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wrapped doxygen comment for thermal conductivity to 88 characters
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wandadars committed Aug 18, 2024
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151 changes: 83 additions & 68 deletions include/cantera/transport/HighPressureGasTransport.h
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Expand Up @@ -64,32 +64,33 @@ class HighPressureGasTransport : public MixTransport
* marked as such. The equations referenced here are the same as the ones from
* the papers from Ely and Hanley.
*
* The thermal conductivity is assumed to have two components: a translational/collisional
* part and an internal part that is related to the transfer of energy to internal degrees
* of freedom.
* The thermal conductivity is assumed to have two components: a
* translational/collisional part and an internal part that is related to the
* transfer of energy to internal degrees of freedom.
*
* @f[
* \lambda_{mix}(\rho, T) = \lambda^{''}_{mix}(T) + \lambda^{'}_{mix}(\rho, T)
* @f]
*
* The first term on the right-hand side is the internal part and the second term is the
* translational part. The internal part is only a function of temperature, and the
* translational part is a function of both temperature and density.
* The first term on the right-hand side is the internal part and the second term
* is the translational part. The internal part is only a function of temperature,
* and the translational part is a function of both temperature and density.
*
* For the internal component of the thermal conductivity the following equations are
* used.
* For the internal component of the thermal conductivity the following equations
* are used.
*
* @f[
* \lambda^{''}_{mix}(T) = \sum_i \sum_j X_i X_j \lambda^{''}_{ij}(T)
*
* \quad \text{(Equation 2)}
* @f]
*
* The mixing rule for combining pure-species internal thermal conductivity components
* is given by:
* The mixing rule for combining pure-species internal thermal conductivity
* components is given by:
*
* @f[
* (\lambda^{''}_{ij})^{-1} = 2[(\lambda^{''}_{i})^{-1} + (\lambda^{''}_{j})^{-1}]
* (\lambda^{''}_{ij})^{-1} = 2[(\lambda^{''}_{i})^{-1}
* + (\lambda^{''}_{j})^{-1}]
*
* \quad \text{(Equation 3)}
* @f]
Expand All @@ -103,34 +104,36 @@ class HighPressureGasTransport : public MixTransport
* @f]
*
* Where @f$ \eta_i^* @f$ is the referred to as the dilute gas viscosity and is the
* component that is temperature dependent described in @cite ely-hanley1981. @f$ MW_i @f$
* is the molecular weight of species i, @f$ f_int @f$ is a constant and is 1.32, @f$ C_{p,i}^0 @f$
* is the ideal gas heat capacity of species i, and R is the gas constant.
* component that is temperature dependent described in @cite ely-hanley1981.
* @f$ MW_i @f$ is the molecular weight of species i, @f$ f_int @f$ is a constant
* and is 1.32, @f$ C_{p,i}^0 @f$ is the ideal gas heat capacity of species i,
* and R is the gas constant (J/kmol/K).
*
* For the translational component of the thermal conductivity the following equations are
* used.
* For the translational component of the thermal conductivity the following
* equations are used.
*
* @f[
* \lambda^{'}_{mix}(\rho, T) = \lambda^{'}_{0}(\rho_0, T_0) F_{\lambda}
*
* \quad \text{(Equation 5)}
* @f]
*
* Where @f$ \lambda^{'}_{0}(\rho_0, T_0) @f$ is the internal component of the thermal
* conductivity of a reference fluid that is at a reference temperature and density,
* @f$ T_0 @f$ and @f$ \rho_0 @f$. These reference conditions are not the same as the
* conditions that the mixture is at (@f$ T @f$ and @f$ \rho @f$). The subscript 0 refers
* to the reference fluid.
* Where @f$ \lambda^{'}_{0}(\rho_0, T_0) @f$ is the internal component of the
* thermal conductivity of a reference fluid that is at a reference temperature
* and density, @f$ T_0 @f$ and @f$ \rho_0 @f$. These reference conditions are not
* the same as the conditions that the mixture is at (@f$ T @f$ and @f$ \rho @f$).
* The subscript 0 refers to the reference fluid.
