University of Jordan Chemical Engineering Department Chemical Engineering Thermodynamics 905721
Dr. Ali Khalaf Al-Matar
[email protected]
Exam I Solution 20/11/2003
1
Derivation of the fugacity coefficient for the Peng-Robinson EOS
The derivation is similar to the derivation given in class for the van der Waal’s EOS. We need to get the equation of state in its mathematical form combined with its associated mixing rules. One of the tricks in the question is that regardless of the complexity of the combining rules given, the derivation is done once. This is a result of the combining rules being composition independent. Consequently, we need only to derive the expression for the fugacity coefficient once (not three times for every case). You can solve this problem using a simple or a hard approach. The simple approach includes using the residual Helmholtz free energy and its relation to fugacity. The hard approach involves the brute force calculation of the derivatives required within the volume integral for fugacity. I am solving using the residual Helmholtz free energy approach.
1.1
Obtain the EOS
The Peng-Robinson EOS is given as P =
RT a RT a − = − v − b v(v + b) + b(v − b) v − b v2 + 2vb − b2
(1)
The mixing rules to be used are a =
b =
m X m X i=1 j=1 m X m X
xi xj aij = xi xj bij =
i=1 j=1
m m 1 XX ni nj aij n2T i=1 j=1
m m 1 XX ni nj bij n2T i=1 j=1
(2)
(3)
With the cross parameters given by the appropriate combining rule. Notice that the formulation in the expressions for the a and b parameters is similar which reduces the amount of derivatives to be evaluated. 1
1.2
Convert the EOS to Density and Compressibility Notation
The equation of state to be used (regardless of its complexity) needs to be converted to the density (specific volume) and compressibility factor. Consequently, the Peng-Robinson EOS becomes Z=
1.3
1 1 aρ − 1 − bρ RT 1 + 2bρ − b2 ρ2
(4)
Obtain an Expression for Helmholtz Free Energy
We have an equation of state that we can apply for a pure component to obtain the residual Helmholtz free energy. At a constant temperature and volume we can use Zρ Zbρ (a − aIG ) Z −1 Z −1 = dρ = d(bρ) (5) RT ρ bρ 0
0
From the Peng-Robinson EOS Z −1 = =
1 1 − bρ aρ 1 − − 2 2 1 − bρ RT 1 + 2bρ − b ρ 1 − bρ 1 bρ aρ − 1 − bρ RT 1 + 2bρ − b2 ρ2
(6)
Substitute this expression into the residual Helmholtz free energy (a − aIG ) = RT
Zbρ 0
a 1 d(bρ) + 1 − bρ bRT
Zbρ 0
1 d(bρ) 1 + 2bρ − b2 ρ2
The first term to the right hand side of the equal sign is easy to integrate. However, the second can be obtained from the tables of integration formulas as ¯ ¯ Z ¯ 2a0 x + b0 − ¡b02 − 4a0 c0 ¢1/2 ¯ 1 1 ¯ ¯ dx = ln ¯ ¯ (7) 1/2 1/2 ¯ 02 0 0 0 0 02 0 0 ¯ a0 x2 + b0 x + c0 (b − 4a c ) 2a x + b + (b − 4a c )
This result applies when the discriminator is negative i.e., 4a0 c0 − b02 < 0 In our case we have a0 = −b2 , b0 = 2b, c0 = 1 Which upon substitution
√ √ ¢1/2 ¡ 02 = (8b2 )1/2 = 8b = 2 2b b − 4a0 c0 2
(8)
Substituting back and carrying out the first integral to get " # √ a (a − aIG ) 1 + (1 + 2)bρ √ ln = − ln(1 − bρ) − √ RT 8bRT 1 + (1 − 2)bρ
(9)
We can use the compressibility factor, and the A and B factors to reduce this equation into dimensionless form. Notice that the A factor is not the Helmholtz free energy to avoid confusion between symbols. " # √ (a − aIG ) Z + (1 + 2)B A √ ln (10) = Z − 1 − ln(Z − B) − √ RT 8B Z + (1 − 2)B
1.4
Obtain an Expression for the Fugacity Coefficient
Using the relationship between fugacity coefficient and residual Helmholtz free energy we have µ ¶ ∂ (A − AIG ) ln φi = − ln Z (11) ∂ni RT T,V,nj Consequently, we need to put the residual Helmholtz free energy into its extensive form by multiplying by the total number of moles ´ ³ ´i h ³ √ √ an2T (A − AIG ) ln 1 + (1 + 2)bρ − ln 1 + (1 − 2)bρ = −nT ln(1−bρ)− √ RT 8bRT nT (12) Carry out the differentiation with respect to the number of moles of any arbitrary component µ ¶ ¶ µ ∂(A − AIG )/RT n ∂(bρ) = − ln(1 − bρ) + (13) ∂ni (1 − bρ) ∂ni T,V,nj √ ³ ∂(bρ) ´ √ ³ ∂(bρ) ´ 2 2) 2) ∂ni (1 + (1 − ∂ni anT √ √ − −√ 8bRT nT 1 + (1 + 2)bρ 1 + (1 − 2)bρ ´ ³ # ³ ∂(an2T ) ´ " ∂(nT b) √ 2 ∂ni ∂ni an 1 + (1 + 2)bρ √ √ −√ T − ln 2 (n 1 + (1 − 2)bρ 8bRT nT 8RT T b) This contribution can be evaluated if the derivatives with respect to the number of moles are obtain. The next section derives general expressions for the quadratic mixing rules.
