Question

roblem 5: When can we neglect axial conduction? Consider steady, fully developed, laminar flow inside a round tube, with the energy equation: The two terms on the right hand side represent conduction effects in the radial and axial directions, respectively. In convection analysis we usually neglect the axial conduction term, resulting in Eq. 8.48. Let's investigate the suitability of this approximation, by comparing the relative magnitudes of those two terms for a tube flow with constant wall temperature The full T(x,r) can be obtained from a series solution in even powers of r, or by an iterative approach mentioned briefly below Eq. 8.54. The series solution is given in more advanced references [e.g. Kays & Crawford, Convective Heat and Mass Transfer, 3rd ed., pg. 117] and can be written as where T, (x) is given by Eq. 8.42; and the first few c, are c,-1 , ?,--1.828397 , c4=1.292857 , ?6=-O634099 For simplicity, it is fine to keep only the first two terms in the f(r) series, i.e., only co and C2" (A note about convergence: n dies out exponentially with n, for large n.) Deliverables: () above, evaluate condaxial (b) Similarly, evaluate ondaarar (c) Evaluate the estimated error from neglecting axial conduction, defined as eond aal /M.cond, rudal If you find your expression for depends on x and/or r, you may pick some representative (or worst case) value(s). (d) Interpret your result for ?: Give a simple criterion to ensure that the axial term is no more than -1% of the radial term. Does your criterion depend on Re, Pr, etc.?

Answer 1

a) T(x, r) = T_s + (T_m (x) - T_s) F(F) for a title flow of Come wall Temp T_s - T_m (x)/T_s - T_m,1 = exp (-px/mcp n) Therefore T_m (x) - T_s = (T_m,1 - T_s) exp (-pxb/mC_p) F(r) = elementof _n^n times n C_n r^~n = C_0 (r/r_0)^0 + C_2 (r/r_0)^2 = 1 - 1.828397 (r/r_0)^2 Therefore T(x, r) = T_s + (T_m, l - T_s) [exp (-pxh/mC_p)] (1 - 1.828397) (r/r_0)^2 q_Cond, oral = partial differential/partial differential x^2 q_Con, axis = (T_m, r - T_s) (P^2 h^-2/m^2 Cp^2 [exp (-pxh/mCp)] (1 - 1.828397) (r/r_0)^2 b) q_Cond, adicl = 1/r partial differential/partial differential r (r partial differential T/partial differential r) 1/r partial differential/partial differential r [(r (T_m,1 - T_S) exp (-Pxh/mCP) (-1.828397 times 2/r^2_0 r)] 1/r partial differential/partial differential r((T_m,1 - T_S) exp (-Pxh/mCP) (-1.828397 times 2/r^2_0)) \r\n 1/r (T_m,1 - T_S) exp (-Pxh/mCP) [-7.313588 times r/r^2_0]