For a 2-D steady-state thermal model, evaluate temperature gradients at the nodal locations and at the points specified byxandycoordinates.

Create a thermal model for steady-state analysis.

thermalmodel = createpde("thermal");

Create the geometry and include it in the model.

R1 = [3,4,-1,1,1,-1,1,1,-1,-1]'; g = decsg(R1,'R1',('R1')'); geometryFromEdges(thermalmodel,g); pdegplot(thermalmodel,"EdgeLabels","on") xlim([-1.5 1.5]) axisequal

Assuming that this geometry represents an iron plate, the thermal conductivity is $$79。5W/(mK)$$。

thermalProperties(thermalmodel,"ThermalConductivity",79.5,"Face",1);

Apply a constant temperature of 300 K to the bottom of the plate (edge 3). Also, assume that the top of the plate (edge 1) is insulated, and apply convection on the two sides of the plate (edges 2 and 4).

thermalBC(thermalmodel,"Edge",3,"Temperature",300); thermalBC(thermalmodel,"Edge",1,"HeatFlux",0); thermalBC(thermalmodel,"Edge",[2 4],。.."ConvectionCoefficient",25,。.."AmbientTemperature",50);

Mesh the geometry and solve the problem.

generateMesh(thermalmodel); results = solve(thermalmodel)

结果= SteadyStateThermalResults properties: Temperature: [1541x1 double] XGradients: [1541x1 double] YGradients: [1541x1 double] ZGradients: [] Mesh: [1x1 FEMesh]

The solver finds the temperatures and temperature gradients at the nodal locations. To access these values, useresults.Temperature,results.XGradients, and so on. For example, plot the temperature gradients at nodal locations.

figure; pdeplot(thermalmodel,。.."FlowData",[results.XGradients results.YGradients]);

Create a grid specified byxandycoordinates, and evaluate temperature gradients to the grid.

v = linspace(-0.5,0.5,11); [X,Y] = meshgrid(v); [gradTx,gradTy] =。..evaluateTemperatureGradient(results,X,Y);

Reshape thegradTxandgradTyvectors, and plot the resulting temperature gradients.

gradTx = reshape(gradTx,size(X)); gradTy = reshape(gradTy,size(Y)); figure quiver(X,Y,gradTx,gradTy)

Alternatively, you can specify the grid by using a matrix of query points.

querypoints = [X(:) Y(:)]'; [gradTx,gradTy] =。..evaluateTemperatureGradient(results,querypoints); gradTx = reshape(gradTx,size(X)); gradTy = reshape(gradTy,size(Y)); figure quiver(X,Y,gradTx,gradTy)

For a 3-D steady-state thermal model, evaluate temperature gradients at the nodal locations and at the points specified byx,y, andzcoordinates.

Create a thermal model for steady-state analysis.

thermalmodel = createpde("thermal");

Create the following 3-D geometry and include it in the model.

importGeometry(thermalmodel,"Block.stl"); pdegplot(thermalmodel,"FaceLabels","on","FaceAlpha",0.5) title("Copper block, cm") axisequal

Assuming that this is a copper block, the thermal conductivity of the block is approximately $$4W/(cmK)$$。

thermalProperties(thermalmodel,"ThermalConductivity",4);

Apply a constant temperature of 373 K to the left side of the block (edge 1) and a constant temperature of 573 K to the right side of the block.

thermalBC(thermalmodel,"Face",1,"Temperature",373); thermalBC(thermalmodel,"Face",3,"Temperature",573);

Apply a heat flux boundary condition to the bottom of the block.

thermalBC(thermalmodel,"Face",4,"HeatFlux",-20);

Mesh the geometry and solve the problem.

generateMesh(thermalmodel); thermalresults = solve(thermalmodel)

thermalresults = SteadyStateThermalResults with properties: Temperature: [12691x1 double] XGradients: [12691x1 double] YGradients: [12691x1 double] ZGradients: [12691x1 double] Mesh: [1x1 FEMesh]

The solver finds the values of temperatures and temperature gradients at the nodal locations. To access these values, useresults.Temperature,results.XGradients, and so on.

Create a grid specified byx,y, andzcoordinates, and evaluate temperature gradients to the grid.

[X,Y,Z] = meshgrid(1:26:100,1:6:20,1:11:50); [gradTx,gradTy,gradTz] =。..evaluateTemperatureGradient(thermalresults,X,Y,Z);

Reshape thegradTx,gradTy, andgradTzvectors, and plot the resulting temperature gradients.

gradTx = reshape(gradTx,size(X)); gradTy = reshape(gradTy,size(Y)); gradTz = reshape(gradTz,size(Z)); figure quiver3(X,Y,Z,gradTx,gradTy,gradTz) axisequalxlabel("x") ylabel("y") zlabel("z")

Alternatively, you can specify the grid by using a matrix of query points.

querypoints = [X(:) Y(:) Z(:)]'; [gradTx,gradTy,gradTz] =。..evaluateTemperatureGradient(thermalresults,querypoints); gradTx = reshape(gradTx,size(X)); gradTy = reshape(gradTy,size(Y)); gradTz = reshape(gradTz,size(Z)); figure quiver3(X,Y,Z,gradTx,gradTy,gradTz) axisequalxlabel("x") ylabel("y") zlabel("z")

Solve a 2-D transient heat transfer problem on a square domain and compute temperature gradients at the convective boundary.

Create a transient thermal model for this problem.

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

Create the geometry and include it in the model.

g = @squareg; geometryFromEdges(thermalmodel,g); pdegplot(thermalmodel,"EdgeLabels","on") xlim([-1.2 1.2]) ylim([-1.2 1.2]) axisequal

Assign the following thermal properties: thermal conductivity is $$100W/(m^{\circ }C)$$, mass density is $$7800kg/m^3$$, and specific heat is $$500J/(k{g}^{\circ }C)$$。

thermalProperties(thermalmodel,"ThermalConductivity", 100,。.."MassDensity",7800,。.."SpecificHeat",500);

Apply insulated boundary conditions on three edges and the free convection boundary condition on the right edge.

thermalBC(thermalmodel,"Edge",[1 3 4],"HeatFlux",0); thermalBC(thermalmodel,"Edge",2,。.."ConvectionCoefficient",5000,。.."AmbientTemperature",25);

Set the initial conditions: uniform room temperature across domain and higher temperature on the left edge.

thermalIC(thermalmodel,25); thermalIC(thermalmodel,100,"Edge",4);

Generate a mesh and solve the problem using0:1000:200000as a vector of times.

generateMesh(thermalmodel); tlist = 0:1000:200000; thermalresults = solve(thermalmodel,tlist);

Define a line at convection boundary and compute temperature gradients across that line.

X = -1:0.1:1; Y = ones(size(X)); [gradTx,gradTy] = evaluateTemperatureGradient(thermalresults,。..X,Y,1:length(tlist));

Plot the interpolated gradient componentgradTxalong thexaxis for the following values from the time intervaltlist。

figure t = [51:50:201];fori = t p(i) = plot(X,gradTx(:,i),"DisplayName",。..strcat("t=",num2str(tlist(i)))); holdonendlegend(p(t)) xlabel("x") ylabel("gradTx")