Thermodynamics: Energy Transformation, Equilibrium, and Properties, Summaries of Thermodynamics

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THERMODYNAMICS (ETDy) Module - 1: Preliminaries and the Zeroth-Law of Thermodynamics  Lecture 1  Lecture 2  Lecture 3  Lecture 4  Lecture 5 Module - 2: First-Law of Thermodynamics and Analysis of Closed Systems  Lecture 6  Lecture 7  Lecture 8 Module-3: Thermodynamic Properties of Pure Substances  Lecture 9  Lecture 10  Lecture 11 Module - 4: First-Law of Thermodynamics for the Flow Processes  Lecture 12  Lecture 13  Lecture 14  Lecture 15 Module - 5: Second Law of Thermodynamics, Entropy and Availability.  Lecture 16  Lecture 17  Lecture 18  Lecture 19  Lecture 20  Lecture 21  Lecture 22  Lecture 23  Lecture 24  Lecture 25  Lecture 26 Module 6: Power and Refrigeration Cycles  Lecture 27  Lecture 28  Lecture 29  Lecture 30  Lecture 31 Module 7: Thermodynamic Relations  Lecture 32  Lecture 33 Module 8 : Gas Vapor Mixtures and Adiabatic Saturation  Lecture 34  Lecture 35  Lecture 36 Macroscopic Approach Consider a certain amount of gas in a cylindrical container. The volume (V) can be measured by measuring the diameter and the height of the cylinder. The pressure (P) of the gas can be measured by a pressure gauge. The temperature (T) of the gas can be measured using a thermometer. The state of the gas can be specified by the measured P, V and T . The values of these variables are space averaged characteristics of the properties of the gas under consideration. In classical thermodynamics, we often use this macroscopic approach. The macroscopic approach has the following features.  The structure of the matter is not considered.  A few variables are used to describe the state of the matter under consideration.  The values of these variables are measurable following the available techniques of experimental physics. Microscopic Approach On the other hand, the gas can be considered as assemblage of a large number of particles each of which moves randomly with independent velocity. The state of each particle can be specified in terms of position coordinates ( xi , yi , zi ) and the momentum components ( pxi , pyi , pzi ). If we consider a gas occupying a volume of 1 cm3 at ambient temperature and pressure, the number of particles present in it is of the order of 1020 . The same number of position coordinates and momentum components are needed to specify the state of the gas. The microscopic approach can be summarized as:  A knowledge of the molecular structure of matter under consideration is essential.  A large number of variables are needed for a complete specification of the state of the matter. SI Units SI is the abbreviation of Système International d' Unités. The SI units for mass, length, time and force are kilogram, meter, second and newton respectively. The unit of length is meter, m, defined as 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the orange-red line of the spectrum of Krypton-86. The unit of time is second, s. The second is defined as the duration of 9 192 631 770 cycles of the radiation associated with a specified transition of the Cesium 133 atom. The unit of mass is kilogram, kg. It is equal to the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures. The amount of substance can also be expressed in terms of the mole (mol). One kilomole of a substance is the amount of that substance in kilograms numerically equal to its molecular weight. The number of kilomoles of a substance, n , is obtained by dividing the mass (m) in kilograms by the moleculare weight (M), in kg/ kmol. The unit for temperature is Kelvin, K . One K is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Quite often the Celsius, oC , is used to express the temperature of a substance. The SI unit of force, called the newton, N is a secondary unit. The, N , is the force required to accelerate a mass of 1 kilogram at the rate of 1 meter per (second)2 . 1 N = (1kg) (1m/s2 )= 1kg m/s2 The smaller or bigger quantities are expressed using the following prefixes Factor Prefix Symbol Factor Prefix Symbol 1012 tera T 10-2 centi c 109 giga G 10-3 milli m 106 mega M 10-6 micro μ 103 kilo k 10-9 nano n 102 hecto h 10-12 pico p Pressure Pressure is the normal force exerted by a system against unit area of the boundary surface. where δA approaches zero. The unit for pressure in SI is pacsal, Pa 1 Pa = 1 N/m2 Two other units are widely used 1 bar = 105 Pa = 100 kPa = 0.1 MPa and the standard atmosphere, where 1 atm = 101.325 kPa = 1.01325 bar = pressure exerted by a columan of 760 mm of Hg Energy Energy is the capacity to exert a force through a distance. In SI, the unit of energy is Newton-meter, N m or Joule, J. Power The rate of energy transfer or storage is called power. The unit of power is watt, W. 1 W = 1 J/s = 1 N m/s and 1 kW = 1000 W. Apart from these, the following units are used for various parameters of interest Frequency, Hertz = Hz = s-1 Electric current, Ampere = A Electric charge, Coulomb, C = As Electric potential, Volt = V = W/A Magnetic flux, Weber, Wb = Vs Magnetic flux density, Tesla, T = Wb/m2 where, If , then dP is said to be an exact differential, and P is a point function. A thermodynamic property is a point function and not a path function. Pressure, temperature, volume or molar volume are some of the quantities which satisfy these requirements. Intensive and Extensive Properties There are certain properties which depend on the size or extent of the system, and there are certain properties which are independent of the size or extent of the system. The properties like volume, which depend on the size of the system are called extensive properties. The properties, like temperature and pressure which are independent of the mass of the system are called intensive