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 [], independent variables in [] and time variable in [].
- 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 [], or equivalently in [].
- The following parameters pars can be given:
-
parameter default symbol "HeatConvectionVelocity" {0,…} , flow velocity [] "HeatSource" 0 , heat source [] "MassDensity" 1 - , density []
"Material" Automatic "ModelForm" "NonConservative" none "RegionSymmetry" None "SpecificHeatCapacity" 1 , specific heat capacity [] "ThermalConductivity" IdentityMatrix , thermal conductivity [ - 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 equation 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
open allclose allBasic Examples (4)
Scope (7)
Basic Examples (2)
1D (1)
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 :
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:
3D (1)
Model a temperature field with two heat conditions at the sides and an orthotropic thermal conductivity :
Set up the heat transfer model variables :
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:
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 :
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 the equation with a thermal heat flux of applied at the left end for the first 300 seconds:
Time-Dependent Nonlinear (1)
Model a temperature field with a nonlinear heat conductivity term with:
Set up the heat transfer model variables :
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 the equation with a thermal heat flux of applied at the left end for the first 300 seconds:
Applications (7)
Boundary Conditions (5)
Compute the temperature field with model variables and parameters with a thermal surface of at the left boundary:
Visualize the solution and note the sinusoidal temperature change on the left:
Compute the temperature field with model variables parameters :
Set up the equation with a thermal outflow boundary at the right end:
Define the initial temperature field:
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 :
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 :
Model a temperature field with heat transfer boundary:
Set up the heat transfer model variables :
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 :
Model a temperature field and a thermal insulation and a thermal heat flux boundary with:
Set up the heat transfer model variables :
Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :
Coupled Equations (2)
Solve a coupled heat and mass transport model:
Set up the heat transfer mass transport model variables :
Specify heat transfer and mass transport model parameters, heat source , thermal conductivity , mass diffusivity and mass source :
Set up the model and initial conditions:
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 :
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 :
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:
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