TAILIEUCHUNG - Free convection flow in a vertical thin cylinder of finite height with power law fluids
In this paper we consider free convection flow in a vertical thin cylinder of finite height with given external temperature (see Fig. The problem is solved by a finite difference scheme. The calculation result when the height is much bigger than the diameter is compared with asymptotic one. A condition of neglecting the thickness is shown. | T~p chi C G,g - generalized Prantl and Grashof numbers, £1k - kinematics viscosity, p - density, Cp - specific heat coefficient, \ - thermal conductivity, g- acceleration of gravity, f3 - thermal expansion coefficient. Boundary conditions: v,G, z) =v,G, -z) =0; _ _ a;;. _ aT _ v,(O,z)= ar(O,z)= ar(O,z)=O; TrG+b',z) =Tw; T,G ,z) =TG ,z)' >.,aT, (~ -z) 8¥2' p'(o) iJ,(r, 0) = () >.aT(~ -z). . Gr2'' = v,(r, o) = T(r, o) = o; = Vzo; P'(1) = 0. Because of the smallness of 8 in comparison with H; ( ~) . Two quantities of particular interest are the average velocity along the channel Vzo and the total heat transfer from the wall Q, which is characterized. by average Nusaelt number N 3. NUMERICAL SOLUTIONS First, we can exclude T, by integrating () combining with (), and we get following boundary condition for T at r = ~ : T 2 , z·)) = ( -(1 2¢ 1 - aT(1 a-r 2 ,-) z >., >.ln(1 + 28) where tf; = :-:---:'-'~= Mter T has found Tr can be calculated as ()- (), (), () is a closed system for il,(r, z), 40 v.(r, z), T(r,z), p'(z), v,,. () We solve this system by a finite difference method. The finite difference equation {see Fig. 2) (drop signs . for convenience) •+I )1+1 ( r v r k+l - (r H1 )J+1 v r k + • ) J+l ( Vz k )J+1 ( H1 v z k az )J Vz k + )J ( )J rvz k+I _- rvz k = )J+1 _ ('+1 )J+1 (v' ) J+I ('+1 v z k v z k-1 ,). r k •+liJ+ 1 P = - 0 () IJ - p !: + f'Jk+I/2 + T • ( - 'J+1 (('+1 )J+1_ ('+1 )J+1)- f]k-l/2 'J+1 (('+1 )J+1_ ('+1 )J+l) V z k+I -V z k V z k . V z k-1 •+1 J+l + Gro ( - ( r •+I )J+I ( •+1 )J+l v z k+l + r tJ 11 1c (2\.r)2 . () () where '1k+ 1 / 2 , , taken equal to I Uk+l - Uk 'lk- 1 12 lS ~n-1 ; I Uk- Uk-1 ~n-1 . This is a non-linear system. The truncation errors is of O(, "). The Von Neuman stability condition is satisfied .
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