Question

3. A liquid enters the bottom of a large insulated pipe of diameter D and length L at a temperature T., flows slowly upward with velocity v and then spills over the top in a thin layer (see figure). A radiation heat flux is incident on the upper surface of the liquid and is absorbed within the liquid such that the flux at any point within the liquid is given by UR = 4R0e=BX The heat transfer between the upper surface of the liquid and the ambient is negligible. a) Develop a model to determine temperature profile in the liquid along the pipe. b) Find the temperature of the liquid spilling over the top of the container.

Answer 1

The solution methodology is explained below:

Solution consider a control volume of thick ness an at a distance & from top. TT sn T Energy balance for control volume. Accumulation = Input - Output + gen.com. o assume steady state) Input: L o due to radiation flux gela. A {Ais cross section) L due to connection: If Av) Cp Tludy ffidensity of fluid. p. Specific heat. output: Lo due to radiation flux gplatan A. y due to connection: HAV) (p. The - 0 = 1 gel. A + HAV) Lp T + dx) - galerian A + (PAW) Cp T) o= - Algalaidu getan) + HAV() (larda - Thad. Divide by dr.

fyspunt. Iluxdu - Tha) du o= - Igelutdu-gelut - Taking lim duso o = - dige) + pvcp di R = frodipl-ßu). en du. du Ro ot feeliff***) +.cpdI de foro ß explaß a) tufv.cp.d I du dI - - froßexpli .... PVC Integrating faT = (-qro ß expl-Bn). dn... fVCp..... T(n = Iho expl=ß....... Cape fVCp Boundary condition, @ L I I .......... Ion fro apl-ß mat. Come PVCp - C Tigo - fre exp(-BL). IVE

Plugging in Tlu, T(n) = are explßn) + Io - proexp (BL) PVCp Vcp T(n) = Tot pro e vep expl-Bn) - expl-BL) 4 bo Temperature of liquid spilling from top & T @ = 0, Tetop = To + gre of explo)- exp(-BL), - qucp T = Is + The | 1- exp(-82) evle