Assumptions and Description of Each Tab:
Hourly Solar Flux: Goal of this tab is to calculate how much energy can be produced by the panel if there
was no snow falling. This spreadsheet takes into account the panel that will be prototyped.
Column Description:
Column A: Date taken from TMY3 data
Column B: Time taken from TMY3 data
Column C: The average of the hour is best calculated in between hours.
𝐷𝑎𝑦 𝐻𝑜𝑢𝑟 = (𝑇𝑖𝑚𝑒 − 𝐼𝑁𝑇(𝑇𝑖𝑚𝑒)) ∗ 24 − 0.5
Column D: DNI (W/m2) Direct beam data taken from TMY3
Column E: DHI (W/m2) Diffuse beam data taken from TMY3
Column F: n is the day number of the year. Uses excel formulas to calculate.
𝑛 = 𝐷𝑎𝑡𝑒 − 𝐷𝐴𝑇𝐸(𝑌𝐸𝐴𝑅(𝐷𝑎𝑡𝑒), 1,0)
Column G: B is a parameter needed to find the solar time.
360
𝐵=
(𝑛 − 81)
365
Column H: E(n) is a parameter needed to find the solar time
𝐸(𝑛) = 9.87 SIN(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(2𝐵)) − 7.53𝐶𝑂 𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐵)) − 1.5𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐵))
Column I: ST is the solar time
4(75−77.63)+𝐸(𝑛)
𝑆𝑇 = (
) + 𝐷𝑎𝑦 𝐻𝑜𝑢𝑟
60
NOTE: 77.63 is the longitude at Rochester and 77 is the local time meridian
Column J: H is the hour angle
𝐻 = (12 − 𝑆𝑇)15
Column K: δ is the solar declination angle
360
𝛿 = 23.45𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆((365)(𝑛 − 81)
Column L: β is the solar altitude angle
𝛽 = 𝐷𝐸𝐺𝑅𝐸𝐸𝐷(𝐴𝑆𝐼𝑁(𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐿))𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛿))𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐻)) +
𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐿))𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛿)))
Column M: φ is the solar azimuth angle
𝜙 = 𝐼𝐹(𝐶𝑂𝑆(𝐻) >= (𝑇𝐴𝑁(δ)/𝑇𝐴𝑁(𝐿)), 𝐷𝐸𝐺𝑅𝐸𝐸𝑆(𝐴𝑆𝐼𝑁((𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(δ)) ∗
𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐻)))/𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(β)))),180 − 𝐷𝐸𝐺𝑅𝐸𝐸𝑆(𝐴𝑆𝐼𝑁((𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(δ)) ∗
𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝐻)))/𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(β)))))
Column N, S, X, AC, AH, AM: cosθ is needed to calculate the direct beam on the panel
𝐶𝑜𝑠𝜃 == 𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(β)) ∗ 𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(φ)) ∗ 𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛴)) +
𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(β)) ∗ 𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛴))
Column O, T, Y, AD, AI, AN: IBN is the direct beam from the sun that hits the panel
I𝐵𝑁 == 𝐼𝐹(𝐶𝑜𝑠𝜃 > 0, 𝐷𝑁𝐼 ∗ 𝐶𝑜𝑠𝜃 ∗ 𝜂, 0)
Column P, U, Z, AE, AJ, AO: IDC is the diffuse radiation that hits the panel
𝐼𝐷𝐶 = 𝐷𝐻𝐼 ∗ ((1 + 𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛴)))/2) ∗ 𝜂
Column Q, V, AA, AF, AK, AP: IRC is the reflected radiation that hits the panel
𝐼𝑅𝐶 = 0.8 ∗ (𝐷𝑁𝐼 ∗ 𝑆𝐼𝑁(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛽)) + 𝐷𝐻𝐼) ∗ ((1 − 𝐶𝑂𝑆(𝑅𝐴𝐷𝐼𝐴𝑁𝑆(𝛴)))/2) ∗ 𝜂
Column R, W, AB, AG, AL, AQ: IC is the total radiation that hits the panel
𝐼𝐶 == 𝑆𝑈𝑀(𝐼𝐵𝑁, 𝐼𝐷𝐶 , 𝐼𝑅𝐶 )
Column AR: Average Ic for tilts for panel takes the average IC for all panel tilts calculated. This is the IC
that is used for all following tabs.
Assumptions/Parameter:
- Uses data for Rochester, NY
- TMY3 Data utilizes weather data from the past 30 year. It takes the month that most closely
resembles an average of that data.
- The panel efficiency, η, was assumed to be the same as the cell efficiency. Prototyped panel will
be unique and there are no specs from a 2 by 2 array of the cell being designed around.
