HeatTransferPDEComponent [vars,pars]
yields a heat transfer PDE term with variables vars and parameters pars.
HeatTransferPDEComponent
HeatTransferPDEComponent [vars,pars]
yields a heat transfer PDE term with variables vars and parameters pars.
Details
- HeatTransferPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
- HeatTransferPDEComponent models the generation and propagation of thermal energy in physical systems by mechanisms such as convection, conduction and radiation.
- HeatTransferPDEComponent models heat transfer phenomena with dependent variable temperature in [TemplateBox[{InterpretationBox[, 1], "K", kelvins, "Kelvins"}, QuantityTF]], independent variables in [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]] and time variable in [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]].
- Stationary variables vars are vars={Θ[x1,…,xn],{x1,…,xn}}.
- Time-dependent variables vars are vars={Θ[t,x1,…,xn],t,{x1,…,xn}}.
- The non-conservative time-dependent heat transfer model HeatTransferPDEComponent is based on a convection-diffusion model with mass density , specific heat capacity , thermal conductivity , convection velocity vector and heat source :
- The non-conservative stationary heat transfer PDE term is given by:
- The implicit default boundary condition for the non-conservative model is a HeatOutflowValue .
- The difference between the non-conservative model and the conservative model is the treatment of a convection velocity .
- The units of the heat transfer model terms are in [TemplateBox[{InterpretationBox[, 1], {"W", , "/", , {"m", ^, 3}}, watts per meter cubed, {{(, "Watts", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]], or equivalently in [TemplateBox[{InterpretationBox[, 1], {"J", , "/(", , {"m", ^, 3}, , "s", , ")"}, joules per meter cubed second, {{(, "Joules", )}, /, {(, {{"Meters", ^, 3}, , "Seconds"}, )}}}, QuantityTF]].
- The following parameters pars can be given:
-
parameter default symbol"HeatConvectionVelocity" {0,…} , flow velocity [TemplateBox[{InterpretationBox[, 1], {"m", , "/", , "s"}, meters per second, {{(, "Meters", )}, /, {(, "Seconds", )}}}, QuantityTF]]"HeatSource" 0 , heat source [TemplateBox[{InterpretationBox[, 1], {"W", , "/", , {"m", ^, 3}}, watts per meter cubed, {{(, "Watts", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]]"MassDensity" 1
- , density [TemplateBox[{InterpretationBox[, 1], {"kg", , "/", , {"m", ^, 3}}, kilograms per meter cubed, {{(, "Kilograms", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]]
"Material" Automatic"ModelForm" "NonConservative" none"RegionSymmetry" None"SpecificHeatCapacity" 1 , specific heat capacity [TemplateBox[{InterpretationBox[, 1], {"J", , "/(", , "kg", , "K", , ")"}, joules per kilogram kelvin, {{(, "Joules", )}, /, {(, {"Kilograms", , "Kelvins"}, )}}}, QuantityTF]]"ThermalConductivity" IdentityMatrix , thermal conductivity [TemplateBox[{InterpretationBox[, 1], {"W", , "/(", , "m", , "K", , ")"}, watts per meter kelvin, {{(, "Watts", )}, /, {(, {"Meters", , "Kelvins"}, )}}}, QuantityTF] - All parameters may depend on any of , and , as well as other dependent variables.
- The number of independent variables determines the dimensions of and the length of .
- Sometimes the heat equation is specified with a thermal diffusivity. The thermal diffusivity is the thermal conductivity divided by the density and the specific heat capacity at constant pressure.
- The thermal convection velocity specifies the velocity with which a fluid transports heat. If no fluid is present, the thermal convection velocity is 0.
- A heat source models thermal energy that is introduced (positive) or removed (negative) from the system.
- A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
- "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
-
dimension reduction equation1D2D
- The input specification for the parameters is exactly the same as for their corresponding operator terms.
- Coupled equations can be generated with the same input specification as with the corresponding operator terms.
- If no parameters are specified, the default heat transfer PDE is:
- If the HeatTransferPDEComponent depends on parameters that are specified in the association pars as …,keypi…,pivi,…, the parameters are replaced with .
