PARTO PRELUDE: REVIEW OF UNIFIED ENGINEERING THERMODYNAMICS
PART 0 PRELUDE: REVIEW OF "UNIFIED ENGINEERING THERMODYNAMICS
PARTO- PRELUDE: REVIEW OF UNIFIED ENGINEERING THERMODYNAMICS [AW pp 2-22, 32-41(see IAW for detailed SB&vw references): VN Chapter 1] 0.1 What it's all about The focus of thermodynamics in 16.050 is on the production of work, often in the form of kinetic energy(for example in the exhaust of a jet engine)or shaft power, from different sources of heat. For the most part the heat will be the result of combustion processes, but this is not always the case. The course content can be viewed in terms of a"propulsion chain"as shown below, where we see a progression from an energy source to useful propulsive work(thrust power of a jet engine). In terms of the different blocks, the thermodynamics in Unified Engineering and in this course are mainly about how to progress from the second block to the third, but there is some examination of the pro esented by the other arrows as well. The course content, objectives, and lecture outline are described in detail in handout #1 Energy source Heat Mechanical Useful propulsive chemical (combustion work rk(thrust nuclear. etc process) electric power→·| power) 0.2 Definitions and Fundamental Ideas of Thermodynamics As with all sciences, thermodynamics is concerned with the mathematical modeling of the real world. In order that the mathematical deductions are consistent, we need some precise definitions of the basic concepts A continuum is a smoothed-out model of matter neglecting the fact that real substances are composed of discrete molecules. Classical thermodynamics is concerned only with continua. If statistical mechanics and kinetic theor? we wish to describe the properties of matter at a molecular level, we must use the techniques of a closed system is a fixed quantity of matter around which we can draw a boundary. Everything outside the boundary is the surroundings. Matter cannot cross the boundary of a closed system and hence the principle of the conservation of mass is automatically satisfied whenever we employ a closed system analysis The thermodynamic state of a system is defined by the value of certain properties of that system For fluid systems, typical properties are pressure, volume and temperature. More complex systems batte require the specification of more unusual properties. As an example, the state of an electric battery requires the specification of the amount of electric charge it contains Properties may be extensive or intensive. Extensive properties are additive. Thus, if the system is divided into a number of sub-systems, the value of the property for the whole system is equal to the sum of the values for the parts. Volume is an extensive property. Intensive properties do not depend on the quantity of matter present. Temperature and pressure are intensive properties. Specific properties are extensive properties per unit mass and are denoted by For example specific volume=Wm=ν 0-1
0-1 PART 0 - PRELUDE: REVIEW OF “UNIFIED ENGINEERING THERMODYNAMICS” [IAW pp 2-22, 32-41 (see IAW for detailed SB&VW references); VN Chapter 1] 0.1 What it’s All About The focus of thermodynamics in 16.050 is on the production of work, often in the form of kinetic energy (for example in the exhaust of a jet engine) or shaft power, from different sources of heat. For the most part the heat will be the result of combustion processes, but this is not always the case. The course content can be viewed in terms of a “propulsion chain” as shown below, where we see a progression from an energy source to useful propulsive work (thrust power of a jet engine). In terms of the different blocks, the thermodynamics in Unified Engineering and in this course are mainly about how to progress from the second block to the third, but there is some examination of the processes represented by the other arrows as well. The course content, objectives, and lecture outline are described in detail in Handout #1. 0.2 Definitions and Fundamental Ideas of Thermodynamics As with all sciences, thermodynamics is concerned with the mathematical modeling of the real world. In order that the mathematical deductions are consistent, we need some precise definitions of the basic concepts. A continuum is a smoothed-out model of matter, neglecting the fact that real substances are composed of discrete molecules. Classical thermodynamics is concerned only with continua. If we wish to describe the properties of matter at a molecular level, we must use the techniques of statistical mechanics and kinetic theory. A closed system is a fixed quantity of matter around which we can draw a boundary. Everything outside the boundary