a parallelogram on T-s coordinates. A Car- not refrigeration cycle operating between T0 and TR would lie between states 1, a, 3, and b. The Lorenz cycle has a smaller refrig- erating effect than the Carnot cycle and more work is required. However this cycle is a more practical reference to use than the Car- not cycle when a refrigeration system operates between two single phase fluids such as air or water. The energy transfers in a Lorenz refrigeration cycle are as fol- lows where ∆T is the temperature change of the refrigerant during the two heat exchange processes. Table 1 Thermodynamic Property Data for Example 2 State t , °C p, kPa v, m3/kg h, kJ/kg s , kJ/(kg·K) 1 −20.0 132.68 0.14744 386.66 1.7417 2 37.8 770.08 0.02798 423.07 1.7417 3 30.0 770.08 0.00084 241.65 1.1432 4 −20.0 132.68 0.04749 241.65 1.1689 s4 sf x4 sg sf–( )+ 0.9009 0.3187 1.7417 0.9009–( )+= = 1.16886 kJ/(kg·K)= COP 386.66 241.65– 423.07 386.66– -------------------------------------- 3.98= = ηR COP T3 T1–( ) T1 --------------------------------- 3.98( ) 50( ) 253.15 -------------------------- 0.79 or 79%= = = m· h1 h4–( ) Q · i 50 kW= = and m· Q · i h1 h4–( ) --------------------- 50 386.66 241.65–( ) ------------------------------------------- 0.345 kg/s= = = Fig. 10 Areas on T-s Diagram Representing Refrigerating Effect and Work Supplied for Theoretical Single-Stage Cycle Fig. 11 Processes of Lorenz Refrigeration Cycle 1.10 1997 ASHRAE Fundamentals Handbook (SI) Thus by Equation (15), (42) Example 3. Determine the entropy change, the work required, and the coefficient of performance for the Lorenz cycle shown in Figure 11 when the temperature of the refrigerated space is TR = 250 K, the ambi- ent temperature is T0 = 300 K, the ∆T of the refrigerant is 5 K and the refrigeration load is 125 kJ. Solution: Note that the entropy change for the Lorenz cycle is larger than for the Carnot cycle at the same temperature levels and the same capacity (see Example 1). That is, the heat rejection is larger and the work requirement is also larger for the Lorenz cycle. This difference is caused by the finite temperature difference between the working fluid in the cycle compared to the bounding temperature reservoirs. However, as discussed previously, the assumption of constant tem- perature heat reservoirs is not necessarily a good representation of an actual refrigeration system because of the temperature changes that occur in the heat exchangers. THEORETICAL SINGLE-STAGE CYCLE USING ZEOTROPIC REFRIGERANT MIXTURE A practical method to approximate the Lorenz refrigeration cycle is to use a fluid mixture as the refrigerant and the four system components shown in Figure 8. When the mixture is not azeotropic and the phase change processes occur at constant pressure, the tem- peratures change during the evaporation and condensation pro- cesses and the theoretical single-stage cycle can be shown on T-s coordinates as in Figure 12. This can be compared with Figure 10 in which the system is shown operating with a pure simple sub- stance or an azeotropic mixture as the refrigerant. Equations (14), (15), (39), (40), and (41) apply to this cycle and to conventional cycles with constant phase change temperatures. Equation (42) should be used as the reversible cycle COP in Equation (17). For zeotropic mixtures, the concept of constant saturation tem- peratures does not exist. For example, in the evaporator, the refrig- erant enters at T4 and exits at a higher temperature T1. The temperature of saturated liquid at a given pressure is the bubble point and the temperature of saturated vapor at a given pressure is called the dew point. The temperature T3 on Figure 12 is at the bub- ble point at the condensing pressure and T1 is at the dew point at the evaporating pressure. An analysis of areas on a T-s diagram representing additional work and reduced refrigerating effect from a Lorenz cycle operating between the same two temperatures T1 and T3 with the same value for ∆T can be performed. The cycle matches the Lorenz cycle most closely when counterflow heat exchangers are used for both the condenser and the evaporator. In a cycle that has heat exchangers with finite thermal resistances and finite external fluid capacity rates, Kuehn and Gronseth (1986) showed that a cycle which uses a refrigerant mixture has a higher coefficient of performance than a cycle that uses a simple pure sub- stance as a refrigerant. However, the improvement in COP is usu- ally small. The performance of the cycle that uses a mixture can be improved further by reducing the thermal resistance of the heat exchangers and passing the fluids through them in a counterflow arrangement. MULTISTAGE VAPOR COMPRESSION REFRIGERATION CYCLES Multistage vapor compression refrigeration is used when several evaporators are needed at various temperatures such as in a supermar- ket or when the temperature of the evaporator becomes very low. Low evaporator temperature indicates low evaporator pressure and low refrigerant density into the compressor. Two small compressors in series have a smaller displacement and are usually operate more effi- ciently than one large compressor that covers the entire pressure range from the evaporator to the condenser. This is especially true in refrigeration systems that use ammonia because of the large amount of superheating that occurs during the compression process. The thermodynamic analysis of multistage cycles is similar to the analysis of single stage cycles. The main difference is that the mass flow differs through various components of the system. A careful mass balance and energy balance performed on individual components or groups of components ensures the correct applica- tion of the first law of thermodynamics. Care must also be exercised when performing second law calculations. Often the refrigerating load is comprised of more than one evaporator, so the total system capacity is the sum of the loads from all evaporators. Likewise the total energy input is the sum of the work into all compressors. For multistage cycles the expression for the coefficient of performance given in Equation 15 should be written as (43) When compressors are connected in series, the vapor between stages should be cooled to bring the vapor to saturated conditions Q0 T0 T∆+ 2⁄( ) S2 S3–( )= Qi TR T∆– 2⁄( ) S1 S4–( ) TR T∆– 2⁄( ) S2 S3–( )= = Wnet Q0 QR–= COP TR ∆T 2⁄( )– TO TR ∆T+– --------------------------------= S∆ Qi T ----- 4 1 ∫ Qi TR T∆ 2⁄( )– ------------------------------ 125 247.5 ------------ 0.5051 kJ K⁄= = = = QO TO T∆ 2⁄( )+[ ] S∆ 300 2.5+( )0.5051 152.78 kJ= = = Wnet QO QR– 152.78 125– 27.78 kJ= = = COP TR T∆ 2⁄( )– TO TR– T∆+ ------------------------------- 250 5 2⁄( )– 300 250– 5+ -------------------------------- 247.5 55 ------------ 4.50= = = = Fig. 12 Areas on T-s Diagram Representing Refrigerating Effect and Work Supplied for Theoretical Single-Stage Cycle Using Zeotropic Mixture as Refrigerant COP Qi∑ Wnet ------------= Thermodynamics and Refrigeration Cycles 1.13 Second law Suction Line: Energy balance Second law Compressor: Energy balance Second law Discharge Line: Energy balance Second law Condenser: Energy balance Second law Liquid Line: Energy balance Second law Expansion Device: Energy balance Second law These results are summarized in Table 4. For the Carnot cycle The Carnot power requirement for the 7 kW load is The actual power requirement for the compressor is This result is within computational error of the measured power input to the compressor of 2.5 kW. The analysis demonstrated in Example 5 can be applied to any actual vapor compression refrigeration system. The only required information for the second law analysis is the refrigerant thermody- namic state points and mass flow rates and the temperatures in which the system is exchanging heat. In this example, the extra Fig. 15 Pressure-Enthalpy Diagram of Actual System and Theoretical Single-Stage System Operating Between Same Inlet Air Temperatures TR and T0. I1 · 7 m· s1 s7–( ) Q1 · 7 TR --------–= 0.04322 1.7810 1.1561–( ) 7.0 263.15 ---------------–= 0.4074 W/K= Q2 · 1 m· h2 h1–( )= 0.04322 406.25 402.08–( ) 0.1802 kW= = I2 · 1 m· s2 s1–( ) Q2 · 1 TO --------–= 0.04322 1.7984 1.7810–( ) 0.1802 303.15⁄–= 0.1575 W/K= Q3 · 2 m· h3 h2–( ) W3 · 2+= 0.04322 454.20 406.25–( ) 2.5–= 0.4276 kW–= I3 · 2 m· s3 s2–( ) Q3 · 2 TO --------–= 0.04322 1.8165 1.7984–( ) 0.4276– 303.15⁄( )–= 2.1928 kW= Q4 · 3 m· h4 h3–( )= 0.04322 444.31 454.20–( ) 0.4274 kW–== I4 · 3 m· s4 s3–( ) Q4 · 3 TO --------–= 0.04322 1.7891 1.8165–( ) 0.4274– 303.15⁄( )–= 0.2258W/K= Table 4 Energy Transfers and Irreversibility Rates for Refrigeration System in Example 5 Component , kW , kW , W/K , % Evaporator 7.0000 0 0.4074 9 Suction line 0.1802 0 0.1575 3 Compressor −0.4276 2.5 2.1928 46 Discharge line −0.4274 0 0.2258 5 Condenser −8.7698 0 0.8747 18 Liquid line −0.0549 0 0.0039 ≈0 Expansion device 0 0 0.8730 18 Totals −2.4995 2.5 4.7351 Q · W · I · I · I · total⁄ Q5 · 4 m· h5 h4–( )= 0.04322 241.4 444.31–( ) 8.7698 kW–== I5 · 4 m· s5 s4–( ) Q5 · 4 TO --------–= 0.04322 1.1400 1.7891–( ) 8.7698– 303.15⁄( )–= 0.8747 W/K= Q6 · 5 m· h6 h5–( )= 0.04322 240.13 241.40–( ) 0.0549 kW–== I6 · 5 m· s6 s5–( ) Q6 · 5 TO --------–= 0.04322 1.1359 1.1400–( ) 0.0549– 303.15⁄( )–= 0.0039 W/K= Q7 · 6 m· h7 h6–( ) 0= = I7 · 6 m· s7 s6–( )= 0.04322 1.1561 1.1359–( ) 0.8730 W/K= = COPCarnot TR To TR– ----------------- 263.15 40 --------------- 6.579= = = W · Carnot Q · E COPCarnot ------------------------- 7.0 6.579 ------------ 1.064 kW= = = W · comp W · Carnot I · totalTo+ 1.064 4.7351 303.15( )+ 2.4994 kW= = = 1.14 1997 ASHRAE Fundamentals Handbook (SI) compressor power required to overcome the irreversibility in each component is determined. The component with the largest loss is the compressor. This loss is due to motor inefficiency, friction losses, and irreversibilities due to pressure drops, mixing, and heat transfer between the compressor and the surroundings. The unrestrained expansion in the expansion device is also a large loss. This loss could be reduced by using an expander rather than a throttling pro- cess. An expander may be economical on large machines. All heat transfer irreversibilities on both the refrigerant side and the air side of the condenser and evaporator are included in the anal- ysis. The refrigerant pressure drop is also included. The only items not included are the air-side pressure drop irreversibilities of the two heat exchangers. However these are equal to the fan power require- ments as all the fan power is dissipated as heat. An overall second law analysis, such as in Example 5, shows the designer those components with the most losses, and it helps deter- mine which components should be replaced or redesigned to improve performance. However, this type of analysis does not iden- tify the nature of the losses. A more detailed second law analysis in which the actual processes are analyzed in terms of fluid flow and heat transfer is required to identify the nature of the losses (Liang and Kuehn 1991). A detailed analysis will show that most irrevers- ibilities associated with heat exchangers are due to heat transfer, while pressure drop on the air side causes a very small loss and the refrigerant pressure drop causes a negligible loss. This finding indi- cates that promoting refrigerant heat transfer at the expense of increasing the pressure drop usually improves performance. ABSORPTION REFRIGERATION CYCLES Absorption cycles are primarily heat-operated cycles in which heat is pumped with a minimum of work input. As with vapor com- pression cycles, absorption cycles can be operated in either a heat- ing or cooling mode. This discussion is restricted to cooling applications because such applications dominate the market. Absorption cooling machines are available in sizes ranging from 10 to 7000 kW of refrigeration. These machines are configured for direct-fired operation as well as for waste heat or heat integration applications. Figure 16 is a simple schematic of a heat engine and a heat pump together in a single package. Such a combination interacts with the surroundings at three temperature levels, which is typical of an absorption cycle. In the cooling mode, the driving heat must be supplied at the highest temperature in the cycle. The refrigera- tion effect is provided at the lowest temperature in the cycle. The sum of these heat inputs to the cycle is then rejected at the interme- diate temperature. From a thermodynamic standpoint, the refriger- ation effect can be accomplished with zero work input. In fact such cycles, called diffusion-absorption cycles, are widely used to refrigerate food in recreational vehicles and hotel rooms. Although such cycles work effectively at the loads required for a food refrig- erator (40 W of cooling), larger machines generally use mechani- cally driven pumps to circulate the internal fluid. Thus, for most absorption cycles, a small work input of about 1% of the heat input must be supplied as electric power input. Frequently, this work input is ignored when describing the thermal performance of an absorption machine. In practice however, the design, operation, and maintenance associated with the pumps must be considered. Absorption technology competes with engine-driven vapor compression and desiccant refrigeration systems for the gas-fired market. A key difference between such gas-fired technologies and electric-driven technologies is that the fuel powering the unit is burned locally. FLOW DESCRIPTION Key processes in the absorption cycle are the absorption and desorption of refrigerant. The cycle has five main components as shown in Figure 17: the generator (sometimes called desorber), the condenser, the evaporator, the absorber, and the solution heat exchanger. Starting with state point 4 at the generator exit, the stream consists of absorbent-refrigerant solution, which flows to the absorber via the heat exchanger. From points 6 to 1, the solution absorbs refrigerant vapor (10) from the evaporator and rejects heat to the environment. The solution rich in refrigerant (1) flows via the heat exchanger to the generator (3). In the generator thermal energy is added and refrigerant (7) boils off the solution. The refrigerant vapor (7) flows to the condenser, where heat is rejected as the refrigerant condenses. The condensed liquid (8) flows through a flow restrictor to the evaporator. In the evaporator, the heat from the load evaporates the refrigerant, which then flows (10) to the absorber. A portion of the refrigerant leaving the evaporator leaves as liquid spillover (11). The state points of absorption cycles are usually represented in a Dühring chart (Figure 18). In this chart, refrigerant saturation tem- perature and its corresponding pressure are plotted versus the solu- tion temperature. The lines of constant solution concentration are straight lines of decreasing slopes for increasing concentrations. In this schematic, the lines represent constant aqueous lithium bromide concentration, with water as the refrigerant. The solution at the exit of the generator (point 4) is cooled to point 5 in the heat exchanger. In the absorber, the solution concentration decreases to that of 1. The solution is then pumped to the generator via a heat exchanger, where its temperature is raised to that of 3. In the generator the solu- tion is reconcentrated to yield 4 again. The refrigerant from the gen- erator condenses at 8 and evaporates at 10 to return to the absorber. Because absorption machines are thermally activated, large amounts of power input are not required. Hence, where power is expensive or unavailable, and gas, waste, geothermal or solar heat is available, absorption machines provide reliable and quiet cooling. Fig. 16 Absorption Refrigeration Machine as Combination of Heat Engine and Heat Pump Fig. 17 Single-Effect Lithium Bromide/Water Absorption Cycle Thermodynamics and Refrigeration Cycles 1.15 The usual figure of performance of an absorption cycle is the COP (coefficient of performance), which is defined as the ratio of the evaporator heat to the generator heat. Because it takes about the same amount of heat to boil the refrigerant in both the generator and evaporator, it might be assumed that single effect cycles are capable of a COP of 1. Yet, the best single effect machines reach COPs of only 0.5 to 0.7. The losses responsible for the COP degradation are traced to the following four phenomena: 1. Circulation loss. When the cold solution from the absorber (1), is heated in the solution heat exchanger (3), the temperature at 3 is always less than the saturation temperature corresponding to the generator pressure and solution concentration, even for cycles with high heat exchanger effectiveness. Hence, heat must be added to boil the solution, which increases the generator heat input. 2. Heat of mixing. Separating the refrigerant from the solution requires about 15% more thermal energy than merely boiling the refrigerant. This additional energy must be supplied to break the intermolecular bonds formed between the refrigerant and absor- bent in solution. The heat of mixing also increases the generator heat input. 3. Expansion loss. As the refrigerant expands from the condenser to the evaporator, a mixture of liquid and vapor enters the evap- orator. Not all of the refrigerant is available as liquid because some vapor was already produced by the expansion process. Thus, the evaporator heat transfer is reduced when vapor forms in the expansion process. This loss can be reduced by subcooling the liquid from the condenser. 4. Reflux condenser loss. In the ammonia-water cycle (Figure 21) another loss is introduced due to the volatility of water. In this cycle the refrigerant is ammonia and the absorbent is water. In the generator water vapor evaporates along with the ammonia. However, for proper operation, the water vapor must be removed from the ammonia vapor. The water vapor is separated in a dis- tillation column, which has a reflux coil that condenses some ammonia-water. The heat removed in the reflux coil must be added to the generator, thus decreasing the COP. In addition, other losses occur during transient operating condi- tions. For instance, if more refrigerant is produced than can be han- dled by the evaporator, the refrigerant is directly returned to the absorber via a spillover (point 11 in Figures 17 and 18). Liquid refrig- erant returned directly to the absorber is a loss, and machines of recent design are tightly controlled to avoid this loss during transients. To attain higher COPs, a double effect cycle is used. In this cycle an additional generator and condenser are added to a single effect cycle. The heat input to the high temperature generator is used to drive off refrigerant, which on condensing drives a lower tempera- ture generator to produce yet more refrigerant. In this way the heat input to the higher temperature generator is used twice, and the arrangement is called double effect. Typical COPs of double effect machines range from 1.0 to 1.2. CHARACTERISTICS OF REFRIGERANT-ABSORBENT PAIRS Few solutions work as suitable absorbent-refrigerant pairs. The materials that make up the refrigerant-absorbent pair should meet the following requirements to be suitable for absorption refrigeration: Absence of Solid Phase. The refrigerant-absorbent pair should not form a solid phase over the range of composition and tempera- ture to which it might be subjected. If a solid forms, it presumably would stop flow and cause equipment to shut down. Volatility Ratio. The refrigerant should be much more volatile than the absorbent so the two can be separated easily. Otherwise, cost and heat requirements can prohibit separation. Affinity. The absorbent should have a strong affinity for the refrigerant under conditions in which absorption takes place. This affinity (1) causes a negative deviation from Raoult’s law and results in an activity coefficient of less than unity for the refrigerant; (2) allows less absorbent to be circulated for the same refrigerating effect so sensible heat losses are less; and (3) requires a smaller liq- uid heat exchanger to transfer heat from the absorbent to the pres- surized refrigerant-absorbent solution. However, calculations by Jacob et al. (1969) indicate that strong affinity has some disadvan- tages. This affinity is associated with a high heat of dilution; conse- quently, extra heat is required in the generator to separate the refrigerant from the absorbent. Pressure. Operating pressures, largely established by physical properties of the refrigerant, should be moderate. High pressures require the use of heavy-walled equipment, and significant electri- cal power may be required to pump the fluids from the low-pressure side to the high-pressure side. Low pressure (vacuum) requires the use of large volume equipment and special means of reducing pres- sure drop in refrigerant vapor flow. Stability. High chemical stability is required because fluids are subjected to severe conditions over many years of service. Instabil- ity could cause the undesirable formation of gases, solids, or corro- sive substances. Corrosion. Because absorption fluids can corrode materials used in constructing equipment, corrosion inhibitors are used. Safety. Fluids must be nontoxic and nonflammable if they are in an occupied dwelling. Industrial process refrigeration is less critical in this respect. Transport Properties. Viscosity, surface tension, thermal diffu- sivity, and mass diffusivity are important characteristics of the refrigerant and absorbent pair. For example, a low fluid viscosity promotes heat and mass transfer and reduces pumping power. Latent Heat. The refrigerant’s latent heat should be high so the circulation rate of the refrigerant and absorbent can be kept at a minimum. Environmental Soundness. The working pairs must be safe, nonflammable, and devoid of lasting environmental effects. No known refrigerant-absorbent pair meets all requirements listed. However, lithium bromide-water and ammonia-water offer excellent thermodynamic performance and they have little long- term environmental effect. The ammonia-water pair meets most requirements, but its volatility ratio is low, and it requires high oper- ating pressures. Furthermore, ammonia is a Safety Code Group 2 fluid (ASHRAE Standard 15), which restricts its use indoors. Advantages of the water-lithium bromide pair include high safety, high volatility ratio, high affinity, high stability, and high latent heat. However, this pair tends to form solids. Because the refrigerant turns to ice at 0°C, the pair cannot be used for low-tem- perature refrigeration. Lithium bromide crystallizes at moderate concentrations, especially when it is air cooled, which typically limits the pair to applications where the absorber is water cooled. However, using a combination of salts as the absorbent can reduce this crystallizing tendency enough to permit air cooling (Macriss 1968). Other disadvantages of the water-lithium bromide pair Fig. 18 Single-Effect Lithium Bromide/Water Absorption Cycle Superimposed on Dühring Plot 1.18 1997 ASHRAE Fundamentals Handbook (SI) using the inputs and assumptions listed in Table 7. The results are shown in Table 8. Note that the COP value obtained is high com- pared to what is obtained in practice. The COP is quite sensitive to several inputs and assumptions. In particular, the effectiveness of the solution heat exchangers and the driving temperature difference between the high temperature condenser and the low temperature generator are two parameters that influence the COP strongly. Ammonia-Water Cycle An ammonia-water single-stage refrigeration cycle (Figure 21) resembles a lithium bromide-water refrigeration cycle with the exception of two components unique to ammonia/water: (1) a rec- tifier and (2) a refrigerant heat exchanger. Both components are needed because the vapor pressure of the absorbent (water) is suffi- ciently high that the water content of the vapor becomes a design issue. The rectifier removes water vapor from the vapor leaving the generator by a fractional distillation process. Distillation occurs in a counterflow mass transfer device such as a packed tower, a bubble tower, or a tower with sieve trays. Reflux, provided by a partial con- denser at the top of the column, purifies the rising ammonia vapors coming from the generator. The reflux, after performing its purifi- cation function, must be returned to the generator. The generator supplies the rectification heat, which reduces the cycle COP. Even with a sophisticated rectifier, pure ammonia vapor cannot be obtained at the outlet. A small water fraction, on the order of 0.1% by mass, is enough to cause the refrigerant in the evaporator to exhibit a temperature glide of 20 K (this assumes a single-pass evap- orator where the temperature changes along the length of the flow path, such as an in-tube design). The refrigerant heat exchanger uses the energy in the liquid stream coming from the condenser to evap- orate the high temperature fraction of the refrigerant. This heat exchanger accounts for the water content in the refrigerant without requiring a blow-down (spillover) system. Inclusion of the refriger- ant heat exchanger allows the rectifier to be less effective without any penalty on performance. In smaller systems, the ammonia vapor purity off the top of the tower (or rectifier/analyzer) is generally less than that in the exam- ple. As a result, the tower and condenser pressure are less, but water contamination of the refrigerant is larger. An alternative to provid- ing a refrigerant heat exchanger is to constantly bleed liquid as blow-down (spillover) from the evaporator to the absorber. In large systems with blow-down (spillover), a vertical liquid leg under the evaporator provides a relatively inactive area and accu- mulates ammonia that is rich in water. The blow-down (spillover) line taps into this liquid leg. At an evaporator pressure of 520 kPa, a pool evaporator containing 10% by mass water increases the evap- orator temperature from 5.1 to 7.7°C for a 2.6 K penalty (Jennings and Shannon 1938). However, if allowed to accumulate without blow-down, the water content will exceed 10% eventually and reach a point where the evaporator temperature is too high to pro- duce the desired cooling. In lithium bromide-water systems, the cooling tower water is fed in series to the absorber and then to the condenser. In ammonia- water systems, the cooling tower water is fed first to the condenser to keep the high-end pressure as low as possible. In both cases, the cooling water can be piped in parallel to improve efficiencies; how- ever, this requires high coolant flow rates and excessively large cooling towers. In the ammonia-water cycle, the reflux for the rectifier can be created by a separate condenser or by the main condenser. Reflux Table 7 Inputs and Assumptions for Double-Effect Lithium Bromide/Water Model Inputs Capacity 1760 kW Evaporator temperature t10 5.1°C Desorber solution exit temperature t14 170.7°C Condenser/absorber low temperature t1 = t8 42.4°C Solution heat exchanger effectiveness ε 0.6 Assumptions Steady state Refrigerant is pure water No pressure changes except through the flow restrictors and the pump States at points 1, 4, 8, 11, 14 and 18 are saturated liquid State at point 10 is saturated vapor Temperature difference between high temperature condenser and low temperature generator is 5 K Parallel flow Both solution heat exchangers have same effectiveness Upper loop solution flow rate is selected such that the upper condenser heat exactly matches the lower generator heat requirement Flow restrictors are adiabatic Pumps are isentropic No jacket heat losses No liquid carryover from evaporator to absorber Vapor leaving both generators is at the equilibrium temperature of the entering solution stream Table 8 State Point Data for Double-Effect Lithium Bromide/Water Cycle of Figure 20 No. h J/g kg/s p kPa Q Fraction t °C x % LiBr 1 117.7 9.551 0.88 0.0 42.4 59.5 2 117.7 9.551 8.36 42.4 59.5 3 182.3 9.551 8.36 75.6 59.5 4 247.3 8.797 8.36 0.0 97.8 64.6 5 177.2 8.797 8.36 58.8 64.6 6 177.2 8.797 0.88 0.004 53.2 64.6 7 2661.1 0.320 8.36 85.6 0.0 8 177.4 0.754 8.36 0.0 42.4 0.0 9 177.4 0.754 0.88 0.063 5.0 0.0 10 2510.8 0.754 0.88 1.0 5.0 0.0 11 201.8 5.498 8.36 0.0 85.6 59.5 12 201.8 5.498 111.8 85.6 59.5 13 301.2 5.498 111.8 136.7 59.5 14 378.8 5.064 111.8 0.0 170.7 64.6 15 270.9 5.064 111.8 110.9 64.6 16 270.9 5.064 8.36 0.008 99.1 64.6 17 2787.3 0.434 111.8 155.7 0.0 18 430.6 0.434 111.8 0.0 102.8 0.0 19 430.6 0.434 8.36 0.105 42.4 0.0 COPc = 1.195 ∆t = 5°C ε = 0.600 = 2328 kW = 1023 kW = 905 kW = 1760 kW = 1472 kW = 617 kW = 546 kW = 0.043 kW = 0.346 kW Q · e m· Q · a Q · cg Q · c Q · e Q · gh Q · shx1 Q · shx2 W · p1 W · p2 Fig. 21 Single-Effect Ammonia/Water Absorption Cycle Thermodynamics and Refrigeration Cycles 1.19 can flow by gravity or it can be pumped to the top of the tower. Ammonia-water machines do not experience crystallization as lith- ium bromide-water machines do, so controls can be simpler. Also, the corrosion characteristics of ammonia-water solutions are less severe, although inhibitors are generally used for both systems. Lithium bromide-water systems use combinations of steel, copper, and copper-nickel materials for shells and heat transfer surfaces; but no copper-bearing materials can be used in ammonia-water systems because ammonia rapidly destroys copper. The mass and energy balances around the cycle shown in Figure 21 were calculated based on the assumptions and input values in Table 9. The conditions were selected to approximately match the conditions chosen for the lithium bromide/water cycle discussed in the previous section. Note that the ammonia/water working fluid allows significantly lower evaporator temperature than that found in this example. The cycle solution is summarized in Table 10 where state point data for all connecting points are given. For most com- ponents, the mass and energy balances are performed in an identical fashion as that discussed earlier in relation to the lithium bro- mide/water example. However, because the ammonia/water cycle has two new components, some additional discussion is needed. The rectifier