 #1
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 Homework Statement:

Hello, I have a question which I am really struggling with attached below. I have tried to answer all sections, and areas in which there is an absence of solutions I have tried to suggest possible methods or approaches I could take. I am very uncertain though and would thus be very grateful of any help. I know there are quite a few questions here but they are related to one another, therefore, where I am uncertain about aspects this may be because I have made a mistake earlier, so if anyone could offer possible suggestions to help I would be very grateful. I am most stuck on question 1 b and c, and the same for 2 b and c which is essentially repeating the method but for the winter reading. This is supposedly an ALevel question I am completing for revision purposes yet I have never come across solar radiation flux density, which is causing me great confusion.
A company invests in 14 solar panels to generate electricity in the northeast of England. Each panel has an area of 1.38 m^2.
The average energy produced per day during the summer is 16.1 kWh and the average energy produced per day during the winter is 2.1 kWh. The panels are all oriented South and the roof is inclined at 35° (which gives an angle of 55° between the panel and the radiation).
Question 1:
a) Calculate the power generated in the summer
b) The solar panels are 100 % efficient, find the solar radiation flux density? Consider the angle of inclination in your calculation.
c) It is found that the solar panels are only 20.8 % efficient, meaning the calculated value of the solar radiation flux density in part c is an underestimate. Find the actual solar radiation flux density in northeastern England during the summer?
Question 2: Repeat for the winter reading.
Question 3: It is often very cold and cloudy in the northeast of England, considering this is the textbook’s value of 300 Wm^2 a reasonable example?
 Relevant Equations:

P=E/t
Intensity (Wm^2)= Power (W)/ Area (m^2)
Question 1;
a) P=E/t
E=5.796*10^7 J energy produced per day during the summer
However, I am not certain how to calculate the time period, since although this concerns the energy produced per day, the sun does not shine for the entire duration of this 24 hour period. Also, I am unsure of the average length of daylight hours in the northeast of England in the summer, and I do not want to insert such a value as it has not been specified by the question. Consequently, I have proceeded taking the time as one day but I do not think this calculation will produce the correct result for the power generated in the day during the summer.
t= 24 hours * 60 minutes * 60 seconds = 86400 s
P=5.796*10^7/86400
P=4025/6=640.8333… ~ 641 W to three significant figures
b) This is where I am having significant difficulty as I am utterly unfamiliar with the concept of solar radiation flux density. I have researched the term further but have had little fortune in finding anything helpful. I do not have anyone else to ask therefore I would be very grateful for any help or governance to assist my understanding.
The concepts I have come across neglect the angle of inclination also which is why I do not think I have arrived at the correct conclusion.
I did however come across the equation for the Solar Flux Density (Sd), which is the amount of solar energy per unit area on a sphere centred at the Sun with a distance d:
Sd =L/(4πd^2) W/m^2
Or perhaps the formula for the Solar Constant (S), which is defined as the solar energy density at the mean distance of Earth from the sun (1.5 x 1011 m):
S = L / (4 π d^2) S=(3.9*10^26 W)/[4*3.14*(1.5 *10^11 m)^2] = 1370 W/m^2
I do not know whether either of these are applicable here?
c) Again, I am not certain how to calculate the solar radiation flux density, regardless of the efficiency of the solar panels, which is why I have not answered this section.
Question 2;
a) P=E/t
E=7.56*10^6 J per day
t=? Again, I am uncertain of the time period, as the sun does not shine for the entire duration of a 24 hour period, and the average length of daylight hours in the northeast of England in the winter is shorter than the summer. However, taking the energy reading as per day I shall also take the time period as one day;
t= 86400 s
P=7.56*10^6/86400
P=175/2=87.5 W
b) and c) I do not know how to find the solar radiation flux density of the solar panels in the winter when the solar panels are 100% efficient or 20.8% efficient.
