N 0 1 WN 2 3 4 5 T(z) 15 24 29 23 13 z=5m z=0m...

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N 0 1 WN 2 3 4 5 T(z) 15 24 29 23 13 z=5m z=0m T(z) 5 2 2 Ay 3 Az T2 3 Ay perfect insulation material The above figure shows a cross-section of a copper alloy material embedded in a perfectly insulating material. Two sides are open and subjected to imposed temperatures. On the left side the temperature is measured at the points given in the table above. Unfortunately one of the temperature sensors was corrupt and did not give a reliable measure (blank). Also on the bottom side, the temperature was measured, but the measurements indicated a nearly constant value of 10C. You can assume that the above 2D-figure is representative for the situation. Questions: 1. You know that the Laplace equation can be used to calculate the temperature in the copper alloy body and that the above problem is a boundary value problem. Explain what this means and demonstrate this by identifying the different types of boundaries for the example above and the boundary values you will impose when you solve the Laplace equation for the equipotential (temperature) and for stream function. 2. Assume that Az-Ay. Assume that you discretize the domain using 14y as standard unit in the y- direction and 14z as standard unit in the z-direction. Set-up the matrix system to solve the Laplace equation for the stream function. You do not need to write down all coefficients of the matrix, but make sure that you show one point for each of the different numerical stencils that you will need to use. Make sure that you include the point at z=3m on the left boundary. Also make sure that you give the correct dimension of the matrix. 3. Without doing any calculations, make a sketch of what you expect the solution will look like. N 0 1 WN 2 3 4 5 T(z) 15 24 29 23 13 z=5m z=0m T(z) 5 2 2 Ay 3 Az T2 3 Ay perfect insulation material The above figure shows a cross-section of a copper alloy material embedded in a perfectly insulating material. Two sides are open and subjected to imposed temperatures. On the left side the temperature is measured at the points given in the table above. Unfortunately one of the temperature sensors was corrupt and did not give a reliable measure (blank). Also on the bottom side, the temperature was measured, but the measurements indicated a nearly constant value of 10C. You can assume that the above 2D-figure is representative for the situation. Questions: 1. You know that the Laplace equation can be used to calculate the temperature in the copper alloy body and that the above problem is a boundary value problem. Explain what this means and demonstrate this by identifying the different types of boundaries for the example above and the boundary values you will impose when you solve the Laplace equation for the equipotential (temperature) and for stream function. 2. Assume that Az-Ay. Assume that you discretize the domain using 14y as standard unit in the y- direction and 14z as standard unit in the z-direction. Set-up the matrix system to solve the Laplace equation for the stream function. You do not need to write down all coefficients of the matrix, but make sure that you show one point for each of the different numerical stencils that you will need to use. Make sure that you include the point at z=3m on the left boundary. Also make sure that you give the correct dimension of the matrix. 3. Without doing any calculations, make a sketch of what you expect the solution will look like.

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It seems that youve provided information regarding a heat transfer problem involving a copper alloy material with certain boundary conditions and are referring to using the Laplace equation to solve f... View the full answer