Fundamentals of Thermodynamic Property Relations
THERMODYNAMIC PROPERTY
RELATIONS
6.1 INTRODUCTION
Determination of properties is very fundamental to
thermodynamic analysis. While some can be
directly measured such as P, v, T, others such as u,
h, s, are determined from measurable properties.
For this the latter properties have to be expressed
mathematically in terms of the former. This will
require formulation of thermodynamic relations
between properties. Also approximation
techniques
1
will be developed as an alternative where sufficient
data are not available to calculate properties.
6.2 FUNDAMENTALS OF PARTIAL
DERIVATIVES
Properties are point functions expressed as functions
of two independent variables. This qualifies them to
be exact differentials. The functional relationship
may be expressed as
x=x(y,z) or f(x,y,z) = 0
And the total differential is expressed as
2
fig-chp6\fig6.1.pptx
shows the basis of partial derivative
formulation.
Rewriting as dx=Mdy + Ndz
Exact differentials satisfy
They also satisfy the cyclic relation
6.3 SOME FUNDAMENTAL RELATIONS
1
st
law:
δ
q=du+
δ
w
For a reversible process Tds=du+Pdv
1
st
Tds equation
.
More useful form:
du=Tds-Pdv u=u(s,v)
Using the definition of enthalpy h=u+Pv
3
Differentiation gives
dh=du+Pdv+vdp or du +Pdv=dh-vdP
Substitution in the 1
st
Tds equation gives the
2
nd
Tds
equation
:
Tds=dh-vdP also
dh=Tds+vdP h=h(s,P)
Helmholtz function: a=u-Ts
Differentiation gives
da=du-Tds-sdT Using 1
st
Tds equation gives
da=-Pdv-sdT a=a(T,v)
Gibbs function: g=h-Ts
Differentiation gives
dg=dh-Tds-sdT Using the 2
nd
Tds equation gives
dg=vdP-sdT g=g(T,P)
4
Applying the test of exactness gives the famous
Maxwell relations
Also
6.4 GENERALIZED RELATIONS FOR
CHANGES IN S, U,AND H
General equations indifferent to the type and phase of
5
of the substance are derived. Usually the two pairs of
independent variables used are (T,v) and (T,P).
Beginning with entropy as s=s(T,v) & 1
st
Tds equation
Also starting with s=s(T,P) & 2
nd
Tds equation
The following derived from Tds equations have been
used.
6
With u=u(T,v)
Using the 1
st
Tds equation and Maxwell relation
,after integration
Substitution gives
Similarly beginning with h=h(T,P)
7
Using the 2
nd
Tds equation and Maxwell relation
Substitution gives
The determination of changes in s, u and h requires the
knowledge of PvT behavior and experimental
information on the relationship between specific
heats and temperature.
8
6.5 GENERALIZED RELATIONS FOR c
P
AND
c
v
Two general relations derived were
Rewriting the ds equations
The test of exactness will give
9
And
The above two equations give the variation of specific
heats with volume and pressure at a given
temperature. As an example, integration of the
second equation from zero pressure gives
10
where c
P,0
is the zero pressure, or ideal gas, specific
heat at the given temperature. The integration
requires the knowledge of PvT behavior.
If we equate the two ds expression (c
P
-c
v
)is
determined as follows:
Division by dP and imposing v=c yields
11
Using the cyclic relation for (∂P/∂ T)
v
will give
1.
c
p
- c
v
must always be positive or zero. It becomes
zero at T=0 and
2.
It also becomes zero when (∂v/∂ T)
P
is ever zero.
Eg. Maximum density of water at 4
o
C.
3.
Since (∂v/∂ T)
P
is very small for liquids and solids-
gives c
p
-c
v
≈0
4.
For an ideal gas c
p
-c
v
=R
12
Other properties dealing with expansivity and
compressibility are
Substitution of the above in the c
p
-c
v
expression gives
The values of the coefficients are assumed to be
constant in many calculations
13
Table 6.1
β
, K
T
, and
ρ
at 1 bar vs. temperature for
(a) copper and (b) water
6.6 RESIDUAL PROPERTY FUNCTIONS
This is an alternate method to determine changes in
properties such as u, h, and s in states other than the
ideal-gas state. For any specific property y a
residual function y
R
is defined as
y
R
≡y
*
- y or y
R
≡ y - y
*
y is the desired value at T, P and y
*
is the property of
an ideal gas at the same T, P (hypothetical). The
change in property can be written as
y
2
-y
1
= (y
2
*
-y
2
R
)- (y
1
*
- y
1
R
) = y
1
R
- y
2
R
+ (y
2
*
- y
1
*
)
With the above definitions
Δ
h can be determined as
15
Referring to
fig-chp6\fig6.2.pptx
Using the definition of residual functions
Similarly
Δ
s can be determined as
The first two are the residual terms.
