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Modeling the Efficiency of a Photovoltaic Solar Cell
Erica Marcinek
United Technologies Corporation
Abstract:
The efficiency of a photovoltaic (PV) cell is largely based on temperature. When the cell is
heated by the sun, the electrical efficiency of the PV cell is reduced. Three mathematical
models were created to determine the electrical efficiency increase at various heat transfer
coefficients: Lump parameter model in Excel, Analytical model in Maple, and a 3D finite
element analysis using COMSOL software. The heat transfer coefficients simulate natural
convection in air, and forced convection with water. Without active cooling, the efficiency of
the cell is 9.8%. The efficiency reached by adding cooling from the surface facing away from the
sun increased the efficiency to 12.7%, with a cell temperature of 302.5 K, thus drawing almost
all of the heat away from the cell.
Keywords: Photovoltaic, Heat Transfer, Efficiency, COMSOL
1. Introduction
Electromagnetic radiation from the sun brings life to the earth. The burning of fossil fuels has
brought a much desired new technologically advanced world along with negative effects. These
greenhouse gases, such as carbon dioxide, get trapped within the atmosphere and thus create
this global warming effect. Areas have resulted in excess drying leading to wildfires, while
others with excess rainfall leading to excess flooding. Renewable technologies have been
extensively researched because electricity can be produced without the burning of fossil fuels.
One of those technologies is solar. Solar thermal technology uses thermal systems, such
collector plates and mirrors to concentrate solar irradiance whereas photovoltaic (PV) cells use
direct sunlight to produce electricity. The first PV cell was constructed of a thin wafer of
selenium in 1883 by Charles Fritts and had an efficiency of less than 1%. In 1953 Bell Labs
developed a PV cell from silicon, created by doping semiconductors which had an efficiency of
6%. To produce enough power for a particular application, groups of PV cells are connected
together forming modules, and those modules are connected together forming arrays. Today,
the best silicon PV modules have an efficiency of 17% [1]. Because not all the energy from the
sun can be absorbed by the panel, the rest is converted into heat. This temperature increase
reduces the efficiency of the PV cell. Mathematical models were created to simulate drawing
heat away from the cell and thus increasing the efficiency is the focus of this paper.
2. Approach
The efficiency of a PV cell is largely determined by the temperature of the solar panel. The face
of the panel exposed to air is subjected to passive cooling, where some heat can be drawn
away from the cell naturally. This is known as natural convection. The act of forcing heat away
from the panel is called forced convection. Using the heat for another application, such as
heating water for homes or swimming pools is of particular interest because instead of wasting
heating, it can be utilized. Three modeling methods are studied in improving the efficiency of
the PV cell.
a)
Uni-directional, numerical, lump parameter model using Microsoft Excel
b)
Analytical model using Maple software
c)
Finite Element Modeling (FEM) COMSOL software
3. Assumptions/Material Properties
The following assumptions were made for each model.
1.) The thickness of a solar cell is 0.270 mm [2].
2.) The solar irradiance is uniform over the entire surface of cell – 1000 W/m2 [3].
3.) The ambient temperature surrounding the cell is 298.15 K.
4.) Material Properties of Silicon:
Density
2300 kg/m3
Specific Heat
700 J/kg K
Thermal Conductivity
130 W/m K
5.) All the solar energy not converted to electricity is converted into heat.
6.) The heat transfer coefficients [4]:
Free convection in air
5-25 W/m2K
Forced convection in air
10-200 W/m2K
Free convection in water
20-100 W/m2K
Forced convection in water
50-10000 W/m2K
4. Governing Equations
The energy coming into the cell, Insol, is the product of the area of the cell exposed to the sun
and the solar irradiance. The amount of converted to electricity is given by multiplying Insol by
the cell efficiency (eta).