is the surroundings. Matter cannot cross the boundary of a closed system and hence the principle of the conservation of mass is automatically satisfied whenever we employ a closed system analysis. The thermodynamic state of a system is defined by the value of certain properties of that system. For fluid systems, typical properties are pressure, volume and temperature. More complex systems may require the specification of more unusual properties. As an example, the state of an electric battery requires the specification of the amount of electric charge it contains. Properties may be extensive or intensive. Extensive properties are additive. Thus, if the system is divided into a number of sub-systems, the value of the property for the whole system is equal to the sum of the values for the parts. Volume is an extensive property. Intensive properties do not depend on the quantity of matter present. Temperature and pressure are intensive properties. Specific properties are extensive properties per unit mass and are denoted by lower case letters. For example: specific volume = V/m = v. Energy source chemical nuclear, etc. Heat (combustion process) Mechanical work, electric power... Useful propulsive work (thrust power)
Specific properties are intensive because they do not depend on the mass of the system a simple system is a system having uniform properties throughout. In general, however, Sub-dividing it(either conceptually or in practice)into a number of simple systems in each of oy properties can vary from point to point in a system. We can usually analyze a general system which the properties are assumed to be uniform If the state of a system changes, then it is undergoing a process. The succession of states through which the system passes defines the path of the process. If, at the end of the process, the properties have returned to their original values, the system has undergone a cyclic process. Note that although the system has returned to its original state, the state of the surroundings may have Muddy points Specific properties(MP o1 What is the difference between extensive and intensive properties?(MP 0. 2) 0.3 Review of Thermodynamic Concepts The following is a brief discussion of some of the concepts introduced in Unified Engineering, which we will need in 16.050. Several of these will be further amplified in the lectures and in other handouts. If you need additional information or examples concerning these topics, they are described clearly and in-depth in the Unified Notes of Professor Waitz, where detailed references to the relevant sections of the text(sB& vw) are given. They are also covered although in a less detailed manner, in Chapters I and 2 of the book by Van Ness 1)Thermodynamics can be regarded as a generalization of an enormous body of empirical evidence. It is extremely general, and there are no hypotheses made concerning the structure and type of matter that we deal with. 2)Thermodynamic system a quantity of matter of fixed identity. Work or heat(see below) can be transferred across the system boundary, but mass cannot. Gas Fluid 3)Thermodynamic properties: For engineering purposes, we want averaged"information, i.e., macroscopic not microscopic(molecular) description.(Knowing the position and velocity of each of 1020+ molecules that we meet in typical engineering applications is generally not useful. 4) The thermodynamic state is defined by specifying values of a(small) set of measured properties which are sufficient to determine all the remaining properties 0-2
0-2 Specific properties are intensive because they do not depend on the mass of the system, A simple system is a system having uniform properties throughout. In general, however, properties can vary from point to point in a system. We can usually analyze a general system by sub-dividing it (either conceptually or in practice) into a number of simple systems in each of which the properties are assumed to be uniform. If the state of a system changes, then it is undergoing a process. The succession of states through which the system passes defines the path of the process. If, at the end of the process, the properties have returned to their original values, the system has undergone a cyclic process. Note that although the system has returned to its original state, the state of the surroundings may have changed. Muddy points Specific properties (MP 0.1) What is the difference between extensive and intensive properties? (MP 0.2) 0.3 Review of Thermodynamic Concepts The following is a brief discussion of some of the concepts introduced in Unified Engineering, which we will need in 16.050. Several of these will be further amplified in the lectures and in