model assumes that the rectification process is reversible. This provides a thermodynamic lower bound for the heat transfer required in the rectifier (and thus gives an upper bound on the COP). The key aspect of the reversible model is the assumption that the reflux leaving the rectifier (14) is in equilibrium with the vapor entering the rectifier (13). In a real rectifier, potential differ- ences would be needed to drive the purification process and these potential differences would be evident at the bottom of the column. Another aspect of the reversible model is the requirement that heat be extracted from the column along the entire length. In a more typical column design, the body of the column would be adiabatic and all the heat transfer would occur in the reflux condenser at the top of the column. These differences between the reversible model and a real rectifier are significant, but the simplified model is used here because it shows the overall trends. Bogart (1981) provides a more detailed discussion of rectifier design. The refrigerant heat exchanger uses a single phase fluid on the hot side (8–9) and an evaporating mixture on the cold side (11–12). Because the flow rate is the same on both sides, the temperature pro- files are not expected to match. However, because the composition of the evaporating fluid changes as it passes through the heat exchanger, the temperature also changes significantly. As a result the temperature profiles match reasonably well. Table 10 shows that the terminal temperature differences are 11.8 K and 7.3 K at the 9– 11 and 8–12 ends respectively. The fact that the temperature profiles match well is one of the reasons why this heat exchanger works so effectively. While operating under nearly identical conditions the overall COP of the ammonia/water cycle is 0.57 as compared to 0.70 for the lithium bromide/water example. The reduction in COP for the ammonia/water cycle can be traced to two major factors: (1) the solution heat exchanger and (2) the rectifier. The solution heat exchanger in the ammonia/water cycle carries more load due to the larger specific heat of the liquid solution as compared to aqueous lithium bromide. Typical specific heat values at the conditions of interest are 4.6 kJ/(kg·K) for liquid ammonia/water and 1.9 kJ/(kg·K) for aqueous lithium bromide, as computed for the solu- tion heat exchanger examples in Table 6 and Table 10. This differ- ence between the two working fluids implies greater sensitivity to solution heat exchanger design for ammonia/water cycles. The rectifier heat loss is overwhelmed in this example by the solution heat exchanger loss. An estimate can be obtained by com- paring the heat transfer rate in the solution heat exchangers from the two examples. In general, the losses in such a device are approxi- mately proportional to the heat transfer rate and the average temper- ature difference driving the heat transfer. The fact that the heat transfer duty in the ammonia/water cycle is much greater due to the high specific heat of the liquid is the key to the difference. NOMENCLATURE cp specific heat at constant pressure COP coefficient of performance g local acceleration of gravity h enthalpy, kJ/kg I irreversibility irreversibility rate m mass mass flow, kg/s p pressure Q heat energy, kJ rate of heat flow, kJ/s Table 9 Inputs and Assumptions for Single-Effect Ammonia/Water Model of Figure 21 Inputs Capacity 1760 kW High side pressure phigh 1461 kPa Low side pressure plow 515 kPa Absorber exit temperature t1 40.6°C Generator exit temperature t4 95°C Rectifier vapor exit temperature t7 55°C Solution heat exchanger eff. εshx 0.692 Refrigerant heat exchanger eff. εrhx 0.629 Assumptions Steady state No pressure changes except through the flow restrictors and the pump States at points 1, 4, 8, 11, and 14 are saturated liquid States at point 12 and 13 are saturated vapor Flow restrictors are adiabatic Pump is isentropic No jacket heat losses No liquid carryover from evaporator to absorber Vapor leaving the generator is at the equilibrium temperature of the enter- ing solution stream Q · e Table 10 State Point Data for Single-Effect Ammonia/Water Cycle of Figure 21 No. h kJ/kg kg/s p kPa Q Fraction t °C x, Frac- tion NH3 1 −57.2 10.65 515.0 0.0 40.56 0.50094 2 −56.0 10.65 1461 40.84 0.50094 3 89.6 10.65 1461 72.81 0.50094 4 195.1 9.09 1461 0.0 95.00 0.41612 5 24.6 9.09 1461 57.52 0.41612 6 24.6 9.09 515.0 0.006 55.55 0.41612 7 1349 1.55 1461 1.000 55.00 0.99809 8 178.3 1.55 1461 0.0 37.82 0.99809 9 82.1 1.55 1461 17.80 0.99809 10 82.1 1.55 515.0 0.049 5.06 0.99809 11 1216 1.55 515.0 0.953 6.00 0.99809 12 1313 1.55 515.0 1.000 30.57 0.99809 13 1429 1.59 1461 1.000 79.15 0.98708 14 120.4 0.04 1461 0.0 79.15 0.50094 COPc = 0.571 ∆trhx = 7.24 K ∆tshx = 16.68 K εrhx = 0.629 εshx = 0.692 = 2869 kW = 1816.2 kW = 1760 kW = 3083 kW = 149 kW = 170 kW = 1550 kW = 12.4 kW m· Q · a Q · c Q · e Q · g Q · rhx Q · r Q · shx W · I · m· Q · 1.20 1997 ASHRAE Fundamentals Handbook (SI) R ideal gas constant s entropy, kJ/(kg·K) S total entropy t temperature, °C T absolute temperature, K u internal energy W mechanical or shaft work rate of work, power v specific volume, m3/kg V velocity of fluid x mass fraction (of either lithium bromide or ammonia) x vapor quality (fraction) z elevation above horizontal reference plane Z compressibility factor ε heat exchanger effectiveness η efficiency Subscripts a absorber c condenser or cooling mode C condensing conditions cg condenser to generator d desorber (generator) e evaporator fg fluid to vapor gh high temperature generator o, 0 reference conditions, usually ambient p pump R refrigerating or evaporator conditions rhx refrigerant heat exchanger shx solution heat exchanger REFERENCES Benedict, M., G.B. Webb, and L.C. Rubin. 1940. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures. Journal of Chemistry and Physics 4:334. Benedict, M. 1937. Pressure, volume, temperature properties of nitrogen at high density, I and II. Journal of American Chemists Society 59(11): 2224. Cooper, H.W. and J.C. Goldfrank. 1967. B-W-R Constants and new corre- lations. Hydrocarbon Processing 46(12):141. Hirschfelder, J.O. et al. 1958. Generalized equation of state for gases and liq- uids. Industrial and Engineering Chemistry 50:375. Howell, J.R. and R.O. Buckius. 1992. Fundamentals of Engineering Ther- modynamics, 2nd Ed. McGraw-Hill, New York. Hust, J.G. and R.D. McCarty. 1967. Curve-fitting techniques and applica- tions to thermodynamics. Cryogenics 8:200. Hust, J.G. and R.B. Stewart. 1966. Thermodynamic property computations for system analysis. ASHRAE Journal 2:64. Jacob, X., L.F. Albright, and W.H. Tucker. 1969. Factors affecting the coef- ficient of performance for absorption air-conditioning systems. ASHRAE Transactions 75(1):103. Jennings, B.H. and F.P. Shannon. 1938. The thermodynamics of absorption refrigeration. Refrigerating Engineering 35(5):338. Kuehn, T.H. and R.E. Gronseth. 1986. The effect of a nonazeotropic binary refrigerant mixture on the performance of a single stage refrigeration cycle. Proceedings International Institute of Refrigeration Conference, Purdue University, p. 119. Liang, H. and T.H. Kuehn. 1991. Irreversibility analysis of a water to water mechanical compression heat pump. Energy 16(6):883. Macriss, R.A. 1968. Physical properties of modified LiBr solutions. AGA Symposium on Absorption Air-Conditioning Systems, February. Macriss, R.A. and T.S. Zawacki. 1989. Absorption fluid data survey: 1989 update. Oak Ridge National Laboratories Report ORNL/Sub84-47989/4. Martin, J.J. and Y. Hou. 1955. Development of an equation of state for gases. AICHE Journal 1:142. Martz, W.L., C.M. Burton, and A.M. Jacobi. 1996a. Liquid-vapor equilibria for R-22, R-134a, R-125, and R-32/125 with a polyol ester lubricant: Measurements and departure from ideality. ASHRAE Transactions 102(1):367-74. Martz, W.L., C.M. Burton and A.M. Jacobi. 1996b. Local composition mod- eling of the thermodynamic properties of refrigerant and oil mixtures. International Journal of Refrigeration 19(1):25-33. McNeely, L.A. 1979. Thermodynamic properties of aqueous solution of lith- ium bromide. ASHRAE Transactions 85(1):413. NASA. 1971. SP-273. US Government Printing Office, Washington, D.C. Stewart, R.B., R.T. Jacobsen, and S.G. Penoncello. 1986. ASHRAE Thermo- dynamic properties of refrigerants. ASHRAE, Atlanta, GA. Strobridge, T.R. 1962. The thermodynamic properties of nitrogen from 64 to 300 K, between 0.1 and 200 atmospheres. National Bureau of Standards Technical Note 129. Stoecker, W.F. 1989. Design of thermal systems, 3rd Ed. McGraw-Hill, New York. Tassios, D.P. 1993. Applied chemical engineering thermodynamics. Springer-Verlag, New York. Thome, J.R. 1995. Comprehensive thermodynamic approach to modeling refrigerant-lubricant oil mixtures. International Journal of Heating, Ven- tilating, Air Conditioning and Refrigeration Research 1(2):110. BIBLIOGRAPHY Alefeld, G. and R. Radermacher. 1994. Heat conversion systems. CRC Press, Boca Raton. Bogart, M. 1981. Ammonia absorption refrigeration in industrial processes. Gulf Publishing Co., Houston, TX. Briggs, S.W. 1971. Concurrent, crosscurrent, and countercurrent absorption in ammonia-water absorption refrigeration. ASHRAE Transactions 77(1):171. Herold, K.E., R. Radermacher, and S.A. Klein. 1996. Absorption chillers and heat pumps. CRC Press, Boca Raton. Jain, P.C. and G.K. Gable. 1971. Equilibrium property data for aqua-ammo- nia mixture. ASHRAE Transactions 77(1):149. Moran, M.J. and Shapiro, H. 1995. Fundamentals of engineering thermody- manics, 3rd Ed. John Wiley and Sons, Inc. New York. Stoecker, W.F. and L.D. Reed. 1971. Effect of operating temperatures on the coefficient of performance of aqua-ammonia refrigerating systems. ASHRAE Transactions 77(1):163. Van Wylen, C.J. and R.E. Sonntag. 1985. Fundamentals of classical ther- modynamics, 3rd ed. John Wiley and Sons, Inc., New York. W · Fluid Flow 2.3 or, dividing by g, in the form as (6b) The factors EM and EL are defined as positive, where gHM = EM represents energy added to the conduit flow by pumps or blowers, and gHL = EL represents energy dissipated, that is, converted into heat as mechanically nonrecoverable energy. A turbine or fluid motor thus has a negative HM or EM . For conduit systems with branches involving inflow or outflow, the total energies must be treated, and analysis is in terms of and not π. When real-fluid effects of viscosity or turbulence are included, the continuity relation in Equation (2b) is not changed, but V must be evaluated from the integral of the velocity profile, using time- averaged local velocities. In fluid flow past fixed boundaries, the velocity at the boundary is zero and shear stresses are produced. The equations of motion then become complex and exact solutions are difficult to find, except in simple cases. Laminar Flow For steady, fully developed laminar flow in a parallel-walled conduit, the shear stress τ varies linearly with distance y from the centerline. For a wide rectangular channel, where τw = wall shear stress = b (dp/ds) 2b = wall spacing s = flow direction Because the velocity is zero at the wall (y = b), the integrated result is This is the Poiseuille-flow parabolic velocity profile for a wide rectangular channel. The average velocity V is two-thirds the max- imum velocity (at y = 0), and the longitudinal pressure drop in terms of conduit flow velocity is (7) The parabolic velocity profile can also be derived for the axisym- metric conduit (pipe) of radius R but with a different constant. The average velocity is then half the maximum, and the pressure drop relation is (8) Turbulence Fluid flows are generally turbulent, involving random perturba- tions or fluctuations of the flow (velocity and pressure), character- ized by an extensive hierarchy of scales or frequencies (Robertson 1963). Flow disturbances that are not random, but have some degree of periodicity, such as the oscillating vortex trail behind bod- ies, have been erroneously identified as turbulence. Only flows involving random perturbations without any order or periodicity are turbulent; the velocity in such a flow varies with time or locale of measurement (Figure 2). Turbulence can be quantified by statistical factors. Thus, the velocity most often used in velocity profiles is the temporal average velocity v, and the strength of the turbulence is characterized by the root-mean-square of the instantaneous variation in velocity about this mean. The effects of turbulence cause the fluid to diffuse momentum, heat, and mass very rapidly across the flow. The Reynolds number Re, a dimensionless quantity, gives the relative ratio of inertial to viscous forces: where L = characteristic length ν = kinematic viscosity In flow through round pipes and tubes, the characteristic length is the diameter D. Generally, laminar flow in pipes can be expected if the Reynolds number, which is based on the pipe diameter, is less than 2000. Fully turbulent flow exists when ReD > 10 000. Between 2000 and 10 000, the flow is in a transition state and predictions are unreliable. In other geometries, different criteria for the Reynolds number exist. BASIC FLOW PROCESSES Wall Friction At the boundary of real-fluid flow, the relative tangential veloc- ity at the fluid surface is zero. Sometimes in turbulent flow studies, velocity at the wall may appear finite, implying a fluid slip at the wall. However, this is not the case; the difficulty is in velocity mea- surement (Goldstein 1938). Zero wall velocity leads to a high shear stress near the wall boundary and a slowing down of adjacent fluid layers. A velocity profile develops near a wall, with the velocity increasing from zero at the wall to an exterior value within a finite lateral distance. Laminar and turbulent flow differ significantly in their velocity profiles. Turbulent flow profiles are flat compared to the more pointed profiles of laminar flow (Figure 3). Near the wall, velocities of the turbulent profile must drop to zero more rapidly than those of the laminar profile, so the shear stress and friction are much greater in the turbulent flow case. Fully developed conduit flow may be characterized by the pipe factor, which is the ratio of average to maximum (centerline) velocity. Viscous velocity profiles result in pipe factors of 0.667 and 0.50 for wide rectangular and axisymmet- ric conduits. Figure 4 indicates much higher values for rectangular and circular conduits for turbulent flow. Due to the flat velocity pro- files, the kinetic energy factor α in Equation (6) ranges from 1.01 to 1.10 for fully developed turbulent pipe flow. p ρg ----- αV 2 2g ----- z+ +    1 HM+ p ρg ----- αV 2 2g ----- z+ +    2 HL+= m· π τ y b --    τw µ dv dy -----= = v b 2 y 2 – 2µ ----------------    dp ds -----= dp ds ----- 3µV b 2 ----------   –= dp ds ----- 8µV R 2 ----------   –= Fig. 2 Velocity Fluctuation at Point in Turbulent Flow Re VL ν⁄= 2.4 1997 ASHRAE Fundamentals Handbook (SI) Boundary Layer In most flows, the friction of a bounding wall on the fluid flow is evidenced by a boundary layer. For flow around bodies, this layer (which is quite thin relative to distances in the flow direction) encom- passes all viscous or turbulent actions, causing the velocity in it to vary rapidly from zero at the wall to that of the outer flow at its edge. Boundary layers are generally laminar near the start of their forma- tion but may become turbulent downstream of the transition point (Figure 5). For conduit flows, spacing between adjacent walls is gen- erally small compared with distances in the flow direction. As a result, layers from the walls meet at the centerline to fill the conduit. A significant boundary-layer occurrence exists in a pipeline or conduit following a well-rounded entrance (Figure 5). Layers grow from the walls until they meet at the center of the pipe. Near the start of the straight conduit, the layer is very thin (and laminar in all prob- ability), so the uniform velocity core outside has a velocity only slightly greater than the average velocity. As the layer grows in thickness, the slower velocity near the wall requires a velocity increase in the uniform core to satisfy continuity. As the flow pro- ceeds, the wall layers grow (and the centerline velocity increases) until they join, after an entrance length Le. Application of the Ber- noulli relation of Equation (5) to the core flow indicates a decrease in pressure along the layer. Ross (1956) shows that although the entrance length Le is many diameters, the length in which the pres- sure drop significantly exceeds those for fully developed flow is on the order of 10 diameters for turbulent flow in smooth pipes. In more general boundary-layer flows, as with wall layer devel- opment in a diffuser or for the layer developing along the surface of a strut or turning vane, pressure gradient effects can be severe and may even lead to separation. The development of a layer in an adverse-pressure gradient situation (velocity v1 at edge y = δ of layer decreasing in flow direction) with separation is shown in Figure 6. Downstream from the separation point, fluid backflows near the wall. Separation is due to frictional velocity (thus local kinetic energy) reduction near the wall. Flow near the wall no longer has energy to move into the higher pressure imposed by the decrease in v1 at the edge of the layer. The locale of this separation is difficult to predict, especially for the turbulent boundary layer. Analyses verify the experimental observation that a turbulent boundary layer is less subject to separation than a laminar one because of its greater kinetic energy. Flow Patterns with Separation In technical applications, flow with separation is common and often accepted if it is too expensive to avoid. Flow separation may be geometric or dynamic. Dynamic separation is shown in Figure 6. Geometric separation (Figures 7 and 8) results when a fluid stream passes over a very sharp corner, as with an orifice; the fluid gener- ally leaves the corner irrespective of how much its velocity has been reduced by friction. For geometric separation in orifice flow (Figure 7), the outer streamlines separate from the sharp corners and, because of fluid inertia, contract to a section smaller than the orifice opening, the vena contracta, with a limiting area of about six-tenths of the ori- fice opening. After the vena contracta, the fluid stream expands rather slowly through turbulent or laminar interaction with the fluid along its sides. Outside the jet, fluid velocity is small compared with that in the jet and is very disturbed. Strong turbulence vorticity helps spread out the jet, increases the losses, and brings the velocity dis- tribution back to a more uniform profile. Finally, at a considerable distance downstream, the velocity profile returns to the fully devel- oped flow of Figure 3. Other geometric separations (Figure 8) occur at a sharp entrance to a conduit, at an inclined plate or damper in a conduit, and at a sud- den expansion. For these, a vena contracta can be identified; for sud- den expansion, its area is that of the upstream contraction. Ideal- fluid theory, using free streamlines, provides insight and predicts contraction coefficients for valves, orifices, and vanes (Robertson 1965). These geometric flow separations are large loss-producing devices. To expand a flow efficiently or to have an entrance with minimum losses, the device should be designed with gradual con- tours, a diffuser, or a rounded entrance. Fig. 3 Velocity Profiles of Flow in Pipes Fig. 4 Pipe Factor for Flow in Conduits Fig. 5 Flow in Conduit Entrance Region Fig. 6 Boundary Layer Flow to Separation Fluid Flow 2.5 Flow devices with gradual contours are subject to separation that is more difficult to predict, because it involves the dynamics of boundary layer growth under an adverse pressure gradient rather than flow over a sharp corner. In a diffuser, which is used to reduce the loss in expansion, it is possible to expand the fluid some distance at a gentle angle without difficulty (particularly if the boundary layer is turbulent). Eventually, separation may occur (Figure 9), which is frequently asymmetrical because of irregularities. Down- stream flow involves flow reversal (backflow) and excess losses exist. Such separation is termed stall (Kline 1959). Larger area expansions may use splitters that divide the diffuser into smaller divisions less likely to have separations (Moore and Kline 1958). Another technique for controlling separation is to bleed some low- velocity fluid near the wall (Furuya et al. 1976). Alternatively, Heskested (1965, 1970) shows that suction at the corner of a sudden expansion has a strong positive effect on geometric separation. Drag Forces on Bodies or Struts Bodies in moving fluid streams are subjected to appreciable fluid forces or drag. Conventionally expressed in coefficient form, drag forces on bodies can be expressed as (9) where A is the projected (normal to flow) area of the body. The drag coefficient CD depends on the body’s shape and angularity and on the Reynolds number of the relative flow in terms of the body’s characteristic dimension. For Reynolds numbers of 103 to above 105, the CD of most bodies is constant due to flow separation, but above 105, the CD of rounded bodies drops suddenly as the surface boundary layer undergoes transition to turbulence. Typical CD values are given in Table 1; Hoerner (1965) gives expanded values. For a strut crossing a conduit, the contribution to the loss of Equation (6b) is (10) where Ac = conduit cross-sectional area A = area of the strut facing the flow Cavitation Liquid flow with gas- or vapor-filled pockets can occur if the absolute pressure is reduced to vapor pressure or less. In this case, a cavity or series of cavities forms, because liquids are rarely pure enough to withstand any tensile stressing or pressures less than vapor pressure for any length of time (John and Haberman 1980, Knapp et al. 1970, Robertson and Wislicenus 1969). Robertson and Wislicenus (1969) indicate significant occurrences in various tech- nical fields, chiefly in hydraulic equipment and turbomachines. Initial evidence of cavitation is the collapse noise of many small bubbles that appear initially as they are carried by the flow into regions of higher pressure. The noise is not deleterious and serves as a warning of the occurrence. As flow velocity further increases or pressure decreases, the severity of cavitation increases. More bub- bles appear and may join to form large fixed cavities. The space they occupy becomes large enough to modify the flow pattern and alter performance of the flow device. Collapse of the cavities on or near solid boundaries becomes so frequent that the cumulative impact in time results in damage in the form of cavitational erosion of the sur- face or excessive vibration. As a result, pumps can lose efficiency or their parts may erode locally. Control valves may be noisy or seri- ously damaged by cavitation. Cavitation in orifice and valve flow is indicated in Figure 10. With high upstream pressure and a low flow rate, no cavitation occurs. As pressure is reduced or flow rate increased, the minimum pressure in the flow (in the shear layer leaving the edge of the ori- fice) eventually approaches vapor pressure. Turbulence in this layer causes fluctuating pressures below the mean (as in vortex cores) and small bubble-like cavities. These are carried downstream into the region of pressure regain where they collapse, either in the fluid or Fig. 7 Geometric Separation, Flow Development, and Loss in Flow Through Orifice Fig. 8 Examples of Geometric Separation Encountered in Flows in Conduits D CDρAV 2 2⁄= Table 1 Drag Coefficients Body Shape 103 < Re < 2 × 105 Re > 3 × 105 Sphere 0.36 to 0.47 ~0.1 Disk 1.12 1.12 Streamlined strut 0.1 to 0.3 < 0.1 Circular cylinder 1.0 to 1.1 0.35 Elongated rectangular strut 1.0 to 1.2 1.0 to 1.2 Square strut ~2.0 ~2.0 Fig. 9 Separation in Flow in Diffuser HL CD A Ac -----    V 2 2g -----   = 2.8 1997 ASHRAE Fundamentals Handbook (SI) flow control devices to avoid flow dependence on downstream conditions. FLOW ANALYSIS Fluid flow analysis is used to correlate pressure changes with flow rates and the nature of the conduit. For a given pipeline, either the pressure drop for a certain flow rate, or the flow rate for a certain pressure difference between the ends of the conduit, is needed. Flow analysis ultimately involves comparing a pump or blower to a con- duit piping system for evaluating the expected flow rate. Generalized Bernoulli Equation Internal energy differences are generally small and usually the only significant effect of heat transfer is to change the density ρ. For gas or vapor flows, use the generalized Bernoulli equation in the pressure-over-density form of Equation (6a), allowing for the ther- modynamic process in the pressure-density relation: (25a) The elevation changes involving z are negligible and are dropped. The pressure form of Equation (5b) is generally unacceptable when appreciable density variations occur, because the volumetric flow rate differs at the two stations. This is particularly serious in fric- tion-loss evaluations where the density usually varies over consid- erable lengths of conduit (Benedict and Carlucci 1966). When the flow is essentially incompressible, Equation (25a) is satisfactory. Example 1. Specify the blower to produce an isothermal airflow of 200 L/s through a ducting system (Figure 12). Accounting for intake and fitting losses, the equivalent conduit lengths are 18 and 50 m and the flow is isothermal. The pressure at the inlet (station 1) and following the dis- charge (station 4), where the velocity is zero, are the same. The fric- tional losses HL are evaluated as 7.5 m of air between stations 1 and 2, and 72.3 m between stations 3 and 4. Solution: The following form of the generalized Bernoulli relation is used in place of Equation (25a), which also could be used: (25b) The term can be calculated as follows: The term can be calculated in a similar manner. In Equation (25b), HM is evaluated by applying the relation between any two points on opposite sides of the blower. Because condi- tions at stations 1 and 4 are known, they are used, and the location- specifying subscripts on the right side of Equation (25b) are changed to 4. Note that p1 = p4 = p, ρ1 = ρ4 = ρ, and V1 = V4 = 0. Thus, so HM = 82.2 m of air. For standard air (ρ = 1.20 kg/m3), this corre- sponds to 970 Pa. The pressure difference measured across the blower (between sta- tions 2 and 3), is often taken as the HM. It can be obtained by calculat- ing the static pressure at stations 2 and 3. Applying Equation (25b) successively between stations 1 and 2 and between 3 and 4 gives where α just ahead of the blower is taken as 1.06, and just after the blower as 1.03; the latter value is uncertain because of possible uneven discharge from the blower. Static pressures p1 and p4 may be taken as zero gage. Thus, The difference between these two numbers is 81 m, which is not the HM calculated after Equation (25b) as 82.2 m. The apparent discrep- ancy results from ignoring the velocity at stations 2 and 3. Actually, HM is the following: The required blower energy is the same, no matter how it is evalu- ated. It is the specific energy added to the system by the machine. Only when the conduit size and velocity profiles on both sides of the machine are the same is EM or HM simply found from ∆p = p3 − p2. Conduit Friction The loss term EL or HL of Equation (6a) or (6b) accounts for fric- tion caused by conduit-wall shearing stresses and losses from con- duit-section changes. HL is the loss of energy per unit weight (J/N) of flowing fluid. In real-fluid flow, a frictional shear occurs at bounding walls, gradually influencing the flow further away from the boundary. A lateral velocity profile is produced and flow energy is converted into heat (fluid internal energy), which is generally unrecoverable (a loss). This loss in fully developed conduit flow is evaluated through the Darcy-Weisbach equation: (26) where L is the length of conduit of diameter D and f is the friction factor. Sometimes a numerically different relation is used with the Fanning friction factor (one-quarter of f ). The value of f is nearly constant for turbulent flow, varying only from about 0.01 to 0.05. dp ρ ----- 1 2 ∫ α1 V1 2 2 ----- EM+ + α2 V2 2 2 ----- EL+= Fig. 12 Blower and Duct System for Example 1 p1 ρ1g⁄( ) α1 V1 2 2g⁄( ) z1 HM+ + + p2 ρ2g⁄( ) α2 V2 2 2g⁄( ) z2 HL+ + += V1 2 2g⁄ A1 π D 2 ---    2 π 0.250 2 ------------    2 0.0491 m 2 === V1 Q A1⁄ 0.200 0.0491 --------------- 4.07 m s⁄== = V1 2 2g⁄ 4.07( )2 2 9.8( )⁄ 0.846 m= = V2 2 2g⁄ p ρg⁄( ) 0 0.61 HM+ + + p ρg⁄( ) 0 3 7.5 72.3+( )+ + += p1 ρg⁄( ) 0 0.61 0+ + + p2 ρg⁄( ) 1.06 0.846×( ) 0 7.5+ + += p3 ρg⁄( ) 1.03 2.07×( )+ 0 0+ + p4 ρg⁄( ) 0 3 72.3+ + += p2 ρg⁄ 7.8 m of air–= p3 ρg⁄ 73.2 m of air= HM p3 ρg⁄( ) α3 V3 2 2g⁄( ) p2 ρg⁄( ) α2 V2 2 2g⁄( )+[ ]–+= 73.2 1.03 2.07×( )+ 7.8 1.06 0.846×( )+–[ ]–= 75.3 6.9–( )– 82.2 m== HL( ) f f L D ---    V 2 2g -----   = Fluid Flow 2.9 For fully developed laminar-viscous flow in a pipe, the loss is evaluated from Equation (8) as follows: (27) where Thus, for laminar flow, the friction factor varies inversely with the Reynolds number. With turbulent flow, friction loss depends not only on flow con- ditions, as characterized by the Reynolds number, but also on the nature of the conduit wall surface. For smooth conduit walls, empir- ical correlations give (28a) (28b) Generally, f also depends on the wall roughness ε. The mode of variation is complex and best expressed in chart form (Moody 1944) as shown in Figure 13. Inspection indicates that, for high Reynolds numbers and relative roughness, the friction factor becomes inde- pendent of the Reynolds number in a fully-rough flow regime. Then (29a) Values of f between the values for smooth tubes and those for the fully-rough regime are represented by Colebrook’s natural rough- ness function: (29b) A transition region appears in Figure 13 for Reynolds numbers between 2000 and 10 000. Below this critical condition, for smooth walls, Equation (27) is used to determine f ; above the critical con- dition, Equation (28b) is used. For rough walls, Figure 13 or Equa- tion (29b) must be used to assess the friction factor in turbulent flow. To do this, the roughness height ε, which may increase with conduit use or aging, must be evaluated from the conduit surface (Table 2). Fig. 13 Relation Between Friction Factor and Reynolds Number (Moody 1944) HL( ) f L ρg ----- 8µV R 2 ----------    32LνV D 2 g ---------------- 64 VD ν⁄ -------------- L D ---    V 2 2g -----   = = = Re VD ν and f 64 Re.