properties. The test for an intensive property is to observe how it is affected when a given system is combined with some fraction of exact replica of itself to create a new system differing only by size. Intensive properties are those which are unchanged by this process, whereas those properties whose values are increased or decreased in proportion to the enlargement or reduction of the system are called extensive properties. Assume two identical systems S1 and S2 as shown in Figure 2.1 . Both the systems are in identical states. Let S3 be the combined system. Is the value of property for S3 same as that for S1 (and S2 )? Figure 2.1  If the answer is yes, then the property is intensive  If the answer is no, then the property is extensive The ratio of the extensive property to the mass is called the specific value of that property specific volume, v = V/m = 1/ ρ ( ρ is the density) specific internal energy, u = U/m Similarly, the molar properties are defined as the ratios of the properties to the mole number (N) of the substance Molar volume = = V/N Molar internal energy = = U/N Lecture 3 : Energy and Processes The lecture deals with  Energy  Macroscopic modes of Energy  Microscopic modes of Energy  Thermodynamic Equilibrium  Process Energy Energy is often defined as the capacity to produce work. However, this "capacity" has a special significance. The capacity represents a combination of an effort and the change brought about by the effort. However, the effort is exerted in overcoming resistance to a particular type of change. The effort involved is measured quantitatively as a "driving force" in thermodynamics. A driving force is a property which causes and also controls the direction of change in another property. The quantitative value of this change is called a "displacement". The product of a driving force and its associated displacement represents a quantity of energy, but in thermodynamics this quantity has meaning only in relation to a specifically defined system. Relative to a particular system there are generally two ways of locating a driving force and the displacement it produces. In one way, the driving force and the displacement are properties of the system and are located entirely within it. The energy calculated from their product represents a change in the internal energy of the system. Similarly, both the driving force and its displacement could be located entirely within the surroundings so that the calculated energy is then a change in the total energy of the surroundings. In another way, the displacement occurs within the system but the driving force is a property of the surroundings and is applied externally at the system boundary. By definition, the boundary of a system is a region of zero thickness containing no matter at all so that the energy calculated in this way is not a property of matter either in the system or in its surroundings but represents a quantity of energy in transition between the two. In any quantitative application of thermodynamics it is always important to make a careful Equilibrium At the state of equilibrium, the properties of the system are uniform and only one value can be assigned to it. In thermodynamics, equilibrium refers to a state of equilibrium with respect to all possible changes, thermal, mechanical and chemical. a. Thermal equilibrium A state of thermal equilibrium can be described as one in which the temperature of the system is uniform. b. Mechanical equilibrium Mechanical equilibrium means there is no unbalanced force. In other words, there is no pressure gradient within the system. c. Chemical equilibrium The criterian for chemical equilibrium is the equality of chemical potential Superscripts A and B refers to systems and subscript i refers to component If Gibbs function is given by G, G = U + PV � TS where ni is the number of moles of substance i . The composition of a system does not undergo any change because of chemical reaction Process In thermodynamics we are mainly concerned with the systems which are in a state of equilibrium. Whenever a system undergoes a change in its condition, from one equilibrium state to another equilibrium state, the system is said to undergo a process. Consider a certain amount of gas enclosed in a piston-cylinder assembly as our system. Suppose the piston moves under such a condition that the opposing force is always infinitesimally smaller than the force exerted by the gas. The process followed by the system is reversible . A process is said to be reversible if the system and its surroundings are restored to their respective initial states by reversing the direction of the process. A reversible process has to be quasi-static, but a quasi - static process is not necessarily quasi-static. Figure 3.2 The process is irreversible if it does not fulfil the criterion of reversibility. Many processes are characterized by the fact that some property of the system remains constant. These processes are: A process in which the volume remains constant  constant volume process. Also called isochoric process / isometric process A process in which the pressure of the system remains constant.  constant pressure process. Also called isobaric process A process in which the temperature of the system is constant.  constant temperature process. Also called isothermal process A process in which the system is enclosed by adiabatic wall.  