thermodynamics and transport reference, Cheat Sheet of Chemical Thermodynamics

Download thermodynamics and transport reference and more Cheat Sheet Chemical Thermodynamics in PDF only on Docsity! ab Xe, CHEMICAL ENGINEERING THERMODYNAMICS Andrew S. Rosen “6 SyMpoL Dictionary | 1 TABLE OF CONTENTS Symbol Dictionary . 1. Measured Thermodynamic Properties and Other Basic Concepts .. 1.1 Preliminary Concepts — The Language of Thermodynamics 1.2. Measured Thermodynamic Properties 1.2.1 Volume . Temperature .. 1.2.3 Pressure. 1.3 Equilibrium .. 1.3.1 Fundamental Definitions 1.3.2 Independent and Dependent Thermodynamic Propertie: 1.3.3 Phases... 2. The First Law of Thermodynamics 2.1 Definition of the First Law. 2.2 Fundamental Definitions 2.3 24 2.5 Reversible and Irrev 2.6 Closed Systems. 2.6.1 Integral Balance 2.6.2 Differential Balance 2.7 Isolated Systems 2.8 Open System: 2.9 Open-System Energy Balance on Process Equipment. 2.9.1 Introduction... 2.9.2 Nozzles and Diffusers 2.9.3. Turbines and Pumps. 2.9.4 Heat Exchangers . 2.9.5 Throttling Devices ..... 2.10 Thermodynamic Data for U and H 2.10.1 Heat Capacity .. 2.10.2 Latent Heat .. 2.10.3 Enthalpy of Reaction 2.11 Calculating First-Law Quanti 2.11.1 Starting Poin 2.11.2 Reversible, Isobaric Proce: 2.11.3 Reversible, Isochoric Process 2.11.4 Reversible, Isothermal Proc 2.11.5 Reversible, Adiabatic Proces 2.11.6 Irreversible, Adiabatic Expansion into a Vacuum . 2.12 Thermodynamic Cycles and the Carnot Cycle.... 3. Entropy and the Second Law of Thermodynamics . 3.1 Definition of Entropy and the Second Law... The Second Law of Thermodynamics for Closed Systems Reversible, Adiabatic Processe: Reversible, Isothermal Processes Reversible, Isobaric Processes Reversible, Isochoric Proces: Reversible Phase Change at Constant T and P Irreversible Processes for Ideal Gases 2.) Entropy Change of Mixing ...... 3.3. The Second Law of Thermodynamics for Open Systems SYMBOL Dictionary | 4 Vi Activity coefficient of species i n Efficiency factor Mi Chemical potential of species i Tl Osmotic pressure p Density Vi Stoichiometric coefficient o Pitzer acentric factor & Extent of reaction MEASURED THERMODYNAMIC PROPERTIES AND OTHER BASIC CONCEPTS | 5 1. MEASURED THERMODYNAMIC PROPERTIES AND OTHER BASIC CONCEPTS 1.1 PRELIMINARY CONCEPTS - THE LANGUAGE OF THERMODYNAMICS In order to accurately and precisely discuss various aspects of thermodynamics, it is essential to have a well-defined vernacular. As such, a list of some foundational concepts and their definitions are shown below: ¢ Universe — all measured space e System — space of interest ¢ Surroundings — the space outside the system ¢ Boundary — the system is separated by the surroundings via a boundary ¢ Open System — a system that can have both mass and energy flowing across the boundary ¢ Isolated System — a system that can have neither mass nor energy flowing across the boundary © Closed System — a system that can have energy but not mass flowing across the boundary ¢ Extensive Property — a property that depends on the size of the system ¢ Intensive Property — a property that does not depend on the size of the system e State — the condition in which one finds a system at any given time (defined by its intensive properties) © Process — what brings the system from one state to another ¢ Adiabatic Process — a process that has no heat transfer (Q = 0) ¢ Isothermal Process — a process that has a constant temperature (AT = 0) ¢ Isobaric Process — a process that has a constant pressure (AP = 0) ¢ Isochoric Process — a process that has a constant volume (AV = 0) ¢ Isenthalpic Process — a process that has a constant enthalpy (AH = 0) ¢ Isentropic Process — a process that has a constant entropy (AS = 0) ¢ State Function — a quantity that depends only on the current state of a system ¢ Path Function — a quantity that depends on the path taken 1.2 MEASURED THERMODYNAMIC PROPERTIES With this set of clearly defined vocabulary, we can now discuss how thermodynamic properties are measured. 