Equation of state is used to derive variety of thermodynamic properties. This article illustrate calculation of specific heat ratio from Peng Robinson Equation of state.
Calculate specific heat ratio ( γ = Cp/Cv ) for methane gas at 11 Bar & 300 °K. Critical constants for Methane are as following
- Critical temperature, Tc : 190.6°K
- Critical Pressure, Pc : 46.002 bar
- Accentric Factor, ω : 0.008
Ideal gas specific heat constants CpIG = A + B.T + C.T² + D.T³ are as following
- A = 4.5980
- B = 0.0125
- C = 2.86 x 10-6
- D = -2.7 x 10-9
where Cp is in cal/mol-K
Peng Robinson equation of state is defined as
P = RT / (V - b) - a / [V(V + b) + b(V - b)]
ac = 0.45723553 R²Tc²/Pc
b = 0.077796074 RTc/Pc
m = 0.37464 + 1.54226ω - 0.26992ω²
a = ac[1 + m(1 - (T/Tc)0.5)]²
Above equation is translated into polynomial form and solved for values of Z using Newton-Raphson method.
Z³ - (1 - B)Z² + Z (A - 2B - 3B²) - (AB - B² - B³) = 0
Z = PV/RT
A = aP/ (RT)²
B = bP/ RT
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Following partial derivatives are required for calculating thermodynamic properties. First derivative is obtained by differentiation of P with respect to V at constant T.
(δP/ δV)T = -RT/(v - b)² + 2a(v + b)/[v(v + b) + b(v - b)]²
(δP/ δV)T = -0.00485 bar/(cm3/mol)
Second derivative is obtained by differentiation of P with respect to T at constant V.
(δP/ δT)V = R/(v - b) - a'/[v(v + b) + b(v - b)]
(δa/ δT)V = -mac/[(TTc)0.5(1 + m( 1 - (T/Tc)0.5))]
(δP/ δT)V = 0.039 bar/K
(δT/ δP)V = 25.814 K/bar
Third derivative is obtained by differentiation of V with respect to T at constant P.
(δV/ δT)P = (R/P)[ T(δZ/δT)P + Z]
(δZ/ δT)P = Num / Denom
Num = (δA/δT)P (B-Z) + (δB/δT)P(6BZ+2Z-3B²-2B+A-Z²)
Denom = 3Z² + 2(B-1)Z + (A-2B-3B²)
(δA/δT)P = (P/(RT)²)(a' - 2a/T)
(δB/δT)P = -bP/(RT²)
Calculation of Heat Capacities
Ideal gas heat capacity CpIG is calculated at 300 °K from polynomial equation provided above. Specific heat at constant volume for ideal gas, CvIG is calculated using following relation.
CvIG = CpIG - R
Residual heat capacity at constant volume Cv R is calculated from internal energy U R as following.
CvR = (δUR/δT)V
UR = [(Ta'-a)/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]
CvR = [Ta"/b(8)0.5] ln[(Z+B(1+20.5))/(Z+B(1-20.5))]
a" = ac m (1 + m)(Tc/T)0.5/ (2TTc)
Specific heat capacity at constant pressure and volume is calculated using following equation.
CpR = CvR + T(δP/δT)V(δV/δT)P - R
Cp = CpIG + CpR
Cv = CvIG + CvR
Specific heat ratio is obtained as :
γ = Cp / Cv
γ = 1.338