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Boundary work for an isothermal process

Boundary work for a constant-pressure process

Boundary work for a polytropic process

Energy balance for closed systems

Energy balance for a constant-pressure expansion or compression process

Specific heats

Constant-pressure specific heat, cp

Constant-volume specific heat, cv

Internal energy, enthalpy and specific heats of ideal gases

Energy balance for a constant-pressure expansion or compression process

Internal energy, enthalpy and specific heats of incompressible substances (Solids and liquids)

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- 1. ENERGY ANALYSIS OF CLOSED SYSTEMS Lecture 7 Keith Vaugh BEng (AERO) MEngReference text: Chapter 5 - Fundamentals of Thermal-Fluid Sciences, 3rd Edition Yunus A. Cengel, Robert H. Turner, John M. Cimbala McGraw-Hill, 2008 KEITH VAUGH
- 2. OBJECTIVES } KEITH VAUGH
- 3. OBJECTIVESExamine the moving boundary work or P dV workcommonly encountered in reciprocating devicessuch as automotive engines and compressors. } KEITH VAUGH
- 4. OBJECTIVESExamine the moving boundary work or P dV workcommonly encountered in reciprocating devicessuch as automotive engines and compressors. }Identify the ﬁrst law of thermodynamics as simply astatement of the conservation of energy principle forclosed (ﬁxed mass) systems. KEITH VAUGH
- 5. OBJECTIVESExamine the moving boundary work or P dV workcommonly encountered in reciprocating devicessuch as automotive engines and compressors. }Identify the ﬁrst law of thermodynamics as simply astatement of the conservation of energy principle forclosed (ﬁxed mass) systems.Develop the general energy balance applied toclosed systems. KEITH VAUGH
- 6. OBJECTIVESExamine the moving boundary work or P dV workcommonly encountered in reciprocating devicessuch as automotive engines and compressors. }Identify the ﬁrst law of thermodynamics as simply astatement of the conservation of energy principle forclosed (ﬁxed mass) systems.Develop the general energy balance applied toclosed systems.Deﬁne the speciﬁc heat at constant volume and thespeciﬁc heat at constant pressure. KEITH VAUGH
- 7. }KEITH VAUGH
- 8. Relate the speciﬁc heats to the calculation of thechanges in internal energy and enthalpy of idealgases. } KEITH VAUGH
- 9. Relate the speciﬁc heats to the calculation of thechanges in internal energy and enthalpy of idealgases. }Describe incompressible substances and determinethe changes in their internal energy and enthalpy. KEITH VAUGH
- 10. Relate the speciﬁc heats to the calculation of thechanges in internal energy and enthalpy of idealgases. }Describe incompressible substances and determinethe changes in their internal energy and enthalpy.Solve energy balance problems for closed (ﬁxedmass) systems that involve heat and workinteractions for general pure substances, ideal gases,and incompressible substances. KEITH VAUGH
- 11. MOVING BOUNDARY WORK Moving boundary work (PdV work) The work associated with a moving boundary is called boundary work } The expansion and compression work in a piston cylinder device is given byA gas does a differential amountof work δWb as it forces the pistonto move by a differential amountds. KEITH VAUGH
- 12. MOVING BOUNDARY WORK Moving boundary work (PdV work) The work associated with a moving boundary is called boundary work } The expansion and compression work in a piston cylinder device is given by δ Wb = Fds = PAds = PdVA gas does a differential amountof work δWb as it forces the pistonto move by a differential amountds. KEITH VAUGH
