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Thermo-Photo-Voltaic Cell
Introduction
The following example illustrates an application that maximizes surface-to-surface radiative fluxes and minimizes conductive heat fluxes.
A thermo-photo-voltaic (TPV) cell generates electricity from the combustion of fuel and through radiation. Figure 1 depicts the general operating principle. The fuel burns inside an emitting device that radiates intensely. Photo-voltaic (PV) cells—almost like solar cells—capture the radiation and convert it to electricity. The efficiency of a TPV device ranges from 1 % to 20 %. In some cases, TPVs are used in heat generators to co-generate electricity, and the efficiency is not so critical. In other cases TPVs are used as electric power sources, for example in automobiles (Ref. 1). In those cases efficiency is a major concern.
Figure 1: Operating principle of a TPV device (Ref. 2), and an image of a prototype system (Ref. 3).
TPV systems, unlike typical electronic systems, must maximize radiation heat transfer to improve efficiency. However, inherent radiation losses—radiation not converted to electric power—contributes to the PV cells’ increased temperature. Further, heat transfer through conduction results in increased cell temperature. PV cells have a limited operating temperature range that depends on the type of material used. Solar cells are limited to temperatures below 80 °C, whereas high-efficiency semiconductor materials can withstand as much as 1000 °C. Photovoltaic efficiency is often a function of temperature with a maximum at some temperature above ambient.
To improve system efficiency, engineers prefer to use high-efficiency PV cells, which however can be quite expensive. To reduce system costs, engineers work with smaller-area PV cells and then use mirrors to focus the radiation on them. However, there is a limit for how much you can focus the beams; if the radiation intensity becomes too high, the cells can overheat. Thus engineers must optimize system geometry and operating conditions to achieve maximum performance at minimum material costs.
The following application, which uses the Heat Transfer with Surface-to-Surface Radiation interface, investigates the influence of operating conditions (flame temperature) on system efficiency and the temperature of components in a typical TPV system. The application can also assess the influence of geometry changes.
Model Definition
Figure 2 depicts the geometry and dimensions of the system under study. To reduce the temperature, the PV cells are water cooled on their back side (at the interface with the insulation).
Figure 2: Geometry and dimensions of the modeled TPV system.
Conduction is always present on the different boundaries. The model simulates the emitter with a specific temperature, Theater, on the inner boundary. At the outer emitter boundary, it takes radiation (surface-to-surface) into account in the boundary condition. It simulates the mirrors by taking radiation into account on all boundaries and applying a low emissivity. The inner boundaries of the PV cells and of the insulation also make use of radiation boundary conditions. However, the PV cells have a high emissivity and the insulation a low emissivity. Further, the PV cells convert a fraction of the irradiation to electricity instead of heat. Heat sinks on their inner boundaries simulate this effect by accounting for a boundary heat source, q, defined by
where G is the irradiation flux (W/m2) and ηpv is the PV cell’s voltaic efficiency. The latter depends on the local temperature, with a maximum of 0.2 at 800 K:
Figure 3 illustrates this expression for temperatures above 1000 K.
Figure 3: PV cell voltaic efficiency versus temperature.
At the outer boundary of the PV cells, the model applies convective water cooling by setting h to 50 W/(m2·K), and Tamb to 273 K. Finally, at the outer boundary of the insulation it applies convective cooling with h set to 5 W/(m2·K) and Tamb to 293 K.
Table 1 summarizes the material properties.
Table 1: Material properties
Component
 k [W/(m·K)] 
 ρ [kg/m3] 
 Cp [J/(kg·K)] 
 ε 
Emitter
10
2000
900
0.99
Mirror
10
5000
840
0.01
PV Cell
93
2000
840
0.99
Insulation
0.05
700
100
0.1
The model calculates the stationary solution for a range of emitter temperatures (1000 K to 2000 K) using the parametric solver.
Results and Discussion
The results shows that the device experiences a significant temperature distribution that varies with operating conditions. Figure 4 depicts the stationary distribution at operating conditions with an emitter temperature of 2000 K.
