# ChtMultiRegionFoam

**ChtMultiRegionFoam**

Solver for steady or transient fluid flow and solid heat conduction, with conjugate heat transfer between regions, buoyancy effects, turbulence, reactions and radiation modelling.

## Contents

## 1 Equations

For each region defined as fluid, the according equation for the fluid is solved and the same is done
for each solid region. The regions are coupled by a thermal boundary condition. A short description of
the solver can be found also in ^{[1]}

### 1.1 Equations Fluid

For each fluid region the compressible Navier Stokes equation are solved.

#### 1.1.1 Mass conservation

The variable-density continuity equation is

| (1) |

The source code can be found in src/finiteVolume/cfdTools/compressible/rhoEqn.H:

{ fvScalarMatrix rhoEqn ( fvm::ddt(rho) + fvc::div(phi) == fvOptions(rho) ); fvOptions.constrain(rhoEqn); rhoEqn.solve(); fvOptions.correct(rho); }

#### 1.1.2 Momentum conservation

| (2) |

represent the velocity, the gravitational acceleration, the pressure minus the hydrostatic pressure and and are the viscose and turbulent stresses.

The source code can be found in Ueqn.H:

// Solve the Momentum equation MRF.correctBoundaryVelocity(U); tmp<fvVectorMatrix> tUEqn ( fvm::ddt(rho, U) + fvm::div(phi, U) + MRF.DDt(rho, U) + turbulence.divDevRhoReff(U) == fvOptions(rho, U) ); fvVectorMatrix& UEqn = tUEqn.ref(); UEqn.relax(); fvOptions.constrain(UEqn); if (pimple.momentumPredictor()) { solve ( UEqn == fvc::reconstruct ( ( - ghf*fvc::snGrad(rho) - fvc::snGrad(p_rgh) )*mesh.magSf() ) ); fvOptions.correct(U); K = 0.5*magSqr(U); } fvOptions.correct(U);

#### 1.1.3 Energy conservation

The energy equation can be found in: https://cfd.direct/openfoam/energy-equation/

The total energy of a fluid element can be seen as the sum of kinetic energy and internal energy . The rate of change of the kinetic energy within a fluid element is the work done on this fluid element by the viscous forces, the pressure and eternal volume forces like the gravity:

| (3) |

The rate of change of the internal energy of a fluid element is the heat transferred to this fluid element by diffusion and turbulence plus the heat source term plus the heat source by radiation :

| (4) |

The change rate of the total energy is the sum of the above two equations:

| (5) |

Instead of the internal energy there is also the option to solve the equation for the enthalpy :

| (5) |

The source code can be found in EEqn.H:

{ volScalarField& he = thermo.he(); fvScalarMatrix EEqn ( fvm::ddt(rho, he) + fvm::div(phi, he) + fvc::ddt(rho, K) + fvc::div(phi, K) + ( he.name() == "e" ? fvc::div ( fvc::absolute(phi/fvc::interpolate(rho), U), p, "div(phiv,p)" ) : -dpdt ) - fvm::laplacian(turbulence.alphaEff(), he) == rho*(U&g) + rad.Sh(thermo, he) + Qdot + fvOptions(rho, he) ); EEqn.relax(); fvOptions.constrain(EEqn); EEqn.solve(); fvOptions.correct(he); thermo.correct(); rad.correct(); Info<< "Min/max T:" << min(thermo.T()).value() << ' ' << max(thermo.T()).value() << endl; }

#### 1.1.4 Species conservation

In order to account for the chemical reactions occurring between different chemical species a conservation equation for each species k has to be solved:

| (6) |

is the reaction rate of the species k.

The source code can be found in YEqn.H:

tmp<fv::convectionScheme<scalar>> mvConvection(nullptr); if (Y.size()) { mvConvection = tmp<fv::convectionScheme<scalar>> ( fv::convectionScheme<scalar>::New ( mesh, fields, phi, mesh.divScheme("div(phi,Yi_h)") ) ); } { reaction.correct(); Qdot = reaction.Qdot(); volScalarField Yt ( IOobject("Yt", runTime.timeName(), mesh), mesh, dimensionedScalar("Yt", dimless, 0) ); forAll(Y, i) { if (i != inertIndex && composition.active(i)) { volScalarField& Yi = Y[i]; fvScalarMatrix YiEqn ( fvm::ddt(rho, Yi) + mvConvection->fvmDiv(phi, Yi) - fvm::laplacian(turbulence.muEff(), Yi) == reaction.R(Yi) + fvOptions(rho, Yi) ); YiEqn.relax(); fvOptions.constrain(YiEqn); YiEqn.solve(mesh.solver("Yi")); fvOptions.correct(Yi); Yi.max(0.0); Yt += Yi; } } if (Y.size()) { Y[inertIndex] = scalar(1) - Yt; Y[inertIndex].max(0.0); } }

### 1.2 Equations Solid

For the solid regions only the energy equation has to be solved. The energy equation states that the temporal change of enthalpy of the solid is equal to the divergence of the heat conducted through the solid:

| (7) |

is the specific enthalpy, the density and is the thermal diffusivity which is defined as the ratio between the thermal conductivity and the specific heat capacity . Note that can be also anisotropic.

## 2 Source Code

# 3 References

- ↑ EL ABBASSIA, M.; LAHAYE, D. J. P.; VUIK, C. MODELLING TURBULENT COMBUSTION COUPLED WITH CONJUGATE HEAT TRANSFER IN OPENFOAM.