Moving boundary work 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)

1. ENERGY ANALYSIS OF
CLOSED SYSTEMS
Lecture 7
Keith Vaugh BEng (AERO) MEng
Reference 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. OBJECTIVES
Examine the moving boundary work or P dV work
commonly encountered in reciprocating devices
such as automotive engines and compressors.
}
KEITH VAUGH

4. OBJECTIVES
Examine the moving boundary work or P dV work
commonly encountered in reciprocating devices
such as automotive engines and compressors.
}
Identify the first law of thermodynamics as simply a
statement of the conservation of energy principle for
closed (fixed mass) systems.
KEITH VAUGH

5. OBJECTIVES
Examine the moving boundary work or P dV work
commonly encountered in reciprocating devices
such as automotive engines and compressors.
}
Identify the first law of thermodynamics as simply a
statement of the conservation of energy principle for
closed (fixed mass) systems.
Develop the general energy balance applied to
closed systems.
KEITH VAUGH

6. OBJECTIVES
Examine the moving boundary work or P dV work
commonly encountered in reciprocating devices
such as automotive engines and compressors.
}
Identify the first law of thermodynamics as simply a
statement of the conservation of energy principle for
closed (fixed mass) systems.
Develop the general energy balance applied to
closed systems.
Define the specific heat at constant volume and the
specific heat at constant pressure.
KEITH VAUGH

7. }
KEITH VAUGH

8. Relate the specific heats to the calculation of the
changes in internal energy and enthalpy of ideal
gases.
}
KEITH VAUGH

9. Relate the specific heats to the calculation of the
changes in internal energy and enthalpy of ideal
gases.
}
Describe incompressible substances and determine
the changes in their internal energy and enthalpy.
KEITH VAUGH

10. Relate the specific heats to the calculation of the
changes in internal energy and enthalpy of ideal
gases.
}
Describe incompressible substances and determine
the changes in their internal energy and enthalpy.
Solve energy balance problems for closed (fixed
mass) systems that involve heat and work
interactions 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 by
A gas does a differential amount
of work δWb as it forces the piston
to move by a differential amount
ds.
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 = PdV
A gas does a differential amount
of work δWb as it forces the piston
to move by a differential amount
ds.
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 amount
of work δWb as it forces the piston
to move by a differential amount
ds.
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 process
of work δWb as it forces the piston
to move by a differential amount
A process during which the system remains
ds. nearly in equilibrium at all times.
KEITH VAUGH

15. KEITH VAUGH

16. The area under the process curve on
a P-V diagram represents the
boundary work.
2 2
Area = 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 on
a P-V diagram represents the
boundary work.
2 2
Area = 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 done
a P-V diagram represents the
during a cycle is the
boundary work.
difference between the
2 2 work done by the
Area = A = ∫ dA = ∫ P dV system and the work
1 1
done on the system.
KEITH VAUGH

19. POLYTROPIC, ISOTHERMAL
& ISOBARIC PROCESSES
P = CV − n Polytropic process: C, n (polytropic exponent) constants }
KEITH VAUGH

20. POLYTROPIC, ISOTHERMAL
& ISOBARIC PROCESSES
P = CV − n Polytropic process: C, n (polytropic exponent) constants
V2− n+1 − V1− n+1 P2V2 − P1V1
}
2 2 Polytropic
Wb = ∫ P dV = ∫ CV − n dV = C = process
1 1 −n + 1 1− n
KEITH VAUGH

21. POLYTROPIC, ISOTHERMAL
& ISOBARIC PROCESSES
P = CV − n Polytropic process: C, n (polytropic exponent) constants
V2− n+1 − V1− n+1 P2V2 − P1V1
}
2 2 Polytropic
Wb = ∫ P dV = ∫ CV − n dV = C = process
1 1 −n + 1 1− n
mR (T2 − T1 ) Polytropic and
Wb = for ideal gas
1− n
KEITH VAUGH

22. POLYTROPIC, ISOTHERMAL
& ISOBARIC PROCESSES
P = CV − n Polytropic process: C, n (polytropic exponent) constants
V2− n+1 − V1− n+1 P2V2 − P1V1
}
2 2 Polytropic
Wb = ∫ P dV = ∫ CV − n dV = C = process
1 1 −n + 1 1− n
mR (T2 − T1 ) Polytropic and
Wb = for ideal gas
1− n
2 2
−1 ⎛ V2 ⎞ When n=1
Wb = ∫ P dV = ∫ CV dV = PV ln ⎜ ⎟
1 1 ⎝ V1 ⎠ (Isothermal process)
KEITH VAUGH

23. POLYTROPIC, ISOTHERMAL
& ISOBARIC PROCESSES
P = CV − n Polytropic process: C, n (polytropic exponent) constants
V2− n+1 − V1− n+1 P2V2 − P1V1
}
2 2 Polytropic
Wb = ∫ P dV = ∫ CV − n dV = C = process
1 1 −n + 1 1− n
mR (T2 − T1 ) Polytropic and
Wb = for ideal gas
1− n
2 2
−1 ⎛ V2 ⎞ When n=1
Wb = ∫ P dV = ∫ CV dV = PV ln ⎜ ⎟
1 1 ⎝ V1 ⎠ (Isothermal process)
2 2 Constant pressure process
Wb = ∫ 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 transfer
by 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 form
Rate 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 form
Rate 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 form
Rate 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 form
Rate 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 cycle
Energy balance when sign convention is used (i.e. heat input and
work 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 cycle
Energy balance when sign convention is used (i.e. heat input and
work 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 cycle
Energy balance when sign convention is used (i.e. heat input and
work 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 no
process in nature is known to have violated the first
law, therefore this can be taken as sufficient proof
KEITH VAUGH

