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:

$ρ c \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q$

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 $h t c (T - T_{\infty})$ , specify the ambient temperature $T_{\infty}$ and the convective heat transfer coefficient htc. For radiative heat flux $ε σ (T^{4} - T_{\infty}^{4})$ , 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

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Thermal Model Setup

`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 structural analysis, heat transfer, or electromagnetic analysis problem
`assembleFEMatrices`	Assemble finite element matrices
`reduce`	Reduce structural or thermal model (Since R2019b)
`reconstructSolution`	Recover full-model transient solution from reduced-order model (ROM) (Since R2019b)
`linearize`	Linearize structural or thermal model (Since R2021b)
`linearizeInput`	Specify inputs to linearized model (Since R2021b)
`linearizeOutput`	Specify outputs of linearized model (Since R2021b)

Solutions at Nodal and Custom Locations

`interpolateTemperature`	Interpolate temperature in thermal result at arbitrary spatial locations
`evaluateTemperatureGradient`	Evaluate temperature gradient of thermal solution at arbitrary spatial locations
`evaluateHeatFlux`	Evaluate heat flux of thermal solution at nodal or arbitrary spatial locations
`evaluateHeatRate`	Evaluate integrated heat flow rate normal to specified boundary

Visualization

`pdeplot`	Plot solution or mesh for 2-D problem
`pdeplot3D`	Plot solution or surface mesh for 3-D problem
`pdegplot`	Plot PDE geometry
`pdemesh`	Plot PDE mesh
`pdeviz`	Create and plot PDE visualization object (Since R2021a)

Thermal Model Properties

`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

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.