*
* @f[
* F_{\lambda} = \left( \frac{MW_0}{MW_m} \right)^{1/2} f_{m,0}^{1/2} h_{m,0}^{-2/3}
* F_{\lambda} = \left( \frac{MW_0}{MW_m} \right)^{1/2} f_{m,0}^{1/2}
* h_{m,0}^{-2/3}
*
* \quad \text{(Equation 6)}
* @f]
*
* Where @f$ MW_0 @f$ is the molecular weight of the reference fluid, @f$ MW_m @f$ is the
* molecular weight of the mixture, @f$ f_{m,0} @f$ and @f$ h_{m,0} @f$ are called
* reducing ratios. The @f$ h_{m,0} @f$ quantity is defined as:
* Where @f$ MW_0 @f$ is the molecular weight of the reference fluid, @f$ MW_m @f$
* is the molecular weight of the mixture, @f$ f_{m,0} @f$ and @f$ h_{m,0} @f$ are
* called reducing ratios. The @f$ h_{m,0} @f$ quantity is defined as:
*
* @f[
* h_{m,0} = \sum_i \sum_j X_i X_j h_{ij}
Expand Down Expand Up @@ -161,24 +164,26 @@ class HighPressureGasTransport : public MixTransport
* factor @f$ \phi_i @f$ is defined in @cite ely-hanley1981 as follows:
*
* @f[
* \phi_i(T_R^i, V_R^i, \omega_i) = \frac{Z_0^c}{Z_i^c} [1 + (\omega_i - \omega_0)G(T_R^i, V_R^i)]
* \phi_i(T_R^i, V_R^i, \omega_i) = \frac{Z_0^c}{Z_i^c} [1 + (\omega_i -
* \omega_0)G(T_R^i, V_R^i)]
*
* \quad \text{(Equation 12 of @cite ely-hanley1981)}
* @f]
*
* Where @f$ Z_0^c @f$ is the critical compressibility of the reference fluid, @f$ Z_i^c @f$
* is the critical compressibility of species i, @f$ \omega_0 @f$ is the acentric factor of
* the reference fluid, and @f$ G(T_R^i, V_R^i) @f$ is a function of the reduced temperature
* and reduced volume of species i. The function @f$ G(T_R^i, V_R^i) @f$ is defined as:
* Where @f$ Z_0^c @f$ is the critical compressibility of the reference fluid,
* @f$ Z_i^c @f$ is the critical compressibility of species i, @f$ \omega_0 @f$
* is the acentric factor of the reference fluid, and @f$ G(T_R^i, V_R^i) @f$ is a
* function of the reduced temperature and reduced volume of species i. The
* function @f$ G(T_R^i, V_R^i) @f$ is defined as:
*
* @f[
* G(T_R^i, V_R^i) = a_2(V_i^+ + b_2) + c_2(V_i^+ + d_2)ln(T_R^i)
*
* \quad \text{(Equation 14 of @cite ely-hanley1981)}
* @f]
*
* Where @f$ T_i^+ @f$ and @f$ V_i^+ @f$ are limited values of the reduced temperature
* and pressure and are defined as:
* Where @f$ T_i^+ @f$ and @f$ V_i^+ @f$ are limited values of the reduced
* temperature and pressure and are defined as:
*
* @f[
* T_i^+ = min(2, max(T_R^i, 0.5))
Expand Down Expand Up @@ -212,8 +217,8 @@ class HighPressureGasTransport : public MixTransport
* @f]
*
*
* The value of @f$ h_{ij} @f$ is the same as the one defined earlier. The combining
* rule for @f$ f_{ij} @f$ is given by:
* The value of @f$ h_{ij} @f$ is the same as the one defined earlier. The
* combining rule for @f$ f_{ij} @f$ is given by:
*
* @f[
* f_{ij} = (f_i f_j)^{1/2} (1-\kappa_{ij})
Expand All @@ -222,39 +227,40 @@ class HighPressureGasTransport : public MixTransport
* @f]
*
* @f$ \kappa_{ij} @f$ is binary interaction parameters and is assumed to be zero
* as was done in the work of Ely and Hanley. The species-specific value of @f$ f_i @f$
* is defined as:
* as was done in the work of Ely and Hanley. The species-specific value of
* @f$ f_i @f$ is defined as:
*
* @f[
* f_i = \frac{T_c^i}{T_c^0} \theta_i(T_R^i, V_R^i, \omega_i)
*
* \quad \text{(Equation 12 @cite ely-hanley1981)}
