HYBRID SOLAR PANEL SIMULATION OF FINS
PERPENDICULAR TO FLOW TO OPTIMIZE PERFORMANCE
by
Robert P. Collins
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 7, 2013
© Copyright 2013
by
Robert P. Collins
All Rights Reserved
ii
ACKNOWLEDGMENT
Thank you father, mother, David, Matthew, and Alex for the example and inspiration
you provide in my life. I would not be who, or where I am today without your influence
on my life, and for that I am grateful.
Also, thank you Ernesto for the time and interest you took in my project as my advisor.
iii
CONTENTS
HYBRID SOLAR PANEL SIMULATION OF FINS PERPENDICULAR TO FLOW
TO OPTIMIZE PERFORMANCE ............................................................................... i
ACKNOWLEDGMENT .................................................................................................. iii
LIST OF TABLES ............................................................................................................. 3
LIST OF FIGURES ........................................................................................................... 4
TABLE OF SYMBOLS .................................................................................................... 5
KEYWORDS ..................................................................................................................... 7
1. ABSTRACT ................................................................................................................ 8
2. INTRODUCTION/BACKGROUND .......................................................................... 9
2.1
Solar Photovoltaic Cells ..................................................................................... 9
2.2
Solar Hot Water Heater .................................................................................... 11
2.3
Hybrid Solar Panel (PV/T) ............................................................................... 12
3. METHODOLOGY/APPROACH .............................................................................. 14
3.1
Materials ........................................................................................................... 15
3.2
Model Arrangement ......................................................................................... 16
3.3
Test Arrangements ........................................................................................... 19
3.4
Model Theory and Relevant Equations ............................................................ 20
3.5
Finite Element Model ....................................................................................... 24
3.6
Expected Results .............................................................................................. 25
3.7
Model Limitations ............................................................................................ 25
4. RESULTS AND DISCUSSION ................................................................................ 26
4.1
PV/T Module Results ....................................................................................... 26
4.2
PV/T Array Results .......................................................................................... 36
4.3
Other Considerations ........................................................................................ 38
5. CONCLUSIONS ....................................................................................................... 39
6. REFERENCES .......................................................................................................... 40
1
7. APPENDIX A: CALCULATIONS ........................................................................... 43
7.1
Conservation of Energy.................................................................................... 43
7.2
Electrical Efficiency Verification .................................................................... 44
7.3
Thermal Efficiency Verification ...................................................................... 44
7.4
Volume Flow Rate ........................................................................................... 45
7.5
Net Energy Collected ....................................................................................... 45
2
LIST OF TABLES
Table 1: PV/T Model Materials ....................................................................................... 16
Table 2: Module Parameters ............................................................................................ 18
Table 3: Model Variables ................................................................................................ 19
Table 4: Fin Test Arrangements ...................................................................................... 20
Table 5: “Coarser” Mesh Solution Data and PC Specifications ...................................... 24
Table 6: Module Results .................................................................................................. 31
Table 7: Array Results ..................................................................................................... 37
3
LIST OF FIGURES
Figure 1: Electrons Absorbing Incident Sunlight [1] ........................................................ 9
Figure 2: Band Gap [2] .................................................................................................... 10
Figure 3: Active Secondary Loop Solar Hot Water Heater System [7] .......................... 12
Figure 4: Hybrid Solar Panel Control Volume ................................................................ 14
Figure 5: Model Isometric Cross Section View .............................................................. 15
Figure 6: Fin Labyrinth .................................................................................................... 17
Figure 7: PV/T Module Landscape View ........................................................................ 17
Figure 8: Hybrid Panel Cross Section View .................................................................... 18
Figure 9: Hydraulic Diameter .......................................................................................... 22
Figure 10: Rectangular Orifice [16] ................................................................................ 23
Figure 11: Averaged Streamlines and Contours of Turbulent Kinetic Energy [16] ........ 23
Figure 12: “Coarser” Mesh .............................................................................................. 24
Figure 13: Fin Velocity Disruption.................................................................................. 27
Figure 14: Velocity Distribution in a Labyrinth Arrangement ........................................ 28
Figure 15: Temperature Distribution Around a Fin ......................................................... 29
Figure 16: Temperature Contours in a Labyrinth Arrangement ...................................... 30
Figure 17: PV Surface Temperature Distribution ............................................................ 31
Figure 18: All Fin Configuration Efficiency ................................................................... 32
Figure 19: Top Fin Arrangement Efficiency ................................................................... 34
Figure 20: Bottom Fin Arrangement Efficiency .............................................................. 35
Figure 21: Labyrinth Arrangement Efficiency ................................................................ 36
Figure 22: PV/T Array [17] ............................................................................................. 37
4
TABLE OF SYMBOLS
Symbol
𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
Description
Units
Collector Area Exposed to Solar Radia-
𝑚2
tion
𝐴𝑓𝑙𝑜𝑤
Flow Cross Sectional Area
𝑚2
𝐽
𝐶𝑣
Constant Volume Specific Heat
𝐶𝑝
Constant Pressure Specific Heat
𝐷ℎ
Hydraulic Diameter
𝐺
Radiative Power Per Unit Area
h
Heat Transfer Coefficient
ℎ1
Half of the Flow Path Height
𝑚
𝐻𝑓𝑙𝑜𝑤
Flow Path Height
m
𝐼𝑚𝑝𝑝
Current Maximum Power Point
𝐴
𝑘𝑔𝐾
𝐽
𝑘𝑔𝐾
𝑚
𝑊
𝑚2
𝑊
𝑚2 𝐾
2
𝑘
Thermal Conductivity
𝑚̇
Mass Flow Rate
𝑊
𝑚𝐾
𝑘𝑔
𝑠
𝑃𝐸 𝑜𝑢𝑡
Electric Power Out of Panel
𝑊
𝑃𝑇 𝑜𝑢𝑡
Thermal Power Out of Panel
𝑊
𝑄̇𝑐𝑜𝑛𝑣
Energy Imparted on the Fluid
𝑊
𝑄̇𝑟𝑎𝑑
Energy Radiated to the Atmosphere
𝑊
Re
𝑇𝑎𝑚𝑏
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
Reynolds Number
Ambient Air Temperature
𝐾
𝑇𝑖
Inlet Working Fluid Temperature
𝐾
𝑇𝑜
Average Outlet Working Fluid
𝐾
Temperature
𝑇𝑟𝑜𝑜𝑚
u
Room Temperature
25 °C
𝑚
Fluid Velocity
𝑠
5
𝑉𝑚𝑝𝑝
w
𝛽𝑟𝑒𝑓
Voltage Maximum Power Point
𝑉
Panel Width
m
PV Cell Temperature Coefficient
1
℃
𝜀
Emissivity
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝐸
Panel Electrical Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
Panel Electrical Efficiency at Room
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝐸𝑟𝑒𝑓
Temperature
𝜂𝑂
Panel Overall Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝑇
Panel Thermal Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
µ
Dynamic Viscosity
𝑘𝑔
𝑚∗𝑠
ν
𝑚2
Kinematic Viscosity
𝑠
ρ
Density
σ
The Stefan-Boltzmann Constant
∇
Gradient
𝑘𝑔
𝑚3
𝑊
5.67 × 10−8 𝑚−2 𝐾−4
𝑂𝑝𝑒𝑟𝑎𝑡𝑜𝑟
6
KEYWORDS
Finite Element Method
Heat Transfer
Hybrid Solar Panel
Solar Energy
Turbulent Flow
7
1. ABSTRACT
A photovoltaic (PV) cell is coupled with a solar hot water heater in an arrangement
called a hybrid solar panel, or PV/Thermal (PV/T) panel. This hybrid solar panel
concept explores the used of fins perpendicular to the flow direction to increase flow
mixing and to reduce boundary layer thickness and therefore increase heat transfer
between the PV cells and solar hot water heater. The hybrid panel is designed with solar
cells attached to a copper reservoir using a thermal paste, with an insulated boundary
between the bottom of the fluid reservoir and the atmosphere. A two dimensional (2-D)
model is used to simulate the temperature distribution and the outlet water temperature
in the PV/T module, where the number of fins and the flow rate in the reservoir are
varied. The module efficiency is compared, with the highest efficiency module arrangement consisting of many, large fins, on both sides of the flow path. The model of
the highest efficiency PV/T module is run three times, with the outlet water temperature
carried from one model to the next in order to simulate a larger, PV/T array, resulting in
a water temperature rise of 10.3°C, and a net efficiency of 57%, 2.6% more efficient
than the PV/T array modeled with no fins.