- Panel faces directly solar South
- Area of the panel is equal 4 cell areas. The cells are assumed to be perfect squares when in
reality they have the corners cut.
- L is the latitude of Rochester NY
- Σ is the tilt angle of the panel
Monthly Solar Flux: The goal of this tab is to just condense the data. Sums up all the IBN, IDC, IRC and IC for
different tilt angles based on month. One row gives the yearly total. An average yearly total is there
based on an average tilt angle. A 10% yearly average was also taken.
Fin Analysis Cell: The goal of this tab is to demonstrate the qfin value of the different possible sections of
a cell. A fin analysis is done on for the glass with dimensions that match the cell.
Column Description:
Columns with T∞ - Values that were chosen from 253k to 273k. This is the ambient temperature.
Columns with Tb – This is the base temperature. It is the temperature that the ink needs to heat up to
𝑇𝑏 = ((𝑇(𝑥) − 𝑇∞ ) ∗ 𝐶𝑂𝑆𝐻(𝑚 ∗ 𝑑𝑥)) + 𝑇∞
NOTE: Usually this equation is divided by cosh(m(L-x)), however for each instance x=L so this
reduces to a value of 1
Columns with qfin- This is the calculated heat needed to ensure that in between ink traces is at 5°C at
different ambient temperatures with different convection coefficients.
𝑞𝑓𝑖𝑛 = 𝑆𝑄𝑅𝑇(ℎ𝑃𝑘𝐴𝑐 )(𝑇𝑏 − 𝑇∞ )𝑇𝐴𝑁𝐻(𝑚𝑥)
Assumptions/Parameters:
- The thermal conductivity of glass, k, is assumed to be 1.4 W/mk
- The length of a cell is 0.156m taken from the spec sheet
- The thickness of the glass is 0.003175m, this is an assumed value
- It is assumed that the panel is insulated on the bottom of the glass.
- T(x) is the temperature of the glass in the center between the ink lines. This values is assumed to
be 278k.
- The perimeter, P, is the perimeter where the heat is traveling through. This is constant for all
sections of the cell
- The cross sectional area, Ac is the area that the heat travels through. This is constant for all
sections of the cell.
- The convection coefficient, h, values were chosen. They range from 5 to 28 W/m2k
- This fin analysis is used assuming an adiabatic tip
- X is half the distance between inks. The possible ink distances were taken from the cell spec
sheet.
- m is a parameter that is used to find both Tb and qfin.
o
ℎ𝑃
𝑘𝐴𝑐
𝑚=√
Heat Generation Cell: The goal of this tab is to look at the total energy required to conduct heat through
the glass, melt the snow and take into account heat that is radiating from the glass to the outside.
Column Description:
Columns with T∞ - Values that were chosen from 258k to 273k. This is the ambient temperature.
Columns with qfin- This is the calculated heat needed to ensure that in between ink traces is at 5°C at
different ambient temperatures with different convection coefficients. These values were taken from
the Fin Analysis Cell tab.
Columns with qmelt- This is the energy required to melt the snow in one section of a cell.
𝑞𝑚𝑒𝑙𝑡 = 𝑞𝑚𝑒𝑙𝑡 (𝑊)𝐿𝑑𝑥
Columns with qrad- This is the energy lost due to radiation.
𝑞𝑟𝑎𝑑 = 𝜎𝜀(𝑇𝑠 4 − 𝑇∞ 4 )𝐿𝑑𝑥
NOTE: The surface temperature, Ts, is assumed to be the temperature in between the inks,
278k.
Columns with qGen – This is the energy needed to be generated to melt the snow, have the panel heat to
a good temperature and compensate the amount of energy radiating out of the panel.
𝑞𝐺𝑒𝑛 = 𝑞𝑓𝑖𝑛 + 𝑞𝑚𝑒𝑙𝑡 + 𝑞𝑟𝑎𝑑
Assumptions/Parameters:
- The thermal conduction of the glass, kGlass, is 1.4W/mk
- σ is the Stefan-Boltzmann constant
-
-
ε is the emissivity of glass, which is assumed to be 0.94.
dx is half the distance between inks. The possible ink distances were taken from the cell spec
sheet.
The convection coefficient, h, values were chosen. They range from 5 to 28 W/m2k
The length of a cell is 0.156m taken from the spec sheet
An average daily snowfall rate was found by taking the average amount of snow that falls on a
snow day from the Snowfall tab. It was assumed that the snow would fall for four hour each day
it snows. The average daily snowfall was divided by (4hours*3600sec). This gave a snowfall rate
in m/s.
