Question

A plate (k = 10 W/m.K) is stiffened by a series of longitudinal ribs having a rectangular cross section with length L = 8 mm and width W = 4 mm. The base of the plate is maintained at a uniform temperature Tb = 75°C, while the rib surfaces are exposed to air at a temperature of T = 25°C and a convection coefficient of h = 600 W/m2.K. Using a finite-difference method with Ax = 4y = 2 mm and a total of 5 x 3 nodal points and regions, estimate the temperature in nodes 1, 3, 4, 6, and 8 and the heat rate from the base per unit depth of the rib. 2 3 4 Rib Plate 1 9 10 11 12 Ay = 2mm +Ax = 2mm A partial solution for the temperature distribution in the interior of the rib obtained by the finite difference method is given by: 75 T1 51.82 T3 T4 75 62.5 T6 47.42 T8 w - 75 T1 51.82 T3 T4

Answer 1

1. Given Data: Tooh 3 4 k rx Toaded . Joihing 4x = 2mm K = 10W/m.k. Th=756 ho Goo Wimkok., Too =25°C sy= 2 mm. 1-8mm W = 4 mm., Now, Due to Symmetry. It is T = To ; T2 = To, Tzati, TA=7,2. So, we have to find only to solve for nodes . 1,2,3,4,5,6,788. For these above & nades heat transfer is due to Conduction from 3 sides and Connection from one side. News fre node 1 : :=7255 + Tz+To+ 2h by Tan -2 (hoxet2 )-7, = 0... → 275 +T2.-4.24-1, +81 = 0 + How Smilady apply for nodes. 2,3 8 8: 2T6 + Tz + T, + 6 - 4.24 T2 =0

For node, 3: = 277 + T +74 +G - 4.24 73 = For node 8! = 277 + ++ + T12 + 6 -4.24 Tp =0 @ Then for node 4 heat transfer by two sides from Connection and two sides conduction. - Too K6 Now, apply fom for node 4, then we get. (13478) + 2MAX! Te-2 (hax +2) T4 = 0. Tz +Tg :+6 - 224 T4 = 0 - how for modes 5,6.2 7 conduction from all Therefore, four Sides. For node 5 : > T6 + T+.T6 + Tq -4 T5 20 75+ T, +To+Tq-475 -0. Fer node 6 . . .. © .. = T5+ 72 +T7 T Tio - 476 -0 Por node 8: a Tz + Te + Tiit To - 477=0 8

2 T2 = 51.82 % Ts = 62.5c. ven Tz = 47.42) we have to determine remaining temperatures ac By using all the equation qwe get above a given values: T, = 60-8066°C.:.:. TG = 53. 39°C. T3 = 46,1302°C T4 = 42.925°C. To = 44.03°C.