* @f]
*
* Where @f$ \theta_i @f$ is a shape factor of Leach and Leland, @f$ T_c^i @f$ is the
* critical temperature of species i, and @f$ T_c^0 @f$ is the critical temperature of
* the reference fluid. The shape factor @f$ \theta_i @f$ is defined in @cite ely-hanley1981
* as:
* Where @f$ \theta_i @f$ is a shape factor of Leach and Leland, @f$ T_c^i @f$ is
* the critical temperature of species i, and @f$ T_c^0 @f$ is the critical
* temperature of the reference fluid. The shape factor @f$ \theta_i @f$ is defined
* in @cite ely-hanley1981 as:
*
* @f[
* \theta_i(T_R^i, V_R^i, \omega_i) = 1 + (\omega_i - \omega_0)F(T_R^i, V_R^i)]
*
* \quad \text{(Equation 11 of @cite ely-hanley1981)}
* @f]
*
* Where @f$ Z_0^c @f$ is the critical compressibility of the reference fluid, @f$ Z_i^c @f$
* is the critical compressibility of species i, @f$ \omega_0 @f$ is the acentric factor of
* the reference fluid, and @f$ G(T_R^i, V_R^i) @f$ is a function of the reduced temperature
* and reduced volume of species i. The function @f$ G(T_R^i, V_R^i) @f$ is defined as:
* Where @f$ Z_0^c @f$ is the critical compressibility of the reference fluid,
* @f$ Z_i^c @f$ is the critical compressibility of species i, @f$ \omega_0 @f$ is
* the acentric factor of the reference fluid, and @f$ G(T_R^i, V_R^i) @f$ is a
* function of the reduced temperature and reduced volume of species i. The
* function @f$ G(T_R^i, V_R^i) @f$ is defined as:
*
* @f[
* F(T_R^i, V_R^i) = a_1 + b_1 ln(T_R^i) + (c_1 + d_1/T_R^i) (V_R^i - 0.5)
*
* \quad \text{(Equation 13 of @cite ely-hanley1981)}
* @f]
*
* Where @f$ T_i^+ @f$ and @f$ V_i^+ @f$ are limited values of the reduced temperature
* and pressure and have the same definition as was shown earlier.
* Where @f$ T_i^+ @f$ and @f$ V_i^+ @f$ are limited values of the reduced
* temperature and pressure and have the same definition as was shown earlier.
*
* The values of @f$ a_1 @f$, @f$ b_1 @f$, @f$ c_1 @f$, and @f$ d_1 @f$ are
* given in Table II of @cite ely-hanley1981. They are:
Expand All @@ -263,19 +269,22 @@ class HighPressureGasTransport : public MixTransport
* a_1 = 0.090569, b_1 = -0.862762, c_1 = 0.316636, d_1 = -0.465684
* @f]
*
* With the definitions above, the value of @f$ h_{m,0} @f$ can be computed from the
* equation that defined the value of @f$ f_{m,0} h_{m,0} @f$. The value of @f$ h_{m,0} @f$
* must be computed first and then the value of @f$ f_{m,0} @f$ can be computed.
* With the definitions above, the value of @f$ h_{m,0} @f$ can be computed from
* the equation that defined the value of @f$ f_{m,0} h_{m,0} @f$. The value of
* @f$ h_{m,0} @f$ must be computed first and then the value of @f$ f_{m,0} @f$
* can be computed.
*
* The value of @f$ MW_m @f$ is the molecular weight of the mixture and is computed using
* the following equation. This expression is somewhat complex, but the method to obtain the
* value of @f$ MW_m @f$ is to compute the right-hand-side first with the definitions of
* @f$ MW_{ij} @f$, @f$ f_{ij} @f$, and @f$ h_{ij} @f$ . Then once the right-hand-side is
* known, along with the previously computed values of @f$ f_{m,0} @f$ and @f$ h_{m,0} @f$,
* we can use simple algebra to isolate @f$ MW_m @f$.