- 13. MOVING BOUNDARY WORK Moving boundary work (PdV work) The work associated with a moving boundary is called boundary work } The expansion and compression work in a piston cylinder device is given by δ Wb = Fds = PAds = PdV 2 Wb = ∫ P dV 1 ( kJ )A gas does a differential amountof work δWb as it forces the pistonto move by a differential amountds. KEITH VAUGH
- 14. MOVING BOUNDARY WORK Moving boundary work (PdV work) The work associated with a moving boundary is called boundary work } The expansion and compression work in a piston cylinder device is given by δ Wb = Fds = PAds = PdV 2 Wb = ∫ P dV 1 ( kJ )A gas does a differential amount Quasi-equilibrium processof work δWb as it forces the pistonto move by a differential amount A process during which the system remainsds. nearly in equilibrium at all times. KEITH VAUGH
- 15. KEITH VAUGH
- 16. The area under the process curve ona P-V diagram represents theboundary work. 2 2Area = A = ∫ dA = ∫ P dV 1 1 KEITH VAUGH
- 17. The boundary work done during a process depends on the path followed as well as the end states.The area under the process curve ona P-V diagram represents theboundary work. 2 2Area = A = ∫ dA = ∫ P dV 1 1 KEITH VAUGH
- 18. The boundary work done during a process depends on the path followed as well as the end states.The area under the process curve on The net work donea P-V diagram represents the during a cycle is theboundary work. difference between the 2 2 work done by theArea = A = ∫ dA = ∫ P dV system and the work 1 1 done on the system. KEITH VAUGH
- 19. POLYTROPIC, ISOTHERMAL & ISOBARIC PROCESSESP = CV − n Polytropic process: C, n (polytropic exponent) constants } KEITH VAUGH
- 20. POLYTROPIC, ISOTHERMAL & ISOBARIC PROCESSESP = CV − n Polytropic process: C, n (polytropic exponent) constants V2− n+1 − V1− n+1 P2V2 − P1V1 } 2 2 PolytropicWb = ∫ P dV = ∫ CV − n dV = C = process 1 1 −n + 1 1− n KEITH VAUGH
- 21. POLYTROPIC, ISOTHERMAL & ISOBARIC PROCESSESP = CV − n Polytropic process: C, n (polytropic exponent) constants V2− n+1 − V1− n+1 P2V2 − P1V1 } 2 2 PolytropicWb = ∫ P dV = ∫ CV − n dV = C = process 1 1 −n + 1 1− n mR (T2 − T1 ) Polytropic andWb = for ideal gas 1− n KEITH VAUGH
- 22. POLYTROPIC, ISOTHERMAL & ISOBARIC PROCESSESP = CV − n Polytropic process: C, n (polytropic exponent) constants V2− n+1 − V1− n+1 P2V2 − P1V1 } 2 2 PolytropicWb = ∫ P dV = ∫ CV − n dV = C = process 1 1 −n + 1 1− n mR (T2 − T1 ) Polytropic andWb = for ideal gas 1− n 2 2 −1 ⎛ V2 ⎞ When n=1Wb = ∫ P dV = ∫ CV dV = PV ln ⎜ ⎟ 1 1 ⎝ V1 ⎠ (Isothermal process) KEITH VAUGH
- 23. POLYTROPIC, ISOTHERMAL & ISOBARIC PROCESSESP = CV − n Polytropic process: C, n (polytropic exponent) constants V2− n+1 − V1− n+1 P2V2 − P1V1 } 2 2 PolytropicWb = ∫ P dV = ∫ CV − n dV = C = process 1 1 −n + 1 1− n mR (T2 − T1 ) Polytropic andWb = for ideal gas 1− n 2 2 −1 ⎛ V2 ⎞ When n=1Wb = ∫ P dV = ∫ CV dV = PV ln ⎜ ⎟ 1 1 ⎝ V1 ⎠ (Isothermal process) 2 2 Constant pressure processWb = ∫ P dV = P0 ∫ dV = P0 (V2 − V1 ) 1 1 (Isobaric process) KEITH VAUGH
- 24. ENERGY BALANCE FOR CLOSED SYSTEMS } KEITH VAUGH
- 25. ENERGY BALANCE FOR CLOSED SYSTEMS Ein − Eout 14 2 4 3 Net energy transferby heat, work and mass = ΔEsystem 123 Change in internal, kinetic, potential, etc... energies ( kJ ) Energy balance for any system undergoing any process } KEITH VAUGH