8
2. INTRODUCTION/BACKGROUND
The amount of electricity and hot water used in the world is increasing as the middle
class, and therefore the average quality of life, increases. This, coupled with the rise in
fossil fuel prices, creates a growing need for a cheap, energy efficient, and environmentally friendly method for providing electricity and hot water. Household solar hot water
heaters are a method of heating water, whereas photovoltaic (PV) cells are employed to
convert solar radiation into electricity. Both methods stated above provide cheap and
environmentally friendly alternatives to fossil fuels converted from the sun’s light, and
are explored below.
2.1 Solar Photovoltaic Cells
The conversion of solar radiation into electrical power is accomplished in PV cells by
the use of semiconducting materials, with silicon as the most common semiconductor in
PV cells. During conversion of solar radiation to electrical power, photons of light are
absorbed by the valence electrons surrounding the nucleus of the silicon atoms. These
absorbed photons excite the electrons, raising them to a higher energy state, or to a
higher electron orbital Figure 1.
Figure 1: Electrons Absorbing Incident Sunlight [1]
The electron orbitals discussed are classified as valence band orbitals, the orbitals where
electrons are bound to an individual atom at resting state, and conduction band orbitals,
where electrons can move freely between different atoms in a material. The difference
in energy between the electron orbitals in the valence and conduction bands, where no
9
electron states can exist, is known as the band gap, and is a quantifiable number for a
given material.
Materials with small or no band gap are classified as conductors,
whereas materials with a large band gap are classified as insulators. Materials such as
silicon, which have an intermediate band gap, are semiconductors. Figure 2, shows a
schematic representation of the band structure of a semiconductor [2].
Figure 2: Band Gap [2]
This band gap is the step, or wall, that the electrons must overcome to move from the
valence band to the conduction band. In other words, the electrons need to be excited by
photons of a certain, minimum energy to jump the band gap. The band gap can be
considered proportional to the open circuit voltage of the semiconductor. With an
increase in temperature, the electrons have a higher resting energy state, effectively
reducing the band gap. With a reduction in band gap, the open circuit voltage of the
semiconductor of the PV cell decreases, while the current remains largely the same.
Because of Watt’s Law, P= 𝐼𝑉, the power output of the semiconductor, or PV cell
decreases for the same amount of power in from solar radiation. The electrical efficiency of a PV Cell is therefore decreased, as shown by the below equation [3]:
𝜂𝐸 =
𝑉𝑚𝑝𝑝 ∗ 𝐼𝑚𝑝𝑝
𝑃𝐸 𝑜𝑢𝑡
=
𝑃𝑖𝑛
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
This phenomenon is well documented, with the relationship between temperature and
%
efficiency explored in [3], which for some models, predicts a 0.41℃ drop in efficiency
above room temperature. In hot, high sunlight conditions, temperatures of 50°C can be
10
reached, severely dropping the efficiency of the panel, and also risking permanent
structural damage to the PV cell from the thermal stress [4].
𝜂𝐸 = 𝜂𝑇𝑟𝑒𝑓 (1 − 𝛽𝑟𝑒𝑓 (𝑇 − 𝑇𝑟𝑜𝑜𝑚 ))
The efficiency of the most common types of solar panels, mono-crystalline silicon PV
cells, typically ranges from 13-20% at room temperature [5], with that percentage of
power in sunlight converted into electrical power. The rest of the energy reflected off
the PV cells, or converted into heat. If that atmosphere is unable to accept the heat from
the PV cells, the temperature of the PV cells rises. As the temperature of the cells rise,
the efficiency of the cells decreases, and on hot, sunny days, PV cells can have a drop in
efficiency of up to 10%. Different methods are therefore used to cool PV cells in order
to maintain their electrical efficiency.
2.2 Solar Hot Water Heater
A solar collector in a solar hot water heater is an enclosed volume that the working fluid
flows through to collect the sun’s energy in the form of heat. This volume is very
insulated and optimized to capture the solar radiation. Solar hot water heaters have a
greater efficiency when the collector volume is hot, and there is a large driving temperature delta between the collector and the working fluid flowing through the collector. The
working fluid is typically water in a single loop solar hot water heater, which directly
feeds water for household usage. Other fluids such as a water/propylene glycol mix are
used to transfer the heat to the household water supply through a secondary heat exchanger in a secondary loop solar hot water heater. The working fluid can be supplied
actively, by using a pump, or passively using natural convection of the fluid. Passive,
natural convection flow is typical of primary loop solar hot water heaters, whereas active
loops are used in both primary and secondary solar hot water heaters. An active, secondary solar hot water heating arrangement, typical to the arrangement used in this
study, is shown below in Figure 3. [6]
11
Figure 3: Active Secondary Loop Solar Hot Water Heater System [7]
The thermal efficiency of a solar hot water heater is given by the below equation [3]:
𝜂𝑇 =
𝑚̇ ∗ 𝐶𝑝 (𝑇𝑜 − 𝑇𝑖 )
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
2.3 Hybrid Solar Panel (PV/T)
In order to avoid the PV cells drop in efficiency, and capture the majority of the waste
heat from the PV cells, the PV cells are coupled with a solar hot water heater in a hybrid
solar panel. This design provides a novel method for cooling the PV cells, whose
efficiency diminishes with increasing temperature, and uses the heat extracted from the
PV cells to heat household or commercial hot water cost effective and environmentally
friendly. The efficiency of the hybrid solar panel is the sum of the efficiency of the PV
cells and the solar hot water heater [3].