Figure 4: Temperature distribution in the TPV system when the emitter temperature is 2000 K.
As the upper plot in Figure 5 shows, the PV cells reach a temperature of approximately 1800 K. This is significantly higher than their maximum operating temperature of 1600 K, above which their photovoltaic efficiency is zero (see Figure 3).
It is interesting to investigate what the optimal operating temperature is. The lower plot in Figure 5 investigates at what temperature the system achieves the maximum electric power output. The optimal emitter temperature for this configuration seems to be between 1600 K and 1700 K, where the electric power (irradiation multiplied by voltaic efficiency) is maximum.
Figure 5: PV cell temperature (top) and electric output power (bottom) versus operating temperature.
The next step is to look at the temperature distribution at the optimal operating conditions (Figure 6).
Figure 6: Temperature distribution and surface irradiation flux in the system at an operating emitter temperature of 1600 K.
When the emitter is at 1600 K, the PV cells reach a temperature of approximately 1200 K, which they can withstand without any problems. Note that the insulation reaches a temperature of approximately 800 K on the outside, suggesting that the system transfers a significant amount of heat to the surrounding air.
The plot also depicts the irradiative flux, which varies significantly along the circumference of the PV cell and insulation jacket. To further investigate this effect, Figure 7 plots the irradiative flux along a quarter of the circumference separately at this operating condition. Clearly the variation it shows is related to the positions of the mirrors and is an effect of shadowing.
Figure 7: Irradiation flux along the TPV cell and insulation inner surface for one quarter of the device circumference.
This plot can help optimize the mirror geometry as well as help decide how large the PV cells should be and where they should be placed.
A general conclusion is that this type of modeling can shortcut the prototype development time and optimize the operating conditions for the finalized TPV device.
References
1. S. Christ and M. Seal, “Viking 27—A Thermophotovoltaic Hybrid Vehicle Designed and Built at Western Washington University”, SAE Technical Paper 972650, 1997.
2. Courtesy of E. Fontes, Catella Generics AB, Sweden.
3. Courtesy of Dr. D. Wilhelm, Paul Sherrer Institute, Switzerland.
Application Library path: Heat_Transfer_Module/Thermal_Radiation/tpv_cell
Modeling Instructions
From the File menu, choose New.
New
In the New window, click Model Wizard.
Model Wizard
1
In the Model Wizard window, click 2D.
2
In the Select Physics tree, select Heat Transfer>Radiation>Heat Transfer with Surface-to-Surface Radiation (ht).
3
Click Add.
4
Click Study.
5
In the Select Study tree, select Preset Studies>Stationary.
6
Click Done.
Global Definitions
Parameters
1
On the Home toolbar, click Parameters.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
T_heater
1000[K]
1000 K
Temperature, emitter inner boundary
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Circle 1 (c1)
1
On the Geometry toolbar, click Primitives and choose Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 40.
4
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (mm)
Layer 1
5
Layer 2
24
Layer 3
1
5
Click Build Selected.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
2
On the object c1, select Boundaries 1–12 only.
3
In the Settings window for Delete Entities, click Build Selected.
Rectangle 1 (r1)
1
On the Geometry toolbar, click Primitives and choose Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 2.
4
In the Height text field, type 10.
5
Locate the Position section. From the Base list, choose Center.
6
In the x text field, type 21.
7
Click Build Selected.
Rectangle 2 (r2)
1
On the Geometry toolbar, click Primitives and choose Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 6.
4
In the Height text field, type 2.
5
Locate the Position section. From the Base list, choose Center.
6
In the x text field, type 23.
7
Click Build Selected.
Union 1 (uni1)
1
On the Geometry toolbar, click Booleans and Partitions and choose Union.
2
Select the objects r1 and r2 only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries check box.
5
Click Build Selected.
Fillet 1 (fil1)
1
On the Geometry toolbar, click Fillet.
2
On the object uni1, select Points 1, 4, 5, and 8–10 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type 0.5.
5
Click Build Selected.
Circle 2 (c2)
1
On the Geometry toolbar, click Primitives and choose Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 37.