1.2.1 VOLUME Even though volume, V, is an extensive property, we can define intensive forms. If we divide the volume by the number of moles, n, we get a molar volume which is simply the inverse of density, p. 1.2.2 TEMPERATURE Temperature, T,, is an intensive property and is proportional to the average kinetic energy of the individual atoms or molecules in a system. Over time, the speed of all molecules in a given system becomes a well- MEASURED THERMODYNAMIC PROPERTIES AND OTHER BASIC CONCEPTS | 6 defined distribution; this is referred to as the Maxwell-Boltzmann distribution, an example of which is shown in Figure 1. s § 3 —T=1000K £ L L 0 500 1000 1500 2000 2500 cca Figure 1. A schematic showing a Maxwell-Boltzmann distribution. From the kinetic theory of gases, one can show that ee eolecutar — — mi72 2 and molecular _ ek = where m is the mass of an individual molecule, V is the mean velocity, and kg is the Boltzmann constant. This means that 3kpT m su Thinking about temperature in terms of molecular motion, we can define an absolute temperature scale where 0 is equivalent to no molecular motion. One such absolute scale is the Kelvin scale, which is related to the temperature in Celsius via T[K] = T[°C] + 273.15 Another absolute temperature scale — the Rankine scale — can be used to convert between SI and English systems: 9 TPR] = <7 IK] T[°R] = T[°F] + 459.67 Of course, one can then write that 9 TPF] = gh +32 1.2.3. PRESSURE Pressure, P, is also an intensive property. It is defined as the (normal) force, F, per unit area, A: ‘THE FIRST LAW OF THERMODYNAMICS | 9 2.2 FUNDAMENTAL DEFINITIONS ¢ Kinetic energy —energy of motion, defined as Ex = $mV? © Potential energy — energy associated with the bulk position of a system in a potential field, denoted Ep ¢ Internal energy -— energy associated with the motion, position, and chemical-bonding configuration of the individual molecules of the substances within a system, denoted U e Sensible heat — a change in internal energy that leads to a change in temperature ¢ Latent heat — a change in internal energy that leads to a phase transformation ¢ Heat — the transfer of energy via a temperature gradient, denoted Q © Work - all forms of energy transfer other than heat, denoted W 2.3 WORK The work, W, can be described as w= | Fax where F is the external force and dx is the displacement. Work can also be related to the external pressure via w =~ Pav which is typically referred to as PV work and can be computed by taking the area underneath a P vs. V curve for a process (and then negating it). In this context, a positive value of W means that energy is transferred from the surroundings to the system whereas a negative value means that energy is transferred from the system to the surroundings. The same sign-convention is chosen for heat (see below). 2.4 HYPOTHETICAL PATHS It is important to note that hypothetical paths can be used to find the value of a state function. Consider the processes in Figure 3. The actual path is not easy to use for calculations, as both T and v are changing. However, one can proposed alternative hypothetical paths to get from State 1 to State 2 that take advantage of the fact that the path does not matter when computing a state function. All three paths (the real one as well as the two hypothetical ones) will produce the same answer for Au. ‘THE First LAW OF THERMODYNAMICS | 10 AU pypethotical Path b Molar volume (m®/mol) Temperature (K) Figure 3. Plot of a process that takes a system from State 1 to State 2. Three alternative paths are shown: the real path as well as two convenient hypothetical paths. 2.5 REVERSIBLE AND IRREVERSIBLE PROCESSES A process is reversible if, after the process occurs, the system can be returned to its original state without any net effect on the surroundings. This result occurs only when the driving force is infinitesimally small. Otherwise, the process is said to be irreversible. All real processes are irreversible; however, reversible processes are essential for approximating reality. The efficiency of expansion is typically given by n _ Wirrev lexp= = Wrev and the efficiency of compressions is typically given by n _ Wrev comp ~ ——_ Wirrev 2.6 CLOSED SYSTEMS 2.6.1 INTEGRAL BALANCE The first law of thermodynamics can be written as AU + AEy + AEp =Q+W However, the kinetic and potential macroscopic energies can often be neglected such that AU=Q+W Since the mass of a closed system stays constant, one can divide by the total number of moles if no chemical reactions take place to yield the intensive form: Au=q+t+w ‘THE First LAW OF THERMODYNAMICS | 11 2.6.2 DIFFERENTIAL BALANCE Oftentimes in chemical engineering thermodynamics we must consider how various properties change as a function of time. In this case, differential balances are necessary. The first law can be written similarly as dU + dE +dEp =6Q +5W or dU = 6Q+6W if we ignore kinetic and potential energy contributions.” Of course, for a closed system we can write the equivalent intensive form of the equation as well. With this differential balance, we can differentiate with respect to time to yield: we 64w dt 2.7 ISOLATED SYSTEMS Since isolated systems do not allow for energy transfer, AU = 0 for this case. As such, Q + W = 0. 