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Equations of State: Auxiliary Equations for Programming
Appendix C
Equations of State: Auxiliary Equations for Programming For some popular equations of state, we present auxiliary information to facilitate programming. Conventions: l
l
All equations of state for real gases have at least two parameters, a characteristic volume (size parameter, covolume, molecular size), v ∗ , and a characteristic temperature, T ∗ = /kB , where denotes a pair potential well depth or similar (energetic) measure of the attraction between pairs of molecules. We write all equations in terms of reduced (dimensionless) densities, ξ = v ∗ /Vm , and reduced temperatures, T˜ = T/T ∗ .
C.1 THE van der WAALS EQUATION OF STATE The original van der Waals attraction parameter is identified with avdW = 2πv ∗ RT ∗ . 1 2π − ξ 1−ξ T˜ Arm 2π = − ln(1 − ξ ) − ξ RT T˜ Z=
(C.1) (C.2)
Cubic polynomial needed for inversion: ˜ − p˜ = 0 2πξ 3 − 2πξ 2 + (˜p + T)ξ pv ∗ with p˜ = RT ∗
(C.3)
Critical point: T˜ c = 1.861685, ξc = 0.333333, Zc = 0.375.
High-Pressure Fluid Phase Equilibria. DOI: 10.1016/B978-0-444-56347-7.00012-8 Copyright © 2012 Elsevier BV. All rights reserved.
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APPENDIX | C Equations of State: Auxiliary Equations for Programming
C.2 THE REDLICH–KWONG EQUATION OF STATE The original attraction parameter is identified with aRK = 8v ∗ R(T ∗ )3/2 . The factor 8 is arbitrary; this choice ensures that kB T ∗ is approximately equal to the Lennard-Jones energy parameter. Z=
8ξ −3/2 1 T˜ − 1−ξ 1+ξ
(C.4)
Arm = − ln(1 − ξ ) − 8 ln(1 + ξ )T˜ −3/2 RT
(C.5)
Cubic polynomial needed for inversion: ˜ − p˜ = 0 8T˜ −1/2 ξ 3 + (T˜ + p˜ − 8T˜ −1/2 )ξ 2 + Tξ pv ∗ with p˜ = RT ∗
(C.6)
Critical point: T˜ c = 1.380160, ξc = 0.259921, Zc = 0.333333.
C.3 THE REDLICH–KWONG–SOAVE EQUATION OF STATE The original attraction parameter is identified with ac = 8v ∗ RT ∗ ; again, the constant 8 is arbitrary. Z=
˜ 1 8ξ α(T) − 1 − ξ 1 + ξ T˜
˜ Arm α(T) = − ln(1 − ξ ) − 8 ln(1 + ξ ) RT T˜
(C.7) (C.8)
Cubic polynomial needed for inversion: 8ξ 3 + (τ + p˜ − 8)ξ 2 + τ ξ − p˜ = 0 pv ∗ T˜ with p˜ = ,τ= ∗ ˜ ˜ RT α(T) α(T)
(C.9)
Critical point: T˜ c = 1.380160, ξc = 0.259921, Zc = 0.333333.
C.4 THE PENG–ROBINSON EQUATION OF STATE The original attraction parameter is identified with ac = 8v ∗ RT ∗ ; the factor 8 is arbitrary. Z=
˜ 1 8ξ α(T) − 1 − ξ 1 + 2ξ − ξ 2 T˜
(C.10)
APPENDIX | C Equations of State: Auxiliary Equations for Programming
√ √ ˜ Arm 1 + ξ( 2 + 1) α(T) = − ln(1 − ξ ) − 2 2 ln √ RT 1 − ξ( 2 − 1) T˜
303
(C.11)
Cubic polynomial needed for inversion: (8 − τ − p˜ )ξ 3 + (2τ + 3˜p − 8)ξ 2 + (τ − p˜ )ξ − p˜ = 0 pv ∗ T˜ with p˜ = ,τ= ˜ ˜ RT ∗ α(T) α(T)
(C.12)
Critical point: T˜ c = 1.361155, ξc = 0.253077, Zc = 0.307401.
