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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 Fig.1). 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 (3.5), too. Substitution iuto (3.6) 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: (4.1) (4 ) 2 T=am(r)+b Te,-Tel a= m(rs/r.) · b=T,,m(r3 )-T,,m(rz) ln(rs/rz) (4.3) r r 1 1 1 Vz = (G,9 Jb*l/2) insignlb*l/lwl /n signlwldr = (G,9 Ib*l/2) insignib*l/ Wdr (4.4) r, where b* = b- 0.5a w(r) = -r- (a/b*)r m (r) W = lwll/n sign (w) 7 + (c/r) (4.5) (4.6) . ~ Constant c is chosen to satisfy the condition JW dr = 0 ,, (4.7) (4.8) if T,, = T., (symmetric external temperatures) then T,, = T,, = 1; a= 0; b = 1 T= 1 (4.1) becomes: (4.6) becomes: w = For comparison we take Prg = 100; Grg = (4.9) (c/r) - r 4. 795 x u)- 2 ; (4.10) n = 0.66; .:\ 1 = 4; r1 = 1; Te 2 = 1.5; T., =0.5. w-•; 5.70 X w-•; The formulae (4.7), (4.8) give Vzo = 5.77 X N.D = 3.19 X 10- 2 Numerical results are Vzo = N.D = 3.15 X 10- 2 The differences are smaller 1.2% 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 -' = 0.597W/mK Vk Cp f3 = 4.18 x 103 jfkgK = 1.8 X 10- 4 1/K n = 7.35 X 10- 8 m 2 / s 2 -n = 0.66 The calculation results are Vzo = 4.34 X 10- 2 (that's 1.36 X 10- 1 cm/s) NuD = 4.18 The distribution ofT, v, are shown in Fig. 3, 4 U.Qo 2.00 1.50 /.00 f. Z: 2.!i.IO-s;, 2.z =O.!iH f. Z:: 21 5.10-SH 0.50 2. Z: O.!iH 3. Z: H Channel WitlfiJ a oo ll-::::-:~::::::::::-~.lJ aoo~,~~~~~~-J J/z ehsfri/;ufion /IS r ¢:::0 T .