# internalHeatSource

Specify internal heat source for a thermal model

## Syntax

## Description

`internalHeatSource(`

specifies an internal heat source for the thermal model. This syntax declares that
the entire geometry is a heat source.`thermalmodel`

,`heatSourceValue`

)

**Note**

Use `internalHeatSource`

for specifying **internal heat generators**, that is, for specifying
heat sources that belong to the geometry of the model. To specify a heat
influx from an external source, use the `thermalBC`

function with the
`HeatFlux`

parameter.

`internalHeatSource(`

specifies geometry regions of type `thermalmodel`

,`heatSourceValue`

,`RegionType`

,`RegionID`

)`RegionType`

with ID numbers
in `RegionID`

as heat sources. Always specify
`heatSourceValue`

first, then specify
`RegionType`

and `RegionID`

.

`internalHeatSource(___,"Label",`

adds a label for the internal heat source to be used by the `labeltext`

)`linearizeInput`

function. This function lets you pass internal heat
sources to the `linearize`

function that extracts sparse linear models for use with Control System Toolbox™.

returns the heat source object.`heatSource`

= internalHeatSource(___)

## Examples

### Specify Internal Heat Generation on Entire Geometry

Create a transient thermal model.

thermalmodel = createpde("thermal","transient");

Import the geometry.

`gm = importGeometry(thermalmodel,"SquareBeam.stl");`

Set thermal conductivity to `0.2`

, mass density to `2700e-9`

, and specific heat to `920`

.

thermalProperties(thermalmodel,"ThermalConductivity",0.2, ... "MassDensity",2700e-9, ... "SpecificHeat",920)

ans = ThermalMaterialAssignment with properties: RegionType: 'cell' RegionID: 1 ThermalConductivity: 0.2000 MassDensity: 2.7000e-06 SpecificHeat: 920

Specify that the entire geometry generates heat at the rate `2e-4`

.

internalHeatSource(thermalmodel,2e-4)

ans = HeatSourceAssignment with properties: RegionType: 'cell' RegionID: 1 HeatSource: 2.0000e-04 Label: []

### Specify a Face of a 2-D Geometry as a Heat Source

Create a steady-state thermal model.

thermalModel = createpde("thermal","transient");

Create the geometry.

SQ1 = [3; 4; 0; 3; 3; 0; 0; 0; 3; 3]; D1 = [2; 4; 0.5; 1.5; 2.5; 1.5; 1.5; 0.5; 1.5; 2.5]; gd = [SQ1 D1]; sf = 'SQ1+D1'; ns = char('SQ1','D1'); ns = ns'; dl = decsg(gd,sf,ns); geometryFromEdges(thermalModel,dl);

Set thermal conductivity to 50, mass density to 2500, and specific heat to 600.

thermalProperties(thermalModel,"ThermalConductivity",50, ... "MassDensity",2500, ... "SpecificHeat",600);

Specify that face 1 generates heat at 25.

`internalHeatSource(thermalModel,25,"Face",1)`

ans = HeatSourceAssignment with properties: RegionType: 'face' RegionID: 1 HeatSource: 25 Label: []

### Specify Nonconstant Internal Heat Source

Use a function handle to specify an internal heat source that depends on coordinates.

Create a thermal model for transient analysis and include the geometry. The geometry is a rod with a circular cross section. The 2-D model is a rectangular strip whose *x*-dimension extends from the axis of symmetry to the outer surface, and whose *y*-dimension extends over the actual length of the rod.

thermalmodel = createpde("thermal","transient"); g = decsg([3 4 0 0 .2 .2 -1.5 1.5 1.5 -1.5]'); geometryFromEdges(thermalmodel,g); pdegplot(thermalmodel.Geometry)

The heat is generated within the rod due to the radioactive decay. Therefore, the entire geometry is an internal nonlinear heat source and can be represented by a function of the *x*-coordinate, for example, $\mathit{q}=2000\mathit{x}$.

q = @(location,state)2000*location.x;

Specify the internal heat source for the transient model.

internalHeatSource(thermalmodel,q)

ans = HeatSourceAssignment with properties: RegionType: 'face' RegionID: 1 HeatSource: @(location,state)2000*location.x Label: []

### Specify Time-Dependent Internal Heat Source

Use a function handle to specify an internal heat source that depends on time.

