Heat transfer in solids

This page derives the weak formulation for heat transfer in solids and describes the available boundary conditions and couplings in Allsolve.

Strong formulation

Heat transfer in solids is a thermal conduction process that is governed by the equation

\begin{align*} \rho C_p \frac{\partial T}{\partial t} = -\nabla \cdot \boldsymbol{q} + Q \tag{1} \end{align*}

where

$\rho$ is the material density ( $kg/m^3$ )
$C_p$ is specific heat capacity of the material ( $J/kg \cdot K$ )
$\boldsymbol{q}$ is the heat flux density ( $W/m^2$ )
$Q$ is the volumetric heat source ( $W/m^3$ )
$T$ is the Temperature field ( $K$ ).

Constitutive equation

Heat flux density due to conduction is given by Fick’s law of diffusion

\begin{align*} \boldsymbol{q} = -\kappa \nabla T \tag{2} \end{align*}

where $\kappa$ is the thermal conductivity of the material ( $W/m^2 K$ ).

The governing equation $\text{(1)}$ then becomes

\begin{align*} -\rho C_p \frac{\partial T}{\partial t} + \nabla \cdot (\kappa \nabla T) + Q = 0 \tag{3} \end{align*}

Weak formulation

To obtain the weak form, multiply $\text{(3)}$ by the test function $T^\prime$ and integrate over the domain $\Omega$ :

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T) \; T^{\prime} \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Applying Leibniz rule on the divergence term, we get

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T \; T^{\prime}) \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Applying Divergence theorem on the divergence term, we get

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (\kappa \nabla T \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Substituting $\text{(2)}$ into the boundary term, we get the final weak formulation

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (-\boldsymbol{q} \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Interactions

Constraint

Applies a fixed temperature $T$ to a node or region, enforcing a Dirichlet boundary condition for the heat equation. Use this to define fixed-temperature boundaries, such as a heat sink held at a constant temperature or a surface in contact with a thermal reservoir.


How to use	Provide a scalar temperature value in Kelvins (K).
Example	`300` applies a fixed temperature of 300 K to the selected node or region. See also Adding Heat-solid Constraint.
Unit	Temperature in Kelvins (K)

Heat source

Applies a heat source $Q$ to a region, acting as a source term in the heat equation. Used to model internally generated heat, such as resistive heating in conductors. Can be specified as a constant value or a spatially varying field. In Quanscient Allsolve, zero heat flux Neumann boundary condition $Q = 0$ is automatically applied to all boundaries where no other condition is set. There is no need to define it manually.


Example	`1000` applies a heat source of 1000 W/m^regdim to the target region. See also Adding Heat source constraint.
Unit	Heat source in Watts/m^regdim. The `regdim` is the dimension of the target regions. For volume targets, the unit is W/m^3 and for surface targets it is W/m^2.

Periodicity

Imposes periodic boundary conditions on the temperature field $T$ between two boundaries. Reduces the computational domain size for geometrically symmetric problems, avoiding the need to model the full geometry.


Example	Periodicity follows the same principles, regardless of which physics module it belongs to. See how periodicity is used in magnetism as a reference: Periodicity in electric motors

Convection

Applies a convective heat flux boundary condition on a surface or a curve, modeling heat exchange between the solid and a surrounding fluid. The heat flux is proportional to the difference between the surface temperature and the ambient fluid temperature, defined by Newton’s law of cooling:

$q = h(T - T_{\infty})$

where $h$ is the heat transfer coefficient and $T_{\infty}$ is the surrounding fluid temperature.


How to use	Provide the heat transfer coefficient $h$ and the surrounding fluid temperature $T_{\infty}$ .
Example	$h = 25\: \frac{W}{m²K}$ and $T_{\infty} = 293\:K$ models natural air convection at room temperature.
Unit	Heat transfer coefficient in Watts per square meter Kelvin $\frac{W}{m²K}$ , fluid temperature in Kelvins (K)

Lump T/Φ

Applies a lumped temperature $T$ or heat power $\Phi$ to a specific region. Used to model simplified thermal elements where the detailed temperature distribution is not explicitly resolved, but replaced with an equivalent lumped temperature or heat flux.


How to use	Specify the target region. From `Actuation mode`, select either temperature, heat power or circuit coupling. Fill in the value.
Example	$T = 100$ applies a temperature of $100\:K$ to the specified region.
Unit	Temperature in Kelvins (K) or heat power in Watts (W)

Couplings to Other Physics

This formulation supports the following couplings:

Joule heating
- Current flow
- Magnetism H