2.8 OPEN SYSTEMS In open systems, mass can flow into and out of the system. This can be expressed via a mole balance as dn . . a yt — > fou in out assuming no chemical reactions. For a stream flowing through a cross-sectional area A with a velocity Vv. the molar flow rate can be written as i . A n=— v A system at steady-state has all differentials with respect to time being zero, so for steady-state: Yitin = > Hout in out In addition to this balance, we also must write an energy balance. The energy balance is dU dE, dE, — 4K, oP dt dt dt = yt (ut ex + ep)in — Y tout (u+ex + epdout +2 in out We + Yin Pv)in + Y Hout CPP) fn out + In steady-state, this reads ? The d differential operator is used for state functions whereas the 6 differential operator is used for path functions. ‘THE First LAW OF THERMODYNAMICS | 14 From this, it is clear that dh = J Cp aT For liquids and solids, Cp © C, [liquids and solids] For ideal gases, Cp — Cy = R [ideal gas] 2.10.2 LATENT HEAT When a substance changes phases, there is a substantial change in internal energy due to the latent heat of transformation. These are typically reported as enthalpies (e.g. enthalpy of vaporization) at 1 bar, which is the normal boiling point, T,. Therefore, the enthalpy of heating water originally at T, to steam at a temperature T, where T, < T, < Tz, for example, would be Tp T, Ah = J ch aT + Ahyap.r, + J cp dT Ty Tp If it is desired to know the enthalpy of a phase transformation at a pressure other than | bar, one can construct a hypothetical path like that shown in Figure 4. The sum of the enthalpy changes from Step 1, Step 2, and Step 3 is equal to that of the actual path. Temperature Phase Figure 4, Hypothetical path to calculate Ahyap at a temperature T’ from data available at 7, and heat capacity data, 2.10.3 ENTHALPY OF REACTION Typically, the standard? enthalpies of formation, Ah;, of individual species are tabulated. The enthalpy of formation is defined as the enthalpy difference between a given molecule and its reference state, which is typically chosen as the pure elemental constituents as found in nature. As a result, the enthalpy of formation of a pure element is always zero. The standard enthalpy of a reaction can be computed as 3 “Standard” refers to a particular reference state, usually 298.15 K and 1 bar. It is indicated by the © symbol. ‘THE First LAW OF THERMODYNAMICS | 15 Ahi = » vAhy Here, v; is the stoichiometric coefficient. For a balanced reaction aA — bB, the stoichiometric coefficient of A would be v, = —a and the stoichiometric coefficient of B would be vg = D. A reaction that releases heat is called exothermic and has a negative enthalpy of reaction, whereas a reaction that absorbs heat is called endothermic and has a positive enthalpy of reaction. An example of using hypothetical paths to calculate the enthalpy of reaction at a temperature other than the reference temperature is shown in Figure 5. T Reactants f ae Abiyn 7 Ah Ahg eT Reactants Products Figure 5. Hypothetical path to calculate hyn at an arbitrary temperature T. 2.11 CALCULATING FIRST-LAW QUANTITIES IN CLOSED SYSTEMS 2.11.1 STARTING POINT When calculating first-law quantities in closed systems for reversible processes, it is best to always start with the following three equations, which are always true: w =~ Pav AU=Q+W AH = AU + A(PV) U= J Cy aT H= J Cp aT If ideal gas conditions can be assumed then, Cp —Cy = mR AH = AH(T) AU = AU(T) AW OF THERMODYNAMICS | 16 2.11.2 REVERSIBLE, ISOBARIC PROCESS Since pressure is constant:* W = —PAV We then have the following relationships for enthalpy: Qp = AH H= J Cp dT AH = AU + PAV 2.11.3 REVERSIBLE, ISOCHORIC PROCESS Since volume is constant: w=0 We then have the following relationships for the internal energy: Qy = AU u={ car AH = AU + VAP 2.11.4 