𝜂𝑜 = 𝜂𝐸 + 𝜂𝑇
Hybrid solar panels, also known as PV/T panels, have been explored in several previous
studies. In a similar study Fountenault [8] varied flow rates and flow channel thickness12
es in a laminar flow, hybrid solar panel. The exploration showed that, when all else was
constant, the average driving temperature difference between the PV cells and the fluid
in the flow channel drives both the thermal and electrical efficiencies. A large temperature delta was achieved by using a high mass flow rate of water in a large channel, which
led to lower temperature changes in the fluid when compared to lower flow rates in
smaller channels. As a result of a lower temperature change in the fluid, a larger average
temperature delta between the PV cells and the working fluid was maintained.
A study by Yang Et al. [9] explored a model and prototype hybrid solar panel with a
functionally graded material (FGM). The FGM is a material with a property gradient.
𝑊
In this case, the thermal conductivity of the material is higher, 1.13𝑚𝐾, near the interface
𝑊
with the PV cells, and much lower, 0.26𝑚𝐾, near the bottom, freely convecting surface.
The FGM is intended transfer heat from the PV cells to the water, but also act as an
insulator between the water and the atmosphere, as the properties reflect. The study
showed that a combined efficiency of 70% could be achieved with hybrid solar panels,
and that there are several novel, although sometimes less practical, designs being explored to optimize hybrid solar panels as a means of harnessing the sun’s power.
13
3. METHODOLOGY/APPROACH
A hybrid solar panel is designed, with a set control volume that encapsulates the panel as
shown below in Figure 4. The sunlight and the cold working fluid will be the two
defined inputs into the control volume, and therefore solar panel. Heat will be transferred from the panel with the mass flow rate of the hot working fluid out, convectively
to the ambient air, and through reflection/radiation from the body of the body of the
panel. There is a current and voltage across the PV cell, which is also accounted for, and
a net electrical power out.
Radiation
Sunlight
Convective Heat
Transfer
Cold Primary Fluid
HYBRID SOLAR
Hot Primary Fluid
PANEL
(CONTROL VOLUME)
Figure 4: Hybrid Solar Panel Control Volume
The net energy balance for the hybrid solar panel control volume is:
𝐺 − 𝑃𝑇 𝑜𝑢𝑡 − 𝑄̇𝑐𝑜𝑛𝑣 − 𝑄̇𝑟𝑎𝑑 − 𝑃𝐸 𝑜𝑢𝑡 = 0
Within the control volume that is the hybrid solar panel, there is heat transfer between
the different material layers. Conduction heat transfer exists between and within the
solid layers of hybrid solar panel which will be constrained by the material conductivity.
Choice of highly conductive materials, such as copper will maximize the heat transfer
away from the solar panel to the walls of the cooling fluid reservoir. The heat transfer
out to the atmosphere is also minimized with a layer of insulation added to the bottom of
the hybrid solar panel.
14
Heat is conducted from the PV cells through highly conductive solids until it reaches the
solid/liquid boundary, where the heat is transferred to the working fluid in the fluid
reservoir. With known solid and liquid properties, the limiting factors explored in this
model are the surface area at the solid liquid boundary, boundary layers, and the flow
mixing. Boundary layers form in duct flow, creating a hot layer of the working fluid
along the solid/liquid interface, where the bulk fluid temperature is much lower. In a
fluid such as water, conduction is a slower method of heat transfer than convection. In
order to increase the surface area, minimize boundary layer formation, and increase heat
transfer within the fluid by inducing mixing, fins are added perpendicular to the flow. A
cross sectional unit thickness (not to scale) of the model is shown below in Figure 5,
which shows the material layers, and the orientation of the fins to the flow.
- PV Cells
- Thermal Paste
- Copper
Working Fluid
- Copper
- Insulation
Figure 5: Model Isometric Cross Section View
3.1 Materials
Hybrid solar panel materials vary from the standard materials used by Fontenault [8] to
the FGM panel explored by Yang Et al. Commonly available materials and those that
maximize heat transfer within the panel were chosen for this study. The materials used
in the model are listed below, with their reference and relevant material properties also
shown. COMSOL Multiphysics has built in materials, which re used for water, copper,
and silicon. All material properties below are constant in the model except for the
properties of water, which vary with temperature, with the water properties shown in
Table 1 are taken at 10°C.
15
Table 1: PV/T Model Materials
Material
Property
Value
Reference
ρ
2329
k
130
𝐶𝑝
700
ε
.60
[10]
ρ
3500
[13]
k
2.87
𝐶𝑝
.7
ρ
8700
k
400
𝐶𝑝
385
ρ
999.8
k
.585
𝐶𝑝 = 𝐶𝑣
4.193
μ
1.307 x 10-3
Extruded
ρ
25.9
Polystyrene
k
.038
𝐶𝑝
1300
PV Cell
Thermal Paste
Copper
Water
COMSOL – Silicon
COMSOL – Copper
COMSOL – Water @ 10°C
[14]
3.2 Model Arrangement
A 2-D model of this scenario is created in COMSOL Multiphysics in order to simulate
this hybrid solar panel design. The number of fins on the top wall (0 or 9), the number
of fins on the bottom of the wall (0, 9, or 18), and the fin length (1⁄4, 1⁄2, or 3⁄4 flow
path height) are varied for a single flow rate to see their effect on the efficiency of the
hybrid solar panel, with the. The fins on the top of the flow path are expected to have a
two-fold effect on the heat transfer; increased flow mixing and increased surface area for
heat transfer on the “hot” wall. Fins on the bottom of the flow path do not increase the
surface area for heat transfer on the “hot” wall, but will instead be tested for their ability
to disrupt boundary layer flow along the top wall. Top and bottom fins are also be tested
together, as shown in Figure 6, which creates a labyrinth design, which will effectively
increase the flow path in the reservoir.
16
Figure 6: Fin Labyrinth
The PV/T module consists of 12 PV cells arranged in a 4x3 rectangle, with the flow
through the short direction of the rectangle, as shown in Figure 7. The model orientation
is along the shown in Figure 8, which illustrates a short cross section of the model.
PV
Flow
Cell
xc
Figure 7: PV/T Module Landscape View
17
Figure 8: Hybrid Panel Cross Section View
The 2-D model’s dimensions, inlets and initial conditions, and other non-material
properties are below in Table 2. The PV cell properties are taken from [10], which is a
standard size for mono-crystalline silicon PV cell. Figure 8 shows a small section view
of the to-scale model. The inlet water temperature of 11°C is taken from [11], and is
approximately the average groundwater temperature in the Northeast United States.