16
Selecting state 1 as the reference state (ideal gas) at T
o
and P
o
with h
o
*
and s
o
*
, then the residual functions at
one are zero. This will give
Required are PvT data, ideal gas specific heat data and
knowledge of the residual function.
17
6.7 RESIDUAL PROPERTIES AND THE GIBBS
FUNCTION
Gibbs function given by dg=vdP-sdT is the convenient
function to develop the residual function since it is
expressed as g=g(P,T) and it will use those PvT
relations that are explicit in v.
Also dimensionless reduced function will be used as
d(g
R
/RT) where
For an ideal gas
18
Subtraction of the first from the second gives the
fundamental residual property relation as
Integrating at constant T from P=0 (ideal gas) to
system pressure gives
The same equation at constant P gives h
R
as
19
Also using g
R
=h
R
-Ts
R
Using the compressibility factor Z, v/RT=Z/P and
substitution gives
20
This will give
Similarly it can easily be shown
For a two parameter equation in terms of reduced
properties
21
Similarly
The above require Z=Z(P
r
,T
r
)-charts A-39 and A-40
For internal energy one can start from (u=h-Pv)
22
Using the three parameter equation with
ω
in the form
of Z = Z
(0)
+
ω
Z
(1)
Since
Substitution in the h
R
/R
u
T
c
expression gives
23
Similarly
Tables A-26 through A-29 give the enthalpy and
entropy residual functions.
Example 6.1
example.docx
24
6.8 RESIDUAL PROPERTIES AND THE
HELMHOLTZ FUNCTION
As most equations are of the form P=P(v,T), the
Helmholtz function, a=a(v,T) also looks to be
convenient to derive residual property relations.
da=-Pdv-sdT or da=-Pdv at T=C
Infinite v indicates an ideal gas situation, hence
25
To eliminate the difficulty of infinite limits on the
lower and upper bound add and subtract the integral
of RT/v as follows:
Since Z was also defined as Z=v
act
/v
ideal
then Z=v/v*
and this will give
26
ad
To make it dimensionless divide by R
u
T
To get the residual functions for s and h, start with
da==Pdv-sdT and noting that s=(∂a/∂T)
v
, hence
Substitution of (a*-a) and taking the derivative with
respect to T at constant v finally gives
27
For enthalpy residual function, start with
h = u + Pv = a+Ts +Pv since a=u-Ts
Then
h
R
= h
*
- h = (a
*
- a) + T(s
*
-s) + P
*
v
*
- Pv
where P
*
v
*
=RT
Substitutions for (a
*
-a) and (s
*
-s) will finally give
28
Using g=a+Pv and u=h-Pv the residual functions for g
and u can be determined from
g
R
/R
u
T=a
R
/R
u
T + 1-Z and u
R
/R
u
T=h
R
/R
u
T + Z-1
All the above equations require P=P(v,T)
6.9 FLOW AVAILABILITY FROM RESIDUAL
FUNCTIONS
ψ
R
=(
ψ
*
-
ψ
)
T,P
=[(h
*
-h)-(h
o
*
-h
o
)]-T
o
[(s
*
-s)-(s
o
*
-s
o
)]
In dimensionless form
29
This can be rearranged to give
Change in exergy will be
Example 6.2
example.docx
6.10 PROPERTIES OF THE SATURATION STATE
Deals with liquid-vapor equilibrium of pure substances.
Fundamental relationships among and approximate
evaluation techniques for the basic properties P, T, u, h,
s, a, and g are sought.
30
During phase change the heat supplied is called latent
heat of evaporation, h
fg
and the change in entropy is
given by h
fg
/T
sat
. The combination gives
h
fg
-Ts
fg
=0 or
Δ
h-T
Δ
s=0 (Phase change)
From the definition of Gibbs function, g=h-Ts it
follows
Δ
g
T
=
Δ
h - T
Δ
s = 0 for a phase change or
g
α
=g
β
or for the liquid-vapor phase
g
f
=g
g
The equality of g for each phase is the criterion for
phase quilibrium.
From the relationship dg=vdP-sdT
31
We see that
Since v and s change discontinuously, the derivatives
also change discontinuously (
fig-chp6\fig6.3.pptx
).
c
P
=T(∂s/∂T)
P
in the mixture region is infinite while
it has finite values at single phase points.
For changes of dT and dP on the two phase (
fig-
chp6\fig6.4.pptx
) equilibrium system at the initial
state (i) g
i
L
=g
i
V
.
At the final equilibrium position
g
i
L
+dg
L
=g
i
V
+dg
V
Thus for the change dT dg
L
= dg
V
32
Using the expression for dg will give
v
L
dP – s
L
dT = v
V
dP – s
V
dT
And upon rearranging
Since
Δ
h = T
Δ
s for a phase change
33
The above is called Clapeyron equation- Used to
determine enthalpy of vaporization. Applicable to
sublimation and melting too.