other handouts. If you need additional information or examples concerning these topics, they are described clearly and in-depth in the Unified Notes of Professor Waitz, where detailed references to the relevant sections of the text (SB&VW) are given. They are also covered, although in a less detailed manner, in Chapters 1 and 2 of the book by Van Ness. 1) Thermodynamics can be regarded as a generalization of an enormous body of empirical evidence. It is extremely general, and there are no hypotheses made concerning the structure and type of matter that we deal with. 2) Thermodynamic system : A quantity of matter of fixed identity. Work or heat (see below) can be transferred across the system boundary, but mass cannot. Gas, Fluid System Boundary 3) Thermodynamic properties : For engineering purposes, we want "averaged" information, i.e., macroscopic not microscopic (molecular) description. (Knowing the position and velocity of each of 1020+ molecules that we meet in typical engineering applications is generally not useful.) 4) State of a system : The thermodynamic state is defined by specifying values of a (small) set of measured properties which are sufficient to determine all the remaining properties
5) Equilibrium The state of a system in which properties have definite(unchanged) values as lon external conditions are unchanged is called an equilibrium state. Properties(P, pressure T temperature, p, density) describe states only when the system is in equilibrium Mechanical Equilibrium Thermal Equilibrium T1 Gas at Copper Partition Mg +PoA=PA Pressure,P Over time,T1→T 6) Equations of state: the stan for a simple compressible substance(e.g. air, water)we need to know two properties to set P=P(v, T, or v=v(P, T, or T= T(P, v) where v is the volume per unit mass, 1/p that are typically of interest for aerospace applications /s. proximation to real gases at conditong s Any of these is equivalent to an equation f(P, v, T=0 which is known as an equation of state equation of state for an ideal gas, which is a very good RT, where v is the volume per mol of gas and r is the "Universal Gas Constant",8.31 k/kmol-K A form of this equation which is more useful in fluid flow problems is obtained if we divide by the molecular weight, M Py=RT or P= OrT where r is r/M. which has a different value for different gases For air at room conditions.r is 0.287 kJ/kg-K 7) Quasi-equilibrium processes: A system in thermodynamic equilibrium satisfies: a)mechanical equilibrium(no unbalanced forces) b)thermal equilibrium(no temperature differences) c)chemical equilibrium For a finite, unbalanced force, the system can pass through non-equilibrium states. We wish to describe processes using thermodynamic coordinates, so we cannot treat situations in which such imbalances exist. An extremely useful idealization, however, is that only infinitesimal unbalanced forces exist, so that the process can be viewed as taking place in a series of"quas equilibrium"states. (The term quasi can be taken to mean"as if you will see it used in a number of contexts such as quasi-one-dimensional, quasi-steady, etc. For this to be true the process must be slow in relation to the time needed for the system to come to equilibrium internally. For 0-3
0-3 5) Equilibrium : The state of a system in which properties have definite (unchanged) values as long as external conditions are unchanged is called an equilibrium state. Properties (P, pressure, T, temperature, ρ, density) describe states only when the system is in equilibrium. Mg + P oA = PA Gas at Pressure, P Mass Mechanical Equilibrium Po Insulation Copper Partition Thermal Equilibrium Gas T1 Over time, T1 → T2 Gas T2 6) Equations of state: For a simple compressible substance (e.g., air, water) we need to know two properties to set the state. Thus: P = P(v,T), or v = v(P, T), or T = T(P,v) where v is the volume per unit mass, 1/ρ. Any of these is equivalent to an equation f(P,v,T) = 0 which is known as an equation of state. The equation of state for an ideal gas, which is a very good approximation to real gases at conditions that are typically of interest for aerospace applications is: Pv– = RT, where v – is the volume per mol of gas and R is the "Universal Gas Constant", 8.31 kJ/kmol-K. A form of this equation which is more useful in fluid flow problems is obtained if we divide by the molecular weight, M: Pv = RT, or P = ρRT where R is R/M, which has a different value for different gases. For air at room conditions, R is 0.287 kJ/kg-K. 7) Quasi-equilibrium processes: A system in thermodynamic equilibrium satisfies: a) mechanical equilibrium (no unbalanced forces) b) thermal equilibrium (no temperature differences) c) chemical equilibrium. For a finite, unbalanced force, the system can pass through non-equilibrium states. We wish to describe processes using thermodynamic coordinates, so we cannot treat situations in which such imbalances exist. An extremely useful idealization, however, is that only "infinitesimal" unbalanced forces exist, so that the process can be viewed as taking place in a series of "quasiequilibrium" states. (The term quasi can be taken to mean "as if"; you will see it used in a number of contexts such as quasi-one-dimensional, quasi-steady, etc.) For this to be true the process must be slow in relation to the time needed for the system to come to equilibrium internally. For a gas
