# Heat Transfer

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

## Live Editor Tasks

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) |

## Topics

### 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.**Reduced-Order Model for Thermal Behavior of Battery**

Generate a reduced-order model (ROM) to enable analysis of the spatial thermal behavior of a battery cell during a fast-charging scenario.**Battery Module Cooling Analysis and Reduced-Order Thermal Model**

Perform a battery cooling analysis to determine effect of a cooling system.

### General PDE Workflow

**Nonlinear Heat Transfer in Thin Plate**

Perform a heat transfer analysis of a thin plate.