Table 2: Module Parameters
Name
Expression
H_Flow
H_Wall
H_Paste
H_PV
H_Insulation
W_PVCell
H_Fin
1/2*H_Flow
W_Fin
U_Flow
T_Amb
T_Init
T_Inlet
Emissivity
HX_Silicon
HX_Insulation
W_Panel
3*W_PVCell
PVEFF0
PVdeg
T_Room
Unit
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[m/s]
[K]
[K]
[K]
Description
5
Flow Channel Thickness
2
Copper Wall Thickness
0.5
Thermal Paste Thickness
0.2
PV Cell Thickness
10
Insulation Thickness
125
Panel Width
Fin Height
1
Fin Width
0.002
Flow Inlet Velocity
298.15
Ambient Air Temperature
298.15
Initial Cell Temperature
284.15
Inlet Water Temperature
0.91
Emissivity Silicon
23.29 [W/(m^2*K)] Silicon/Air Heat Transfer Coefficient
[W/(m^2*K)] Insulation/Air Heat Transfer Coefficient
Width of Three PV Cells
0.182
PV Cell Efficiency at Room Temperature
0.0041 [1/K]
PV Cell Degradation With Temperature
298.15
Room Temperature
18
Ltenth
Q_Sun
P_in
W_Panel/10
1000 [W/m^2]
Q_Sun*W_Panel*1[m]
Fin Spacing (9 fins)
Sun Incident Radiation
Power In
The heat transfer coefficients for the top and bottom of the panel are from [5], with the
top heat transfer coefficient as a forced convection value of 23.29
heat transfer coefficient chosen as 4.17
𝑊
𝑚2 𝐾
𝑊
𝑚2 𝐾
and the bottom
. The heat transfer coefficients above are
both taken with a wind speed of 5.76 m/s, which is the closest available data point in [6]
to the east coast of the United States average wind speed of about 5 m/s [12]. These
vales for the heat transfer coefficients are higher, but expected to be more fitting than for
the north east rather than those used by Fontenault [8] and Yang Et al. [9], which are
based on an arbitrary wind speed of 1 m/s.
INSERT SKETCH OF SYSTEM
The model variables in Table 3 are evaluated by the model for each step in the solver,
and are used iteratively to find the steady state condition for the model. An important
variable to note is Q_heat, which is the sun’s radiation that is not converted to electrical
power by the solar panel, which varies with temperature.
Table 3: Model Variables
Name
PVEFF
Q_Heat
mdot
ThermEFF
EFF_Net
Expression
PVEFF0*(1-PVdeg*(T-T_Room))
Q_Sun*PVEFF
nitf.rho*H_Flow*U_Flow*1[m]
mdot*4173[J/(kg*K)]*(T-T_Inlet)/P_in
PVEFF+ThermEFF
Units
W
kg/s
-
Description
PV Cell Efficiency Temp. Dependence
Sun’s Energy Converted to Heat
Mass Flow Rate Water (per unit depth)
Thermal Efficiency
Net Efficiency
3.3 Test Arrangements
A 2-D, stationary COMSOL model is used to model the hybrid solar panel. Because of
the multiple avenues of heat transfer within, and in and out of the control volume, a
19
turbulent flow, conjugate heat transfer model is used. Altering the number of fins, and
the flow rate will change the flow distribution in the fluid reservoir. The fin test arrangements are shown below in Table 4.
Table 4: Fin Test Arrangements
Flow Velocity (u)
# Top Fins
# Bottom Fins
Fin Lengths
.002 m/s
0
0
0
002 m/s
9
0
¼, ½, ¾
002 m/s
18
0
¼, ½, ¾
002 m/s
0
9
¼, ½, ¾
002 m/s
0
18
¼, ½, ¾
002 m/s
9
9
¼, ½, ¾
002 m/s
9
9
¼, ½, ¾
002 m/s
18
18
¼, ½, ¾
002 m/s
27
27
¼, ½, ¾
Twenty five fin arrangements are outlined in Table 4. These conditions are expected to
show a useful correlation between the different fin arrangements and the outlet thermal
properties and efficiencies of a PV/T module.
3.4 Model Theory and Relevant Equations
The relevant equations used in this model, and determining the panel’s overall efficiency
are below. The thermal, electrical, and net efficiency of the hybrid panel are mentioned
above in the introduction. The relevant equations in the COMSOL conjugate heat
transfer model are shown below. A simplified computation was used to verify the
model’s relevance for modeling the conditions, with vector quantities shown in bold.
Steady state heat transfer in solids is described by the conservation of energy equation
∇ ∙ (𝑘∇𝑇) = 0
20
where k is the thermal conductivity of the solid.
Steady state heat transfer in liquids is described by the conservation of energy equation.
𝜌𝐶𝑝 𝒖 ∙ ∇𝑇 = ∇ ∙ (𝑘∇𝑇)
where 𝜌𝐶𝑝 𝒖 ∙ ∇𝑇 is the rate of convective heat transfer in the fluid.
The heat flux from solar irradiance into the PV cell is given by
−𝐧 ∙ (−𝑘∇𝑇) = 𝑞𝑜
Where 𝑞𝑜 is a defined value of the incident heat flux and 𝐧 is the vector normal to the
heat transfer surface. This equation is also used to describe a perfectly insulated boundary. Thermal Insulation in the model means there is no heat transfer across a given
boundary, which essentially means the temperature gradient leading up to and across the
boundary is zero.
−𝐧 ∙ (−𝑘∇𝑇) = 0
A no slip boundary condition is used, with the fluid velocity set as zero along the walls
of the flow path.
𝐮=0
The velocity profile of the fluid is given by:
𝐮 = −𝑈𝑜 𝒏
𝑈𝑜 is the initial average velocity, which is a defined test condition. Free convection
between the atmosphere and the hybrid solar panel is based on the heat transfer coeffi-
21
cient and the temperature delta between the atmosphere and the surface of the panel.
The convective heat loss from the panel to the atmosphere is given by
−𝐧 ∙ (−𝑘∇𝑇) = h ∙ (𝑇𝑎𝑚𝑏 − 𝑇)
where h is the heat transfer coefficient.
Radiative heat transfer was in included in the model, but was found to be negligible for
the temperatures in the hybrid solar panel.
−𝐧 ∙ (−𝑘∇𝑇) = 𝜀σ(𝑇𝑎𝑚𝑏 4 − 𝑇 4 )
The Reynolds number is evaluated at the tip of each fin to evaluate mixing, with the
hydraulic diameter as one half the flow path, as shown in Figure 9.
𝑅𝑒 =
𝑢𝐷ℎ
𝜈
ℎ1
2
Figure 9: Hydraulic Diameter
𝐷ℎ = 4ℎ1
2
For flow between two parallel plates with the model geometry in Table 2, the Reynolds
number is calculated below.