- 26. ENERGY BALANCE FOR CLOSED SYSTEMS Ein − Eout 14 2 4 3 Net energy transfer by heat, work and mass = ΔEsystem 123 Change in internal, kinetic, potential, etc... energies dEsystem ( kJ ) Energy balance for any system undergoing any process } & & Ein − Eout = ( kW ) Energy balance 14 2 4 3 dt in the rate formRate of net energy transfer 123 by heat, work and mass Rate of change in internal, kinetic, potential, etc... energies KEITH VAUGH
- 27. ENERGY BALANCE FOR CLOSED SYSTEMS Ein − Eout 14 2 4 3 Net energy transfer by heat, work and mass = ΔEsystem 123 Change in internal, kinetic, potential, etc... energies dEsystem ( kJ ) Energy balance for any system undergoing any process } & & Ein − Eout = ( kW ) Energy balance 14 2 4 3 dt in the rate formRate of net energy transfer 123 by heat, work and mass Rate of change in internal, kinetic, potential, etc... energies The total quantities are & & ⎛ dE ⎞ Q = QΔt, W = W Δt, and ΔE = ⎜ ⎟ Δt ⎝ dt ⎠ ( kJ ) related to the quantities per unit time KEITH VAUGH
- 28. ENERGY BALANCE FOR CLOSED SYSTEMS Ein − Eout 14 2 4 3 Net energy transfer by heat, work and mass = ΔEsystem 123 Change in internal, kinetic, potential, etc... energies dEsystem ( kJ ) Energy balance for any system undergoing any process } & & Ein − Eout = ( kW ) Energy balance 14 2 4 3 dt in the rate formRate of net energy transfer 123 by heat, work and mass Rate of change in internal, kinetic, potential, etc... energies The total quantities are & & ⎛ dE ⎞ Q = QΔt, W = W Δt, and ΔE = ⎜ ⎟ Δt ⎝ dt ⎠ ( kJ ) related to the quantities per unit time ein − eout = Δesystem ( kg) kJ Energy balance per unit mass basis KEITH VAUGH
- 29. ENERGY BALANCE FOR CLOSED SYSTEMS Ein − Eout 14 2 4 3 Net energy transfer by heat, work and mass = ΔEsystem 123 Change in internal, kinetic, potential, etc... energies dEsystem ( kJ ) Energy balance for any system undergoing any process } & & Ein − Eout = ( kW ) Energy balance 14 2 4 3 dt in the rate formRate of net energy transfer 123 by heat, work and mass Rate of change in internal, kinetic, potential, etc... energies The total quantities are & & ⎛ dE ⎞ Q = QΔt, W = W Δt, and ΔE = ⎜ ⎟ Δt ⎝ dt ⎠ ( kJ ) related to the quantities per unit time ein − eout = Δesystem ( kg) kJ Energy balance per unit mass basis δ Ein − δ Eout = dEsystem or δ ein − δ eout = desystem Energy balance in differential form KEITH VAUGH
- 30. & &Wnet , out = Qnet , in or Wnet , out = Qnet , in Energy balance for a cycle KEITH VAUGH
- 31. & & Wnet , out = Qnet , in or Wnet , out = Qnet , in Energy balance for a cycleEnergy balance when sign convention is used (i.e. heat input andwork output are positive; heat output and work input are negative Qnet ,in − Wnet ,out = ΔEsystem For a cycle ΔE = 0 or Q − W = ΔE thus Q = W KEITH VAUGH
- 32. & & Wnet , out = Qnet , in or Wnet , out = Qnet , in Energy balance for a cycleEnergy balance when sign convention is used (i.e. heat input andwork output are positive; heat output and work input are negative Qnet ,in − Wnet ,out = ΔEsystem For a cycle ΔE = 0 or Q − W = ΔE thus Q = W Q = Qnet ,in = Qin − Qout W = Wnet ,out = Wout − Win KEITH VAUGH
- 33. & & Wnet , out = Qnet , in or Wnet , out = Qnet , in Energy balance for a cycleEnergy balance when sign convention is used (i.e. heat input andwork output are positive; heat output and work input are negative Qnet ,in − Wnet ,out = ΔEsystem For a cycle ΔE = 0 or Q − W = ΔE thus Q = W Q = Qnet ,in = Qin − Qout W = Wnet ,out = Wout − Win General Q − W = ΔE Stationary systems Q − W = ΔU Per unit mass q − w = Δe Various forms of the ﬁrst-law relation for closed systems Differential form δ q = δ w = de when sign convention is used. KEITH VAUGH
