Steady state heat transfer in solid materials - NAFEMS Benchmark
Model definition
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 |
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 |
bottom boundary | temperature | 373.15 | K |
right boundary | heat flux | -h (T - T) | W/m |
top boundary | heat flux | -h (T - T) | W/m |
ambient (T) | temperature | 273.15 | K |
natural convection | heat transfer coefficient (h) | 750.0 | W/(m K) |
front boundary | heat flux | 0.0 | W/m |
back boundary | heat flux | 0.0 | W/m |
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
.
- Click on
Apply
and thenConfirm 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 toShared regions
and selectsurface
to create a new surface. Name the region as Insulated. Select the surface highlighted in red and click onApply
to set this surface as the Insulated region. -
Click again on the
+
icon next toShared regions
and selectsurface
to create another new surface. Name the region as ConstantTemperature. Select the surface highlighted in red and click onApply
to set this surface as the ConstantTemperature region. -
Click again on the
+
icon next toShared regions
and selectsurface
to create another new surface. Name the region as NaturalConvection. Select the surfaces highlighted in red and click onApply
to set this surface as the NaturalConvection region. -
Click on the
+
icon next toMaterials
and selectNew material
from the list and clickConfirm
.
-
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 clickApply
. -
Scroll down the middle pane and click on the
+
icon next toProperties
. -
Select
Density
from the list. Repeat the previous step to addHeat capacity
andThermal conductivity
. Assign the values as given in the material data. -
Click on the
+
icon next toShared expressions
and add a new expression of typeExpression
from theExpression 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 onApply
to confirm. -
Click on the
+
icon next toShared expressions
and add another new expression of typeExpression
from theExpression 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 onApply
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. SelectHeat 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 toHeat solid
and a list of interactions appears. -
Select
Constraint
interaction and choose the predefined regionConstantTemperature
fromSelect region
as the target region. Click onApply
. -
Assign the value of 373.15 to the constraint and click on
Apply
. -
Add another interaction by clicking on the
+
icon next toHeat solid
and SelectHeat source
interaction. Choose the predefined regionNaturalConvection
fromSelect region
as the target region and click onApply
. -
Assign an expression of -h * (T - Tamb) to the source and click on
Apply
.
Step 4 - Meshing the geometry
-
Proceed to the
Simulations
tab and add click on+
icon next to theMeshes
to add a new mesh. -
Under the
MESH SETTINGS
, click onMesh quality
to view the dropdown menu and change the selection fromDefault
toExpert Settings
. -
Scroll down to
Structured meshing
and click onAdd structured entity
, which opens a list of geometric entity. SelectVolume
entity from the list. -
Click in the input box
Segments
forA
. This highlights the edges that are defined by this segment. Assign 1 as the value and click onApply
, which will create one element in Z-direction to mimic two dimensional domain. -
Click in the input box
Segments
forB
. This highlights the edges that are defined by this segment. Assign 50 as the value and click onApply
, which will create elements of length of 0.02 m in Y-direction. -
Click in the input box
Segments
forC
. This highlights the edges that are defined by this segment. Assign 30 as the value and click onApply
, 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 toMesh results
underMESH SETTINGS
and click onShow 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
, selectSteady state
and click onApply
. -
Scroll down under
SIMULATION SETTINGS
and setNode type
to 1 CPU, 16 GB andNode count
to 1 for the current simulation. -
Click on
Mesh
underSimulation 1
and selectMesh 1
to set this mesh for the current simulation. -
Click on
+
icon next to theOutputs
. UnderField outputs
selectTemperature field
. The outputs allow to visualize the results of the simulation on the target region. -
Click on
+
icon next to theOutputs
. UnderValue outputs
selectCustom
. 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 theOutput expression
to interpolate(reg.solid_target, T, [0.3, -0.3, 0]) and click onApply
.
-
Now click on
Run simulation
button. The simulation status changes fromNot run
toRunning
. -
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 theScript
underSimulation 1
and click onScripting mode
. -
Scroll down the script and change
fld.T
(highlighted) in theHeat source interaction
toqs.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 toVisualizations
. On this added visualization, click on+
icon and select fieldT
. -
Click on
Activate current visualization
. The steady-state temperature field distribution becomes visible. -
To check the
ProbeTemperature
, click onSummary
underResults
. 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.