⁄=⁄= f 0.3164 Re 0.25 ---------------= for Re 10 5< f 0.0032 0.221 Re 0.237 ----------------+= for 10 5 Re 3< < 10 6× 1 f -------- 1.14 2 log D ε⁄( )+= 1 f -------- 1.14 2 log D ε⁄( )+= 2 log 1 9.3 Re ε D⁄( ) f --------------------------------+– 2.10 1997 ASHRAE Fundamentals Handbook (SI) Although the preceding discussion has focused on circular pipes and ducts, air ducts are often rectangular in cross section. The equiv- alent circular conduit corresponding to the noncircular conduit must be found before Figure 13 or Equations (28) or (29) can be used. Based on turbulent flow concepts, the equivalent diameter is determined by (30) where A = flow area Pw = wetted perimeter of the cross section For turbulent flow, Deq is substituted for D in Equation (26) and the Reynolds number definition in Equation (27). Noncircular duct friction can be evaluated to within 5% for all except very extreme cross sections. A more refined method for finding the equivalent circular duct diameter is given in Chapter 32. With laminar flow, the loss predictions may be off by a factor as large as two. Section Change Effects and Losses Valve and section changes (contractions, expansions and diffus- ers, elbows or bends, tees), as well as entrances, distort the fully developed velocity profiles (Figure 3) and introduce extra flow losses (dissipated as heat) into pipelines or duct systems. Valves produce such extra losses to control flow rate. In contractions and expansions, flow separation as shown in Figures 8 and 9 causes the extra loss. The loss at rounded entrances develops as the flow accel- erates to higher velocities. The resulting higher velocity near the wall leads to wall shear stresses greater than those of fully devel- oped flow (Figure 5). In flow around bends, the velocity increases along the inner wall near the start of the bend. This increased veloc- ity creates a secondary motion, which is a double helical vortex pat- tern of flow downstream from the bend. In all these devices, the disturbance produced locally is converted into turbulence and appears as a loss in the downstream region. The return of disturbed flow to a fully developed velocity profile is quite slow. Ito (1962) showed that the secondary motion follow- ing a bend takes up to 100 diameters of conduit to die out but the pressure gradient settles out after 50 diameters. With laminar flow following a rounded entrance, the entrance length depends on the Reynolds number: (31) At Re = 2000, a length of 120 diameters is needed to establish the parabolic profile. The pressure gradient reaches the developed value of Equation (26) much sooner. The extra drop is 1.2V2/2g; the change in profile from uniform to parabolic results in a drop of 1.0V2/2g (since α = 2.0), and the rest is due to excess friction. With turbulent flow, 80 to 100 diameters following the rounded entrance are needed for the velocity profile to become fully developed, but the friction loss per unit length reaches a value close to that of the fully developed flow value more quickly. After six diameters, the loss rate at a Reynolds number of 105 is only 14% above that of fully developed flow in the same length, while at 107, it is only 10% higher (Robertson 1963). For a sharp entrance, the flow separation (Figure 8) causes a greater disturbance, but fully developed flow is achieved in about half the length required for a rounded entrance. With sudden expansion, the pressure change settles out in about eight times the diameter change (D2 = D1), while the velocity profile takes at least a 50% greater distance to return to fully developed pipe flow (Lipstein 1962). These disturbance effects are assumed compressed (in the flow direction) into a point, and the losses are treated as locally occur- ring. Such a loss is related to the velocity by the fitting loss coeffi- cient K: (32) Chapter 33 and the Pipe Friction Manual (Hydraulic Institute 1961) have information for pipe applications. Chapter 32 gives infor- mation for airflow. The same type of fitting in pipes and ducts may give a different loss, because flow disturbances are controlled by the detailed geometry of the fitting. The elbow of a small pipe may be a threaded fitting that differs from a bend in a circular duct. For 90 screw-fitting elbows, K is about 0.8 (Ito 1962), whereas smooth flanged elbows have a K as low as 0.2 at the optimum curvature. Table 3 gives a list of fitting loss coefficients. These values indi- cate the losses, but there is considerable variance. Expansion flows, such as from one conduit size to another or at the exit into a room or reservoir, are not included. For such occurrences, the Borda loss pre- diction (from impulse-momentum considerations) is appropriate: (33) Such expansion loss is reduced by avoiding or delaying separa- tion using a gradual diffuser (Figure 9). For a diffuser of about 7° total angle, the loss is minimal, about one-sixth that given by Equa- tion (33). The diffuser loss for total angles above 45 to 60° exceeds that of the sudden expansion, depending somewhat on the diameter ratio of the expansion. Optimum design of diffusers involves many factors; excellent performance can be achieved in short diffusers with splitter vanes or suction. Turning vanes in miter bends produce the least disturbance and loss for elbows; with careful design, the loss coefficient can be reduced to as low as 0.1. For losses in smooth elbows, Ito (1962) found a Reynolds num- ber effect (K slowly decreasing with increasing Re) and a minimum loss at a bend curvature (bend radius to diameter ratio) of 2.5. At this optimum curvature, a 45° turn had 63%, and a 180° turn approxi- mately 120%, of the loss of a 90° bend. The loss does not vary lin- early with the turning angle because secondary motion occurs. Use of coefficient K presumes its independence of the Reynolds number. Crane Co. (1976) found a variation with the Reynolds num- ber similar to that of the friction factor; Kittridge and Rowley (1957) observed it only with laminar flow. Assuming that K varies with Re similarly to f , it is convenient to represent fitting losses as adding to the effective length of uniform conduit. The effective length of a fitting is then (34) where fref is an appropriate reference value of the friction factor. Deissler (1951) uses 0.028, and the air duct values in Chapter 32 are based on an fref of about 0.02. For rough conduits, appreciable errors can occur if the relative roughness does not correspond to that used when fref was fixed. It is unlikely that the fitting losses involving separation are affected by pipe roughness. The effective length method for fitting loss evaluation is still useful. When a conduit contains a number of section changes or fittings, the values of K are added to the fL/D friction loss, or the Leff /D of the fittings are added to the conduit length L/D for evaluating the Table 2 Effective Roughness of Conduit Surfaces Material ε, µm Commercially smooth brass, lead, copper, or plastic pipe 1.5 Steel and wrought iron 46 Galvanized iron or steel 150 Cast iron 250 Deq 4A Pw⁄= Le D⁄ 0.06 Re≈ Loss of section K V 2 2g -----   = Loss at expansion V1 V2–( )2 2g ------------------------- V1 2 2g ----- 1 A1 A2 -----–    2 = = Leff D⁄ K fref⁄= Fluid Flow 2.13 (40) where R is the radius of curvature of the bend. Again, a discharge coefficient Cd is needed; as in Figure 18, this drops off for the lower Reynolds numbers (below 105). These devices are calibrated in pipes with fully developed velocity profiles, so they must be located far enough downstream of sections that modify the approach velocity. Unsteady Flow Conduit flows are not always steady. In a compressible fluid, the acoustic velocity is usually high and the conduit length is rather short, so the time of signal travel is negligibly small. Even in the incompressible approximation, system response is not instanta- neous. If a pressure difference ∆p is applied between the conduit ends, the fluid mass must be accelerated and wall friction overcome, so a finite time passes before the steady flow rate corresponding to the pressure drop is achieved. The time it takes for an incompressible fluid in a horizontal con- stant-area conduit of length L to achieve steady flow may be esti- mated by using the unsteady flow equation of motion with wall friction effects included. On the quasi-steady assumption, friction is given by Equation (26); also by continuity, V is constant along the conduit. The occurrences are characterized by the relation (41) where θ = time s = distance in the flow direction Since a certain ∆p is applied over the conduit length L, (42) For laminar flow, f is given by Equation (27), and (43) Equation (43) can be rearranged and integrated to yield the time to reach a certain velocity: (44) and (45a) For long times (θ → ∞), this indicates steady velocity as (45b) as by Equation (8). Then, Equation (45a) becomes (46) where The general nature of velocity development for starting-up flow is derived by more complex techniques; however, the temporal variation is as given above. For shutdown flow (steady flow with ∆p = 0 at θ > 0), the flow decays exponentially as e−θ. Turbulent flow analysis of Equation (41) also must be based on the quasi-steady approximation, with less justification. Daily et al. (1956) indicate that the frictional resistance is slightly greater than the steady-state result for accelerating flows, but appreciably less for decelerating flows. If the friction factor is approximated as constant, and, for the accelerating flow, or Because the hyperbolic tangent is zero when the independent variable is zero and unity when the variable is infinity, the initial (V = 0 at θ = 0) and final conditions are verified. Thus, for long times (θ → ∞), which is in accord with Equation (26) when f is constant (the flow regime is the fully rough one of Figure 13). The temporal velocity variation is then (47) Fig. 18 Flowmeter Coefficients Qtheor πD 2 4 --------- R 2D ------- 2 p∆ ρ ---------   = dV dθ ------ 1 ρ --    dp ds ------ fV 2 2D -------+ + 0= dV dθ ------ p∆ ρL ------ fV 2 2D -------–= dV dθ ------ p∆ ρL ------ 32µV ρD 2 -------------– A BV–= = θ θd∫ Vd A BV– ----------------∫ 1 B -- A BV–( )ln–= = = V p∆ L ----- D 2 32µ ---------    1 ρL p∆ ------ 32νθ– D 2 ---------------   exp–= V∞ p∆ L ----- D 2 32µ ---------    p∆ L ----- R 2 8µ ------   = = V V∞ 1 ρL p∆ ------ f∞V∞θ– 2D -------------------   exp–= f∞ 64ν V∞D -----------= dV dθ ------ p∆ ρL ------ fV 2 2D -------– A BV 2–= = θ 1 AB ------------- tanh 1– V B A ----   = V A B ---- tanh θ AB( )= V∞ A B ---- p∆ ρL⁄ f∞ 2D⁄ ---------------- p∆ ρL ------ 2D f∞ ------   = = = V V∞ f∞ V∞θ 2D⁄( )tanh= 2.14 1997 ASHRAE Fundamentals Handbook (SI) In Figure 19, the turbulent velocity start-up result is compared with the laminar one in Figure 19, where initially the turbulent is steeper but of the same general form, increasing rapidly at the start but reaching V∞ asymptotically. NOISE FROM FLUID FLOW Noise in flowing fluids results from unsteady flow fields and can be at discrete frequencies or broadly distributed over the audible range. With liquid flow, cavitation results in noise through the col- lapse of vapor bubbles. The noise in pumps or fittings (such as valves) can be a rattling or sharp hissing sound. It is easily elimi- nated by raising the system pressure. With severe cavitation, the resulting unsteady flow can produce indirect noise from induced vibration of adjacent parts. See Chapter 46 of the 1999 ASHRAE Handbook—Applications for more information on sound control. The disturbed laminar flow behind cylinders can be an oscillat- ing motion. The shedding frequency f of these vortexes is charac- terized by a Strouhal number St = fd/V of about 0.21 for a circular cylinder of diameter d, over a considerable range of Reynolds num- bers. This oscillating flow can be a powerful noise source, particu- larly when f is close to the natural frequency of the cylinder or some nearby structural member so that resonance occurs. With cylinders of another shape, such as impeller blades of a pump or blower, the characterizing Strouhal number involves the trailing edge thickness of the member. The strength of the vortex wake, with its resulting vibrations and noise potential, can be reduced by breaking up the flow with downstream splitter plates or boundary-layer trip devices (wires) on the cylinder surface. Noise produced in pipes and ducts, especially from valves and fittings, is associated with the loss through such elements. The sound pressure of noise in water pipe flow increases linearly with the pressure loss; the broad-band noise increases, but only in the lower frequency range. Fitting-produced noise levels also increase with fitting loss (even without cavitation) and significantly exceed noise levels of the pipe flow. The relation between noise and loss is not surprising because both involve excessive flow perturbations. A valve’s pressure-flow characteristics and structural elasticity may be such that for some operating point it oscillates, perhaps in reso- nance with part of the piping system, to produce excessive noise. A change in the operating point conditions or details of the valve geometry can result in significant noise reduction. Pumps and blowers are strong potential noise sources. Turbo- machinery noise is associated with blade-flow occurrences. Broad- band noise appears from vortex and turbulence interaction with walls and is primarily a function of the operating point of the machine. For blowers, it has a minimum at the peak efficiency point (Groff et al. 1967). Narrow-band noise also appears at the blade- crossing frequency and its harmonics. Such noise can be very annoying because it stands out from the background. To reduce this noise, increase clearances between impeller and housing, and space impeller blades unevenly around the circumference. REFERENCES Baines, W.D. and E.G. Peterson. 1951. An investigation of flow through screens. ASME Transactions 73:467. Baker, A.J. 1983. Finite element computational fluid mechanics. McGraw- Hill, New York. Ball, J.W. 1957. Cavitation characteristics of gate valves and globe values used as flow regulators under heads up to about 125 ft. ASME Transac- tions 79:1275. Benedict, R.P. and N.A. Carlucci. 1966. Handbook of specific losses in flow systems. Plenum Press Data Division, New York. Binder, R.C. 1944. Limiting isothermal flow in pipes. ASME Transactions 66:221. Bober, W. and R.A. Kenyon. 1980. Fluid mechanics. John Wiley and Sons, New York. Colborne, W.G. and A.J. Drobitch. 1966. An experimental study of non-iso- thermal flow in a vertical circular tube. ASHRAE Transactions 72(4):5. Crane Co. 1976. Flow of fluids. Technical Paper No. 410. New York. Daily, J.W., et al. 1956. Resistance coefficients for accelerated and deceler- ated flows through smooth tubes and orifices. ASME Transactions 78:1071. Deissler, R.G. 1951. Laminar flow in tubes with heat transfer. National Advisory Technical Note 2410, Committee for Aeronautics. Furuya, Y., T. Sate, and T. Kushida. 1976. The loss of flow in the conical with suction at the entrance. Bulletin of the Japan Society of Mechanical Engineers 19:131. Goldstein, S., ed. 1938. Modern developments in fluid mechanics. Oxford University Press, London. Reprinted by Dover Publications, New York. Groff, G.C., J.R. Schreiner, and C.E. Bullock. 1967. Centrifugal fan sound power level prediction. ASHRAE Transactions 73(II): V.4.1. Heskested, G. 1965. An edge suction effect. AIAA Journal 3:1958. Heskested, G. 1970. Further experiments with suction at a sudden enlarge- ment. Journal of Basic Engineering, ASME Transactions 92D:437. Hoerner, S.F. 1965. Fluid dynamic drag, 3rd ed. Published by author, Mid- land Park, NJ. Hydraulic Institute. 1961. Pipe friction manual. New York. Ito, H. 1962. Pressure losses in smooth pipe bends. Journal of Basic Engi- neering, ASME Transactions 4(7):43. John, J.E.A. and W.L. Haberman. 1980. Introduction to fluid mechanics, 2nd ed. Prentice Hall, Englewood Cliffs, NJ. Kittridge, C.P. and D.S. Rowley. 1957. Resistance coefficients for laminar and turbulent flow through one-half inch valves and fittings. ASME Transactions 79:759. Kline, S.J. 1959. On the nature of stall. Journal of Basic Engineering, ASME Transactions 81D:305. Knapp, R.T., J.W. Daily, and F.G. Hammitt. 1970. Cavitation. McGraw-Hill, New York. Lipstein, N.J. 1962. Low velocity sudden expansion pipe flow. ASHRAE Journal 4(7):43. Moody, L.F. 1944. Friction factors for pipe flow. ASME Transactions 66:672. Moore, C.A. and S.J. Kline. 1958. Some effects of vanes and turbulence in two-dimensional wide-angle subsonic diffusers. National Advisory Committee for Aeronautics, Technical Memo 4080. Murdock, J.W., C.J. Foltz, and C. Gregory. 1964. Performance characteris- tics of elbow flow meters. Journal of Basic Engineering, ASME Trans- actions 86D:498. Olson, R.M. 1980. Essentials of engineering fluid mechanics, 4th ed. Harper and Row, New York. Robertson, J.M. 1963. A turbulence primer. University of Illinois (Urbana, IL), Engineering Experiment Station Circular 79. Robertson, J.M. 1965. Hydrodynamics in theory and application. Prentice- Hall, Englewood Cliffs, NJ. Robertson, J.M. and G.F. Wislicenus, ed. 1969 (discussion 1970). Cavitation state of knowledge. American Society of Mechanical Engineers, New York. Ross, D. 1956. Turbulent flow in the entrance region of a pipe. ASME Trans- actions 78:915. Schlichting, H. 1979. Boundary layer theory, 7th ed. McGraw-Hill, New York. Streeter, V.L. and E.B. Wylie. 1979. Fluid mechanics, 7th ed. McGraw-Hill, New York. Wile, D.D. 1947. Air flow measurement in the laboratory. Refrigerating Engineering: 515. Fig. 19 Temporal Increase in Velocity Following Sudden Application of Pressure CHAPTER 3 HEAT TRANSFER Heat Transfer Processes ........................................................... 3.1 Steady-State Conduction ........................................................... 3.1 Overall Heat Transfer ............................................................... 3.2 Transient Heat Flow ................................................................. 3.4 Thermal Radiation .................................................................... 3.6 Natural Convection ................................................................. 3.11 Forced Convection .................................................................. 3.12 Extended Surface ..................................................................... 3.17 Symbols ................................................................................... 3.22 EAT is energy in transit due to a temperature difference. TheHthermal energy is transferred from one region to another by three modes of heat transfer: conduction, convection, and radia- tion. Heat transfer is among a group of energy transport phenomena that includes mass transfer (see Chapter 5), momentum transfer or fluid friction (see Chapter 2), and electrical conduction. Transport phenomena have similar rate equations, in which flux is propor- tional to a potential difference. In heat transfer by conduction and convection, the potential difference is the temperature difference. Heat, mass, and momentum transfer are often considered together because of their similarities and interrelationship in many common physical processes. This chapter presents the elementary principles of single-phase heat transfer with emphasis on heating, refrigerating, and air condi- tioning. Boiling and condensation are discussed in Chapter 4. More specific information on heat transfer to or from buildings or refrig- erated spaces can be found in Chapters 24 through 30 of this volume and in Chapter 12 of the 1998 ASHRAE Handbook—Refrigeration. Physical properties of substances can be found in Chapters 18, 22, 24, and 36 of this volume and in Chapter 8 of the 1998 ASHRAE Handbook—Refrigeration. Heat transfer equipment, including evaporators, condensers, heating and cooling coils, furnaces, and radiators, is covered in the 2000 ASHRAE Handbook—Systems and Equipment. For further information on heat transfer, see the section on Bibliography. HEAT TRANSFER PROCESSES Thermal Conduction. This is the mechanism of heat transfer whereby energy is transported between parts of a continuum by the transfer of kinetic energy between particles or groups of particles at the atomic level. In gases, conduction is caused by elastic collision of molecules; in liquids and electrically nonconducting solids, it is believed to be caused by longitudinal oscillations of the lattice structure. Thermal conduction in metals occurs, like electrical con- duction, through the motion of free electrons. Thermal energy trans- fer occurs in the direction of decreasing temperature, a consequence of the second law of thermodynamics. In solid opaque bodies, ther- mal conduction is the significant heat transfer mechanism because no net material flows in the process. With flowing fluids, thermal conduction dominates in the region very close to a solid boundary, where the flow is laminar and parallel to the surface and where there is no eddy motion. Thermal Convection. This form of heat transfer involves energy transfer by fluid movement and molecular conduction (Burmeister 1983, Kays and Crawford 1980). Consider heat transfer to a fluid flowing inside a pipe. If the Reynolds number is large enough, three different flow regions exist. Immediately adjacent to the wall is a laminar sublayer where heat transfer occurs by thermal conduction; outside the laminar sublayer is a transition region called the buffer layer, where both eddy mixing and conduction effects are significant; beyond the buffer layer and extending to the center of the pipe is the turbulent region, where the dominant mechanism of transfer is eddy mixing. In most equipment, the main body of fluid is in turbulent flow, and the laminar layer exists at the solid walls only. In cases of low- velocity flow in small tubes, or with viscous liquids such as oil (i.e., at low Reynolds numbers), the entire flow may be laminar with no transition or turbulent region. When fluid currents are produced by external sources (for exam- ple, a blower or pump), the solid-to-fluid heat transfer is termed forced convection. If the fluid flow is generated internally by non- homogeneous densities caused by temperature variation, the heat transfer is termed free convection or natural convection. Thermal Radiation. In conduction and convection, heat trans- fer takes place through matter. In thermal radiation, there is a change in energy form from internal energy at the source to elec- tromagnetic energy for transmission, then back to internal energy at the receiver. Whereas conduction and convection are affected primarily by temperature difference and somewhat by temperature level, the heat transferred by radiation increases rapidly as the tem- perature increases. Although some generalized heat transfer equations have been mathematically derived from fundamentals, they are usually obtained from correlations of experimental data. Normally, the cor- relations employ certain dimensionless numbers, shown in Table 1, that are derived from dimensional analysis or analogy. STEADY-STATE CONDUCTION For steady-state heat conduction in one dimension, the Fourier law is (1) where q = heat flow rate, W k = thermal conductivity, W/(m·K) A = cross-sectional area normal to flow, m2 dt/dx = temperature gradient, K/m Equation (1) states that the heat flow rate q in the x direction is directly proportional to the temperature gradient dt/dx and the cross- sectional area A normal to the heat flow. The proportionality factor is the thermal conductivity k. The minus sign indicates that the heat flow is positive in the direction of decreasing temperature. Conduc- tivity values are sometimes given in other units, but consistent units must be used in Equation (1). The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow. q kA( ) dt dx -----–= 3.4 1997 ASHRAE Fundamentals Handbook (SI) Calculations using Equation (14) and ∆tm are convenient when terminal temperatures are known. In many cases, however, the tem- peratures of the fluids leaving the exchanger are not known. To avoid trial-and-error calculations, an alternate method involves the use of three nondimensional parameters, defined as follows: 1. Exchanger Heat Transfer Effectiveness ε (16) where Ch = ( cp)h = hot fluid capacity rate, W/K Cc = ( cp)c = cold fluid capacity rate, W/K Cmin = smaller of capacity rates Ch and Cc th = terminal temperature of hot fluid, °C. Subscript i indicates enter- ing condition; subscript o indicates leaving condition. tc = terminal temperature of cold fluid, °C. Subscripts i and o are the same as for th. 2. Number of Exchanger Heat Transfer Units (NTU) (17) where A is the area used to define overall coefficient U. 3. Capacity Rate Ratio Z (18) Generally, the heat transfer effectiveness can be expressed for a given exchanger as a function of the number of transfer units and the capacity rate ratio: (19) The effectiveness is independent of the temperatures in the exchanger. For any exchanger in which the capacity rate ratio Z is zero (where one fluid undergoes a phase change; e.g., in a condenser or evaporator), the effectiveness is (20) Heat transferred can be determined from (21) Combining Equations (16) and (21) produces an expression for heat transfer rate in terms of entering fluid temperatures: (22) The proper mean temperature difference for Equation (14) is then given by (23) The effectiveness for parallel flow exchangers is (24) For Z = 1, (25) The effectiveness for counterflow exchangers is (26) (27) Incropera and DeWitt (1996) and Kays and London (1984) show the relations of ε, NTU, and Z for other flow arrangements. These authors and Afgan and Schlunder (1974) present graphical repre- sentations for convenience. TRANSIENT HEAT FLOW Often, the heat transfer and temperature distribution under unsteady-state (varying with time) conditions must be known. Examples are (1) cold storage temperature variations on starting or stopping a refrigeration unit; (2) variation of external air temperature and solar irradiation affecting the heat load of a cold storage room or wall temperatures; (3) the time required to freeze a given material under certain conditions in a storage room; (4) quick freezing of objects by direct immersion in brines; and (5) sudden heating or cooling of fluids and solids from one temperature to a different temperature. The equations describing transient temperature distribution and heat transfer are presented in this section. Numerical methods are the simplest means of solving these equations because numerical data are easy to obtain. However, with some numerical solutions and off-the-shelf software, the physics that drives the energy trans- port can be lost. Thus, analytical solution techniques are also included in this section. The fundamental equation for unsteady-state conduction in sol- ids or fluids in which there is no substantial motion is (28) where thermal diffusivity α is the ratio k/ρcp; k is thermal conduc- tivity; ρ, density; and cp, specific heat. If α is large (high conductiv- ity, low density and specific heat, or both), heat will diffuse faster. One of the most elementary transient heat transfer models pre- dicts the rate of temperature change of a body or material being held at constant volume with uniform temperature, such as a well-stirred reservoir of fluid whose temperature is changing because of a net rate of heat gain or loss: (29) where M is the mass of the body, and cv is its specific heat at con- stant volume. qnet is algebraic, with positive being into the body and negative being out of the body. If the heating occurs at constant pressure, cv should be replaced by cp; however, for liquids and sol- ids, cv and cp are nearly equal, and cp can be used with negligible error. The term qnet may include heat transfer by conduction, con- vection, or radiation and is the difference between the heat transfer rates into and out of the body. From Equations (28) and (29), it is possible to derive expressions for temperature and heat flow variations at different instants and different locations. Most common cases have been solved and ε thi tho–( ) thi tci–( ) -----------------------= when Ch Cmin= ε tco tci–( ) thi tci–( ) ----------------------= when Cc Cmin= m· m· NTU AUavg Cmin --------------- 1 Cmin ----------- U A∫ dA== Z Cmin Cmax ------------= ε f NTU, Z, flow arrangement( )= ε 1 exp NTU–( )–= q Ch thi tho–( ) Cc tco tci–( )== q εCmin thi tci–( )= tm∆ thi tci–( )ε NTU ------------------------= ε 1 exp NTU–[– 1 Z+( ) ] 1 Z+ --------------------------------------------------------= ε 1 exp 2 NTU–( )– 2 ------------------------------------------= ε 1 exp NTU– 1 Z–( )[ ]– 1 Z exp NTU– 1 Z–( )[ ]– -------------------------------------------------------------= ε NTU 1 NTU+ --------------------- for Z 1== ∂t ∂τ ----- α ∂2 t ∂x 2 ------- ∂2 t ∂y 2 ------- ∂2 t ∂z 2 -------+ +       = qnet Mcv( ) dt dτ -----= Heat Transfer 3.5 presented in graphical forms (Jakob 1957, Schneider 1964, Myers 1971). In other cases, it is simpler to use numerical methods (Croft and Lilley 1977, Patankar 1980). When convective boundary con- ditions are required in the solution of Equations (28) and (29), h val- ues based on steady-state correlations are often used. However, this approach may not be valid when rapid transients are involved. Estimating Cooling Times Cooling times for materials can be estimated (McAdams 1954) by Gurnie-Lurie charts (Figures 2, 3, and 4), which are graphical solutions for the heating or cooling of infinite slabs, infinite cylin- ders, and spheres. These charts assume an initial uniform tempera- ture distribution and no change of phase. They apply to a body exposed to a constant temperature fluid with a constant surface con- vection coefficient of h. Using Figures 2, 3, and 4, it is possible to estimate both the tem- perature at any point and the average temperature in a homogeneous mass of material as a function of time in a cooling process. It is pos- sible to estimate cooling times for rectangular-shaped solids, cubes, cylinders, and spheres. From the point of view of heat transfer, a cylinder insulated on its ends behaves like a cylinder of infinite length, and a rectangular solid insulated so that only two parallel faces allow heat transfer behaves like an infinite slab. A thin slab or a long, thin cylinder may be also considered infinite objects. Consider a slab having insulated edges being cooled. If the cool- ing time is the time required for the center of the slab to reach a tem- perature of t2, the cooling time can be calculated as follows: 1. Evaluate the temperature ratio (tc − t2)/(tc − t1). where tc = temperature of cooling medium t1 = initial temperature of product t2 = final temperature of product at center Note that in Figures 2, 3, and 4, the temperature ratio (tc − t2)/ (tc − t1) is designated as Y to simplify the equations. 