C.5 THE CARNAHAN–STARLING–van der WAALS EQUATION OF STATE The van der Waals attraction parameter is identified with avdW = 8v ∗ RT ∗ , where f is an arbitrary numerical constant. 4ξ − 2ξ 2 8 − ξ (1 − ξ )3 T˜ Arm 4ξ − 3ξ 2 8 =− − ξ RT (1 − ξ )2 T˜
Z =1+
(C.13) (C.14)
Fifth-order polynomial needed for inversion: ˜ + p˜ = 0 −8ξ 5 + (T˜ + 24)ξ 4 − (T˜ + p˜ + 24)ξ 3 + (3˜p − T˜ + 8)ξ 2 − (3˜p + T)ξ pv ∗ with p˜ = RT ∗ (C.15) Critical point: T˜ c = 0.754630, ξc = 0.130444, Zc = 0.358956.
C.6 THE SIMPLIFIED PERTURBED–HARD–CHAIN EQUATION OF STATE Z =1+c
4ξ − 2ξ 2 18cξ f − (1 − ξ )3 1 + ξ f (C.16)
1 1 with f = exp −1 , τ 2T˜
π√ τ= 2 6
Arm 4ξ − 3ξ 2 =c − 18c ln(1 + ξ f ) RT (1 − ξ )2
(C.17)
304
APPENDIX | C Equations of State: Auxiliary Equations for Programming
Fifth-order polynomial needed for inversion: ˜ ξ 5 + T(3f ˜ −56cf −1) + p˜ f ξ 4 + T(−3f ˜ (18c −1)Tf + 58cf − 2c + 3) 3 ˜ +˜p(1−3f )) ξ + T(4c − 3 + f − 18cf ) + 3˜p(f − 1) ξ 2 (C.18) + T˜ + p˜ (3 − f ) ξ − p˜ = 0 The roots of this equation can then be calculated with a suitable polynomial root finder. But the application of a general nonlinear equation solver to Eq. (C.16) might be more efficient. The location of the critical point depends on the parameter c in a complicated way. The following expressions are merely approximations: T˜ c ≈ 1.253540 − 0.4642899(c − 1), ξc ≈ 0.157179 − 0.033608(c − 1), Zc ≈ 0.348018 − 0.001302(c − 1).
Equations of State: Auxiliary Equations for Programming For some popular equations of state, we present auxiliary information to facilitate programming. Conventions: l
l
All equations of state for real gases have at least two parameters, a characteristic volume (size parameter, covolume, molecular size), v ∗ , and a characteristic temperature, T ∗ = /kB , where denotes a pair potential well depth or similar (energetic) measure of the attraction between pairs of molecules. We write all equations in terms of reduced (dimensionless) densities, ξ = v ∗ /Vm , and reduced temperatures, T˜ = T/T ∗ .
C.1 THE van der WAALS EQUATION OF STATE The original van der Waals attraction parameter is identified with avdW = 2πv ∗ RT ∗ . 1 2π − ξ 1−ξ T˜ Arm 2π = − ln(1 − ξ ) − ξ RT T˜ Z=
(C.1) (C.2)
Cubic polynomial needed for inversion: ˜ − p˜ = 0 2πξ 3 − 2πξ 2 + (˜p + T)ξ pv ∗ with p˜ = RT ∗
(C.3)
Critical point: T˜ c = 1.861685, ξc = 0.333333, Zc = 0.375.
High-Pressure Fluid Phase Equilibria. DOI: 10.1016/B978-0-444-56347-7.00012-8 Copyright © 2012 Elsevier BV. All rights reserved.
301
302
APPENDIX | C Equations of State: Auxiliary Equations for Programming
C.2 THE REDLICH–KWONG EQUATION OF STATE The original attraction parameter is identified with aRK = 8v ∗ R(T ∗ )3/2 . The factor 8 is arbitrary; this choice ensures that kB T ∗ is approximately equal to the Lennard-Jones energy parameter. Z=
8ξ −3/2 1 T˜ − 1−ξ 1+ξ
(C.4)
Arm = − ln(1 − ξ ) − 8 ln(1 + ξ )T˜ −3/2 RT
(C.5)
Cubic polynomial needed for inversion: ˜ − p˜ = 0 8T˜ −1/2 ξ 3 + (T˜ + p˜ − 8T˜ −1/2 )ξ 2 + Tξ pv ∗ with p˜ = RT ∗
(C.6)
Critical point: T˜ c = 1.380160, ξc = 0.259921, Zc = 0.333333.