4
In the Sector angle text field, type 360/24.
5
Locate the Rotation Angle section. In the Rotation text field, type 360/24.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (mm)
Layer 1
2
7
Click Build Selected.
Delete Entities 2 (del2)
1
Right-click Geometry 1 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object c2, select Domain 1 only.
Rotate 1 (rot1)
1
On the Geometry toolbar, click Transforms and choose Rotate.
2
Select the objects fil1 and del2 only.
3
In the Settings window for Rotate, locate the Rotation Angle section.
4
Click Range.
5
In the Range dialog box, type 0 in the Start text field.
6
In the Step text field, type 360/8.
7
In the Stop text field, type 360*7/8.
8
Click Replace.
9
On the Geometry toolbar, click Build All.
The model geometry is now complete.
Definitions
Next, create selections for the mirror domains and boundaries. These selections will be useful when you implement material settings and boundary conditions. You could create selections for other domain and boundary groups in the same manner but that is not assumed in the following instructions.
Explicit 1
1
On the Definitions toolbar, click Explicit.
2
In the Settings window for Explicit, type Mirror Domains in the Label text field.
3
Select Domains 5–7, 10, 11, and 14–16 only.
Adjacent 1
1
On the Definitions toolbar, click Adjacent.
2
In the Settings window for Adjacent, type Mirror Boundaries in the Label text field.
3
Locate the Input Entities section. Under Input selections, click Add.
4
In the Add dialog box, select Mirror Domains in the Input selections list.
5
Click OK.
Materials
Material 1 (mat1)
1
On the Materials toolbar, click Blank Material.
2
In the Settings window for Material, type Insulation in the Label text field.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Name
Value
Unit
Property group
Thermal conductivity
k
0.05
W/(m·K)
Basic
Density
rho
700
kg/m³
Basic
Heat capacity at constant pressure
Cp
100
J/(kg·K)
Basic
Material 2 (mat2)
1
On the Materials toolbar, click Blank Material.
2
In the Settings window for Material, type PV Cell in the Label text field.
3
Select Domains 2, 3, 8, 9, 12, 13, 17, and 18 only.
These are the PV-cell domains. Note that you can select these domains by copying the text in the modeling instructions and then clicking the Paste Selection button or pressing Ctrl+V. When you have selected the domains you can also create a named selection by clicking the Create Selection button.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Name
Value
Unit
Property group
Thermal conductivity
k
93
W/(m·K)
Basic
Density
rho
2000
kg/m³
Basic
Heat capacity at constant pressure
Cp
840
J/(kg·K)
Basic
Material 3 (mat3)
1
On the Materials toolbar, click Blank Material.
2
In the Settings window for Material, type Mirror in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Mirror Domains.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Name
Value
Unit
Property group
Thermal conductivity
k
10
W/(m·K)
Basic
Density
rho
5000
kg/m³
Basic
Heat capacity at constant pressure
Cp
840
J/(kg·K)
Basic
Material 4 (mat4)
1
On the Materials toolbar, click Blank Material.
2
In the Settings window for Material, type Emitter in the Label text field.
3
Select Domain 19 only.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Name
Value
Unit
Property group
Thermal conductivity
k
10
W/(m·K)
Basic
Density
rho
2000
kg/m³
Basic
Heat capacity at constant pressure
Cp
900
J/(kg·K)
Basic
Add Material
1
On the Materials toolbar, click Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-In>Air.
4
Click Add to Component in the window toolbar.
Materials
Air (mat5)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Air (mat5).
2
Select Domains 4 and 20 only.
3
On the Materials toolbar, click Add Material to close the Add Material window.
Heat Transfer with Surface-to-Surface Radiation (ht)
Fluid 1
1
On the Physics toolbar, click Domains and choose Fluid.
2
Select Domain 4 only.
3
In the Settings window for Fluid, locate the Thermodynamics, Fluid section.
4
From the γ list, choose User defined.
Diffuse Surface 1
1
On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2
In the Settings window for Diffuse Surface, locate the Boundary Selection section.
