Question

Problem 3. A plane wall of thickness 2L = 40 mm and thermal conductivity k = 5 W/m.K experiences uniform volumetric heat generation at a rate ġ, while convection heat transfer occurs at both of its surfaces (x = -1, + L), each of which is exposed to a fluid of temperature Too = 20 °C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) = a + bx + cx? where a = 82.0°C, b = -210°C/m, c = -2 x 104 °C/m², and x is in meters. The origin of the x-coordinate is at the midplane of the wall. In order to get the full credit, start with energy balance and show all the details of your calculation. T = a + bx + cx2 Fluid To = 20°C, hi Fluid To = 20°C, h, -L (a) What is the volumetric rate of heat generation ġ in the wall (b) Determine the surface heat fluxes, qx"(-L) and qx"(+L). (c) What are the convection coefficients for the surface at x = -L and x = +L

Answer 1

@ rate of heat generation, d²I= q dx2 K K 9 = (6+20) = 2 = -210 - 2x2x10h _ -q 20105o w/m3 6 Surface heat flux x=1, 2 1 (+6)= - 4 2 7 1 x = 2 9 x (+2) = -k[b+ 2c L at 2" x (+2) = -54-210 - 2x2x114 x 20 x 10 3] 9 (+2) = 5050 W/m2 at X=-L, g (2 L) = 2(+2) + 9 (-2) - 201050xo.oH=5050t glx (-2) go"x (-1) = 299 2 W/m2) © convechon coefficients at X=th, 2e +L) = hy [To-Too) hr 282-210x - 2x10 4 x ² - To] 505o = hr [82-210 (0.02) - 2x10" (0.02) 2 - 2o] hr= 101.406 W/m2K at x=-L, 9! (x) (-2) = he [Te - To] 2992= heſ82210(0.02) - 2x10% (0.02)2-20] he = 51.409 W/m2K 5050=