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[email protected] ‫ اﻟﺠﺎﻡﻌﺔ اﻻردﻥﻴﺔ‬,‫ ﻗﺴﻢ اﻟﻬﻨﺪﺳﺔ اﻟﻜﻴﻤﻴﺎﺋﻴﺔ‬,‫ ﻋﻠﻲ ﺧﻠﻒ اﻟﻤﻄﺮ‬.‫د‬
©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
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Lecture 05
Independent Variables V and T
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.
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„
„
„
„
„
„
„
„
„
Why use V-T instead of P-T.
Pressure effect on S and U.
Derivation of thermodynamic properties.
Required information.
The pure component limit.
Transforming to density dependence.
Deriving fugacity for vdW EOS.
Analysis of EOS capabilities.
Phase equilibrium from volumetric properties.
Adv. Thermo -Lecture 05: V-T Specified
Why use V-T instead of P-T?
Pressure Effects
„ At a constant temperature and composition, we can
„ The P-T description is the most natural.
(EOS) are P-explicit.
„ Consequently, it is more convenient to use V-T
description instead of P-T description.
3
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
„ However, the majority of equations of state
Adv. Thermo -Lecture 05: V-T Specified
2
use one of Maxwell relations to obtain the effect of
volume on entropy and entropy.
  ∂P 

dU = T 
− P  dV

  ∂T V , nT

(5-1)
 ∂P 
dS = 
dV

 ∂T V , nT
(5-2)
„ From the entropy and energy we can obtain by
simple integration all other relevant thermodynamic
properties
Adv. Thermo -Lecture 05: V-T Specified
4
1
Thermodynamic Properties
Required Information
(5-3)
m

 ∂P  
0
H = ∫ P − T 
  dV + PV + ∑ ni ui
 ∂T V ,nT 
i =1

V 
(5-4)
∞
 n R  ∂P  
V
+ ∑ ni si0
S = ∫ T −
  dV + R ∑ ni ln
ni RT i =1
 ∂T V ,nT 
i =1
 V
V 
∞
m
m
(5-5)
∞
m
m
n RT 
V

+ ∑ ni (ui0 − Tsi0 )
A = ∫ P − T
dV − RT ∑ ni ln
V 
n RT

i =1
V
i
i =1
(5-6)
∞
m
m
n RT 
V

G = ∫ P − T
dV − RT ∑ ni ln
+ PV + ∑ ni (ui0 − Tsi0 )
V 
ni RT
i =1
i =1
V 
∞
 ∂P 
RT 
V
 dV − RT ln
−
+ RT + ui0 − Tsi0

V 
ni RT
 ∂n
 i T ,V ,ni≠ j

(5-7)
µ i = ∫ 
(5-8)
∞
 ∂P 
RT 
 dV − RT ln z
RT ln φi = ∫ 
−

V 
∂ni T ,V ,n
V 
i≠ j


(5-9)
V
Adv. Thermo -Lecture 05: V-T Specified
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
∞
m

 ∂P  
0
U = ∫ P − T 
  dV + ∑ ni ui
 ∂T V ,nT 
i =1

V 
Volumetric behavior in the form of an equation
of state that is P-explicitP = P(T ,V , n)
„
(5-10)
In the P-T case we ended up requiring the
partial molar volume. In the V-T case, we end
up requiring  ∂P  which is not a partial molar
 ∂n 
quantity!
T ,V , ni ≠ j
6
Transforming to Density: Prelude
„ For a pure component the fugacity coefficient
„ It is desirable to work in terms of density (specific
(5-11)
„ Notice that this is not a convenient form
compared to the expression obtained from
the P-T description.
P
(4-12)
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
is given by:
Adv. Thermo -Lecture 05: V-T Specified
„
Adv. Thermo -Lecture 05: V-T Specified
Pure Component Limit
RT 
 f