at conditions of interest to us, a given molecule can undergo roughly 10 molecular collisions per second, so that, if ten collisions are needed to come to equilibrium, the equilibration time is on the order of 10-9 seconds. This is generally much shorter than the time scales associated with the bulk properties of the flow(say the time needed for a fluid particle to move some significant fraction of the lighten of the device of interest). Over a large range of parameters, therefore, it is a very good approximation to view the thermodynamic processes as consisting of such a succession of equilibrium states 8)Reversible process For a simple compressible substance Work If we look at a simple system, for example a cylinder of gas and a piston, we see that there can be two pressures, Ps, the system pressure and Px, the external pressure The work done by the system on the environment is Work=Pdv This can only be related to the system properties if Px=Ps. For this to occur, there cannot be any friction, and the process must also be slow enough so that pressure differences due to accelerations are not significant P with friction ∫PxdV≠0 but jPs dV=0 Work during an irreversible Under these conditions, we say that the process is reversible. The conditions for reversibility are a) If the process is reversed, the system and the surroundings will be returned to the b) To reverse the process we need to apply only an infinitesimal dP. a reversible process can be altered in direction by infinitesimal changes in the external conditions (see Van Ness, Chapter 2) 9)Work For simple compressible substances in reversible processes, the work done by the system on the environment is PdV. This can be represented as the area under a curve in a Pressure volume diagram 0-4
0-4 at conditions of interest to us, a given molecule can undergo roughly 1010 molecular collisions per second, so that, if ten collisions are needed to come to equilibrium, the equilibration time is on the order of 10-9 seconds. This is generally much shorter than the time scales associated with the bulk properties of the flow (say the time needed for a fluid particle to move some significant fraction of the lighten of the device of interest). Over a large range of parameters, therefore, it is a very good approximation to view the thermodynamic processes as consisting of such a succession of equilibrium states. 8) Reversible process For a simple compressible substance, Work = ∫PdV. If we look at a simple system, for example a cylinder of gas and a piston, we see that there can be two pressures, Ps, the system pressure and Px, the external pressure. Ps Px The work done by the system on the environment is Work = ∫PxdV. This can only be related to the system properties if Px ≈ Ps. For this to occur, there cannot be any friction, and the process must also be slow enough so that pressure differences due to accelerations are not significant. Work during an irreversible process ≠ ∫Ps dV ∫PxdV ≠ 0 but ∫Ps dV = 0 Ps (V) Px with friction P Vs ➀ ➀ ➀ ➁ ➁ Under these conditions, we say that the process is reversible. The conditions for reversibility are that: a) If the process is reversed, the system and the surroundings will be returned to the original states. b) To reverse the process we need to apply only an infinitesimal dP. A reversible process can be altered in direction by infinitesimal changes in the external conditions (see Van Ness, Chapter 2). 9) Work: For simple compressible substances in reversible processes , the work done by the system on the environment is ∫PdV. This can be represented as the area under a curve in a Pressurevolume diagram:
