CMPS Theory Manual › Solid Energy, Conjugate Heat Transfer and Thermal Coupling

Solid Energy, Conjugate Heat Transfer and Thermal Coupling

Solid energy equation

Solid zones solve temperature as the active unknown. In differential form the conductive problem is

\[\rho_s c_s\frac{\partial T_s}{\partial t}=\nabla\cdot(\mathbf K\nabla T_s)+\dot q_v.\]

For isotropic solids, \(\mathbf K=k\mathbf I\). CMPS also supports directional thermal conductivity through a conductivity tensor.

Interior solid-face flux

For an interior solid face, CMPS forms an effective conductivity in the face-normal direction. For anisotropic material,

\[k_{n}=\mathbf n^T\mathbf K\mathbf n.\]

The owner and neighbor conductivities are combined harmonically. With interpolation weight \(g=V_P/(V_P+V_F)\),

\[k_f=\frac{k_Pk_F}{(1-g)k_P+gk_F}.\]

The orthogonal conductive face flux is then

\[\dot Q_f=-k_f\,\frac{A_f}{d_{PF}}(T_F-T_P),\]

with the geometric coefficient supplied by the finite-volume face metrics.

Fluid–solid thermal coupling

At a fluid–solid interface, the wall temperature and heat flux are obtained from the gas-side thermal treatment and the adjacent solid conduction resistance. For turbulent gas flow, the gas-side coefficient can use the wall-function thermal law; for laminar flow the corresponding convective coefficient is used directly. The resulting heat flux is inserted with equal and opposite roles in the coupled fluid and solid energy balances.

External wall thermal conditions

The wall data model includes adiabatic, prescribed temperature, convective heat transfer, prescribed heat flux, external radiation, combined external heat transfer, and one-dimensional charring treatment. Representative flux closures are

\[q''_w=0\qquad\text{(adiabatic)},\]

\[T_w=T_{specified}\qquad\text{(prescribed temperature)},\]

\[q''_{conv}=h_{ext}(T_\infty-T_w),\]

\[q''_{rad}=\varepsilon\sigma\left(T_{sur}^4-T_w^4\right),\]

\[q''_{mixed}=q''_{conv}+q''_{rad}.\]

Fluid–solid walls additionally support adiabatic, non-adiabatic and charring modes. At a conservative interface, the two sides use equal and opposite heat transfer,

\[q''_{f\rightarrow s}=-q''_{s\rightarrow f}.\]

Transient solid conduction

Transient solid cells use the same implicit first/second-order physical-time framework as the coupled flow solver. The solid physical-time contribution is restricted to the temperature row; non-solid flow rows remain zero for solid cells.

Charring-material coupling

The one-dimensional charring model is a specialized wall/solid thermal response and is documented separately. At a fluid–solid charring interface, the material response supplies a solid-outward heat flux while the gas-side state provides the thermal driving condition.

VOF limitation

Although CMPS supports non-adiabatic fluid–solid coupling for the single-carrier flow path, the currently audited homogeneous VOF path requires adiabatic ordinary, interior and fluid–solid walls. This prevents applying an incomplete phase-wall heat-transfer closure.