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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: 1D lumped parameter model in Excel, analytical model in Maple, and a 3D finite
element analysis using COMSOL software. The heat transfer coefficients simulate both natural
and forced convection in air, and forced convection with water. Without active cooling, the
efficiency of the cell is 9.8%. The added cooling from the surface facing away from the sun drew
almost all of the heat away from the cell generating a cell temperature of 302.5 K, and
increasing the efficiency to 12.7%.
Keywords: Photovoltaic, Heat Transfer, Efficiency, COMSOL
1. Introduction
The burning of fossil fuels has been integral in bringing a much desired technologically
advanced world, however alongside the improvements in daily life there have come devastating
negative effects. Use of fossil fuel has generated so called “greenhouse gases,” such as carbon
dioxide which get trapped within the atmosphere and create a dramatic global warming effect.
The devastating results of this have been varied across different areas of the globe with some
areas seeing excess drying and extreme droughts leading to wildfires, while others experience
extreme rainfall leading to excess flooding. There is, however, a possible way to have the
benefits of a modern society without the devastating effects of global warming: renewable
technologies have been extensively researched as a means of generating electricity without the
burning of fossil fuels. One of the best known, and most promising of those technologies is
solar. Solar power generation comes in two flavors. Thermal technology uses heat-based
systems, for example the use of collector plates and mirrors to concentrate solar irradiance and
generate extreme amounts of heat which can be tapped for power production. The second, and
more widespread, utilizes photovoltaic (PV) cells which use the energy of photons from 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%, while today
the best silicon PV modules have an efficiency of 17% [1]. To produce enough power for most
applications, groups of PV cells are connected together forming modules, and those modules
are connected together forming arrays. Not all the energy from the sun can be absorbed by the
panel; the rest is converted into heat. The resulting temperature increase reduces the efficiency
of the PV cell. Mathematical models have been created to simulate drawing heat away from the
cell and this paper focuses on utilizing those models to explore how various cooling
methodologies can increase PV cell efficiency.
2. Approach
The efficiency of a PV cell is largely determined by the temperature of the solar panel. The
upper face of the panel exposed to air is subjected to passive cooling, where some heat can be
drawn away from the cell naturally, a process known as natural convection. The act of forcing
heat away from the panel through active cooling is called forced convection. Using the heat for
another application, such as heating water for homes or swimming pools is of particular
interest because the waste heat can itself be utilized. Three modeling methods are used to
study the improvement of PV cell efficiency:
a)
Uni-directional, numerical, lumped 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 air 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 (W), is the product of the area of the cell exposed to the
sun (m2) and the solar irradiance (W/m2). The amount of energy converted to electricity is given
by multiplying Insol by the cell efficiency (η).
𝑃𝑒𝑙𝑒𝑐 = 𝐼𝑛𝑠𝑜𝑙 ∗ 𝜂(𝑇)
(1)
The remaining energy 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
temperature change with respect to time:
𝑑𝑇
𝑃ℎ𝑒𝑎𝑡 = 𝜌𝑐𝑝 𝑉 𝑑𝑡
(3)
Equation 3 is used to determine the increase in cell temperature,
𝑃
ℎ𝑒𝑎𝑡
𝑇𝑡+∆𝑡 = 𝑇(𝑡) + ∆𝑡 [ 𝑉𝜌𝑐
]
(4)
𝑝
The electrical efficiency of the PV cell, η, is given as a function of temperature by the following
equation:
𝜂 = 𝜂𝑟𝑒𝑓 [1 − 𝛽(𝑇 − 𝑇𝑟𝑒𝑓 )]
(5)
where ηref is the efficiency at reference temperature (0.13), Tref is the reference temperature
(298.15 K), and β is the temperature coefficient (0.0054) for mono-crystalline silicon [2].
Combining Equations 2 and 5, the equation for cell temperature including losses becomes:
𝐼𝑛𝑠𝑜𝑙[𝜂𝑟𝑒𝑓 [1−𝛽(𝑇−𝑇𝑟𝑒𝑓 )]] –HTC[T−Tref ]
𝑇𝑡+∆𝑡 = 𝑇(𝑡) + ∆𝑡 [
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 Lumped Parameter Model
The uni-directional numerical Excel model simplifies heat extraction, taking into account heat
flow from only 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
(HTC). 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%.
Lumped 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%.
Lumped 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%.
Lumped 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 Lumped parameter model above.
5.2.1 Results
The code used in the Maple software for a HTC of 20 W/m2K is shown below:
The temperature plots for HTC of 20 W/m2K, 100 W/m2K, and 200 W/m2K are 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
Variations in mesh size were 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 were not
affected by the mesh size, however this process is important especially in models that are more
complex because smaller mesh sizes, while more accurate, and 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 modeled 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, ηref, and 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 set 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 average temperature of the cell 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 inclusion 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 show how cooling increases the efficiency of
PV solar panels. Various practical heat transfer coefficients were examined to determine a
feasible method of drawing heat away from the cell. Forced water cooling yielded the best
results, bringing the cell temperature nearly back to ambient air temperature. This method is a
promising solution to the PV heat/efficiency problem as it provides the opportunity for the
heated water to be used for another application, thus increasing the overall energy generation
of the PV array.
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