2. Determine the radius ratio r/rm designated as n in Figures 2, 3, and 4. where r = distance from centerline rm = half thickness of slab 3. Evaluate the resistance ratio k/hrm designated as m in Figures 2, 3, and 4. where k = thermal conductivity of material h = heat transfer coefficient 4. From Figure 2 for infinite slabs, select the appropriate value of kτ/ρcp rm 2 designated as Fo in Figures 2, 3, and 4. where τ = time elapsed cp = specific heat ρ = density 5. Determine τ from the value of kτ/ρcprm 2. Temperature Distribution in Finite Objects Finite objects can be formed from the intersection of infinite objects. For example, the solid of intersection of an infinite cylinder and an infinite slab is a finite cylinder with a length equal to the thickness of the slab and a radius equal to that of the cylinder (Fig- ure 5). Intersection of three infinite slabs with the same thickness produces a cube; intersection of three dissimilar slabs forms a finite rectangular solid. The temperature in the finite object can be calculated from the temperature ratio Y of the infinite objects that intersect to form the finite object. The product of the temperature ratios of the infinite Fig. 2 Transient Temperatures for Infinite Slab 3.6 1997 ASHRAE Fundamentals Handbook (SI) objects is the temperature ratio of the finite object; for example, for the finite cylinder of Figure 5, (30) where Yfc = temperature ratio of finite cylinder Yis = temperature ratio of infinite slab Yic = temperature ratio of infinite cylinder For a finite rectangular solid, (31) where Yfrs = temperature ratio of finite rectangular solid, and sub- scripts 1, 2, and 3 designate three infinite slabs. Heat Exchanger Transients Determination of the transient behavior of heat exchangers is becoming increasingly important in evaluating the dynamic behav- ior of heating and air-conditioning systems. Many studies of the transient behavior of counterflow and parallel flow heat exchangers have been conducted; some are listed in the section on Bibliography. THERMAL RADIATION Radiation, one of the basic mechanisms for energy transfer between different temperature regions, is distinguished from con- duction and convection in that it does not depend on an interme- diate material as a carrier of energy but rather is impeded by the presence of material between the regions. The radiation energy transfer process is the consequence of energy-carrying electro- magnetic waves that are emitted by atoms and molecules due to changes in their energy content. The amount and characteristics of radiant energy emitted by a quantity of material depend on the nature of the material, its microscopic arrangement, and its abso- lute temperature. Although rate of energy emission is indepen- dent of the surroundings, the net energy transfer rate depends on the temperatures and spatial relationships of the surface and its surroundings. Blackbody Radiation The rate of thermal radiant energy emitted by a surface depends on its absolute temperature. A surface is called black if it can absorb all incident radiation. The total energy emitted per unit time per unit area of black surface Wb to the hemispherical region above it is given by the Stefan-Boltzmann law. Fig. 3 Transient Temperatures for Infinite Cylinder Yfc YisYic= Yfrs Yis( ) 1 Yis( ) 2 Yis( ) 3 = Heat Transfer 3.9 Certain flat black paints also exhibit emittances of 98% over a wide range of conditions. They provide a much more durable sur- face than gold or platinum black and are frequently used on radia- tion instruments and as standard reference in emittance or reflectance measurements. Kirchhoff’s law relates emittance and absorptance of any opaque surface from thermodynamic considerations; it states that for any surface where the incident radiation is independent of angle or where the surface is diffuse, ελ = αλ. If the surface is gray, or the incident radiation is from a black surface at the same temperature, then ε = α as well, but many surfaces are not gray. For most surfaces listed in Table 3, absorptance for solar radiation is different from emittance for low-temperature radiation. This is because the wave- length distributions are different in the two cases, and ελ varies with wavelength. The foregoing discussion relates to total hemispherical radiation from surfaces. Energy distribution over the hemispherical region above the surface also has an important effect on the rate of heat transfer in various geometric arrangements. Lambert’s law states that the emissive power of radiant energy over a hemispherical surface above the emitting surface varies as the cosine of the angle between the normal to the radiating surface and the line joining the radiating surface to the point of the hemi- spherical surface. This radiation is diffuse radiation. The Lambert emissive power variation is equivalent to assuming that radiation from a surface in a direction other than normal occurs as if it came from an equivalent area with the same emissive power (per unit area) as the original surface. The equivalent area is obtained by pro- jecting the original area onto a plane normal to the direction of radi- ation. Black surfaces obey the Lambert law exactly. The law is approximate for many actual radiation and reflection processes, especially those involving rough surfaces and nonmetallic materi- als. Most radiation analyses are based on the assumption of gray dif- fuse radiation and reflection. In estimating heat transfer rates between surfaces of different geometries, radiation characteristics, and orientations, it is usually assumed that • All surfaces are gray or black • Radiation and reflection are diffuse • Properties are uniform over the surfaces • Absorptance equals emittance and is independent of the temper- ature of the source of incident radiation • The material in the space between the radiating surfaces neither emits nor absorbs radiation These assumptions greatly simplify problems, although results must be considered approximate. Angle Factor The distribution of radiation from a surface among the surfaces it irradiates is indicated by a quantity variously called an intercep- tion, a view, a configuration, or an angle factor. In terms of two surfaces i and j, the angle factor Fij from surface i to surface j is defined as the fraction of diffuse radiant energy leaving surface i that falls directly on j (i.e., is intercepted by j). The angle factor from j to i is similarly defined, merely by interchanging the roles of i and j. This second angle factor is not, in general, numerically equal to the first. However, the reciprocity relation Fij Ai = Fji Aj, where A is the surface area, is always valid. Note that a concave surface may “see itself” (Fii ≠ 0), and that if n surfaces form an enclosure, (40) The angle factor F12 between two surfaces is (41) where dA1, and dA2 are elemental areas of the two surfaces, r is the distance between dA1 and dA2, and φ1 and φ2 are the angles between the respective normals to dA1 and dA2 and the connecting line r. Numerical, graphical, and mechanical techniques can solve this equation (Siegel and Howell 1981, Modest 1993). Numerical values of the angle factor for common geometries are given in Figure 6. Calculation of Radiant Exchange Between Surfaces Separated by Nonabsorbing Media A surface radiates energy at a rate independent of its surroundings and absorbs and reflects incident energy at a rate dependent on its surface condition. The net energy exchange per unit area is denoted by q or qj for unit area Aj. It is the rate of emission of the surface minus the total rate of absorption at the surface from all radiant effects in its surroundings, possibly including the return of some of its own emission by reflection off its surroundings. The rate at which energy must be supplied to the surface by other exchange processes if its temperature is to remain constant is q; therefore, to define q, the total radiant surroundings (in effect, an enclosure) must be specified. Several methods have been developed to solve certain problems. To calculate the radiation exchange at each surface of an enclosure of n opaque surfaces by simple, general equations convenient for machine calculation, two terms must be defined: G = irradiation; total radiation incident on surface per unit time and per unit area J = radiosity; total radiation that leaves surface per unit time and per unit area The radiosity is the sum of the energy emitted and the energy reflected: (42) Because the transmittance is zero, the reflectance is Thus, (43) The net energy lost by a surface is the difference between the radi- osity and the irradiation: (44) Substituting for G in terms of J from Equation (43), (45) Consider an enclosure of n isothermal surfaces with areas of A1, A2, …, An, emittances of ε1, ε2, …, εn, and reflectances of ρ1, ρ2, …, ρn, respectively. The irradiation of surface i is the sum of the radiation incident on it from all n surfaces: or Fij j 1= n ∑ 1= F12 1 A1 ----- cos φ1cos φ2 πr 2 ------------------------------ A1 A2dd A2∫A1∫= J εWb ρG+= ρ 1 α 1 ε–=–= J εWb 1 ε–( )G+= q A⁄ J G εWb 1 ε–( )G G–+=–= q Wb J– 1 ε–( ) εA⁄ --------------------------= GiAi FjiJjAj FijJjAi j 1= n ∑= j 1= n ∑= Gi FijJj j 1= n ∑= 3.10 1997 ASHRAE Fundamentals Handbook (SI) Substituting in Equation (44) yields the following simultaneous equations when each of the n surfaces is considered: (46) Equation (46) can be solved manually for the unknown Js if the number of surfaces is small. The solution for more complex enclo- sures requires a computer. Once the radiosities (Js) are known, the net radiant energy lost by each surface is determined from Equation (45) as If the surface is black, Equation (45) becomes indeterminate, and an alternate expression must be used, such as or (47) since All diffuse radiation processes are included in the aforemen- tioned enclosure method, and surfaces with special characteristics are assigned consistent properties. An opening is treated as an equivalent surface area Ae with a reflectance of zero. If energy enters the enclosure diffusely through the opening, Ae is assigned an equivalent temperature; otherwise, its temperature is taken as zero. If the loss through the opening is desired, q2 is found. A window in the enclosure is assigned its actual properties. A surface in radiant balance is one for which radiant emission is balanced by radiant absorption; heat is neither removed from nor supplied to the surface. Reradiating surfaces (insulated surfaces with qnet = 0), can be treated in Equation (46) as being perfectly reflective (i.e., ε = 0). The equilibrium temperature of such a sur- face can be found from Fig. 6 Radiation Angle Factor for Various Geometries Ji εiWbi 1 εi–( ) FijJj j 1= n ∑+= i 1 2 … n, , ,= qi Wbi Ji– 1 εi–( ) εiAi⁄ -------------------------------= qi JiAiFij JjAjFji– j 1= n ∑= qi FijAi Ji Jj–( ) j 1= n ∑= Fij Ai Fji Aj= Heat Transfer 3.11 once Equation (46) has been solved for the radiosities. Use of angle factors and radiation properties as defined assumes that the surfaces are diffuse radiators—a good assumption for most nonmetals in the infrared region, but a poor assumption for highly polished metals. Subdividing the surfaces and considering the vari- ation of radiation properties with angle of incidence improves the approximation but increases the work required for a solution. Radiation in Gases Elementary gases such as oxygen, nitrogen, hydrogen, and helium are essentially transparent to thermal radiation. Their absorption and emission bands are confined mainly to the ultravio- let region of the spectrum. The gaseous vapors of most compounds, however, have absorption bands in the infrared region. Carbon monoxide, carbon dioxide, water vapor, sulfur dioxide, ammonia, acid vapors, and organic vapors absorb and emit significant amounts of energy. Radiation exchange by opaque solids is considered a surface phenomenon. Radiant energy does, however, penetrate the surface of all materials. The absorption coefficient gives the rate of expo- nential attenuation of the energy. Metals have large absorption coef- ficients, and radiant energy penetrates less than 100 nm at most. Absorption coefficients for nonmetals are lower. Radiation may be considered a surface phenomenon unless the material is transparent. Gases have small absorption coefficients, so the path length of radi- ation through gas becomes very significant. Beer’s law states that the attenuation of radiant energy in a gas is a function of the product pgL of the partial pressure of the gas and the path length. The monochromatic absorptance of a body of gas of thickness L is then given by (48) Because absorption occurs in discrete wavelengths, the absorp- tances must be summed over the spectral region corresponding to the temperature of the blackbody radiation passing through the gas. The monochromatic absorption coefficient αλ is also a function of temperature and pressure of the gas; therefore, detailed treatment of gas radiation is quite complex. Estimated emittance for carbon dioxide and water vapor in air at 24°C is a function of concentration and path length (Table 4). The values are for a hemispherically shaped body of gas radiating to an element of area at the center of the hemisphere. Among others, Mod- est (1993), Siegel and Howell (1981), and Hottel and Sarofim (1967) describe geometrical calculations in their texts on radiation transfer. Generally, at low values of pgL, the mean path length L (or equiva- lent hemispherical radius for a gas body radiating to its surrounding surfaces) is four times the mean hydraulic radius of the enclosure. A room with a dimensional ratio of 1:1:4 has a mean path length of 0.89 times the shortest dimension when considering radiation to all walls. For a room with a dimensional ratio of 1:2:6, the mean path length for the gas radiating to all surfaces is 1.2 times the shortest dimen- sion. The mean path length for radiation to the 2 by 6 face is 1.18 times the shortest dimension. These values are for cases where the partial pressure of the gas times the mean path length approaches zero (pgL ≈ 0). The factor decreases with increasing values of pgL. For average rooms with approximately 2.4 m ceilings and relative humidity ranging from 10 to 75% at 24°C, the effective path length for carbon dioxide radiation is about 85% of the ceiling height, or 2.0 m. The effective path length for water vapor is about 93% of the ceiling height, or 2.3 m. The effective emittance of the water vapor and carbon dioxide radiating to the walls, ceiling, and floor of a room 4.9 m by 14.6 m with 2.4 m ceilings is in the following tabulation. The radiation heat transfer from the gas to the walls is then (49) The examples in Table 4 and the preceding text indicate the importance of gas radiation in environmental heat transfer prob- lems. Gas radiation in large furnaces is the dominant mode of heat transfer, and many additional factors must be considered. Increased pressure broadens the spectral bands, and interaction of different radiating species prohibits simple summation of the emittance fac- tors for the individual species. Departures from blackbody condi- tions necessitate separate calculations of the emittance and absorptance. McAdams (1954) and Hottel and Sarofim (1967) give more complete treatments of gas radiation. NATURAL CONVECTION Heat transfer involving motion in a fluid due to the difference in density and the action of gravity is called natural convection or free convection. Heat transfer coefficients for natural convection are gen- erally much lower than those for forced convection, and it is therefore important not to ignore radiation in calculating the total heat loss or gain. Radiant transfer may be of the same order of magnitude as nat- ural convection, even at room temperatures, because wall tempera- tures in a room can affect human comfort (see Chapter 8). Natural convection is important in a variety of heating and refrigeration equipment: (1) gravity coils used in high-humidity cold storage rooms and in roof-mounted refrigerant condensers, (2) the evaporator and condenser of household refrigerators, (3) baseboard radiators and convectors for space heating, and (4) cooling panels for air conditioning. Natural convection is also involved in heat loss or gain to equipment casings and intercon- necting ducts and pipes. Consider heat transfer by natural convection between a cold fluid and a hot surface. The fluid in immediate contact with the surface is heated by conduction, becomes lighter, and rises because of the dif- ference in density of the adjacent fluid. The viscosity of the fluid resists this motion. The heat transfer is influenced by (1) gravita- tional force due to thermal expansion, (2) viscous drag, and (3) ther- mal diffusion. Gravitational acceleration g, coefficient of thermal expansion β, kinematic viscosity v = µ/ρ, and thermal diffusivity α = k/ρcp affect natural convection. These variables are included in the dimensionless numbers given in Equation (1) in Table 5. The Nusselt number Nu is a function of the product of the Prandtl num- ber Pr and the Grashof number Gr. These numbers, when combined, depend on the fluid properties, the temperature difference ∆t between the surface and the fluid, and the characteristic length L of the surface. The constant c and the exponent n depend on the phys- ical configuration and the nature of flow. Tk Jk σ ----    0.25 = αλL 1 e αλL– –= Relative Humidity, % εg 10 0.10 50 0.19 75 0.22 Table 4 Emittance of CO2 and Water Vapor in Air at 24°C Path Length, CO2, % by Volume Relative Humidity, % 0.1 0.3 1.0 10 50 100 0.03 0.06 0.09 0.06 0.17 0.22 30 0.09 0.12 0.16 0.22 0.39 0.47 300 0.16 0.19 0.23 0.47 0.64 0.70 q σAwεg Tg 4 Tw 4–( )= 3.14 1997 ASHRAE Fundamentals Handbook (SI) Table 6 Equations for Forced Convection Description Reference EquationAuthor Page Eq. No. I. Generalized correlations Jakob 491 (23-36) (1) (a) Turbulent flow inside tubes (1) Using fluid properties based on bulk temperature t McAdams 219 (9-10a) (See Note a) (2) (2) Same as (1), except µ at surface temperature ts McAdams 219 (9-10c) (3) (3) Using fluid properties based on film temperature tf = 0.5(ts + t), except cp in Stanton modulus McAdams 219 (9-10b) (4) (4) For viscous fluids (viscosities higher than twice water), using viscosity µ at bulk temperature t and µs at surface temperature ts Jakob 547 (26-12) (5) (b) Laminar flow inside tubes (6)(1) For large D or high ∆t, the effect of natural convection should be included Jakob 544 (26-5) When (2) For very long tubes (c) Annular spaces, turbulent flow All fluid properties at bulk temperature except µs at surface temperature ts McAdams 242 (9-32c) (7) II. Simplified equations for gases, turbulent flow inside tubes [K in W/(m2 ·K), cp in kJ/(kg·K), G in kg/(m2 ·s), D in m] (a) Most common gases, turbulent flow (assuming µ = 18.8 µPa·s and µcp/k = 0.78) Obtained from Eq. (2) (8) (b) Air at ordinary temperatures Obtained from Eq. (2) (See Note b) (9) (c) Fluorinated hydrocarbon refrigerant gas at ordinary pressures Obtained from Eq. (2) (See Note b) (10) (d) Ammonia gas at approximately 65°C, 2 MPa Obtained from Eq. (2) (11) At −18°C, 165 kPa (gage) Obtained from Eq. (2) (12) III. Simplified equations for liquids, turbulent flow inside tubes [h in W/(m2 ·K), G in kg/(m2 ·s), V in m/s, D in m, t in °C, µ in N·s/m2] (a) Water at ordinary temperatures, 4 to 93°C. V is velocity in m/s, D is tube ID in metres. McAdams 228 (9-19) (13) (b) Fluorinated hydrocarbon refrigerant liquid Obtained from Eq. (2) (See Note b) (14) (c) Ammonia liquid at approximately 38°C Obtained from Eq. (2) (15) (d) Oil heating, approximate equation Brown and Marco 146 (7-15) (16) (e) Oil cooling, approximate equation Brown and Marco 146 (7-15) (17) IV. Simplified equations for air (a) Vertical plane surfaces, V of 5 to 30 m/s (room temperature)c McAdams 249 (9-42) (18) (b) Vertical plane surfaces, V < 5 m/s (room temperature)c McAdams 249 (9-42) (19) (c) Single cylinder cross flow (film tempera- ture = 93°C) 1000 < GD/µf < 50 000 McAdams 261 (10-3c) (20) (d) Single sphere 17 < GD/µf < 70 000 McAdams 265 (10-6) (21) V. Gases flowing normal to pipes (dimensionless) (a) Single cylinder Re from 0.1 to 1000 McAdams 260 (10-3) (22) Re from 1000 to 50 000 McAdams 260 (10-3) (23) (b) Unbaffled staggered tubes, 10 rows. Approxi- mate equation for turbulent flowd McAdams 272 (10-11a) (24) (c) Unbaffled in-line tubes, 10 rows. Approximate equation for turbulent flowd (GmaxD/µf) from 2000 to 32 000 McAdams 272 (10-11a) (25) aMcAdams (1954) recommends this equation for heating and cooling. Others recom- mend exponents of 0.4 for heating and 0.3 for cooling, with a change in constant. bTable 7 in Chapter 2 of the 1981 ASHRAE Handbook—Fundamentals lists values for c. ch′ is expressed in W/(m2 ·K) based on initial temperature difference. dGmax is based on minimum free area. Coefficients for tube banks depend greatly on geometrical details. These values approximate only. hD k ------- c GD µ --------    m µcp k --------    n = hD k ------- 0.023 GD µ --------    0.8 µcp k --------    0.4 = h cpG --------- cpµ k --------    2 3⁄ µs µ ----    0.14 0.023 GD µ⁄( )0.2 --------------------------= h cpG --------- cpµ k --------    f 2 3⁄ 0.023 GD µf⁄( )0.2 ---------------------------- j== hD k ------- 0.027 GD µ --------    0.8 µcp k --------    1 3⁄ µ µs ----    0.14 = hD k ------- 1.86 GD µ --------    cpµ k --------    D L ---    1 3⁄ µ µs ----    0.14 = GD µ --------    cpµ k --------    D L ---    20 Eq. (6) should not be used,< h cpG --------- cpµ k --------    2 3⁄ µs µ ----    0.14 0.023 DeG µ⁄( )0.2 -----------------------------= h 3.031 cpG 0.8 D 0.2⁄( )= h 155.2c G 0.8 D 0.2⁄( )= h 155.2c G 0.8 D 0.2⁄( )= h 6.663 G 0.8 D 0.2⁄( )= h 5.323 G 0.8 D 0.2⁄( )= h 1057 1.352 0.0198 t+( )V 0.8 D 0.2 -----------------------------------------------------------------= h 155.2c G 0.8 D 0.2⁄( )= h 13.75 G 0.8 D 0.2⁄( )= h 0.0047V µf 0.63⁄= h 0.0035V µf 0.63⁄= h ′ 7.2V 0.78= h ′ 5.62 3.9V+= h 4.83 G 0.6 D 0.4⁄( )= h 0.37 kf D --- GD µf --------    0.6 = hD kf ------- 0.32 0.43 GD µ --------    0.52 += hD kf ------- 0.24 GD µf --------    0.6 = hD kf ------- 0.33 GmaxD µf ----------------    0.6 µcp k --------    f 1 3⁄ = hD kf ------- 0.26 GmaxD µf ----------------    0.6 µcp k --------    f 1 3⁄ = Heat Transfer 3.15 When augmentation is used, the dominant thermal resistances in Equation (9) should be considered; that is, do not invest in reducing an already low thermal resistance (increasing an already high heat transfer coefficient). Additionally, heat exchangers with a large number of heat transfer units (NTU) show relatively small gains in effectiveness with augmentation [see Equations (26) and (27)]. Finally, the increased friction factor that accompanies the heat transfer augmentation must be considered. Passive Techniques. Several examples of tubes with internal roughness or fins are shown in Figure 12. Rough surfaces of the spi- ral repeated rib variety are widely used to improve in-tube heat transfer with water, as in flooded chillers. The roughness may be produced by spirally indenting the outer wall, forming the inner wall, or inserting coils. Longitudinal or spiral internal fins in tubes can be produced by extrusion or forming and give a substantial increase in the surface area. The fin efficiency (see the section on Fin Efficiency) can usually be taken as unity. Twisted strips can be inserted as original equipment or as retrofit devices. The increased friction factor may not require increased heat loss or pumping power if the flow rate can be adjusted or if the length of the heat exchanger can be reduced. Nelson and Bergles (1986) dis- cuss this issue of performance evaluation criteria, especially for HVAC applications. Of concern in chilled water systems is the fouling that in some cases may seriously reduce the overall heat transfer coefficient U. In general, fouled enhanced tubes perform better than fouled plain tubes, as shown in recent studies of scaling of cooling tower water (Knudsen and Roy 1983) and particulate fouling (Somerscales et al. 1991). Fire-tube boilers are frequently fitted with turbulators to improve the turbulent convective heat transfer coefficient constitut- ing the dominant thermal resistance. Also, due to the high gas tem- peratures, radiation from the convectively heated insert to the tube wall can represent as much as 50% of the total heat transfer. (Note, however, that the magnitude of the convective contribution decreases as the radiative contribution increases because of the reduced temperature difference.) Two commercial bent-strip inserts, a twisted-strip insert, and a simple bent-tab insert are depicted in Figure 13. Design equations, for convection only, are included in Table 7. Beckermann and Goldschmidt (1986) present procedures to include radiation, and Junkhan et al. (1985, 1988) give friction factor data and performance evaluations. Several enhanced surfaces for gases are depicted in Figure 14. The offset strip fin is an example of an interrupted fin that is often found in compact plate fin heat exchangers used for heat recovery from exhaust air. Design equations are included in Table 7. These equations are comprehensive in that they apply to laminar and tran- sitional flow as well as to turbulent flow, which is a necessary fea- ture because the small hydraulic diameter of these surfaces drives the Reynolds number down. Data for other surfaces (wavy, spine, louvered, etc.) are given in the section on Bibliography. Active Techniques. Among the various active augmentation techniques, several mechanical aids, including stirring of the fluid by mechanical means, rotation of the heat transfer surface, and use of electrostatic fields have significantly increased the forced con- vective heat transfer. While mechanical aids are used in appropriate applications (e.g., surface scraping, baking, and drying processes), the electrostatic technique has been demonstrated only on prototype heat exchangers. The electrostatic or electrohydrodynamic (EHD) augmentation technique uses electrically induced secondary motions to destabilize the thermal boundary layer near the heat transfer surface, thereby substantially increasing the heat transfer coefficients at the wall. The magnitude and nature of enhancements are a function of (1) electric field parameters such as field potential, field polarity, pulse versus steady discharge, electrode geometry and electrode spacing; (2) flow field parameters such as mass flow rate, temperature, density, and electrical permittivity of the working fluid; and (3) the heat transfer surface type such as smooth, porous, or integrally finned/grooved configurations. The EHD effect is generally applied by placing wire or plate electrodes parallel and adjacent to the heat transfer surface. Figure 15 presents four electrode configurations for augmentation of forced-convection heat transfer in tube flows. A high-voltage elec- tric field charges the electrode and establishes the electrical body forces required to initiate and sustain augmentation. Fig. 11 Heat Transfer Coefficient for Turbulent Flow of Water Inside Tubes Fig. 12 Typical Tube-Side Enhancements 3.16 1997 ASHRAE Fundamentals Handbook (SI) Table 7 Equation for Augmented Forced Convection Description Equation I. Turbulent in-tube flow of liquids (a) Spiral repeated riba where w = 0.67 − 0.06(p/d) − 0.49(α/90) x = 0.37 − 0.157(p/d) y = −1.66 × 10−6(GD/µ) − 0.33(α/90) z = 4.59 + 4.11 × 10-6(GD/µ) − 0.15(p/d) (b) Finsb Note that in computing the Reynolds number for (b) and (c) there is allowance for the reduced cross- sectional area. (c) Twisted-strip insertsc (d) Twisted-strip inserts for an evaporator (cooling)c II. Turbulent in-tube flow of gases (a), (b) Bent-strip insertsd (c) Twisted-strip insertsd Note that in computing the Reynolds number there is no allowance for the flow blockage of the insert.