Question

Question 1 (4 marks) maintained at a constantytemperature Ti-0°C, while the fourth A square plate has three sides a side is maintained at a constant temperature T2·100℃. Using the exact infinite series solution produced by the method of separation of variables, calculate the temperature at the midpoint of the plate. Show full details of your calculations. 17

Answer 1

For 2 - D heat flow, partial differential^2 T/partial differential x^2 + partial differential^2 T/partial differential y^2 = 0 (A) T = T(x, y) and is temperature For given case, we have given Boundary conditions as (I) T(x, 0) = 0 degree c = 0 (II) T(0, y) = 0 degree c = 0 (iii) T(x, a_= 100 degree c = 100 (IV) T(a, y) = 0 degree c = 0 a is side of given square plate writing, T(x, y) = X(x) Y(y) eqn (A), becomes, partial differential^2/partial differential x^2 (X(x), Y/(y) + partial differential^2/partial differential y^2 (X(x) Y(y)) = 0 implies Y(y) partial differential^2/partial differential x^2 X(x) + X(x) partial differential^2/partial differential y^2 Y(y) = 0 = 1/x partial differential^2 X/partial differential x^2 + 1/y partial differential^2 y/partial differential y^2 = 0 implies 1/X partial differential^2 X/partial differential x^2 = -1/Y partial differential^2 y/partial differential y^2 - - p^2 (say) implies 1/X partial differential^2 X/partial differential x^2 = -p^2 1/y partial differential^2 y/partial differential y^2 = p^2 implies partial differential^2 X/partial differential x^2 = -X p^2 X partial differential^2 y/partial differential y^2 = P^2 y