At low pressures ideal gas behavior can be considered
and also v
f
<<v
g
= RT/P. This will give
All the above are known as Clausius-Clapeyron
equations.
For small temperature change h
fg
remains
approximately constant. Integration gives
34
The above equation shows linearity between ln P
sat
and 1/T for a small change in T at low pressures.
Actual observation is that the linearity holds from
the triple state to the critical state. This can be seen
by inserting
Δ
v=(Z
g
– Z
f
)RT/P =
Δ
Z (RT/P) in the
Clapeyron equation and gives the modified
Clapeyron equation as
The ratio h
fg
/Z
fg
tends to remain constant with a
minimum value at T
r
= 0.85. Using Watson’s
h
fg
=
α
(1-T
r
)
0.38
and Liley’s Z
fg
=
β
(1-T
r
)
0.38
35
Results in
6.10.1 Vapor-Pressure Correlations
According to Clausius-Clapeyron equation vapor
pressure can be fitted as
Modification is made on the above equation by using
the normal boiling point T
b
(at 1bar or 1 atm) and T
c
and P
c
to determine the constants A and B
36
This will give B=At
b
giving
A=T
c
ln P
c
/(T
c
-T
b
) and the final equation becomes
(reasonably accurate over a fairly wide range of
temperatures)
To improve the accuracy over a wider range of
temperatures, the change in h
fg
must be
incorporated. A Typical empirical vapor pressure
correlations in use is
37
If a two parameter corresponding state is used on the
Clausius-Clapeyron equation (ln P
sat
= A-B/T)
It can easily be converted to
This gives a single straight line approximating the
slope for all substances which is approximately true
for simple fluids (
ω
<0.05) as shown in
fig-
chp6\fig6.5.pptx
.
A second parameter is used to include the normal
fluids in the form of P
sat
=f(T
sat
,
ω
)
Pitzer’s proposal
38
The values for the right hand expressions are given in
A-30
Other proposals of the form ln P
r
sat
=f
(0)
+
ω
f
(1)
due to
Lee-Kessler
ω=
α
/
β
(recommended) where
α
=-ln P
c
-5.92714+6.09648
Θ
-1
+1.28862 ln
Θ
-0.169347
Θ
6
β
=15.2518-15.6875
Θ
-1
-13.4721 ln
Θ
+0.43577
Θ
6
Θ
=T
b
/T
c
, P
c
(atm)
39
Satisfies definition of
ω
; curve passes through the
critical state; curvature of the saturation line is zero
at the critical state.
Dong and Lienhard proposal:
Its major advantage over the Lee-Kessler equation is
that it predicts better over a wider range of
ω
.
Satisfies the definition of
ω
, as well as the critical state
condition.
40
6.10.2 Estimation of h
fg
Data
Fishtine’s correlation using the normal boiling point T
b
given by
k=1 non-polar and 1<k<1.38 for polar and
hydrogen bonded compounds
Reidel’s proposal
41
Watsons for non associating liquids requiring a
knowledge of h
fg
at one temperature
Application of three-parameter corresponding states
uses Clapeyron equation dP/dT=
Δ
s/
Δ
v and since
v=ZRT/P, then
42
Rewriting as
P
r
=f(T
r
,
ω
) (slides 39 &40) and Table A-30 can be used
for the above.
Analytical correlation for
Δ
Z from Haggenmacher
(1946) is also useful
43
A suitable three-parameter (vaporization) correlation
has the form
Δ
s = s
fg
=
Δ
s
v
(0)
+
ωΔ
s
v
(1)
Table A-30 gives the right hand expressions.
Since h
fg
=Ts
fg
then
(gives reasonable values for normal fluids)
In dimensionless form
44
fig-chp6\fig6.6.pptx
shows a linear function in the
form of
According to Reid for 0.6 < Tr ≤ 1.0 the curves are
well represented by
which is linear in
ω
for fixed T
r
.
45
Finally a dimensionless correlation for temperature
range T
c
to triple state T
t
according to Torquato and
Stell and Torquato and Smith
The equation represents the solid line of
fig-
chp6\fig6.7.pptx
which fits the data for water. The
above method requires h
fg
at the triple point (a major
disadvantage).
Example 6.3
example.docx
46
6.10.3 Phase Equilibrium Properties from
Equations of State
For liquid vapor equilibrium g
L
=g
V
In terms of residual functions
If P(v,T) is given, then
47
Using the expression for a
R
yields
As an example for RK fluid
By first assuming P
sat
(using appropriate equations),
RK equation (cubic in v) is solved for v
f
and v
g
and
then Z
g
and Z
f
. Then check the equality of the phase
equilibrium condition. If deviation is large, assume
another P
sat
and repeat the procedure.