W12≠W12 II Volume Work depends on the path Work is area under curve of P(v) a)Work is path dependent; b) Properties only depend on states: c) Work is not a property, not a state variable d) when we say W1-2, the work between states I and 2, we need to specify the path e)For irreversible(non-reversible)processes, we cannot use PdV; either the work must be given or it must be found by another method Muddy points How do we know when work is done?(MP 0.3 10) Heat Heat is energy transferred due to temperature differences a) Heat transfer can alter system states b)Bodies dont"contain"heat; heat is identified as it comes across system boundaries c) The amount of heat needed to go from one state to another is path dependent d) Heat and work are different modes of energy transfer ferred e) Adiabatic processes are ones in which no heat is transfer I1) First Law of Thermodynamics For a system AE=Q-w E is the energy of the system Q is the heat input to the system, and w is the work done by the system E=U(thermal energy)+ Ekinetic Potential+ If changes in kinetic and potential energy are not important a)U arises from molecular motion b)U is a function of state, and thus au is a function of state(as is 4E c)Q and w are not functions Comparing(b)and(c) we have the striking result that d )40 is independent of path even though Q and w are not! 0-5
0-5 P Volume Work is area under curve of P(V) V1 V2 W1-2 ≠ P Work depends on the path V I I II W1-2 II a) Work is path dependent; b) Properties only depend on states; c) Work is not a property, not a state variable; d) When we say W1-2, the work between states 1 and 2, we need to specify the path; e) For irreversible (non-reversible) processes, we cannot use ∫PdV; either the work must be given or it must be found by another method. Muddy points How do we know when work is done? (MP 0.3) 10) Heat Heat is energy transferred due to temperature differences. a) Heat transfer can alter system states; b) Bodies don't "contain" heat; heat is identified as it comes across system boundaries; c) The amount of heat needed to go from one state to another is path dependent; d) Heat and work are different modes of energy transfer; e) Adiabatic processes are ones in which no heat is transferred. 11) First Law of Thermodynamics For a system, ∆E QW = − E is the energy of the system, Q is the heat input to the system, and W is the work done by the system. E = U (thermal energy) + Ekinetic + Epotential + .... If changes in kinetic and potential energy are not important, ∆U = Q −W . a) U arises from molecular motion. b) U is a function of state, and thus ∆U is a function of state (as is ∆E). c) Q and W are not functions of state. Comparing (b) and (c) we have the striking result that: d) ∆U is independent of path even though Q and W are not!
Muddy points What are the conventions for work and heat in the first law?(MP 0.4) When does E->U?(MP 0.5 12) Enthalpy: A useful thermodynamic property, especially for flow processes, is the enthalpy. Enthal s usually denoted by h, or h for enthalpy per unit mass, and is defined by H=U+PV In terms of the specific quantities, the enthalpy per unit mass is h=u+ Pv=u+P/p 13)Specific heats -relation between temperature change and heat input For a change in state between two temperatures, the"specific heat" is: Specific heat =@/(Tfinal- Tinitial) We must, however, specify the process, i. e, the path, for the heat transfer. Two useful processes are constant pressure and constant volume. The specific heat at constant pressure is denoted as Cp and that at constant volume as Cv, or cp and cyper unit mass and c For an ideal gas dh= cpdr and du=Cudr. The ratio of specific heats, Cp/cv is denoted by y This ratio is 1. 4 for air at room conditions The specific heats Cv and cp have a basic definition as derivatives of the energy and enthalpy Suppose we view the internal energy per unit mass, L, as being fixed by specification of T, the temperature and v, the specific volume, 1. e, the volume per unit mass.( For a simple compressible substance, these two variables specify the state of the system. Thus u=u(T, v) The difference in energy between any two states separated by small temperature and specific volume differences dt and dv is du The derivative(Ou/), represents the slope of a line of constant v on a u-T plane. The derivative is also a function of state, 1. e, a thermodynamic property, and is called the specific heat at constant volume, Cy. The name specific heat is perhaps unfortunate in that only for special circumstances is the derivative related to energy transfer as heat. If a process is carried out slowly at constant volume, n work will be done and any energy increase will be due only to energy transfer as heat. For such a 0-6