(d) Bent-tab insertsd III. Offset strip fins for plate-fin heat exchangers where h/cpG, fh , and GDh/µ are based on the hydraulic diameter, given by Dh = 4shl/[2(sl + hl + th) + ts] References: bCarnavos (1979) dJunkhan et al. (1985) aRavigururajan and Bergles (1985) cLopina and Bergles (1969) eManglik and Bergles (1990) Table 8 Electrohydrodynamic Heat Transfer Enhancement in Heat Exchangers Source Maximum Reported Enhancement, % Test Fluid Heat Transfer Wall/ Electrode Configuration Process Fernandez and Poulter (1987) 2 300 Transformer oil Tube/wire Forced convection Ohadi et al. (1991) 320 Air Tube/wire or rod Forced convection Ohadi et al. (1992) 480 R-123 Tube/wire Boiling Sunada et al. (1991) 600 R-123 Vertical wall/plate Condensation Uemura et al. (1990) 1 400 R-113 Plate/wire mesh Film boiling Yabe and Maki (1988) 10 000 96% (by mass) R-113, 4% ethanol Plate/ring Natural convection ha hs ---- 1 2.64 GD µ --------    0.036 e d --    0.212 p d --    0.21– α 90 -----    0.29 cpµ k --------    0.024– + 7       = 1 7⁄ fa fs --- 1 29.1 GD µ --------    w e d --    x p d --    y α 90 -----    z 1 2.94 n ---------+    sin β 15 16⁄ +       16 15⁄ = hDh k --------- 0.023 cpµ k --------    0.4 GDh µ ----------    0.8 AF AFi -------    0.1 Ai A ----    0.5 sec α( )3 = fh 0.046 GDh µ ----------    0.2– AF AFi -------    0.5 sec α( )0.75 = hDh k --------- F= 0.023 1 π 2y -----    2 + 0.4 GDh µ ----------    0.8 cpµ k --------    0.4 0.193 GDh µy ----------    2 Dh Di ------ ρ∆ ρ ------ cpµ k --------    1 3⁄ +       fh iso, 0.127 0.406– GDh µ ----------    0.2– = hDh k --------- 0.023F 1 π 2y -----    2 + 0.4 GDh µ ----------    0.8 cpµ k --------    0.4 = hD k ------- Tw Tb -----    0.45 0.258 GD µ --------    0.6 = hD k ------- Tw Tb -----    0.45 0.208 GD µ --------    0.63 = hD k ------- Tw Tb -----    0.45 0.122 GD µ --------    0.65 = hD k ------- Tw Tb -----    0.45 0.406 GD µ --------    0.54 = h cpG --------- 0.6522 GDh µ ----------    0.5403– α 0.1541– δ0.1499γ 0.0678– 1 5.269 10 5– GDh µ ----------    1.340 × α0.504δ0.456γ 1.055– + 0.1 = fh 9.6243 GDh µ ----------    0.7422– α 0.1856– δ 0.3053– γ 0.2659– 1 7.669 10 8– GDh µ ----------    4.429 × α0.920δ3.767γ0.236 + 0.1 = Heat Transfer 3.19 19 show curves and equations for annular fins, straight fins, and spines. For constant thickness square fins, the efficiency of a constant thickness annular fin of the same area can be used. More accuracy, particularly with rectangular fins of large aspect ratio, can be obtained by dividing the fin into circular sectors (Rich 1966). Rich (1966) presents results for a wide range of geometries in a compact form for equipment designers by defining a dimensionless unit thermal resistance Φ: (54) (54a) where Φ = dimensionless thermal resistance φ = fin efficiency to = fin thickness at fin base l = length dimension = rt − ro for annular fins = W for rectangular fins Rich (1966) also developed expressions for Φmax, the maximum limiting value of Φ. Figure 20 gives Φmax for annular fins of con- stant and tapered cross section as a function of R = rt /ro (i.e., the ratio of the fin tip-to-root radii). Figure 21 gives Φmax for rectangu- lar fins of a given geometry as determined by the sector method. Figure 22 gives correction factors (Φ/Φmax) for the determination of Φ from Φmax for both annular and rectangular fins. Example. This example illustrates the use of the fin resistance number for a rectangular fin typical of that for an air-conditioning coil. Given: L = 18 mm to = 0.15 mm W = 12 mm h = 60 W/(m2 ·K) ro = 6 mm k = 170 W/(m·K) Solution: From Figure 21 at W/ro = 2.0 and L/W = 1.5, The correction factor Φ/Φmax, which is multiplied by Rf (max) to give Rf, is given in Figure 22 as a function of the fin efficiency. As a first approximation, the fin efficiency is calculated from Equation (54a) assuming Rf = Rf (max). Interpolating between L/W = 1 and L/ W = 2 at W/ro = 2 gives Therefore, The above steps may now be repeated using the corrected value of fin resistance. φ = 0.745 Φ/Φmax = 0.90.75 Rf = 0.00569 m2 ·K/W Note that the improvement in accuracy by reevaluating Φ/Φmax is less than 1% of the overall thermal resistance (environment to fin base). The error produced by using Rf(max) without correction is less than 3%. For many practical cases where greater accuracy is not warranted, a single value of Rf, obtained by estimating Φ/Φmax, can be used over a range of heat transfer coefficients for a given fin. For approximate cal- culations, the fin resistance for other values of k and to can be obtained by simple proportion if the range covered is not excessive. Schmidt (1949) presents approximate, but reasonably accurate, analytical expressions (for computer use) for circular, rectangular, and hexagonal fins. Hexagonal fins are the representative fin shape for the common staggered tube arrangement in finned-tube heat exchangers. Schmidt’s empirical solution is given by where and Φ is given by For circular fins, Fig. 18 Efficiency of Several Types of Straight Fin Fig. 19 Efficiency of Four Types of Spine Φ Rf tok l 2 ------------= Rf 1 h --    1 φ -- 1–   = Φmax Rf max( ) tok W 2⁄ 1.12= = Rf max( ) 1.12 12 2× 0.15 170× 1000× ------------------------------------------- 0.00632 m 2 K⋅ W⁄= = φ 1 1 hRf+( )⁄ 0.72≈= Φ Φmax⁄ 0.88= Rf 0.88 0.00632× 0.00556 m 2 · K W⁄= = φ tanh mriΦ( ) mriΦ ----------------------------= m 2h kt⁄= Φ re ri⁄( ) 1–[ ] 1 0.35 re ri⁄( )ln+[ ]= re ri⁄ ro ri⁄= 3.20 1997 ASHRAE Fundamentals Handbook (SI) Fig. 20 Maximum Fin Resistance Number of Annular Fins (Gardner 1945) Fig. 21 Maximum Fin Resistance Number of Rectangular Fins Determined by Sector Method Fig. 22 Variation of Fin Resistance Number with Efficiency for Annular and Rectangular Fins (Gardner 1945) Heat Transfer 3.21 For rectangular fins, where M and L are defined by Figure 23 as a/2 or b/2, depending on which is greater. For hexagonal fins, where ψ and β are defined as above and M and L are defined by Figure 24 as a/2 or b (whichever is less) and respectively. The section on Bibliography lists other sources of information on finned surfaces. Thermal Contact Resistance Fins can be extruded from the prime surface (e.g., the short fins on the tubes in flooded evaporators or water-cooled condensers) or they can be fabricated separately, sometimes of a different material, and bonded to the prime surface. Metallurgical bonds are achieved by furnace-brazing, dip-brazing, or soldering. Nonmetallic bonding materials, such as epoxy resin, are also used. Mechanical bonds are obtained by tension-winding fins around tubes (spiral fins) or expanding the tubes into the fins (plate fins). Metallurgical bonding, properly done, leaves negligible thermal resistance at the joint but is not always economical. Thermal resistance of a mechanical bond may or may not be negligible, depending on the application, quality of manufacture, materials, and temperatures involved. Tests of plate fin coils with expanded tubes have indicated that substantial losses in performance can occur with fins that have cracked collars; but negligible thermal resistance was found in coils with continuous collars and properly expanded tubes (Dart 1959). Thermal resistance at an interface between two solid materials is largely a function of the surface properties and characteristics of the solids, the contact pressure, and the fluid in the interface, if any. Eckels (1977) models the influence of fin density, fin thickness, and tube diameter on contact pressure and compared it to data for wet and dry coils. Shlykov (1964) shows that the range of attain- able contact resistances is large. Sonokama (1964) presents data on the effects of contact pressure, surface roughness, hardness, void material, and the pressure of the gas in the voids. Lewis and Sauer (1965) show the resistance of adhesive bonds, and Kaspareck (1964) and Clausing (1964) give data on the contact resistance in a vacuum environment. Finned-Tube Heat Transfer The heat transfer coefficients for finned coils follow the basic equations of convection, condensation, and evaporation. The arrangement of the fins affects the values of constants and the expo- nential powers in the equations. It is generally necessary to refer to test data for the exact coefficients. For natural-convection finned coils (gravity coils), approximate coefficients can be obtained by considering the coil to be made of tubular and vertical fin surfaces at different temperatures and then applying the natural-convection equations to each. This calculation is difficult because the natural-convection coefficient depends on the temperature difference, which varies at different points on the fin. Fin efficiency should be high (80 to 90%) for optimum natural-con- vection heat transfer. A low fin efficiency reduces the temperature near the tip. This reduces ∆t near the tip and also the coefficient h, which in natural convection depends on ∆t. The coefficient of heat transfer also decreases as the fin spacing decreases because of interfering convec- tion currents from adjacent fins and reduced free-flow passage; 50 to 100 mm spacing is common. Generally, high coefficients result from large temperature differences and small flow restriction. Edwards and Chaddock (1963) give coefficients for several cir- cular fin-on-tube arrangements, using fin spacing δ as the charac- teristic length and in the form Nu = f (GrPrδ/Do), where Do is the fin diameter. Forced-convection finned coils are used extensively in a wide variety of equipment. The fin efficiency for optimum performance is smaller than that for gravity coils because the forced-convection coefficient is almost independent of the temperature difference between the surface and the fluid. Very low fin efficiencies should be avoided because an inefficient surface gives a high (uneconom- ical) pressure drop. An efficiency of 70 to 90% is often used. As fin spacing is decreased to obtain a large surface area for heat transfer, the coefficient generally increases because of higher air velocity between fins at the same face velocity and reduced equiv- alent diameter. The limit is reached when the boundary layer formed on one fin surface (Figure 8) begins to interfere with the boundary layer formed on the adjacent fin surface, resulting in a decrease of the heat transfer coefficient, which may offset the advantage of larger surface area. Selection of the fin spacing for forced-convection finned coils usu- ally depends on economic and practical considerations, such as foul- ing, frost formation, condensate drainage, cost, weight, and volume. Fins for conventional coils generally are spaced 1.8 to 4.2 mm apart, except where factors such as frost formation necessitate wider spacing. Several means are used to obtain higher coefficients with a given air velocity and surface, usually by creating air turbulence, generally re ri⁄ 1.28ψ β 0.2– ,= ψ M ri⁄ ,= β L M 1≥⁄= Fig. 23 Rectangular Tube Array re ri⁄ 1.27ψ β 0.3–= Fig. 24 Hexagonal Tube Array 0.5 a2 2⁄( )2 b 2 ,+ 3.24 1997 ASHRAE Fundamentals Handbook (SI) Gray, D.L. and R.L. Webb. 1986. Heat transfer and friction correlations for plate finned-tube heat exchangers having plain fins. Proceedings of Eighth International Heat Transfer Conference, San Francisco, CA. Wavy Beecher, D.T. and T.J. Fagan. 1987. Fin patternization effects in plate finned tube heat exchangers. ASHRAE Transactions 93:2. Yashu, T. 1972. Transient testing technique for heat exchanger fin. Reito 47(531):23-29. Spine Abbott, R.W., R.H. Norris, and W.A. Spofford. 1980. Compact heat exchangers for general electric products—Sixty years of advances in design and manufacturing technologies. Compact heat exchangers— History, technological advancement and mechanical design problems. R.K. Shah, C.F. McDonald, and C.P. Howard, eds. Book No. G00183, pp. 37-55. American Society of Mechanical Engineers, New York. Moore, F.K. 1975. Analysis of large dry cooling towers with spine-fin heat exchanger elements. ASME Paper No. 75-WA/HT-46. American Soci- ety of Mechanical Engineers, New York. Rabas, T.J. and P.W. Eckels. 1975. Heat transfer and pressure drop perfor- mance of segmented surface tube bundles. ASME Paper No. 75-HT-45. American Society of Mechanical Engineers, New York. Weierman, C. 1976. Correlations ease the selection of finned tubes. Oil and Gas Journal 9:94-100. Louvered Hosoda, T. et al. 1977. Louver fin type heat exchangers. Heat Transfer Jap- anese Research 6(2):69-77. Mahaymam, W. and L.P. Xu. 1983. Enhanced fins for air-cooled heat ex- changers—Heat transfer and friction factor correlations. Y. Mori and W. Yang, eds. Proceedings of the ASME-JSME Thermal Engineering Joint Conference, Hawaii. Senshu, T. et al. 1979. Surface heat transfer coefficient of fins utilized in air- cooled heat exchangers. Reito 54(615):11-17. Circular Jameson, S.L. 1945. Tube spacing in finned tube banks. ASME Transactions 11:633. Katz, D.L. and Associates. 1954-55. Finned tubes in heat exchangers; Cool- ing liquids with finned coils; Condensing vapors on finned coils; and Boiling outside finned tubes. Bulletin reprinted from Petroleum Refiner. Heat Exchangers Gartner, J.R. and H.L. Harrison. 1963. Frequency response transfer func- tions for a tube in crossflow. ASHRAE Transactions 69:323. Gartner, J.R. and H.L. Harrison. 1965. Dynamic characteristics of water-to- air crossflow heat exchangers. ASHRAE Transactions 71:212. McQuiston, F.C. 1981. Finned tube heat exchangers: State of the art for the air side. ASHRAE Transactions 87:1. Myers, G.E., J.W. Mitchell, and R. Nagaoka. 1965. A method of estimating crossflow heat exchangers transients. ASHRAE Transactions 71:225. Stermole, F.J. and M.H. Carson. 1964. Dynamics of flow forced distributed parameter heat exchangers. AIChE Journal 10(5):9. Thomasson, R.K. 1964. Frequency response of linear counterflow heat exchangers. Journal of Mechanical Engineering Science 6(1):3. Wyngaard, J.C. and F.W. Schmidt. Comparison of methods for determining transient response of shell and tube heat exchangers. ASME Paper 64- WA/HT-20. American Society of Mechanical Engineers, New York. Yang, W.J. Frequency response of multipass shell and tube heat exchangers to timewise variant flow perturbance. ASME Paper 64-HT-18. Ameri- can Society of Mechanical Engineers, New York. Heat Transfer, General Bennet, C.O. and J.E. Myers. 1984. Momentum, heat and mass transfer, 3rd ed. McGraw-Hill, New York. Chapman, A.J. 1981. Heat transfer, 4th ed. Macmillan, New York. Holman, J.D. 1981. Heat transfer, 5th ed. McGraw-Hill, New York. Kern, D.Q. and A.D. Kraus. 1972. Extended surface heat transfer. McGraw- Hill, New York. Kreith, F. and W.Z. Black. 1980. Basic heat transfer. Harper and Row, New York. Lienhard, J.H. 1981. A heat transfer textbook. Prentice Hall, Englewood Cliffs, NJ. McQuiston, F.C. and J.D. Parker. 1988. Heating, ventilating and air-condi- tioning, analysis and design, 4th ed. John Wiley and Sons, New York. Rohsenow, W.M. and J.P. Hartnett, eds. 1973. Handbook of heat transfer. McGraw-Hill, New York. Sissom, L.E. and D.R. Pitts. 1972. Elements of transport phenomena. McGraw-Hill, New York. Todd, J.P. and H.B. Ellis. 1982. Applied heat transfer. Harper and Row, New York. Webb, R.L. and A.E. Bergles. 1983. Heat transfer enhancement, second generation technology. Mechanical Engineering 6:60-67. Welty, J.R. 1974. Engineering heat transfer. John Wiley and Sons, New York. Welty, J.R., C.E. Wicks, and R.E. Wilson. 1972. Fundamentals of momen- tum, heat and mass transfer. John Wiley and Sons, New York. Wolf, H. 1983. Heat transfer. Harper and Row, New York. CHAPTER 4 TWO-PHASE FLOW Boiling ............................................................................................................................................ 4.1 Condensing ..................................................................................................................................... 4.8 Pressure Drop .............................................................................................................................. 4.12 Enhanced Surfaces ....................................................................................................................... 4.14 Symbols ........................................................................................................................................ 4.14 WO-PHASE flow is encountered extensively in the air-con-Tditioning, heating, and refrigeration industries. A combination of liquid and vapor refrigerant exists in flooded coolers, direct- expansion coolers, thermosiphon coolers, brazed and gasketed plate evaporators and condensers, and tube-in-tube evaporators and con- densers, as well as in air-cooled evaporators and condensers. In the pipes of heating systems, steam and liquid water may both be present. Because the hydrodynamic and heat transfer aspects of two-phase flow are not as well understood as those of single-phase flow, no single set of correlations can be used to predict pressure drops or heat transfer rates. Instead, the correlations are for specific thermal and hydrodynamic operating conditions. This chapter presents the basic principles of two-phase flow and provides information on the vast number of correlations that have been developed to predict heat transfer coefficients and pressure drops in these systems. BOILING Commonly used refrigeration evaporators are (1) flooded evapo- rators, where refrigerants at low fluid velocities boil outside or inside tubes; and (2) dry expansion shell-and-tube evaporators, where refrig- erants at substantial fluid velocities boil outside or inside tubes. Two-phase heat and mass transport are characterized by various flow and thermal regimes, whether vaporization takes place under natural convection or in forced flow. As in single-phase flow sys- tems, the heat transfer coefficient for a two-phase mixture depends on the flow regime, the thermodynamic and transport properties of the vapor and the liquid, the roughness of the heating surface, the wetting characteristics of the surface-liquid pair, and other parame- ters. Therefore, it is necessary to consider each flow and boiling regime separately to determine the heat transfer coefficient. Accurate data defining limits of regimes and determining the effects of various parameters are not available. The accuracy of cor- relations in predicting the heat transfer coefficient for two-phase flow is in most cases not known beyond the range of the test data. Boiling and Pool Boiling in Natural Convection Systems Regimes of Boiling. The different regimes of pool boiling described by Farber and Scorah (1948) verified those suggested by Nukiyama (1934). The regimes are illustrated in Figure 1. When the temperature of the heating surface is near the fluid saturation tem- perature, heat is transferred by convection currents to the free sur- face where evaporation occurs (Region I). Transition to nucleate boiling occurs when the surface temperature exceeds saturation by a few degrees (Region II). In nucleate boiling (Region III), a thin layer of superheated liq- uid is formed adjacent to the heating surface. In this layer, bubbles nucleate and grow from spots on the surface. The thermal resistance of the superheated liquid film is greatly reduced by bubble-induced Fig. 1 Characteristic Pool Boiling Curve The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow. 4.2 1997 ASHRAE Fundamentals Handbook (SI) agitation and vaporization. Increased wall temperature increases bubble population, causing a large increase in heat flux. As heat flux or temperature difference increases further and as more vapor forms, the flow of the liquid toward the surface is inter- rupted, and a vapor blanket forms. This gives the maximum or critical heat flux (CHF) in nucleate boiling (point a, Figure 1). This flux is often termed the burnout heat flux or boiling crisis because, for constant power-generating systems, an increase of heat flux beyond this point results in a jump of the heater temper- ature (to point c, Figure 1), often beyond the melting point of a metal heating surface. In systems with controllable surface temperature, an increase beyond the temperature for CHF causes a decrease of heat flux den- sity. This is the transitional boiling regime (Region IV); liquid alternately falls onto the surface and is repulsed by an explosive burst of vapor. At sufficiently high surface temperature, a stable vapor film forms at the heater surface; this is the film boiling regime (Regions V and VI). Because heat transfer is by conduction (and some radiation) across the vapor film, the heater temperature is much higher than for comparable heat flux densities in the nucleate boiling regime. Free Surface Evaporation. In Region I, where surface temper- ature exceeds liquid saturation temperature by less than a few degrees, no bubbles form. Evaporation occurs at the free surface by convection of superheated liquid from the heated surface. Correla- tions of heat transfer coefficients for this region are similar to those for fluids under ordinary natural convection [Equations (1) through (4) in Table 1]. Nucleate Boiling. Much information is available on boiling heat transfer coefficients, but no universally reliable method is available for correlating the data. In the nucleate boiling regime, heat flux density is not a single, valued function of the temperature but depends also on the nucleating characteristics of the surface, as illustrated by Figure 2 (Berenson 1962). The equations proposed for correlating nucleate boiling data can be put in a form that relates heat transfer coefficient h to temperature difference (tw − tsat): (1) Exponent a is normally between 1 and 3; the constant depends on the thermodynamic and transport properties of the vapor and the liq- uid. Nucleating characteristics of the surface, including the size dis- tribution of surface cavities and the wetting characteristics of the surface-liquid pair, affect the value of the multiplying constant and the value of the exponent a in Equation (1). For example, variations in exponent a from 1 to 25 can be produced by polishing the surface with different grades of emery paper. A generalized correlation cannot be expected without consider- ation of the nucleating characteristics of the heating surface. A sta- tistical analysis of data for 25 liquids by Hughmark (1962) shows that in a correlation not considering surface condition, deviations of more than 100% are common. In the following sections, correlations and nomographs for pre- diction of nucleate and flow boiling of various refrigerants are given. For most cases, these correlations have been tested for refrig- erants, such as R-11, R-12, R-113, and R-114, that have now been identified as environmentally harmful and are no longer being used in new equipment. Although extensive research on the thermal and fluid characteristics of alternative refrigerants/refrigerant mixtures has taken place in recent years and some correlations have been sug- gested, the test databases are not yet comprehensive enough to rec- ommend any particular equations among those recently developed. In the absence of quantitative nucleating characteristics, Rohse- now (1951) devised a test that evaluated surface effects for a given surface-liquid combination, with the liquid at atmospheric pressure. The effect of pressure can be determined by using the dimensionless groups in Equation (5) in Table 1. Values of the coefficient Csf found by Blatt and Adt (1963) for some liquid-solid combinations are pre- sented in Section II of Table 1. The nomographs of Figures 3 and 4 (Stephan 1963b) can be used to estimate the heat transfer coefficients for various refrigerants in nucleate boiling from a horizontal plate (Figure 3) and from the out- side of a horizontal cylinder with OD = 30.0 mm (Figure 4). Pres- sures range from 100 to 300 kPa. Stephan’s correlation (Stephan 1963c) is subject to previously mentioned limitations (particularly the heat transfer surface micro- structure and nucleation characteristics) because its form is that of Equation (1), with exponent a equal to 4 for horizontal plates and 2.33 for horizontal cylinders. Data show variations of a from 2 to 25, depending on surface conditions. The nomographs of Figures 3 and 4 are based on experimental data and can be used for estimating the heat transfer coefficient within the range tested (Stephan 1963a). Equation (6) in Table 1 presents an extensively used correlation (Kutateladze 1963). It includes the effect of the diameter of the heat- ing surface (Gilmour 1958) in the last term on the right side. This equation predicts the heat transfer coefficients in nucleate boiling from horizontal and vertical plates and cylinders. In addition to correlations dependent on thermodynamic and transport properties of the vapor and the liquid, Borishansky et al. (1962) and Lienhard and Schrock (1963) documented a correlating method based on the law of corresponding states. The properties can be expressed in terms of fundamental molecular parameters, leading to scaling criteria based on the reduced pressure, pr = p/pc, where pc is the critical thermodynamic pressure for the coolant. An example of this method of correlation is shown in Figure 5. The reference pressure p* was chosen as p* = 0.029pc. This correlation provides a simple method for scaling the effect of pressure if data are available for one pressure level. It also has an advantage if the thermodynamic and particularly the transport properties used in several equations in Table 1 are not accurately known. In its present form, this correlation gives a value of a = 2.33 for the exponent in Equation (1) and con- sequently should apply for typical aged metal surfaces. There are explicit heat transfer coefficient correlations based on the law of corresponding states for various substances (Borishansky and Kosyrev 1966), halogenated refrigerants (Danilova 1965), and flooded evaporators (Starczewski 1965). Other investigations examined the Fig. 2 Effect of Surface Roughness on Temperature in Pool Boiling of Pentane h constant tw tsat–( )a = Two-Phase Flow 4.5 The minimum heat flux density (point b, Figure 1) in film boiling from a horizontal surface and a horizontal cylinder can be predicted by Equations (8) and (9) in Table 1. The numerical factors 0.09 and 0.114 were adjusted to fit experimental data; values predicted by two analyses were approximately 30% higher. Equation (10) in Table 1 predicts the temperature difference at minimum heat flux of film boiling. The heat transfer coefficient in film boiling from a horizontal surface can be predicted by Equation (11) in Table 1; and from a horizontal cylinder by Equation (12) in Table 1 (Bromley 1950), which has been generalized to include the effect of surface tension and cylinder diameter, as shown in Equations (13), (14), and (15) in Table 1 (Breen and Westwater 1962). Frederking and Clark (1962) found that for turbulent film boil- ing, Equation (16) in Table 1 agrees with data from experiments at reduced gravity (Rohsenow 1963, Westwater 1963, Kutateladze 1963, Jakob 1949 and 1957). Flooded Evaporators Equations in Table 1 merely approximate heat transfer rates in flooded evaporators. One reason is that vapor entering the evaporator combined with vapor generated within the evaporator can produce significant forced convection effects superimposed on those caused by nucleation. Nonuniform distribution of the two-phase, vapor-liq- uid flow within the tube bundle of shell-and-tube evaporators or the tubes of vertical-tube flooded evaporators is also important. Myers and Katz (1952) investigated the effect of vapor generated by the bottom rows of a tube bundle on the heat transfer coefficient for the upper rows. Improvement in coefficients for the upper tube rows is greatest at low temperature differences where nucleation effects are less pronounced. Hofmann (1957) summarizes other data for flooded tube bundles. Typical performance of vertical tube natural circulation evapo- rators, based on data for water, is shown in Figure 6 (Perry 1950). Low coefficients are at low liquid levels because insufficient liquid covers the heating surface. The lower coefficient at high levels is the result of an adverse effect of hydrostatic pressure on temperature difference and circulation rate. Perry (1950) noted similar effects in horizontal shell-and-tube evaporators. Forced-Convection Evaporation in Tubes Flow Mechanics. When a mixture of liquid and vapor flows inside a tube, a number of flow patterns occur, depending on the mass fraction of liquid, the fluid properties of each phase, and the flow rate. In an evaporator tube, the mass fraction of liquid decreases along the circuit length, resulting in a series of changing vapor-liquid flow patterns. If the fluid enters as a subcooled liquid, the first indications of vapor generation are bubbles forming at the heated tube wall (nucleation). Subsequently, bubble, plug, churn (or semiannular), annular, spray annular, and mist flows can occur as the vapor content increases for two-phase flows in horizontal tubes. Idealized flow patterns are illustrated in Figure 7A for a horizontal tube evaporator. Because nucleation occurs at the heated surface in a thin sub- layer of superheated liquid, boiling in forced convection may begin while the bulk of the liquid is subcooled. Depending on the nature of the fluid and the amount of subcooling, the bubbles formed can either collapse or continue to grow and coalesce (Figure 7A), as Gouse and Coumou (1965) observed for R-113. Bergles and Rohsenow (1964) developed a method to determine the point of incipient surface boiling. After nucleation begins, bubbles quickly agglomerate to form vapor plugs at the center of a vertical tube, or, as shown in Figure 7A, vapor plugs form along the top surface of a horizontal tube. At the point where the bulk of the fluid reaches saturation temperature, which corresponds to local static pressure, there will be up to 1% vapor quality because of the preceding surface boiling (Guerrieri and Talty 1956). Further coalescence of vapor bubbles and plugs results in churn, or semiannular flow. If the fluid velocity is high enough, a continu- ous vapor core surrounded by a liquid annulus at the tube wall soon forms. This annular flow occurs when the ratio of the tube cross sec- tion filled with vapor to the total cross section is approximately 85%. With common refrigerants, this equals a vapor quality of about 3 to 5%. Vapor quality is the ratio of mass (or mass flow rate) of vapor to total mass (or mass flow rate) of the mixture. The usual flowing vapor quality or vapor fraction is referred to throughout this discussion. Static vapor quality is smaller because the vapor in the core flows at a higher average velocity than the liquid at the walls (see Chapter 2). If two-phase mass velocity is high [greater than 200 kg/(s·m2) for a 12 mm tube], annular flow with small drops of entrained liquid in the vapor core (spray) can persist over a vapor quality range from a few percentage points to more than 90%. Refrigerant evaporators are fed from an expansion device at vapor qualities of approxi- mately 20%, so that annular and spray annular flow predominate in most tube lengths. In a vertical tube, the liquid annulus is distributed uniformly over the periphery, but it is somewhat asymmetric in a horizontal tube (Figure 7A). As vapor quality reaches about 90%, the surface dries out, although there are still entrained droplets of liquid in the vapor (mist). Chaddock and Noerager (1966) found that in a horizontal tube, dryout occurs first at the top of the tube and later at the bottom (Figure 7A). Fig. 5 Correlation of Pool Boiling Data in Terms of Reduced Pressure Fig. 6 Boiling Heat Transfer Coefficients for Flooded Evaporator 4.6 1997 ASHRAE Fundamentals Handbook (SI) If two-phase mass velocity is low [less than 200 kg/(s·m2) for a 12 mm horizontal tube], liquid occupies only the lower cross section of the tube. This causes a wavy type of flow at vapor qualities above about 5%. As the vapor accelerates with increasing evaporation, the interface is disturbed sufficiently to develop annular flow (Figure 7B). Liquid slugging can be superimposed on the flow configura- tions illustrated; the liquid forms a continuous, or nearly continu- ous, sheet over the tube cross section. The slugs move rapidly and at irregular intervals. Heat Transfer. It is difficult to develop a single relation to describe the heat transfer performance for evaporation in a tube over the full quality range. For refrigerant evaporators with several per- centage points of flash gas at entrance, it is less difficult because annular flow occurs in most of the tube length. The reported data are accurate only within geometry, flow, and refrigerant conditions tested; therefore, a large number of methods for calculating heat transfer coefficients for evaporation in tubes is presented in Table 2 (also see Figures 8 through 11). Figure 8 gives heat transfer data obtained for R-12 evaporating in a 14.5 mm copper tube (Ashley 1942). The curves for the tube diameters shown are approximations based on an assumed depen- dence, as in Table 2. Heat transfer coefficient dependence on the vapor fraction can be understood better from the data in Figure 9 (Gouse and Coumou 1965). At low mass velocities [below 200 kg/(s·m2)], the wavy flow regime shown in Figure 7B probably exists, and the heat transfer coefficient is nearly constant along the tube length, dropping at the tube exit as complete vaporization occurs. At higher mass velocities, the flow pattern is usually annu- lar, and the coefficient increases as vapor accelerates. As the sur- face dries and the flow reaches a 90% vapor quality, the coefficient drops sharply. Fig. 7 Flow Regimes in Typical Smooth Horizontal Tube Evaporator Fig. 8 Boiling Heat Transfer Coefficients for R-12 Inside Horizontal Tubes Fig. 9 Heat Transfer Coefficient Versus Vapor Fraction for Partial Evaporation Two-Phase Flow 4.7 Table 2 Equations for Forced Convection Evaporation in Tubes Equations Comments and References HORIZONTAL TUBES Graphical presentation in Figure 8 h versus qm where qm = whfg(1 + x)/2 Average coefficients for complete evaporation of R-12 at 4.4°C in a 14.6 mm ID copper tube, 10 m long; the curves for other diameters in Figure 8 are based on the assumption that h varies inversely as the square of the tube diameter (Ashley 1942). Graphical presentation in Figure 9 h versus x2 where x2 = leaving vapor fraction Average coefficients for R-22 evaporating at 4.4°C in a 16.9 mm ID copper tube, 2290 mm long. Vapor fraction varied from 20% to 100%. Average coefficients plotted are for vapor fraction changes of 0.20 (or 0.10). For average coefficients (at the same heat flux) over larger vapor fraction ranges, the curves can be integrated (Anderson et al. 1966). (1) Average coefficients for R-12 and R-22 evaporating in copper tubes of 12.0 and 18.0 mm ID, from 4.1 to 9.5 m long, and at evaporating temperatures from −20 to 0°C. Vapor fraction varied from 0.08 to 6 K superheat (Pierre 1955, 1957).where C1 = 0.0009 and n = 0.5 for exit qualities ≤ 90%; and C1 = 0.0082 and n = 0.4 for 6 K superheat at exit Equation (1) with Average coefficients for R-22 evaporating at temperatures from 4.4 to 26.7°C in a 8.7 mm ID tube, 2.4 m long. Coefficients were determined for approximately 15% vapor quality changes. The range investigated was x = 0.20 to superheat (Altman et al. 1960b). c1 = 0.0225 and n = 0.375 (2) Local coefficients for R-12 and R-22 evaporating in a 18.6 mm ID tube 305 mm long at saturation temperatures from 23.9 to 32.2°C. Location of transition from annular to mist flow is established, and a heat transfer equation for the mist flow regime is presented (Lavin and Young 1964). where C2 = 6.59 (3) h = 1.85 hL[Bo × 104 + (1/Xtt) 0.67]0.6 (4) Local coefficients for R-12 evaporating in an 11.7 mm ID stainless steel tube with a uni- form wall heat flux (electric heating) over a length of 1934 mm, and an evaporating tem- perature of 11.7°C. Vapor fraction range was 0.20 to 0.88. Equation (4) is a modified form of the Schrock and Grossman equation for vertical tube evaporation [Equation (10)] (Chaddock and Noerager 1966). where Bo = q/Ghfg (5) (6) (7) Best agreement was with Equation (11) for vertical tubes. Local coefficients for R-113 evaporating in a 10.9 mm ID transparent tube with a uni- form wall heat flux over a length of 3.8 m; evaporating temperature approximately 49°C. Report includes photographs of subcooled surface boiling, bubble, plug, and annular flow evaporation regimes (Gouse and Coumou 1965). VERTICAL TUBES h = 3.4hl(1/Xtt) 0.45 (8) Equations (8) and (9) were fitted to experimental data for vertical upflow in tubes. Both relate to forced-convection evaporation regions where nucleate boiling is suppressed (Guerrieri and Talty 1956, Dengler and Addoms 1956). A multiplying factor is recom- mended when nucleation is present. h = 3.5hL(1/Xtt) 0.5 (9) where hl is from (3), Xtt from (7), hL from (6) h = 0.74hL[Bo × 104 + (1/Xtt) 0.67] (10) Local coefficients for water in vertical upflow in tubes with diameters from 3.0 to 11.0 mm and lengths of 380 to 1020 mm The boiling number Bo accounts for nucleation effects, and the Martinelli parameter Xtt , for forced-convection effects (Schrock and Grossman 1962). where Bo is from (5), hL from (6), Xtt from (7) h = hmic + hmac (11) Chen developed this correlation reasoning that the nucleation transfer mechanism (rep- resented by hmic) and the convective transfer mechanism (represented by hmac) are addi- tive. hmac is expressed as a function of the two-phase Reynolds number after Martinelli, and hmic is obtained from the nucleate boiling correlation of Forster and Zuber (1955). Sc is a suppression factor for nucleate boiling (Chen 1963). where hmac = hlFc hmic = 0.00122 (Sc)(E )(∆t)0.24(∆p)0.75 (12) (13) Fc and Sc from Figures 10 and 11 Equation (2) with C2 = 3.79 (14) See comments for Equation (2). Note the superior performance of the horizontal versus vertical configuration (C2 = 6.59 versus 3.79) from this investigation, which used the same apparatus and test techniques for both orientations (Lavin and Young 1964). Note: Except for dimensionless equations, inch-pound units (lbm, h, ft, °F, and Btu) must be used. h C1 kl d ---    GD µl --------    2 J xhfg∆ L ----------------    n = h C2hl 1 x+ 1 x– -----------     1.16 q Ghfg -----------    0.1 = hl 0.023kl d ----------------- DG 1 x–( ) µl ------------------------- 0.8 Pr( )l 0.4= hL 0.023kl d ----------------- DG µl --------    0.8 Pr( )l 0.4= Xtt 1 x– x ----------     0.9 ρv ρl ----     0.5 µl µv ----     0.1 = E kl 0.79 cp( )l 0.45ρl 0.49gc 0.25 σt 0.50µl 0.29hfg 0.24ρv 0.24 -----------------------------------------------------= 4.10 1997 ASHRAE Fundamentals Handbook (SI) Condensation on Outside Surface of Horizontal Tubes For a bank of N tubes, Nusselt’s equations, increased by 10% (Jakob 1949 and 1957), are given in Equations (5) and (6) in Table 3. Experiments by Short and Brown (1951) with R-11 suggest that drops of condensation falling from row to row cause local turbu- lence and increase heat transfer. For condensation on the outside surface of horizontal finned tubes, Equation (7) in Table 3 is used for liquids that drain readily from the surface (Beatty and Katz 1948). For condensing steam out- side finned tubes, where liquid is retained in the spaces between the tubes, coefficients substantially lower than those given by Equation (7) in Table 3 were reported. For additional data on condensation outside finned tubes, see Katz et al. (1947). Simplified Equations for Steam For film-type steam condensation at atmospheric pressure and film temperature drops of 5 to 85 K, McAdams (1954) recommends Equations (8) and (9) in Table 3. Condensation on Inside Surface of Vertical Tubes Condensation on the inside surface of tubes is generally affected by appreciable vapor velocity. The measured heat transfer coeffi- cients are as much as 10 times those predicted by Equation (4) in Table 3. For vertical tubes, Jakob (1949 and 1957) gives theoretical derivations for upward and downward vapor flow. For downward vapor flow, Carpenter and Colburn (1949) suggest Equation (10) in Table 3. The friction factor f ′ for vapor in a pipe containing conden- sate should be taken from Figure 13. Condensation on Inside Surface of Horizontal Tubes For condensation on the inside surface of horizontal tubes (as in air-cooled condensers, evaporative condensers, and some shell-and- tube condensers), the vapor velocity and resulting shear at the vapor-liquid interface are major factors in analyzing heat transfer. Hoogendoorn (1959) identified seven types of two-phase flow pat- terns. For semistratified and laminar annular flow, use Equations (11) and (12) in Table 3 (Ackers and Rosson 1960). Ackers et al. (1959) recommend Equation (13) in Table 3 for turbulent annular flow (vapor Reynolds number greater than 20 000 and liquid Rey- nolds number greater than 5000). Equation (14) in Table 3 correlates the local heat transfer coefficients for R-22 condensing on the inside surface of pipes (Altman et al. 1960b); R-22 and several other fluids take the same form. The two-phase pressure drop in Equation (14) is determined by the method proposed by Martinelli and Nelson (1948); see also Altman et al. (1960a). A method for using a flow regime map to predict the heat transfer coefficient for condensation of pure components in a horizontal tube is presented in Breber et al. (1980). Noncondensable Gases Condensation heat transfer rates reduce drastically if one or more noncondensable gases are present in the condensing vapor/gas mix- ture. In mixtures, the condensable component is termed vapor and the noncondensable component is called gas. As the mass fraction of gas increases, the heat transfer coefficient decreases in an approx- Table 4 Values of Condensing Coefficient Factors for Different Refrigerants (from Chapter 18) Refrigerant Film Temperature, °C tf = tsat − 0.75∆t F1 F2 Refrigerant 11 24 80.7 347.7 38 80.3 344.7 52 79.2 339.7 Refrigerant 12 24 69.8 284.3 38 64.0 257.2 52 58.7 227.6 Refrigerant 22 24 80.3 347.7 38 75.5 319.4 52 69.2 285.5 Sulfur Dioxide 24 152.1 812.2 38 156.8 846.0 52 166.8 917.9 Ammonia 24 214.5 1285.9 38 214.0 1283.8 52 214.0 1281.7 Propane 24 83.4 359.6 38 82.3 357.4 52 80.7 353.6 Butane 24 81.8 355.3 38 81.8 356.6 52 82.3 357.4 F1 kf 3ρf 2g µf --------------    0.25 = Units: W 3 kg⋅ s m 7 K 3⋅ ⋅ ------------------------ 0.25 F2 kf 3ρf 2g µf --------------    1 3⁄ = Units: W 3 kg⋅ s m 7 K 3⋅ ⋅ ------------------------ 1 3⁄ Fig. 12 Film-Type Condensation Two-Phase Flow 4.11 imately linear manner. In a steam chest with 2.89% air by volume, Othmer (1929) found that the heat transfer coefficient dropped from about 11.4 to about 3.4 kW/(m2 ·K). Consider a surface cooled to some temperature ts below the saturation temperature of the vapor (Figure 14). In this system, accumulated condensate falls or is driven across the condenser surface. At a finite heat transfer rate, a temperature profile develops across the condensate that can be esti- mated from Table 3; the interface of the condensate is at a temper- ature tif > ts. In the absence of gas, the interface temperature is the vapor saturation temperature at the pressure of the condenser. The presence of noncondensable gas lowers the vapor partial pressure and hence the saturation temperature of the vapor in equi- librium with the condensate. Further, the movement of the vapor toward the cooled surface implies similar bulk motion of the gas. At the condensing interface, the vapor is condensed at temperature tif and is then swept out of the system as a liquid. The gas concentra- tion rises to ultimately diffuse away from the cooled surface at the same rate as it is convected toward the surface (Figure 14). If gas (mole fraction) concentration is Yg and total pressure of the system is p, the partial pressure of the bulk gas is (2) The partial pressure of the bulk vapor is (3) As opposing fluxes of convection and diffusion of the gas increase, the partial pressure of gas at the condensing interface is pgif > pg∞. By Dalton’s law, assuming isobaric condition, (4) Hence, pvif < pv∞. Sparrow et al. (1967) noted that thermodynamic equilibrium exists at the interface, except in the case of very low pressures or liquid metal condensation, so that (5) where psat(t) is the saturation pressure of the vapor at temperature t. The available ∆t for condensation across the condensate film is reduced from (t∞ − ts) to (tif − ts), where t∞ is the bulk temperature of the condensing vapor-gas mixture, caused by the additional non- condensable resistance. The equations in Table 3 are still valid for the condensate resis- tance, but the interface temperature tif must be found. The noncon- densable resistance, which accounts for the temperature difference (t∞ − tif), depends on the heat flux (through the convecting flow to the interface) and the diffusion of gas away from the interface. In simple cases, Sparrow et al. (1967), Rose (1969), and Spar- row and Lin (1964) found solutions to the combined energy, diffu- sion, and momentum problem of noncondensables, but they are cumbersome. Fig. 13 Friction Factors for Gas Flow Inside Pipes with Wetted Walls Curve parameter = Γ/ρs, where Γ = liquid flow rate, ρ = liquid density, and s = surface tension of liquid relative to water; values of gas velocity used in calculating f and Re are calculated as though no liquid were present. Fig. 14 Origin of Noncondensable Resistance pg∞ Yg∞p= pv∞ 1 Yg∞–( )p Yv∞p= = pgif pvif+ p= pvif psat tif( )= 4.12 1997 ASHRAE Fundamentals Handbook (SI) A general method given by Colburn and Hougen (1934) can be used over a wide range if correct expressions are provided for the rate equations—add the contributions of the sensible heat transport through the noncondensable gas film and the latent heat transport via condensation: (6) where h is from the appropriate equation in Table 3. The value of the heat transfer coefficient for the stagnant gas depends on the geometry and flow conditions. For flow parallel to a condenser tube, for example, (7) where j is a known function of Re = GD/µgv. The mass transfer coefficient KD is (8) The calculation method requires substitution of Equation (8) into Equation (6). For a given flow condition, G, Re, j, Mm, pg∞, hg, and h (or U ) are known. Assume values of tif ; calculate psat(tif ) = pvif and hence pgif . If ts is not known, use the overall coefficient U to the coolant and tc in place of h and ts in Equation (6). For either case, at each location in the condenser, iterate Equation (6) until it balances, giving the condensing interface temperature and, hence, the thermal load to that point (Colburn and Hougen 1934, Colburn 1951). Other Impurities Vapor entering the condenser often contains a small percentage of impurities such as oil. Oil forms a film on the condensing surfaces, creating additional resistance to heat transfer. Some allowance should be made for this, especially in the absence of an oil separator or when the discharge line from the compressor to the condenser is short. PRESSURE DROP Total pressure drop for two-phase flow in tubes consists of fric- tion, acceleration, and gravitational components. It is necessary to know the void fraction (the ratio of gas flow area to total flow area) to compute the acceleration and gravitational components. To com- pute the frictional component of pressure drop, either the two- phase friction factor or the two-phase frictional multiplier must be determined. The homogeneous model provides a simple method for comput- ing the acceleration and gravitational components of pressure drop. The homogeneous model assumes that the flow can be character- ized by average fluid properties and that the velocities of the liquid and vapor phases are equal (Collier 1972, Wallis 1969). Martinelli and Nelson (1948) developed a method for predicting the void fraction and two-phase frictional multiplier to use with a separated flow model. This method predicts the pressure drops of boiling refrigerants reasonably well. Other methods of computing the void fraction and two-phase frictional multiplier used in a sep- arated flow model are given in Collier (1972) and Wallis (1969). The general nature of annular gas-liquid flow in vertical, and to some extent horizontal, pipe is indicated in Figure 15 (Wallis 1970), which plots the effective gas friction factor versus the liquid fraction (1 − a). Here a is the void fraction, or fraction of the pipe cross section taken up by the gas or vapor. The effective gas friction factor is defined as (9) hg t∞ tif–( ) KDMvhlv pv∞ pvif–( )+ h tif ts–( ) U tif tc–( )= = j hg cp( ) g G ----------------       cp( ) g µgv KDg --------------------       2 3⁄ = KD Mm ------- pg∞ pgif– pg∞ pgif⁄( )ln ------------------------------- µgv ρgD ---------    2 3⁄ j= Fig. 15 Qualitative Pressure Drop Characteristics of Two-Phase Flow Regime feff a 5 2⁄ D 2ρg 4Qg πD 2⁄( ) 2 ----------------------------------------- dp ds ----- –   = Two-Phase Flow 4.15 S = distance along flow direction Sc = suppression factor (Table 2 and Figure 11) t = temperature U = overall heat transfer coefficient V = linear velocity x = quality (i.e., vapor fraction = Mv/M); or distance in dt/dx Xtt = Martinelli parameter [Figure 10, Table 2, and Equation (14)] x,y,z = lengths along principal coordinate axes Yg = mole fraction of gas [Equations (2) and (3)] Yv = mole fraction of vapor [Equation (3)] α = thermal diffusivity = k/ρcp α(λ) = ratio of two-phase friction factor to single-phase friction factor at two-phase Reynolds number [Equation (21)] β = ratio of two-phase density to no-slip density [Equation (22)] ∆ = difference between values ε = roughness of interface Γ = mass rate of flow of condensate per unit of breadth (see section on Condensing) Λ = special coefficient [Equations (13) through (15) in Table 1] λ = ratio of liquid volumetric flow rate to total volumetric flow rate [Equation (25)] µ = absolute (dynamic) viscosity µl = dynamic viscosity of saturated liquid µNS = dynamic viscosity of two-phase homogeneous mixture [Equation (24)] µv = dynamic viscosity of saturated vapor ν = kinematic viscosity ρ = density ρl = density of saturated liquid ρNS = density of two-phase homogeneous mixture [Equation (23)] ρv = density of saturated vapor phase σ = surface tension φg = fin efficiency, Martinelli factor [Equation (11)] φv = Lockhart-Martinelli parameter [Equation (15)] ψ = void fraction Subscripts and Superscripts a = exponent in Equation (1) b = bubble c = critical or cold (fluid) cg = condensing e = equivalent eff = effective f = film or fin g = gas h = horizontal or hot (fluid) or hydraulic i = inlet or inside if = interface L = liquid l = liquid m = mean mac = convective mechanism [Equations (11) through (13) in Table 2] max = maximum mic = nucleation mechanism [Equations (11) through (13) in Table 2] min = minimum o = outside or outlet or overall r = root (fin) or reduced pressure s = surface or secondary heat transfer surface sat = saturation (pressure) t = temperature or terminal temperature of tip (fin) v = vapor or vertical w = wall ∞ = bulk * = reference REFERENCES Ackers, W.W., H.A. 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The effect of pressure, geometry and the equation of state upon peak and minimum boiling heat flux. ASME Journal of Heat Transfer 85:261. Lienhard, J.H. and P.T.Y. Wong. 1963. The dominant unstable wave length and minimum heat flux during film boiling on a horizontal cylinder. ASME Paper No. 63-HT-3. ASME-AIChE Heat Transfer Conference, Boston, August. Lockhart, R.W. and R.C. Martinelli. 1949. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engi- neering Progress 45(1):39-48. Luu, M. and A.E. Bergles. 1980. Augmentation of in-tube condensation of R-113. ASHRAE Research Project RP-219. Martinelli, R.C. and D.B. Nelson. 1948. Prediction of pressure drops during forced circulation boiling of water. ASME Transactions 70:695. McAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York. Myers, J.E. and D.L. Katz. 1952. Boiling coefficients outside horizontal plain, and finned tubes. Refrigerating Engineering (January):56. 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Schrock, V.E. and L.M. Grossman. 1962. Forced convection boiling in tubes. Nuclear Science and Engineering 12:474. Short, B.E. and H.E. Brown. 1951. Condensation of vapors on vertical banks of horizontal tubes. American Society of Mechanical Engineers, New York. Silver, R.S. and G.B. Wallis. 1965-66. A simple theory for longitudinal pres- sure drop in the presence of lateral condensation. Proceedings of Institute of Mechanical Engineering, 180 Part I(1):36-42. Soliman, M., J.R. Schuster, and P.J. Berenson. 1968. A general heat transfer correlation for annular flow condensation. Journal of Heat Transfer 90:267-76. Sparrow, E.M. and S.H. Lin. 1964. Condensation in the presence of a non- condensable gas. ASME Transactions, Journal of Heat Transfer 86C:430. Sparrow, E.M., W.J. Minkowycz, and M. Saddy. 1967. Forced convection condensation in the presence of noncondensables and interfacial resis- tance. International Journal of Heat and Mass Transfer 10:1829. Starczewski, J. 1965. 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Chemical Engineering (British) 4(January):78. Wallis, G.B. 1969. One-dimensional two-phase flow. McGraw-Hill, New York. Wallis, G.C. 1970. Annular two-phase flow, Part I: A simple theory, Part II: Additional effect. ASME Transactions, Journal of Basic Engineering 92D:59 and 73. Webb, R.L. 1981. The evolution of enhanced surface geometrics for nucle- ate boiling. Heat Transfer Engineering 2(3-4):46-69. Westwater, J.W. 1963. Things we don’t know about boiling. In Research in Heat Transfer, ed. J. Clark. Pergamon Press, New York. Worsoe-Schmidt, P. 1959. Some characteristics of flow-pattern and heat transfer of Freon-12 evaporating in horizontal tubes. Ingenieren, Inter- national edition, 3(3). Worsoe-Schmidt, P. 1960. ASME Transactions (August):197. Zivi, S.M. 1964. Estimation of steady-state steam void-fraction by means of the principle of minimum entropy production. Journal of Heat Transfer 86:247-52. Zuber, N. 1959. Hydrodynamic aspects of boiling heat transfer. U.S. Atomic Energy Commission, Technical Information Service, Report AECU 4439. Oak Ridge, TN. Zuber, N., M. Tribus, and J.W. Westwater. 1962. The hydrodynamic crisis in pool boiling of saturated and subcooled liquids. Proceedings of the Inter- national Heat Transfer Conference 2:230, and discussion of the papers, Vol. 6. CHAPTER 5 MASS TRANSFER Molecular Diffusion ....................................................................................................................... 5.1 Convection of Mass ........................................................................................................................ 5.5 Simultaneous Heat and Mass Transfer Between Water-Wetted Surfaces and Air ....................... 5.11 Symbols ........................................................................................................................................ 5.15 ASS transfer by either molecular diffusion or convectionM is the transport of one component of a mixture relative to the motion of the mixture and is the result of a concentration gradient. In an air-conditioning process, water vapor is added or removed from the air, with a simultaneous transfer of heat and mass (water vapor) between the airstream and a wetted surface. The wetted sur- face can be water droplets in an air washer, wetted slats of a cooling tower, condensate on the surface of a dehumidifying coil, surface presented by a spray of liquid absorbent, or wetted surfaces of an evaporative condenser. The performance of equipment with these phenomena must be calculated carefully because of the simulta- neous heat and mass transfer. This chapter addresses the principles of mass transfer and pro- vides methods of solving a simultaneous heat and mass transfer problem involving air and water vapor. Emphasis is on air-condi- tioning processes involving mass transfer. The formulations pre- sented can help in analyzing the performance of specific equipment. For a discussion on the performance of air washers, cooling coils, evaporative condensers, and cooling towers, see Chapters 19, 21, 35, and 36, respectively, of the 2000 ASHRAE Handbook—Systems and Equipment. This chapter is divided into (1) the principles of molecular diffu- sion, (2) a discussion on the convection of mass, and (3) simultaneous heat and mass transfer and its application to specific equipment. MOLECULAR DIFFUSION Most mass transfer problems can be analyzed by considering the diffusion of a gas into a second gas, a liquid, or a solid. In this chap- ter, the diffusing or dilute component is designated as component B, and the other component as component A. For example, when water vapor diffuses into air, the water vapor is component B and dry air is component A. Properties with subscripts A or B are local proper- ties of that component. Properties without subscripts are local prop- erties of the mixture. The primary mechanism of mass diffusion at ordinary tempera- ture and pressure conditions is molecular diffusion, a result of density gradient. In a binary gas mixture, the presence of a concen- tration gradient causes transport of matter by molecular diffusion; that is, because of random molecular motion, gas B diffuses through the mixture of gases A and B in a direction that reduces the concen- tration gradient. Fick’s Law The basic equation for molecular diffusion is Fick’s law. Expressing the concentration of component B of a binary mixture in terms of the mass fraction ρB/ρ or mole fraction CB/C , Fick’s law is (1a) (1b) The minus sign indicates that the concentration gradient is neg- ative in the direction of diffusion. The proportionality factor Dv is the mass diffusivity or the diffusion coefficient. The diffusive mass flux JB and the diffusive molar flux J * B are (2a) (2b) where (vB − v) is the velocity of component B relative to the velocity of the mixture and v* is the molar average velocity. Bird et al. (1960) present an analysis of Equations (1a) and (1b). Equations (1a) and (1b) are equivalent forms of Fick’s law. The equation used depends on the problem and individual preference. This chapter emphasizes mass analysis rather than molar analysis. However, all results can be converted to the molar form using the relation CB ≡ ρB /MB . Fick’s Law for Dilute Mixtures In many mass diffusion problems, component B is dilute; the den- sity of component B is small compared to the density of the mixture, and the variation in the density of the mixture throughout the prob- lem is about ρB or less. In this case, Equation (1a) can be written as (3) when ρB << ρ, ∆ρ < ρB. Equation (3) can be used without significant error for water vapor diffusing through air at atmospheric pressure and a temperature less than 27°C. In this case, ρB < 0.02ρ, where ρB is the density of water vapor and ρ is the density of moist air (air and water vapor mixture). The error in JB caused by replacing ρ[d(ρB/ρ)/dy] with dρB/dy is less than 2%. At temperatures below 60°C where ρB < 0.10ρ, Equation (3) can still be used if errors in JB as great as 10% are tolerable. Fick’s Law for Mass Diffusion Through Solids or Stagnant Fluids Fick’s law can be simplified for cases of dilute mass diffusion in solids, stagnant liquids, or stagnant gases. In these cases, ρB << ρ and v = 0, which yields the following approximate result: (4) Therefore, Fick’s law reduces to (5) when ρB << ρ, ∆ρ < ρB, vA = 0. The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow. JB ρ– Dv d ρB ρ⁄( ) dy ---------------------= JB * CDv d CB C⁄( ) dy ----------------------–= JB ρB vB v–( )≡ JB * CB vB v*–( )≡ JB Dv– dρB dy ---------= JB ρB vB v–( ) ρB vB ρBvB ρ -----------–    ρBvB≈ m· B ″= = = m· B ″ Dv – dρB dy ---------= 5.4 1997 ASHRAE Fundamentals Handbook (SI) The partial pressure gradient of the water vapor causes a partial pressure gradient of the air such that or (19) Air, then, diffuses toward the liquid water interface. Because it can- not be absorbed there, a bulk velocity v of the gas mixture is estab- lished in a direction away from the liquid surface, so that the net transport of air is zero (i.e., the air is stagnant). (20) The bulk velocity v transports not only air but also water vapor away from the interface. Therefore, the total rate of water vapor dif- fusion is (21) Substituting for the velocity v from Equation (20) and using Equations (18b) and (19) gives (22) Integration yields (23a) or (23b) where (24) PAm is the logarithmic mean density factor of the stagnant air. The pressure distribution for this type of diffusion is illustrated in Figure 2. Stagnant refers to the net behavior of the air; it does not move because the bulk flow exactly offsets diffusion. The term PAm in Equation (23b) approximately equals unity for dilute mixtures such as water vapor in air at near atmospheric conditions. This con- dition makes it possible to simplify Equation (23) and implies that in the case of dilute mixtures, the partial pressure distribution curves in Figure 2 are straight lines. Example 1. A vertical tube of 25 mm diameter is partially filled with water so that the distance from the water surface to the open end of the tube is 60 mm, as shown in Figure 1. Perfectly dried air is blown over the open tube end, and the complete system is at a constant temperature of 15°C. In 200 h of steady operation, 2.15 g of water evaporates from the tube. The total pressure of the system is 101.325 kPa. Using these data, (a) calculate the mass diffusivity of water vapor in air, and (b) compare