48
Then use
to determine h
g
R
and h
f
R
.
h
fg
= R
u
T(h
g
R
– h
f
R
)
Finally
s
fg
=h
fg
/T
For building up the table for h
f
, h
g
, s
f
,s
g
in a
saturation table requires a reference state.
49
6.11 THE JOULE-THOMSON COEFFICIENT
It has been seen that 1
st
law applied on throttling
devices resulted in an isenthalpic process h
1
=h
2
.
Also called
Joule-Thomson effect
.
This usually results in a cold temperature, where the
end result can be a two phase fluid and separation
occurs.
Other properties such as specific volumes, specific
heats, and enthalpies may be evaluated from
measurements of the Joule-Thomson effect.
50
During throttling T may increase, decrease or stay the
same while the enthalpy remains the same.
Isenthalpic processes can be constructed as shown in
fig-chp6\fig6.8.pptx
.
The Joule-Thomson coefficient is defined using this
isenthalpic curve as
The line formed by connection of the maximum points
is called the inversion curve and the temperatures at
these points are called inversion temperatures.
51
μ
JT
is –ve to the right of the inversion line and +ve to
the left of the inversion line.
At high temperatures, many constant enthalpy lines
may never pass through the inversion line:
μ
JT
is
always negative. Such fluids must be artificially
cooled before throttling (hydrogen and helium).
Data for Joule-Thomson coefficient is frequently
plotted against temperature for various pressures.
Such a plot (or tabular data) shows the regions of
pressure and temperature where a
cooling effect
is
possible.
fig-chp6\fig6.9.pptx
is for argon and
nitrogen while
fig-chp6\fig6.10.pptx
is for air.
52
Using the cyclic relation
And substitution in
μ
JT
gives
Recalling
53
The term in brackets is (∂h/∂P)
T
. This will give
This permits evaluation of
μ
JT
if an equation of state
explicit in v is at hand. The inversion curve can also
be evaluated by setting
μ
JT
=0 and this will give the
general criterion as
-very sensitive test for an equation of state.
54
Using the equation of state Pv=ZRT and substitution
of v=ZRT/P and (∂v/∂T)
P
= ZR/P+RT (∂Z/∂T)
P
/P will
give
Cooling occurs when the sign of (∂Z/∂T)
P
is positive.
On the basis of the Z chart
1.
For P
r
<10, Z increases with increase in T
r
until T
r
reaches 5, beyond which the opposite occurs, ie.
rise in T upon throttling.
2.
For P
r
>10, Z always decreases with increasing T
r
55
The above general analysis is in agreement with the
generalized P
r
– T
r
diagram shown in
fig-
chp6\fig6.11.pptx
, based on experimental data.
Least squares fit of experimental data according to
Miller
With respect to relationships to other thermodynamic
properties, as an example to residual enthalpy
56
Determining thermodynamic properties is crucial for analysis, involving direct and derived properties expressed through formulations and relations. Partial derivatives, fundamental relations like the 1st and 2nd laws, and functions such as enthalpy, Helmholtz, and Gibbs are explored. Maxwell relations and generalized relations for changes in S, U, and H are discussed based on different pairs of independent variables.
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Presentation Transcript
THERMODYNAMIC PROPERTY RELATIONS 6.1 INTRODUCTION Determination of properties is very fundamental to thermodynamic analysis. directly measured such as P, v, T, others such as u, h, s, are determined from measurable properties. For this the latter properties have to be expressed mathematically in terms of the former. This will require formulation of thermodynamic relations between properties. techniques While some can be Also approximation 1
will be developed as an alternative where sufficient data are not available to calculate properties. 6.2 FUNDAMENTALS OF PARTIAL DERIVATIVES Properties are point functions expressed as functions of two independent variables. This qualifies them to be exact differentials. The functional relationship may be expressed as x=x(y,z) or f(x,y,z) = 0 And the total differential is expressed as x dy y z x = + dx dz z y 2
