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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:


A typical programmatic workflow for solving a heat transfer problem includes the following 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(TT), specify the ambient temperature T and the convective heat transfer coefficient htc. For radiative heat flux εσ(T4T4), specify the ambient temperature T, 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.

Temperature distribution on a sphere


expand all

createpdeCreate model
thermalPropertiesAssign thermal properties of a material for a thermal model
internalHeatSourceSpecify internal heat source for a thermal model
thermalBCSpecify boundary conditions for a thermal model
thermalICSet initial conditions or initial guess for a thermal model
solveSolve heat transfer, structural analysis, or electromagnetic analysis problem
assembleFEMatricesAssemble finite element matrices
linearizeLinearize structural or thermal model
linearizeInputSpecify inputs to linearized model
linearizeOutputSpecify outputs of linearized model
interpolateTemperatureInterpolate temperature in a thermal result at arbitrary spatial locations
evaluateTemperatureGradientEvaluate temperature gradient of a thermal solution at arbitrary spatial locations
evaluateHeatFluxEvaluate heat flux of a thermal solution at nodal or arbitrary spatial locations
evaluateHeatRateEvaluate integrated heat flow rate normal to specified boundary
pdeplotPlot solution or mesh for 2-D problem
pdeplot3DPlot solution or surface mesh for 3-D problem
pdegplotPlot PDE geometry
pdemeshPlot PDE mesh
pdevizCreate and plot PDE visualization object
findThermalPropertiesFind thermal material properties assigned to a geometric region
findHeatSourceFind heat source assigned to a geometric region
findThermalBCFind thermal boundary conditions assigned to a geometric region
findThermalICFind thermal initial conditions assigned to a geometric region


ThermalModelThermal model object
SteadyStateThermalResultsSteady-state thermal solution and derived quantities
TransientThermalResultsTransient thermal solution and derived quantities


ThermalMaterialAssignment PropertiesThermal material properties assignments
HeatSourceAssignment PropertiesHeat source assignments
ThermalBC PropertiesBoundary condition for thermal model
NodalThermalICs PropertiesInitial temperature at mesh nodes
GeometricThermalICs PropertiesInitial temperature over a region or region boundary
PDESolverOptions PropertiesAlgorithm options for solvers
PDEVisualization PropertiesPDE visualization of mesh and nodal results


Thermal Analysis Workflow

Heat Transfer in Block with Cavity

Solve a heat equation that describes heat diffusion in a block with a rectangular cavity.

Temperature Distribution in Heat Sink

Perform a 3-D transient heat transfer analysis of a heat sink.

Heat Distribution in Circular Cylindrical Rod

Analyze a 3-D axisymmetric model by using a 2-D model.

Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux

Perform a 3-D transient heat conduction analysis of a hollow sphere made of three different layers of material, subject to a nonuniform external heat flux.

Inhomogeneous Heat Equation on Square Domain

Solve the heat equation with a source term.

Heat Transfer Problem with Temperature-Dependent Properties

Solve the heat equation with a temperature-dependent thermal conductivity.

Thermal Analysis of Disc Brake

Use Partial Differential Equation Toolbox™ and Simscape™ Driveline™ to simulate a brake pad moving around a disc and analyze temperatures when braking.

Axisymmetric Thermal and Structural Analysis of Disc Brake

Simplify analysis of a disc brake by using an axisymmetric model for thermal and thermal stress computations.

General PDE Workflow

Nonlinear Heat Transfer in Thin Plate

Perform a heat transfer analysis of a thin plate.