- 34. KEITH VAUGH
- 35. The ﬁrst law cannot be proven mathematically, but noprocess in nature is known to have violated the ﬁrstlaw, therefore this can be taken as sufﬁcient proof KEITH VAUGH
- 36. Energy balance for a constant pressureexpansion or compression processGeneral analysis for a closed systemundergoing a quasi-equilibrium constant-pressure process Q is to the system andW is from the systemExample 5-5 page 168 Ein − Eout = ΔEsystem 14 2 4 3 123 Net energy transfer Change in internal, kinetic, by heat, work and mass potential, etc... energies KEITH VAUGH
- 37. Energy balance for a constant pressureexpansion or compression process Z 0 Z 0General analysis for a closed system Q − W = ΔU + Δ KE + Δ PEundergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1pressure process Q is to the system and Q − Wother − P0 (V2 − V1 ) = U 2 − U1W is from the system Q − Wother = (U 2 + P2V2 ) − (U1 − P1V1 )Example 5-5 page 168 H = U + PV Ein − Eout = ΔEsystem 14 2 4 3 123 Q − Wother = H 2 − H 1 Net energy transfer Change in internal, kinetic, by heat, work and mass potential, etc... energies KEITH VAUGH
- 38. Energy balance for a constant pressureexpansion or compression process Z 0 Z 0General analysis for a closed system Q − W = ΔU + Δ KE + Δ PEundergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1pressure process Q is to the system and Q − Wother − P0 (V2 − V1 ) = U 2 − U1W is from the system Q − Wother = (U 2 + P2V2 ) − (U1 − P1V1 )Example 5-5 page 168 H = U + PV Ein − Eout = ΔEsystem 14 2 4 3 123 Q − Wother = H 2 − H 1 Net energy transfer Change in internal, kinetic, by heat, work and mass potential, etc... energies For a constant pressure expansion or compression process: ΔU + Wb = ΔH KEITH VAUGH
- 39. Energy balance for a constant pressureexpansion or compression process Z 0 Z 0General analysis for a closed system Q − W = ΔU + Δ KE + Δ PEundergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1pressure process Q is to the system and Q − Wother − P0 (V2 − V1 ) = U 2 − U1W is from the system Q − Wother = (U 2 + P2V2 ) − (U1 − P1V1 )Example 5-5 page 168 H = U + PV Ein − Eout = ΔEsystem 14 2 4 3 123 Q − Wother = H 2 − H 1 Net energy transfer Change in internal, kinetic, by heat, work and mass potential, etc... energies For a constant pressure expansion or compression process: ΔU + Wb = ΔH We,in − Qout − Wb = ΔU We,in − Qout = ΔH = m ( h2 − h1 ) KEITH VAUGH
- 40. SPECIFIC HEATThe energy required to raise the temperature of a unit mass of asubstance by one degree } Speciﬁc heat is the energy required to raise the temperature of a unit mass of a substance by one degree in a speciﬁed way. KEITH VAUGH
- 41. SPECIFIC HEATThe energy required to raise the temperature of a unit mass of asubstance by one degree } Speciﬁc heat is the energy required to raise the temperature of a unit mass of a substance by one degree in a speciﬁed way.In thermodynamics we are interested in two kinds of speciﬁc heats ✓ Speciﬁc heat at constant volume, cv ✓ Speciﬁc heat at constant pressure, cp KEITH VAUGH
- 42. Speciﬁc heat at constant volume, cvThe energy required to raise the temperature of the unit mass of a substance byone degree as the volume is maintained constant.Speciﬁc heat at constant pressure, cpThe energy required to raise the temperature ofthe unit mass of a substance by one degree as thepressure is maintained constant. Constant-volume and constant-pressure speciﬁc heats cv and cp (values are for helium gas). KEITH VAUGH
