Solar Panel Lab - APPM 2350 - Spring 2010

Items to be turned in:

1. FORMAL WRITTEN REPORT (cover page/intro/conclusion style), containing all answers within the appropriate paragraphs.  Click here for an example.
2. Appendix, containing references, scripts, and any hand-written work (hand-written work is not recommended as the most efficient way to do this particular lab).

Helpful files:

CLICK HERE for a Mathematica notebook on functions, derivatives, and plots.  Everything you need for this lab can be found in this file. 

I. Instructions

This lab is due Monday, March 15 in lecture. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you can work alone. If you do work in a group, your partners can be from any of the Calculus 3 lectures. Turn in only one lab per group.

On your title page, clearly mark

• Names
• Student ID numbers
• Recitation number and professor
• Time and TA for recitation
• If you do not include this information, as much as 10% will be deducted from your final score
  

For general directions for your report, please look at a sample report.

II. Lab Goals

The goal with this lab is to apply the concept of optimization over a compact domain to a physical problem. The function you qualitatively optimize depends on two variables. Your result should be critically interpreted. The lab will also provide practice in report writing.  You will also learn some facts about solar radiation and how the solar radiation received on the ground depends on the season, time of the day and weather conditions.

III. Background

In order to solve the problem you are assigned later in this instruction, it is important to read the background below!

In this lab you will analyze how the efficiency of a solar panel depends on the season and its orientation.  The solar radiation falling on a tilted plane (such as a solar panel) depends on numerous factors: the orientation of the plane, the time of the day, the season, the weather, etc. In the next few sections you will learn more about how this affects the power we can receive from solar panels.

III.a The Earth and the Sun

The Earth's orbit about the sun is almost circular so for this lab we will assume that the distance to the Sun is constant. However, the axis of the Earth is tilted relative to the plane in which the Earth moves around the Sun. The angle between the axis and a normal to this plane is roughly . As indicated in Fig. 1, the north pole points away from the Sun during the northern hemisphere winter, and points towards the Sun during the northern hemisphere summer.Now let us imagine that we stand on the equator. We introduce the angle s as illustrated in Fig. 2. In words, the angle s determines the angle between you (if you are standing straight up on the equator at noon) and a line from the center of the Earth to the center of the Sun. By "noon" we mean the time of the day when the Sun is at its highest position in the sky. As we can see from Fig. 2, on December 21 (winter solstice) and on June 21 (summer solstice). On March 20 (vernal equinox) and September 22 (autumnal equinox) . Hence, we can think of s as describing the time of the year. This angle s will be of great importance in the lab so let us summarize its values in a table. 

IMPORTANT: The functions we use later will require that we convert degrees to radians.  For example, 23 degrees is 23/180*Pi radians.

Note that .

III.b Radiation falling on a tilted plane

In this lab we will consider the solar radiation falling on a tilted plane.  We will consider a solar panel that maintains a fixed angle with respect to the ground throughout the entire day.  Let u be the angle between the plane and the ground. If , then the plane is lying on the ground and if the plane is standing up vertically. (If u is negative, the solar panel is facing south. If u is positive, the solar panel is facing north. Understanding this might give you a better idea of what's going on when you look at your 3D plot in Question 2.) 
Introduce t to be an angle proportional to the time of the day such that at 6 A.M., at noon and at 6 P.M.. If I₀ is the intensity of solar radiation (measured in W/m², where W=J/s) falling on the ground, then the radiation on the tilted plane, Iplane is given by the function:

There are more factors that affect the radiation on the plane. There is also an absorption factor which is strongly dependent on the distance the sun rays have traveled through the atmosphere before it reaches the ground. The law describing absorption is called Beer's law (sic!). Let us describe the absorption with a function A(s,t) such that . Here A(s,t)=1 means that basically all the sunlight gets through the atmosphere (no light absorption) and that the absorption of light by the atmosphere increases with decreasing values of A(s,t) so that a low value () of A(s,t) means that almost no sunlight makes it through (very high absorption). In addition, we should also take different weather patterns into account since the solar radiation on the ground varies with the cloudiness. Let us describe the cloudiness with a function C(s,t) such that . Here C(s,t)=1 means that there are no clouds and that the cloudiness increases with decreasing values of C(s,t) so that a low value () of C(s,t) means that the sky is covered with thick black thunder clouds. The total amount of energy received per square meter and day is now given by the following equation.

Here tmin and tmax are the angles that we talked about above.  These angle limits represent the time of sunrise and sunset respectively. These times vary over the year. However, on the equator these times are relatively constant and to simplify the mathematics we will assume that for our purpose we can use and throughout the year. (This is an approximation. For a more careful analysis of a problem, we should let these times depend on the season, i.e., s.)