fig-chp6\fig6.1.pptx shows the basis of partial derivative formulation. Rewriting as dx=Mdy + Ndz Exact differentials satisfy N z M = y y z They also satisfy the cyclic relation y z y z x y x z = 1 x 6.3 SOME FUNDAMENTAL RELATIONS 1st law: q=du+ w For a reversible process Tds=du+Pdv 1st Tds equation. More useful form: du=Tds-Pdv u=u(s,v) Using the definition of enthalpy h=u+Pv 3
Differentiation gives dh=du+Pdv+vdp or du +Pdv=dh-vdP Substitution in the 1st Tds equation gives the 2nd Tds equation : Tds=dh-vdP also dh=Tds+vdP h=h(s,P) Helmholtz function: a=u-Ts Differentiation gives da=du-Tds-sdT Using 1st Tds equation gives da=-Pdv-sdT a=a(T,v) Gibbs function: g=h-Ts Differentiation gives dg=dh-Tds-sdT Using the 2nd Tds equation gives dg=vdP-sdT g=g(T,P) 4
Applying the test of exactness gives the famous Maxwell relations T s v T P v P s v s = = = = P s T v T P s v s P v T P T Also u u h h = = = = T P T v s v s P v s P s a a g g = = = = P s v s v T P T T v T P 6.4 GENERALIZED RELATIONS FOR CHANGES IN S, U,AND H General equations indifferent to the type and phase of 5
of the substance are derived. Usually the two pairs of independent variables used are (T,v) and (T,P). Beginning with entropy as s=s(T,v) & 1st Tds equation s dT T T v dT c du v = dh = c s P = + = + v ds dv dT dv v T T v c dT p Also starting with s=s(T,P) & 2nd Tds equation + s s c v = = P ds dT dP dT dP T P T T P T P The following derived from Tds equations have been used. and T s s = = c T c T v P T 6 v P
With u=u(T,v) u u u = + = + du dT dv c dT dv v T v v v T T Using the 1st Tds equation and Maxwell relation = ,after integration = = du Tds pdv u Ts pv u s P = T P T P v v T T T v Substitution gives du = P + c dT T P dv v T v Similarly beginning with h=h(T,P) 7
h h h = + = + dh dT dP c dT dP P T P P P T T Using the 2nd Tds equation and Maxwell relation h s v = + = + T v T v P P T T T P Substitution gives v = + dh c dT v T dP P T P The determination of changes in s, u and h requires the knowledge of PvT behavior and experimental information on the relationship between specific heats and temperature. 8
6.5 GENERALIZED RELATIONS FOR cP AND cv Two general relations derived were s T c = s = and c T v P T T v P Rewriting the ds equations c ds = P c v + = v P dT dv ds dT dP T T T T v P The test of exactness will give c v T 2 2 c P 2 P 2 = = v Or T v T v T T v v T 9
And c P T 2 2 v 2 c v 2 = = P Or T P T P T T P P T The above two equations give the variation of specific heats with volume and pressure at a given temperature. As an example, integration of the second equation from zero pressure gives 2 v 2 P = c c T dP 0 , P P T 0 P 10
where cP,0 is the zero pressure, or ideal gas, specific heat at the given temperature. The integration requires the knowledge of PvT behavior. If we equate the two ds expression (cP-cv)is determined as follows: c P c v + = v P dT dv dT dP T T T T v P c c P v = + P v or dT dv dP T T T v P Division by dP and imposing v=c yields T T c c v v P = = P v or c c T P v P T T T 11 v P P v
Using the cyclic relation for (P/ T)v will give v T c c 2 P = P v T v P T 1. cp- cv must always be positive or zero. It becomes zero at T=0 and 2. It also becomes zero when ( v/ T)P is ever zero. Eg. Maximum density of water at 4oC. 3. Since ( v/ T)P is very small for liquids and solids- gives cp-cv 0 4. For an ideal gas cp-cv =R 12
Other properties dealing with expansivity and compressibility are = 1 v exp volumetric ansion coefficien t 1 v T P v = K isothermal coefficien t of compressib ility T v P T Substitution of the above in the cp-cv expression gives vT c c = 2 P v K T The values of the coefficients are assumed to be constant in many calculations 13
Table 6.1 , KT, and at 1 bar vs. temperature for (a) copper and (b) water
6.6 RESIDUAL PROPERTY FUNCTIONS This is an alternate method to determine changes in properties such as u, h, and s in states other than the ideal-gas state. For any specific property y a residual function yR is defined as yR y*- y or yR y - y* y is the desired value at T, P and y* is the property of an ideal gas at the same T, P (hypothetical). The change in property can be written as y2-y1 = (y2*-y2R)- (y1*- y1R) = y1R - y2R + (y2* - y1*) With the above definitions h can be determined as h ( ) h h ( h h 2 T , P 1 1 1 2 1 1 = = + + * * * P * P h ) h ( h ) 2 P T , 2 T , 2 T , 1 2 2 1 15
