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MATHS 3021 Capstone Project in Mathematical Sciences III

RAA for Tianhaoyi He a1754272

You are to re-work and extend Section 3.3 of your previous project report for application to Modelling of a Closed-Loop Geothermal Heat Pump.

Specifically you are to consider a direct exchange system in which an antifreeze solution is pumped through copper tubing in the ground and heat is exchanged between the antifreeze solution and the ground to either cool or heat the antifreeze solution. You may

● restrict your consideration to heat exchange between the the ground and the antifreeze solution in the pipe buried in the ground,

● assume the antifreeze solution to be water,

● assume the temperature of the  antifreeze solution to enter the pipe in the ground at a constant temperature,

● assume the ground temperature to be constant.

In your report you must:

1。Give a complete description of the problem you are modelling;

2。Include your own picture to aid you in describing your problem and mathematical model;

3。Demonstrate/explain how equation (29) from your first report is an appropriate model, using a “word

equation”, stating assumptions that are necessary in order to get/use this equation, and defining all variables and parameters;

4。Give additional conditions that complete the mathematical model and enable complete solution of the

equation;

5。Completely solve your mathematical model yourself, showing/explaining your working and defining

all notation that you introduce;

6。Explain, using appropriate diagrams generated by you, what your solution tells you and how the

various parameters in your model affect the change in temperature of the antifreeze solution from the start to the end of the pipe; ensure that your solution makes physical sense;

7。Give some conclusions that relate to your model and solutions;

8。Cite all sources that you use and give details in your Reference list such that they can be found by

others.

Your report should include the following sections

• Problem description,

• Mathematical model development,

• Solution,

• Results,

• Conclusion,

• References.