Page 217 - Modelling in Transport Phenomena A Conceptual Approach

7.4. CONSERVATION OF MOMENTUM 197

where

0.31 Re;
I = LRe” (1.067 x lo6 - 18 Rep -3.114Reg657 - ) d Rep
1 + 16,300 Re;’.’’
The value of I can be determined by using one of the numerical techniques given in
Section A.8-4 in Appendix A. The use of the Gauss-Legendre quadrature is shown
below. According to Eq. (A.8-13)

1021
Rep = - + 1)
(u
2
and the five-point quadrature is given by

where the function F(u) is given by
1
F(u) =
80,789 (u + 1)2
1.067 x lo6 - 9189 (U + 1) - 95602 (U + l)1.657 - + 18.22 (u + 1)-l.09

The values of wi and F(ui) are given up to three decimals in the following table:

i Ui wi qui) x 106 WiF(Ui) x 106
0 0.000 0.569 1.044 0.594
1 +0.538 0.479 1.187 0.569
2 -0.538 0.479 0.966 0.463
3 +0.906 0.237 1.348 0.319
4 -0.906 0.237 0.940 0.223

Therefore, the value of I can be calculated from Eq. (2) as

I=- lo2’ (2.17 x 10-6) = 1.11 x 10-3
2

Substitution of this value into Eq. (1) gives
(11,307)(1.5 x 10-3)2 (1.11 x
t= = 1.44s
19.57 x