Question

Consider a rectangular bar of thermal conductivity k W/m-K and total length 2L, as shown in the figure, is connected to a hot surface that is at a temperature T1. The connection between the bar and the surface is imperfect and results in a thermal contact resistance of R’’ m2-K/W. The width of the rod into the depth of the paper is W meters and the thickness of the rod is t meters. The first section of the rod of length L meters is perfectly insulated. The second portion of length L meters of the bar is exposed to a convective environment with a heat transfer coefficient of h W/m2-K and a temperature of Tinf.

Assume: The thickness is small and thermal conductivity of the bar is large. The width into the page can be considered to be large.

(a) Derive an expression for the interface temperature of the bar T2 in terms of known/given parameters in the problem statement. Express the result in an explicit form.

(b) Determine the heat transfer rate from the rectangular bar While formulating your solution, show the following required steps (i) control volume/control surface/resistance network and (ii) assumptions

n, int T2 = ? Thickness t, width into page w h, Tint Rectangular bar, conductivityk

Answer 1

Given, y Tink Control volume i ir Paz Thermal conductivity of bar = kw/mk Total length = 2L contact resistance =R" mlk-w Heat transfer coefficient = h w/mk (a) Assuming steady state condition.

So, for control volume Ein = Yout 9.in = (Ti-T) . Rth Tomm met here, ? at A - Area of Cross Section A = ut in In The Assuming k is very large din = (T1-72) S. Ru Now, part . half exposed to of bar which is environment will be act fin. as

Assuming Fin is finite - afin = Yout= thoka (T2-Tent) tanh (MLC) 2 where, p= perimeter of fin pa 26c+t) A = Wit ma The 2c = (2+1/2) From Equation (1) and (2) ТЯ, °im - mut Ti-Tz = Thora (T2-Ting tanh (mic) R" T-T2 Ta-Time Re' Thaka tanh (mbe) Adding both sides 2+1 Ti-Tinf = R"Wapka tank (Lc) +1 T2-Time

12 = Tine - Tinf + (g" hºa anh (ml)+1) (6) Heat transfer Rate:- : 2 = T, -T2 lanet I-inf (R"Shaka tanhimle) +1). Р)