3
1.5
Obtain Derivatives of the Mixing Rules
The following derivatives are required to be obtained from the mixing rules # # " " m m P m m P P P ni nj aij ni nj aij ∂ ∂ ¶ µ i=1 j=1 i=1 j=1 ∂(n2T a) = = ∂ni ∂ni ∂ni nj µ
∂(n2T a) ∂ni
nj
¶
= 2
nj
m X
nj
nk aik .
(14)
k=1
Also, µ
∂(nT b) ∂ni
¶
"
nj
∂ = =
=
µ
∂(nT b) ∂ni
¶
∂ 1 nT
nj
#
m P m P
ni nj bij ∂ni
i=1 j=1
"
k=1
ni nj bij ∂ni
i=1 j=1
"
m X
#
m m P P
∂ 1 nT
= 2 nj
1 nT
−
m m 1 XX ni nj bij n2T i=1 j=1
nj
m P
i=1
#
m P
ni
nj bij ∂ni j=1
−b
nj
xk bik − b.
(15)
For a single summation: µ
∂(nT b) ∂ni
¶
nj
=
∂(
m P
ni bi )
i=1
∂ni
nj
4
= bi
(16)
1.6
Plug into the Fugacity Coefficient Expression
The derivatives are ready to be plugged into the fugacity coefficient expressions as follows ¶ µ m P xk bik − b ρ 2 k=1 ln φi = − ln Z − ln(1 − bρ) + (1 − bρ) ¶ µ m P # √ √ xk bik − b " ρa 2 (1 − 2) (1 + 2) k=1 √ √ √ − − 8bRT 1 + (1 + 2)bρ 1 + (1 − 2)bρ ¶ µ m P m P # " √ 2 xk aik xk bik − b 2 a 1 + (1 + 2)bρ k=1 k=1 √ √ √ − − ln b 1 + (1 − 2)bρ 8bRT 8bRT Simplifying this equation further by taking a common factor in the last term to obtain ¶ µ m P xk bik − b ρ 2 k=1 ln φi = − ln Z − ln(1 − bρ) + (1 − bρ) ¶ µ m P # √ √ xk bik − b " ρa 2 (1 + 2) (1 − 2) k=1 √ √ √ − − 8bRT 1 + (1 + 2)bρ 1 + (1 − 2)bρ ¶ µ m m P " # 2 P xk aik √ xk bik − b 2 a 1 + (1 + 2)bρ k=1 k=1 √ ln − −√ a b 8bRT 1 + (1 − 2)bρ To simplify the notation further define à m ! X bi = 2 xk bik − b
(17)
k=1
ai
=
m X
xk aik
k=1
From which ln φi
bi ρ = − ln Z − ln(1 − bρ) + (1 − bρ) " # √ √ ρabi (1 + 2) (1 − 2) √ √ −√ − 8bRT 1 + (1 + 2)bρ 1 + (1 − 2)bρ " #· √ ¸ a 1 + (1 + 2)bρ 2ai bi √ −√ ln − a b 8bRT 1 + (1 − 2)bρ 5
(18)
There is another simplification that can be carried out " # √ √ (1 − 2) ρabi bi ρ (1 + 2) √ √ − −√ (1 − bρ) 8bRT 1 + (1 + 2)bρ 1 + (1 − 2)bρ µ ¶¸ · bi 1 bi bρ ρab = = (Z − 1) −√ b (1 − bρ) b 8bRT 1 + 2bρ − b2 ρ2 Consequently, ln φi
bi ρ bi = − ln Z − ln(1 − bρ) + + (Z − 1) (1 − bρ) b " #· √ ¸ 1 + (1 + 2)bρ 2ai bi a √ ln − . −√ a b 8bRT 1 + (1 − 2)bρ
(19)
To have a more compact notation, define the variables in terms of the reducing variables A, B, and Z as follows bρ =
B ; Z
a A = ; bRT B
ai Ai = ; a A
bi Bi = . b B
Applying these transformations, we end up with the desired expression for the fugacity coefficient using the Peng-Robinson equation of state. This expression is derived for quadratic mixing rules for the co-volume and the energy parameters. It simplifies a little computationally if we use arithmetic averages for the covolume. " #· √ ¸ Bi Z + (1 + 2)B 2Ai A Bi √ ln φi = − ln(Z−B)+ (Z−1)− √ ln − . (20) B A B 8B Z + (1 − 2)B where B
=
Bi
=
bP ; RT ¶ µ m P xk bik − b P 2 k=1
RT RTc b = 0.07780 Pc aP A = ; (RT )2 ¶ µm P xk aik P k=1 Ai = ; (RT )2 (RTc )2 α(T ); a(T ) = 0.45724 Pc 6