Adiabatic process Lecture 4 : Work and Heat The lecture deals with  Work  Thermodynamic Definition of Work  Heat Work Work is one of the basic modes of energy transfer. The work done by a system is a path function, and not a point function. Therefore, work is not a property of the system, and it cannot be said that the work is possessed by the system. It is an interaction across the boundary. What is stored in the system is energy, but not work. A decrease in energy of the system appears as work done. Therefore, work is energy in transit and it can be indentified only when the system undergoes a process. Work must be regarded only as a type of energy in transition across a well defined, zero thickness, boundary of a system. Consequently work, is never a property or any quantity contained within a system. Work is energy driven across by differences in the driving forces on either side of it. Various kinds of work are identified by the kind of driving force involved and the characteristic extensive property change which accompanied it. Work is measured quantitatively in the following way. Any driving force other than temperature, located outside the system on its external boundary, is multiplied by a transported extensive property change within the system which was transferred across the system boundary in response to this force. The result is the numerical value of the work Situation in which W ≠ P dV Figure 4.4  Let the initial volume be V1 and pressure P1  Let the final volume be V2 and pressure P2 What should be the work done in this case? Is it equal to ∫P dV ? ∫P dV = area under the curve indicating the process on P-V diagram. The expansion process may be carried out in steps as shown in figure 4.4. It is possible to draw a smooth curve passing through the points 1bcde2 . Does the area under the curve (figure 4.5) represent work done by the system? The answer is no, because the process is not reversible. The expansion of the gas is not restrained by an equal and opposing force at the moving boundary. W ≠ ∫ P dV Figure 4.5 No external force has moved through any distance in this case, the work done is zero. Therefore, we observe that W = ∫ P dV only for reversible process W ≠ ∫ P dV for an irreversible process Another exceptional situation !  The piston is held rigid using latches ! (Figure 4.6)  dV = 0  Work done on the gas is equal to the decrease in the potential energy of mass m  A situation where dV = 0 and yet dW is not zero  such work can be done in one direction only. Work is done on the system by the surroundings Figure 4.6 Heat Heat is energy transfer which occurs by virtue of temperature difference across the boundary. Heat like work, is energy in transit. It can be identified only at the boundary of the system. Heat is not stored in the body but energy is stored in the body. Heat like work, is not a property of the system and hence its differential is not exact. Heat and work are two different ways of transferring energy across the boundary of the system. The displacement work is given by (figure 4.7) Figure 4.7 sufficient number of properties are specified, the rest of the properties assume certain values automatically. The number of properties required to fix the state of a system is given by the state postulate: The state of a simple compressible system is completely specified by two independent properties. A system is called a simple compressible system in the absence of electrical, magnetic, gravitational, motion, and surface tension effects. These effects are due to external force fields and are negligible for most engineering problems. Otherwise, an additional property needs to be specified for each effect which is significant. If the gravitational effects are to be considered, for example, the elevation z needs to be specified in addition to the two properties necessary to fix the state. The state postulate is also known as the two-property rule. The state postulate requires that the two properties specified be independent to fix the state. Two properties are independent if one property can be varied while the other one is held constant. Temperature and specific volume, for example, are always independent properties, and together they can fix the state of a simple compressible system (Fig. 5.1). Figure 5.1 For a simple compressible substance (gas), we need the following properties to fix the state of a system: ρ or PT or T or uP or u . Why not u and T? They are closely related. For ideal gas u=u(t) Temperature and pressure are dependent properties for multi-phase systems. At sea level ( P = 1 atm), water boils at 100oC, but on a mountain-top where the pressure is lower, water boils at a lower temperature. That is, T = f (P) during a phase-change process, thus temperature and pressure are not sufficient to fix the state of a two-phase system. Therefore, by specifying any two properties we can specify all other state properties. Let us choose P and Then T = T(P, ) u = u(P, ) etc. Can we say W = W(p, ) No. Work is not a state property, nor is the heat added (Q) to the system. Zeroth Law of Thermodynamics Statement: If a body 1 is in thermal equilibrium with body 2 and body 3, then the body 2 and body 3 are also in thermal equilibrium with each other Figure 5.2 Two systems 1 and 2 with independent variables ( U1 , V1 , N1 ) and (U2 , V2 , N2 ) are brought into contact with each other through a diathermal wall (figure 5.2). Let the system 1 be hot and system 2 be cold. Because of interaction, the energies of both the systems, as well as their independent properties undergo a change. The hot body becomes cold and the cold body becomes hot. After sometime, the states of the two systems do not undergo any further change and they assume fixed values of all thermodynamic properties. These two systems are then said to be in a state of thermal equilibrium with each other. The two bodies which are in thermal equilibrium with each other have a common characteristic called temperature. Therefore temperature is a property which has the same value for all the bodies in thermal equilibrium. Suppose we have three systems 1, 2 and 3 placed in an adiabatic enclosure as shown in figure 5.3. Figure 