HeatTransferPDEComponent

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`HeatTransferPDEComponent`

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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,…,x_n],{x₁,…,x_n}}.
Time-dependent variables vars are vars={Θ[t,x₁,…,x_n],t,{x₁,…,x_n}}.
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 []
"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 []
"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 …,keyp_i…,p_iv_i,…], the parameters are replaced with .

Examples

open allclose all

Basic Examples (4)Summary of the most common use cases

Define a time-independent heat transfer model:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-gratei

Out[1]=1

Define a time-dependent heat transfer model:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-1a2ea6

Out[1]=1

Set up a time-dependent heat transfer model with particular material parameters:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-k1ns9q

Out[1]=1

Model a temperature field with a heat source in a rod:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-t36opy

Out[1]=1

Solve the PDE:

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rksc17

Visualize the solution:

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-khqjar

Out[3]=3

Scope (7)Survey of the scope of standard use cases

Basic Examples (2)

Set up a time-dependent heat transfer model for a particular material:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-y30p7j

Out[1]=1

Set up a time-dependent heat transfer model for several material regions:

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-zjwnmy

Out[1]=1

1D (1)

Model a temperature field with two heat conditions at the sides:

Set up the heat transfer model variables :

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-vc1i8f

Set up a region :

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ywex3f

Specify heat transfer model parameter thermal conductivity :

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-kzw74j

Specify heat surface conditions:

In[4]:=4

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rg1afc

Out[4]=4

Set up the equation:

In[5]:=5

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-38wl2z

Out[5]=5

Solve the PDE:

In[6]:=6

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-d4x4w2

Visualize the solution:

In[7]:=7

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-fpq186

Out[7]=7

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 :

In[1]:=1

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-m21j9s

Set up a rectangular domain with a width of and a height of :

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-0ob17d

Specify thermal conductivity :

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-6n11ho

Set up temperature surface boundary conditions at the left and right boundaries:

In[4]:=4

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-1a3zuy

Out[4]=4

Set up a heat transfer boundary condition on the top surface:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-4lussn

Out[5]=5

Also set up a thermal radiation boundary condition on the top surface:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-jrz2i2

Out[6]=6

Set up the equation:

In[7]:=7

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-7ecwjm

Out[7]=7

Solve the PDE:

In[8]:=8

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-qc09l9

Visualize the solution:

In[9]:=9

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-nl09e2

Out[9]=9

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 :

In[1]:=1

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-mriezn

Set up a region :

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-vvom6e

Specify an orthotropic thermal conductivity :

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-cgktag

Specify heat surface conditions:

In[4]:=4

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-0mlvs2

Out[4]=4

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

In[5]:=5

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-x8o7sh

Out[5]=5

Solve the PDE:

In[6]:=6

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-fki728

Visualize the solution:

In[7]:=7

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-lqb6x0

Out[7]=7

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 :

In[1]:=1

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-34j9jk

Set up a region :

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-r7tzll

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-nh5hwt

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-qhujyr

Set up initial conditions:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-mo5420

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-q9buq4

Out[6]=6

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-hdh9t

Visualize the solution:

In[8]:=8

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-9rw1lg

Out[8]=8

Time-Dependent Nonlinear (1)

Model a temperature field with a nonlinear heat conductivity term with:

Set up the heat transfer model variables :

In[1]:=1

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-89v7o3

Set up a region :

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-pwm7no

Specify heat transfer model parameters mass density , specific heat capacity and a nonlinear thermal conductivity :

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-52zc5m

Specify a thermal heat flux of applied at the left end for the first 300 seconds:

In[4]:=4

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-olua29

Set up initial conditions:

In[5]:=5

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-07dtcx

Set up the equation with a thermal heat flux of applied at the left end for the first 300 seconds:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-q79fer

Out[6]=6

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ba1s25

Solve a linear version of the PDE:

In[8]:=8

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-vk4bd5

Visualize the solutions:

In[11]:=11

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-q1rikc

Out[11]=11

Applications (7)Sample problems that can be solved with this function

Boundary Conditions (5)

Compute the temperature field with model variables and parameters with a thermal surface of at the left boundary:

In[4]:=4

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-7u3cv4

Set up the equation:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-oksi8h

Out[6]=6

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-fib84z

Visualize the solution and note the sinusoidal temperature change on the left:

In[9]:=9

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-1wx5br

Out[9]=9

Compute the temperature field with model variables parameters :

In[1]:=1

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-6dumtk

Set up the equation with a thermal outflow boundary at the right end:

In[2]:=2

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-m5d5sp

Out[2]=2

Define the initial temperature field:

In[3]:=3

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https://wolfram.com/xid/0jvrfas5pq4yornuf19e-9p9r14

Solve the PDE:

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-c3ji41

Visualize the solution and note how the energy leaves the domain through the thermal outflow boundary on the right:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-5hqogr

Out[5]=5

Model a temperature field and a thermal radiation boundary with:

Set up the heat transfer model variables :

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-4dpxxz

Set up a region :

In[2]:=2

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ne9b6c

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

In[3]:=3

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-7my2xp

Specify boundary condition parameters with a constant ambient temperature of and a surface emissivity of :

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-dd4ukq

Specify the equation:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-lp5vnt

Out[5]=5

Set up initial conditions:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-iecz68

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-jl65z5

Visualize the solution:

In[8]:=8

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-bhb4ri

Out[8]=8

Model a temperature field with heat transfer boundary:

Set up the heat transfer model variables :

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-h8inxy

Set up a region :

In[2]:=2

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rwwjwk

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

In[3]:=3

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-niwhpg

Specify boundary condition parameters with an external flow temperature of and a heat transfer coefficient of :

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ugxrli

Specify the equation:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-trtpk1

Out[5]=5

Set up initial conditions:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ldz1rm

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-yzdc2u

Visualize the solution:

In[8]:=8

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ncr9ny

Out[8]=8

Model a temperature field and a thermal insulation and a thermal heat flux boundary with:

Set up the heat transfer model variables :

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-58b0vw

Set up a region :

In[2]:=2

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-pxkjuo

Specify heat transfer model parameters mass density , specific heat capacity and thermal conductivity :

In[3]:=3

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-hxcnyd

Specify boundary condition parameters for a heat flux of :

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-ml9b8u

Specify the equation:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-8fooqp

Out[5]=5

Set up initial conditions:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rmaw60

Solve the PDE:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-nnhj0f

Visualize the solution:

In[8]:=8

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-31aqe5

Out[8]=8

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 :

In[1]:=1

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rc8kak

Set up a region :

In[3]:=3

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-4hrepw

Specify heat transfer and mass transport model parameters, heat source , thermal conductivity , mass diffusivity and mass source :

In[4]:=4

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-rhdean

Set up the model and initial conditions:

In[5]:=5

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-o2zn4n

Out[5]=5

Set up initial conditions:

In[6]:=6

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-sk1jic

Solve the model:

In[7]:=7

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-d8tpd9

Visualize the solution:

In[8]:=8

✖

https://wolfram.com/xid/0jvrfas5pq4yornuf19e-tue3fc

Out[8]=8

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 ))