MECH 202 Jupyter Notebook Written Report – Engine cooling
MECH 202
Jupyter Notebook Written Report
Engine cooling
Due date: Sunday of Week 8
Grading & Weight: This assignment is out of 100 marks, as further specified in the rubric at the
end of this document, and is worth 8.5% of your overall final grade in the course
Late Penalty: Late submissions will be penalized at 10% each day for up to 5 days in which
case a grade of zero will be given.
1. Overview
For this assignment, you will predict an engine’s temperature using numerical methods to solve
an ordinary differential equation. As explained in class, there are two parts to this project.
PART I : Calculate the time-dependence of the temperature of an engine after it is shut off.
PART II : Calculate the top speed that a motor vehicle can reach before overheating.
You will submit a report in the form of a Jupyter notebook.
This assignment directly aligns with the following Course Learning Outcomes (CLO):
CLO (3): Solve first-order ordinary differential equations analytically
CLO (5): Solve ordinary differential equations numerically, coding in Python
1.1 Time for completion
This assignment will take approximately six (6) hours to complete. You will begin working on
this assignment in the two hour tutorial of Week 7, and finish on your own time before the
deadline.
2. Instructions
For this assignment, you will implement the ODE solving techniques that we saw in the online
content and in-class. You will submit a report in the form of a Jupyter notebook. Your Jupyter
notebook must contain the following information:
• Introduction
o In your introduction, include pertinent theoretical background. This can be
completed in about three paragraph (200-300 words). Include references as you
judge pertinent. You might want to refer to some of those presented in class.
• Completion of the tasks and discussion
o Include code required to correctly complete the task
MECH 202 Jupyter Notebook Written Report – Engine cooling
o Include comments to explain your thought process and code logic
o Discuss the results of what was found and answer the questions in the task list
• Conclusion
o Discuss what you learned, what surprised you, and how you could extend this
work. This can be completed in about 250 words.
• Optional reference list
o Include a references if sources are used
o This can be included at the end of the intro, or at the end of the document. This
esthetic choice is up to you!
• Format
o See the report template in onQ to format your report.
Part 1 Engine cooling on a cold winter day
Context:
The normal operating range for a car engine is in the region of 80-90oC. Here, you will model the
temperature of the engine using Newton’s heat equation. Set the initial engine temperature at 90oC and
outdoor temperature at -20oC. Suppose a cooling rate of 0.07 min-1
. In all cases, you will estimate the
temperature evolution over 20 minutes.
Steps:
1. Implement Euler’s method to solve this problem. Use a timestep of 0.2. Plot your answer and
show the final temperature value. (5 marks)
2. Implement Runge-Kutta’s method (4th order) to solve this problem. Use a timestep of 0.2. Plot
your answer and show the final temperature value. (5 marks)
3. Implement the implicit trapezoid (implicit R-K) method to solve this problem. Use a timestep of
0.2. Plot your answer and show the final temperature value. (5 marks)
PS: see some code to help you out attached in OnQ!
4. Solve the problem using the odeint or solve_ivp method pre-built in Python. Make sure to note
the number of function codes (‘nfe’ in the dictionnary). Plot your answer and show the final
temperature value. (5 marks)
5. Solve the problem analytically. Plot your answer and show the final temperature value. (5
marks)
6. Plot the difference, as a function of time, between the output of each numerical method (i.e.
tasks 1 through 4) and the analytical solution. (5 marks)
7. In tasks 1 through 3, a different total number of function calls were made for each method. For
example, Euler’s method required 1/4 of the function calls required by the R-K 4th order method.
Here, re-solve the problem, using the same number of function calls. For each scenario, provide
the engine temperature after 20 minutes (no need to plot the temperature as a function of
time)
a. Perform 24 Euler steps (i.e. 24 function calls). (4 marks)
b. Perform 6 Runge-Kutta 4th order steps (i.e. 24 function calls). (5 marks)
MECH 202 Jupyter Notebook Written Report – Engine cooling
c. Perform 8 implicit trapezoid steps (i.e. 24 function calls). (5 marks)
d. Which method provides the best accuracy, when performing the same number of
function calls? (4 marks)
8. What stepsize should you use to calculate the engine temperature after 20 minutes, within
0.01% of the analytical solution?
a. Use Euler’s method. How many function calls were made? (4 marks)
b. Use Runge-Kutta’s 4th order method. How many function calls were made? (4 marks)
c. Use the implicit trapezoid method. How many function calls were made? (4 marks)
PS: By now, we hope that you are convinced that Euler does not perform well, compared to
alternatives. However, the situation is worst than what you uncovered up until now. If the
function being calculated fluctuates rapidly in time—i.e. rapid oscillations—then Euler will
perform even worse. In fact, even Runge-Kutta 4th order will behave badly if the solution varies
too rapidly (technically, these are called stiff equations). In these cases, implicit solvers are best.
We will come back to these concepts in module 6, where you will solve a second-order ODE,
that involves a stiff equation.
Part 2 Top speed before overheating
Consider the following scenario:
• The velocity-independent cooling rate is k = 0.07 min-1
.
• The velocity-dependent cooling rate is 0.005 min-1
/ (m.s-1
). (in other words, the faster the car
goes, the more cooling is provided to the engine)
• The engine block weighs 200 kg. Its specific heat capacity is 900 J / (kg.K).
• At maximum acceleration, the engine generates 450kW of waste heat.
• Maximum acceleration is 3 m/s2. • The car starts at rest, and the engine is initially at 80oC.
• The outside temperature is -20oC.
Steps:
1. How fast is the car going when it can no longer accelerate without overheating the engine?
Assume that engine overheat occurs when temperature exceeds 110 ∘ C. Ignore terminal
velocity from drag. (20 marks)
2. Plot the engine temperature as a function of time. You’ll notice that the temperature increase is
non-linear. Why is this? (5 marks)
3. What could be done to the car so that the temperature plateaus at a lower level, before the
car’s maximum velocity is reached (i.e. make drag force the limiting factor, rather than engine
temperature)? (5 marks)
MECH 202 Jupyter Notebook Written Report – Engine cooling
Your assignment will be evaluated using the following criterion:
Criteria Mastery (A+) High Quality (A) Developing (B) Marginal (C) Not