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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:

`$\rho c\frac{\partial T}{\partial t}-\nabla \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\left(k\nabla T\right)=Q$`

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 $htc\left(T-{T}_{\infty }\right)$, 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.

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

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 heat transfer or structural analysis problem `assembleFEMatrices` Assemble finite element matrices
 `interpolateTemperature` Interpolate temperature in a thermal result at arbitrary spatial locations `evaluateTemperatureGradient` Evaluate temperature gradient of a thermal solution at arbitrary spatial locations `evaluateHeatFlux` Evaluate heat flux of a 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 geometry `pdeplot3D` Plot solution or surface mesh for 3-D geometry
 `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

Objects

 `ThermalModel` Thermal model object `SteadyStateThermalResults` Steady-state thermal solution and derived quantities `TransientThermalResults` Transient thermal solution and derived quantities

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

Apps

 PDE Modeler Solve partial differential equations in 2-D regions

Topics

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.

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