0-6 Muddy points What are the conventions for work and heat in the first law? (MP 0.4) When does E->U? (MP 0.5) 12) Enthalpy: A useful thermodynamic property, especially for flow processes, is the enthalpy. Enthalpy is usually denoted by H, or h for enthalpy per unit mass, and is defined by: H = U + PV. In terms of the specific quantities, the enthalpy per unit mass is h = u + Pv = u P + /ρ. 13) Specific heats - relation between temperature change and heat input For a change in state between two temperatures, the “specific heat” is: Specific heat = Q/(Tfinal - Tinitial) We must, however, specify the process, i.e., the path, for the heat transfer. Two useful processes are constant pressure and constant volume. The specific heat at constant pressure is denoted as Cp and that at constant volume as Cv, or cp and cv per unit mass. c h T c u T p p v v = = ∂ ∂ ∂ ∂ and For an ideal gas dh = cpdT and du = cvdT. The ratio of specific heats, cp/cv is denoted by γ. This ratio is 1.4 for air at room conditions. The specific heats cv. and cp have a basic definition as derivatives of the energy and enthalpy. Suppose we view the internal energy per unit mass, u, as being fixed by specification of T, the temperature and v, the specific volume, i.e., the volume per unit mass. (For a simple compressible substance, these two variables specify the state of the system.) Thus, u = u(T,v). The difference in energy between any two states separated by small temperature and specific volume differences, dT and dv is du = ∂u ∂T v dT + ∂u ∂v T dv The derivative ∂u ∂T v represents the slope of a line of constant v on a u-T plane. The derivative is also a function of state, i. e., a thermodynamic property, and is called the specific heat at constant volume, cv. The name specific heat is perhaps unfortunate in that only for special circumstances is the derivative related to energy transfer as heat. If a process is carried out slowly at constant volume, no work will be done and any energy increase will be due only to energy transfer as heat. For such a
process, Cy does represent the energy increase per unit of temperature(per unit of mass)and consequently has been called the"specific heat at constant volume". However, it is more useful to think of Cy in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than a quantity related to energy transfer as heat in the special constant volume ocess the enthalpy as a function of T and P, that is view the imple compressible substance we can regard that define the state. Thus h=h(, P) Taking the differential dh d7 dp The derivative(ah/aT)p is called the specific heat at constant pressure, denoted by Cp The derivatives cy and cp constitute two of the most important thermodynamic derivative functions. Values of these properties have been experimentally determined as a function of the thermodynamic state for an enormous number of simple compressible substances 14)Ideal gases The equation of state for an ideal gas is PV= NRT where n is the number of moles of gas in the volume V. Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a substance as an ideal gas constitutes a model of the actual physical situation, and the limits of model validity must always be kept in mind One of the other important features of an ideal gas is that its internal energy depends only upon its temperature.( For now, this can be regarded as another aspect of the model of actual systems that the perfect gas represents, but it can be shown that this is a consequence of the form of the equation of state Since u depends only on T du= Cy(r)dT In the above equation we have indicated that Cy can depend on t. Like the internal energy, the enthalpy is also only dependent on T for an ideal gas. (If u is a function of T, then, using the perfect gas equation of state, u Pv is also Therefore dh= cp(T)dT. Further, dh= du d(Pv)=CvdT+r dt. Hence, for an ideal gas, Cu=C In general, for other substances, u and h depend on pressure as well as on temperature. In this respect, the ideal gas is a very special model 0-7
0-7 process, cv does represent the energy increase per unit of temperature (per unit of mass) and consequently has been called the "specific heat at constant volume". However, it is more useful to think of cv in terms of its definition as a certain partial derivative, which is a thermodynamic property, rather than a quantity related to energy transfer as heat in the special constant volume process. The enthalpy is also a function of state. For a simple compressible substance we can regard the enthalpy as a function of T and P, that is view the temperature and pressure as the two variables that define the state. Thus, h = h(T,P). Taking the differential, dh = ∂h ∂T P dT + ∂h ∂P T dP The derivative ∂h ∂T P is called the specific heat at constant pressure, denoted by cp. The derivatives cv and cp constitute two of the most important thermodynamic derivative functions. Values of these properties have been experimentally determined as a function of the thermodynamic state for an enormous number of simple compressible substances. 