Difference between revisions of "Contrib/CompressibleMixingPhaseChangeFoam"
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== Model Equations Derivation == | == Model Equations Derivation == | ||
− | * Equation of state | + | * '''Equation of state''' |
Low-compressible fluid: <math> | Low-compressible fluid: <math> | ||
\rho = \rho_0 + \frac{\partial \rho}{\partial T} \Delta T + \frac{\partial \rho}{\partial p} \Delta p | \rho = \rho_0 + \frac{\partial \rho}{\partial T} \Delta T + \frac{\partial \rho}{\partial p} \Delta p | ||
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without index - mixture variable (or all variables local to some phase) | without index - mixture variable (or all variables local to some phase) | ||
− | * Liquid volume transport | + | * '''Liquid volume transport''' |
Let us consider transport of liquid (heavy phase) volume fraction <math>\alpha_l</math>: | Let us consider transport of liquid (heavy phase) volume fraction <math>\alpha_l</math>: | ||
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</math> | </math> | ||
− | * General rule for converting from mass to volume fluxes in transport equation | + | * '''General rule for converting from mass to volume fluxes in transport equation''' |
<math> | <math> | ||
\frac{\partial \alpha_i \rho_i \zeta}{\partial t} | \frac{\partial \alpha_i \rho_i \zeta}{\partial t} | ||
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</math> | </math> | ||
− | * Momentum equation (velocity prediction) | + | * '''Momentum equation (velocity prediction)''' |
<math> | <math> | ||
\frac {\partial \rho \textbf{U}}{\partial t} | \frac {\partial \rho \textbf{U}}{\partial t} | ||
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\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> | </math> | ||
+ | |||
by substituting piezometric pressure <math> \hat p = p - \rho \textbf{g} \textbf{x} </math> | by substituting piezometric pressure <math> \hat p = p - \rho \textbf{g} \textbf{x} </math> | ||
we get: | we get: | ||
+ | |||
<math> | <math> | ||
\frac {\partial \rho \textbf{U}}{\partial t} | \frac {\partial \rho \textbf{U}}{\partial t} | ||
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</math> | </math> | ||
− | * Pressure equation obtained by summation of equation for volume phase fraction of liquid and gas phases: | + | * '''Pressure equation''' obtained by summation of equation for volume phase fraction of liquid and gas phases: |
<math> | <math> | ||
\nabla \cdot \left ( \textbf{U} \right ) | \nabla \cdot \left ( \textbf{U} \right ) | ||
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</math> | </math> | ||
− | * Energy equation | + | * '''Energy equation''' |
Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider energy equation for phase-1: | Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider energy equation for phase-1: | ||
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</math> | </math> | ||
− | * Linking liquid volume transport to pressure equation is done by introducing <math>+\alpha_1 \nabla \cdot \textbf{U}</math> and <math>-\alpha_1 \nabla \cdot \textbf{U}</math> at r.h.s of volume fraction balance equation. Then, <math>-\nabla \cdot \textbf{U}</math> replaced by value from pressure equation | + | * '''Linking liquid volume transport to pressure equation''' is done by introducing <math>+\alpha_1 \nabla \cdot \textbf{U}</math> and <math>-\alpha_1 \nabla \cdot \textbf{U}</math> at r.h.s of volume fraction balance equation. Then, <math>-\nabla \cdot \textbf{U}</math> replaced by value from pressure equation |
<math> | <math> | ||
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+\alpha_1 \alpha_2 \frac{d p}{d t}\left ( -\frac{\psi_1}{\rho_1} + \frac{\psi_2}{\rho_2} \right ) | +\alpha_1 \alpha_2 \frac{d p}{d t}\left ( -\frac{\psi_1}{\rho_1} + \frac{\psi_2}{\rho_2} \right ) | ||
</math> | </math> | ||
− | |||
* Phase change model | * Phase change model | ||
+ | |||
+ | == Model Equations Summary == | ||
[http://www.os-cfd.ru/compressibleMixingPhaseChangeFoam/Solver.tgz Solver sources and tutorials located here] | [http://www.os-cfd.ru/compressibleMixingPhaseChangeFoam/Solver.tgz Solver sources and tutorials located here] |
Revision as of 17:26, 30 December 2012
Solver for two fluids with phase change (for example - water <---> steam), pressure and temperature density dependence
1 Model Equations Derivation
- Equation of state
Low-compressible fluid:
Ideal gas:
By combining this equations, we can get general relation:
where computed with respect to previous formulations
mixture density calculated as
Here, and afterthere indices:
1,l,f - for liquid (heavy media with low compressibility)
2,g,s - for gas (light media (like steam) with big compressibility)
without index - mixture variable (or all variables local to some phase)
- Liquid volume transport
Let us consider transport of liquid (heavy phase) volume fraction :
By converting to volume fluxes we get:
Using equation of state, we can reformulate substantial derivative for density in terms of pressure for any phase:
- General rule for converting from mass to volume fluxes in transport equation
- Momentum equation (velocity prediction)
by substituting piezometric pressure we get:
- Pressure equation obtained by summation of equation for volume phase fraction of liquid and gas phases:
- Energy equation
Energy equation for mixture temperature obtained from sum of energy equations for each phase. Consider energy equation for phase-1:
by converting to enthalpies we get:
By substituting temperature instead of enthalpy, after conversion to volume fluxes we get equation for temperature (divided by )
By combining equations of phases, we get energy balance for mixture:
- Linking liquid volume transport to pressure equation is done by introducing and at r.h.s of volume fraction balance equation. Then, replaced by value from pressure equation
- Phase change model