Steady state heat transfer in solid materials - NAFEMS Benchmark

Model definition

INTRO_nafems2dheattransfer — Fig. 1: Problem setup.

This model simulates NAFEMS benchmark test for thermal analysis. A steady-state thermal analysis is performed with different boundary conditions (BCs) including adiabatic (insulated or zero heat flux), Dirichlet BCs, and finite heat flux by natural convection specified by a known heat transfer coefficient. The result is the steady-state temperature distribution in the spatial domain. Fig. 1 shows the schematic of the spatial domain with the BCs.

Output Results:

Temperature at coordinates (0.3, -0.3, 0.0) m, which is on the right boundary exposed to ambient. The benchmark value is 291.4 K.

Material Data

Property	Value	Unit
Density	1.0	kg/m $^3$
Specific heat capacity	1.0	J/(kg K)
Thermal conductivity	52.0	W/(m K)

Boundary conditions

Name	Type	Value	Unit
left boundary	heat flux	0.0	W/m $^2$
bottom boundary	temperature	373.15	K
right boundary	heat flux	-h (T - T $_{\text{amb}}$ )	W/m $^2$
top boundary	heat flux	-h (T - T $_{\text{amb}}$ )	W/m $^2$
ambient (T $_{\text{amb}}$ )	temperature	273.15	K
natural convection	heat transfer coefficient (h)	750.0	W/(m $^2$ K)
front boundary	heat flux	0.0	W/m $^2$
back boundary	heat flux	0.0	W/m $^2$

Step-by-step guide

Here you’ll find a step-by-step tutorial on how to simulate this in Quanscient Allsolve

Step 1 - Create geometry

Start with a new project.
Click on box icon under Create a geometry.
In the geometry settings, provide the size of the box in meters: X = 0.6; Y = 1.0; Z = 0.0.

Note: The origin of the coordinate system is at the center of the solid block.

Click on Apply and then Confirm model changes.

Step 2 - Define regions, materials, and shared expressions

Proceed to the Properties tab to define regions and materials.
Click on the + icon next to Shared regions and select surface to create a new surface. Name the region as Insulated. Select the surface highlighted in red and click on Apply to set this surface as the Insulated region.
Click again on the + icon next to Shared regions and select surface to create another new surface. Name the region as ConstantTemperature. Select the surface highlighted in red and click on Apply to set this surface as the ConstantTemperature region.
Click again on the + icon next to Shared regions and select surface to create another new surface. Name the region as NaturalConvection. Select the surfaces highlighted in red and click on Apply to set this surface as the NaturalConvection region.
Click on the + icon next to Materials and select New material from the list and click Confirm.

Note: We are creating a user-defined material as this specific material is not available in the predefined materials library.

Rename the material to Solid and optionally add the description NAFEMS material. To apply this material on the geometry, Click on Add volume, select the solid volume and click Apply.
Scroll down the middle pane and click on the + icon next to Properties.
Select Density from the list. Repeat the previous step to add Heat capacity and Thermal conductivity. Assign the values as given in the material data.
Click on the + icon next to Shared expressions and add a new expression of type Expression from the Expression type dropdown menu. Name the expression h and add the description (optional) Heat transfer coefficient (W/m^2/K). Assign the value of 750 to the expression and click on Apply to confirm.
Click on the + icon next to Shared expressions and add another new expression of type Expression from the Expression type dropdown menu. Name the expression Tamb and add the description (optional) Ambient temperature (K). Assign the value of 273.15 to the expression and click on Apply to confirm.

Step 3 - Define the physics and apply boundary conditions

Proceed to the Physics tab to define physics and interactions.
Click on the + icon to add a new physics. Select Heat solid.
Under PHYSICS SETTINGS, if no specific volume is added, this physics is by default applied to the whole geometry.
To add interactions, click on the + icon next to Heat solid and a list of interactions appears.
Select Constraint interaction and choose the predefined region ConstantTemperature from Select region as the target region. Click on Apply.
Assign the value of 373.15 to the constraint and click on Apply.
Add another interaction by clicking on the + icon next to Heat solid and Select Heat source interaction. Choose the predefined region NaturalConvection from Select region as the target region and click on Apply.
Assign an expression of -h * (T - Tamb) to the source and click on Apply.