Referring to fig-chp6\fig6.2.pptx h ( h h a 1 2 = = + + h ) h ( h ) h ( h ) 1 b 2 b a Using the definition of residual functions T R R = + 2 h h h h c dT 2 1 1 2 , P o T 1 Similarly s can be determined as s = = + + * 1 * 2 * P * P s s s ( s ) s ( s ) s ( s ) 2 1 1 P T , 1 2 P T , 2 T T 1 2 2 2 1 1 The first two are the residual terms. c P T R R , = + P o 2 ln 2 s s s s dT R 2 1 1 2 T P T 1 1 16
Selecting state 1 as the reference state (ideal gas) at To and Po with ho* and so*, then the residual functions at one are zero. This will give T * = + R h h c dT h and , o P o T o c P T * , = + P T o R ln s s dT R s o P T o o Required are PvT data, ideal gas specific heat data and knowledge of the residual function. 17
6.7 RESIDUAL PROPERTIES AND THE GIBBS FUNCTION Gibbs function given by dg=vdP-sdT is the convenient function to develop the residual function since it is expressed as g=g(P,T) and it will use those PvT relations that are explicit in v. Also dimensionless reduced function will be used as d(gR/RT) where v RT RT RT ( = = g d = g dg gdT h = = d dP dT 2 2 RT RT ) g h Ts and dg vdp v sdT were h 2 used * * * dP dT For an ideal gas RT RT RT 18
Subtraction of the first from the second gives the fundamental residual property relation as R R R g v h = d dP dT 2 RT RT RT Integrating at constant T from P=0 (ideal gas) to system pressure gives v RT R R 1 P g v P P = = = dP ( ) dP T C RT RT 0 0 The same equation at constant P gives hR as RT g T RT R R ( / ) h = = P C T 19 P
Also using gR=hR-TsR R R R R R ( / ) s h g g RT g = = T R RT RT T RT P Using the compressibility factor Z, v/RT=Z/P and substitution gives R g dP ( ) P = = 1 ( ) Z T C RT P RT 0 1 R R ( / ) h g Z P = = T T dP RT T T P 0 P P 20
This will give h = = R Z dP P = = T ( T C ) RT T P 0 P Similarly it can easily be shown Z T R R s dP dP ( ) P P = = 1 ( ) Z T C T P P 0 0 P For a two parameter equation in terms of reduced properties Z T RT P r c r R h P 2 = = r ln ( ) d P T C r r r T 0 21
Similarly R R R R s s h h dP dP ( ( ) ) P P = = = = r r 1 1 ( ( ) ) r r Z Z T T C C r r R R RT RT T c T c P P 0 0 r r r r The above require Z=Z(Pr,Tr)-charts A-39 and A-40 For internal energy one can start from (u=h-Pv) R R u h = = + + = = ( Z ) 1 ( T C ) RT RT 22
Using the three parameter equation with in the form of Z = Z(0) + Z(1) ) 0 ( ) 1 ( ) 0 ( ) 1 ( R R R R R R h h h s s s = + = + and R T R T R T R R R u c u c u c u u u Since ) 0 ( ) 1 ( Z Z Z = + T T T r r r P P P r r r Substitution in the hR/RuTc expression gives P r c T RT ) 0 ( ) 1 ( R h Z dP Z dP P P 2 2 = + r r r r T T r r P T P 0 0 r r r P r r 23
Similarly ) 0 ( R s Z dP P = + r ) 0 ( 1 r T Z r R T P 0 u r r P r ) 1 ( Z dP P + + r ) 0 ( 1 r T Z r T P 0 r r P r Tables A-26 through A-29 give the enthalpy and entropy residual functions. Example 6.1 example.docx 24
6.8 RESIDUAL PROPERTIES AND THE HELMHOLTZ FUNCTION As most equations are of the form P=P(v,T), the Helmholtz function, a=a(v,T) also looks to be convenient to derive residual property relations. da=-Pdv-sdT or da=-Pdv at T=C * * v v = = = = * R a a a Pdv Pdv Pdv T C v v Infinite v indicates an ideal gas situation, hence RT * * v v = = = = * R a a a Pdv dv Pdv T C v v v 25
To eliminate the difficulty of infinite limits on the lower and upper bound add and subtract the integral of RT/v as follows: RT dv v RT RT RT * v = = * R a a a P dv dv dv v v v v v RT RT v * v v = = + ln P dv P dv RT * v v v v v Since Z was also defined as Z=vact/videal then Z=v/v* and this will give RT v = = + = * R ln a a a P dv RT Z T C v 26
ad To make it dimensionless divide by RuT a a T R u u * R 1 R T a v = = + = ln u P dv Z T C R T R T v u To get the residual functions for s and h, start with da==Pdv-sdT and noting that s=( a/ T)v, hence = * * ( ) s s a a v T Substitution of (a*-a) and taking the derivative with respect to T at constant v finally gives * 1 R s s P v = ln u dv R Z R R T v u u v 27
For enthalpy residual function, start with h = u + Pv = a+Ts +Pv since a=u-Ts Then hR = h*- h = (a*- a) + T(s*-s) + P*v*- Pv where P*v*=RT Substitutions for (a*-a) and (s*-s) will finally give T u * 1 h h P v = + 1 T P dv Z R R T T u v 28