𝑃𝑒𝑙𝑒𝑐 = 𝐼𝑛𝑠𝑜𝑙 ∗ 𝜂(𝑇)
(1)
The remaining is converted to heat (Pheat), thus
𝑃ℎ𝑒𝑎𝑡 = 𝐼𝑛𝑠𝑜𝑙(1 − 𝜂(𝑇))
(2)
The energy converted to heat is the product of the density, the specific heat, volume, and the
change in temperature with respect to time:
𝑑𝑇
𝑃ℎ𝑒𝑎𝑡 = 𝜌𝑐𝑝 𝑉 𝑑𝑡
(3)
Equation 3 is used to determine the cell’s raise in temperature,
𝑃
ℎ𝑒𝑎𝑡
𝑇𝑡+∆𝑡 = 𝑇(𝑡) + ∆𝑡 [ 𝑉𝜌𝑐
]
𝑝
(4)
The electrical efficiency of the PV cell, ηTref, at reference temperature, Tref, is given by the
following equation [2]:
𝜂𝑃𝑉 = 𝜂𝑇𝑟𝑒𝑓 [1 − 𝛽𝑟𝑒𝑓 (𝑇𝑃𝑉 − 𝑇𝑟𝑒𝑓 )]
(5)
For mono-crystalline silicon, the efficiency, ηTref, at reference temperature 298.15 K is 13%,
and the temperature coefficient, βref, is 0.54% [2].
Using the material based equation for the efficiency of the PV cell, from Equation 5, and the
energy converted into heat, from Equation 2, equation for cell temperature becomes:
𝑇𝑡+∆𝑡 = 𝑇(𝑡) + ∆𝑡 [
𝑃𝑠𝑢𝑛 [𝜂𝑇𝑟𝑒𝑓 [1−𝛽𝑟𝑒𝑓 (𝑇𝑃𝑉 −𝑇𝑟𝑒𝑓 )]]
𝑉𝜌𝑐𝑝
]
(6)
5. Test Models
All models developed use simplified heat transfer coefficients typical to real life to demonstrate
the removal of heat from the PV cell.
5.1 Lump parameter model
The uni-directional numerical Excel model created simplifies heat extraction only from the top
surface of the cell. Exact solutions for cell temperature and efficiency, using Equations 4 and 5,
can be calculated for any given value for heat transfer coefficient. A HTC of 20 W/m2K is plotted
below to demonstrate heat removal by natural convection in air, a HTC of 100 W/m2K to
demonstrate forced convection with air, and a HTC of 200 W/m2K to demonstrate forced
convection with water.
5.1.1 Results
A heat transfer coefficient simulating natural convection in air results in a cell temperature of
343 K, and an efficiency of 9.8%.
Lump parameter model - HTC 20 W/m^2 K
345
340
335
Temperature (K)
330
325
320
315
310
305
300
295
0
20
40
60
80
100
120
140
160
180
Time (sec)
A heat transfer coefficient simulating forced convection with water results in a cell temperature
of 307 K, and an efficiency of 12.4%.
Lump parameter model - HTC 100 W/m^2 K
310
Temperature (K)
308
306
304
302
300
298
0
20
40
60
80
100
120
140
160
180
Time (sec)
A heat transfer coefficient simulating forced convection in water results in a cell temperature of
302.5 K, and an efficiency of 12.7%.
Lump parameter model - HTC 200 W/m^2 K
304
Temperature (K)
303
302
301
300
299
298
0
20
40
60
80
100
120
140
160
180
Time (sec)
5.2 Analytical model
The analytical modeling using Maple software yields the same results at the same heat transfer
coefficients shown in the Lump parameter model above.
5.2.1 Results
The code used in the Maple software is shown below:
5.3 Finite Element model
Dividing the domain into smaller domains, or elements, is called meshing. The domains can be
sliced into many different shapes, most commonly triangular, but curved domains can be more
beneficial for certain domain shapes. Partial differential equations are broken into simpler
equations, where each element is solved separately and then combined back together.
Several cases using the COMSOL 3D model were used to determine the efficiencies using
multiple heat transfer coefficients.