(21)
(22) (23) (24)
(25) (26)
à r ! p T α(T ) = 1 + κ 1 − ; Tc
(27)
κ = 0.37464 + 1.5422ω − 0.26992ω 2 .
2
(28)
Algorithm for using the Fugacity Expression
The expression for the fugacity coefficient and its associated variables and equations needs to be solved to obtain a value for the fugacity. This is not hard computationally. However, one of the main points encountered frequently is the solution for the roots of a cubic equation. The provided notes are implemented in the Excel worksheet to obtain the roots of the cubic at any given temperature and pressure.
3
Mixing Rules
Three mixing rules were used to solve the exam statement. • Lorentz-Berthelot
1 (bii + bjj ), 2
(29)
aij = (aii ajj )1/2 .
(30)
bij =
• Waldman-Hagler bij =
Ã
b2ii + b2jj 2 Ã 1/2
aij = (aii ajj ) • MADAR-1
2 1X bij = 3
Ã
!1/2 bii bjj b2ij
, !
0.25 (bii + bjj )2 L/3 L/3
(31)
.
1 ! 2−2L/3
bii bjj à ! bii bjj 1/2 aij = (aii ajj ) . b2ij L=0
4
(32)
(33)
(34)
Isobutane and Carbon Dioxide System
The exam asked to generate plots for the fugacity as a function of composition and pressure for the system: isobutane and carbon dioxide. The required input information are given below for the two components. The fugacity coefficient as obtained from the Peng-Robinson equation of state was evaluated as a function of pressure and composition using Microsoft 7
Table 1: Critical parameters, and the acentric factors of i-butane. Component Tc (K) Pc (MPa) ω (-) Carbon dioxide 304.14 7.375 0.239 Isobutane 407.8 3.604 0.183
carbon dioxide and Zc (-) 0.274145 0.275296
Excel. The Excel worksheet is obtained and modified from the textbook of Elliot and Lira. Figures 1 through 3 present the final plots of fugacity for isobutane and carbon dioxide respectively as a function of composition and pressure. Figure 1 is plotted using the Lorentz-Berthelot set of mixing rules, Figure 2 is plotted using the Waldman-Hagler, while Figure 3 is plotted using the MADAR-1 set of mixing rules. Additionally, Figure 4 is plotted to show the effect of mixing rules on the same graph at pressures of 1, 2, and 4 MPa.
4.1
Pure Component Limit
From the first three figures, it is evident that at the pure component limit, the fugacity of CO2 approaches the pressure of the system. This is to be expected since CO2 is supercritical at the temperature of the system. However, for the isobutane it seems that the fugacity is almost an order of magnitude lower than the system pressure. Isobutane being a liquid at the given temperature explains the low fugacity of at the pure component limit. Furthermore, there is a linear composition dependence at any given pressure as the pure component limit is approached.
