Download Part III

Thermodynamic
Properties

Property Table -from
direct measurement
 Equation of State -any equations that relates
P,v, and T of a substance
Ideal -Gas Equation of State
 Any
relation among the pressure,
temperature, and specific volume of a
substance is called an equation of state. The
simplest and best-known equation of state is
the ideal-gas equation of state, given as
where R is the gas constant. Caution should be
exercised in using this relation since an ideal
gas is a fictitious substance. Real gases exhibit
ideal-gas behavior at relatively low pressures
and high temperatures.
Universal Gas Constant
Universal gas constant is given on
Ru = 8.31434 kJ/kmol-K
= 8.31434 kPa-m3/kmol-k
= 0.0831434 bar-m3/kmol-K
= 82.05 L-atm/kmol-K
= 1.9858 Btu/lbmol-R
= 1545.35 ft-lbf/lbmol-R
= 10.73 psia-ft3/lbmol-R
Example
Determine the particular gas constant
for air and hydrogen.
kJ
8.1417
kJ
R
kmol

K
u
 0.287
Rair  M 
kg
kg

K
28.97
kmol
kJ
8.1417
kJ
kmol

K
 4.124
Rhydrogen 
kg
kg  K
2.016
kmol
Ideal Gas “Law” is a simple
Equation of State
PV  mRT
Pv  RT
PV  NRuT
P v  RuT, v  V/N
P1V1 P2V2

T1
T2
Percent error for applying ideal gas
equation of state to steam
Question …...
Under what conditions is it
appropriate to apply the
ideal gas equation of state?
Ideal Gas Law

Good approximation for P-v-T
behaviors of real gases at low
densities (low pressure and high
temperature).
 Air, nitrogen, oxygen, hydrogen,
helium, argon, neon, carbon
dioxide, …. ( < 1% error).
Compressibility Factor
 The
deviation from ideal-gas behavior
can be properly accounted for by using
the compressibility factor Z, defined as
Z represents the volume ratio
or compressibility.
Ideal
Gas
Real
Gases
Z=1
Z > 1 or
Z<1
Real Gases
 Pv
= ZRT or
 Pv = ZRuT, where v is volume
per unit mole.


Z is known as the compressibility factor.
Real gases, Z < 1 or Z > 1.
Compressibility factor

What is it really doing?
 It accounts mainly for two things
• Molecular structure
• Intermolecular attractive forces
Principle of
corresponding states
The
compressibility factor Z is
approximately the same for all
gases at the same reduced
temperature and reduced pressure.
Z = Z(PR,TR) for all gases
Reduced Pressure
and Temperature
P
T
PR 
; TR 
Pcr
Tcr
where:
PR and TR are reduced values.
Pcr and Tcr are critical properties.
Compressibility factor for ten substances
(applicable for all gases Table A-3)
Where do you find critical-point properties?
Table A-7
Mol
(kg-Mol)
R
(J/kg.K)
Tcrit
(K)
Pcrit
(MPa)
Ar
28,97
287,0
(---)
(---)
O2
32,00
259,8
154,8
5,08
H2
2,016
4124,2
33,3
1,30
H2 O
18,016
461,5
647,1
22,09
CO2
44,01
188,9
304,2
7,39
Reduced Properties

This works great if you are given a
gas, a P and a T and asked to find
the v.
 However, if you are given P and v
and asked to find T (or T and v and
asked to find P), trouble lies ahead.
 Use pseudo-reduced specific volume.
Pseudo-Reduced
Specific Volume
 When
either P or T is unknown, Z can
be determined from the compressibility
chart with the help of the pseudoreduced specific volume, defined as
not vcr !
Ideal-Gas Approximation

The compressibility chart shows
the conditions for which Z = 1 and
the gas behaves as an ideal gas:
 (a)
PR < 0.1 and TR > 1
Other Thermodynamic Properties:
Isobaric (c. pressure) Coefficient
v
P
1  v 
   0
v  T  P
 v 
 
 T  P
T
Other Thermodynamic Properties:
Isothermal (c. temp) Coefficient
v
 v 
 
 P  T
1  v 
   0
v  P  T
T
P
Other Thermodynamic Properties:
We can think of the volume as being a function
of pressure and temperature, v = v(P,T).
Hence infinitesimal differences in volume are
expressed as infinitesimal differences in P and
T, using  and  coefficients
 v 
 v 
dv    dT    dP   vdT  vdP
 T  P
 P  T
If  and  are constant, we can integrate for v:
 v 
Ln   T  T0    P  P0 
 v0 
Other Thermodynamic Properties:
Internal Energy, Enthalpy and
Entropy
u  uT, v 
h  hT, P   u  Pv
s  su, v 
Other Thermodynamic Properties:
Specific Heat at Const. Volume
u
v
 u 
Cv     0
 T  v
 u 
 
 T  v
T
Other Thermodynamic Properties:
Specific Heat at Const. Pressure
h
P
 h 
CP     0
 T  P
 h 
 
 T  P
T
Other Thermodynamic Properties:
Ratio of Specific Heat
 Cp 

  
 Cv 
Other Thermodynamic Properties:
Temperature
v
s
1

s
 
T  0
 u  v
1
T
u
Ideal Gases: u = u(T)
0
 u 
 u 
du  
 dT    dv
 T  v
 v  T
Therefore,
 u 
du  
 dT  C v ( T )dT
 T  v
We can start with du and
integrate to get the change in u:
du  C v dT, and
u  u 2  u1  
T2
T1
C v (T) dT
Note that Cv does change with temperature
and cannot be automatically pulled from
the integral.
Let’s look at enthalpy
for an ideal gas:

h = u + Pv where Pv can be replaced
by RT because Pv = RT.