- 43. Formal deﬁnitions of cv and cp ⎛ ∂u ⎞ cv = ⎜ ⎝ ∂T ⎟ v ⎠ ⎛ ∂h ⎞ c p = ⎜ ⎝ ∂T ⎟ p ⎠ KEITH VAUGH
- 44. Formal deﬁnitions of cv and cp } ⎛ ∂u ⎞ the change in internal energy cv = ⎜ with temperature at constant ⎝ ∂T ⎟ v ⎠ volume } ⎛ ∂h ⎞ the change in internal enthalpy c p = ⎜ ⎝ ∂T ⎟ p ⎠ with temperature at constant pressure KEITH VAUGH
- 45. Formal deﬁnitions of cv and cp } ⎛ ∂u ⎞ the change in internal energy cv = ⎜ with temperature at constant ⎝ ∂T ⎟ v ⎠ volume } ⎛ ∂h ⎞ the change in internal enthalpy c p = ⎜ ⎝ ∂T ⎟ p ⎠ with temperature at constant pressureThe equations in the ﬁgure are valid for anysubstance undergoing any process.cv and cp are properties.cv is related to the changes in internal energyand cp to the changes in enthalpy.A common unit for speciﬁc heats is kJ/kg · °C or The speciﬁc heat of a substancekJ/kg · K changes with temperature KEITH VAUGH
- 46. INTERNAL ENERGY, ENTHALPY &SPECIFIC HEATS OD IDEAL GASES An Ideal gas can be deﬁned as a gas whose temperature, pressure and speciﬁc volume are related by } PV = RT Joule demonstrated mathematically and experimentally in 1843, that for an Ideal gas the internal energy is a function of the temperature only; u = u (T ) KEITH VAUGH
- 47. Using the deﬁnition of enthalpy and the equation of state of an ideal gas h = u + PV ⎫ ⎬ h = u + RT PV = RT ⎭For ideal gases, u, h, u = u (T ) , h = h (T ) , du = cv (T ) dT ,cv, and cp vary withtemperature only. dh = c p (T ) dT KEITH VAUGH
- 48. Using the deﬁnition of enthalpy and the equation of state of an ideal gas h = u + PV ⎫ ⎬ h = u + RT PV = RT ⎭For ideal gases, u, h, u = u (T ) , h = h (T ) , du = cv (T ) dT ,cv, and cp vary withtemperature only. dh = c p (T ) dT Internal energy change of an ideal gas 2 Δu = u2 − u1 = ∫ cv (T ) dT 1 KEITH VAUGH
- 49. Using the deﬁnition of enthalpy and the equation of state of an ideal gas h = u + PV ⎫ ⎬ h = u + RT PV = RT ⎭For ideal gases, u, h, u = u (T ) , h = h (T ) , du = cv (T ) dT ,cv, and cp vary withtemperature only. dh = c p (T ) dT Internal energy change of an ideal gas 2 Δu = u2 − u1 = ∫ cv (T ) dT 1 Internal energy change of an ideal gas 2 Δh = h2 − h1 = ∫ c p (T ) dT 1 KEITH VAUGH
- 50. At low pressures, all real gasesapproach ideal-gas behaviour, andtherefore their speciﬁc heats dependon temperature only.The speciﬁc heats of real gases at lowpressures are called ideal-gas speciﬁcheats, or zero-pressure speciﬁc heats,and are often denoted cp0 and cv0. Ideal-gas constant-pressure speciﬁc heats for some gases (see Table A–2c for cp equations). KEITH VAUGH
- 51. At low pressures, all real gases u and h data for a number ofapproach ideal-gas behaviour, and gases have been tabulated.therefore their speciﬁc heats dependon temperature only. These tables are obtained by choosing an arbitrary referenceThe speciﬁc heats of real gases at low point and performing thepressures are called ideal-gas speciﬁc integrations by treating state 1 asheats, or zero-pressure speciﬁc heats, the reference state.and are often denoted cp0 and cv0. Ideal-gas constant-pressure speciﬁc heats for some gases (see In the preparation of ideal-gas Table A–2c for cp tables, 0 K is chosen as the equations). reference temperature KEITH VAUGH