𝑅𝑒 =
𝑚
. 005𝑚
(. 002 𝑠 ) (4 ( 2 ))
1.307𝑥10−6
22
𝑚2
𝑠
= 15.3
The flow through the “no-fin” model has a Reynolds number well below that of turbulent flow, with the transition between laminar and turbulent flow occurring at a Reynolds
number between 2300 and 4000 [15].
The mixing in the model, is therefore accounted for by low-Reynolds number turbulent
flow. This behavior is typical of flow through an orifice or diffuser. Typically known as
diffuser stall, which is a term used in aerofoil aerodynamics, this term denotes boundary
layer separation. “The expanding-area diffuser produces low velocity and increasing
pressure, an adverse gradient. It the diffuser angle is too large, the adverse gradient is
excessive, and the boundary layer will separate at one or both walls, with backflow, and
poor pressure recovery” [15]. It is this boundary layer separation or disruption which is
relied upon to increase fluid mixing and therefore heat transfer in the model. Figures 10,
below, shows the rectangular orifice modeled by Tsukahara, Kawase, and Kawaguchi
[16], with the turbulent kinetic energy of a Newtonian fluid shown in Figure 11. The
simulation was carried out with a Reynolds number of 100, and shows that the reduction
in area through the orifice disrupts the normal laminar flow boundary layers and introduces turbulent kinetic energy in the form of flow mixing.
Figure 10: Rectangular Orifice [16]
Figure 11: Averaged Streamlines and Contours of Turbulent Kinetic Energy [16]
23
Figure 11 shows that the flow has increased energy as a result of the expanding area,
causing enhanced mixing, which is the behavior that is expected in this model. Similar
to Tsukahara, Kawase, and Kawaguchi [16], sharp edged bodies, or fins, are used, which
are insensitive to Reynolds number and “cause flow separation regardless of the character of the boundary layer” [15].
3.5 Finite Element Model
The hybrid solar panel model is meshed using the “Coarser” Physics Controlled Mesh in
COMSOL Multiphysics. COMSOL uses a segregated solver, with two groups that
converge to a single solution, for the turbulent flow k-ε. The segregated solver is
computationally complex, and even for a “Coarser mesh”, approximately 50,000 elements are created for the more simple models with fewer fins. Solutions with more fins
are more computationally demanding, but the “Coarser” mesh was still used to maintain
the integrity of the results. An example mesh is shown below in Figure 12, with the
finite element mesh data shown in Table 5.
Figure 12: “Coarser” Mesh
Table 5: “Coarser” Mesh Solution Data and PC Specifications
Objects, Domains, Boundaries, Vertices
41
Elements
Domain: 53,098
Boundary: 6,954
Degrees of Freedom
Solver 1: 27,804
Solver 2: 177,715
24
41
196
156
Solution Time
6 minutes, 18 seconds
PC Type
Lenovo PG101
PC Processor
Intel i3 – 3220; 3.30 GHz
PC RAM
4 GB
Studies involving 27 fins on the top and bottom of the flow path require the use of, an
“Extra Coarse” mesh, because the segregated solver runs out of memory during its
Lower/Upper matrix factorization for a “Coarser” mesh. With the “Coarser” mesh, the
number of degrees of freedom approached 250,000 for the second solver.
3.6 Expected Results
The expectation is that heat transfer will be maximized when the surface area in contact
with the fluid is maximized with the top fins, the boundary layer and the extent of the
dead flow zones in front and behind the fins is interrupted by the bottom fins, and the
flow path is extended with the labyrinth arrangement. In summary, the expectation is
that the greatest heat transfer will occur with the largest number of large fins on the top
and bottom of the flow path. Altogether, the efficiency of both entities in the panel, the
PV cells and the solar hot water heater, are expected to reach their peak when heat
transfer between the two components of the hybrid solar panel are maximized.
3.7 Model Limitations
A general proof of concept calculation, in Appendix A shows that the model, without
heat transfer to the surrounding atmosphere, is 85% accurate. This is largely due to the
model’s wall functions, which are used to model flow and heat transfer along the fluid/wall boundary. This limitation is especially noticeable under the low flow, small flow
height conditions, which this model is simulating. Higher flow rates, or larger flow
heights were not used, because they would not yield the useful temperature delta that is a
design requirement for a hybrid solar panel.
25
4. RESULTS AND DISCUSSION
4.1 PV/T Module Results
The 2-D model was run for the PV/T module with the flow length of three PV Cells for
each of the test arrangements outlined in Table 4. The parameters used in the model are
outlined in Table 2, which represent typical hybrid solar panel dimensions, properties,
and expected operating conditions for a hybrid solar panel in the northeastern United
States. COMSOL Multiphysics iteratively solves the finite element mesh of the PV/T
module using the equations stated in Section 3.4. Outlet water temperatures and PV cell
surface temperatures are averaged by COMSOL for each test arrangement. COMSOL
also uses the equations for the variables shown in Table 3 to calculate the electrical,
thermal, and net efficiency of the PV/T module; 𝜂𝑒 , 𝜂𝑡 , and 𝜂𝑜 respectively. The
electrical and thermal efficiency values were also calculated by hand in sections 7.2 and
7.3 using the inputs from Table 2 the COMSOL output temperatures. The calculation of
the electrical efficiency differed slightly from the COMSOL model result; the hand
calculation in 7.2 used the averaged cell temperature to calculate the efficiency once,
whereas the model result calculated the efficiency at each element of the cell and then
averaged, which is a more accurate method. The net efficiency is a simple addition of
the thermal and electrical efficiencies and is visually checked for each solution.
The COMSOL 2-D model result for flow around fins perpendicular to the flow path are
shown in Figure 8. The no slip boundary condition is noted along the walls of the flow
path, where the velocity is zero at the walls. The fins introduced into the flow disrupt
the developed flow profile exhibited in between fins. Water is accelerated as the flow
path height decreases at the tips of the fins, and the flow begins to decelerate as the area
suddenly increases after each fin. As the flow decelerates, the energy kinetic energy is
converted to pressure, creating an adverse gradient, flow separation, and therefore
increased mixing [15]. The use of sharp edged fins assures separation despite the low
bulk Reynolds number of the fluid.
26
Figure 13: Fin Velocity Disruption
It should be noted from Figure 13 that the flow quickly re-establishes itself after travelling past a fin. Figure 14 illustrates that, as the fins are moved closer together, the flow
spends less time at a constant velocity, and is instead constantly accelerated across each
fin and decelerated in between the fins. The flow path length is also increased as more
fins are added and the spacing between fins decreases. Flow no longer travels directly
across the panel, but instead crisscrosses between a labyrinth of fins. This arrangement
increases distance the flow travels, without increasing the length of the PV/T module.
27
Figure 14: Velocity Distribution in a Labyrinth Arrangement
Heat is transferred from the PV cells, through the highly conductive thermal paste and
copper wall, and into the fluid reservoir. Lines of constant temperature around a fin are
illustrated in Figure 15. The lines of constant temperature contour around the fin,
illustrating that the fin is a heat source to the fluid. The fins add surface area to the “hot”
top wall, which increases the heat conducted to the fluid, raising the water temperature
and cooling the PV cells.