3
From the Selection list, choose Mirror Boundaries.
By default, the radiation direction is controlled by the opacity of the domains. The solid parts are automatically defined as opaque while the fluid parts are transparent. You can change these settings by modifying the Opacity subnode under the Solid and Fluid features.
4
Locate the Ambient section. In the Tamb text field, type T.
5
Locate the Surface Emissivity section. From the ε list, choose User defined. In the associated text field, type 0.01.
Heat Flux 1
1
On the Physics toolbar, click Boundaries and choose Heat Flux.
2
Select Boundaries 97, 98, 141, and 148 only.
These are the outer boundaries of the modeling domain.
3
In the Settings window for Heat Flux, locate the Heat Flux section.
4
Click the Convective heat flux button.
5
In the h text field, type 5.
Diffuse Surface 2
1
On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2
Select Boundaries 97, 98, 141, and 148 only.
3
In the Settings window for Diffuse Surface, locate the Surface-to-Surface Radiation section.
4
Clear the Include surface-to-surface radiation check box.
5
Locate the Surface Emissivity section. From the ε list, choose User defined. In the associated text field, type 0.1.
Diffuse Surface 3
1
On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2
Select Boundaries 101, 102, 105, 106, 133, 134, 142, 147, 167, 168, 183, and 184 only.
These are the arc-shaped boundaries connecting the PV cells.
3
In the Settings window for Diffuse Surface, locate the Ambient section.
4
In the Tamb text field, type T.
5
Locate the Surface Emissivity section. From the ε list, choose User defined. In the associated text field, type 0.1.
Boundary Heat Source 1
1
On the Physics toolbar, click Boundaries and choose Boundary Heat Source.
2
Select Boundaries 99, 100, 117, 118, 157, 158, 181, and 182 only.
These are the outward-facing PV-cell boundaries.
3
In the Settings window for Boundary Heat Source, locate the Boundary Heat Source section.
4
In the Qb text field, type 50[W/(m^2*K)]*(273.15[K]-T).
Diffuse Surface 4
1
On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2
Select Boundaries 103, 104, 123, 124, 155, 156, 179, and 180 only.
These are the inward-facing PV-cell boundaries.
3
In the Settings window for Diffuse Surface, locate the Ambient section.
4
In the Tamb text field, type T.
5
Locate the Surface Emissivity section. From the ε list, choose User defined. In the associated text field, type 0.99.
Boundary Heat Source 2
1
On the Physics toolbar, click Boundaries and choose Boundary Heat Source.
2
Select Boundaries 103, 104, 123, 124, 155, 156, 179, and 180 only.
3
In the Settings window for Boundary Heat Source, locate the Boundary Heat Source section.
4
In the Qb text field, type -q_out.
Diffuse Surface 5
1
On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2
Select Boundaries 127, 128, 143, and 146 only.
These are the outward-facing emitter boundaries.
3
In the Settings window for Diffuse Surface, locate the Ambient section.
4
In the Tamb text field, type T.
5
Locate the Surface Emissivity section. From the ε list, choose User defined. In the associated text field, type 0.99.
Temperature 1
1
On the Physics toolbar, click Boundaries and choose Temperature.
2
Select Boundaries 131, 132, 144, and 145 only.
These are the inward-facing emitter boundaries.
3
In the Settings window for Temperature, locate the Temperature section.
4
In the T0 text field, type T_heater.
Definitions
Variables 1
1
On the Home toolbar, click Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
eta_pv
if(T<1600[K],0.2*(1-(T/800[K]-1)^2),0)
 
Voltaic efficiency, PV cell
q_out
ht.Gm*eta_pv
W/m²
Electric output power
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Mesh Settings section.
3
From the Element size list, choose Coarser.
4
On the Mesh toolbar, click Free Triangular.
Size 1
1
In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Mirror Boundaries.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
Select the Minimum element size check box.
8
Select the Maximum element growth rate check box.
9
Select the Curvature factor check box.
10
In the Maximum element size text field, type 1.
