# AMME3060: You will Design and Numerically Model an Air-cooled Heat Exchanger fin for a Motorcycle Engine Cylinder Head: Engineering Methods Assignment, Sydney University, Australia

University | Sydney University |

Subject | AMME3060: Engineering Methods |

**Overview**

In this assignment, you will design and numerically model an air-cooled heat exchanger fin for a motorcycle engine cylinder head. The problem is depicted in figure 1, where an engine cylinder block (sketched as a vertical grey cylinder) is shown. Four cooling fins are shown. The engine exhaust header pipe (shown as a red cylinder) is also cooled by the fins. The cooling fins are made of Aluminium alloy with a conductivity of κ = 200 (W/mK) and each fin has a thickness of B = 4 mm. A two-dimensional sketch of each fin is shown in figure 2.

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The center of the plate contains a diameter D = 70 mm hole (cylinder location) indicated by the large red circle on the figure. The cylinder head is required to reject a constant heat flux of QC (W/m2 ) through each cooling fin. The small black circle indicates the location of the exhaust pipe with dimensions d = 14 mm, R = 40 mm. Ambient air at Tα = 25◦C flows over the outer surface of the fin at high speed producing a constant heat transfer coefficient of 72W/m2K. The exhaust gasses flowing through the exhaust pipe have a temperature of Tǫ = 250◦C and exchange heat with the pipe walls with a heat transfer coefficient of 120W/m2K

The outer shape of the cooling fin is undefined and forms part of this assignment brief.

**Objective and Constraints**

Your objective is to design the two-dimensional fin shape so that the outer edge of the fin is no hotter than 90◦C. The fin should be efficiently sized, so the temperature of the outer edge of the fin should be quite close to this limit (ideally within the range of 80-90◦C). The fin must fit within the dashed boundary shown in figure 2 i.e. a box centered on (x, y) = (0, 0) with limits x ± Lx/2 and y ± Ly/2 where Lx = Ly = 140 mm.

The heat flux at the cylinder is unique to each student. The heat flux is determined as QC = Qc/A (W/m2) where A is the contact area of the fin at the cylinder i.e A = πDB ( m2 ), and where the total heat Qc (W) is Qc = 75(1 + γ/1000) where γ is the last three numbers of your SID.

You may modify your shape in any way which meets these constraints. You may include additional holes and cutouts for example. The thermal boundary conditions for these should be treated as the same as the outer edge of the fin i.e. convective boundary conditions.

**Approach**

In this assignment, you will design the shape of the fin and test your design using numerical simulations. You will first develop and test your design using the ANSYS Steady-State Thermal solver and then you will write your own simple Matlab code to solve this problem and obtain solutions. The assignment is arranged into two parts corresponding to these two numerical solutions:

In Part A you will use ANSYS Steady State thermal solver to sketch a simple three-dimensional fin and obtain the temperature. You will iterate your fin shape design until you have a solution that satisfies the constraints above. You will then undertake some accuracy tests by repeating your solutions in a range of grid sizes.

In Part B you will treat the problem as two-dimensional. You will use the Galerkin Finite Element method to obtain discrete equations for this problem using linear triangular elements and then write a Matlab script to solve this problem. You will use the mesh generator in Matlab to sketch your fin shape design.

You will present your results in a short report. In this report, you must briefly present your design and solutions. In particular, you must show how you determined your solution is accurate by giving tables of results and showing the convergence of results with element size. You will also briefly present your method, the discrete equations solved, how boundary conditions were applied, explain what simulations were performed, what settings were used.

**PART A**

In Part A you are to use the ANSYS Steady State thermal solver to obtain a solution for a single three-dimensional cooling fin. You can initially model the fin similar to the one shown in figure 2. The fin should have a constant thickness of B. You can begin by using the Lab 1 sheet as a guide.

1. Sketch a simple heat exchanger design (for a single fin). Draw the problem using the ANSYS Design Modeller in the Steady State Thermal Solver. Mesh the object. Apply the correct thermal boundary conditions. Make sure the material properties are correct. Use your judgment to select all other settings.

2. Once you have obtained a solution, decide if you need to refine your shape to meet the design objectives. If so then repeat the design, mesh and simulate steps. You can change the object dimensions easily in ANSYS and can re-mesh and re-calculate the solution without having to re-do the boundary conditions. Repeat until you are satisfied with the result.

3. Obtain solutions on a range of grid sizes. You can export the temperature on the outer edges of the object and also export the maximum temperature values as shown in the Lab 1 sheet.

4. Write a short report presenting your solution with surface plots/contour plots. Describe all non-default settings used.

**PART B**

The heat transfer along the fin can be approximated as two-dimensional if the fin is thin. The two-dimensional steady heat equation for this problem is,

where θ(x) = T(x) − Tα. The equation can be re-written,

where β = 2h/Bκ. For this problem Tα = 25◦C

The internal edge of the large cylinder hole in figure 2 has a heat flux boundary condition of QC W/m2.

The outer edge and edge associated with the inlet port have convective heat transfer boundary conditions where qn = h(T −Tα) = hθ W/m2 and n is the direction normal to the edge (pointing outwards). The edge associated

with the exhaust port has qn = h(T −Tǫ) = h(θ−225) W/m2.

The heat transfer coefficients are given on page 1.

Your task is to solve this problem use the finite element Galerkin method with two-dimensional linear triangular elements and determine the temperature distribution around the outer edge of the fin.

1. Derive the element stiffness matrix and load vector for this problem. Present your derivation of each term and present the equations in matrix form i.e. [k][θ] = [f]. Derive the discrete form of the boundary conditions for each type of boundary condition in this problem.

2. Draw the fin using the Matlab tool package. Mesh the object you have drawn and export the mesh files : p, e, t arrays. Save the mesh to file:

3. Write a Matlab script to solve the set of equations for an arbitrary number of elements. Your code should include: the import of the mesh (you don’t have to automate the mesh generation step unless you want to!), the assembly process, apply the correct boundary conditions and solve the global system of equations. To import the mesh you might like to use: >> load(‘lastnameSIDMesh.mat’);

4. Run the code and produce a contour plot of the temperature on the fin. You might like to use the Matlab trisurf function.

5. Refine your mesh and run your code again. Repeat this until you are satisfied with your solution.

6. Present your results with line plots and contour plots. Plot the temperature around the outer edge of the fin.

7. Using the above results write a short report presenting your solution T(x, y) and in particular the temperature around on the outer edge of the fin. You should briefly present your numerical method including the discrete equations and boundary conditions. Your derivations of the element equations can be handwritten (neatly) and included as an appendix to your report

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