4.2
Influence of System Pressure
The pressure affects the fugacity of both components appreciably. However, the influence of the pressure may be divided into two main regions: • At pressures below approximately 1 MPa, the fugacity is a linear function of the composition in all the composition range. • At pressures above approximately 1 MPa, the fugacity begins to show some sort of going through a maximum, rising again, going through a minimum, then it begins to increase again. This is due to the mixture being a vapor approaching its dew pressure at compositions rich in CO2 that is being enriched with isobutane tending to lower the fugacity. Consequently, condensing the mixture to a liquid phase at isobutane rich areas. At very low pressures, the ideal gas mixture limit is approached. This leads to the fugacity of any component being equal to its partial pressure. This happens at a pressure of 0.001 MPa which is sufficiently low to guarantee the application of the ideal gas mixture limit. This pressure was given to check
8
the results obtained from the Excel sheet. The ideal gas mixture and pure component limits are useful safeguards for checking and debugging the codes used.
4.3
Effect of Different Mixing Rules
The figures provided; indicate that there is an effect of the mixing rules used. There are minor differences between the Waldman-Hagler and MADAR-1 rules. However, both rules feature some major differences with the Lorentz-Berthelot rules. These differences increase close to regions where there is a liquid phase. The Lorentz-Berthelot rules provides lower values of the fugacity compared to the Waldman-Hagler and MADAR-1 rules. This may be explained by the overestimation of repulsive forces using the Lorentz-Berthelot rules. The differences between the Lorentz-Berthelot rules and the two other rules is quantitative and qualitative. The trends predicted and locations of phase changes are contradictory among these rules. These differences increase with the pressure i.e., increase when the Lorentz-Berthelot rules results are not as good as the gas limits. The difference in the values of fugacity amounts to threefold difference. Consider P = 4 MPa, the maximum fugacity of isobutane predicted by the Lorentz-Berthelot rules is about 0.2 MPa at x = 0.15, while that predicted by Waldman-Hagler and MADAR-1 rules is 0.57 MPa at x = 0.37! Additionally, the Lorentz-Berthelot rules show an almost constant value of carbon dioxide fugacity over the region where minimum and maximum fugacities in isobutane fugacity occurs. This is some sort of an inconsistency compared to the inverted trends predicted by the Waldman-Hagler and MADAR-1 rules.
9
0.8
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
Fugacity of isobutane (MPa)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole fraction of isobuatne
Fugacity of carbon dioxide (MPa)
3.5
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole fraction of isobutane
Figure 1: Fugacities of isobuatne and carbon dioxide as a function of composition and pressure using the Lorentz-Berthelot mixing rules.
10
0.8
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
Fugacity of isobutane (MPa)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.7
0.8
0.9
1.0
Mole fraction of isobuatne
Fugacity of carbon dioxide (MPa)
4
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
3
2
1
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction of isobutane
Figure 2: Fugacities of isobuatne and carbon dioxide as a function of composition and pressure using the Waldman-Hagler mixing rules.
11
0.8
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
Fugacity of isobutane (MPa)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.7
0.8
0.9
1.0
Mole fraction of isobuatne
Fugacity of carbon dioxide (MPa)
4
0.001 MPa 0.100 MPa 1.000 MP 2.000 MPa 4.000 MPa
3
2
1
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction of isobutane
Figure 3: Fugacities of isobuatne and carbon dioxide as a function of composition and pressure using the MADAR-1 mixing rules.
12
0.8
L-B, 1 MPa L-B, 2 MPa L-B, 4 MPa W-H, 1 MPa W-H, 2 MPa W-H, 4 MPa MADAR-1, 1 MPa MADAR-1, 2 MPa MADAR-1, 4 MPa
Fugacity of isobutane (MPa)
0.6
0.4
0.2
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.7
0.8
0.9
1.0
Mole fraction of isobutane
Fugacity of carbon dioxide (MPa)
4
L-B, 1 MPa L-B, 2 MPa L-B, 4 MPa W-H, 1 MPa W-H, 2 MPa W-H, 4 MPa MADAR-1, 1 MPa MADAR-1, 2 MPa MADAR-1, 4 MPa
3
2
1
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction of isobutane
Figure 4: The effect of different mixing rules on the fugacity of isobutane and carbon dioxide at pressures of 1, 2, and 4 MPa.
13