5.3 The systems 1 and 2 do not interact with each other but they interact separately with systems 3 through a diathermal wall. Then system 1 is in thermal equilibrium with system 3 and system 2 is also in thermal equilibrium with system 3. By intuition we can say that though system 1 and 2 are not interacting, they are in thermal equilibrium with each other. Suppose system 1 and 2 are brought into contact with each other by replacing the adiabatic wall by a diathermal wall as shown in figure 5.3 (B). Further they are isolated from system 3 by an adiabatic wall. Then one observes no change in the state of the systems 1 and 2. Temperature Scale Based on zeroth law of thermodynamics, the temperature of a group of bodies can be compared by bringing a particular body (a thermometer) into contact with each of them in turn. To quantify the measurement, the instrument should have thermometric properties. These properties include: The length of a mercury column in a capillary tube, the resistance (electrical) of a wire, the pressure of a gas in a closed vessel, the emf generated at the junction of two dissimilar metal wires etc. are commonly used thermometric properties. To assign numerical values to the thermal state of a system, it is necessary to establish a temperature scale on which temperature of a system can be read. Therefore, the Figure 5.5 When these curves are extrapolated to zero pressure, all of them yield the same intercept. This behaviour can be expected since all gases behave like ideal gas when their pressure approaches zero. The correct temperature of the system can be obtained only when the gas behaves like an ideal gas, and hence the value is to be calculated in limit Ptp 0. Therefore , as A constant pressure thermometer also can be used to measure the temperature. In that case, ; when Here Vtp is the volume of the gas at the triple point of water and V is the volume of the gas at the system temperature. Lecture 6 : The lecture deals with  First Law of Thermodynamics  Heat is a Path Function  Energy is a Property of the System  A Perpetual Motion Machine of First Kind  Analysis of Closed Systems First Law of Thermodynamics A series of Experiments carried out by Joule between 1843 and 1848 from the basis for the First Law of Thermodyanmics The following are the observations during the Paddle Wheel experiment shown in Fig. 6.1. Figure 6.1  Work done on the system by lowering the mass m through = change in PE of m  Temperature of the system was found to increase  System was brought into contact with a water bath  System was allowed to come back to initial state  Energy is transferred as heat from the system to the bath The system thus executes a cycle which consists of work input to the system followed by the transfer of heat from the system. (6.1) Whenever a system undergoes a cyclic change, however complex the cycle may be, the algebraic sum of the work transfer is equal to the algebraic sum of the energy transfer as heat (FIRST LAW OF THERMODYNAMICS). Sign convention followed in this text:  Work done by a system on its surroundings is treated as a positive quantity.  Energy transfer as heat to a system from its surroundings is treated as a positive quantity (6.2) or, Heat is Path Function Lets us consider following two cycles: 1a2b1 and la2cl and apply the first law of thermodynamics Eq (6.2) to get (6.11) An imaginary device which would produce work continuously without absorbing any energy from its surroundings is called a Perpetual Motion Machine of the First kind, (PMMFK). A PMMFK is a device which violates the first law of thermodynamics. It is impossible to devise a PMMFK (Figure 6.3) Figure 6.3 The converse of the above statement is also true, i.e., there can be no machine which would continuously consume work without some other form of energy appearing simultaneously. Analysis of Closed System Let us consider a system that refers to a definite quantity of matter which remains constant while the system undergoes a change of state. We shall discuss the following elementary processes involving the closed systems. Constant Volume Process Our system is a gas confined in a rigid container of volume V (Refer to Figure 6.4) Figure 6.4  Let the system be brought into contact with a heat source.  The energy is exchanged reversibly. The expansion work done (PdV) by the system is zero.  Applying the first law of thermodynamics, we get (6.12) or, (6.13)  Hence the heat interaction is equal to the change in the internal energy of the system. Constant Volume Adiabatic Process Refer to Figure 6.5 where a change in the state of the system is brought about by performing paddle wheel work on the system. Figure 6.5 The process is irreversible. However, the first law gives (6.14) or (6.14) Interaction of heat and irreversible work with the system is same in nature. (-W) represents the work done on the system by the surroundings Specific Heat at Constant Volume By definition it is the amount of energy required to change the temperature of a unit mass of the substance by one degree. Energy transfer as heat takes place reversibly. The work is done by system when it changes from the initial state (1) to the final state (2). (7.1) Applying the first law, we get (7.2) or (7.3) or (7.4) The quantity is known as enthalpy, H (a property) of the system. The specific enthalpy h is defined as (7.5) The molar enthalpy is , where N is the mole number of the substance. Constant Pressure Process-2 Let us assume paddle wheel work is done on the system figure 7.2. Also, consider adiabatic walls, so that Figure 7.2 Now the application of first law enables us to write (7.6) or (7.7) or (7.8) Therefore, the increase in the enthalpy of the system is equal to the amount of shaft work done on the system. Specific Heat at Constant Pressure Let us focus on Figure 7.3 Figure 7.3  System changes its state from 1 to 2 following a constant pressure process.  