Page 843 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

798 Appendix D: Fluid Mechanics—Reviews of Selected Topics

h L (header)
HGL(header) c

b
a

g HGL(lateral) d
Q(header) h L (lateral)
Q(orifice) e h L (orifice) Water surface h
Q(lateral 2)
i Lateral 3 Lateral 4

f
Lateral 1
FIGURE D.6 Hydraulic grade line for header and lateral 2 of under-drain system; water surface ‘‘h–i’’ and is terminus for the HGL for any
orifice.

D.2.5.3 Calculation Algorithm and to achieve this equality,
As noted in the previous paragraph, the role of a manifold is
to distribute flow and to do so uniformly (to the extent Q(orifice) i, j¼1 ¼ Dh(orifice) i, j¼2 ¼ Dh(orifice) i, j¼3
feasible). With a spreadsheet model, any changes can be
¼ ¼ Dh(orifice)m, n (D:24)
incorporated and detailed calculations made so that pressure
changes can be shown along the header pipe and each
lateral pipe. The equations applicable are delineated in the in which
following:
First, the flow through any orifice is Dh(orifice) i, j ¼ headloss across any orifice designated
by lateral i and orifice j (m)
Q(orifice) ¼ A(orifice)C D [2gDh] 0:5 (D:22)
The only means to achieve this uniform headloss for each
orifice, i.e., Equation D.21, is by changing the size of the
in which
3
Q(orifice) is the flow through any orifice (m =s) orifice along the lateral to compensate for the declining
2
A(orifice) is the area of any orifice (m ) Dh(orifice) i along the length of the lateral. This is not a
practical solution. A ‘‘quick and dirty’’ means to approximate
C D is the orifice coefficient—dimensionless
the conditions of Equations D.23 and D.24 is to use a large
Dh is the headloss between two sides of orifice (m)
header pipe and then smaller but still large lateral pipes. The
idea is that if the headlosses in these pipes are negligible, then
For equal flow from each orifice, the mathematical state-
Dh(orifice) 1 > Dh(orifice) n , but only slightly. A spreadsheet
ment is
solution to the problem, as seen in Figure D.6, provides a
means to refine the solution (i.e., with regard to the problem of
Q(orifice) i, j¼1 ¼ Q(orifice) i, j¼2 ¼ Q(orifice) i, j¼3 pipe sizing and orifice sizing). The needed equations for a
computer solution are enumerated as follows, which comprise
¼ ¼ Q(orifice)m, n (D:23)
a solution ‘‘algorithm.’’
in which
1. Continuity applies to the header–lateral relation,
n
Q(orifice) ¼ flow through any orifice designated by X
i, j
Q(lateral) i (D:25)
3 Q(header) ¼
lateral i and orifice j m =s 1