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Heat Transfer

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

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.

For 2-D geometry problems, you also can use the PDE Modeler app. The app includes geometry creation and preset modes for applications.


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 or structural analysis problem
assembleFEMatricesAssemble finite element matrices
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 geometry
pdeplot3DPlot solution or surface mesh for 3-D geometry
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


PDE ModelerSolve partial differential equations in 2-D regions


Programmatic Workflow

Heat Transfer in Block with Cavity

Use command-line functions to solve a heat equation that describes heat diffusion in a metal block with a rectangular cavity.

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.

Nonlinear Heat Transfer in Thin Plate

Perform a heat transfer analysis of a thin plate.

PDE Modeler App Workflow

Heat Transfer in Block with Cavity: PDE Modeler App

Use the PDE Modeler app to solve a heat equation that describes heat diffusion in a block with a rectangular cavity.

Heat Distribution in Circular Cylindrical Rod: PDE Modeler App

Solve a 3-D parabolic PDE problem by reducing it to 2-D using coordinate transformation.

Heat Transfer Between Two Squares Made of Different Materials: PDE Modeler App

Solve a heat transfer problem with different material parameters.