MATLAB Programming Project, Numerical Methods

Project Instructions is given in the files attached. It is mostly MATLAB with a couple of handwritten questions.

The data collected during the lab is also attached in a file below.

Semester and Year
Spring 2020
Due Dates
Mini project 1 – Thursday, March 26 at 3:30PM
Title
Cooling the Aluminum Cylinder Experiment to Illustrate Use of Numerical Methods
Points
100 points
Background
A solid aluminum cylinder treated as a lumped-mass1 system is immersed in a bath of iced
water. Let us develop the mathematical model for the problem to find how the temperature
of the cylinder would behave as a function of time.
When the cylinder is placed in the iced-water bath, the cylinder loses heat to its
surroundings by convection.
Rate of heat loss due to convection =ℎ𝐴(𝜃(𝑡) − 𝜃𝑎 ).
(1)
where
𝜃(𝑡) = the temperature of cylinder as a function of time t, oC
ℎ = the average convective cooling coefficient, W/(m2-oC)
𝐴 =surface area, m2
𝜃𝑎 =the ambient temperature of iced water, oC
The energy stored in the mass is given by
Energy stored by mass at a particular time = mC𝜃(𝑡)
(2)
where
m = mass of the cylinder, kg
C = specific heat of the cylinder, J/(kg- oC)
From an energy balance,
The rate at which heat is gained ─ Rate at which heat is lost
= Rate at which heat is stored
gives
1
It implies that the internal conduction in the trunnion is large enough that the temperature throughout the
trunnion is uniform. This allows us to assume that the temperature is only a function of time and not of the
location in the trunnion. This means that if a differential equation governs this physical problem, it would
be an ordinary differential equation for a lumped system and a partial differential equation for a nonlumped system. In your heat transfer course, you will learn when a system can be considered lumped or
non-lumped. In simplistic terms, this distinction is based on the material, geometry, and heat exchange
factors of the ball with its surroundings.
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0 − ℎ𝐴(𝜃(𝑡) − 𝜃𝑎 ) = 𝑚𝐶
−ℎ𝐴(𝜃(𝑡) − 𝜃𝑎 ) = 𝑚𝐶
𝑑𝜃(𝑡)
𝑑𝑡
𝑑𝜃(𝑡)
(3)
𝑑𝑡
The ordinary differential equation is subjected to
𝜃(0) = 𝜃0
where
𝜃0 =the initial temperature of cylinder, oC
Assuming the convective cooling coefficient, h to be a constant function of temperature,
the exact solution to the differential equation (3) is
ℎ𝐴𝑡
𝜃(𝑡) = 𝜃𝑎 + (𝜃0 − 𝜃𝑎 )𝑒 − 𝑚𝐶
It can now also be written in a normalized form as
𝜃(𝑡)−𝜃𝑎
𝜃0 −𝜃𝑎
(4a)
ℎ𝐴𝑡
= 𝑒 −𝑚𝐶
(4b)
Grading
This project is part of the Special Assignment/Project grade. Your solution will be graded
on the following categories:
• The merit of the conceptual portion
• The merit of programming portion
• The format of submission
Learning Objectives
• Identify and implement the correct procedure for a given problem
• Improve existing programming skills
• Reinforce prerequisite knowledge
• Solve real-world problems
Formatting
•
•
•
Follow the sample project format including cell formatting, published html
format, commenting, etc.
http://www.eng.usf.edu/~kaw/class/EML3041/homework/sample_experimental.ht
ml
Use MATLAB to solve all the problems, unless mentioned otherwise.
Use comments, display commands and fprintf statements, sensible variable names
and units to explain your work. Use SI system of units throughout.
What to do in the laboratory
1. Fill the ice-cooler with half-water and half-ice. It is better to use the water from
the water-cooler, as it is cooler than the tap water. Keep stirring the ice, so that ice
cubes are not stuck to each other.
2. Take the thermocouple wires and connect them properly (+ to +, – to -) to the
temperature indicator. Two thermocouples are attached to illustrate the concept of
a lumped system.
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3. Turn the temperature indicator on and wait for a few seconds to record the initial
temperature of the cylinder.
4. Record the temperature of the iced-water using a temperature indicator.
5. Immerse the aluminum cylinder in a bath of iced water and start the stopwatch
simultaneously. Every five to ten seconds, record the temperature of the cylinder
as a function of time.
Figure 1. Cooling the Aluminum Cylinder
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What to submit
For the hard copy, staple all the work in the following sequence.
1. Signed typed affidavit sheet (Your printed name will be considered to be the
signature)
(http://www.eng.usf.edu/~kaw/class/EML3041/homework/affidavit_sheet_individ
ual_projects.pdf) ;
(http://www.eng.usf.edu/~kaw/class/EML3041/homework/affidavit_sheet_individ
ual_projects.doc)
2. Attach completed checklist given at end of this assignment. Check-mark the
boxes you have accommodated in your assignment. Do not do this blindly.
3. Published mfile in html format
4. Answers on plain white paper or engineering graph paper when asked for in
problems of “Project Exercises” section. Each answer needs to be on a fresh page.
5. Any typed pages when asked for in problems of “Project Exercises” section. Each
answer needs to be on a fresh page.
For the soft copy submission on CANVAS
Submit the mfile of the computer simulation under the CANVAS assignment.
Name it as lastname_firstinitial_conv_spring20_x.m, where x is one or two
depending on the mini-project number. For example, if your name is Abraham
Lincoln, the name of your file would be Lincoln_A_conv_spring20_one.m for
mini-project 1 and Lincoln_A_conv_spring20_two.m for mini-project 2.
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Project Exercises
Mini project 1 (100 points) – Thursday, March 26, at class start time
1. On plain white paper or engineering graph paper, handwrite the data of temperature
vs. time we collected in class and the following data.
Diameter of cylinder = 49 mm
Length of cylinder = 100 mm
Density of aluminum = 2700 kg/m3
Specific heat of aluminum = 902 J/(kg-oC)
Thermal conductivity of aluminum = 240 W/(m-oC)
