# Heat Transfer

Solve conduction-dominant heat transfer problems with convection and radiation occurring at boundaries

Address challenges with thermal management by analyzing the temperature distributions of components based on material properties, external heat sources, and internal heat generation for steady-state and transient problems.

The heat transfer equation is a parabolic partial differential equation that describes the distribution of temperature in a particular region over given time:

`$\rho c\frac{\partial T}{\partial t}-\nabla \cdot \left(k\nabla T\right)=Q$`

A typical programmatic workflow for solving a heat transfer problem includes these steps:

• Create a special thermal model container for a steady-state or transient thermal model.

• Define 2-D or 3-D geometry and mesh it.

• Assign thermal properties of the material, such as thermal conductivity k, specific heat c, and mass density ρ.

• Specify internal heat sources Q within the geometry.

• Specify temperatures on the boundaries or heat fluxes through the boundaries. For convective heat flux through the boundary $htc\left(T-{T}_{\infty }\right)$, specify the ambient temperature ${T}_{\infty }$ and the convective heat transfer coefficient htc. For radiative heat flux $\epsilon \sigma \left({T}^{4}-{T}_{\infty }{}^{4}\right)$, specify the ambient temperature ${T}_{\infty }$, emissivity ε, and Stefan-Boltzmann constant σ.

• Set an initial temperature or initial guess.

• Solve and plot results, such as the resulting temperatures, temperature gradients, heat fluxes, and heat rates.

• Approximate dynamic characteristics of a thermal model by using reduced order modeling (ROM).

## Functions

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 `createpde` Create model `thermalProperties` Assign thermal properties of a material for a thermal model `internalHeatSource` Specify internal heat source for a thermal model `thermalBC` Specify boundary conditions for a thermal model `thermalIC` Set initial conditions or initial guess for a thermal model `solve` Solve structural analysis, heat transfer, or electromagnetic analysis problem `assembleFEMatrices` Assemble finite element matrices `reduce` Reduce structural or thermal model (Since R2019b) `reconstructSolution` Recover full-model transient solution from reduced-order model (ROM) (Since R2019b) `linearize` Linearize structural or thermal model (Since R2021b) `linearizeInput` Specify inputs to linearized model (Since R2021b) `linearizeOutput` Specify outputs of linearized model (Since R2021b)
 `interpolateTemperature` Interpolate temperature in thermal result at arbitrary spatial locations `evaluateTemperatureGradient` Evaluate temperature gradient of thermal solution at arbitrary spatial locations `evaluateHeatFlux` Evaluate heat flux of thermal solution at nodal or arbitrary spatial locations `evaluateHeatRate` Evaluate integrated heat flow rate normal to specified boundary
 `pdeplot` Plot solution or mesh for 2-D problem `pdeplot3D` Plot solution or surface mesh for 3-D problem `pdegplot` Plot PDE geometry `pdemesh` Plot PDE mesh `pdeviz` Create and plot PDE visualization object (Since R2021a)
 `findThermalProperties` Find thermal material properties assigned to a geometric region `findHeatSource` Find heat source assigned to a geometric region `findThermalBC` Find thermal boundary conditions assigned to a geometric region `findThermalIC` Find thermal initial conditions assigned to a geometric region

 Visualize PDE Results Create and explore visualizations of PDE results in the Live Editor (Since R2022b)

## Objects

 `ThermalModel` Thermal model object `ReducedThermalModel` Reduced-order thermal model (Since R2022a) `SteadyStateThermalResults` Steady-state thermal solution and derived quantities `TransientThermalResults` Transient thermal solution and derived quantities `ModalThermalResults` Modal thermal solution (Since R2022a)

## Properties

 ThermalMaterialAssignment Properties Thermal material properties assignments HeatSourceAssignment Properties Heat source assignments ThermalBC Properties Boundary condition for thermal model NodalThermalICs Properties Initial temperature at mesh nodes GeometricThermalICs Properties Initial temperature over a region or region boundary PDESolverOptions Properties Algorithm options for solvers PDEVisualization Properties PDE visualization of mesh and nodal results (Since R2021a)