There will be an accompanying change in temperature. Specific heat at constant pressure is defined as the quantity of energy required to change the temperature of a unit mass of the substance by one degree during a constant pressure process. (7.9) The total heat interaction for a change in temperature from T1to T2 can be calculated from (7.10) The molar specific heat at constant pressure can be defined as (7.11) (7.18) or (7.19) or For an ideal gas, Therefore, (7.20) The ratio of specific heats is given by (7.21) or, (7.22) Thus, (7.23) Therefore when an ideal gas expands reversibly and adiabatically from the initial state to final state , the work done per mole of the gas is given by the above expression. Lecture 8 : The lecture contains  Characterisation of Reversible Adiabatic Process  Polytropic Process  Ideal Gas Model Characterisation of Reversible Adiabatic Process Let us find out the path followed by the system in reaching the final sate starting from the initial state. We have already seen that for an ideal gas (8.1) or, (8.2) or, (8.3) or, (8.4) or = constant (8.5) Since, (8.6) From (8.4) and (8.6) we get (8.7) or, (8.8) or, =constant (8.9) Polytropic Process Ideal gas undergoes a reversible-adiabatic process; the path followed by the system is given by =constant (8.10) and the work done per kg of gas Lecture 9 : Thermodynamic Properties of Fluids The lecture deals with  Thermodynamic Properties of Fluids  Pure substance  Equations of State  Ideal Gas Thermodynamic Properties of fluids Most useful properties:  Properties like pressure, volume and temperature which can be measured directly. Also viscosity, thermal conductivity, density etc can be measured.  Properties like internal energy, enthalpy etc. which cannot be measured directly Pure Substance A pure substance is one which consists of a single chemical species. Concept of Phase Substances can be found in different states of aggregation. Ice, water and steam are the three different physical states of the same species . Based on the physical states of aggregation the substances are generally classified into three states as a. Solid b. liquid c. gas The different states of aggregation in which a pure substance can exist, are called its phases. Mixtures also exist in different phases. For example, a mixture of alcohal and water can exist in both liquid and vapour phases. The chemical composition of the vapour phase is generally different from that of the liquid phase. Generalizing the above observations, it can be said that a system which is uniform throughout both in chemical composition and physical state, is called a homogeneous substance or a phase. All substances,-solids, liquids and gases, change their specific volume depending on the range of applied pressure and temperature. When the changes in the specific volume is negligible as in the case of liquids and solids, the substance may be treated as incompressible. The isothermal compressibility of a substance κ is defined as The coefficient of volume expansion of a substance is defined as Equations of State Equations that relate the pressure, temperature and specific volume or molar volume of a substance. They predict the relationship of a �gas� reasonably well within selected regions. Boyle's law: at constant pressure or, Constant at constant temperature Charles' Law: At constant pressure At constant volume Boyle's law and Charles' law can be combined to yield Since, (9.3) (9.4) Van der Waal's equation of state: The gases at low pressure and high temperatures follow ideal gas law. Gases that do not follow ideal gas law are required to be represented by a similar set of mathematical relations. The first effort was from clausius. Clausius Proposed (9.5) Van der Waals, by applying the laws of mechanics to individual molecules, introduced two correction terms in the equation of ideal gas. Van der Waals equation of state is given by. (9.6) The term accounts for the intermolecular forces and b accounts for the volume occupied by the gas molecules. Since Van der Waal's equation takes into account the inter-molecular forces, it is expected to hold good for liquid phase also. Now a days, the Van der Waal's equation is used to predict the phase equilibrium data. The constants and b are determined from experimental data. However, rearranging the Van der Waal's equation, we get (9.7) Therefore, a plot of P versus T at constant volume gives a straight line with a slope and an intercept . An interesting feature can be obtained from the following form (9.8) For low temperature, three positive real roots exist for a certain range of pressure. As the temperature increases, the three real roots approach one another and at the critical temperature they become equal. For only one real value of exists. three real values of exists, where is the critical temperature of the gas. Figure 9.2 Refer to Fig 9.2. The curve ABDEF is predicted by the Van der Waals equation of state. The curve ABDEF is practically not realizable. The physically feasible curve is the straight line path from A to F. It can be shown that the straight line AF is such that the area ABDA is exactly equal to the area DEFD for a stable system. Lecture 10 : The Constants of Van der Waals Equation and Compressibility Chart The lecture deals with  The Van der Waals Constants  Virial Equation od State  Compressibility Chart The Van der Waals Constants The constants and are different for different substances. We can get the estimates of and of a substance by knowing the critical values of that substance. Before we proceed further, let us look at the mathematical function For the maximum value of x: and For the minimum value of