- 52. Internal energy and enthalpy changewhen speciﬁc heat is taken constantat an average value u2 − u1 = cv,avg (T2 − T1 ) kJ kg h2 − h1 = c p,avg (T2 − T1 ) kJ kgThe relation Δ u = cv ΔT is valid for anykind of process, constant-volume or not. KEITH VAUGH
- 53. Internal energy and enthalpy changewhen speciﬁc heat is taken constantat an average value u2 − u1 = cv,avg (T2 − T1 ) kJ kg h2 − h1 = c p,avg (T2 − T1 ) kJ kg For small temperature intervals, the speciﬁc heats may be assumed to vary linearly with temperature.The relation Δ u = cv ΔT is valid for anykind of process, constant-volume or not. KEITH VAUGH
- 54. Three ways of calculating Δu and Δh KEITH VAUGH
- 55. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available. KEITH VAUGH
- 56. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available. KEITH VAUGH
- 57. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available.By using the cv or cp relations (Table A-2c) as afunction of temperature and performing theintegrations. This is very inconvenient for handcalculations but quite desirable for computerisedcalculations. The results obtained are veryaccurate. KEITH VAUGH
- 58. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available.By using the cv or cp relations (Table A-2c) as afunction of temperature and performing theintegrations. This is very inconvenient for handcalculations but quite desirable for computerisedcalculations. The results obtained are veryaccurate. KEITH VAUGH
- 59. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available.By using the cv or cp relations (Table A-2c) as afunction of temperature and performing theintegrations. This is very inconvenient for handcalculations but quite desirable for computerisedcalculations. The results obtained are veryaccurate.By using average speciﬁc heats. This is verysimple and certainly very convenient whenproperty tables are not available. The resultsobtained are reasonably accurate if thetemperature interval is not very large. KEITH VAUGH
- 60. Three ways of calculating Δu and ΔhBy using the tabulated u and h data. This is theeasiest and most accurate way when tables arereadily available.By using the cv or cp relations (Table A-2c) as afunction of temperature and performing theintegrations. This is very inconvenient for hand Δu = u2 − u1 ( table )calculations but quite desirable for computerised 2calculations. The results obtained are very Δu = ∫ c (T ) dT 1 vaccurate. Δu ≅ cv,avg ΔTBy using average speciﬁc heats. This is very Three ways of calculating Δusimple and certainly very convenient whenproperty tables are not available. The resultsobtained are reasonably accurate if thetemperature interval is not very large. KEITH VAUGH
- 61. Speciﬁc Heat Relations of Ideal Gases h = u + RT dh = du + RdT dh = c p dT and du = cv dT KEITH VAUGH
- 62. Speciﬁc Heat Relations of Ideal Gases The relationship between cp, cv and R } h = u + RT dh = du + RdT c p = cv + R ( kJ kg ⋅ K ) dh = c p dT and du = cv dT KEITH VAUGH
- 63. Speciﬁc Heat Relations of Ideal Gases The relationship between cp, cv and R } h = u + RT dh = du + RdT c p = cv + R ( kJ kg ⋅ K ) dh = c p dT and du = cv dT On a molar basis c p = cv + Ru ( kJ kmol ⋅ K ) KEITH VAUGH
- 64. Speciﬁc Heat Relations of Ideal Gases The relationship between cp, cv and R } h = u + RT dh = du + RdT c p = cv + R ( kJ kg ⋅ K ) dh = c p dT and du = cv dT On a molar basis c p = cv + Ru ( kJ kmol ⋅ K ) cp Speciﬁc k= cv heat ratio KEITH VAUGH