28
Figure 15: Temperature Distribution Around a Fin
Lines of constant temperature are shown in Figure 16 below for a labyrinth fin arrangement. In the scenario below, the flow is mixing, causing the fluid temperature to be
more evenly distributed. In Figure 15, above, there are many lines of constant temperature that show layers of water with different temperatures. By contrast, Figure 16 shows
a flow with a more evenly distributed temperature, with fewer lines of constant temperature and the lines disappearing as the flow mixes and weaves through the labyrinth of
fins.
29
Figure 16: Temperature Contours in a Labyrinth Arrangement
The panel surface temperature has a non-linear distribution as shown in Figure 17. Heat
transfer between the hot solid layers at the top of the PV/T module and the cooling flow
is directly proportional to the driving temperature difference between the two. The
water temperature increases as it travels through the PV/T module, and, with a smaller
temperature difference between the panel and the cooling fluid, there is less heat transfer
to the fluid. Also, heat is transferred from the hot fluid outlet end of the panel through
the highly conductive PV, thermal paste, and copper layers to the cold fluid inlet end.
As a result, there is a greater amount of heat transfer and therefore a larger temperature
rise at the cold fluid inlet end of the panel, and there is a downward concavity to the
curve in Figure 17.
30
Figure 17: PV Surface Temperature Distribution
The results of each model run are shown in Table 6, related to their fin arrangements
proposed in Table 4. The water inlet temperature for each condition listed below is 11°C
(284.15 K), and the other parameters that remain constant are listed in Table 2.
Table 6: Module Results
Number
Fin Length*
𝑇𝑜 (K)
𝑇 𝐶𝑒𝑙𝑙𝑎𝑣𝑔 (K)
𝜂𝐸
𝜂𝑇
𝜂𝑜
None
0
0
288.05
287.42
19.00
43.48
62.48
9
¼
288.05
287.43
19.00
43.45
62.45
9
½
288.09
287.39
19.00
43.91
62.91
9
¾
288.19
287.33
19.01
44.95
63.96
18
¼
288.07
287.41
19.00
43.66
62.66
18
½
288.13
287.41
19.01
44.36
63.37
18
¾
288.25
287.27
19.01
45.73
64.74
9
¼
288.07
287.40
19.00
43.46
62.64
9
½
288.08
287.35
19.01
43.79
62.80
9
¾
288.12
287.25
19.02
44.17
63.19
18
¼
288.07
287.41
19.00
43.66
62.66
18
½
288.11
287.30
19.01
44.15
63.16
Bottom
Top
Fins
31
(Top and Bottom)
Labyrinth
18
¾
288.19
287.17
19.02
45.02
64.04
9
¼
288.07
287.39
19.00
43.65
62.65
9
½
288.12
287.32
19.01
44.20
63.21
9
¾
288.21
287.20
19.02
45.24
64.26
18
¼
288.10
287.37
19.00
43.96
62.96
18
½
288.19
287.26
19.01
44.91
63.92
18
¾
288.31
287.11
19.02
46.34
65.36
27
¼
288.13
287.33
19.01
44.32
63.33
27
½
288.24
287.20
19.02
45.58
64.60
27
¾
288.40
287.05
19.03
47.37
66.40
All results are plotted on the same chart in Figure 18 for comparison of each condition’s
efficiency; however, the subsequent plots show a less cluttered comparison of the
different conditions. The bottom fin conditions, top fin conditions, and labyrinth flow
conditions are all explored individually in the subsequent charts, Figures 20 – 22 respectively.
Fin Arrangement vs. Overall Efficiency
66.5
66
65.5
27 Labyrinth
Efficiency
65
18 Labyrinth
64.5
18 Top Fins
64
9 Labyrinth
63.5
18 Bottom Fins
9 Bottom Fins
63
9 Top Fins
62.5
62
1/4
1/2
3/4
Fin Length
Figure 18: All Fin Configuration Efficiency
32
The electrical efficiency of the PV cell varied only slightly with each case, with a lowest
efficiency of 19.00% at the no-fin condition and only 19.03% for the most efficient,
many, large fin labyrinth condition. This is behavior is caused by the large driving
temperature difference between the PV Cells and the cooling water as well as the already
low thermal resistance between the two. Adding fins perpendicular to the flow path has
a slightly more dramatic effect on the thermal efficiency of the hybrid solar panel, as the
flow separation does cause the fluid to mix more, and therefore accept more energy from
to PV cell. The thermal efficiency varies between 43.5% and 47.4%, a 3.9% difference
between the arrangement thermal efficiencies. The thermal efficiency is correlated with
electrical efficiency in Figure 19 below
PV/T Module Efficiency Correlation
48
47.5
Thermal Efficiency
47
46.5
46
45.5
45
44.5
44
43.5
43
18.995
19
19.005
19.01
19.015
19.02
19.025
19.03
19.035
Electrical Efficiency
Figure 19: PV/T Module Thermal and Electrical Efficiency Correlation
As expected, the electrical and mechanical efficiencies are correlated. As the thermal
efficiency increases, more heat is transferred away from the PV cells, keeping the cells
at a lower operating temperature. Consistent with semiconductor properties, PV cell
33
efficiency, or hybrid solar panel electrical efficiency, is inversely related to the operating
temperature. Due to the small changes in electrical efficiency, net or overall efficiency
of the PV/T module is largely governed by the thermal efficiency.
Efficiency of the PV/T module is dependent on the fin length for fins both on the top and
the bottom of the flow channel. Longer fins perpendicular to the flow path increase the
efficiency of the hybrid solar panels when compared to small fins. The large fins create
the largest flow disruption, mixing the fluid.
Fin Arrangement vs. Overall Efficiency
66.5
66
65.5
Efficiency
65
64.5
9 Top Fins
64
18 Top Fins
63.5
63
62.5
62
1/4
1/2
3/4
Fin Length
Figure 20: Top Fin Arrangement Efficiency
The top fins are shown to have a more dramatic affect than bottom fins, as the top fins
increase surface area of the “hot” top wall while still causing separation of boundary
layers in the flow. By contrast, the bottom fins only contribute to flow mixing and
disruption of boundary layers. This is evident when comparing the efficiency graphs of
the top vs. bottom fins; Figures 20 and 21 respectively.
34
Fin Arrangement vs. Overall Efficiency
66.5
66
65.5
Efficiency
65
64.5
9 Bottom Fins
64
18 Bottom Fins
63.5
63
62.5
62
1/4
1/2
3/4
Fin Length
Figure 21: Bottom Fin Arrangement Efficiency
As expected, top and bottom fins together yield the highest efficiencies for the same
number of fins. Flow is mixed due to the addition of top and bottom fins, the surface
area of the “hot” boundary is increased with the addition of top fins, and the flow path
length is increased as the flow has to “crisscross” over top and bottom fins as illustrated
in Figure 14.