The qmelt in W/m2 was found from:
o 𝑞𝑚𝑒𝑙𝑡 = 𝑆𝑛𝑜𝑤𝑓𝑎𝑙𝑙 𝑅𝑎𝑡𝑒 ∗ 𝜌 ∗ ℎ
ρ is assumed to be 60kg/m3
h, the heat fusion is assumed to be the same as for water, 334,000 J/kg
Configurations: The goal of this tab is to compare 4 different possible configurations under assumed
conditions. The layout was picked based on these calculations.
Column Description:
Columns with T∞ - Values that were chosen from 258k to 273k. This is the ambient temperature.
Columns with Configuration 1: This configuration has 8 0.013 sections and 8 0.052 sections.
Corresponding (h and T∞) values of qGen were taken from the Heat Generation Cell tab.
Columns with Configuration 2: This configuration has 8 0.039 sections. Corresponding (h and T∞) values
of qGen were taken from the Heat Generation Cell tab.
Columns with Configuration 3: This configuration has 4 0.078 sections. Corresponding (h and T∞) values
of qGen were taken from the Heat Generation Cell tab.
Columns with Configuration 4: This configuration has 4 0.013 sections, 4 0.052 sections and 2 0.029
sections. Corresponding (h and T∞) values of qGen were taken from the Heat Generation Cell tab.
Assumptions/Parameters:
- The different configurations can be viewed from the link at the top of the spreadsheet.
- The convection coefficient, h, values were chosen. They range from 5 to 28 W/m2k
Day Snow: The goal of this tab is on days it snows compare the net energy output of the prototype
panel if snow was prevented vs if nothing was done to the snow on a panel.
Column Description:
Total snow: This is the total amount of snow that falls on that particular day taken from the Snowfall
tab.
Column A: Time taken from TMY3 data
Column B: Ic (W) is the amount of power that the panel will produce if there was no snow This was taken
from the Hourly Solar Flux tab.
Column C: Ic (J) This is column B converted into joules (multiply by 3600).
Column D: Actual Ic is how much energy the panel will produce based on the percentage of light that will
get through the snow when it accumulates on the panel. The equation to get the percentage was taken
from the chart on the % Radiation tab.
𝐴𝑐𝑢𝑡𝑎𝑙 𝐼𝑐 = 𝐼𝑐 (𝐽) ∗ ((−10.53 ∗ 𝐿𝑁(𝑆𝑛𝑜𝑤 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 ∗ 100) + 27.269)/100)
Column E: Snowfall (m) this column distributes the total snow in that day over four hours during the
peak sun hours.
Column F: Snow Accumulation (m) is how much snow would accumulate on the panel to that point in
the day.
Column G: Snow(kg) converts the Snowfall(m) into kg.
𝑆𝑛𝑜𝑤 (𝑘𝑔) = 𝑆𝑛𝑜𝑤𝑓𝑎𝑙𝑙(𝑚) ∗ 𝜌
Column H: Energy to melt (J) is thee energy required to melt the snow at that particular hour.
𝐸𝑛𝑒𝑟𝑔𝑦 𝑡𝑜 𝑚𝑒𝑙𝑡 (𝐽) = 𝑆𝑛𝑜𝑤(𝑘𝑔)ℎ
Column I: h(W/m2k) is the convection coefficient for that particular hour. These values were taken from
the convection coeff tab.
Column J: m is a parameter needed for calculating Tb and qfin
𝑚 = 𝑆𝑄𝑅𝑇((ℎ𝑃)/(𝑘𝐴𝑐 ))
Column K: T∞ is the ambient temperature taken from TMY3 data
Column L: Is the base temperature, Tb, for one section of a cell.
𝑇𝑏 = ((𝑇(𝑥) − 𝑇∞ ) ∗ 𝐶𝑂𝑆𝐻(𝑚 ∗ 𝑑𝑥)) + 𝑇∞
NOTE: Usually this equation is divided by cosh(m(L-x)), however for each instance x=L so this
reduces to a value of 1
Column M: This is the calculated heat needed to ensure that in between ink traces (x=0.039m) is at 5°C
using specific data for that day.
𝑞𝑓𝑖𝑛 = 𝑆𝑄𝑅𝑇(ℎ𝑃𝑘𝐴𝑐 )(𝑇𝑏 − 𝑇∞ )𝑇𝐴𝑁𝐻(𝑚𝑥)
Column N: qmelt (W) is the energy in Watts to melt the snow for that hour.