- 65. Speciﬁc Heat Relations of Ideal Gases The relationship between cp, cv and R } h = u + RT dh = du + RdT c p = cv + R ( kJ kg ⋅ K ) dh = c p dT and du = cv dT On a molar basis c p = cv + Ru ( kJ kmol ⋅ K ) cp Speciﬁc k= cv heat ratioThe speciﬁc ratio varies with temperature, but this variation is very mild.For monatomic gases (helium, argon, etc.), its value is essentially constant at 1.667.Many diatomic gases, including air, have a speciﬁc heat ratio of about 1.4 at roomtemperature. KEITH VAUGH
- 66. INTERNAL ENERGY, ENTHALPY &SPECIFIC HEATS OF SOLIDS & LIQUIDS Incompressible substance A substance whose speciﬁc volume (or density) is constant. Solids and liquids are incompressible substances. } The speciﬁc volumes of The cv and cp values of incompressible incompressible substances substances are identical and are remain constant during a denoted by c. process. KEITH VAUGH
- 67. Internal energy changes du = cv dT = c (T ) dT 2 Δu = u2 − u1 = ∫ c (T ) dT 1 ( kg) kJ Δu ≅ cv,avg (T2 − T1 ) ( kg) kJ KEITH VAUGH
- 68. Enthalpy changes h = u + PV KEITH VAUGH
- 69. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP KEITH VAUGH
- 70. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP Δh = Δu + V ΔP ≅ cavg ΔT + V ΔP ( kg) kJ KEITH VAUGH
- 71. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP Δh = Δu + V ΔP ≅ cavg ΔT + V ΔP ( kg) kJFor solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔTFor liquids, two specical cases are commonly encountered1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT2. Constant temperature processes as in pumps ( ΔT ≅ 0 ); Δh = V ΔP KEITH VAUGH
- 72. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP Δh = Δu + V ΔP ≅ cavg ΔT + V ΔP ( kg) kJFor solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔTFor liquids, two specical cases are commonly encountered1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT2. Constant temperature processes as in pumps ( ΔT ≅ 0 ); Δh = V ΔP The enthalpy of a h@P,T ≅ h f @T + V f @T ( P − Psat @T ) compressed liquid KEITH VAUGH
- 73. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP Δh = Δu + V ΔP ≅ cavg ΔT + V ΔP ( kg) kJFor solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔTFor liquids, two specical cases are commonly encountered1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT2. Constant temperature processes as in pumps ( ΔT ≅ 0 ); Δh = V ΔP The enthalpy of a h@P,T ≅ h f @T + V f @T ( P − Psat @T ) compressed liquid A more accurate relation than KEITH VAUGH
- 74. Enthalpy changes h = u + PV Z 0 dh = du + VdP + P dV = du + VdP Δh = Δu + V ΔP ≅ cavg ΔT + V ΔP ( kg) kJFor solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔTFor liquids, two specical cases are commonly encountered1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT2. Constant temperature processes as in pumps ( ΔT ≅ 0 ); Δh = V ΔP The enthalpy of a h@P,T ≅ h f @T + V f @T ( P − Psat @T ) compressed liquid A more accurate relation than h@P,T ≅ h f @T KEITH VAUGH
- 75. Moving boundary work Wb for an isothermal process Wb for a constant-pressure process Wb for a polytropic processEnergy balance for closed systems Energy balance for a constant-pressure expansion or compression processSpeciﬁc heats Constant-pressure speciﬁc heat, cp Constant-volume speciﬁc heat, cvInternal energy, enthalpy and speciﬁc heats of ideal gases Energy balance for a constant-pressure expansion or compression processInternal energy, enthalpy and speciﬁc heats of incompressible substances(solids and liquids) KEITH VAUGH

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