Using g=a+Pv and u=h-Pv the residual functions for g and u can be determined from gR/RuT=aR/RuT + 1-Z and uR/RuT=hR/RuT + Z-1 All the above equations require P=P(v,T) 6.9 FLOW AVAILABILITY FROM RESIDUAL FUNCTIONS R=( *- )T,P=[(h*-h)-(ho*-ho)]-To[(s*-s)-(so*-so)] In dimensionless form * * R R * * * R T h h T s s h h s s = = o o o o o o RT RT RT T RT T c c c c c c , T P 29
This can be rearranged to give * R R R R R T T h s h s = = o o RT RT RT T R RT T R c c c c c c , T P , . T P oP T o Change in exergy will be R R R R h h s s 2 = + 2 1 2 1 RT RT 1 c o RT RT R R c c Example 6.2 example.docx 6.10 PROPERTIES OF THE SATURATION STATE Deals with liquid-vapor equilibrium of pure substances. Fundamental relationships among and approximate evaluation techniques for the basic properties P, T, u, h, s, a, and g are sought. 30
During phase change the heat supplied is called latent heat of evaporation, hfg and the change in entropy is given by hfg/Tsat. The combination gives hfg-Tsfg=0 or h-T s=0 (Phase change) From the definition of Gibbs function, g=h-Ts it follows gT = h - T s = 0 for a phase change or g =g or for the liquid-vapor phase gf=gg The equality of g for each phase is the criterion for phase quilibrium. From the relationship dg=vdP-sdT 31
We see that = g g = v and s P T T P Since v and s change discontinuously, the derivatives also change discontinuously (fig-chp6\fig6.3.pptx ). cP=T( s/ T)P in the mixture region is infinite while it has finite values at single phase points. For changes of dT and dP on the two phase (fig- chp6\fig6.4.pptx ) equilibrium system at the initial state (i) giL=giV. At the final equilibrium position giL +dgL=giV+dgV Thus for the change dT dgL = dgV 32
Using the expression for dg will give vL dP sL dT = vV dP sV dT And upon rearranging s dT s V L dP s fg = = V L v v v sat fg Since h = T s for a phase change h V L dP h ( h h fg = = = V L ) dT T v v T v Tv sat fg 33
The above is called Clapeyron equation- Used to determine enthalpy of vaporization. Applicable to sublimation and melting too. At low pressures ideal gas behavior can be considered and also vf<<vg = RT/P. This will give dT h P sat sat ln dP d P fg = = or h R fg 2 / 1 ( d ) RT T All the above are known as Clausius-Clapeyron equations. For small temperature change hfg remains approximately constant. Integration gives h P sat h 1 T 1 T P fg fg = + = sat 2 ln ln const or sat RT R P 34 2 1 1
The above equation shows linearity between ln Psat and 1/T for a small change in T at low pressures. Actual observation is that the linearity holds from the triple state to the critical state. This can be seen by inserting v=(Zg Zf)RT/P = Z (RT/P) in the Clapeyron equation and gives the modified Clapeyron equation as T RZ fg h 1 fg = sat ln d P d The ratio hfg/Zfg tends to remain constant with a minimum value at Tr= 0.85. Using Watson s hfg= (1-Tr)0.38 and Liley s Zfg= (1-Tr)0.38 35
Results in 1 dT dT = = = sat ln d P d 2 2 R T T T 6.10.1 Vapor-Pressure Correlations According to Clausius-Clapeyron equation vapor pressure can be fitted as B A P ln = = = sat tan / A cons t B h R fg T Modification is made on the above equation by using the normal boiling point Tb (at 1bar or 1 atm) and Tc and Pc to determine the constants A and B 36
= ln 1 ( A / ) P T T This will give B=Atb giving A=Tc ln Pc/(Tc-Tb) and the final equation becomes P ln T P ln b c b T = = sat 1 [ P ( bars , ) T ( K )] c c b T T T (reasonably accurate over a fairly wide range of temperatures) To improve the accuracy over a wider range of temperatures, the change incorporated. A Typical empirical vapor pressure correlations in use is in hfg must be B = + + + 2+ Psat ln ln A CT DT E T T 37
If a two parameter corresponding state is used on the Clausius-Clapeyron equation (ln Psat = A-B/T) It can easily be converted to = = sat r r T T ' 1 B sat ln ' ' 1 P A A r sat This gives a single straight line approximating the slope for all substances which is approximately true for simple fluids ( <0.05) as shown in fig- chp6\fig6.5.pptx . A second parameter is used to include the normal fluids in the form of Psat=f(Tsat, ) Pitzer s proposal r P = log log P sat + ) 0 ( (log ) r P r r T 38