Size 2
1
Right-click Component 1 (comp1)>Mesh 1>Free Triangular 1>Size 1 and choose Duplicate.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
Click Clear Selection.
4
Select Boundaries 127, 128, 143, and 146 only.
Size 3
1
In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 101–106, 123, 124, 133, 134, 142, 147, 155, 156, 167, 168, 179, 180, 183, and 184 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
Select the Maximum element growth rate check box.
8
In the Maximum element size text field, type 2.
9
In the Model Builder window, click Mesh 1.
10
In the Settings window for Mesh, click Build All.
Study 1
Step 1: Stationary
Set up an auxiliary continuation sweep for the parameter T_heater.
1
In the Settings window for Stationary, click to expand the Study extensions section.
2
Locate the Study Extensions section. Select the Auxiliary sweep check box.
3
Click Add.
4
Click Range.
5
In the Range dialog box, type 1000 in the Start text field.
6
In the Step text field, type 100.
7
In the Stop text field, type 2000.
8
Click Replace.
9
In the Settings window for Stationary, locate the Study Extensions section.
10
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
T_heater
range(1000,100,2000)
K
Solution 1 (sol1)
1
On the Study toolbar, click Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
Using constant prediction for the continuation sweep improves convergence when the solution is very nonlinear in the swept parameter.
3
In the Model Builder window, expand the Study 1>Solver Configurations>Solution 1 (sol1)>Stationary Solver 1 node, then click Parametric 1.
4
In the Settings window for Parametric, click to expand the Continuation section.
5
From the Predictor list, choose Constant.
6
On the Study toolbar, click Compute.
Results
Temperature (ht)
1
Click the Zoom Extents button on the Graphics toolbar.
The first default surface plot shows the TPV-cell temperature for the last value in the sweep over operating temperatures.
Isothermal Contours (ht)
The second default plot shows isothermal contours.
Radiosity (ht)
The third default plot shows radiosity.
Reproduce the plots in Figure 5 with the following steps:
1D Plot Group 4
1
On the Home toolbar, click Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type PV Cell Temperature in the Label text field.
3
On the PV Cell Temperature toolbar, click Point Graph.
Point Graph 1
1
In the Model Builder window, under Results>PV Cell Temperature click Point Graph 1.
2
Select Point 6 only.
3
On the PV Cell Temperature toolbar, click Plot.
1D Plot Group 5
1
On the Home toolbar, click Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Electric Output Power in the Label text field.
3
On the Electric Output Power toolbar, click Point Graph.
Point Graph 1
1
In the Model Builder window, under Results>Electric Output Power click Point Graph 1.
2
Select Point 6 only.
3
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Model>Component 1>Definitions>Variables>q_out - Electric output power.
4
On the Electric Output Power toolbar, click Plot.
As this last plot shows, the electric output power has a maximum near 1,600 K. To see the temperature distribution at this operating temperature, go back to the first plot group and change the parameter value.
Temperature (ht)
1
In the Model Builder window, under Results click Temperature (ht).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (T_heater (K)) list, choose 1600.
4
On the Temperature (ht) toolbar, click Plot.
Finally, reproduce the surface irradiation plot in Figure 7 as follows:
1D Plot Group 6
1
On the Home toolbar, click Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Surface Irradiation in the Label text field.
3
Locate the Data section. From the Parameter selection (T_heater) list, choose From list.
4
In the Parameter values (T_heater (K)) list, select 1600.
5
On the Surface Irradiation toolbar, click Line Graph.
Line Graph 1
1
In the Model Builder window, under Results>Surface Irradiation click Line Graph 1.
2
Select Boundaries 102, 104, 106, 124, and 134 only.
3
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Model>Component 1>Heat Transfer with Surface-to-Surface Radiation>Radiation>Mutual surface irradiation>ht.Gm_gp - Mutual surface irradiation, Gauss point evaluation.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type atan2(y,x)*180/pi.
6
Click to expand the Quality section. From the Smoothing list, choose Everywhere.
7
From the Resolution list, choose No refinement.
8
On the Surface Irradiation toolbar, click Plot.