Table 1. Coefficient of thermal expansion vs. temperature for aluminum
(http://www.llnl.gov/tid/lof/documents/pdf/322526.pdf)
2.
3.
4.
5.
Temperature Coefficient of thermal expansion
(oC)
(μm/m/oC)
-10
5.8
77
9.3
127
13.9
177
25.5
227
32.6
277
34.1
327
36.1
377
38.9
427
39.8
Assign all the required input data (experimental data and other data that is needed
for Mini project 1) to variables as MATLAB statements at the beginning of the
mfile as one section. Do not change the units of the inputs – enter them as given.
Any changes in the input data should not require one to change any part of the rest
of the program, and that is what is called “avoiding hardcoding”. Of course,
fprintf/sprintf/disp the input data using the variables.
Change the units of input variables, if needed, to the SI system in a new section.
There is no need to fprintf/sprintf/disp in this section.
Find the convective cooling coefficient h by a crude method as follows. Use the
value of the temperature of the aluminum cylinder at the 3rd data point in your
readings of temperature vs. time, and solve the nonlinear equation (4a) (do not
simplify by hand) to calculate the value of h. Hint: Use the solve command.
Only using the experimental temperature vs time data, estimate the rate of change
of temperature with respect to time at the time corresponding to the 3rd data point.
Page 5 of 8
6. Use the rate of change of temperature with respect to time from #5 and the righthand-side of equation (3) to estimate the rate of change of heat stored in the cylinder
at the time corresponding to the 3rd data point.
7. Use the value of convective cooling coefficient h you found in problem#4 and the
left-hand-side of equation (3) to estimate the rate of change of heat loss due to
convection at the time corresponding to the 3rd data point.
8. In 50 words or so, do the following. You obtained values in problems#6 and #7;
what did you expect and why; if it did not turn out as expected, what are the possible
reasons? The work needs to be professional, typed on a separate sheet(s) of paper,
with all the variables defined, and with appropriate equation editors of your word
processor used.
9. Estimate the change in the diameter of the aluminum cylinder if it was placed in
iced water for several hours, and the iced water temperature is assumed to stay
constant.
How to approach solving problems on paper
This following is meant to help students approach engineering problems effectively and
efficiently. Without the proper approach, engineering problems can be very confusing. The
following guidelines are written with common correct and incorrect approaches in mind.
Remembering and implementing these approaches can not only help you find a solution
faster, but it can increase your understanding of the problem and its conceptual basis. Most
of these guidelines are not relegated to this class; you can use them in any engineering
class!
•
Start with what you know. If you do not know where to start, start with what you
know. It’s a little bit like connecting the dots. You cannot connect the dots until you
have put some down.
o
Look at the information you’re given.
o
Look at the applicable equations.
▪
o
Be methodical in your approach.
▪
•
•
What are the restrictions on these equations?
Often students will say, “I don’t know anything about this!”
Typically, this is because they don’t know what they know and what
they don’t know. Start with what you know!
Use dimensional analysis as a hint.
o
If you can’t find a mistake in your work, check the unit consistency in the
problem.
o
If you don’t know how to solve a problem, determine the units of the
solution and then look to see what units you’re missing in the solution.
Don’t cut corners! This WILL hurt you sooner or later.
Page 6 of 8
How to approach programming
• Start with what you know.
o
If you’re having trouble programming a problem, start by working through
the problem on paper.
o
Don’t try to think up the whole program in your head and then type it out!
•
When translating the problem solution into a program, display each part of the code.
Fix one piece at a time.
•
Avoid using “;” at the end of statements while debugging the program. You can
add the “;” later when the program is finalized.
•
Look at the ‘How do I do that in MATLAB series’.
•
Use the MATLAB help site (http://www.mathworks.com/help/matlab/) to look up
error codes, syntax, etc.
o
If you’re looking for syntax examples, click the “example” links on the right
side of MathWorks sections for a sample program.
Common mistakes in programming
• Hard coding
• Incorrect format
• Misunderstanding the conceptual (paper) solution
• Inefficient program debugging
• The printer cuts off published file lines.
• Unit errors/no units
Look at the checklist on the next page that needs to be attached to the hard copy of
your submission.
Page 7 of 8
Checklist for submission
 I followed the general format as given in the sample project.
 I uploaded the mfile.
 I attached the affidavit sheet.
 I wrote the code only by myself.
 I did not show my code to anyone else.
 I attached any handwritten or typed pages if asked for.
 I followed the section format as given in the sample project.
 I published the mfile in published format – HTML format.
 I wrote proper and reasonable comments.
 I put the comments on their own lines as seen in the sample project mfile (not
at the end of a code line).
 I CLEARLY identified my methods for each problem.
 I suppressed all statements.
 I showed input and output variables using fprintf and disp statements for all
exercises unless specified otherwise.
 I checked for cut off errors in the hard copy of the published file.
 I avoided all hard-coding (i.e., the program should still work if ANY of the
input data is changed).
Page 8 of 8
Time (s)
O 5
10
15 20 25
30 60
Temperature
22 18.3 15.2 13.4 12 10.9 10.3 8.6
(°C)
Temperature of iced water = 2.8 °C