𝑆𝑛𝑜𝑤𝑓𝑎𝑙𝑙(𝑚)
) ℎ𝜌)𝐿𝑊
3600
𝑞𝑚𝑒𝑙𝑡 = ((
Column O: qrad (W) is how much energy is radiated out of the panel
𝑞𝑟𝑎𝑑 = 𝜎𝜀(𝑇𝑠 4 − 𝑇∞ 4 )𝐿𝑑𝑥
Column P: qgen (W) is the total energy the panel needs to prevent snow on one section of a cell
𝑞𝐺𝑒𝑛 = 𝑞𝑓𝑖𝑛 + 𝑞𝑚𝑒𝑙𝑡 + 𝑞𝑟𝑎𝑑
Column Q: qgen (W) is the total energy the panel need to prevent snow on the entire panel based on
configuration 2. This is column P*8
Column R: qgen (J) is the total energy the panel need to prevent snow on the entire panel based on
configuration 2. In joules This is column Q*3600.
Column S: Energy total prevention is the net energy of the panel if snow is prevented.
𝐸𝑛𝑒𝑟𝑔𝑦 𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑒𝑣𝑒𝑛𝑡𝑖𝑜𝑛 = 𝐼𝑐 − 𝑞𝑔𝑒𝑛 (𝐽)
Column T: Energy Total Accumulation (J) is the same as column D.
Columns S and T are graphed for each day.
Assumptions/Parameters:
- On each day it snows for four hours during the peak sun hours at a constant rate
- It is assumed that each day has no snow on the panel to start with.
- The analysis was only done on days it snowed.
- When snow accumulates on a panel it is assumed that it stays there for the rest of the day.
- This analysis doesn’t take into account the fact that when a cell or parts of a cell on solar panel
are shaded, the shaded area heats up, which would help melt the snow
- These calculation are based on configuration 2 for a full prototype panel
- Total energy for accumulation is the sum of all the net energy if snow was left alone on the
panel for an entire year (sum of column T)
- Total energy for prevention is the sum of all the net energy if snow was melt right away on the
panel for an entire year (sum of column S)
- The average power output when snows is the average energy needed to melt snow for four
hours in one day.
- The density of snow, ρ, is assumed to be 60 kg/m3
h, the heat fusion is assumed to be the same as for water, 334,000 J/kg
- The thermal conduction of the glass, kGlass, is 1.4W/mk
- The thickness of the glass is 0.003175m, this is an assumed value
- The length of a cell is 0.156m taken from the spec sheet
- The width of the cell section is 0.039m based on configuration 2
- σ is the Stefan-Boltzmann constant
- ε is the emissivity of glass, which is assumed to be 0.94.
- The section perimeter is 0.3185 based on the length of the cell and the thickness of the glass.
- T(x) is the temperature of the glass in the center between the ink lines. This values is assumed to
be 278k.
- The cross sectional area, Ac is the area that the heat travels through.
Convection Coeff: The goal of this tab is to calculate the convection coefficient based on daily weather
data.
Column Description:
Column A: Date taken from TMY3 data
Column B: Time taken from TMY3 data
Column C: The dry bulb temperature was taken from TMY3 data
Column D: The absolute temp, T, is the conversion of column C from Celsius to Kalvin
Column E: The pressure is taken from TMY3 data.
Column F: The pressure, P, from column E is converted from mbar into atm.
Column G: The wind speed, V, was taken from the TMY3 data
Column H: The density of the air is calculated for every hour
𝑃∗𝑀𝑊
𝜌 = 𝑅𝑇
Column I: This is the viscosity, μ, inputted based on temperature.
Column J: The Reynolds Number, Re, was calculated. All values are below 5*105 so a laminar flow model
can be used.
𝜌𝐿𝑉
𝑅𝑒 =
𝜇
Column K: The Prandtl Number, Pr, was inputted based on temperature
Column L: The Nusselt Number, Nu, was calculated based on a forced convection and laminar flow.
𝑁𝑢 = 0.453𝑅𝑒1/2 𝑃𝑟1/3
Column M: The thermal conduction, k, of the air was inputted based on temperature.
Column N: The convection coefficient, h, was calculated
ℎ = (𝑁𝑢𝑘)/𝐿
Assumptions/Parameters:
- This calculation does not take into account the angle of the of the panel
- The direction of the wind compared to the direction of the panel is not taken into account
- Calculations only done on days it snows
% Radiation: The goal of this tab is to find a relationship between the snow depth and the percentage of
light that passes through the snow to the panel. The relationship is based on two points found during
research. A best fit logarithmic curve was put in based on research saying this is a log relationship.
Snowfall: The goal of this tab is to compile the daily snowfall data.