Heat solid

Strong formulation

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

$$\begin{equation} \rho C_p \frac{\partial T}{\partial t} = -\nabla \cdot \boldsymbol{q} + Q, \end{equation}$$

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

$$\begin{equation} \boldsymbol{q} = -\kappa \nabla T, \end{equation}$$

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

The governing equation $(1)$ then becomes

$$\begin{equation} -\rho C_p \frac{\partial T}{\partial t} + \nabla \cdot (\kappa \nabla T) + Q = 0. \end{equation}$$

Weak formulation

The weak form of solid mechanics is obtained by multiplying the partial differential equation in $(3)$ with the test function of Temperature field $T^\prime$, and integrating over the whole domain $\Omega$ to get

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

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 $(2)$ into the boundary term, we get the final weak formulation

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

Available 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.

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. The unit depends on the dimension of the target region.

Adding Heat source constraint

Unit: Heat source in Watts/m^regdim. The regdim is the dimension of the target regions. For volume target, the unit is in 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 is similar in every physics section. This example is from $\varphi$-formulation, but workflows are identical:

Periodicity in electric motor

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 or Magnetism H)
Models resistive heat generation caused by electrical currents.

Use cases

• Heat solid: Steady-state heat transfer
• Heat solid: Manifold microchannel heat sink
• Heat solid: 2D Taylor-Couette flow
• Heat solid - Current flow coupling in AC system