Question

1. In class, we examined in detail case "C" of table 3.4 on page 150 of your text. Prove the expressions provided in the table for cases A, B, and D. Specifically, start from the general equation 3.67, and apply at x-L the boundary condition on the second column of Table 3.4 for each of the cases. Then, solve the differential equation and acquire the information on the third and the fourth column. Hint In some cases, you will need to write the general solution of 3.64 in a smart way, possibly using appropriate forms of the functions cosh and sinh, so that the calculation of the constants with the use of the boundary conditions is in a neat form.

Answer 1

Page: let us take a longitudinal fine of non-uniform cross-section on a plane surface : LIRAEA C) - As = As (*) = P(x) dx 6 = åbetda + alcons. a sy deal de & alcone -> LAGOK ? dze + Ag(x) ho (T-Too) 1 k da TET Q = Tf (33) → she [Alksten] =P holt-Toc) Let a= T- To = Q(x) = T(x) - Too The e cake - Photo

Case A : Pin bir having uniform and circular como A= TR² = Const. P=25R = 4 s tr. 7- -> the- m?O=0 ; mathe so Case - Alo Long fin infinitely long 8(x) = e, emx + ca e us ; oca400 BCs. :- lis T(x=0) = To = 8x=0) = TB-Too = e. (ii) T is finite at x x G and Cy can be found out by using Bez. Q(x) x) = cemo te em h will be finite only when og so. la can be found out by using Bes(i).

SIDIR Date : Page: - mx : 062) = T(m)To ze me x= x b P 8(x) dx Q. 1 easier - to perform Both the equations com be used o Phak

- Case-A2 fin of fint length with insulated tipos BCs ci) T(x=0) = T2 => (x=0) = @ cil de 30 => , = 0 x=L - Let 8(x) = C, Cosh [m CL-x)] + G Sinh [m (L-2] 1) Cash [m (Lx) T(*) - To to - Cosh [ml] - To - To - a= 6, P.Al tenn [L - dalx=LR Case - A3 8 fin of finite length (L) with convective It.T. at tip BCS Tit Ta=o) = To >> O G=0)=0 - (ij) -R dI hate it-To) > - k do = hate 0(X) - Cash [m (1-x)] + me sinh [m (1-x)] , [(x) - To - Cash [ML] + hat Sinh [mail] . To-To 8 = D. V PR Ak Sinh (mL] + hotel Cash [ml] Cash [mL] + hoot sinh [mL]