Examples
open all close allBasic Examples (4)
Define a time-independent heat transfer model:
Define a time-dependent heat transfer model:
Set up a time-dependent heat transfer model with particular material parameters:
Model a temperature field with a heat source in a rod:
Solve the PDE:
Visualize the solution:
Scope (7)
Basic Examples (2)
Set up a time-dependent heat transfer model for a particular material:
Set up a time-dependent heat transfer model for several material regions:
1D (1)
Model a temperature field with two heat conditions at the sides:
del .(-k del Theta(x))^(︷^( heat transfer model)) =0
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameter thermal conductivity :
Specify heat surface conditions:
Set up the equation:
Solve the PDE:
Visualize the solution:
2D (1)
Model a ceramic strip that is embedded in a high-thermal-conductive material. The side boundaries of the strip are maintained at a constant temperature . The top surface of the strip is losing heat via both heat convection and heat radiation to the ambient environment at . The bottom boundary is assumed to be thermally insulated:
Model a temperature field and the thermal radiation and thermal transfer with:
Set up the heat transfer model variables vars:
Set up a rectangular domain with a width of and a height of :
Specify thermal conductivity :
Set up temperature surface boundary conditions at the left and right boundaries:
Set up a heat transfer boundary condition on the top surface:
Also set up a thermal radiation boundary condition on the top surface:
Set up the equation:
Solve the PDE:
Visualize the solution:
3D (1)
Model a temperature field with two heat conditions at the sides and an orthotropic thermal conductivity :
del .(-k del Theta(x,y,z))^(︷^( heat transfer model)) =0
Set up the heat transfer model variables vars:
Set up a region :
Specify an orthotropic thermal conductivity :
Specify heat surface conditions:
Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:
Solve the PDE:
Visualize the solution:
Time Dependent (1)
Model a temperature field and a thermal heat flux through part of the boundary with:
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :
Specify a thermal heat flux of applied at the left end for the first 300 seconds:
Set up initial conditions:
Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:
Solve the PDE:
Visualize the solution:
Time-Dependent Nonlinear (1)
Model a temperature field with a nonlinear heat conductivity term with:
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameters mass density , specific heat capacity and a nonlinear thermal conductivity :
Specify a thermal heat flux of applied at the left end for the first 300 seconds:
Set up initial conditions:
Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:
Solve the PDE:
Solve a linear version of the PDE:
Visualize the solutions:
Applications (7)
Boundary Conditions (5)
Compute the temperature field with model variables vars and parameters pars with a thermal surface of at the left boundary:
Set up the equation:
Solve the PDE:
Visualize the solution and note the sinusoidal temperature change on the left:
Compute the temperature field with model variables vars parameters pars:
Set up the equation with a thermal outflow boundary at the right end:
Define the initial temperature field:
Solve the PDE:
Visualize the solution and note how the energy leaves the domain through the thermal outflow boundary on the right:
Model a temperature field and a thermal radiation boundary with:
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :
Specify boundary condition parameters with a constant ambient temperature of and a surface emissivity of :
Specify the equation:
Set up initial conditions:
Solve the PDE:
Visualize the solution:
Model a temperature field with heat transfer boundary:
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :
Specify boundary condition parameters with an external flow temperature of and a heat transfer coefficient of :
Specify the equation:
Set up initial conditions:
Solve the PDE:
Visualize the solution:
Model a temperature field and a thermal insulation and a thermal heat flux boundary with:
Set up the heat transfer model variables vars:
Set up a region :
Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :
Specify boundary condition parameters for a heat flux of :
Specify the equation:
Set up initial conditions:
Solve the PDE:
Visualize the solution:
Coupled Equations (2)
Solve a coupled heat and mass transport model:
Set up the heat transfer mass transport model variables vars:
Set up a region :
Specify heat transfer and mass transport model parameters, heat source , thermal conductivity , mass diffusivity and mass source :
Set up the model and initial conditions:
Set up initial conditions:
Solve the model:
Visualize the solution:
Solve a coupled heat transfer and mass transport model with a thermal transfer value and a mass flux value on the boundary:
Set up the heat transfer mass transport model variables vars:
Set up a region :
Specify heat transfer and mass transport model parameters, heat source , thermal conductivity , mass diffusivity and mass source :
Specify boundary condition parameters for a thermal convection value with an external flow temperature of 1000 K and a heat transfer coefficient of :
Specify the equation:
Set up initial conditions:
Solve the model:
Visualize the solution:
Possible Issues (1)
For symbolic computation, the "ThermalConductivity" parameter should be given as a matrix:
For numeric values, the "ThermalConductivity" parameter is automatically converted to a matrix of proper dimensions:
This automatic conversion is not possible for symbolic input:
Not providing the properly dimensioned matrix will result in an error:
Tech Notes
Related Guides
Text
Wolfram Research (2020), HeatTransferPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html (updated 2022).
CMS
Wolfram Language. 2020. "HeatTransferPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html.
APA
Wolfram Language. (2020). HeatTransferPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html
BibTeX
@misc{reference.wolfram_2025_heattransferpdecomponent, author="Wolfram Research", title="{HeatTransferPDEComponent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html}", note=[Accessed: 09-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_heattransferpdecomponent, organization={Wolfram Research}, title={HeatTransferPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html}, note=[Accessed: 09-January-2026]}