36. Energy balance for a constant pressure
expansion or compression process
General analysis for a closed system
undergoing a quasi-equilibrium constant-
pressure process Q is to the system and
W is from the system
Example 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 pressure
expansion or compression process
Z 0 Z 0
General analysis for a closed system Q − W = ΔU + Δ KE + Δ PE
undergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1
pressure process Q is to the system and
Q − Wother − P0 (V2 − V1 ) = U 2 − U1
W 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 pressure
expansion or compression process
Z 0 Z 0
General analysis for a closed system Q − W = ΔU + Δ KE + Δ PE
undergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1
pressure process Q is to the system and
Q − Wother − P0 (V2 − V1 ) = U 2 − U1
W 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 pressure
expansion or compression process
Z 0 Z 0
General analysis for a closed system Q − W = ΔU + Δ KE + Δ PE
undergoing a quasi-equilibrium constant- Q − Wother − Wb = U 2 − U1
pressure process Q is to the system and
Q − Wother − P0 (V2 − V1 ) = U 2 − U1
W 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 HEAT
The energy required to raise the temperature of a unit mass of a
substance 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 HEAT
The energy required to raise the temperature of a unit mass of a
substance 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, cv
The energy required to raise the temperature of the unit mass of a substance by
one degree as the volume is maintained constant.
Specific heat at constant pressure, cp
The energy required to raise the temperature of
the unit mass of a substance by one degree as the
pressure 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
pressure
The equations in the figure are valid for any
substance undergoing any process.
cv and cp are properties.
cv is related to the changes in internal energy
and cp to the changes in enthalpy.
A common unit for specific heats is kJ/kg · °C or The specific heat of a substance
kJ/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 with
temperature 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 with
temperature 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 with
temperature 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 gases
approach ideal-gas behaviour, and
therefore their specific heats depend
on temperature only.
The specific heats of real gases at low
pressures are called ideal-gas specific
heats, 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 of
approach ideal-gas behaviour, and gases have been tabulated.
therefore their specific heats depend
on temperature only. These tables are obtained by
choosing an arbitrary reference
The specific heats of real gases at low point and performing the
pressures are called ideal-gas specific integrations by treating state 1 as
heats, 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 change
when specific heat is taken constant
at an average value
u2 − u1 = cv,avg (T2 − T1 ) kJ
kg
h2 − h1 = c p,avg (T2 − T1 ) kJ
kg
The relation Δ u = cv ΔT is valid for any
kind of process, constant-volume or not.
KEITH VAUGH

53. Internal energy and enthalpy change
when specific heat is taken constant
at 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 any
kind 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 Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
KEITH VAUGH

56. Three ways of calculating Δu and Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
KEITH VAUGH

57. Three ways of calculating Δu and Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
By using the cv or cp relations (Table A-2c) as a
function of temperature and performing the
integrations. This is very inconvenient for hand
calculations but quite desirable for computerised
calculations. The results obtained are very
accurate.
KEITH VAUGH

58. Three ways of calculating Δu and Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
By using the cv or cp relations (Table A-2c) as a
function of temperature and performing the
integrations. This is very inconvenient for hand
calculations but quite desirable for computerised
calculations. The results obtained are very
accurate.
KEITH VAUGH

59. Three ways of calculating Δu and Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
By using the cv or cp relations (Table A-2c) as a
function of temperature and performing the
integrations. This is very inconvenient for hand
calculations but quite desirable for computerised
calculations. The results obtained are very
accurate.
By using average specific heats. This is very
simple and certainly very convenient when
property tables are not available. The results
obtained are reasonably accurate if the
temperature interval is not very large.
KEITH VAUGH

60. Three ways of calculating Δu and Δh
By using the tabulated u and h data. This is the
easiest and most accurate way when tables are
readily available.
By using the cv or cp relations (Table A-2c) as a
function of temperature and performing the
integrations. This is very inconvenient for hand Δu = u2 − u1 ( table )
calculations but quite desirable for computerised 2
calculations. The results obtained are very
Δu = ∫ c (T ) dT
1
v
accurate. Δu ≅ cv,avg ΔT
By using average specific heats. This is very Three ways of calculating Δu
simple and certainly very convenient when
property tables are not available. The results
obtained are reasonably accurate if the
temperature 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 ratio
The 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 room
temperature.
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)
kJ
For solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔT
For liquids, two specical cases are commonly encountered
1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT
2. 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)
kJ
For solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔT
For liquids, two specical cases are commonly encountered
1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT
2. 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)
kJ
For solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔT
For liquids, two specical cases are commonly encountered
1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT
2. 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)
kJ
For solids, the term V ΔP is insignificant thus, Δh = Δu ≅ cavg ΔT
For liquids, two specical cases are commonly encountered
1. Constant pressure process, as in heaters ( Δp = 0 ) : h = Δu ≅ cavg ΔT
2. 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 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)
KEITH VAUGH