C.3 THE REDLICH–KWONG–SOAVE EQUATION OF STATE The original attraction parameter is identified with ac = 8v ∗ RT ∗ ; again, the constant 8 is arbitrary. Z=
˜ 1 8ξ α(T) − 1 − ξ 1 + ξ T˜
˜ Arm α(T) = − ln(1 − ξ ) − 8 ln(1 + ξ ) RT T˜
(C.7) (C.8)
Cubic polynomial needed for inversion: 8ξ 3 + (τ + p˜ − 8)ξ 2 + τ ξ − p˜ = 0 pv ∗ T˜ with p˜ = ,τ= ∗ ˜ ˜ RT α(T) α(T)
(C.9)
Critical point: T˜ c = 1.380160, ξc = 0.259921, Zc = 0.333333.
C.4 THE PENG–ROBINSON EQUATION OF STATE The original attraction parameter is identified with ac = 8v ∗ RT ∗ ; the factor 8 is arbitrary. Z=
˜ 1 8ξ α(T) − 1 − ξ 1 + 2ξ − ξ 2 T˜
(C.10)
APPENDIX | C Equations of State: Auxiliary Equations for Programming
√ √ ˜ Arm 1 + ξ( 2 + 1) α(T) = − ln(1 − ξ ) − 2 2 ln √ RT 1 − ξ( 2 − 1) T˜
303
(C.11)
Cubic polynomial needed for inversion: (8 − τ − p˜ )ξ 3 + (2τ + 3˜p − 8)ξ 2 + (τ − p˜ )ξ − p˜ = 0 pv ∗ T˜ with p˜ = ,τ= ˜ ˜ RT ∗ α(T) α(T)
(C.12)
Critical point: T˜ c = 1.361155, ξc = 0.253077, Zc = 0.307401.
C.5 THE CARNAHAN–STARLING–van der WAALS EQUATION OF STATE The van der Waals attraction parameter is identified with avdW = 8v ∗ RT ∗ , where f is an arbitrary numerical constant. 4ξ − 2ξ 2 8 − ξ (1 − ξ )3 T˜ Arm 4ξ − 3ξ 2 8 =− − ξ RT (1 − ξ )2 T˜
Z =1+
(C.13) (C.14)
Fifth-order polynomial needed for inversion: ˜ + p˜ = 0 −8ξ 5 + (T˜ + 24)ξ 4 − (T˜ + p˜ + 24)ξ 3 + (3˜p − T˜ + 8)ξ 2 − (3˜p + T)ξ pv ∗ with p˜ = RT ∗ (C.15) Critical point: T˜ c = 0.754630, ξc = 0.130444, Zc = 0.358956.
C.6 THE SIMPLIFIED PERTURBED–HARD–CHAIN EQUATION OF STATE Z =1+c
4ξ − 2ξ 2 18cξ f − (1 − ξ )3 1 + ξ f (C.16)
1 1 with f = exp −1 , τ 2T˜
π√ τ= 2 6
Arm 4ξ − 3ξ 2 =c − 18c ln(1 + ξ f ) RT (1 − ξ )2
(C.17)
304
APPENDIX | C Equations of State: Auxiliary Equations for Programming
Fifth-order polynomial needed for inversion: ˜ ξ 5 + T(3f ˜ −56cf −1) + p˜ f ξ 4 + T(−3f ˜ (18c −1)Tf + 58cf − 2c + 3) 3 ˜ +˜p(1−3f )) ξ + T(4c − 3 + f − 18cf ) + 3˜p(f − 1) ξ 2 (C.18) + T˜ + p˜ (3 − f ) ξ − p˜ = 0 The roots of this equation can then be calculated with a suitable polynomial root finder. But the application of a general nonlinear equation solver to Eq. (C.16) might be more efficient. The location of the critical point depends on the parameter c in a complicated way. The following expressions are merely approximations: T˜ c ≈ 1.253540 − 0.4642899(c − 1), ξc ≈ 0.157179 − 0.033608(c − 1), Zc ≈ 0.348018 − 0.001302(c − 1).