Create a thermal model for transient analysis and include the geometry. The geometry is a rectangular strip.

thermalmodel = createpde("thermal","transient"); g = decsg([3 4 -1.5 1.5 1.5 -1.5 0 0 .2 .2]'); geometryFromEdges(thermalmodel,g);

Specify the thermal properties of the rod.

thermalProperties(thermalmodel,"ThermalConductivity",40,... "MassDensity",7800,... "SpecificHeat",500);

Specify the boundary conditions and initial temperature.

thermalBC(thermalmodel,"Edge",2,"Temperature",100); thermalBC(thermalmodel,"Edge",3,... "ConvectionCoefficient",50,... "AmbientTemperature",100); thermalIC(thermalmodel,0);

Specify that the entire geometry generates heat at the rate 20000*t* during the first 500 seconds, and then the heat source turns off. For details, see Time-Dependent Heat Source Function.

internalHeatSource(thermalmodel,@heatSource);

Generate the mesh, solve the model using the solution times from 0 to 50000 seconds, and plot the results.

generateMesh(thermalmodel); tfinal = 50000; tlist = 0:100:tfinal; result = solve(thermalmodel,tlist); T = result.Temperature; figure subplot(2,1,1) pdeplot(thermalmodel,"XYData",T(:,6),"Contour","on") axis equal title(sprintf("Temperature at %g s",tlist(6))) subplot(2,1,2) pdeplot(thermalmodel,"XYData",T(:,end),"Contour","on") axis equal title(sprintf("Temperature at %g s",tfinal))

Always ensure that your function returns a matrix of `NaN`

of the correct size when `state.time`

is `NaN`

. The solver properly recognizes a time-dependent problem by passing `NaN`

state values and looking for returned `NaN`

values. Without this condition, the solver might fail or return incorrect results.

internalHeatSource(thermalmodel,@heatSourceInvalid); result = solve(thermalmodel,tlist); T = result.Temperature; figure subplot(2,1,1) pdeplot(thermalmodel,"XYData",T(:,6),"Contour","on") axis equal title(sprintf("Temperature at %g s",tlist(6))) subplot(2,1,2) pdeplot(thermalmodel,"XYData",T(:,end),"Contour","on") axis equal title(sprintf("Temperature at %g s",tfinal))

**Time-Dependent Heat Source Function**

function Q = heatSource(location,state) Q = zeros(1,numel(location.x)); if(isnan(state.time)) % Returning a NaN when time=NaN tells % the solver that the heat source is a function of time. Q(1,:) = NaN; return end if state.time < 500 Q(1,:) = 20000*state.time; end end function Q = heatSourceInvalid(location,state) % No checks for NaN Q = zeros(1,numel(location.x)); if state.time < 500 Q(1,:) = 20000*state.time; end end

## Input Arguments

`thermalmodel`

— Thermal model

`ThermalModel`

object

Thermal model, specified as a `ThermalModel`

object.
The model contains the geometry, mesh, thermal properties of the material,
internal heat source, boundary conditions, and initial conditions.

**Example: **`thermalmodel = createpde("thermal","steadystate")`

`RegionType`

— Geometric region type

`"Face"`

| `"Cell"`

Geometric region type, specified as `"Face"`

for a 2-D
model or `"Cell"`

for a 3-D model.

**Example: **`internalHeatSource(thermalmodel,25,"Cell",1)`

**Data Types: **`char`

| `string`

`RegionID`

— Geometric region ID

vector of positive integers

Geometric region ID, specified as a vector of positive integers. Find the
region IDs by using `pdegplot`

.

**Example: **`internalHeatSource(thermalmodel,25,"Cell",1:3)`

**Data Types: **`double`

`heatSourceValue`

— Heat source value

number | function handle

Heat source value, specified as a number or a function handle. Use a function handle to specify the internal heat source that depends on space, time, or temperature. For details, see More About.

**Example: **`internalHeatSource(thermalmodel,25)`

**Data Types: **`double`

| `function_handle`

`labeltext`

— Label for internal heat source

character vector | string

Label for the internal heat source, specified as a character vector or a string.

**Data Types: **`char`

| `string`

## Output Arguments

`heatSource`

— Handle to heat source

`HeatSourceAssignment`

object

Handle to heat source, returned as a
`HeatSourceAssignment`

object. See HeatSourceAssignment Properties.

`heatSourceValue`

associates the heat source value with
the geometric region.

