Page 163 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – CARTESIAN GRIDS
The friction factor and Reynolds number are defined as May 20, 2005 12:28
2β H ρ u av 2 H
f = , Re = . (5.119)
2ρ u 2 av µ
It is difficult to specify exact thermal boundary conditions at the blocks in a
real heat exchanger. Nonetheless, we may assume that each block or plate delivers
heat flux q w (say) along its perimeter so that the total heat transfer will be Q =
q w (2 L + 4t/2) and the total bulk temperature rise across the module will be T b =
Q/( ˙ mC p ). Thus, the periodic temperature boundary condition will be
T (0, x 2 ) = T o (0, x 2 ) − 0.5 T b ,

T (2L, x 2 ) = T o (2L, x 2 ) + 0.5 T b ,
T o (0, x 2 ) = T o (2L, x 2 ). (5.120)

In Equations 5.117 and 5.120, all variables must be evaluated at x 1 = 0(I = 1)
and x 1 = 2L(I = IN). This evaluation is done as follows:
(1, J) = (IN, J) = 0.5[ (2, J) + (IN − 1, J)], (5.121)

where = p o , u i , T and it is assumed that the chosen grid disposition is such
that x 1 (IN) − x 1 (IN − 1) = x 1 (2) − x 1 (1). Solution of the temperature equation
enables evaluation of the mean bulk temperature T b = 0.5(T b,AF + T b,CD ), where
the bulk temperatures at the periodic planes are evaluated from

H/2
0 ρ C p u 1 Tdx 2 . (5.122)
H/2
T b,AForCD =
0 ρ C p u 1 dx 2
Finally, the Stanton number St is evaluated as
h av
St = , (5.123)
ρ C p u av
where the average heat transfer coefficient is evaluated from
1 q w
h av = ds, (5.124)
(2L + 2t) T w,s − T b
and s is measured along the heated surfaces. Computations are performed for air
(Pr = 0.7) with a 38(x 1 ) × 36(x 2 ) grid and the results are shown in Figure 5.15.
In all computations, L/H = 1.0 and t/H is varied. Also plotted in the figure are
experimental data of Kays and London as read from reference [54]. These data
have been obtained for t/H = 0.05, L/H = 1.14 (instead of 1 in the present case),
and the (x 3 -direction width)/H = 5.9. Therefore, the geometric data approximate
the present 2D computational domain. It is seen from the figure that the predicted
friction factor data (solid lines) are in very good agreement with the experimental
data (open circles). The predicted St × Pr 2/3 (dashed lines) trend, however, de-
viates from the experimental data (open squares). But, as indicated earlier, it is
difficult to approximate the exact boundary conditions of the experiment, which