Note 1: Whereever a symbol $f_x$ is visible in the input box, instead of numerical values, you can also use expressions in terms of previously defined Shared expressions and Fields available in the simulation.

Note 2: In the finite element formulation, the zero heat flux (zero Neumann) BC is naturally applied; therefore, there is no need to set it additionally.

Step 4 - Meshing the geometry

Proceed to the Simulations tab and add click on + icon next to the Meshes to add a new mesh.
Under the MESH SETTINGS, click on Mesh quality to view the dropdown menu and change the selection from Default to Expert Settings.
Scroll down to Structured meshing and click on Add structured entity, which opens a list of geometric entity. Select Volume entity from the list.
Click in the input box Segments for A. This highlights the edges that are defined by this segment. Assign 1 as the value and click on Apply, which will create one element in Z-direction to mimic two dimensional domain.
Click in the input box Segments for B. This highlights the edges that are defined by this segment. Assign 50 as the value and click on Apply, which will create elements of length of 0.02 m in Y-direction.
Click in the input box Segments for C. This highlights the edges that are defined by this segment. Assign 30 as the value and click on Apply, which will create elements of length of 0.02 m in X-direction.
Click on Mesh to generate the mesh.
Once the mesh status changes to Success scroll down to Mesh results under MESH SETTINGS and click on Show preview to see the generated mesh.

Step 5 - Apply simulation settings

Proceed to the Simulations tab and add a new simulation by clicking the + icon.
In Analysis Type, select Steady state and click on Apply.
Scroll down under SIMULATION SETTINGS and set Node type to 1 CPU, 16 GB and Node count to 1 for the current simulation.
Click on Mesh under Simulation 1 and select Mesh 1 to set this mesh for the current simulation.
Click on + icon next to the Outputs. Under Field outputs select Temperature field. The outputs allow to visualize the results of the simulation on the target region.
Click on + icon next to the Outputs. Under Value outputs select Custom. The value outputs provide a single value. In this case, we will interpolate the value of temperature at coordinates (0.3, -0.3, 0.0) m.
Set the name of the custom output to ProbeTemperature in Name input box. Set the Output expression to interpolate(reg.solid_target, T, [0.3, -0.3, 0]) and click on Apply.

Note: When you hover on the interpolate function call, the usage of the function is displayed.

Now click on Run simulation button. The simulation status changes from Not run to Running.
To check the progress of the simulation, click on Logs. The logs show the relative change at each iteration.
As the relative change is increasing with each iteration, the simulation will not converge to a steady-state simulation. Therefore, we will Abort this simulation.
To solve the convergence issue, we will modify the formulation in the Script. Click on the Script under Simulation 1 and click on Scripting mode.
Scroll down the script and change fld.T (highlighted) in the Heat source interaction to qs.dof(fld.T). This change makes the temperature field as unknown which is solved for in the iteration. The idea here is to linearize the boundary source term, which in turn makes the diagonal of the assembly matrix dominant, resulting in the well-conditioned problem.
Run the simulation again and this time the simulation converges within one iteration as seen from the Logs.

Step 6 - Post-processing of the simulation results

Once the simulation status changes to Success, we can add a visualization. Click on + icon next to Visualizations. On this added visualization, click on + icon and select field T.
Click on Activate current visualization. The steady-state temperature field distribution becomes visible.
To check the ProbeTemperature, click on Summary under Results. The value of the temperature at the probe location is 291.404 K, which is very close to the benchmark value of 291.4 K.

References

[1] A.D. Cameron, J.A. Casey, G.B. Simpson. Benchmark Tests for Thermal Analysis. NAFEMS Documentation.