REVERSIBLE, ISOTHERMAL PROCESS If one is dealing with an ideal gas, AU and AH are only functions of temperature, so AU = AH =0 Due to the fact that AU = Q + W, Q=-w For an ideal gas, integrate the ideal gas law with respect to V to get V2 Po W =—nRT In (2) =nRT In (2) Vy Py 2.11.5 REVERSIBLE, ADIABATIC PROCESS By definition the heat exchange is zero, so: Due to the fact that AU = Q + W, 4 When dealing with thermodynamic quantities, it is important to keep track of units. For instance, computing W = —PAV will get units of [Pressure][ Volume]. To convert this to a unit of [Energy], one must use a conversion factor, such as (8.3145 J/mol K)/(0.08206 L atm/mol K) = 101.32 J/L*atm. ENTROPY AND THE SECOND LAW OF THERMODYNAMICS | 19 3.2 THESECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS 3.2.1 REVERSIBLE, ADIABATIC PROCESSES Since the process is reversible and there is no heat transfer®, As=0, Assur = 0, ASuniv = 0 3.2.2 REVERSIBLE, ISOTHERMAL PROCESSES Since temperature is constant, _ rev A SF If the ideal gas assumption can be made, then Au = 0 such that qrey = Wrey = — J P AV. Plug in the ideal gas law to get Po As = —-Rlin (2) Py Since all reversible processes have no change in the entropy of the universe (i.e. ASyniy = 0), we can say that Asgur, = —As. 3.2.3 REVERSIBLE, ISOBARIC PROCESSES Since 6qp = dh = Cp dT for isobaric processes, Cp As = | —dT Ir Since all reversible processes have no change in the entropy of the universe (i.e. ASyniy = 0), we can say that Asgur, = —As. 3.2.4 REVERSIBLE, ISOCHORIC PROCESSES Since Sqy = du = cy AT for isochoric processes, C As = J ar T Since all reversible processes have no change in the entropy of the universe (i.e. ASuniy = 0), we can say that Asgur, = —As. 3.2.5 REVERSIBLE PHASE CHANGE AT CONSTANT T AND P In this case, dyey is the latent heat of the phase transition. As such, — dp _ Altransition As = = T T Since all reversible processes have no change in the entropy of the universe (i.e. ASyniy = 0), we can say that Assurr = —As. © It will be tacitly assumed any quantity without a subscript refers to that the system. ENTROPY AND THE SECOND LAW OF THERMODYNAMICS | 20 3.2.6 IRREVERSIBLE PROCESSES FOR IDEAL GASES A general expression can be written to describe the entropy change of an ideal gas. Two equivalent expressions are: v, As = [fer +Rin(2) T v, and P As = [fer ~Rn(Z) T P, In order to find the entropy change of the universe, one must think about the conditions of the problem statement. If the real process is adiabatic, then qgurr = 0 and then As,y,; = 0 such that Asyyiy = As. If the real process is isothermal, note that q = w from the First Law of Thermodynamics (i.e. Au = 0) amd that due to conservation of energy dsurr = —. Once qsurr is known, simply use Asgury = a, The entropy change in the universe is then Asyyiy = AS + ASgurr- If the ideal gas approximation cannot be made, try splitting up the irreversible process into hypothetical, reversible pathways that may be easier to calculate. 3.2.7 ENTROPY CHANGE OF MIXING If we assume that we are mixing different inert, ideal gases then the entropy of mixing is Vp ASnix = Ryn, In () i For an ideal gas at constant T and P then P. ASnix = -R Yn in ) = -R Yn, Ing) ‘tot where P; is the partial pressure of species i and y; is the mole fraction of species i. 3.3. THE SECOND LAW OF THERMODYNAMICS FOR OPEN SYSTEMS Since mass can flow into and out of an open system, the Second Law must be written with respect to time: (5) _ (5) + (2) =o dthmiv \dt/sys \dt/surr At steady-state, (S) =0 dt) sys If there is a constant surrounding temperature, then (Fae ™ Data Yad —_ = ) NoutSout — Sin — dt) curr out Sout Nin Sin T. surr ENTROPY AND THE SECOND Law OF THERMODYNAMICS | 21 3.4 THE MECHANICAL ENERGY BALANCE For steady-state, reversible processes with one stream in and one stream out, the mechanical energy balance is W, = | vaP + deg + dep n which can frequently be written as A(V?) | va + MW — + MWgAz Where MW refers to the molecular weight of the fluid. The latter equation is referred to as the Bernoulli Equation. 3.5 VAPOR-COMPRESSION POWER AND REFRIGERATION CYCLES 3.5.1 RANKINE CYCLE The ideal Rankine cycle can be used to convert fuel into electrical power and is shown in Figure 7. The efficiency of a Rankine cycle can be given by ‘means Wwe |t Rankine On Rankine cycle - urbine +L | Fuel 5 a alr pH }___ . s {S] Cooling Boiler af water ta > Condenser 3 i s Compressor Figure 7. The ideal Rankine cycle and its corresponding 7's diagram. 