Question 3;
I am very confused here, firstly I have no idea which textbook the question is referring to. Secondly, I assume the value of 300 Wm^2 is a measure of intensity calculated by; Intensity (Wm^2)= Power (W)/ Area (m^2)
Consequently, to evaluate the reading of 300 Wm^2 would I input the values for the power in the summer and for the winter, to see whether this is a reasonable example? I do not think my values found for the power are correct but I will proceed nonetheless to show how I may continue.
In the summer;
Intensity= 641/ 1.38
Intensity = 464.4927.. ~ 464 Wm^2 to three significant figures
In the winter;
Intensity= 87.5/ 1.38
Intensity=63.40579… ~ 63.4 Wm^2 to three significant figures
I do not think my calculated values for power in the summer or winter are correct, however, if they were evaluating the calculations for the intensity I would assume the textbook’s reading of 300 Wm^2 is not a reasonable example, as this figure is too low for the summer and too high for the winter.
Thank you to anyone who replies
a) P=E/t
E=5.796*10^7 J energy produced per day during the summer
However, I am not certain how to calculate the time period, since although this concerns the energy produced per day, the sun does not shine for the entire duration of this 24 hour period. Also, I am unsure of the average length of daylight hours in the northeast of England in the summer, and I do not want to insert such a value as it has not been specified by the question. Consequently, I have proceeded taking the time as one day but I do not think this calculation will produce the correct result for the power generated in the day during the summer.
t= 24 hours * 60 minutes * 60 seconds = 86400 s
P=5.796*10^7/86400
P=4025/6=640.8333… ~ 641 W to three significant figures
b) This is where I am having significant difficulty as I am utterly unfamiliar with the concept of solar radiation flux density. I have researched the term further but have had little fortune in finding anything helpful. I do not have anyone else to ask therefore I would be very grateful for any help or governance to assist my understanding.
The concepts I have come across neglect the angle of inclination also which is why I do not think I have arrived at the correct conclusion.
I did however come across the equation for the Solar Flux Density (Sd), which is the amount of solar energy per unit area on a sphere centred at the Sun with a distance d:
Sd =L/(4πd^2) W/m^2
Or perhaps the formula for the Solar Constant (S), which is defined as the solar energy density at the mean distance of Earth from the sun (1.5 x 1011 m):
S = L / (4 π d^2) S=(3.9*10^26 W)/[4*3.14*(1.5 *10^11 m)^2] = 1370 W/m^2
I do not know whether either of these are applicable here?
c) Again, I am not certain how to calculate the solar radiation flux density, regardless of the efficiency of the solar panels, which is why I have not answered this section.
Question 2;
a) P=E/t
E=7.56*10^6 J per day
t=? Again, I am uncertain of the time period, as the sun does not shine for the entire duration of a 24 hour period, and the average length of daylight hours in the northeast of England in the winter is shorter than the summer. However, taking the energy reading as per day I shall also take the time period as one day;
t= 86400 s
P=7.56*10^6/86400
P=175/2=87.5 W
b) and c) I do not know how to find the solar radiation flux density of the solar panels in the winter when the solar panels are 100% efficient or 20.8% efficient.
Question 3;
I am very confused here, firstly I have no idea which textbook the question is referring to. Secondly, I assume the value of 300 Wm^2 is a measure of intensity calculated by; Intensity (Wm^2)= Power (W)/ Area (m^2)
Consequently, to evaluate the reading of 300 Wm^2 would I input the values for the power in the summer and for the winter, to see whether this is a reasonable example? I do not think my values found for the power are correct but I will proceed nonetheless to show how I may continue.
In the summer;
Intensity= 641/ 1.38
Intensity = 464.4927.. ~ 464 Wm^2 to three significant figures
In the winter;
Intensity= 87.5/ 1.38
Intensity=63.40579… ~ 63.4 Wm^2 to three significant figures
I do not think my calculated values for power in the summer or winter are correct, however, if they were evaluating the calculations for the intensity I would assume the textbook’s reading of 300 Wm^2 is not a reasonable example, as this figure is too low for the summer and too high for the winter.
Thank you to anyone who replies