IV. Problem Statement

The Boulder company "Solar Power Inc." (invented for this lab) has asked you to do a consulting job for them. They will export solar panels to the pacific island "Suluclac" which is located on the equator. A meteorologist has described their weather by the "cloudiness function" 

Note that this equation is only valid when s and u are given in radians!  Also remember that the angle u of the solar panel is fixed on a given day.  We are not working with a panel that moves throughout the day to follow the sun by the hour.  A panel like this might collect more solar energy, but it also requires a more expensive mechanical set up, so a solar panel with a fixed angle u, which may be changed manually on, say, a daily or weekly basis to at least account for the time of year, is often more practical.The variable s is defined in section III.a and u is defined in section III.b, but recall that s determines the time of year and u is the angle between the solar panel and the ground. The energy equation given above has already incorporated effects due to absorption and cloudiness, so the function W(s,u) is already in the form you will use for this lab.

The company now wants you to answer the following questions.The questions "look" long because they contain lots of hints!

• Refer to the Mathematica plots and functions notebook
• We suggest that all computational work be done in Mathematica for this particular project. Everything you need can be found in the example Mathematica notebook.
• You need to have all your answers in the formal written report.

Question 1

• i) Create a labeled 3D plot of C(s,t), which will help you visualize average cloudiness.  Refer to the given plots and functions notebook (see link above) for information on 3D plots.  MAKE SURE TO PROPERLY LABEL ALL PLOTS, just like you see in the example notebook.  You can even copy and paste code from the notebook, changing just a few key things to build your functions and plots.
• ii) Describe how the weather, or more specifically the cloudiness, varies on Suluclac. In other words, use the information in section III.b along with your plot and analyze what the function C(s,t) (equation 3) can tell us about how the cloudiness varies over the course of the average Suluclac day and over the course of a year.
  • Remember that C(s,t) goes from 0 to 1, and that 1 means no clouds, while 0 means maximum cloudiness.  You can think of the funcation as saying "0 means that no light gets through the clouds at all."  Also remember that t is an angle which is negative during the first half of the day and positive during the last half of the day.

Question 2

In the domain  and where the angles are given in radians, find the critical point(s) of w[s,u] (see definition for critical point and read about the second derivative test for local extrema, starting on page 970 of your Calculus book), and compute the energy at the point(s).

• i) First do a 3D plot of w[s,u] (equation 4) with the given domain to get an idea of what is going on.  (If you copy and paste code, you will have to make sure you change the limits to correctly provide the domain we want.)
• ii) Use a contour plot to find where the gradient of w[s,u] is zero in order to determine the initial guess for a possible critical point.  An example of how to build a gradient function AND do the requested plot is shown in the Mathematica plots and functions notebook.
• iii) Use the Mathematica command FindRoot (explained in detail in the Mathematica plots and functions notebook) to find the point(s) at which the gradient of w[s,u] is the zero vector.  In order to do this, you will need to give the FindRoot function an initial guess for a critical point so that it will search in that local area.  
• iv) Classify the critical point(s) as local maxima, local minima or saddle points.  
• v) Compute the energy at these points.
  • Make sure to explain the significance of what you are doing in each step of this problem (refer to book pages 970-972) as it is very important to be able to do this when writing a formal report.

Note on Questions 3 and 4: You will find all local extreme points of the energy function in its domain (notice that in part 2 above you have already searched for these points in the interior of the domain. All you have left to do is to look for these points on the boundary of the domain of the function.  Don't forget to check corner points.)  The point of doing this is to see if you have any boundary values that are bigger or smaller than the max or min interior values.

Question 3

• i) Find other possible local extreme points by fixing a single variable at the edge of its boundary, forming a single variable function.  
  
  • As an example, s, the angle proportional to the time of year is bounded below by -0.4 at winter solstice and above by 0.4 at summer solstice.  If we fix s to be 0.4, then we will look for the local max of the function w[.4, u] by taking its derivative and setting it to zero.  Notice that when we plug in 0.4 for s in to the two variable function w[s,u], we just end up with a one variable function w[.4, u].  We can then use FindRoot on D[ w[.4,u], u], the first derivative with respect to u to find possible critical points.  If we fix s to be -0.4, we look for the local max of w[-.4,s].  It works the same way for u, which is bounded below by -Pi/2 and above by Pi/2.  If it appears obvious to you that a particular side will have no critical points, then you don't have to do the derivative test on such a side. Explain why you can see this from your 3D plot.
• ii) Use your 3D plot to explain if the points you found in part i) are maxima, minima, saddle points, etc.
  