x: and For the point of inflection, and From the Figure 9.2 we can see that the critical point is a point of inflection. The critical isotherm must show a point of inflection at the critical point (10.1) (10.2) At the critical point, the Van der Waals equation is Compressibility Chart To quantify deviation of real behavior from the ideal gas behavior, we introduce a new term namely, the compressibility factor. The compressibility factor Z is defined as the ratio of the actual volume to the volume predicted by the ideal gas law at a given temperature and pressure. Z = (Actual volume) / (volume predicted by the ideal gas law) (10.10) If the gas behaves like an ideal gas, Z =1 at all temperatures and pressures. A plot of Z as a function of temperature and pressure should reveal the extent of deviation from the ideal gas law. Figure 10.1 shows a plot of Z as a function of temperature and pressure for N2. Figure 10.1 For each substance, a compressibility factor chart or compressibility chart is available. Figure 10.2 It would be very convenient if one chart could be used for all substances. The general shapes of the vapour dome and of the constant temperature lines on the plane are similar. This similarity can be exploited by using dimensionless properties called reduced properties. The reduced pressure is the ratio of the existing pressure to the critical pressure of the substance and the same is followed for the reduced temperature and reduced volume. Then (10.11) At the same temperature and pressure, the molar volumes of different gases are different. However, it is found from experimental data that at the same reduced pressure and reduced temperature, the reduced volumes of different gases are approximately the same. Therefore for all substances (10.12) or, (10.13) Where is called critical compressibility factor. Experimental values of for most substance fall with in a narrow range 0.2-0.3. Therefore from the above equation we can write (10.14) When Z is plotted as a function of reduced pressure and , a single plot, known as general compressibility chart (Figure 10.2) is found to be satisfactory for a wide variety of substances. Although necessarily approximate, the plots are extremely useful in situations where detailed data on a particular gas are lacking but its critical properties are available. Lecture 11 : Phase-Change Process of Pure Substances The lecture deals with  Graphical Representation of Data for Pure Substances  Specific internal energy and enthalpy  Steam Tables There are certain situations when two phases of a pure substance coexist in equilibrium. As a commonly used substance water may be taken up to demonstrate the basic principles involved. However, all pure substances exhibit the same general behaviour. We shall remember the following definitions: Saturated State: A state at which a phase change begins or ends Saturation Temperature: Temperature at which phase change (liquid-vapour) begins or ends at a given pressure Along, 1-T, or T-C, the system is univariant , that is only one thermodynamic property of the system can be varied independently. The system is bivariant in the single phases region. The curve 2-T can be extended indefinitely, the curve T-C terminates at point C which is called the critical point . The critical point represents highest temperature and pressure at which both the liquid phase and vapour phase can coexist in equilibrium. At the critical point , the specific volumes and all other thermodynamic properties of the liquid phase and the vapour phase are indentical. and are called the critical temperature and critical pressure, respectively. If the substance exist as a liquid on the curve T−C, it is called saturated liquid, and if it exists as a vapour, it is called a saturated vapour. Under the constant pressure, the line abc indicates melting, - sublimation and - vaporization. (b) Diagram Refer to the diagram as shown in Figure 11.2. Figure 11.2 The isotherm is at a temperature greater than the critical temperature . The isotherms and are at temperatures less than the critical temperature and they cross the phase boundary. The point C represents the critical point. The curve AC is called the saturated liquid line; and the curve CB is called the saturated vapour line. The area under the curve ACB is the two-phase region where both liquid and vapour phase are present. Left to the curve AC is the liquid region. Region to the right of curve CB is the vapour region The isotherm appears in three segments: DE, EF and FG. DE is almost vertical, because the change in the volume of liquid is very small for a large change in pressure. The segment FG is less steep because vapour is compressible. Segment EF is horizontal, because the phase change from liquid to vapour occurs at constant pressure and constant temperature for a pure substance. EF represents all possible mixtures of saturated liquid and saturated vapour. The total volume of the mixture is the sum of the volumes of the liquid and vapour phases. (11.1) Dividing by the total mass m of the mixture, m an average specific volume for mixture is obtained (11.2) Since the liquid phase is a saturated liquid and the vapour phase is a saturated vapour, (11.3) so, (11.4) Introducing the definition of quality or dryness fraction and noting that the above expression becomes (11.5) The increase in specific volume due to vaporization is often denoted by (11.6) Again we can write (11.7) or, (11.8) or, (11.9) Known as lever rule. At the critical point, the specific volumes and all other thermodynamic properties of the liquid phase and the vapour phase are identical. (c) Diagram Refer to diagram shown in Fig 11.3. Let us consider constant pressure heating of liquid water in a cylinder-piston assembly. If the water is initially at state 1, on heating at constant pressure, the temperature and the specific volume of the water increase and follows the path 1-2 as shown in Fig 11.3(a).  