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:
  • parameterdefaultsymbol
    "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:
  • dimensionreductionequation
    1D
    2D
  • 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

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Basic 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:

del .(-k del Theta(x,y))^(︷^( heat transfer model )) =|_(Gamma_(x=0))epsilon k_B ((Theta_(amb)-Theta_(ref))^4-(Theta(x,y)-Theta_(ref))^( 4))^(︷^( heat radiation boundary ))+|_(Gamma_(x=0))h (Theta_(ext)(x,y)-Theta(x,y))^(︷^( heat transfer boundary ))

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:

rho C_p(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))^(︷^( heat transfer model )) =|_(Gamma_(x=0))q(t,x)^(︷^( heat flux ))

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:

rho C_p(partialTheta(t, x))/(partialt)+del .(-k(Theta) del Theta(t,x))^(︷^( heat transfer model )) =|_(Gamma_(x=0))q(t,x)^(︷^( heat flux ))

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:

rho C_p(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))^(︷^( heat transfer model )) =|_(Gamma_(x=0))epsilon k_B ((Theta_(amb)-Theta_(ref))^4-(Theta(t,x)-Theta_(ref))^( 4))^(︷^( heat radiation 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 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:

rho C_p(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))^(︷^( heat transfer model )) =|_(Gamma_(x=0))h (Theta_(ext)(t,x)-Theta(t,x))^(︷^( 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:

rho C_p(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))^(︷^( heat transfer model )) =|_(Gamma_(x=0))0^(︷^( heat insulation ))+|_(Gamma_(x=1/5))q(t,x)^(︷^( thermal heat flux ))

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:

(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))-Q^(︷^( heat transfer model )) = 0; (partialc(t,x))/(partialt)+del .(-d del c(t,x))-R^(︷^( mass transport model )) = 0

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:

(partialTheta(t, x))/(partialt)+del .(-k del Theta(t,x))-Q^(︷^( heat transfer model )) = |_(Gamma_(x=1))h (Theta_(ext)(t,x)-Theta(t,x))^(︷^( heat transfer boundary )); (partialc(t,x))/(partialt)+del .(-d del c(t,x))-R^(︷^( mass transport model )) = |_(Gamma_(x=0||x=1))q (t,x)^(︷^( mass flux 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:

Wolfram Research (2020), HeatTransferPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html (updated 2022).

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: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_heattransferpdecomponent, organization={Wolfram Research}, title={HeatTransferPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/HeatTransferPDEComponent.html}, note=[Accessed: 22-February-2025 ]}