* The value of @f$ MW_m @f$ is the molecular weight of the mixture and is computed
* using the following equation. This expression is somewhat complex, but the
* method to obtain the value of @f$ MW_m @f$ is to compute the right-hand-side
* first with the definitions of @f$ MW_{ij} @f$, @f$ f_{ij} @f$, and
* @f$ h_{ij} @f$ . Then once the right-hand-side is known, along with the
* previously computed values of @f$ f_{m,0} @f$ and @f$ h_{m,0} @f$,we can use
* simple algebra to isolate @f$ MW_m @f$.
*
* @f[
* MW_m^{1/2} f_{m,0}^{1/2} h_{m,0}^{-4/3} = \sum_i \sum_j X_i X_j MW_{ij}^{-1/2} f_{ij}^{1/2} h_{ij}^{-4/3}
* MW_m^{1/2} f_{m,0}^{1/2} h_{m,0}^{-4/3} = \sum_i \sum_j X_i X_j MW_{ij}^{-1/2}
* f_{ij}^{1/2} h_{ij}^{-4/3}
*
* \quad \text{(Equation 14)}
* @f]
Expand All @@ -288,18 +297,20 @@ class HighPressureGasTransport : public MixTransport
* \quad \text{(Equation 15)}
* @f]
*
* For clarity, if we assume the right-hand-side of the molecular weight mixture equation is A, then
* the mixture molecular weight is given by:
* For clarity, if we assume the right-hand-side of the molecular weight mixture
* equation is A, then the mixture molecular weight is given by:
*
* @f[
* MW_m = A^{-2} f_{m,0} h_{m,0}^{-8/3}
* @f]
*
* All of the above descriptions allow for the calculation of @f$ F_{\lambda} @f$ in the expression for
* the mixture value of the translational component of the thermal conductivity. The reference fluid
* value of the translational component of the thermal conductivity is still needed. The first thing
* that needs to be done is the obtain the temperature and density of the reference fluid. The following
* relations are used to compute the reference fluid temperature and density.
* All of the above descriptions allow for the calculation of @f$ F_{\lambda} @f$
* in the expression for the mixture value of the translational component of the
* thermal conductivity. The reference fluid value of the translational component
* of the thermal conductivity is still needed. The first thing that needs to be
* done is the obtain the temperature and density of the reference fluid. The
* following relations are used to compute the reference fluid temperature and
* density.
*
* @f[
* \rho_0 = \rho h_{m,0}
Expand All @@ -318,15 +329,19 @@ class HighPressureGasTransport : public MixTransport
* computed using the following equation.
*
* @f[
* \lambda^{'}_{0}(\rho_0, T_0) = \frac{15R}{4MW_0} \eta_0^* + \lambda_0^{(1)}\rho_0 + \Delta\lambda_0(\rho_0, T_0)
* \lambda^{'}_{0}(\rho_0, T_0) = \frac{15R}{4MW_0} \eta_0^* +
* \lambda_0^{(1)}\rho_0 +
* \Delta\lambda_0(\rho_0, T_0)
*
* \quad \text{(Equation 18)}
* @f]
*
* Where @f$ \eta_0^* @f$ is the dilute gas viscosity, @f$ \lambda_0^{(1)} @f$ is the first density
* correction to the thermal conductivity, and @f$ \Delta\lambda_0 @f$ is the high-density contribution.
* Ely and Hanley provide a correlation equation to solve for this quantity for the methane reference
* fluid. The details of the correlation and the parameters are shown in Table I of @cite ely-hanley1983.
* Where @f$ \eta_0^* @f$ is the dilute gas viscosity, @f$ \lambda_0^{(1)} @f$ is
* the first density correction to the thermal conductivity, and
* @f$ \Delta\lambda_0 @f$ is the high-density contribution. Ely and Hanley provide
* a correlation equation to solve for this quantity for the methane reference
* fluid. The details of the correlation and the parameters are shown in Table I
* of @cite ely-hanley1983.
*
* For completeness, the relations are reproduced here.
*
Expand Down

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