In the previous section we considered problems where the

Transcription

1 5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient was constant with distance along the pipe. In this section we conside poblems in which the velocity is fully developed at the point whee the heat tansfe stats. Futhemoe, as befoe, we conside two cases of constant wall tempeatue and wall heat flux, both by assuming unifom tempeatue at the inlet. Unde these conditions the heat tansfe coefficient is not constant and vaies along the tube. 1

2 Whiteman and Dake (1980), Lyche and Bid (1956) and Blackwell (1985) studied the case of fully developed flow with themal enty effects fo non-newtonian fluids. Sellas et all (1959) obtained themal enty length solutions fo the case of a Newtonian fluid with constant wall tempeatue and fully developed flow, which ae pesented below.

3 We make the following assumptions in ode to obtain a closed fom solution fo heat tansfe analysis in a cicula tube. 1. Incompessible Newtonian fluid. Lamina flow 3. Two-dimensional steady state 4. Axial conduction and viscous dissipation is neglected 5. Constant popeties p This does not mean that one cannot obtain analytical solutions when one o moe of the above assumptions is valid, but the solution will be much easie by making the above assumptions. 3

4 Since the fully developed velocity was aleady obtained in Section 5., we will focus the emphasis on the solution of the enegy equation and bounday conditions fo a developing tempeatue pofile. 4

5 Constant Wall Tempeatue The descibing dimensionless enegy equation (5.33) and bounday conditions with making the assumptions noted above fo the case of constant wall tempeatue ae u 1 x,0 1 1, x 0 (5.6) 0, x finite o 0, x 0 (5.63) whee T Tw u x/ 0,, u, x T T u Re P o in w m 5

6 Fo a fully developed lamina flow, the paabolic velocity pofile developed befoe is applicable. u u 1 o 1 m u o Substituting the above equation into the enegy equation (5.6), we get 1 1 (5.64) x Since the above patial diffeential equation is linea and homogeneous, one can apply the method of sepaation of vaiables. The sepaation of vaiables solution is assumed of the fom, x R X x (5.65) 6

7 The substitution of the above equation into equation (5.64) yields two odinay diffeential equations (5.66) whee X X 0 1 R R R 1 0 dx d X dr d R X, X, R, R dx dx d d and λ is the sepaation constant o eigenvalue. (5.67) The solution fo equation (5.66) is a simple exponential function of the fom x e while the solution of equation (5.67) is of infinite seies efeed to by Stum-Liouville theoy. 7

8 The final solution can be of the fom, n n exp n n 0 x c R x (5.68) whee λ n ae eigenvalues and R n ae eigenfuncitons it coesponding to equation (5.67) and c n ae constants. 8

9 The local heat flux, dimensionless mean tempeatue, local Nusselt numbe and mean Nusselt numbe can be obtained fom the following equations, using the tempeatue distibution above T Tw Tin qw k k (5.69) Nu x o k T T G x o o 1 w in n exp n n0 T exp m T n x w 8 m Gn Tin Tw n0 n exp Gn n x hx o qw o n0 k T w Tin k m m 1 G exp x / n0 1 1 G exp x o Nu xdx hx Nu ln 8 k x 0 x 1 whee G n c nr n 1 m x n n n0 n n n n (5.70) (5.71) (5.7) 9

10 The fist five tems in equations (5.69) and (5.7) ae fully sufficient i to povide accuate solutions to the above infinite seies. The eigenvalues, λ n and G n, to calculate q w, θ m, Nu x and Nu m fo the above poblem ae pesented in Table 5.3. Table 5.4 povides the vaiation of Nu x, Nu m and θ m with distance along the tube. It can be easily obseved fom Table 5.3 that the fully developed tempeatue pofile stats at appoximately x x / 0 Re P (5.73) Theefoe, (L T,T / D) = 0.05ReP whee L T,T is the themal entance length fo constant wall tempeatue. The themal enty length inceases as the Reynolds numbe and Pandtl numbe incease. A vey long themal enty length is needed d fo fluids with a high Pandtl numbe, such as oil. Theefoe, cae should be taken to make a fully developed tempeatue pofile assumption fo fluids with a high Pandtl numbe

11 Table 5.3 Eigenvalues and Eigenfunctions of a Cicula Duct; Themal Enty Effect with Fully Developed Lamina Flow and Constant Wall Tempeatue (Blackwell 1985) n λ n / G n

12 Table 5.4 Nusselt Solution fo Themal Enty Effect of a Cicula Tube fo Fully Developed Lamina Flow and Constant Wall Tempeatue (Blackwell 1985) x + Nu x Nu m θ m

13 Constant Heat Flux at the Wall The lamina fully developed d themal enty length (developing tempeatue pofile) fo constant wall heat flux is vey simila to constant wall tempeatue, except that dimensionless tempeatue and bounday conditions ae defined as follows Tin T q D / k w (5.74) q k T At w constant (5.75) o 13

14 Siegel et al. (1958) solved the above poblem fo lamina fully developed flow using sepaation of vaiables and the Stum-Liouville theoy of which the esult is pesented below * 7, x 4x 4 4 *, x c exp nrn n x n1 (5.76) (5.77) The eigenvalues λ n, eigenfunctions R n and constants c n ae pesented in Table

15 Table 5.5 Eigenvalues and Eigenfunctions fo Themal Enty Effect of a Cicula Tube fo Fully Developed Lamina Flow and Constant Wall Heat Flux (Siegel et al. 1958) n λ n R n (1) C n

16 The local l Nusselt numbe, based on the above solution, is given below and numeical values ae pesented in Table /11 Nux (5.78) 1 4/11 c exp x R 1 n 0 n n n The themal entance fo constant wall heat flux based on the numeical esults pesented in Table 5.6 is o x 0.05 LT, H 0.05 ReP D (5.79) whee L T,H is the themal enty length fo fully developed flow with constant wall heat flux. 16

17 Table 5.6 Nusselt Numbe fo Themal Enty Effect with Fully Developed Flow of a Cicula Tube with Constant Wall Heat Flux (Siegel et al. 1958) x + Nu x

18 The mean tempeatue vaiation can be obtained fom Nusselt numbe, equation (5.78), by using the following equation: T w T w m hx q q D w h Nu k x (6.80) 18