14) Ideal Gases The equation of state for an ideal gas is PV = NRT, where N is the number of moles of gas in the volume V. Ideal gas behavior furnishes an extremely good approximation to the behavior of real gases for a wide variety of aerospace applications. It should be remembered, however, that describing a substance as an ideal gas constitutes a model of the actual physical situation , and the limits of model validity must always be kept in mind. One of the other important features of an ideal gas is that its internal energy depends only upon its temperature. (For now, this can be regarded as another aspect of the model of actual systems that the perfect gas represents, but it can be shown that this is a consequence of the form of the equation of state.) Since u depends only on T, du = cv (T)dT In the above equation we have indicated that cv can depend on T. Like the internal energy, the enthalpy is also only dependent on T for an ideal gas. (If u is a function of T, then, using the perfect gas equation of state, u + Pv is also.) Therefore, dh = cP(T)dT. Further, dh = du + d(Pv) = cv dT + R dT. Hence, for an ideal gas, cv = cP - R. In general, for other substances, u and h depend on pressure as well as on temperature. In this respect, the ideal gas is a very special model
The specific heats do not vary greatly over wide ranges in temperature, as shown in VwB&s Figure 511. It is thus often useful to treat them as constant. If so (T2-7) h=cp(T2-T1) These equations are useful in calculating internal energy or enthalpy differences, but it should be remembered that they hold only for an ideal gas with constant specific heats In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume. The simple relations between changes in energy(or enthalpy)and temperature are a consequence of the behavior of an ideal gas, specificall the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances Adapted from"Engineering Thermodynamics", Reynolds, w. C and Perkins, H C McGraw-Hill Publishers 1. All ideal gases: (a) The specific heat at constant volume(Cy for a unit mass or Cv for one kmol)is a function of T (b) The specific heat at constant pressure(cp for a unit mass or Cp for one kmol) is a function of T only (c) A relation that connects the specific heats cp Cv, and the gas constant Cy= R where the units depend on the mass considered. For a unit mass of gas, e. g, a kilogram, and cy would be the specific heats for one kilogram of gas and r is as defined above. Forone kmol of gas, the expression takes the form: CP-Cv=R, where Cp and Cv have been used to denote the specific heats for one kmol of gas and r is the universal gas constant (d) The specific heat ratio, y, =Cp/cy(or Cp/Cv), is a function of Tonly and is greater than unity 2. Monatomic gases, such as He, Ne, Ar, and most metallic vapors (a)Cy(or Cv) is constant over a wide temperature range and is very nearly equal to (3/2R [ (3/2R, for one kmol (b) Cp(or Cp)is constant over a wide temperature range and is very nearly equal to(5/2)R [or (5/2)R, for one kmol (c) y is constant over a wide temperature range and is very nearly equal to 5/3 [y= 1.67 0-8
0-8 The specific heats do not vary greatly over wide ranges in temperature, as shown in VWB&S Figure 5.11. It is thus often useful to treat them as constant. If so u2 - u1 = cv (T2 - T1) h2 - h1 = cp (T2 - T1) These equations are useful in calculating internal energy or enthalpy differences, but it should be remembered that they hold only for an ideal gas with constant specific heats. In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume. The simple relations between changes in energy (or enthalpy) and temperature are a consequence of the behavior of an ideal gas, specifically the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances. Adapted from "Engineering Thermodynamics", Reynolds, W. C and Perkins, H. C, McGraw-Hill Publishers 15) Specific Heats of an Ideal Gas 1. All ideal gases: (a) The specific heat at constant volume (cv for a unit mass or CV for one kmol) is a function of T only. (b) The specific heat at constant pressure (cp for a unit mass or CP for one kmol) is a function of T only. (c) A relation that connects the specific heats cp, cv, and the gas constant is cp - cv = R where the units depend on the mass considered. For a unit mass of gas, e. g., a kilogram, cp and cv would be the specific heats for one kilogram of gas and R is as defined above. For one kmol of gas, the expression takes the form: CP - CV = R, where CP and CV have been used to denote the specific heats for one kmol of gas and R is the universal gas constant. (d) The specific heat ratio, γ, = cp/cv (or CP/CV), is a function of T only and is greater than unity. 2. Monatomic gases, such as He, Ne, Ar, and most metallic vapors: (a) cv (or CV) is constant over a wide temperature range and is very nearly equal to (3/2)R [or (3/2)R , for one kmol]. (b) cp (or CP) is constant over a wide temperature range and is very nearly equal to (5/2)R [or (5/2)R , for one kmol]. (c) γ is constant over a wide temperature range and is very nearly equal to 5/3 [γ = 1.67]