## More About

### Specifying Nonconstant Parameters of a Thermal Model

Use a function handle to specify these thermal parameters when they depend on space, temperature, and time:

Thermal conductivity of the material

Mass density of the material

Specific heat of the material

Internal heat source

Temperature on the boundary

Heat flux through the boundary

Convection coefficient on the boundary

Radiation emissivity coefficient on the boundary

Initial temperature (can depend on space only)

For example, use function handles to specify the thermal conductivity, internal heat source, convection coefficient, and initial temperature for this model.

thermalProperties(model,"ThermalConductivity", ... @myfunConductivity) internalHeatSource(model,"Face",2,@myfunHeatSource) thermalBC(model,"Edge",[3,4], ... "ConvectionCoefficient",@myfunBC, ... "AmbientTemperature",27) thermalIC(model,@myfunIC)

For all parameters, except the initial temperature, the function must be of the form:

`function thermalVal = myfun(location,state)`

For the initial temperature the function must be of the form:

`function thermalVal = myfun(location)`

The solver computes and populates the data in the `location`

and
`state`

structure arrays and passes this data to your function. You can
define your function so that its output depends on this data. You can use any names instead of
`location`

and `state`

, but the function must have exactly
two arguments (or one argument if the function specifies the initial temperature).

`location`

— A structure containing these fields:`location.x`

— The*x*-coordinate of the point or points`location.y`

— The*y*-coordinate of the point or points`location.z`

— For a 3-D or an axisymmetric geometry, the*z*-coordinate of the point or points`location.r`

— For an axisymmetric geometry, the*r*-coordinate of the point or points

Furthermore, for boundary conditions, the solver passes these data in the

`location`

structure:`location.nx`

—*x*-component of the normal vector at the evaluation point or points`location.ny`

—*y*-component of the normal vector at the evaluation point or points`location.nz`

— For a 3-D or an axisymmetric geometry,*z*-component of the normal vector at the evaluation point or points`location.nr`

— For an axisymmetric geometry,*r*-component of the normal vector at the evaluation point or points

`state`

— A structure containing these fields for transient or nonlinear problems:`state.u`

— Temperatures at the corresponding points of the location structure`state.ux`

— Estimates of the*x*-component of temperature gradients at the corresponding points of the location structure`state.uy`

— Estimates of the*y*-component of temperature gradients at the corresponding points of the location structure`state.uz`

— For a 3-D or an axisymmetric geometry, estimates of the*z*-component of temperature gradients at the corresponding points of the location structure`state.ur`

— For an axisymmetric geometry, estimates of the*r*-component of temperature gradients at the corresponding points of the location structure`state.time`

— Time at evaluation points

Thermal material properties (thermal conductivity, mass density, and specific heat) and internal heat source get these data from the solver:

`location.x`

,`location.y`

,`location.z`

,`location.r`

Subdomain ID

`state.u`

,`state.ux`

,`state.uy`

,`state.uz`

,`state.r`

,`state.time`

Boundary conditions (temperature on the boundary, heat flux, convection coefficient, and radiation emissivity coefficient) get these data from the solver:

`location.x`

,`location.y`

,`location.z`

,`location.r`

`location.nx`

,`location.ny`

,`location.nz`

,`location.nr`

`state.u`

,`state.time`

Initial temperature gets the following data from the solver:

`location.x`

,`location.y`

,`location.z`

,`location.r`

Subdomain ID

For all thermal parameters, except for thermal conductivity, your function must return a row
vector `thermalVal`

with the number of columns
equal to the number of evaluation points, for example, ```
M =
length(location.y)
```

.

For thermal conductivity, your function must return a matrix
`thermalVal`

with number of rows equal to 1, `Ndim`

,
`Ndim*(Ndim+1)/2`

, or `Ndim*Ndim`

, where
`Ndim`

is 2 for 2-D problems and 3 for 3-D problems. The number of columns
must equal the number of evaluation points, for example, ```
M =
length(location.y)
```

. For details about dimensions of the matrix, see c Coefficient for specifyCoefficients.

If properties depend on the time or temperature, ensure that your function returns a matrix of
`NaN`

of the correct size when `state.u`

or
`state.time`

are `NaN`

. Solvers check whether a problem is
time dependent by passing `NaN`

state values and looking for returned
`NaN`

values.

### Additional Arguments in Functions for Nonconstant Thermal Parameters

To use additional arguments in your function, wrap your function (that takes additional arguments) with an anonymous function that takes only the `location`

and `state`

arguments. For example:

thermalVal = ... @(location,state) myfunWithAdditionalArgs(location,state,arg1,arg2...) thermalBC(model,"Edge",3,"Temperature",thermalVal) thermalVal = @(location) myfunWithAdditionalArgs(location,arg1,arg2...) thermalIC(model,thermalVal)

## Version History

**Introduced in R2017a**

### R2021b: Label to extract sparse linear models for use with Control System Toolbox

Now you can add a label for the internal heat source to be used by the `linearizeInput`

function. This function lets you pass internal heat
sources to the `linearize`

function that extracts sparse linear models for use with Control System Toolbox.

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