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ENCH298: Assignment 2
Can Cooling Dynamics
Due: 9 am, Monday 22 May 2017 at the lab, or in a Level 5 assignment box.
This homework is worth up to 10% of your final grade in ENCH298.
You may complete this assignment in pairs, but you must work with a different person from the last ENCH298 assignment.
Complete the tasks listed below and write a brief report of the results.
Overview
It is a warm summer day (24°C) day and you feel like a cold drink.
You have a can of drink that has been sitting on the kitchen bench in the sun and has warmed up to 20 °C. Unfortunately, your fridge has broken down and will take days to get fixed. But you are able to get some uniformly sized, 25 mm diameter, spherical ice balls and you have a 9 L plastic bucket. You intend to add enough ice balls to a bucket of water initially at 15 °C and immediately add the can of drink. As soon as the ice has melted you remove the can to drink it.
Use your engineering modelling and simulation skills to determine how much ice and water to add to reduce the drink temperature to 8 °C once all the ice has melted. At all stages in the modelling and simulation consider the practical validity of your model.
An important part of this assignment is the physical interpretation of parameters and results. Mathematical solutions in isolation will not be sufficient.
A model is given below.
Objective
To determine the temperature of drink inside a can over time after being added to a bucket containing water and ice balls.
Diagram and Scope
The bucket and contents as shown in the figure
Assumptions
Uniform temperature within the water.
Uniform temperature within the ice, i.e., there is negligible conduction resistance within it.
Uniform temperature within the can.
Constant and uniform ambient conditions.
Constant and uniform specific heat capacities and densities.
The ice is fully submerged (not true).
The can is fully submerged.
The ice is initially 0 °C.
The ice is spherical and all balls melt equally so are the same size at any time. The thermal mass of the bucket wall and can wall are negligible.
No evaporation.
There is heat gain from the ambient air into the bucket through the walls and top. The heat transfer coefficients are constant.
Equations
The required balances are:
The mass of ice
The mass of water
The energy of the water
The energy of the drink
Because we assume that the ice is always at 0 °C and we have a balance for the mass of ice, then the energy of the ice can be calculated, so an ice energy balance is not required.
Balance
Mass of ice
Total mass of liquid water  𝑑𝑡 ^{𝑚𝑒𝑙𝑡𝑖𝑛𝑔} 

 𝑑𝑚𝑤𝑎𝑡𝑒𝑟  (2) 
𝑑𝑚𝑖𝑐𝑒 = −𝑚̇ (1)
=𝑚̇ 𝑚𝑒𝑙𝑡𝑖𝑛𝑔 𝑑𝑡
Energy of liquid water
𝑑𝐸
𝑤𝑎𝑡𝑒𝑟 =𝑄𝑎𝑚𝑏 −𝑄𝑖𝑐𝑒 −𝑄𝑐𝑎𝑛 (3)
𝑑𝑡
Energy of drink in the can
𝑑𝐸
𝑐𝑎𝑛 = 𝑄_{𝑐𝑎𝑛} (4)
𝑑𝑡
Algebraic equations
𝑚𝑖𝑐𝑒 =𝑁𝜌𝑖𝑐𝑒 𝜋𝐷3 (5)
6
𝑄_{𝑖𝑐𝑒 }(6)
𝑚𝑒𝑙𝑡𝑖𝑛𝑔 Δ𝐻𝑓𝑢𝑠𝑖𝑜𝑛 

𝐸𝑤𝑎𝑡𝑒𝑟 = 𝑚𝑤𝑎𝑡𝑒𝑟𝐶𝑝,𝑤𝑎𝑡𝑒𝑟(𝑇𝑤𝑎𝑡𝑒𝑟 −𝑇𝑟𝑒𝑓)  (7) 
𝐸𝑐𝑎𝑛 = 𝑚𝑐𝑎𝑛𝐶𝑝,𝑑𝑟𝑖𝑛𝑘(𝑇𝑐𝑎𝑛 −𝑇𝑟𝑒𝑓)  (8) 
𝑄𝑖𝑐𝑒 = 𝑈𝑖𝑐𝑒𝐴𝑖𝑐𝑒(𝑇𝑤𝑎𝑡𝑒𝑟 −𝑇𝑖𝑐𝑒)  (9) 
𝑄𝑐𝑎𝑛 =𝑈𝑐𝑎𝑛𝐴𝑐𝑎𝑛(𝑇𝑤𝑎𝑡𝑒𝑟 −𝑇𝑐𝑎𝑛)  (10) 
𝑄𝑎𝑚𝑏 =𝑈𝑎𝑚𝑏𝐴𝑎𝑚𝑏(𝑇𝑎𝑚𝑏 −𝑇𝑤𝑎𝑡𝑒𝑟)  (11) 
𝐴𝑖𝑐𝑒 =𝑁𝜋𝐷2  (12) 
𝑚̇ =
Variables
Independent
t time s
Dependent
mice Mass of ice in the bucket kg
mwater Mass of water in the bucket kg
Ecan Energy of the drink in the can J
Ewater Energy of the water J
m& melting Mass flow rate from the ice due to melting kg/s
Qice Flow rate of heat into the ice J/s
Qamb Flow rate of heat into water from ambient air J/s
Qcan Flow rate of heat into the can from the water J/s
Twater Water temperature °C
Tcan Temperature of the drink in the can °C
D Diameter of the ice spheres m
Aice Surface area of the ice m^{2}
Parameters
U_{ice} Overall heat transfer coefficient from water to ice W m2 K1
Uamb Overall heat transfer coefficient for air to the bucket W m^{2} K^{1 }
