Difference between revisions of "Contrib/CompressibleMixingPhaseChangeFoam"

Revision as of 16:14, 30 December 2012

Solver for two fluids with phase change (for example - water <---> steam), pressure and temperature density dependence

Model Equations

Equation of state

Low-compressible fluid: $\rho = \rho_0 + \frac{\partial \rho}{\partial T} \Delta T + \frac{\partial \rho}{\partial p} \Delta p$

Ideal gas: $\rho=\frac{p}{(C_p/C_v)(R/M)T}$

By combining this equations, we can get general relation:

$\rho = \hat \rho + \frac{\partial \rho}{\partial p} \Delta p$

where $\hat \rho$ computed with respect to previous formulations

mixture density $\rho$ calculated as

$\rho = \alpha_l \rho_l + \alpha_v \rho_v$

Liquid volume transport

Let us consider transport of liquid (heavy phase) volume fraction $\alpha_l$ :

$\frac{\partial \alpha_l \rho_l}{\partial t} + \nabla \cdot \left ( \alpha_l \rho_l \mathbf{U} \right ) = \dot m_l$

By converting to volume fluxes we get:

$\frac {\partial \alpha_l}{\partial t} + \nabla \cdot \left ( \alpha_l \textbf{U} \right ) = -\frac {\alpha_l}{\rho_l} \frac {d \rho_l}{dt} + \frac {\dot m_l}{\rho_l}$

Using equation of state, we can reformulate substantial derivative for density in terms of pressure for any phase:

$\frac {d \rho}{dt}= \frac {d \hat \rho} {dt} + \psi \frac {dp} {dt} + p \frac {d \psi}{dt}$

General rule for converting from mass to volume fluxes in transport equation

$\frac{\partial \alpha_i \rho_i \zeta}{\partial t} + \nabla \cdot \left ( \alpha_i \rho_i \textbf{U} \zeta \right ) = \rho_i \alpha_i \left ( \frac {\partial \zeta}{\partial t} + \nabla \cdot \left ( \textbf{U} \zeta \right ) \right ) + \zeta \alpha_l \frac {d \rho_i}{dt} + \zeta \rho_i \frac {d \alpha_i}{dt}$

Momentum equation (velocity prediction)

$\frac {\partial \rho \textbf{U}}{\partial t} + \nabla \cdot \rho \textbf{U} \textbf{U} = \nabla \cdot R^{Eff} - \nabla p + \rho \textbf{g}$

Pressure equation obtained by summation of equation for volume phase fraction of liquid and gas phases:

$\nabla \cdot \left ( \textbf{U} \right ) = - \left ( \frac {\alpha_1 \psi_1}{\rho_1} + \frac {\alpha_2 \psi_2}{\rho_2} \right ) \frac {d p}{d t} - \left ( \frac {\alpha_1}{\rho_1} \frac {d \psi_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \psi_2}{d t} \right ) p - \left ( \frac {\alpha_1}{\rho_1} \frac {d \hat \rho_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \hat \rho_2}{d t} \right )+ \dot m_1 \left ( \frac{1}{\rho_1} - \frac{1}{\rho_2} \right )$

Phase change model
Energy equation

Solver sources and tutorials located here

@@ Line 22: / Line 22: @@
-<math>\rho =  \alpha_l \rho_l + \alpha_v \rho_v </math>
+<math>
+\rho =  \alpha_l \rho_l + \alpha_v \rho_v
+</math>
@@ Line 45: / Line 47: @@
   \alpha_l \textbf{U}
 \right )
-= \frac {\alpha_l}{\rho_l} \frac {d \rho_l}{dt} + \frac {\dot m_l}{\rho_l}
+= -\frac {\alpha_l}{\rho_l} \frac {d \rho_l}{dt} + \frac {\dot m_l}{\rho_l}
 </math>
@@ Line 76: / Line 78: @@
 \frac {\partial \rho \textbf{U}}{\partial t}
 +
-\nabla \cdot \rho \textbf{U} \textbf{U} = \nabla \cdot R^Eff - \nabla p + \rho textbf{g}
+\nabla \cdot \rho \textbf{U} \textbf{U} = \nabla \cdot R^{Eff} - \nabla p + \rho \textbf{g}
+</math>
+* Pressure equation obtained by summation of equation for volume phase fraction of liquid and gas phases:
+<math>
+ \nabla \cdot \left ( \textbf{U} \right )
+=
+- \left ( \frac {\alpha_1 \psi_1}{\rho_1} + \frac {\alpha_2 \psi_2}{\rho_2} \right ) \frac {d p}{d t} -
+\left ( \frac {\alpha_1}{\rho_1} \frac {d \psi_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \psi_2}{d t} \right ) p -
+\left ( \frac {\alpha_1}{\rho_1} \frac {d \hat \rho_1}{d t} + \frac {\alpha_2}{\rho_2} \frac {d \hat \rho_2}{d t} \right )+
+\dot m_1 \left ( \frac{1}{\rho_1} - \frac{1}{\rho_2} \right )
 </math>
- * Phase change model
+* Phase change model
- * Energy equation
+* Energy equation
 [http://www.os-cfd.ru/compressibleMixingPhaseChangeFoam/Solver.tgz Solver sources and tutorials located here]