35
Fin Arrangement vs. Overall Efficiency
66.5
66
65.5
Efficiency
65
64.5
9 Labyrinth
64
18 Labyrinth
63.5
27 Labyrinth
63
62.5
62
1/4
1/2
3/4
Fin Length
Figure 22: Labyrinth Arrangement Efficiency
The highest overall efficiency is achieved for the arrangement with the largest number of
fins with fins on the top and bottom of the flow path and the largest fin size. This is
evidenced in Figure 22, which shows that the case with 27 fins on the top and bottom of
the flow path and fins ¾ as long as the flow path height.
4.2 PV/T Array Results
With the PV/T module results evaluated, the most efficient module case is repeated in a
head to tail fashion to model an array. From section 4.2, the most efficient module was
the case with many, large fins in a labyrinth arrangement. This is compared to an array
with no fins for an overall comparison to the outlet water temperature and efficiency.
The array is shown in Figure 22, which consists of three modules, Figure 7 linked
together. The boundary between each module is assumed to be perfectly insulated, with
only the outlet water temperature carried from one module to the next. A perfectly
insulated boundary assumption allows for each module to be run as a separate entity,
36
removing concerns of conduction from the outlet, hot end of the array affecting the
results already calculated at the cold, inlet end.
Figure 23: PV/T Array [17]
The array results are displayed in Table 7. As an array, the hybrid solar panel with a
𝐿
labyrinth fin arrangement delivers 0.3 𝑚𝑖𝑛, calculated in Appendix 7.4, of water that has
been heated by 10.28 degrees Celsius. Without fins, the model array heats the water to a
temperature of 9.58 degrees, 0.7 degrees less than the arrangement with fins. The
energy transferred to the water has also cooled the PV cells, maintaining, in both cases a
similar cell operating efficiency. There is more heat transferred to the water in the case
with the labyrinth fin arrangement, creating a higher thermal efficiency; however, as the
water temperature rises, the PV cell temperature increases, decreasing the PV cell
temperature towards the end of the array. As a result of this, the average electrical
efficiency of the arrays, with and without fins, is about equal at 18.8%.
Table 7: Array Results
Fins
Number
Fin Length*
𝑇𝑜 (K)
𝑇 𝐶𝑒𝑙𝑙𝑎𝑣𝑔 (K)
𝜂𝑒
𝜂𝑡
𝜂𝑜
None
0
0
293.73
290.48
18.78
35.60
54.38
Labyrinth
27
¾
294.43
290.56
18.77
38.19
56.96
37
The difference in net efficiency is therefore controlled by the thermal efficiency, as
noted in Table 7. The amount of energy recouped from the environment is calculated in
the Appendix, Section 7.5, is a total of 320 W, for the conditions listed in Table 2.
4.3 Other Considerations
Because of this model’s 2-D nature, heat exchange structures such as pins were not
explored. An arrangement of many, small, cylindrical pins are expected to have a
positive effect on heat transfer between the fluid and the PV Cell. For a very space or
weight limited application, where cost doesn’t have as much of an impact, a porous
media heat exchange process might also be explored.
38
5. CONCLUSIONS
The greatest net PV/T module efficiency of 66.4% occurs with the labyrinth arrangement
or the arrangement with 27 top and 27 bottom fins that are ¾ the height of the flow path.
This is an approximate 4% increase in efficiency over the arrangement with no fins.
Sharp edged fins are used to cause flow separation, which mixes the fluid despite the
low Reynolds number and regardless of the boundary layer formation. Not only is the
flow mixing increased, but the flow path has been extended, as the flow crisscrosses
around the fins at the top and the bottom of the flow path. This increases surface area
between the working fluid and the “hot”, upper heat transfer boundary, without increasing the length of the hybrid solar panel.
The thermal efficiency has the greatest
variation, with the PV cell efficiency kept relatively constant due to the small temperature differences of the PV cell temperature between each arrangement.
When connected as an array, three modules linked in a head to tail arrangement, heat the
water by 10.3 degrees Celsius, collecting 320 W from the environment in the form of
usable electrical and thermal energy. The fins in the array provide a 2.6% increase in the
net efficiency over the array without fins, 57.0% vs. 54.4% respectively. Despite the
model limitations, fins perpendicular to the hybrid solar panel flow path are shown in
this model to increase the heat transfer between the PV cells and the cooling water,
increasing the amount of energy collected from the sun’s radiation
As predicted, the efficiency of a hybrid solar panel can be increased with fins perpendicular to the flow path. The efficiency increase is dependent on the number, size, and
arrangement of the fins, with the ideal arrangement consisting of many, large fins,
alternating between the top and bottom of the flow path in the direction of flow.
39
6. REFERENCES
1. NASA Goodard Space Flight Center. X-ray Spectroscopy and the Chemistry of
Supernova Remnants. March 25, 2010. Retrieved from http://imagine.gsfc.nasa.
gov/docs/teachers/lessons/xray_spectra/spectra_unit.html.
2. Wagner, Doris J. Rensselaer Polytechnic Institute. Glossary for Semiconductors.
2004.
Retrieved from http://www.rpi.edu/dept/phys/ScIT/Information Pro-
cessing/ semicond/sc_ glossary/scglossary.htm.
3. Chow, T. T. A review on photovoltaic/thermal hybrid solar technology; Applied
Energy, Volume 87, Issue 2010, Pages 365-379.
4. Skoplaki, E. and Palyvos, J.A. On the temperature dependence of photovoltaic
module electrical performance. Solar Energy, Volume 83, Issue 2009, Pages
614-624.
5. Armstrong, S. and Hurly, W.G. A thermal model for photovoltaic panels under
varying atmospheric conditions. Applied Thermal Engineering, Volume 30, Issue 2010, Pages 1488-1495.
6. National Renewable Energy Laboratory. Solar Water Heating. March 1996.
Retrieved from http://www.nrel.gov/ docs/legosti/fy96/17459.pdf.
7. Green Air Incorporated. “Efficient Solar Energy”. November 29, 2012. Retrieved from http://www.getgreenair.com/solar.
8. Fontenault, Bradley. “Active Forced Convection Photovoltaic/Thermal Panel
Efficiency Optimization Analysis” April 2012.
40
9. 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 (2012) 1076-1082.
10. B. Sopori Et al., Calculation of emissivity of Si wafers, Journal of Electronic Materials. Volume 28, Issue 1999, 1385–1389.
11. Suniva. ARTisun Select Monocrystalline Photovoltaic Cells. February 9, 2012.
Retrieved from http://www.suniva.com/products/ARTisun-Select-02-09-12.
12. United States Environmental Protection Agency. Average Temperature of Shallow Groundwater. January 10, 2013. Retrieved from
http://www.epa.gov/athens/learn2model/part-two/onsite/ex/jne_henrys_
map.html.