The values for the right hand expressions are given in A-30 Other proposals of the form ln Prsat=f(0) + f(1) due to Lee-Kessler = . 1 92714 . 5 ln r T 09648 . 6 6 sat + 28862 ln 169347 . 0 P T T r r r 15 . 6875 T 6 + + 15 2518 . 13 4721 . ln 43577 . 0 T T r r r = / (recommended) where =-ln Pc-5.92714+6.09648 -1+1.28862 ln -0.169347 6 =15.2518-15.6875 -1 -13.4721 ln +0.43577 6 =Tb/Tc , Pc (atm) 39
Satisfies definition of ; curve passes through the critical state; curvature of the saturation line is zero at the critical state. Dong and Lienhard proposal: 1 T 3 6 sat 37270 . 5 = + . 7 ( 68769 . 3 + ln 1 49408 18177 . 11 P T T r r r r + 92998 . 17 ln ) T r Its major advantage over the Lee-Kessler equation is that it predicts better over a wider range of . Satisfies the definition of , as well as the critical state condition. 40
6.10.2 Estimation of hfg Data Fishtine s correlation using the normal boiling point Tb given by h b b , fg b = 36 + ( . . ln ) ( / ) 6 8 31 k T J gmol and K T k=1 non-polar and 1<k<1.38 for polar and hydrogen bonded compounds Reidel s proposal (ln . , , P T RT c 930 0 . ) h 1 093 1 013 fg b = b r c ( , ) K bars . T , b r 41
Watsons for non associating liquids requiring a knowledge of hfg at one temperature . , r fg T h 0 38 h 1 T 2 fg , 2 = r 1 1 , 1 , Application of three-parameter corresponding states uses Clapeyron equation dP/dT= s/ v and since v=ZRT/P, then ZRT ln 1 dP R Z d P = = = ( ) r s Z Z Z g f ( / ) dT P T d T r r 42
Rewriting as 2 ( ) sat RT Z Z dP RTT Z dP g f = = r r h fg P dT P dT r r Pr=f(Tr, ) (slides 39 &40) and Table A-30 can be used for the above. Analytical correlation for Z from Haggenmacher (1946) is also useful 3 1 r T / 1 2 P = r Z Z g f 43
A suitable three-parameter (vaporization) correlation has the form s = sfg = sv(0) + sv(1) Table A-30 gives the right hand expressions. Since hfg=Tsfg then + + = = R ) 0 ( ) 1 ( v s s h RT v fg R (gives reasonable values for normal fluids) In dimensionless form R RT c r ) 0 ( ) 1 ( v h s s fg = = + + T v R 44
fig-chp6\fig6.6.pptx shows a linear function in the form of h + = fg ( ) a b fixed T r RT c According to Reid for 0.6 < Tr 1.0 the curves are well represented by h + = fg 354 . 456 . 0 0 . ( ) 95 . ( ) 7 08 1 10 1 T T r r RT c which is linear in for fixed Tr. 45
Finally a dimensionless correlation for temperature range Tc to triple state Tt according to Torquato and Stell and Torquato and Smith = fg t c h T T ( ) h T T T fg = c and ( ) T t = + + / 79 . 208 . 1 3 0 1 60176 . 45913 . 62671 . 0 3 4 + 2 3 89614 . 10643 . 31522 . 6 1 0 The equation represents the solid line of fig- chp6\fig6.7.pptx which fits the data for water. The above method requires hfg at the triple point (a major disadvantage). Example 6.3 example.docx 46
6.10.3 Phase Equilibrium Properties from Equations of State For liquid vapor equilibrium gL=gV In terms of residual functions g g T R * * g g = R T u u L V If P(v,T) is given, then * * g g a a = + 1 Z R T R T u u 47
Using the expression for aR yields T R T R * 1 R T g g ( ) v = + ln 1 u v P dv Z Z u u As an example for RK fluid Z T R u * 1 + 1 g g b a b = ( ) ln 1ln 1 Z 5 . v bR T v u By first assuming Psat (using appropriate equations), RK equation (cubic in v) is solved for vf and vg and then Zg and Zf. Then check the equality of the phase equilibrium condition. If deviation is large, assume another Psat and repeat the procedure. 48
Then use R + 1 3 h a b = + ( ) 1ln 1 Z 5 . bR 2 R T T v u u to determine hgR and hfR. hfg= RuT(hgR hfR) Finally sfg=hfg/T For building up the table for hf, hg , sf ,sg in a saturation table requires a reference state. 49
6.11 THE JOULE-THOMSON COEFFICIENT It has been seen that 1st law applied on throttling devices resulted in an isenthalpic process h1=h2. Also called Joule-Thomson effect. This usually results in a cold temperature, where the end result can be a two phase fluid and separation occurs. Other properties such as specific volumes, specific heats, and enthalpies may be evaluated from measurements of the Joule-Thomson effect. 50
