An engineer wants to calculate the temperature distribution in a triangular plate with the dimensionless side length 1, which is shown in the following figure
The dimensionless steady-state temperature distribution is given by the energy equation
The dimensionless temperature takes the following values on the three sides of the triangle
Because the engineer can’t find a solution to this problem in his standard text books, he has the following idea: He wants to calculate the steady-state temperature in the following rectangular region.
He thinks that when he is just using the temperature distribution in the lower triangular region, he will obtain the temperature distribution he is looking for.
(a) Solve the temperature distribution in the rectangular region by using the expression H = a0 ) a1~x ) a2~y ) a3~x~y. Which values do the constants ai
take?
(b) Predict now the dimensionless temperature distribution on the side ~y = 1 ~x of the triangle and show that the engineer was not right with his assumption.
(c) Which additional problem needs the engineer to solve in order to solve the original problem?
(d) Use the above given expression for the dimensionless temperature again to solve the new problem and show that the combined solution fulfills now all boundary conditions.