4. Cubic Equation of State¶
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class
scithermo.eos.cubic.Cubic(sigma: float, epsilon: float, Omega: float, Psi: float, dippr_no: str = None, compound_name: str = None, cas_number: str = None, **kwargs)[source]¶ Generic Cubic Equation of State Defined as in [GP07]
\[P = \frac{RT}{V - b} - \frac{a(T)}{(V + \epsilon b)(V + \sigma b)}\]where \(\epsilon\) and \(\sigma\) are pure numbers–the same for all substances. \(a(T)\) and \(b\) are substance-dependent.
- Parameters
R (float, hard-coded) – gas constant, set to SI units
sigma (float) – Parameter defined by specific equation of state, \(\sigma\)
epsilon (float) – Parameter defined by specific equation of state, \(\epsilon\)
Omega (float) – Parameter defined by specific equation of state, \(\Omega\)
Psi (float) – Parameter defined by specific equation of state, \(\Psi\)
tol (float) – tolerance for percent difference between compressilibity factor calculated iteratively, set to 0.01
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G_R_RT_expr(P, V, T, log=None)[source]¶ Dimensionless residual gibbs
\[\frac{G^\mathrm{R}}{RT} = Z - 1 + \ln(Z-\beta) - qI\]- Returns
Expression for residual gibbs free (divided by RT) – dimensionless
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H_R_RT_expr(P, V, T, log=None)[source]¶ Dimensionless residual enthalpy
\[\frac{H^\mathrm{R}}{RT} = Z - 1 + \left[\frac{\mathrm{d} \ln \alpha(T_\mathrm{r})}{\mathrm{d} \ln T_\mathrm{r}} - 1\right]qI\]- Returns
Expression for residual enthalpy (divided by RT) – dimensionless
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H_expr(P, V, T, T_ref, val_ref=0.0, log=None)[source]¶ Expression for fluid enthalpy
- Parameters
P – pressure in Pa
V – molar volume [m**3/mol]
T – temperautre in K
T_ref – reference temperature in K
val_ref – value at reference temperature [J/mol/K]
log – function used for logarithm
- Returns
\(H\) [J/mol/K]
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I_expr(P, V, T, log=None)[source]¶ - \[I = \frac{1}{\sigma - \epsilon}\ln{\left( \frac{Z + \sigma \beta}{Z + \epsilon \beta} \right)}\]
- Parameters
log (callable, optional) – function to be used for natural logarithm, defaults to np.log
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S_R_R_expr(P, V, T, log=None)[source]¶ Dimensionless residual entropy
\[\frac{S^\mathrm{R}}{R} = \ln(Z-\beta) + \frac{\mathrm{d} \ln \alpha(T_\mathrm{r})}{\mathrm{d} \ln T_\mathrm{r}} qI\]- Returns
Expression for residual entropy (divided by R) – dimensionless
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Z_liquid_RHS(Z, beta, q)[source]¶ Compressibility of vapor [GP07]
(1)¶\[\beta + (Z + \epsilon\beta)(Z + \sigma\beta)\left(\frac{1 + \beta - Z}{q\beta}\right)\]
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Z_vapor_RHS(Z, beta, q)[source]¶ Compressibility of vapor [GP07]
(2)¶\[1 + \beta - \frac{q\beta (Z - \beta)}{(Z + \epsilon\beta)(Z + \sigma\beta)}\]- Returns
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a_expr(T)[source]¶ - \[a(T) = \Psi \frac{\alpha(T_r)R^2T_{\mathrm{c}}^2}{P_{\mathrm{c}}}\]
- Parameters
T – temperautre in K
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alpha_expr(T_r)[source]¶ An empirical expression, specific to a particular form of the equation of state
- Parameters
T_r – reduced temperature (T/Tc), dimensionless
- Returns
\(\alpha (T_r)\)
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beta_expr(T, P)[source]¶ - \[\beta = \Omega\frac{P_\mathrm{r}}{T_\mathrm{r}}\]
- Parameters
T – temperature in K
P – pressure in Pa
- Returns
\(\beta\)
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cardano_constants(T, P)[source]¶ - Parameters
T – temperature [T]
P – pressure [Pa]
- Returns
cardano constants p, q
- Return type
tuple
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coefficients(T, P)[source]¶ Polynomial oefficients for cubic equation of state
\[Z^3 c_0 + Z^2c_1 + Z*c_2 + c_3 = 0\]- Returns
(c_0, c_1, c_2, c_3)
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d_ln_alpha_d_ln_Tr(T_r)[source]¶ - Parameters
T_r – reduced temperature [dimensionless]
- Returns
Expression for \(\frac{\mathrm{d} \ln \alpha(T_\mathrm{r})}{\mathrm{d} \ln T_\mathrm{r}}\)
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iterate_to_solve_Z(T, P, phase) → float[source]¶ - Parameters
T – temperature in K
P – pressure in Pa
phase (str) – phase [vapor or liquid]
- Returns
compressibility factor
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num_roots(T, P)[source]¶ Find number of roots
- Parameters
T – temperature in K
P – pressure in Pa
- Returns
number of roots
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plot_Z_vs_P(T, P_min, P_max, phase='vapor', symbol='o', ax=None, **kwargs)[source]¶ Plot compressibility as a function of pressure
- Parameters
T (float) – temperature [K]
P_min (float) – minimum pressure for plotting [Pa]
P_max (float) – maximum pressure for plotting [Pa]
phase (str) – phase type (liquid or vapor), defaults to vapor
symbol (str) – marker symbol, defaults to ‘o’
ax (plt.axis) – matplotlib axes for plotting, defaults to None
kwargs – keyword arguments for plotting
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class
scithermo.eos.cubic.RedlichKwong(**kwargs)[source]¶ Redlich-Kwong Equation of State [RK49]
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class
scithermo.eos.cubic.SoaveRedlichKwong(**kwargs)[source]¶ Soave Redlich-Kwong Equation of State [Soa72]
- Parameters
f_w (float, derived) – empirical expression used in \(\alpha\) [dimensionless?]