RT ln φi = RT ln  
dP
= ∫ vi −
P 
 P  pure i 0 
thermodynamic properties require the
availability of volumetric behavior
i
5
∞
 P RT 
f
RT ln  
= ∫ −
 dV − RT ln z + RT ( z − 1)
P 
 P  pure i V  ni
„ All the equations obtained for the
volume) instead of the total volume of the mixture.
„ We carry out the transformation procedure to end up
with expressions containing derivatives instead of
integrals for the fugacity
„
„
„
Derivatives are easier than integrals.
Not all functions are analytically integrable.
To utilize this transformation, we need to have a model
for the Helmholtz free energy as a function of
temperature, pressure and density, A=A(T,ρ,x).
D. Dimitrelis, and J. M. Prausnitz, 1986, Fluid Phase Equilibria, 31: 1.
R. J. D. Topliss, D. Dimitrelis, and J. M. Prausnitz, 1988, Computers and Chemical Engineering, 12: 483.
7
Adv. Thermo -Lecture 05: V-T Specified
8
2
Transforming to Density: Definitions
„ Start with the definition of the compressibility
z=
P ( ρ , T , x)
ρ RT
(5-12)
„ Define the reduced molar residual Helmholtz
energy
ρ
A − AIG
Ar
z ( ρ , T , x) − 1
A =
dρ
=
=
ρ
nT RT
nT RT ∫0
(5-13)
Adv. Thermo -Lecture 05: V-T Specified
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
factor for a mixture as a function of the mole
(mass) fraction vector, temperature, and
pressure
9
(5-14)
10
Application: vdW EOS-Differentiation
(5-15)
„ The derivative of the residual Helmholtz free
energy with respect to the number of moles
of component i is
(5-16)
11
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
coefficient of component i becomes
Adv. Thermo -Lecture 05: V-T Specified
 DA 
 DA 
=



 Dxi  ρ ,T , x j  Dxi  ρ ,T , xi ≠ x j
„ Find an expression for the fugacity coefficient for the
„ The associated expression for the fugacity
m
 ∂ (nT A ) 
 DA 
 DA 
= A + 
− ∑ xj 




 Dxi  ρ ,T , x j j =1  Dxi  ρ ,T , xi
 ∂ni  ρ ,T ,ni≠ j
indicates that differentiation is carried out
with respect to xi while all other xj are held
constant.
Adv. Thermo -Lecture 05: V-T Specified
Transforming to Density: Derivation
 ∂ (n A ) 
+ ( z − 1) − ln z
ln φi =  T 
 ∂ni  ρ ,T ,ni≠ j
„ Define the differential operator D, which
van der Waal’s equation of state.
RT
a
−
P=
v − b v2
„
Transform the EOS from molar to total basis
P=
„
(5-17)
nT RT
n2 a
− T
V − nT b V 2
Differentiate the transformed EOS w.r.t. the
number of moles of component ni:
 ∂P 
RT
=
+


 ∂ni T ,V ,n j V − nT b
Adv. Thermo -Lecture 05: V-T Specified
 ∂(nT b) 
nT RT 

 ∂ni T ,V ,n j
(V − nT b)2
−
1  ∂ (nT2 a ) 


V 2  ∂ni T ,V ,n
j
12
3
Application: vdW EOS-Integration
„
Application: vdW EOS-Mixing Rules
∞
 ∂(nT b) 
V − nT b 
1 

− nT RT 

V V
 ∂ni T ,V ,n j V − nT b V
∞
RT ln φi = RT ln
∞
 ∂ (nT2 a) 
1
 − RT ln z
+

 ∂ni T ,V ,n j V V
„
Apply the integration limits
RT ln φi = RT ln
 ∂ (nT b) 
V
1
+ nT RT 