3. So-called permanent diatomic gases, namely H2, O2, N2, Air, NO, and Co (a) Cy(or Cv) is nearly constant at ordinary temperatures, being approximately (5/2)R [(5/2R for one kmol], and increases slowly at higher temperatures (b) cp(or Cp )is nearly constant at ordinary temperatures, being approximately (7/2)R [(7/2)R for one kmol], and increases slowly at higher temperatures (c) y is constant over a temperature range of roughly 150 to 600K and is very nearly equal to 775 Ly=1. 4]. It decreases with temperature above this 4. Polyatomic gases and gases that are chemically active, such as CO2, NH3, CH4, and Freons: The specific heats, cv and cp, and vary with the temperature, the variation being different for each gas. The general trend is that heavy molecular weight gases (i.e, more complex gas molecules than those listed in 2 or 3), have values of closer to unity than diatomic gases, which, as can be seen above, are closer to unity than monatomic gases. For example, values of y below 1. 2 are typical of Freons which have molecular weights of over one hundred Adapted from Zemansky, M. w. and Dittman, R.H., Heat and Thermodynamics", Sixth Edition, McGraw-Hill book company, 1981 16) Reversible adiabatic processes for an ideal From the first law, with 0=0, du CudT, and Work= Pd du t pdv=0 () Also, using the definition of enthalpy dh= du+ pdy vdP The underlined terms are zero for an adiabatic process. Re-writing(i) and (ii) ydT=-yPdv dt= vdP Combining the above two equations we obtain - y Pdv= vdP or -y dv/v= dP/P Equation(iii) can be integrated between states I and 2 to give yIn(v2/)=In(P2/Pn), or, equivalently, (P2)PYi )=1 For an ideal gas undergoing a reversible, adiabatic process, the relation between pressure and olume is thus P=constant xp I7 Examples of flow problems and the use of enthalpy a) Adiabatic, steady, throttling of a gas(flow through a valve or other restriction) Figure 0-l shows the configuration of interest. We wish to know the relation between properties upstream of the valve, denoted by l"and those downstream, denoted by 2 0-9
0-9 3. So-called permanent diatomic gases, namely H2, O2, N2, Air, NO, and CO: (a) cv (or CV ) is nearly constant at ordinary temperatures, being approximately (5/2)R [(5/2)R , for one kmol], and increases slowly at higher temperatures. (b) cp (or CP ) is nearly constant at ordinary temperatures, being approximately (7/2)R [(7/2)R , for one kmol], and increases slowly at higher temperatures. (c) γ is constant over a temperature range of roughly 150 to 600K and is very nearly equal to 7/5 [γ = 1.4]. It decreases with temperature above this. 4. Polyatomic gases and gases that are chemically active, such as CO2, NH3, CH4, and Freons: The specific heats, cv and cp, and γ vary with the temperature, the variation being different for each gas. The general trend is that heavy molecular weight gases (i.e., more complex gas molecules than those listed in 2 or 3), have values of γ closer to unity than diatomic gases, which, as can be seen above, are closer to unity than monatomic gases. For example, values of γ below 1.2 are typical of Freons which have molecular weights of over one hundred. Adapted from Zemansky, M. W. and Dittman, R. H., "Heat and Thermodynamics", Sixth Edition, McGraw-Hill book company, 1981 16) Reversible adiabatic processes for an ideal gas From the first law, with Q = 0, du = cvdT, and Work = Pdv du + Pdv = 0 (i) Also, using the definition of enthalpy dh = du + Pdv + vdP. (ii) The underlined terms are zero for an adiabatic process. Re-writing (i) and (ii), γ cvdT = - γ pdT = vdP. Pdv c Combining the above two equations we obtain -γ Pdv = vdP or -γ dv/v = dP/P (iii) Equation (iii) can be integrated between states 1 and 2 to give γln(v2/v1) = ln(P2/P1), or, equivalently, ( )( ) / 1 2 2 1 1 = γ γ P v Pv For an ideal gas undergoing a reversible, adiabatic process, the relation between pressure and volume is thus: Pvγ = constant, or P = constant ×ργ . 17) Examples of flow problems and the use of enthalpy a) Adiabatic, steady, throttling of a gas (flow through a valve or other restriction) Figure 0-1 shows the configuration of interest. We wish to know the relation between properties upstream of the valve, denoted by “1” and those downstream, denoted by “2