Heat interaction in an isobaric process is equal to change in enthalpy of the system.  For a flow process (will be discussed later) the work done by an adiabatic device is equal to the decrease in enthalpy of the flowing fluid.  A throttling process is an isenthalpic process. A throttling process is a process in which a fluid flows from a region of high pressure to a region of low pressure without exchanging energy as heat or work.  A reversible adiabatic process is an isentropic process. Specific internal energy and enthalpy or, [on a unit mass basis] or, [on per mole basis] Data for specific internal energy, u and enthalpy, h can be calculated from the property tables in the same way as for specific volume. (11.10) Increase in specific internal energy is often denoted by The specific enthallpy is (11.11) Increase in specific enthalpy is often denoted by . Steam Tables Two Saturated steam tables  Saturated Steam Pressure Table  Saturated Steam Temperature Table Similar tables can exist for any pure substance (e.g. Freon 12) Saturated Steam: Pressure Table (kJ/kg) P T (m3/kg) h (kJ/kg) s (kJ/kg k) (bar) (0c) 0.01 6.98 0.05 32.90 0.10 45.83 . . . . . . 2.0 120.23 In this table pressure is selected as the independent variable. Saturated Steam: Temperature Table T P h s ( oc) (bar) 0 6.1×10-3 2 4 . . . . . . 374.15 221.2 Superheated Steam Table P(bar) (0.2) Temperature (oc) 100 200 300 400 (m3/kg) 8.5 h (kJ/kg) 2686 s (kJ/kg k) 8.126 The values for a given condition are to be evaluated through interpolation. Lecture 12 : First Law of Thermodynamics for a Continuous System The lecture deals with  Conservation of Mass applied to a control volume  Conservation of Energy applied to a Control Volume Conservation of Mass applied to a control volume Let us consider the law of conservation of mass as applied to the control volume. = specific energy of matter inside the control volume at time . and = Pressure at the inlet and exit ports, respectively. and = Flow velocity at the inlet and exit ports, respectively. and = specific volumes at the inlet and exit ports respectively and = specific energy of the material at the inlet and exit ports respectively. = Rate of energy flow as heat into the control volume = Rate of shaft work done by the control volume The mass contained in the region A which enters the control volume during the time interval The mass contained in the region B which leaves the control volume during the time interval From the mass balance, we can write (12.6) where is the mass entering the control volume during the differential time . To accommodate this, the mass inside the control volume has to be compressed such that its volume decreases by the amount . This is accomplished by the pressure acting on the material entering the control volume. Therefore, the work done = Since the mass has to leave the control volume at the exit port, the work done = , Energy of the system at time Energy of the system at time During the time interval We may account for the following Energy transferred as heat to the system = Shaft work done by the system Energy flow as heat into the control volume and the shaft work delivered by the control volume are taken as positive. By applying the first law of thermodynamics, we get (12.7) or, (12.8) or, (12.9) In the limiting case, (12.10) Where, and elevation of the exit and inlet ports above the datum level. The above expression can now be rearranged as (12.11) Rate of energy accumulation = Rate of energy inflow - Rate of energy outflow compressor at a low pressure and emanates at a higher pressure. If the changes in the kinetic energy and potential energy are ignored, and the energy losses are negligible, then the first law for this flow process reduces to (13.5) In a compressor, is negative. Because the work is being done on the system, . The work done on the compressor per unit mass of the fluid is equal to the increase in enthalpy of the fluid. The compressors discharge the fluid with higher enthalpy, i.e, with higher pressure and temperature. (c) Nozzle: A nozzle is primarily used to increase the flow velocity. Figure 13.2 The first law reduces to (13.6) or, (13.7) If the inlet velocity is negligible and then otherwise, .The velocity is increased at the cost of drop in enthalpy. If an ideal gas is flowing through the nozzle, the exit velocity can be expressed in terms of inlet and outlet pressure and temperatures by making use of the relations: and . Therefore, (13.8) From the relations governing adiabatic expansion, (13.9) We get, (13.10) (d) Diffuser: A diffuser can be thought of as a nozzle in which the direction of flow is reversed. Figure 13.3 For an adiabatic diffuser, and are zero and the first law reduces to (13.11) The diffuser discharges fluid with higher enthalpy. The velocity of the fluid is reduced. (e) Heat Exchangers Figure 13.4 The figure 13.4 explains the working of a simple heat exchanger. The governing equation may be modified for multiple entry and multiple exit of the system as (13.12) (13.13) (13.14) (13.15) Figure 14.3 Such a plot of P versus yields may also be called isenthalpic curve. The slope of the isenthalpic curve is called the Joule-Thomson coefficient and given by The experiments are conducted with different and in order to find out the isenthalpic curves. A family of isenthalpic curves is shown in Figure 14.4 which is typical of all real gases. Figure 14.4 The point at which is called the inversion point. The locus of all the inversion points is the inversion curve. In the region left of the inversion curve, . In the throttling process the down stream pressure is always less than the upstream