V − nT b
 ∂ni T ,V ,n V − nT b
j
 ∂(nT2 a ) 
1
−
− RT ln z

 ∂ni T ,V ,n j V
„
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
Substitute in the fugacity expression (Eq. 5-9)
and carry out the integration
Adv. Thermo -Lecture 05: V-T Specified
vRT
− ln z
(5-18)
15
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
m
j =1
m
i =1
„
The b term can be interpreted as a term
proportional to the strength of attraction
between two molecules.
m
m
14
Application: Analysis
Substitute the derivative of mixing rules in the
fugacity expression to obtain the final form of
the fugacity coefficient for the vdW EOS
fi
b
v
= ln
+ i −
yi P
v −b v −b
m
i =1
b1/ 3 = ∑ yi bi1/ 3 vs. b = ∑ yi bi
Adv. Thermo -Lecture 05: V-T Specified
Application: vdW EOS-Final Form
ln φi = ln
The b term can be interpreted as a term
proportional to the size of the molecule. Under
the assumption of molecules being spherical
i =1 j =1
13
2ai1/ 2 ∑ y j a1/j 2
„
a = ∑∑ yi y j aij , and aij = (aii a jj )1/ 2
Adv. Thermo -Lecture 05: V-T Specified
„
To proceed further we need to specify the
composition dependence of the parameters a
and b (mixing rules).
„ To calculate φ(T,P,x) in a mixture
„ Compute the constants a, and b for the
mixture using any proper mixing rule.
„ Using the constants a, and b and the EOS,
find the molar volume v of the mixture using
„
„
Analytical roots of the cubic equation, or
Numerical solutions.
Calculate the compressibility factor z from the
given T, and P and the calculated v.
„ Find the fugacity f or φ from the proper
expression (Eq. 5-18 for vdW EOS).
„
Adv. Thermo -Lecture 05: V-T Specified
16
4
Phase Equilibrium
Which EOS?
„ There are literally hundreds, if not thousands, of EOS
Adv. Thermo -Lecture 05: V-T Specified
„ The conditions for equilibria requires, thermal,
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
in the literature.
„ A rule of thumb says that the more complicated in
form an EOS is, and the more constants it contains,
the better it represents the PVT behavior of a pure
substance.
„ However, to predict the behavior of mixtures from
pure component data alone, the more constants and
mixing rules which are subject to much uncertainty
are involved in the computations.
„ Consequently, it happens that the use of simple EOS
(e.g. Cubic) performs better than multi-parameter,
complicated EOS.
17
mechanical and chemical equilibrium to be
established among different phases.
T (1) = T (2) = " = T (π )
P (1) = P ( 2) = " = P (π )
(3.17)
f i (1) = f i ( 2) = " = f i (π ) , i = 1," , m
„ Usually, we specify T and P, which leaves us with
finding a logical systematic way to find the fugacities
of each component in every phase.
Adv. Thermo -Lecture 05: V-T Specified
18
Concept Summary
Framework to (Fluid) Phase equilibria
„ Introduce the independent variables (T,V) and understand their
significance.
„ A framework for the solutions to the phase equilibria
„ Derive pressure effects on thermodynamic entropy and energy, and
„
„
„
Equilibrium conditions,
An equation of state.
An expression for the fugacity of species in any phase.
„ The fundamental difficulty is that we do not have an
EOS that is:
„
„
applicable to mixtures, and
performs satisfactorily from zero density (ideal gas
limit) to liquid densities.
Adv. Thermo -Lecture 05: V-T Specified
19
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©2003: Ali Al-Matar, Chemical Engineering Dept., University of Jordan
problem, can in principle be laid with a combination
of:
derive the rest of thermodynamic properties
„ Know the required information for the equations derived, and how to
obtain them
„ Apply logical reasoning and mathematical equations for the derivations
of expressions for the fugacity of the van der Waal’s EOS.
„ Be able to derive an expression for the fugacity from equations of
state. Implement these expressions to obtain physically meaningful
results.
„ Distinguish between different EOS and their limitations.
„ Lay a framework for the phase equilibria using the expressions of
fugacity obtained, and implement it.
„ As a side objective: apply their math and computer skills in obtaining
the roots of cubic or general nonlinear single equation.
Adv. Thermo -Lecture 05: V-T Specified
20
5