TAILIEUCHUNG - Free convection flow in a vertical annulus with power law fluid
In this paper we consider free convection fiow in a vertical annulus offinite height with different external temperatures (see ). The problem is solved by a finite difference scheme. The calculation result when the height is much bigger than the diametter is compared with asymptotic solution. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 4 (4- 10) TPt (), too. Substitution iuto () gives + {1- a)p2, av~1 ) + (1- a)v~2 ); Va are solutions of r, ~Vzo(rs + r2) - / rv12ldr Q = ---:;,.-----'-'"-'- - - I" r(vi 1) - vi2l)dr " 4. Discussion of the results A. The case without channel thickness a. Asymptotic solution. When (H/ D) ---> oo then far from the entrance the problem is one· dimensional and we can find the solution easily: () (4 ) 2 T=am(r)+b Te,-Tel a= m(rs/r.) · b=T,,m(r3 )-T,,m(rz) ln(rs/rz) () r r 1 1 1 Vz = (G,9 Jb*l/2) insignlb*l/lwl /n signlwldr = (G,9 Ib*l/2) insignib*l/ Wdr () r, where b* = b- w(r) = -r- (a/b*)r m (r) W = lwll/n sign (w) 7 + (c/r) () () . ~ Constant c is chosen to satisfy the condition JW dr = 0 ,, () () if T,, = T., (symmetric external temperatures) then T,, = T,, = 1; a= 0; b = 1 T= 1 () becomes: () becomes: w = For comparison we take Prg = 100; Grg = () (c/r) - r 4. 795 x u)- 2 ; () n = ; .:\ 1 = 4; r1 = 1; Te 2 = ; T., =. w-•; X w-•; The formulae (), () give Vzo = X = X 10- 2 Numerical results are Vzo = = X 10- 2 The differences are smaller b. Numerical example. The fluid under consideration is a 1000 wppm solution of water and CMC (carboxy methyl cellulose). The input data are as follows (with dimensions) (see [2)) T= = 15°C T,, = 20'C T,, = 30'C D=2cm H= 20cm p = 1000kg/m3 -' = Vk Cp f3 = x 103 jfkgK = X 10- 4 1/K n = X 10- 8 m 2 / s 2 -n = The calculation results are Vzo = X 10- 2 (that's X 10- 1 cm/s) NuD = The distribution ofT, v, are shown in Fig. 3, 4 /.00 f. Z: 2.!;, =O.!iH f. Z:: 21 2. Z: O.!iH 3. Z: H Channel WitlfiJ a oo ll-::::-:~::::::::::-~.lJ aoo~,~~~~~~-J J/z ehsfri/;ufion /IS r ¢:::0 T .
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