Ucan Overall heat transfer coefficient for the can to water W m^{2} K^{1 }
N_{ }Number of ice spheres
mcan Mass of drink in the can kg
Cp,water Specific heat capacity of water J kg^{1}K^{1 }
Cp,drink Specific heat capacity of the drink J kg^{1}K^{1 }
Tamb Ambient air temperature °C
Tref Reference temperature 0 °C
Hfusion Latent heat of fusion (positive) J kg^{1 }
Initial Conditions
IC’s are required for m_{ice}(t_{0}) or N and D(t_{0})
mwater(t0)
Ewater(t0) or Twater(t0)
Ecan(t0) or Tcan(t0)
Tasks
These tasks require Matlab simulation and engineering thinking. Think about the answers you obtain.
a) Manipulate the model to replace energy with temperature and get a set of 4 differential equations with the additional algebraic equations that can be used to evaluate terms in the differential equations.
b) Write a prediction for the responses of the four differential variables based only on your intuition. (In other words, think about the likely response before you simulate it to help with verification.)
c) Simulate, using your 2^{nd} order RK (also known as Heun’s method or modified Euler’s method) code and Matlab ode23. You should be able to write the equations in a Matlab function that is identical for your 2^{nd} order RK and ode23. For this simulation you can choose any unknown parameter values.
d) Adjust the initial mass of water and the mass of ice (i.e., the number of ice balls) to achieve the desired temperature of 8 °C in the can. This can be done by trial and error but you should explain the strategy you use.
e) What practical combination of water and ice mass gives the fastest cooling?
f) Check the influence of step size on your RK2 method and of tolerances in ode23 on the answers to d). For ode23 you can use the syntax:
options=odeset('RelTol', 1e6, 'AbsTol', 1e6);
[t x]=ode23(@Equations, tspan, x0, options);
The error tolerance used by ode23 is then max(AbsTol, RelTol × x)
Once the ice has melted, some of the variables become zero, but continued simulation should be possible.
g) Check your simulation results to see what solution you get when the mass of ice reduces to zero. Comment on this and seek to improve the results if necessary and possible.
Many assumptions were made in the modelling process.
h) Evaluate the influence of any parameters that you are unsure about. (As a counterexample, there is no need to check the influence of the specific heat capacity of water.)
i) You might be able to carry out simple experiments, and use the results with your Matlab simulation, to get a better estimate of the heat transfer coefficient for ice in water.
j) Do you think it is worthwhile insulating the bucket? (Justify your answer).
k) What is the maximum number of cans (with the same number of each type) that you can cool down to 10 °C in the bucket?
Report
For this project a report should include
• Title page.
• Table of Contents.
• Short introduction to the problem.
• A clear model with a statement of the modelling objective, a diagram of your system (bucket, ice, can).
• Answers for each part with an outline of the part with as much supporting data, graphs, comments, etc. as suitable.
• Appendices clearly showing selected mfiles that were used for each of the parts. You do not need to include complete listings showing all the different code used. Just include one complete mfile and then parts that are changed for other parts.
The main part of the report should contain sufficient information to enable an engineer to understand what you have done, and interpret your answers. Appendices should be used for the detail for someone who wants to reproduce your work. Refer to appendices from the main part of the report.
There is no page limit but if you are approaching 10 pages for the main parts, think about making it more concise. Bored markers tend to give lower marks.