Document 6520963

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Document 6520963

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Outline
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aalmatar@ju.edu.jo ‫ 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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aalmatar@ju.edu.jo

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
However, the majority of equations of state
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
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
∞
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)
is given by:

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
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)
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
coefficient of component i becomes
 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.
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
 ∂(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

Substitute in the fugacity expression (Eq. 5-9)
and carry out the integration
vRT
− ln z
(5-18)
15
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
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

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).

16
4
Phase Equilibrium
Which EOS?
There are literally hundreds, if not thousands, of EOS
The conditions for equilibria requires, thermal,
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.
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.
19
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.
20
5

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