5.3.1 Setup of the 3D COMSOL model
An appropriate mesh size was explored to determine how large a mesh size could be used
without affecting the results. The temperature results in the model of the solar cell was not
affected by the mesh size, but this is an important aspect in models that are complex because
the smaller mesh sizes result in longer computation times. A mesh size of “Normal” was used
for these computations, resulting in multiple rows of slices throughout the thickness of the cell.
The length and width of the solar cell was reduced to 0.0001 mm in the COMSOL model to
reduce computation time. This did not alter the results since heat transfer was though the top
and bottom surfaces only. The material selection for the domain was done by using the built-in
material properties of poly-silicon under the “Material Browser”. The same material properties
were used in all three test models.
The reference temperature, Tref, the efficiency at the reference temperature, ηTref, the
temperature coefficient, β, variables were added under “Global Definitions”. The “Analytic”
function was added to create the expression for efficiency dependant on temperature and the
reference values for mono-crystalline silicon. The solar irradiance, Insol, was also added to
“Global Definitions” for calculation of heat flux.
“Heat transfer in Solids” modeling was added to the model to heat transfer from the solid
element. The inward heat flux from the sun was setup under “Heat Flux” and was expressed
using Equation 2, using the variables and analytic values that were entered into “Global
Definitions”. “Convective Cooling” was added to simulate heat removed from the cell as a
function of the heat transfer coefficient. The face of the cell subject to heat transfer is selected
and added to the “Boundary Selection”, the faces not selected are assumed to be completely
insulated.
5.3.2 Test Cases
1.) Passive Air Cooling: natural convection in air from the top surface of the cell, 20 W/m2K.
Assume the bottom surface is completely insulated.
2.) Active Air Cooling: forced convection in air from the top surface of the cell, 100 W/m2K.
Assume the bottom surface is completely insulated.
3.) Passive Air Cooling/Active Water Cooling: natural convection in air from the top surface, 20
W/m2K, forced convection with water from the bottom surface, 200 W/m2K.
5.3.3 Results
Test Case 1: Yields the same results as the first two models: ~343 K cell temperature, efficiency
9.8%. Running the case with the heat removal from the bottom surface instead of the top
surface yielded very similar result, indicating that the cell thickness is independent of the
direction of heat removal.
Removal from Top
Removal from Bottom
Test Case 2: Cell temperature: ~307 K. Active air cooling increasing the efficiency to 12.4%.
Again, very similar results are achieved regardless of which face subject to heat removal.
Removal from Top
Removal from Bottom
This heat transfer can be achieved on windy days, or by manually blowing over the panel with a
fan.
Test Case 3: Cell temperature: ~302 K. Efficiency is increased to 12.7% with the set up of a
water flow system, i.e. a pipe, flowing underneath the cell. The same results are obtained for
passive air cooling beneath, and active water cooling above, but this seems unnecessarily
complicated to the former.
Removal from Top and Bottom
6. Conclusions
COMSOL finite element software, along with Microsoft Excel and Maple software were used to
model the efficiency of a solar cell in the effort to Increase the efficiency of PV solar panels.
Various practical heat transfer coefficients were examined to determine a feasible solution of
drawing heat away from the cell, since the efficiency is largely impacted by cell temperature.
Forced water cooling yielded the best results, bringing the cell temperature nearly back to
ambient temperature. This is a promising solution because it opens the opportunity for the
heated water to be used for another application.
7. References
[1] Boyle, Godfrey, and Open University. Renewable Energy. 2. Oxford University Press, USA,
2004. 66-68. Print.
[2] D. J. Yang, Z. F. Yuan, P. H. Lee, and H. M. Yin, Simulation and experimental validation of
heat transfer in a novel hybrid solar panel, International Journal of Heat and Mass Transfer, 55,
1076-1082 (2012)
[3] Fay, James A., and Dan S. Golomb. Energy and the Environment . Second Edition. Oxford:
Oxford University Press, 2012. 180. Print.
[4] http://www.engineeringtoolbox.com/convective-heat-transfer-d_430.html