3.5.2 THE VAPOR-COMPRESSION REFRIGERATION CYCLE The ideal vapor-compression refrigeration cycle is shown in Figure 8. The efficiency can be described by the coefficient of performance as Qc _ Ig =hy CoP = —— = We hg he ‘THE THERMODYNAMIC WEB | 24 The following relationship is also true: (5), 9), 53),- 4 0z/y \Ox/, \dy/,. 5.2 DERIVED THERMODYNAMIC QUANTITIES The measured properties of a system are P,v,T and composition. The fundamental thermodynamic properties are u and s, as previously discussed. There are also derived thermodynamic properties. One of which is h. There are also two other convenient derived properties: a, which is Helmholtz free energy, and g, which is Gibbs free energy. The derived thermodynamic properties have the following relationships: h=u+Pv az=u-Ts g=h-Ts Also recall the heat capacity definitions discussed earlier: Ou oh «= (ar), %= (ar), 5.3. FUNDAMENTAL PROPERTY RELATIONS The First Law of Thermodynamics states du = Sdrey + OWrey Now consider enthalpy: dh = du+ d(Pv) Similarly, consider Helmholtz free energy: da = du —d(Ts) Finally, consider Gibbs free energy: dg = dh—d(Ts) Recall that 5q;ey =T ds from the Second Law and 6w,., = —Pdv. With this, we can write a new expression for du and therefore new expressions for dh, da, and dg as well. These are called the fundamental property relations: du=Tds—Pdv dh =Tds+vdP da=-—P dv—sdT dg =vdP—sdT With these expressions, one can write a number of unique relationships by holding certain values constant. By doings so, one yields: () =? (=) = _p ds), - dv), - amy _, (aby _ (33), (),=” ‘THE THERMODYNAMICW | 2 ) _ (52) __p ar), au) p G,-- (Ge), ar)» CNOP)y 5.4 MAXWELL RELATIONS The Maxwell Relations can be derived by applying Euler’s Reciprocity to the derivative of the equation of state. The Euler Reciprocity is 2 dz ax dy dy ax Another useful identity to keep in mind is az ( a (22) ) Ox dy Ox \dy/,. y These mathematical relationships allow one to derive what are called the Maxwell Relations.* These are shown below: au (2) _ (2) a7h : (2) _ (2) asdv'\av/, \ds/, as AP’ \OP/, dsp aa (25) _ (2) ag (2) _ (2) oTav \dv/7 \OT/, OT OP’ \AP/r OT! p By using the thermodynamic property relations in conjunction with the Maxwell Relations, one can also write heat capacities in terms of T and s: =T (=) =T (=) o*\ar), %*\ar), 5.5 DEPENDENT OF STATE FUNCTIONS ONT, P, ANDV With the previous information, one can find the dependence of any state function on T,P, or v quite easily. The procedure to do so can be broken down as follows: 1) Start with the fundamental property relation for du, dh, da, or dg 2) Impose the conditions of constant T, P, or v 3) Divide by dPr,dv7,dT,, or dTp as necessary A . 2&6 . . ae a (ac . ; a . 8 For instance, consider ——. This can be rewritten as = Z () ) using Euler’s Reciprocity. Using the ar ap arap— \ar \ap) 7), appropriate fundamental property relation, © =(2(2)) = (2) Prop Property ” ar aP at \aP) 7), aT) pp” . . . . ai . . . ° For instance, consider trying to find what () can also be written as. Write out the fundamental property relation: esp =f Oe du = T ds — P dv. Then impose constant T and divide by dvr: () = r( T as’ av, - — P. Recognize that (=), from the Maxwell Relations such that (=) = aH P. T ‘THE THERMODYNAMIC WEB | 26 4) Use a Maxwell relation or other identity to eliminate any terms with entropy change in the numerator (if desired) It is useful to know the following identities: _1 (5) _ 1 (3) ~ v\OT/p = oe T where f and x are the thermal expansion coefficient and isothermal compressibility, respectively. 5.6 THERMODYNAMIC WEB A roadmap that outlines the previously discussed relations is shown in Figure 10. EL o LEN — (2 WY (2) Maxwell Relation a8 Qaations In the case of: without T XY JL _ er) ap) 6B A=BCC av OT Is (4) WY (2) 2 WY as) & 05” Relations \¥!T 9817 Relations ‘TIP T without P| without v Relations without s v explicit coe Cir, (irl pr £88 MY Figure 10. Thermodynamic web relating partial derivatives of T, P, s, and v. 5.7 REFORMULATED THERMODYNAMIC STATE FUNCTIONS One can write s(T,v) as ds = Os ar Os a s=(), +(5), v By using the thermodynamic relations and Maxwell relations and integrating, this yields Cy OP as={Sar+ (5) dv T aT), Similarly, one can write s(T,P) as