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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.

`ThermalModel` | Thermal model object |

`SteadyStateThermalResults` | Steady-state thermal solution and derived quantities |

`TransientThermalResults` | Transient thermal solution and derived quantities |

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 |

PDE Modeler | Solve partial differential equations in 2-D regions |

**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.

**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.