**Introduction:**

There is an area of thin mantle at the bottom of the ocean which is substantially raising the temperature of the sea bed due to a breach of molten material. In this assignment, you will use the heat equation to analytically solve for the transient temperature profile within the sea given a fixed Dirichlet boundary condition at the bottom of the ocean.**AMME2000/BMET2960/BMET9960 Heat Equation Assignment 1 Australia.**

The initial 1D temperature versus depth profile, at the instant when the ocean bed breach occurs, can be represented by a piece-wise function: the temperature is a constant T↵ = 25 C from sea-level d0 = 0m to a depth of d1 = 500m; there is then a sharp transition to a constant temperature of T= 5

C from d1 to the ocean floor d2 = 2600m; finally, the exposed mantle causes the temperature at d2 to be fixed at T= 460 C. The thermal diffusivity of sea-water can be approximated to kt = 1.6 ⇥ 10 2m2/s.

**Part 1: Analytic Solution to the Heat Equation**

In order to solve this problem analytically you will need to separate your solution into a homo- geneous and steady-state solution. It is recommended that you keep your boundary and initial conditions expressed as variables.

1.Using the information given in the Introduction, briefly summarise the problem information.(5%). Be sure to include the following:

• Governing equation

• Order and type of the PDE

• Initial condition

• Boundary conditions

• Type of solution you expect to obtain

2.Derive the steady state analytic solution for the temperature as a function of ocean depth, showing all working. Produce a plot comparing the initial condition, steady-state solution and the initial condition that will be used for the homogeneous problem. All three plots should be overlaid.

**AMME2000/BMET2960/BMET9960 Heat Equation Assignment 1 Australia.**

3.Using the method of Separation of Variables you’ve seen in the lectures and tutorials, derive a solution to the heat equation for this problem in terms of unknown Fourier coecients (i.e.do not apply the boundary conditions yet). In your working, clearly indicate: the spatial, F(x), and temporal, G(t), functions and the eigenvalues.

4.By expressing the temperatures and depths as variables (rather than actual values), show that the unknown Fourier coecient can be expressed as

Using the result from equation (1), show that the Fourier coecient can be expressed by Equation 2. Include all working in the integration process.

6.Implement your temperature solution in MATLAB, and use this to generate a plot of the temperature distribution at t = 0, t = 10 days and t = 100 days (show these on one figure for ease of comparison). In your report you should also include a brief explanation of your choice of grid resolution and number of Fourier terms. (10%) Note that your Matlab code should be included in an appendix to your assignment report.

**Part 2: Numerical Solution to the Heat Equation**

1.Briefly outline the approach you will use to solve for the temperature profile numerically (10%). In your answer, be sure to include:

• What stencil you will use (state the name and the formula) and why;

• How you will implement the initial condition;

• Comment on the stability of your chosen stencil and discuss the limitations/conse- quences of this for the solution;

• How you will validate your numerical solution.

2.Implement your numerical solution in MATLAB. Use the analytic solution from Part 1 to validate your numerical solution. You should include:

- A figure showing the temperature distribution for at least three grid sizes, to demon- strate convergence. The exact analytic solution should also be overlaid.
- A table of the error norms for 6 grid sizes. Successive grid sizes should di↵er by a factor of 2.
- And a brief discussion of (i) the convergence be haviour and whether it agrees with what you expect, and (ii) the choice of time-step size for this problem.

**Part 3: Non-uniform Physical Properties**

1.Considering a more realistic model, we will now assume that the thermal diffusivity is a function of temperature:

**AMME2000/BMET2960/BMET9960 Heat Equation Assignment 1 Australia.**

Explain how you will implement the non-uniform thermal diffusivity into your numerical scheme, and also how you will choose an appropriate time-step size.

2.Using an appropriate grid spacing (this should be based on your findings in Part 2), plot:

(a) On one figure, the initial temperature, and the temperature distribution as a function of depth at t = 100 days for both the constant value of kt and kt(T).

(b) On two separate figures, a comparison of the change in temperature over time at x =1500m and x = 2500m for both the constant value of kt and kt(T). (Note: Each figure should compare the temperature between different diffusivity models, not different positions.)

Briefly discuss what difference the temperature dependence makes.

**Assignment Submission Information**

Your assignment is to be presented as a concise report in PDF format. Note that 10% of the assignment marks are awarded based on the presentation, clarity and functionality of your MATLAB code. A further 10% of the assignment marks are allocated based on overall structure, clarity and

presentation of the report.

**AMME2000/BMET2960/BMET9960 Heat Equation Assignment 1 Australia.**

**IMPORTANT:**

- Marks will be deducted for handwritten answers and screenshots of equations and/or figures. See Tutorial 2 for detailed notes on how to insert figures into your assignment.
- The report should not exceed 10 pages; additional pages will not be marked, so aim to be concise with your findings. (Note that this page limit does NOT include your MATLAB code in the appendix.)
- Any figure/table you include in your report must be numbered and must be referred to and discussed in the main text of your report.
- You are strongly encouraged to read through the ”Example Assignment” we have made available on Canvas here. This demonstrates the expectations with respect to assignment layout, explanations and figures/tables.

**AMME2000/BMET2960/BMET9960 Heat Equation Assignment 1 Australia.**

An electronic copy of the report should be submitted to Turnitin by the due date. Late submissions will incur a penalty of 5% per day late.

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