pressure . Therefore, whenever a real gas is subjected to throttling, the temperature of the gas decreases if the initial state lies in the region to the left of the isenthalpic curve. This is explained by a process from to in Figure 14.4. To the right of the inversion curve, . If the initial state of gas lies in the region to the right of the inversion curve, the temperature of the gas increase upon throttling. For almost all the gases, at ordinary range of pressures and temperature, and the maximum inversion temperature is above the room temperature. The exceptions are hydrogen, helium and neon. For hydrogen, the maximum inversion temperature is 200 K and for helium the maximum inversion temperature is 24 K. If hydrogen is throttled at room temperature, the temperature of the gas increases. To produce low temperature by throttling, the initial temperature of hydrogen should be below 200 K. This is usually accomplished by cooling with liquid nitrogen. Similarly, in the production of liquid helium by throttling, the initial temperature of helium should be below 24 K. Hence it is cooled by liquid hydrogen prior to throttling. Suppose an ideal gas is throttled. Since throttling process is isenthalpic, and for an ideal gas enthalpy is a function of temperature only, the temperature of an ideal gas does not change during a throttling process. Hence for an ideal gas. Applications of Throttling (a) Refrigeration The working fluid (called refrigerant) leaving the evaporator as vapour at state 1 undergoes an adiabatic compression in a compressor and leaves as saturated vapour at state 2. This compressed refrigerant enters a condenser where it rejects heat to the ambient atmosphere and leaves as a saturated liquid at high pressure at state 3. Then the liquid refrigerant at high pressure undergoes throttling and leaves as a mixture of liquid and vapour at low pressure at state 4. During throttling, the liquid vaporizes partially and its temperature decreases. The cooled refrigerant then passes through an evaporator where it absorbs energy from the body to be cooled, and leaves as hot vapor at state 1. Finally, the hot refrigerant vapour goes to the compressor, thus completing a cycle. Figure 14.5 Applying the first law of thermodynamics to each of the units (control volume) the following relations are obtained: Compressor (adiabatic) Condenser (constant pressure cooling) Throttling (isenthalpic) Evaporator (constant pressure heat addition) The coefficient of performance (COP) of the refrigerator is defined as the ratio of the energy absorbed at the evaporator to the energy required to perform the task. (b) Liquefaction of Gases When a real gas which is initially at a temperature lower than the maximum inversion temperature is throttled, its temperature decreases. This principle is used by Linde in the Liquefaction of gases. Refer to figure 14.6 for understanding the operating cycle. Charging of a Cylinder (control Volume Analysis) While charging the cylinder from the supply mains, no gas leaves the control volume, and . No shaft work is done; . Let us ignore the changes in the kinetic energy and potential energy of the gas during charging operation. Then it is possible to write from the equation (15.1) (15.2) or, (15.3) Where, denotes the mass of the gas in the control volume and u denotes the specific internal energy of the gas inside the control volume at time . It is assumed that the state of the gas at any instant of time inside the control volume is uniform. The expression above can be integrated from the start of the charging operation to the end of the charging operation , to obtain (15.4) The principle of conservation of mass for the charging process reduces to: (15.5) Rate of increase of mass in Rate of mass entry Integration of the above equation over time gives the change of mass in the control volume during the charging operation (15.6) Now in the equation (15.4), the last term on the RHS can be evaluated as (15.7) and (15.8) Substituting (15.6), (15.7), and (15.8) in (15.4) we get (15.9) If the cylinder is initially empty and the filling operation is carried out under adiabatic conditions the above expression reduces to (15.10) If the gas is treated as ideal and Then, (15.11) Where, is the final temperature of the gas in the cylinder and is the ratio of the specific heats. is the temperature of the gas in supply mains. Discharging of a cylinder (control volume Analysis) Consider a tank containing a gas at known conditions of temperature and pressure (figure 15.2). Figure 15.2 The valve is opened, the gas escapes from the tank and the properties of the gas in the tank change with respect to time. Let, pressure of the gas inside the cylinder at temperature of the gas inside the cylinder at specific internal energy of the gas inside the cylinder at and denote corresponding quantities of the gas left in the cylinder after the discharging operation. velocity of the escaping gas. From the first law, we can write The general expression is: (15.21) We can change the basis of analysis from a control mass to a control volume based analysis. Under such a situation, Let us consider the cylinder volume as the volume of interest. , and ignore the change in KE and PE (15.22) From the principle of conservation of mass (15.23) Therefore (15.24) For a perfect gas, and . Then equation (15.24) can be rewritten as (15.25) or, (15.26) or, (15.27) or, (15.28) By integration between final state and initial state (15.29) As (15.30) or, (15.31) Where is the characteristics gas constant So, (15.32) and (15.33) Invoking these relations, we get (15.34) or, (15.35) (15.36) Therefore the gas that remains in the tank can be considered to have undergone a reversible adiabatic expansion from the initial pressure to the final pressure .