Therefore, h = u + RT => since u is
only a function of T, R is a constant,
then h is also only a function of T

so h = h(T)
Similarly, for a change in
enthalpy for ideal gases:
C p  C p (T )
 h 

0
 P 
&
dh  C p dT, and
h  h2  h1  
T2
T1
C p (T)dT
Summary: Ideal Gases
 For ideal
gases u, h, Cv, and Cp are
functions of temperature alone.
 For ideal gases, Cv and Cp are written
in terms of ordinary differentials as
 du 
 dh 
Cv  
; Cp  


 dT  ideal gas
 dT  ideal gas
For an ideal gas,
h
= u + Pv = u + RT
dh du

R
dT dT
Cp = Cv + R
 kJ 
 kg  K 


Ratio of specific heats
is given the symbol, 

Cp
Cv
Cp

Cp (T)
C v (T)
  (T)
R
 1
  1
Cv
Cv
Other relations with the
ratio of specific heats which
can be easily developed:
R
R
Cv 
and Cp 
-1
-1
For monatomic gases,
5
3
C p  R , Cv  R
2
2
and both are constants.
Argon, Helium, and Neon
For all other gases,

Cp is a function of temperature and it may be
calculated from equations such as those in
Table A-5(c) in the Appendice

Cv may be calculated from Cp=Cv+R.

Next figure shows the temperature behavior
…. specific heats go up with temperature.
Specific Heats for Some Gases

Cp = Cp(T)
a function of
temperature
Three Ways to Calculate
Δu and Δh

Δu = u2 - u1 (table)

2

Δu =

Δu = Cv,av ΔT
1
C v (T) dT

Δh = h2 - h1 (table)

2

Δh =

Δh = Cp,av ΔT
1
C p (T) dT
Isothermal Process
 Ideal
gas: PV = mRT
For ideal gas, PV = mRT
We substitute into the
integral
2
Wb   PdV 
1

2
1
mRT
dV
V
Collecting terms and integrating yields:
Wb  mRT 
2
1
 V2 
dV
 mRTn 
V
 V1 
Polytropic Process

PVn = C
Ideal Gas Adiabatic Process and
Reversible Work


1.
2.
3.
4.
5.
What is the path for process with expand or
contract without heat flux? How P,v and T
behavior when Q = 0?
To develop an expression to the adiabatic
process is necessary employ:
Reversible work mode: dW = PdV
Adiabatic hypothesis: dQ =0
Ideal Gas Law: Pv=RT
Specific Heat Relationships
First Law Thermodynamics: dQ-dW=dU
Ideal Gas Adiabatic Process and
Reversible Work (cont)
First Law:
Using P = MRT/V
dQ

  dW
0
PdV
dU

MC vdT
CV  dT 
 dV 


 
R  T
 V 

1  1
Integrating from
(1) to (2)
 T2   V2 
  
 T1   V1 
1  
Ideal Gas Adiabatic Process and
Reversible Work (cont)
Using the gas law :
Pv=RT, other
relationship amid
T, V and P are
 T2   V1 
  
 T1   V2 
 P2   V1 
  
 P1   V2 
developed
accordingly:
 T2   P2 
  
 T1   P1 
  1

  1 
Ideal Gas Adiabatic Process and
Reversible Work (cont)
An expression for
work is developed
using PV =
constant.
i and f represent
the initial and final
states
 

W   PdV  PV i 
dV
V


PV  i 1  

1  
V  V 
W
   1 f

PV f  PV i 

   1
i
Ideal Gas Adiabatic Process and
Reversible Work (cont)

The path representation
are lines where Pv =
constant.
 For most of the gases, 
1.4
 The adiabatic lines are
always at the righ of the
isothermal lines.
 The former is Pv =
constant (the exponent is
unity)
P
f
f
i
v
Polytropic Process
A frequently encountered process for gases
is the polytropic process:
PVn = c = constant
Since this expression relates P & V, we
can calculate the work for this path.
Wb  
V2
V1
PdV
Polytropic Process
Process

Constant pressure
 Constant volume
 Isothermal & ideal gas
 Adiabatic & ideal gas
Exponent n
0

1
k=Cp/Cv
Boundary work for a gas which
obeys the polytropic equation
Wb 

2
1
PdV  c 
v2
2
1
dV
n
V
1 n
1 n
V 
V2  V1
 c


1 n
 1-n  v1
1-n
n  1
We can simplify it
further
n
The constant c = P1V1 =
n
2
P
2 V (V
Wb 
n
P2V2
)  P1V 1n V 11 n 
1-n
1 n
2
P2V2  P1V1

,
1 n
n 1
Polytropic Process
Wb  
2
1
PdV  
2
1
c
dV
n
V
P2V2  P1V1

, n 1
1 n
V 2 
 , n  1
 PVn
 V1 