Page 389 - Water and wastewater engineering

10-6 WATER AND WASTEWATER ENGINEERING

This is in the transition region and Stokes’ law is not valid. An iterative solution using Newton’s
equation is required. Equation 10-11 is used to estimate the drag coefficient. The Reynolds
number of 7.88 is used for the starting value of R , and a new velocity is calculated. The result-
ing velocity is used to check the Reynolds number. The process is repeated until the value of the
Reynolds number used to calculate the velocity matches the check of the Reynolds number.
d. The Solver program* in a spreadsheet was used to perform the iterations. The spread-
sheet cells are shown in Figure 10-3 . The cell locations used in the figure are identified
by brackets [ ] in the discussion below. For completeness, the spreadsheet solution be-
gins by repeating the Stokes’ law solution.
[B5] through [B10] are input values in the correct units.

e. In cell [B14] write the Stokes’ law solution for the settling velocity
∗ ∗ ∧ ∗
B
(981 ([ 6 ] [ 7 ]) ([ 5 ]) 218 [ 9 ]) 3 77E 02
.
B
B
) (
/
B
.
E
f. In cell [B18] check the Reynolds number
∗
([B5 ] [B14 ]) [B10 ] 7 87 E 00
.
/
g. Begin the set up for the Solver solution by entering the value of the Reynolds number
that has just been computed in cell [B25]. DO NOT type in the cell reference.
4
h. In cell [B29] calculate the Newton drag coefficient for R between 0.5 and 10
∧
.
/
.
(24/ [$B$ 25 ]) ( ($B$ 25 ) 0 5 ) 0 34 4 46E 00
.
3
i. In cell [B31] enter Newton’s equation for settling velocity
∗ ∗ ∗ ∗ ∗ ∗ ∧
E
.
.
.
/
((49 81 ) ([$B$ 6 - $B$ ) 7 $B 5 ) (3 $B$ 29 [$B$77])) 0 5 3 11 02
j. In cell [B35] check the Reynolds number
∗
.
([$B$5 ] [B31 ])/$B$10 6 50
k. Activate the dialog box for Solver and designate the target cell [B35], that is, the last
computation of the Reynolds number.
l. Set Equal to to “Min.”
m. By changing the cell containing the the first estimate of Reynolds number, that is,
[B25].
n. Add the following Constraint in the dialog box: [B35] [B25]
o. Execute solve to find the settling velocity is 2.73E 02 m/s as shown in Figure 10-3 .

®
* Solver is a “tool” in Excel . Other spreadsheets may have a different name for this program.