Question 4

• i) Compute the energy at the points that you found above, as well as at the four corner points.   Plug in the values you just found in Question 3 to the w[s,u] function.  Then plug in the corner values.  For example, check the corner value of w[-.4, Pi/2].
• ii) Specify if these values (points from Question 3, and corner points) are global maxima or global minima in the domain of the function by comparing them to the energy at the critical point(s) found in Question 2.
  • For example, if the value at one of the corners turns out to be smaller than any of the other values observed, then the value at that corner would be the global minimum for the given domain.  Also important to note is that if we were to increase the size of the domain, we might have different global extrema, but for this problem we are only focused on the domain given at the beginning of this lab because it is most relevant to our application.
  • Make sure you've defined your domain correctly.

Question 5

First remember that the angle u refers to the angle that a fixed solar panel makes with the ground.  The solar panel does not move on its own.  It will stay at the same angle until someone manually changes it.  Suppose we hire someone to change the angle u of the solar panel on a regular basis, maybe once at the end of each day, so that on any given day we end up collecting as much energy as we can, based on the angle s that the equator makes with the incoming sun rays.

• i) During what season of the year will we end up collecting the most solar energy?  Explain your answer, i.e. don't just say "When there is more light!" 
  Also remember that the angle s is proportional to the season of the year.  In radians, s goes from 0.4 at summer solstice (the longest day of the year) all the way down 0 at the autumnal equinox, then down to -0.4 at the winter solstice (the shortest day of the year), back up to 0 at the vernal equinox, and back to 0.4 at the summer solstice.

Question 6

• i) Produce a contour plot of w[s,u] (NOT its gradient like in Question 2), similar to the plot shown in the notebook file.
• ii) Explain in words approximately how our employee should change the angle u of the satellite over time in order to maximize the amount of solar energy collected over the course of a year.  You do not need an exact solution for u.  On your contour plot, draw a line (if not on the computer, then IN PEN) to give a qualitative representation of how u should be changed.  
• iii) Does your answer to ii) make sense based on what you see from your 3D graph of w[s,u]?  Explain.
  

Question 7

• i) Suppose that the solar panels need to be replaced once per year, and suppose it takes a significant amount of time to replace them, i.e. days or weeks. 
• ii) Finally, suppose that during the replacement process, we can't collect any solar energy.  What time of year should we replace the panels?  Explain.

Hints

Refer to your book (pgs 970-972) for critical point definition and second derivative test for two variable functions.

Read the definition of local maxima and minima on page 970. Among all interior critical points, boundary candidates, there should be only 3 local extrema. Note that in Questions 2 and 3, your are asked to find all candidates for critical points, i.e., critical points in the interior, ciritical points in the interior of the boundary, and corner points. This will give you a list of all possible candidates for global extrema in the given region--which you will need to find in Question 3.

V. Lab Report

"Solar Power Inc." wants you to describe all your results in a formal written report, containing an introduction to the concept of solar energy collection, relevant equations given, ANSWERS TO ALL QUESTIONS at appropriate places within paragraphs, and a conclusion.

• Explain the purpose of the lab. State the problem.
• State which mathematical tools were used while working on the lab.
• Define the equations used (equations 3 and 4) and define all variables used. You may use Figs. 1 and 2 directly in your report (printed from the web).
• Answer all the questions asked by "Solar Power Inc." Don't forget to list all local extreme points, classify them, and compute their energy output.
• Show all your work. State the equations and how you found the maxima and minima (i.e. derivatives equal zero, initial guesses, etc). 
• Include plots that back up your reasoning.
• Interpret your results! Interpret the results in physical terms. For example, write "the maximum energy output is given at this time of year because..." rather than "W has a maximum at s=X and I have no idea what s even is..."

Sample grading sheet

Remember the following:

• Always include units in your answers;
• Always label plots and refer to them in the text;
• The main body of your paper should NOT include lengthy calculations. These should be included in the appendix, and referred to in the main body.
• Labs must be typed, including the equations in the main body (part of your learning experience is to learn how to use an equation editor). An exception can be made for the curve you might draw in pen for Question 6, as well as any other hand written calculations you might want to include in the appendix (although there probably shouldn't be any for this particular lab because of the way it is designed).
• Your report doesn't have to be long. You need quality, not quantity of work. Of course you cannot omit any important piece of information, but you need not add any extras.
• You must include any plot that supports your conclusions or gives you insight in your investigations.
• Write your report in an organized and logical fashion. Section headers such as Introduction, Background, Problem Statement, Calculations, Results, Conclusion, Appendix, etc... are highly recommended.

During the week prior to the lab's due date, there will be TAs present in ECCR 143 to help you with the labs. Please see the lab hour schedule.

Lab Created by Kristian Sandberg. Modified by John Villavert and Jason Boorn, Oct. 2008. Further Modified by Russell Latterman, March 2009 and then Henry Romero, March 2010.