13. United States Department of Energy. Stakeholder Engagement & Outreach.
September 30, 2013. Retrieved from http://www.windpoweringamerica.
gov/wind_maps.asp October 15, 2013.
14. Brand TC-5026 Thermally Conductive Compound. Dow Corning. 2010. Form
No. 11-1689A-01. Retrieved From http://www.dowcorning.com/content/ publishedlit /11-1689a-01.pdf October 27, 2013.
15. Owens Corning. Foamular 400/600/1000 High Density Extruded Polystyrene
Rigid Insulation. May 2006.
16. White, Frank M. Fluid Mechanics 6th Ed. McGraw-Hill series in mechanical
engineering. 1221 Avenue of the Americas, New York, NY. Copyright 2008.
17. Tsukahara, Takahiro, Kawase, Tomohiro, and Kawaguchi, Yasuo. DNS of Viscoelastic Turbulent Channel Flow with Rectangular Orifice at Low Reynolds
Number. International Journal of Heat and Fluid Flow Volume 32, Issue 2011,
Pages 529-538.
41
18. SMS075-090W Monocrystalline Photovoltaic Module. Iwiss Solar. http://iwisssolar.com/Solar-Module/SMS075-090W-Monocrystalline-Photovoltaic-Module/
42
7. APPENDIX A: CALCULATIONS
7.1 Conservation of Energy
For a control volume, using the conservation of energy, the first law of thermodynamics
𝑄𝑖 = 𝑄𝑜𝑢𝑡
The inlet energy is the energy incident from the sun, taken for a unit length to relate to
the temperature rise in the 2-D model
𝑄𝑖 = 𝑞 ′′ 𝑠𝑢𝑛 𝐴𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
𝑄𝑖 = 1000
𝑊
(. 375𝑚)(1𝑚) = 375 𝑊
𝑚2
The outlet energy is the energy absorbed by the fluid
𝑄𝑜𝑢𝑡 = 𝑚̇𝐶𝑝 ∆𝑇
Where the mass flow rate is
𝑚̇ = 𝜌 𝑢𝑓𝑙𝑜𝑤 𝐴𝑓𝑙𝑜𝑤
Because we are working with a 2-D model, the cross sectional area perpendicular to the
flow is done based on a unit depth into the model
𝑚̇ = 𝜌 𝑢𝑓𝑙𝑜𝑤 𝐻𝑓𝑙𝑜𝑤 1 𝑚
𝑚̇ = 1000
𝑘𝑔
𝑚
𝑘𝑔
∗ .002 ∗ .005𝑚 ∗ 1𝑚 = .01
3
𝑚
𝑠
𝑠
𝑄𝑜𝑢𝑡 = 𝑄𝑖 = 375 𝑊 = (. 01
∆𝑇 = 8.98 𝐾
43
𝑘𝑔
𝑊
) (4175
) (∆𝑇)
𝑠
𝑚𝐾
7.2 Electrical Efficiency Verification
The equation for the electrical efficiency of a hybrid solar panel is dependent on the PV
cell temperature. The COMSOL model evaluates the efficiency at each element of the
PV cell layer in the model and then determines an average efficiency. The simple
calculation to verify the electrical efficiency is accomplished using only the average cell
temperature of the labyrinth arrangement with 27 top and bottom fins of ¾ flow path
height and the below equation [3].
𝜂𝐸 = 𝜂𝑇𝑟𝑒𝑓 (1 − 𝛽𝑟𝑒𝑓 (𝑇 − 𝑇𝑟𝑜𝑜𝑚 )
The values of the room temperature efficiency, 𝜂𝑇𝑟𝑒𝑓 , the temperature coefficient of
mono-crystalline silicon cells, 𝛽𝑟𝑒𝑓 , and the room temperature, 𝑇𝑟𝑜𝑜𝑚 are taken from
Table 2, with the value for the average cell temperature in Table 4.
𝜂𝐸 = .182(1 − .0041
1
(287.05𝐾 − 298.15𝐾) = 19.02%
℃
The a temperature difference is the same in Celsius as Kelvin, with the temperature
values left in Kelvin for convenience. Using the average temperature of the PV cells
under predicts the efficiency in this case, but is a close estimate to the 19.3% calculated
by the COMSOL model for each cell element and then averaged.
7.3 Thermal Efficiency Verification
The thermal efficiency of the labyrinth arrangement with 27 top and bottom fins of ¾
flow path height is calculated using the below equation [3].
𝜂𝑇 =
𝑚̇ ∗ 𝐶𝑝 (𝑇𝑜 − 𝑇𝑖 )
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
The output water temperature value from the PV/T module and inputs for the mass flow
rate, 𝑚̇, and the sun’s inlet radiative power in, 𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 , previously calculated in
44
Section 7.1, and 𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 , or the sun’s power into the panel, is 375 𝑊 are used to
calculate the efficiency. The specific heat value is taken at the outlet water temperature
of approximately 15°C.
𝑘𝑔
𝐽
(288.40K − 284.15𝐾)
. 01 𝑠 ∗ 4185
𝑘𝑔𝐾
𝜂𝑇 =
= 47.4%
375 𝑊
This value closely matches the 47.37% calculated by the model, and can be explained by
rounding error in this hand calculation.
7.4 Volume Flow Rate
The cross sectional area
𝐴𝑓𝑙𝑜𝑤 = 𝐻𝑓𝑙𝑜𝑤 ∗ 𝑤 = 0.5𝑚 ∗ 0.005𝑚 = 0.0025𝑚2
𝑉̇ = 𝐴𝑓𝑙𝑜𝑤 𝑢 = 0.0025𝑚2 ∗ 0.002
𝑉̇ = 0.000005
𝑚
𝑚3
= 0.000005
𝑠
𝑠
𝑚3 60𝑠 1000 𝐿 1000 𝑚𝐿
𝐿
∗
∗
∗
=
0.3
𝑠 1 𝑚𝑖𝑛 1 𝑚3
𝐿
𝑚𝑖𝑛
7.5 Net Energy Collected
For a PV/T cell, the net efficiency for a condition can be used to obtain the energy
collected from the environment.
𝑄𝑜𝑢𝑡 = 𝑄𝑖 ∗ 𝜂𝑜
The inlet energy is the energy incident from the sun using the dimensions of the PV/T
cell array is calculated below
𝑄𝑖 = 𝑞 ′′ 𝑠𝑢𝑛 𝐴𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 = 𝑞 ′′ 𝑠𝑢𝑛 ∗ 𝑙 ∗ 𝑤 = 1000
𝑊
∗ 0.5𝑚 ∗ 1.125𝑚 = 562.5 𝑊
𝑚2
The outlet energy of the cell is then taken using the efficiency of the PV/T cell with
many, large fins, in a labyrinth arrangement.
45
𝑄𝑜𝑢𝑡 = 562.5 𝑊 ∗ 0.5696 = 320.4 𝑊
46