Question

odd 100 and 3. In the initial steady state of an infinite slab of thickness l, the face I = 0 and the face =lis at 100°. From t= 0 on, the x = 0 face is held at 100° = I face at 0°. Find the temperature distribution at time t. Answer: u = 100 - 100x – 400 100 – T- 1e-(na/1)2t sin NTT 2 ne even n 1.1. nne at 0° and one at 20°, are stadia

Answer 1

Taxis Love temperature & -temprature x-axix 1 -ακές temprature 0° temprature lovo ten naarin to the theat equation is t2 uc Loucaut) ) = az at using separation of variable assume that wlaut) can be written as in the form , ucut) = Hn).TCt) .: two explicit I fxns one in Ivariable n and putting in ean another is in t we will get I d²F - I di I 212 = 22 Ī dat Cleasley Here both sides sides the the functions functions are are o of different variables 80 wer have to equal to a constant to both equal sides So suppose d 2F f da2 =-* then t I dI = rk? Now integrating it be * - k2 is pusely negati

- T = x 226² + suppose let For = ama @ (esponontial form fan) dF da = m2 enn memu by ean a 10 d²F = m2 e-ma -k² e-mn da? = -K²=m? m = tik iki - Complex no purely imaginary so fox) = e inn oreiun and now The General solun is sina or Coxx -2²4 2t sinun Henes ulat) = { e l e- x2 u't Cos kr taking boundary condition at 20 uF 100° so we will discard cosken because at 120.-22428 logkn a x 22 u2t 1) = 100 while it is not generally happen a&t. 6-2 26 ²t sinun so et solu a talsing How involbed solun.

it becomes 24?t sinkel zo. klana or kan uerit) - e -<?(QT)'Z sin ( for appropriating it to a=0 we begin with .-(& nulegt gin longon ucket ) = 100 - e" But if we put n=l . it does not provide uso so for this we will tubstrait (100) ucxut) = 100-) e-(ano/) € sin (na ja The most treneral sour ixras - (LTL)?t sin mope ē no! { bn e t ulait) = 100 -10on en wherre x=0 Now we will find (bn' at to side a=0 is at u=0° and there ŭ initial steady state T2 U(d) to (a) =antb Hence (620) where x=d, u = 100° Hence Loo -la ) a= Loole uo - 100 g "Carol - No (N)

and n=1 (9) = loon = 1.00 Jou + - Ein Sin (mu ya > (4 qum) = 100 - 4,00* + 3 min (ut ja 3 Pey - - sinh T). h ! | nz How use it po le -100) sin cost you do . (2 m Bin Mat Ja ) sim (nga Ja dhe si lo ) sin (03) dan a) lo n (6")* site*sin(en ja de (it Cox (not) talsina - ln - 1oo n217² - mesin me In de 7 - 100 l I nu (It Cosnt) + 28 in na 4 1 1 2 1 2

The on 1 then bn = 200 [ not ( 1+ Cognit) + 2 sin nn) then bn = 0, you o ut 272 -u , o, 10,-- - when n is even bn = - 400 nu Hence finally u(out) = 100 - loon - 400 stė (nitet e ty ht) = loo - in evennen if you have any doubt please Comment time was short.