this experimental result with that from Equation (11). Solution: (a) The mass diffusion flux of water vapor from the water surface is The cross-sectional area of a 25 mm diameter tube is π(12.5)2 = 491 mm2. Therefore, = 0.00608 g/(m2 ·s). The partial densities are determined with the aid of the psychrometric tables. Since p = pAL = 101.325 kPa, the logarithmic mean density factor [Equation (24)] is The mass diffusivity is now computed from Equation (23b) as (b) By Equation (11), with p = 101.325 kPa and T = 15 + 273 = 288 K, Neglecting the correction factor PAm for this example gives a dif- ference of less than 1% between the calculated experimental and empirically predicted values of Dv. Molecular Diffusion in Liquids and Solids Because of the greater density, diffusion is slower in liquids than in gases. No satisfactory molecular theories have been developed for calculating diffusion coefficients. The limited measured values of Dv show that, unlike for gas mixtures at low pressures, the diffu- sion coefficient for liquids varies appreciably with concentration. Reasoning largely from analogy to the case of one-dimensional diffusion in gases and employing Fick’s law as expressed by Equa- tion (5), (25) where CB = molal concentration of solute in solvent, mol/m3. dpA dy -------- dpB dy --------–= 1 MA -------    dρA dy --------- 1 MB -------    dρB dy ---------–= m· A ″ D– v dρA dy --------- ρAv+ 0= = m· B ″ D– v dρB dy --------- ρBv+= m· B ″ DvMBp ρARUT -----------------    dρA dy ---------= m· B ″ DvMBp RUT ----------------- ρAL ρA0⁄( )ln yL y0– -------------------------------- = m· B ″ D– vPAm ρBL ρB0– yL y0– -----------------------   = PAm p pAL --------ρAL ρAL ρA0⁄( )ln ρAL ρA0– --------------------------------≡ m· B 2.15 200⁄ 0.01075 g/h= = Fig. 2 Pressure Profiles for Diffusion of Water Vapor Through Stagnant Air m· B ″ ρBL 0; ρB0 12.8 g/m3== ρAL 1.225 kg/m3; ρA0 1.204 kg/m3== PAm 1.225 1.225 1.204⁄( )ln 1.225 1.204– ---------------------------------------- 1.009= = Dv m· B ″– yL y0–( ) PAm ρBL ρB0–( ) -------------------------------------- 0.00608( )– 0.060( ) 106( ) 1.009( ) 0 12.8–( ) -----------------------------------------------------------= = 28.2 mm2 s⁄= Dv 0.926 101.325 ------------------ 2882.5 288 245+ -----------------------    24.1 mm2 s⁄= = m· B ″* Dv CB1 CB2– y1 y2– ------------------------   = Mass Transfer 5.5 Equation (25) expresses the steady-state diffusion of the solute B through the solvent A in terms of the molal concentration difference of the solute at two locations separated by the distance ∆y = y1 − y2. Bird et al. (1960), Hirschfelder et al. (1954), Sherwood and Pigford (1952), Reid and Sherwood (1966), Treybal (1980), and Eckert and Drake (1972) provide equations and tables for evaluating Dv. Hir- schfelder et al. (1954) provide comprehensive treatment of the molecular developments. Diffusion through a solid when the solute is dissolved to form a homogeneous solid solution is known as structure-insensitive dif- fusion (Treybal 1980). This solid diffusion closely parallels diffu- sion through fluids, and Equation (25) can be applied to one- dimensional steady-state problems. Values of mass diffusivity are generally lower than they are for liquids and vary with temperature. The flow of a liquid or gas through the interstices and capillaries of a porous or granular solid is a concern. The fundamental mechanism of transport differs for gaseous diffusion, which is considered struc- ture-sensitive diffusion. Experimental measurements are important because of the complex geometry of the flow passages. Generally a factor , called permeability, is defined by the following equation: (26) For moisture transfer through a porous building material of thick- ness ∆y, the water vapor pressure gradient ∆pB/∆y is expressed as kPa per metre of thickness, and the mass flux as mg/(s·m2), so that the permeability has the units of mg/(s·m·Pa). Chapter 22 has further information. CONVECTION OF MASS Convection of mass involves the mass transfer mechanisms of molecular diffusion and bulk fluid motion. Fluid motion in the region adjacent to a mass transfer surface may be laminar or turbu- lent, depending on geometry and flow conditions. Mass Transfer Coefficient Convective mass transfer is analogous to convective heat trans- fer where geometry and boundary conditions are similar. The anal- ogy holds for both laminar and turbulent flows and applies to both external and internal flow problems. Mass Transfer Coefficients for External Flows. Most external convective mass transfer problems can be solved with an appropri- ate formulation that relates the mass transfer flux (to or from an interfacial surface) to the concentration difference across the boundary layer illustrated in Figure 3. This formulation gives rise to the convective mass transfer coefficient, defined as (27) where hM = local external mass transfer coefficient, m/s = mass flux of gas B from surface, kg/(m2 ·s) ρBi = density of gas B at interface (saturation density), kg/m3 ρB∞ = density of component B outside boundary layer, kg/m3 If ρBi and ρB∞ are constant over the entire interfacial surface, the mass transfer rate from the surface can be expressed as (28) where is the average mass transfer coefficient, defined as (29a) Mass Transfer Coefficients for Internal Flows. Most internal convective mass transfer problems, such as those that occur in chan- nels or in the cores of dehumidification coils, can be solved if an appropriate expression is available to relate the mass transfer flux (to or from the interfacial surface) to the difference between the con- centration at the surface and the bulk concentration in the channel, as illustrated in Figure 4. This formulation leads to the definition of the mass transfer coefficient for internal flows: (29b) where hM = internal mass transfer coefficient, m/s = mass flux of gas B at interfacial surface, kg/(m2 ·s) ρBi = density of gas B at interfacial surface, kg/m3 ρBb ≡ = bulk density of gas B at location x ≡ = average velocity of gas B at location x, m/s Acs = cross-sectional area of channel at station x, m2 uB = velocity of component B in x direction, m/s ρB = density distribution of component B at station x, kg/m3 Often, it is easier to obtain the bulk density of gas B from (30) where = mass flow rate of component B at station x = 0, kg/s A = interfacial area of channel between station x = 0 and station x = x, m2 µ m· B ″ – Dv dρB d RT( ) -------------- µ pB∆ y∆ ---------   = = m· B µ Fig. 3 Nomenclature for Convective Mass Transfer from External Surface at Location x Where Surface Is Impermeable to Gas A hM m· B ″ ρBi ρB∞– -----------------------≡ m· B ″ m· B ″ hM ρBi ρB∞–( )= hM hM 1 A --- hm Ad A∫≡ Fig. 4 Nomenclature for Convective Mass Transfer from Internal Surface Impermeable to Gas A hM m· B ″ ρBi ρBb– ----------------------≡ m· B ″ 1 uBAcs⁄( ) Acs ∫ uBρB dAcs uB 1 Acs⁄( ) A∫ uB dAcs ρBb m· Bo m· B ″ Ad A∫+ uBAcs ------------------------------------= m· Bo 5.6 1997 ASHRAE Fundamentals Handbook (SI) Equation (30) can be derived from the preceding definitions. The major problem is the determination of . If, however, the analysis is restricted to cases where B is dilute and concentration gradients of B in the x direction are negligibly small, . Component B is swept along in the x direction with an average velocity equal to the average velocity of the dilute mixture. Analogy Between Convective Heat and Mass Transfer Most expressions for the convective mass transfer coefficient hM are determined from expressions for the convective heat transfer coefficient h. For problems in internal and external flow where mass transfer occurs at the convective surface and where component B is dilute, it is shown by Bird et al. (1960) and Incropera and DeWitt (1996) that the Nusselt and Sherwood numbers are defined as follows: (31) (32) and (33) (34) where the function f is the same in Equations (31) and (33), and the function g is the same in Equations (32) and (34). The quantities Pr and Sc are dimensionless Prandtl and Schmidt numbers, respec- tively, as defined in the section on Symbols. The primary restric- tions on the analogy are that the surface shapes are the same and that the dimensionless temperature boundary conditions are analogous to the dimensionless density distribution boundary conditions for component B, as indicated in Equations (16) and (17). Several pri- mary factors prevent the analogy from being perfect. In some cases, the Nusselt number was derived for smooth surfaces. Many mass transfer problems involve wavy, droplet-like, or roughened sur- faces. Many Nusselt number relations are obtained for constant temperature surfaces. Sometimes ρBi is not constant over the entire surface because of varying saturation conditions and the possibility of surface dryout. In all mass transfer problems, there is some blowing or suction at the surface because of the condensation, evaporation, or transpira- tion of component B. In most cases, this blowing/suction phenom- enon has little effect on the Sherwood number, but the analogy should be examined closely if vi/u∞ > 0.01 or > 0.01, espe- cially if the Reynolds number is large. Example 2. Use the analogy expressed in Equations (32) and (34) to solve the following problem. An expression for heat transfer from a constant temperature flat plate in laminar flow is (35) Sc = 0.35, Dv = 3.6 × 10−5 m2/s, and Pr = 0.708 for the given condi- tions; determine the mass transfer rate and temperature of the water- wetted flat plate in Figure 5 using the heat/mass transfer analogy. Solution: To solve the problem, properties should be evaluated at film conditions. However, since the plate temperature and the interfacial water vapor density are not known, a first estimate will be obtained assuming the plate ti1 to be at 25°C. The plate Reynolds number is The plate is entirely in laminar flow, since the transitional Reynolds number is about 5 × 105. Using the mass transfer analogy, Equation (35) yields From the definition of the Sherwood number, The psychrometric tables give a humidity ratio W of 0.0121 at 25°C and 60% rh. Therefore [see Equation (51)], From steam tables, the saturation density for water at 25°C is Therefore, the mass transfer rate from the double-sided plate is This mass rate, transformed from the liquid state to the vapor state, requires the following heat rate to the plate to maintain the evaporation: To obtain a second estimate of the wetted plate temperature in this type of problem, the following criteria are used. Calculate the ti neces- sary to provide a heat rate of qi1. If this temperature tiq1 is above the dew-point temperature tid, set the second estimate at ti2 = (tiq1 + ti1)/2. If tiq1 is below the dew-point temperature, set ti2 = (tid + ti1)/2. For this problem, the dew point is tid = 14°C. Obtaining the second estimate of the plate temperature requires an approximate value of the heat transfer coefficient. From the definition of the Nusselt number: Therefore, the second estimate for the plate temperature is This temperature is below the dew-point temperature; therefore, The second estimate of the film temperature is uB uB u≅ Nu f X Y Z Pr Re, , , ,( )= Nu g Pr Re,( )= Sh f X Y Z Sc Re, , , ,( )= Sh g Sc Re,( )= vi u⁄ NuL 0.664Pr1 3⁄ ReL 1 2⁄= ReL1 ρu∞L µ ------------- 1.166 kg m3⁄( ) 10 m s⁄( ) 0.1 m( ) 1.965 10 5– kg m s⋅( )⁄×[ ] -------------------------------------------------------------------------------- 59 340= = = Fig. 5 Water-Saturated Flat Plate in Flowing Airstream ShL1 0.664 Sc1 3⁄ ReL 1 2⁄= 0.664 0.35( )1 3⁄ 59 340( )1 2⁄ 114= = hM1 ShL1Dv L⁄ 114( ) 3.6 10 5– m2 s⁄×( ) 0.1 m( )⁄ 0.04104 m/s= = = ρB∞ 0.0121ρA∞ 0.0121( ) 1.166 kg m3⁄( ) 0.01411 kg/m3= = = ρBi1 0.02352 kg/m3= m· B1 hM1A ρBi ρB∞–( )= 0.04104 m/s( ) 0.1 m 1.5 m 2××( )= 0.02352 kg m3⁄ 0.01411 kg m3⁄–( )× 1.159 4–×10 kg s⁄ 0.1159= g/s= qi1 m· B1hfg 0.1159 g/s( ) 2443 kJ/kg( ) 283.1 W= = = NuL1 0.664Pr1 3⁄ ReL 1 2⁄ 0.664 0.708( )1 3⁄ 59 340( )1 2⁄= = 144.2= h1 NuL1k L⁄ 144.2( ) 0.0261 W m K⋅( )⁄[ ] 0.1 m⁄= = 37.6 W m2 K⋅( )⁄= tiq1 t∞ qi1 h1A( )⁄–= 25°C 283.1 W 37.6 W m2 K⋅( )⁄ 2 0.1 m× 1.5 m×( )×[ ]⁄{ }–= 25°C 25°C– 0°C== ti2 14°C 25°C+( ) 2⁄ 19.5°C= = tf 2 ti2 t∞+( ) 2⁄ 19.5°C 25°C+( ) 2⁄ 22.25°C= = = Mass Transfer 5.9 flat plate (left-hand portion of the solid line) and Goldstein’s solution for a turbulent boundary layer (right-hand portion). The right-hand portion of the solid line also represents McAdams’ (1954) correlation of turbulent flow heat transfer coefficient for a flat plate. A wetted-wall column is a vertical tube in which a thin liquid film adheres to the tube surface and exchanges mass by evaporation or absorption with a gas flowing through the tube. Figure 8 illustrates typical data on vaporization in wetted-wall columns, plotted as jD versus Re. The spread of the points with variation in µ/ρDv results from Gilliland’s finding of an exponent of 0.56, not 2/3, represent- ing the effect of the Schmidt number. Gilliland’s equation can be written as follows: (47) Similarly, McAdams’ (1954) equation for heat transfer in pipes can be expressed as (48) This is represented by the dash-dot curve in Figure 8, which falls below the mass transfer data. The curve f /2 representing friction in smooth tubes is the upper, solid curve. Data for the evaporation of liquids from single cylinders into gas streams flowing transversely to the cylinders’ axes are shown in Figure 9. Although the dash-dot line on Figure 9 represents the data, it is actually taken from McAdams (1954) as representative of a large collection of data on heat transfer to single cylinders placed transverse to airstreams. To compare these data with fric- tion, it is necessary to distinguish between total drag and skin fric- tion. Since the analogies are based on skin friction, the normal pressure drag must be subtracted from the measured total drag. At Re = 1000, the skin friction is 12.6% of the total drag; at Re = 31 600, it is only 1.9%. Consequently, the values of f /2 at a high Reynolds number, obtained by the difference, are subject to con- siderable error. In Figure 10, data on the evaporation of water into air for single spheres are presented. The solid line, which best represents these data, agrees with the dashed line representing McAdams’ correla- tion for heat transfer to spheres. These results cannot be compared Fig. 7 Mass Transfer from Flat Plate Fig. 8 Vaporization and Absorption in Wetted-Wall Column jD 0.023Re 0.17– µ ρDv ---------     0.11 = jH 0.023Re 0.20– cpµ k --------     0.07 = Fig. 9 Mass Transfer from Single Cylinders in Crossflow Fig. 10 Mass Transfer from Single Spheres 5.10 1997 ASHRAE Fundamentals Handbook (SI) with friction or momentum transfer because total drag has not been allocated to skin friction and normal pressure drag. Application of these data to air-water contacting devices such as air washers and spray cooling towers is well substantiated. When the temperature of the heat exchanger surface in contact with moist air is below the dew-point temperature of the air, vapor condensation occurs. Typically, the air dry-bulb temperature and humidity ratio both decrease as the air flows through the exchanger. Therefore, sensible and latent heat transfer occur simultaneously. This process is similar to one that occurs in a spray dehumidifier and can be analyzed using the same procedure; however, this is not gen- erally done. Cooling coil analysis and design are complicated by the problem of determining transport coefficients h, hM, and f . It would be con- venient if heat transfer and friction data for dry heating coils could be used with the Colburn analogy to obtain the mass transfer coeffi- cients. However, this approach is not always reliable, and work by Guillory and McQuiston (1973) and Helmer (1974) shows that the analogy is not consistently true. Figure 11 shows j-factors for a sim- ple parallel plate exchanger for different surface conditions with sensible heat transfer. Mass transfer j-factors and the friction factors exhibit the same behavior. Dry surface j-factors fall below those obtained under dehumidifying conditions with the surface wet. At low Reynolds numbers, the boundary layer grows quickly; the droplets are soon covered and have little effect on the flow field. As the Reynolds number is increased, the boundary layer becomes thin and more of the total flow field is exposed to the droplets. The roughness caused by the droplets induces mixing and larger j-fac- tors. The data in Figure 11 cannot be applied to all surfaces because the length of the flow channel is also an important variable. How- ever, the water collecting on the surface is mainly responsible for breakdown of the j-factor analogy. The j-factor analogy is approxi- mately true when the surface conditions are identical. Under some conditions, it is possible to obtain a film of condensate on the sur- face instead of droplets. Guillory and McQuiston (1973) and Helmer (1974) related dry sensible j- and f-factors to those for wet- ted dehumidifying surfaces. The equality of jH, jD, and f /2 for certain streamline shapes at low mass transfer rates has experimental verification. For flow past bluff objects, jH and jD are much smaller than f /2, based on total pressure drag. The heat and mass transfer, however, still relate in a useful way by equating jH and jD . Example 3. Using solid cylinders of volatile solids (e.g., naphthalene, camphor, dichlorobenzene) with airflow normal to these cylinders, Bedingfield and Drew (1950) found that the ratio between the heat and mass transfer coefficients could be closely correlated by the following relation: For completely dry air at 21°C flowing at a velocity of 9.5 m/s over a wet-bulb thermometer of diameter d = 7.5 mm, determine the heat and mass transfer coefficients from Figure 9 and compare their ratio with the Bedingfield-Drew relation. Solution: For dry air at 21°C and standard pressure, ρ = 1.198 kg/m3, µ = 1.82 × 10−5 kg/(s· m), k = 0.02581 W/(m·K), and cp = 1.006 kJ/(kg·K). From Equation (11), Dv = 25.13 mm2/s. Therefore, From Figure 9 at Reda = 4700, read jH = 0.0089, jD = 0.010. From Equations (45) and (46), From the Bedingfield-Drew relation, The Reynolds analogy, Equation (44b), suggests that h/ρhM = cp = 1006 J/(kg·K). This close agreement is because the ratio Sc/Pr is 0.605/0.709 or 0.85, so that the exponent of these numbers has little effect on the ratio of the transfer coefficients. The extensive developments for calculating heat transfer coeffi- cients can be applied to calculate mass transfer coefficients under similar geometrical and flow conditions using the j-factor analogy. For example, Table 6 of Chapter 3 lists equations for calculating heat transfer coefficients for flow inside and normal to pipes. Each equation can be used for mass transfer coefficient calculations by equating jH and jD and imposing the same restriction to each stated in Table 6 of Chapter 3. Similarly, mass transfer experiments often replace corresponding heat transfer experiments with complex geometries where exact boundary conditions are difficult to model (Sparrow and Ohadi 1987a, 1987b). The j-factor analogy is useful only at low mass transfer rates. As the rate of mass transfer increases, the movement of matter normal to the transfer surface increases the convective velocity [vi in Equa- tion (38) and Figure 6]. For example, if a gas is blown from many small holes in a flat plate placed parallel to an airstream, the bound- ary layer thickens, and resistance to both mass and heat transfer increases with increasing blowing rate. Heat transfer data are usu- ally collected at zero or, at least, insignificant mass transfer rates. Therefore, if such data are to be valid for a mass transfer process, the mass transfer rate (i.e., the blowing) must be low. The j-factor relationship jH = jD can still be valid at high mass transfer rates, but neither jH nor jD can be represented by data at zero mass transfer conditions. Eckert and Drake (1972) and Chap- ter 24 of Bird et al. (1960) have detailed information on high mass transfer rates. Fig. 11 Sensible Heat Transfer j-Factors for Parallel Plate Exchanger h ρhM ---------- 1230 J kg K⋅( )⁄[ ] µ ρDv ---------    0.56 = Reda ρu∞d µ⁄ 1.198 9.5 7.5×× 1000 1.82× 10 5–×( )⁄ 4690= = = Pr cpµ k⁄ 1.006 1.82× 10 5–× 1000× 0.02581( )⁄ 0.709= = = Sc µ ρDv⁄ 1.82= 10 5–× 106× 1.198 25.13×( )⁄ 0.605= = h jHρcpu∞ Pr( )2 3⁄⁄= 0.0089 1.198× 1.006× 9.5× 1000 0.709( )2 3⁄⁄×= 128 W m2 K⋅( )⁄= hM jDu∞ Sc( )⁄ 2 3⁄ 0.010 9.5× 0.605( )2 3⁄⁄= = 0.133 m/s= h ρhM⁄ 128 1.198 0.133×( )⁄ 803 J kg K⋅( )⁄= = h ρhM⁄ 1230 0.605( )0.56 928 J kg K⋅( )⁄= = Mass Transfer 5.11 Lewis Relation Heat and mass transfer coefficients are satisfactorily related at the same Reynolds number by equating the Chilton-Colburn j-fac- tors. From Equations (45) and (46), this leads to or (49) The quantity α/Dv is the Lewis number Le. Its magnitude expresses relative rates of propagation of energy and mass within a system. It is fairly insensitive to temperature variation. For air and water vapor mixtures, the ratio is (0.60/0.71) or 0.845, and (0.845)2/3 is 0.894. At low diffusion rates, where the heat-mass transfer analogy is valid, PAm is essentially unity. Therefore, for air and water vapor mixtures, (50) The ratio of the heat transfer coefficient to the mass transfer coef- ficient is equal to the specific heat per unit volume of the mixture at constant pressure. This relation [Equation (50)] is usually called the Lewis relation and is nearly true for air and water vapor at low mass transfer rates. It is generally not true for other gas mixtures because the ratio Le of thermal to vapor diffusivity can differ from unity. The agreement between wet-bulb temperature and adiabatic saturation temperature is a direct result of the nearness of the Lewis number to unity for air and water vapor. The Lewis relation is valid in turbulent flow whether or not α/Dv equals 1 because eddy diffusion in turbulent flow involves the same mixing action for heat exchange as for mass exchange, and this action overwhelms any molecular diffusion. Deviations from the Lewis relation are, therefore, due to a laminar boundary layer or a laminar sublayer and buffer zone, as in Figure 6, where molecular transport phenomena are the controlling factors. SIMULTANEOUS HEAT AND MASS TRANSFER BETWEEN WATER-WETTED SURFACES AND AIR A simplified method used to solve simultaneous heat and mass transfer problems was developed using the Lewis relation, and it gives satisfactory results for most air-conditioning processes. Ex- trapolation to very high mass transfer rates, where the simple heat- mass transfer analogy is not valid, will lead to erroneous results. Enthalpy Potential The water vapor concentration in the air is the humidity ratio W, defined as (51) A mass transfer coefficient is defined using W as the driving potential: (52) where the coefficient KM is in kg/(s·m2). For dilute mixtures, ρAi ≅ ρA∞ ; that is, the partial mass density of dry air changes by only a small percentage between interface and free stream conditions. Therefore, (53) where ρAm = mean density of dry air, kg/m3. Comparing Equation (53) with Equation (27) shows that (54) The humid specific heat cpm of the airstream is, by definition (Mason and Monchick 1965), (55a) or (55b) where cpm is in kJ/(kgda ·K). Substituting from Equations (54) and (55b) into Equation (50) gives (56) since ρAm ≅ ρA∞ because of the small change in dry-air density. Using a mass transfer coefficient with the humidity ratio as the driv- ing force, the Lewis relation becomes ratio of heat to mass transfer coefficient equals humid specific heat. For the plate humidifier illustrated in Figure 5, the total heat transfer from liquid to interface is (57) Using the definitions of the transfer coefficients [Equations (41b) and (52)], (58) Assuming Equation (56) is valid, (59) The enthalpy of the air is approximately (60) The enthalpy hs of the water vapor can be expressed by the ideal gas law as (61) where the base of enthalpy is taken as saturated water at temperature to. Choosing to = 0°C to correspond with the base of the dry-air enthalpy gives (62) h ρcpu ----------- cpµ k --------    2 3⁄ hMPAm u ----------------- µ ρDv ---------     2 3⁄ = h hMρcp --------------- PAm µ ρDv⁄( ) cpµ k⁄( ) --------------------- 2 3⁄ = PAm α Dv⁄( )2 3⁄= h hMρcp --------------- 1≈ W ρB ρA -----≡ m· B ″ KM Wi W∞–( )= m· B ″ KM ρAm --------- ρBi ρ∞–( )= hM KM ρAm ---------= cpm 1 W∞+( )cp= cpm ρ ρA∞ ---------    cp= hρAm KMρA∞cpm --------------------------- 1 h KMcpm ----------------≈= q″ qA ″ m· B ″ hfg+= q″ h ti t∞–( ) KM Wi W∞–( )hfg+= q″ KM cpm ti t∞–( ) Wi W∞–( )hfg+= h cpat Whs+= hs cps t to–( ) hfgo+= h cpa Wcps+( )t Whfgo+ cpmt Whfgo+= = 5.14 1997 ASHRAE Fundamentals Handbook (SI) 6. Continue in the manner of step 5 until point 2, the final state of the air leaving the chamber, is reached. In this example, six steps are used in the graphical construction with the following results: The final state of the air leaving the washer is ta2 = 22.4°C and h2 = 73.6 kJ/kg (wet-bulb temperature ta2′ = 19.4°C). 7. The final step involves calculating the required length of the spray chamber. From Equation (70), The integral is evaluated graphically by plotting 1/(hi − h) versus h as shown in Figure 15. Any satisfactory graphical method can be used to evaluate the area under the curve. Simpson’s rule with four equal increments of ∆h equal to 8.2 gives The design length is, therefore, l = (1.628/1.33)(0.975) = 1.19 m. The method used in Example 4 can also be used to predict the performance of existing direct-contact equipment and can deter- mine the transfer coefficients when performance data from test runs are available. By knowing the water and air temperatures entering and leaving the chamber and the spray ratio, it is possible, by trial and error, to determine the proper slope of the tie-line necessary to achieve the measured final air state. The tie-line slope gives the ratio hLaH/KMaM ; KMaM is found from the integral relationship in Exam- ple 4 from the known chamber length l. Additional descriptions of air spray washers and general perfor- mance criteria are given in Chapter 19 of the 2000 ASHRAE Hand- book—Systems and Equipment. Cooling Towers A cooling tower is a direct-contact heat exchanger in which waste heat picked up by the cooling water from a refrigerator, air conditioner, or industrial process is transferred to atmospheric air by cooling the water. Cooling is achieved by breaking up the water flow to provide a large water surface for air, moving by natural or forced convection through the tower, to contact the water. Cooling towers may be counterflow, crossflow, or a combination of both. The temperature of the water leaving the tower and the packing depth needed to achieve the desired leaving water temperature are of primary interest for design. Therefore, the mass and energy bal- ance equations are based on an overall coefficient K, which is based on (1) the enthalpy driving force due to h at the bulk water temper- ature and (2) neglecting the film resistance. Combining Equations (70) and (71) and using the parameters described above yields (76) or (77) Chapter 36 of the 2000 ASHRAE Handbook—Systems and Equipment covers cooling tower design in detail. State 1 a b c d 2 tL 35 32.8 30.6 28.3 26.1 23.9 h 41.1 47.7 54.3 60.8 67.4 73.9 ti 29.2 27.9 26.7 25.4 24.2 22.9 hi 114.4 108.0 102.3 96.9 91.3 86.4 ta 18.3 19.3 20.3 21.1 21.9 22.4 Fig. 14 Graphical Solution for Air-State Path in Parallel Flow Air Washer l Ga KMaM -------------- hd hi h–( ) ----------------- 1 2 ∫= N hd hi h–( ) ----------------- 1 2 ∫ h∆ 3⁄( ) y1 4y2 2y3 4y4 y5+ + + +( )≈= N 8.2 3⁄( )[0.0136 4 0.0167×( ) 2 0.0238×( )++= 4 0.0372×( ) 0.0800] 0.975=+ + Fig. 15 Graphical Solution of ∫ dh/(hi − h) GLcL dt KMaM hi h–( )dl Ga dh= = Ka dV h′ ha–( ) Acs -----------------------------------= KaV m· L --------- cL td h′ ha–( ) -------------------- t1 t2∫= Mass Transfer 5.15 Cooling and Dehumidifying Coils When water vapor is condensed out of an airstream onto an ex- tended surface (finned) cooling coil, the simultaneous heat and mass transfer problem can be solved by the same procedure set forth for di- rect-contact equipment. The basic equations are the same, except that the true surface area of the coil A is known and the problem does not have to be solved on a unit volume basis. Therefore, if in Equations (67), (68), and (70) aM dl or aH dl is replaced by dA/Acs, these equa- tions become the basic heat, mass, and total energy transfer equations for indirect-contact equipment such as dehumidifying coils. The en- ergy balance shown by Equation (71) remains unchanged. The heat transfer from the interface to the refrigerant now encounters the com- bined resistances of the condensate film (RL = 1/hL); the metal wall and fins, if any (Rm); and the refrigerant film (Rr = A/hr Ar). If this combined resistance is designated as Ri = RL + Rm + Rr = 1/Ui, Equa- tion (72) becomes, for a coil dehumidifier, (78) (plus sign for counterflow, minus sign for parallel flow). The tie-line slope is then (79) Figure 16 illustrates the graphical solution on a psychrometric chart for the air path through a dehumidifying coil with a constant refrigerant temperature. Because the tie-line slope is infinite in this case, the energy balance line is vertical. The corresponding inter- face states and air states are denoted by the same letter symbols, and the solution follows the same procedure as in Example 4. If the problem is to determine the required coil surface area for a given performance, the area is computed by the following relation: (80) This graphical solution on the psychrometric chart automatically determines whether any part of the coil is dry. Thus, in the example illustrated in Figure 16, the entering air at state 1 initially encounters an interface saturation state 1i, clearly below its dew-point temper- ature td1, so the coil immediately becomes wet. Had the graphical technique resulted in an initial interface state above the dew-point temperature of the entering air, the coil would be initially dry. The air would then follow a constant humidity ratio line (the sloping W = constant lines on the chart) until the interface state reached the air dew-point temperature. Mizushina et al. (1959) developed this method not only for water vapor and air, but also for other vapor-gas mixtures. Chapter 21 of the 2000 ASHRAE Handbook—Systems and Equipment shows another related method, based on ARI Standard 410, of determining air-cooling and dehumidifying coil performance. SYMBOLS a = constant, dimensionless; or surface area per unit volume, m2/m3 A = surface area, m2 Acs = cross-sectional area, m2 b = exponent, dimensionless C = molal concentration of solute in solvent, mol/m3 cL = specific heat of liquid, kJ/(kg·K) cp = specific heat at constant pressure, kJ/(kg·K) cpm = specific heat of moist air at constant pressure, kJ/(kgda ·K) d = diameter, m Dv = diffusion coefficient (mass diffusivity), mm2/s f = Fanning friction factor, dimensionless Fo = Fourier number = ατ/L 2, dimensionless Fom = mass transfer Fourier number Dvτ/L 2 G = mass flux, flow rate per unit of cross-sectional area, kg/(s·m2) h = enthalpy, kJ/kg; or heat transfer coefficient, W/(m2 ·K) hfg = enthalpy of vaporization, kJ/kg hM = mass transfer coefficient, m/s J = diffusive mass flux, kg/(s·m2) J* = diffusive molar flux, mol/(s·m2) jD = Colburn mass transfer group = Sh/(Re·Pr1/3), dimensionless jH = Colburn heat transfer group = Nu/(Re·Pr1/3), dimensionless k = thermal conductivity, W/(m·K) KM = mass transfer coefficient, kg/(s·m2) l = length, m L = characteristic length, m L/G = liquid-to-air mass flow ratio Le = Lewis number = α/Dv, dimensionless = rate of mass transfer, m/s = mass flux, kg/(s·m2) = molar flux, mol/(s·m2) M = relative molecular mass, kg/mol Nu = Nusselt number = hL/k, dimensionless p = pressure, kPa PAm = logarithmic mean density factor, Equation (24) Pr = Prandtl number = cpµ/k, dimensionless q = rate of heat transfer, W = heat flux per unit area, W/m2 Q = volumetric flow rate, m3/s Re = Reynolds number = , dimensionless RU = universal gas constant = 8.314 J/(mol·K) Ri = combined thermal resistance, m2 ·K/W RL = thermal resistance of condensate film, m2 ·K/W Rm = thermal resistance across metal wall and fins, m2 ·K/W Rr = thermal resistance of refrigerant film, m2 ·K/W Sc = Schmidt number = µ/ρDv, dimensionless Sh = Sherwood number = hML/Dv, dimensionless St = Stanton number = , dimensionless Stm = mass transfer Stanton number = , dimensionless t = temperature, °C T = absolute temperature, K u = velocity in x direction, m/s Ui = overall conductance from refrigerant to air-water interface for dehumidifying coil, W/(m2 ·K) v = velocity in y direction, m/s vi = velocity normal to mass transfer surface for component i, m/s m· LcL dtL± Ui tL ti–( )dA= h hi– tL ti– ------------- Ui KM -------+−= Fig. 16 Graphical Solution for Air-State Path in Dehumidifying Coil with Constant Refrigerant Temperature A m· a KM ------- hd hi h–( ) ----------------- 1 2 ∫= m· m· ″ m· ″* q″ ρuL µ⁄ h ρcpu⁄ hM PAm u⁄ 5.16 1997 ASHRAE Fundamentals Handbook (SI) V = fluid stream velocity, m/s W = humidity ratio, kg of water vapor per kg of dry air x, y, z = coordinate direction, m X,Y, Z = coordinate direction, dimensionless α = thermal diffusivity = k/ρcp, m2/s εD = eddy mass diffusivity, m2/s µ = absolute (dynamic) viscosity, kg/(m·s) = permeability, mg/(s·m·Pa) ν = kinematic viscosity, m2/s σ = characteristic molecular diameter, nm θ = dimensionless time parameter ρ = mass density or concentration, kg/m3 τ = time τi = shear stress in the x-y coordinate plane, N/m2 ω = mass fraction, kg/kg ΩD,AB = temperature function in Equation (10) Subscripts a = air property da = dry air property or air-side transfer quantity Am = logarithmic mean A = gas component of binary mixture B = the more dilute gas component of binary mixture c = critical state H = heat transfer quantity i = air-water interface value L = liquid m = mean value or metal M = mass transfer quantity o = property evaluated at 0°C s = water vapor property or transport quantity w = water vapor ∞ = property of main fluid stream Superscripts * = on molar basis − = average value ′ = wet bulb REFERENCES Bedingfield, G.H., Jr. and T.B. Drew. 1950. Analogy between heat transfer and mass transfer—A psychrometric study. Industrial and Engineering Chemistry 42:1164. Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 1960. Transport phenomena. John Wiley and Sons, New York. Chilton, T.H. and A.P. Colburn. 1934. Mass transfer (absorption) coeffi- cients—Prediction from data on heat transfer and fluid friction. Indus- trial and Engineering Chemistry 26 (November):1183. Eckert, E.R.G. and R.M. Drake, Jr. 1972. Analysis of heat and mass transfer. McGraw-Hill, New York. Guillory, J.L. and F.C. McQuiston. 1973. An experimental investigation of air dehumidification in a parallel plate heat exchanger. ASHRAE Trans- actions 79(2):146. Helmer, W.A. 1974. Condensing water vapor—Airflow in a parallel plate heat exchanger. Ph.D. thesis, Purdue University, West Lafayette, IN. Hirschfelder, J.O., C.F. Curtiss, and R.B. Bird. 1954. Molecular theory of gases and liquids. John Wiley and Sons, New York. Incropera, F.P. and D.P. DeWitt. 1996. Fundamentals of heat and mass transfer, 4th ed. John Wiley and Sons, New York. Kusuda, T. 1957. Graphical method simplifies determination of aircoil, wet- heat-transfer surface temperature. Refrigerating Engineering 65:41. Mason, E.A. and L. Monchick. 1965. Survey of the equation of state and transport properties of moist gases. Humidity and Moisture 3. Reinhold Publishing Corpration, New York. McAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York. Mizushina, T., N. Hashimoto, and M. Nakajima. 1959. Design of cooler con- densers for gas-vapour mixtures. Chemical Engineering Science 9:195. Ohadi, M.M. and E.M. Sparrow. 1989. Heat transfer in a straight tube situ- ated downstream of a bend. International Journal of Heat and Mass Transfer 32(2):201-12. Reid, R.C. and T.K. Sherwood. 1966. The properties of gases and liquids: Their estimation and correlation, 2nd ed. McGraw-Hill, New York, pp. 520-43. Reid, R.C., J.M. Prausnitz, and B.E. Poling. 1987. The properties of gases and liquids, 4th ed. McGraw-Hill, New York, pp.21-78. Sherwood, T.K. and R.L. Pigford. 1952. Absorption and extraction. McGraw-Hill, New York, pp. 1-28. Sparrow, E.M. and M.M. Ohadi. 1987a. Comparison of turbulent thermal entrance regions for pipe flows with developed velocity and velocity developing from a sharp-edged inlet. ASME Transactions, Journal of Heat Transfer 109:1028-30. Sparrow, E.M. and M.M. Ohadi. 1987b. Numerical and experimental studies of turbulent flow in a tube. Numerical Heat Transfer 11:461-76. Treybal, R.E. 1980. Mass transfer operations, 3rd ed. McGraw-Hill, New York. BIBLIOGRAPHY Bennett, C.O. and J.E. Myers. 1982. Momentum, heat and mass transfer, 3rd ed. McGraw-Hill, New York. DeWitt, D.P. and E.L. Cussler. 1984. Diffusion, mass transfer in fluid sys- tems. Cambridge University Press, UK. Geankopolis, C.J. 1993. Transport processes and unit operations, 3rd ed. Prentice Hall, Englewood Cliffs, NJ Kays, W.M. and M.E. Crawford. 1993. Convective heat and mass transfer. McGraw-Hill, New York. Mikielviez, J. and A.M.A. Rageb. 1995. Simple theroretical approach to direct-contact condensation on subcooled liquid film. International Journal of Heat and Mass Transfer 38(3):557. Slattery, J.C. 1972. Momentum, energy and mass transfer in continua. McGraw-Hill, New York. µ Psychrometrics 6.3 Table 2 Thermodynamic Properties of Moist Air (Standard Atmospheric Pressure, 101.325 kPa) Temp. t, °C Humidity Ratio, kgw/kgda Ws Volume, m3/kg (dry air) Enthalpy, kJ/kg (dry air) Entropy, kJ/(kg · K) (dry air) Condensed Water Temp. ,°C Enthalpy , kJ/kg hw Entropy, kJ/(kg·K) sw Vapor Pressure, kPa psva vas vs ha has hs sa sas ss −60 0.0000067 0.6027 0.0000 0.6027 −60.351 0.017 −60.334 −0.2495 0.0001 −0.2494 −446.29 −1.6854 0.00108 −60 −59 0.0000076 0.6056 0.0000 0.6056 −59.344 0.018 −59.326 −0.2448 0.0001 −0.2447 −444.63 −1.6776 0.00124 −59 −58 0.0000087 0.6084 0.0000 0.6084 −58.338 0.021 −58.317 −0.2401 0.0001 −0.2400 −442.95 −1.6698 0.00141 −58 −57 0.0000100 0.6113 0.0000 0.6113 −57.332 0.024 −57.308 −0.2354 0.0001 −0.2353 −441.27 −1.6620 0.00161 −57 −56 0.0000114 0.6141 0.0000 0.6141 −56.326 0.028 −56.298 −0.2308 0.0001 −0.2306 −439.58 −1.6542 0.00184 −56 −55 0.0000129 0.6170 0.0000 0.6170 −55.319 0.031 −55.288 −0.2261 0.0002 −0.2260 −437.89 −1.6464 0.00209 −55 −54 0.0000147 0.6198 0.0000 0.6198 −54.313 0.036 −54.278 −0.2215 0.0002 −0.2214 −436.19 −1.6386 0.00238 −54 −53 0.0000167 0.6226 0.0000 0.6227 −53.307 0.041 −53.267 −0.2170 0.0002 −0.2168 −434.48 −1.6308 0.00271 −53 −52 0.0000190 0.6255 0.0000 0.6255 −52.301 0.046 −52.255 −0.2124 0.0002 −0.2122 −432.76 −1.6230 0.00307 −52 −51 0.0000215 0.6283 0.0000 0.6284 −51.295 0.052 −51.243 −0.2079 0.0002 −0.2076 −431.03 −1.6153 0.00348 −51 −50 0.0000243 0.6312 0.0000 0.6312 −50.289 0.059 −50.230 −0.2033 0.0003 −0.2031 −429.30 −1.6075 0.00394 −50 −49 0.0000275 0.6340 0.0000 0.6341 −49.283 0.067 −49.216 −0.1988 0.0003 −0.1985 −427.56 −1.5997 0.00445 −49 −48 0.0000311 0.6369 0.0000 0.6369 −48.277 0.075 −48.202 −0.1944 0.0004 −0.1940 −425.82 −1.5919 0.00503 −48 −47 0.0000350 0.6397 0.0000 0.6398 −47.271 0.085 −47.186 −0.1899 0.0004 −0.1895 −424.06 −1.5842 0.00568 −47 −46 0.0000395 0.6426 0.0000 0.6426 −46.265 0.095 −46.170 −0.1855 0.0004 −0.1850 −422.30 −1.5764 0.00640 −46 −45 0.0000445 0.6454 0.0000 0.6455 −45.259 0.108 −45.151 −0.1811 0.0005 −0.1805 −420.54 −1.5686 0.00721 −45 −44 0.0000500 0.6483 0.0001 0.6483 −44.253 0.121 −44.132 −0.1767 0.0006 −0.1761 −418.76 −1.5609 0.00811 −44 −43 0.0000562 0.6511 0.0001 0.6512 −43.247 0.137 −43.111 −0.1723 0.0006 −0.1716 −416.98 −1.5531 0.00911 −43 −42 0.0000631 0.6540 0.0001 0.6540 −42.241 0.153 −42.088 −0.1679 0.0007 −0.1672 −415.19 −1.5453 0.01022 −42 −41 0.0000708 0.6568 0.0001 0.6569 −41.235 0.172 −41.063 −0.1636 0.0008 −0.1628 −413.39 −1.5376 0.01147 −41 −40 0.0000793 0.6597 0.0001 0.6597 −40.229 0.192 −40.037 −0.1592 0.0009 −0.1584 −411.59 −1.5298 0.01285 −40 −39 0.0000887 0.6625 0.0001 0.6626 −39.224 0.216 −39.007 −0.1549 0.0010 −0.1540 −409.77 −1.5221 0.01438 −39 −38 0.0000992 0.6653 0.0001 0.6654 −38.218 0.241 −37.976 −0.1507 0.0011 −0.1496 −407.96 −1.5143 0.01608 −38 −37 0.0001108 0.6682 0.0001 0.6683 −37.212 0.270 −36.942 −0.1464 0.0012 −0.1452 −406.13 −1.5066 0.01796 −37 −36 0.0001237 0.6710 0.0001 0.6712 −36.206 0.302 −35.905 −0.1421 0.0014 −0.1408 −404.29 −1.4988 0.02005 −36 −35 0.0001379 0.6739 0.0001 0.6740 −35.200 0.336 −34.864 −0.1379 0.0015 −0.1364 −402.45 −1.4911 0.02235 −35 −34 0.0001536 0.6767 0.0002 0.6769 −34.195 0.375 −33.820 −0.1337 0.0017 −0.1320 −400.60 −1.4833 0.02490 −34 −33 0.0001710 0.6796 0.0002 0.6798 −33.189 0.417 −32.772 −0.1295 0.0018 −0.1276 −398.75 −1.4756 0.02772 −33 −32 0.0001902 0.6824 0.0002 0.6826 −32.183 0.464 −31.718 −0.1253 0.0020 −0.1233 −396.89 −1.4678 0.03082 −32 −31 0.0002113 0.6853 0.0002 0.6855 −31.178 0.517 −30.661 −0.1212 0.0023 −0.1189 −395.01 −1.4601 0.03425 −31 −30 0.0002346 0.6881 0.0003 0.6884 −30.171 0.574 −29.597 −0.1170 0.0025 −0.1145 −393.14 −1.4524 0.03802 −30 −29 0.0002602 0.6909 0.0003 0.6912 −29.166 0.636 −28.529 −0.1129 0.0028 −0.1101 −391.25 −1.4446 0.04217 −29 −28 0.0002883 0.6938 0.0003 0.6941 −28.160 0.707 −27.454 −0.1088 0.0031 −0.1057 −389.36 −1.4369 0.04673 −28 −27 0.0003193 0.6966 0.0004 0.6970 −27.154 0.782 −26.372 −0.1047 0.0034 −0.1013 −387.46 −1.4291 0.05175 −27 −26 0.0003533 0.6995 0.0004 0.6999 −26.149 0.867 −25.282 −0.1006 0.0037 −0.0969 −385.55 −1.4214 0.05725 −26 −25 0.0003905 0.7023 0.0004 0.7028 −25.143 0.959 −24.184 −0.0965 0.0041 −0.0924 −383.63 −1.4137 0.06329 −25 −24 0.0004314 0.7052 0.0005 0.7057 −24.137 1.059 −23.078 −0.0925 0.0045 −0.0880 −381.71 −1.4059 0.06991 −24 −23 0.0004762 0.7080 0.0005 0.7086 −23.132 1.171 −21.961 −0.0885 0.0050 −0.0835 −379.78 −1.3982 0.07716 −23 −22 0.0005251 0.7109 0.0006 0.7115 −22.126 1.292 −20.834 −0.0845 0.0054 −0.0790 −377.84 −1.3905 0.08510 −22 −21 0.0005787 0.7137 0.0007 0.7144 −21.120 1.425 −19.695 −0.0805 0.0060 −0.0745 −375.90 −1.3828 0.09378 −21 −20 0.0006373 0.7165 0.0007 0.7173 −20.115 1.570 −18.545 −0.0765 0.0066 −0.0699 −373.95 −1.3750 0.10326 −20 −19 0.0007013 0.7194 0.0008 0.7202 −19.109 1.729 −17.380 −0.0725 0.0072 −0.0653 −371.99 −1.3673 0.11362 −19 −18 0.0007711 0.7222 0.0009 0.7231 −18.103 1.902 −16.201 −0.0686 0.0079 −0.0607 −370.02 −1.3596 0.12492 −18 −17 0.0008473 0.7251 0.0010 0.7261 −17.098 2.092 −15.006 −0.0646 0.0086 −0.0560 −368.04 −1.3518 0.13725 −17 −16 0.0009303 0.7279 0.0011 0.7290 −16.092 2.299 −13.793 −0.0607 0.0094 −0.0513 −366.06 −1.3441 0.15068 −16 −15 0.0010207 0.7308 0.0012 0.7320 −15.086 2.524 −12.562 −0.0568 0.0103 −0.0465 −364.07 −1.3364 0.16530 −15 −14 0.0011191 0.7336 0.0013 0.7349 −14.080 2.769 −11.311 −0.0529 0.0113 −0.0416 −362.07 −1.3287 0.18122 −14 −13 0.0012262 0.7364 0.0014 0.7379 −13.075 3.036 −10.039 −0.0490 0.0123 −0.0367 −360.07 −1.3210 0.19852 −13 −12 0.0013425 0.7393 0.0016 0.7409 −12.069 3.327 −8.742 −0.0452 0.0134 −0.0318 −358.06 −1.3132 0.21732 −12 −11 0.0014690 0.7421 0.0017 0.7439 −11.063 3.642 −7.421 −0.0413 0.0146 −0.0267 −356.04 −1.3055 0.23775 −11 −10 0.0016062 0.7450 0.0019 0.7469 −10.057 3.986 −6.072 −0.0375 0.0160 −0.0215 −354.01 −1.2978 0.25991 −10 −9 0.0017551 0.7478 0.0021 0.7499 −9.052 4.358 −4.693 −0.0337 0.0174 −0.0163 −351.97 −1.2901 0.28395 −9 −8 0.0019166 0.7507 0.0023 0.7530 −8.046 4.764 −3.283 −0.0299 0.0189 −0.0110 −349.93 −1.2824 0.30999 −8 −7 0.0020916 0.7535 0.0025 0.7560 −7.040 5.202 −1.838 −0.0261 0.0206 −0.0055 −347.88 −1.2746 0.33821 −7 −6 0.0022811 0.7563 0.0028 0.7591 −6.035 5.677 −0.357 −0.0223 0.0224 −0.0000 −345.82 −1.2669 0.36874 −6 −5 0.0024862 0.7592 0.0030 0.7622 −5.029 6.192 1.164 −0.0186 0.0243 −0.0057 −343.76 −1.2592 0.40178 −5 −4 0.0027081 0.7620 0.0033 0.7653 −4.023 6.751 2.728 −0.0148 0.0264 −0.0115 −341.69 −1.2515 0.43748 −4 −3 0.0029480 0.7649 0.0036 0.7685 −3.017 7.353 4.336 −0.0111 0.0286 −0.0175 −339.61 −1.2438 0.47606 −3 −2 0.0032074 0.7677 0.0039 0.7717 −2.011 8.007 5.995 −0.0074 0.0310 −0.0236 −337.52 −1.2361 0.51773 −2 −1 0.0034874 0.7705 0.0043 0.7749 −1.006 8.712 7.706 −0.0037 0.0336 −0.0299 −335.42 −1.2284 0.56268 −1 0 0.0037895 0.7734 0.0047 0.7781 −0.000 9.473 9.473 0.0000 0.0364 0.0364 −333.32 −1.2206 0.61117 −0 0* 0.003789 0.7734 0.0047 0.7781 −0.000 9.473 9.473 0.0000 0.0364 0.0364 0.06 −0.0001 0.6112 0 1 0.004076 0.7762 0.0051 0.7813 1.006 10.197 11.203 0.0037 0.0391 0.0427 4.28 0.0153 0.6571 1 2 0.004381 0.7791 0.0055 0.7845 2.012 10.970 12.982 0.0073 0.0419 0.0492 8.49 0.0306 0.7060 2 3 0.004707 0.7819 0.0059 0.7878 3.018 11.793 14.811 0.0110 0.0449 0.0559 12.70 0.0459 0.7581 3 4 0.005054 0.7848 0.0064 0.7911 4.024 12.672 16.696 0.0146 0.0480 0.0627 16.91 0.0611 0.8135 4 5 0.005424 0.7876 0.0068 0.7944 5.029 13.610 18.639 0.0182 0.0514 0.0697 21.12 0.0762 0.8725 5 6 0.005818 0.7904 0.0074 0.7978 6.036 14.608 20.644 0.0219 0.0550 0.0769 25.32 0.0913 0.9353 6 7 0.006237 0.7933 0.0079 0.8012 7.041 15.671 22.713 0.0255 0.0588 0.0843 29.52 0.1064 1.0020 7 8 0.006683 0.7961 0.0085 0.8046 8.047 16.805 24.852 0.0290 0.0628 0.0919 33.72 0.1213 1.0729 8 9 0.007157 0.7990 0.0092 0.8081 9.053 18.010 27.064 0.0326 0.0671 0.0997 37.92 0.1362 1.1481 9 10 0.007661 0.8018 0.0098 0.8116 10.059 19.293 29.352 0.0362 0.0717 0.1078 42.11 0.1511 1.2280 10 11 0.008197 0.8046 0.0106 0.8152 11.065 20.658 31.724 0.0397 0.0765 0.1162 46.31 0.1659 1.3128 11 12 0.008766 0.8075 0.0113 0.8188 12.071 22.108 34.179 0.0433 0.0816 0.1248 50.50 0.1806 1.4026 12 13 0.009370 0.8103 0.0122 0.8225 13.077 23.649 36.726 0.0468 0.0870 0.1337 54.69 0.1953 1.4979 13 *Extrapolated to represent metastable equilibrium with undercooled liquid. 6.4 1997 ASHRAE Fundamentals Handbook (SI) 14 0.010012 0.8132 0.0131 0.8262 14.084 25.286 39.370 0.0503 0.0927 0.1430 58.88 0.2099 1.5987 14 15 0.010692 0.8160 0.0140 0.8300 15.090 27.023 42.113 0.0538 0.0987 0.1525 63.07 0.2244 1.7055 15 16 0.011413 0.8188 0.0150 0.8338 16.096 28.867 44.963 0.0573 0.1051 0.1624 67.26 0.2389 1.8185 16 17 0.012178 0.8217 0.0160 0.8377 17.102 30.824 47.926 0.0607 0.1119 0.1726 71.44 0.2534 1.9380 17 18 0.012989 0.8245 0.0172 0.8417 18.108 32.900 51.008 0.0642 0.1190 0.1832 75.63 0.2678 2.0643 18 19 0.013848 0.8274 0.0184 0.8457 19.114 35.101 54.216 0.0677 0.1266 0.1942 79.81 0.2821 2.1979 19 20 0.014758 0.8302 0.0196 0.8498 20.121 37.434 57.555 0.0711 0.1346 0.2057 84.00 0.2965 2.3389 20 21 0.015721 0.8330 0.0210 0.8540 21.127 39.908 61.035 0.0745 0.1430 0.2175 88.18 0.3107 2.4878 21 22 0.016741 0.8359 0.0224 0.8583 22.133 42.527 64.660 0.0779 0.1519 0.2298 92.36 0.3249 2.6448 22 23 0.017821 0.8387 0.0240 0.8627 23.140 45.301 68.440 0.0813 0.1613 0.2426 96.55 0.3390 2.8105 23 24 0.018963 0.8416 0.0256 0.8671 24.146 48.239 72.385 0.0847 0.1712 0.2559 100.73 0.3531 2.9852 24 25 0.020170 0.8444 0.0273 0.8717 25.153 51.347 76.500 0.0881 0.1817 0.2698 104.91 0.3672 3.1693 25 26 0.021448 0.8472 0.0291 0.8764 26.159 54.638 80.798 0.0915 0.1927 0.2842 109.09 0.3812 3.3633 26 27 0.022798 0.8501 0.0311 0.8811 27.165 58.120 85.285 0.0948 0.2044 0.2992 113.27 0.3951 3.5674 27 28 0.024226 0.8529 0.0331 0.8860 28.172 61.804 89.976 0.0982 0.2166 0.3148 117.45 0.4090 3.7823 28 29 0.025735 0.8558 0.0353 0.8910 29.179 65.699 94.878 0.1015 0.2296 0.3311 121.63 0.4229 4.0084 29 30 0.027329 0.8586 0.0376 0.8962 30.185 69.820 100.006 0.1048 0.2432 0.3481 125.81 0.4367 4.2462 30 31 0.029014 0.8614 0.0400 0.9015 31.192 74.177 105.369 0.1082 0.2576 0.3658 129.99 0.4505 4.4961 31 32 0.030793 0.8643 0.0426 0.9069 32.198 78.780 110.979 0.1115 0.2728 0.3842 134.17 0.4642 4.7586 32 33 0.032674 0.8671 0.0454 0.9125 33.205 83.652 116.857 0.1148 0.2887 0.4035 138.35 0.4779 5.0345 33 34 0.034660 0.8700 0.0483 0.9183 34.212 88.799 123.011 0.1180 0.3056 0.4236 142.53 0.4915 5.3242 34 35 0.036756 0.8728 0.0514 0.9242 35.219 94.236 129.455 0.1213 0.3233 0.4446 146.71 0.5051 5.6280 35 36 0.038971 0.8756 0.0546 0.9303 36.226 99.983 136.209 0.1246 0.3420 0.4666 150.89 0.5186 5.9468 36 37 0.041309 0.8785 0.0581 0.9366 37.233 106.058 143.290 0.1278 0.3617 0.4895 155.07 0.5321 6.2812 37 38 0.043778 0.8813 0.0618 0.9431 38.239 112.474 150.713 0.1311 0.3824 0.5135 159.25 0.5456 6.6315 38 39 0.046386 0.8842 0.0657 0.9498 39.246 119.258 158.504 0.1343 0.4043 0.5386 163.43 0.5590 6.9988 39 40 0.049141 0.8870 0.0698 0.9568 40.253 126.430 166.683 0.1375 0.4273 0.5649 167.61 0.5724 7.3838 40 41 0.052049 0.8898 0.0741 0.9640 41.261 134.005 175.265 0.1407 0.4516 0.5923 171.79 0.5857 7.7866 41 42 0.055119 0.8927 0.0788 0.9714 42.268 142.007 184.275 0.1439 0.4771 0.6211 175.97 0.5990 8.2081 42 43 0.058365 0.8955 0.0837 0.9792 43.275 150.475 193.749 0.1471 0.5041 0.6512 180.15 0.6122 8.6495 43 44 0.061791 0.8983 0.0888 0.9872 44.282 159.417 203.699 0.1503 0.5325 0.6828 184.33 0.6254 9.1110 44 45 0.065411 0.9012 0.0943 0.9955 45.289 168.874 214.164 0.1535 0.5624 0.7159 188.51 0.6386 9.5935 45 46 0.069239 0.9040 0.1002 1.0042 46.296 178.882 225.179 0.1566 0.5940 0.7507 192.69 0.6517 10.0982 46 47 0.073282 0.9069 0.1063 1.0132 47.304 189.455 236.759 0.1598 0.6273 0.7871 196.88 0.6648 10.6250 47 48 0.077556 0.9097 0.1129 1.0226 48.311 200.644 248.955 0.1629 0.6624 0.8253 201.06 0.6778 11.1754 48 49 0.082077 0.9125 0.1198 1.0323 49.319 212.485 261.803 0.1661 0.6994 0.8655 205.24 0.6908 11.7502 49 50 0.086858 0.9154 0.1272 1.0425 50.326 225.019 275.345 0.1692 0.7385 0.9077 209.42 0.7038 12.3503 50 51 0.091918 0.9182 0.1350 1.0532 51.334 238.290 289.624 0.1723 0.7798 0.9521 213.60 0.7167 12.9764 51 52 0.097272 0.9211 0.1433 1.0643 52.341 252.340 304.682 0.1754 0.8234 0.9988 217.78 0.7296 13.6293 52 53 0.102948 0.9239 0.1521 1.0760 53.349 267.247 320.596 0.1785 0.8695 1.0480 221.97 0.7424 14.3108 53 54 0.108954 0.9267 0.1614 1.0882 54.357 283.031 337.388 0.1816 0.9182 1.0998 226.15 0.7552 15.0205 54 55 0.115321 0.9296 0.1713 1.1009 55.365 299.772 355.137 0.1847 0.9698 1.1544 230.33 0.7680 15.7601 55 56 0.122077 0.9324 0.1819 1.1143 56.373 317.549 373.922 0.1877 1.0243 1.2120 234.52 0.7807 16.5311 56 57 0.129243 0.9353 0.1932 1.1284 57.381 336.417 393.798 0.1908 1.0820 1.2728 238.70 0.7934 17.3337 57 58 0.136851 0.9381 0.2051 1.1432 58.389 356.461 414.850 0.1938 1.1432 1.3370 242.88 0.8061 18.1691 58 59 0.144942 0.9409 0.2179 1.1588 59.397 377.788 437.185 0.1969 1.2081 1.4050 247.07 0.8187 19.0393 59 60 0.15354 0.9438 0.2315 1.1752 60.405 400.458 460.863 0.1999 1.2769 1.4768 251.25 0.8313 19.9439 60 61 0.16269 0.9466 0.2460 1.1926 61.413 424.624 486.036 0.2029 1.3500 1.5530 255.44 0.8438 20.8858 61 62 0.17244 0.9494 0.2614 1.2109 62.421 450.377 512.798 0.2059 1.4278 1.6337 259.62 0.8563 21.8651 62 63 0.18284 0.9523 0.2780 1.2303 63.429 477.837 541.266 0.2089 1.5104 1.7194 263.81 0.8688 22.8826 63 64 0.19393 0.9551 0.2957 1.2508 64.438 507.177 571.615 0.2119 1.5985 1.8105 268.00 0.8812 23.9405 64 65 0.20579 0.9580 0.3147 1.2726 65.446 538.548 603.995 0.2149 1.6925 1.9074 272.18 0.8936 25.0397 65 66 0.21848 0.9608 0.3350 1.2958 66.455 572.116 638.571 0.2179 1.7927 2.0106 276.37 0.9060 26.1810 66 67 0.23207 0.9636 0.3568 1.3204 67.463 608.103 675.566 0.2209 1.8999 2.1208 280.56 0.9183 27.3664 67 68 0.24664 0.9665 0.3803 1.3467 68.472 646.724 715.196 0.2238 2.0147 2.2385 284.75 0.9306 28.5967 68 69 0.26231 0.9693 0.4055 1.3749 69.481 688.261 757.742 0.2268 2.1378 2.3646 288.94 0.9429 29.8741 69 70 0.27916 0.9721 0.4328 1.4049 70.489 732.959 803.448 0.2297 2.2699 2.4996 293.13 0.9551 31.1986 70 71 0.29734 0.9750 0.4622 1.4372 71.498 781.208 852.706 0.2327 2.4122 2.6448 297.32 0.9673 32.5734 71 72 0.31698 0.9778 0.4941 1.4719 72.507 833.335 905.842 0.2356 2.5655 2.8010 301.51 0.9794 33.9983 72 73 0.33824 0.9807 0.5287 1.5093 73.516 889.807 963.323 0.2385 2.7311 2.9696 305.70 0.9916 35.4759 73 74 0.36130 0.9835 0.5662 1.5497 74.525 951.077 1025.603 0.2414 2.9104 3.1518 309.89 1.0037 37.0063 74 75 0.38641 0.9863 0.6072 1.5935 75.535 1017.841 1093.375 0.2443 3.1052 3.3496 314.08 1.0157 38.5940 75 76 0.41377 0.9892 0.6519 1.6411 76.543 1090.628 1167.172 0.2472 3.3171 3.5644 318.28 1.0278 40.2369 76 77 0.44372 0.9920 0.7010 1.6930 77.553 1170.328 1247.881 0.2501 3.5486 3.7987 322.47 1.0398 41.9388 77 78 0.47663 0.9948 0.7550 1.7498 78.562 1257.921 1336.483 0.2530 3.8023 4.0553 326.67 1.0517 43.7020 78 79 0.51284 0.9977 0.8145 1.8121 79.572 1354.347 1433.918 0.2559 4.0810 4.3368 330.86 1.0636 45.5248 79 80 0.55295 1.0005 0.8805 1.8810 80.581 1461.200 1541.781 0.2587 4.3890 4.6477 335.06 1.0755 47.4135 80 81 0.59751 1.0034 0.9539 1.9572 81.591 1579.961 1661.552 0.2616 4.7305 4.9921 339.25 1.0874 49.3670 81 82 0.64724 1.0062 1.0360 2.0422 82.600 1712.547 1795.148 0.2644 5.1108 5.3753 343.45 1.0993 51.3860 82 83 0.70311 1.0090 1.1283 2.1373 83.610 1861.548 1945.158 0.2673 5.5372 5.8045 347.65 1.1111 53.4746 83 84 0.76624 1.0119 1.2328 2.2446 84.620 2029.983 2114.603 0.2701 6.0181 6.2882 351.85 1.1228 55.6337 84 85 0.83812 1.0147 1.3518 2.3666 85.630 2221.806 2307.436 0.2729 6.5644 6.8373 356.05 1.1346 57.8658 85 86 0.92062 1.0175 1.4887 2.5062 86.640 2442.036 2528.677 0.2757 7.1901 7.4658 360.25 1.1463 60.1727 86 87 1.01611 1.0204 1.6473 2.6676 87.650 2697.016 2784.666 0.2785 7.9128 8.1914 364.45 1.1580 62.5544 87 88 1.12800 1.0232 1.8333 2.8565 88.661 2995.890 3084.551 0.2813 8.7580 9.0393 368.65 1.1696 65.0166 88 89 1.26064 1.0261 2.0540 3.0800 89.671 3350.254 3439.925 0.2841 9.7577 10.0419 372.86 1.1812 67.5581 89 90 1.42031 1.0289 2.3199 3.3488 90.681 3776.918 3867.599 0.2869 10.9586 11.2455 377.06 1.1928 70.1817 90 Table 2 Thermodynamic Properties of Moist Air (Standard Atmospheric Pressure, 101.325 kPa) (Continued) Temp. t, °C Humidity Ratio, kgw/kgda Ws Volume, m3/kg (dry air) Enthalpy, kJ/kg (dry air) Entropy, kJ/(kg · K) (dry air) Condensed Water Temp. ,°C Enthalpy , kJ/kg hw Entropy, kJ/(kg·K) sw Vapor Pressure, kPa psva vas vs ha has hs sa sas ss Psychrometrics 6.5 Table 3 Thermodynamic Properties of Water at Saturation Temp. t, °C Absolute Pressure kPa p Specific Volume, m3/kg Enthalpy, kJ/kg Entropy, kJ/(kg ·K) Temp., °C Sat. Liquid vf /vf Evap. vfg /vfg Sat. Vapor vg Sat. Liquid hf /hf Evap. hfg /hfg Sat. Vapor hg Sat. Liquid sf /sf Evap. sfg /sfg Sat. Vapor sg −60 0.00108 0.001082 90942.00 90942.00 −446.40 2836.27 2389.87 −1.6854 13.3065 11.6211 −60 −59 0.00124 0.001082 79858.69 79858.69 −444.74 2836.46 2391.72 −1.7667 13.2452 11.5677 −59 −58 0.00141 0.001082 70212.37 70212.37 −443.06 2836.64 2393.57 −1.6698 13.8145 11.5147 −58 −57 0.00161 0.001082 61805.35 61805.35 −441.38 2836.81 2395.43 −1.6620 13.1243 11.4623 −57 −56 0.00184 0.001082 54469.39 54469.39 −439.69 2836.97 2397.28 −1.6542 13.0646 11.4104 −56 −55 0.00209 0.001082 48061.05 48061.05 −438.00 2837.13 2399.12 −1.6464 13.0054 11.3590 −55 −54 0.00238 0.001082 42455.57 42455.57 −436.29 2837.27 2400.98 −1.6386 12.9468 11.3082 −54 −53 0.00271 0.001083 37546.09 37546.09 −434.59 2837.42 2402.83 −1.6308 12.8886 11.2578 −53 −52 0.00307 0.001083 33242.14 33242.14 −432.87 2837.55 2404.68 −1.6230 12.8309 11.2079 −52 −51 0.00348 0.001083 29464.67 29464.67 −431.14 2837.68 2406.53 −1.6153 12.7738 11.1585 −51 −50 0.00394 0.001083 26145.01 26145.01 −429.41 2837.80 2408.39 −1.6075 12.7170 11.1096 −50 −49 0.00445 0.001083 23223.69 23223.70 −427.67 2837.91 2410.24 −1.5997 12.6608 11.0611 −49 −48 0.00503 0.001083 20651.68 20651.69 −425.93 2838.02 2412.09 −1.5919 12.6051 11.0131 −48 −47 0.00568 0.001083 18383.50 18383.51 −424.27 2838.12 2413.94 −1.5842 12.5498 10.9656 −47 −46 0.00640 0.001083 16381.35 16381.36 −422.41 2838.21 2415.79 −1.5764 12.4949 10.9185 −46 −45 0.00721 0.001984 14612.35 14512.36 −420.65 2838.29 2417.65 −1.5686 12.4405 10.8719 −45 −44 0.00811 0.001084 13047.65 13047.66 −418.87 2838.37 2419.50 −1.5609 12.3866 10.8257 −44 −43 0.00911 0.001084 11661.85 11661.85 −417.09 2838.44 2421.35 −1.5531 12.3330 10.7799 −43 −42 0.01022 0.001084 10433.85 10433.85 −415.30 2838.50 2423.20 −1.5453 12.2799 10.7346 −42 −41 0.01147 0.001084 9344.25 9344.25 −413.50 2838.55 2425.05 −1.5376 12.2273 10.6897 −41 −40 0.01285 0.001084 8376.33 8376.33 −411.70 2838.60 2426.90 −1.5298 12.1750 10.6452 −40 −39 0.01438 0.001085 7515.86 7515.87 −409.88 2838.64 2428.76 −1.5221 12.1232 10.6011 −39 −38 0.01608 0.001085 6750.36 6750.36 −508.07 2838.67 1430.61 −1.5143 12.0718 10.5575 −38 −37 0.01796 0.001085 6068.16 6068.17 −406.24 2838.70 2432.46 −1.5066 12.0208 10.5142 −37 −36 0.02004 0.001085 5459.82 5459.82 −404.40 2838.71 2434.31 −1.4988 11.9702 10.4713 −36 −35 0.02235 0.001085 4917.09 4917.10 −402.56 2838.73 2436.16 −1.4911 11.9199 10.4289 −35 −34 0.02490 0.001085 4432.36 4432.37 −400.72 2838.73 2438.01 −1.4833 11.8701 10.3868 −34 −33 0.02771 0.001085 3998.71 3998.71 −398.86 2838.72 2439.86 −1.4756 11.8207 10.3451 −33 −32 0.03082 0.001086 3610.71 3610.71 −397.00 2838.71 2441.72 −1.4678 11.7716 10.3037 −32 −31 0.03424 0.001086 3263.20 3263.20 −395.12 2838.69 2443.57 −1.4601 11.7229 10.2628 −31 −30 0.03802 0.001086 2951.64 2951.64 −393.25 2838.66 2445.42 −1.4524 11.6746 10.2222 −30 −29 0.04217 0.001086 2672.03 2672.03 −391.36 2838.63 2447.27 −1.4446 11.6266 10.1820 −29 −28 0.04673 0.001086 2420.89 2420.89 −389.47 2838.59 2449.12 −1.4369 11.4790 10.1421 −28 −27 0.05174 0.001086 2195.23 2195.23 −387.57 2838.53 2450.97 −1.4291 11.5318 10.1026 −27 −26 0.05725 0.001087 1992.15 1992.15 −385.66 2838.48 2452.82 −1.4214 11.4849 10.0634 −26 −25 0.06329 0.001087 1809.35 1809.35 −383.74 2838.41 2454.67 −1.4137 11.4383 10.0246 −25 −24 0.06991 0.001087 1644.59 1644.59 −381.34 2838.34 2456.52 −1.4059 11.3921 9.9862 −24 −23 0.07716 0.001087 1495.98 1495.98 −379.89 2838.26 2458.37 −1.3982 11.3462 9.9480 −23 −22 0.08510 0.001087 1361.94 1361.94 −377.95 2838.17 2460.22 −1.3905 11.3007 9.9102 −22 −21 0.09378 0.001087 1240.77 1240.77 −376.01 2838.07 2462.06 −1.3828 11.2555 9.8728 −21 −20 0.10326 0.001087 1131.27 1131.27 −374.06 2837.97 2463.91 −1.3750 11.2106 9.8356 −20 −19 0.11362 0.001088 1032.18 1032.18 −372.10 2837.86 2465.76 −1.3673 11.1661 9.7988 −19 −18 0.12492 0.001088 942.46 942.47 −370.13 2837.74 2467.61 −1.3596 11.1218 9.7623 −18 −17 0.13725 0.001088 861.17 861.18 −368.15 2837.61 2469.46 −1.3518 11.0779 9.7261 −17 −16 0.15068 0.001088 787.48 787.49 −366.17 2837.47 2471.30 −1.3441 11.0343 9.6902 −16 −15 0.16530 0.001088 720.59 720.59 −364.18 2837.33 2473.15 −1.3364 10.9910 9.6546 −15 −14 0.18122 0.001088 659.86 659.86 −362.18 2837.18 2474.99 −1.3287 10.9480 9.6193 −14 −13 0.19852 0.001089 604.65 604.65 −360.18 2837.02 2476.84 −1.3210 10.9053 9.5844 −13 −12 0.21732 0.001089 554.45 554.45 −358.17 2836.85 2478.68 −1.3232 10.8629 9.5497 −12 −11 0.23774 0.001089 508.75 508.75 −356.15 2836.68 2480.53 −1.3055 10.8208 9.5153 −11 −10 0.25990 0.001089 467.14 467.14 −354.12 2836.49 2482.37 −1.2978 10.7790 9.4812 −10 −9 0.28393 0.001089 429.21 429.21 −352.08 2836.30 2484.22 −1.2901 10.7375 9.4474 −9 −8 0.30998 0.001090 394.64 394.64 −350.04 2836.10 2486.06 −1.2824 10.6962 9.4139 −8 −7 0.33819 0.001090 363.07 363.07 −347.99 2835.89 2487.90 −1.2746 10.6552 9.3806 −7 −6 0.36874 0.001090 334.25 334.25 −345.93 2835.68 2489.74 −1.2669 10.6145 9.3476 −6 −5 0.40176 0.001090 307.91 307.91 −343.87 2835.45 2491.58 −2.2592 10.4741 9.3149 −5 −4 0.43747 0.001090 283.83 283.83 −341.80 2835.22 2493.42 −1.2515 10.5340 9.2825 −4 −3 0.47606 0.001090 261.79 261.79 −339.72 2834.98 2495.26 −1.2438 10.4941 9.2503 −3 −2 0.51772 0.001091 241.60 241.60 −337.63 2834.72 2497.10 −1.2361 10.4544 9.2184 −2 −1 0.56267 0.001091 223.11 223.11 −335.53 2834.47 2498.93 −1.2284 10.4151 9.1867 −1 0 0.61115 0.001091 206.16 206.16 −333.43 2834.20 2500.77 −1.2206 10.3760 9.1553 0 0 0.6112 0.001000 206.141 206.143 −0.04 2500.81 2500.77 −0.0002 9.1555 9.1553 0 1 0.6571 0.001000 192.455 192.456 4.18 2498.43 2502.61 0.0153 9.1134 9.1286 1 2 0.7060 0.001000 179.769 179.770 8.39 2496.05 2504.45 0.0306 9.0716 9.1022 2 3 0.7580 0.001000 168.026 168.027 12.60 2493.68 2506.28 0.0459 9.0302 9.0761 3 4 0.8135 0.001000 157.137 157.138 16.81 2491.31 2508.12 0.0611 8.9890 9.0501 4 5 0.8725 0.001000 147.032 147.033 21.02 2488.94 2509.96 0.0763 8.9482 9.0244 5 6 0.9353 0.001000 137.653 137.654 25.22 2486.57 2511.79 0.0913 8.9077 8.9990 6 7 1.0020 0.001000 128.947 128.948 29.42 2484.20 2513.62 0.1064 8.8674 8.9738 7 8 1.0728 0.001000 120.850 120.851 33.62 2481.84 2515.46 0.1213 8.8273 8.9488 8 9 1.1481 0.001000 113.326 113.327 37.82 2479.47 2517.29 0.1362 8.7878 8.9245 9 10 1.2280 0.001000 106.328 106.329 42.01 2477.11 2519.12 0.1511 8.7484 8.8995 10 11 1.3127 0.001000 99.812 99.813 46.21 2474.74 2520.95 0.1659 8.7093 8.8752 11