Page 191 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

146 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

chamber-Parshall flume design by permitting exploration 7.2.2.4.2 Calculation of p
of many alternatives. Example 7.6 illustrates the use of If the bottom of a parabolic grit chamber is the same as the floor
such a spreadsheet, that is, Table CD7.6. The formulae
used in the cells and the calculation procedure are docu- of a Parshall flume, v H will remain constant, that is, v H ¼ 0.3
mented in the sections below the four categories of m=s (1.0 ft=s) (Camp, 1942). The match is obtained by solving
calculations. for p of the parabola, obtained by forcing the equality of flows
between the grit chamber and the Parshall flume, for example,

Example 7.5 Design of Grit Chamber=Parshall Q(parabolic grit chamber section) ¼ Q(Parshall flume)
Flume by Spreadsheet
(7:16)
Problem Now substitute Equations 7.15 and 7.6, respectively, after
Explore the design of a Parshall flume and grit chamber multiplying A(parabola) by v H ,
3
combination for a 0.44 m =s (10 mgd) average daily flow.
Solution 8 ffiffiffip p y 3=2 ¼ CH n (7:17)
Set up the protocol for calculation on a computer spread- v H 3 a
sheet, Table CD7.6. The first category of the spreadsheet is
flow and the full range is entered. Next, knowing Q(max), a Now by letting v H ¼ 0.305 m=s (i.e., Camp’s velocity criterion)
0.457 mm (1.50 ft) flume is selected. From these data the and assuming flow, Q, to yield H a by Equation 7.6, and if
n
H a (max) depth is calculated by formula, that is, Q ¼ CH a . y ¼ (set)H a , then p can be calculated. The ‘‘throat’’ width of
The grit chamber calculations start with assumptions for the flume selected should be as narrow as possible to accom-
channel width, w, and with assumption of channel velocity modate the maximum flow, that is, Q(max). As a caveat, the
at maximum flow, v H (max) 0.38 m=s (1.25 ft=s). From width of the flume at maximum depth could be too large to
these data, the channel depth, d(max), is calculated,
along with DZ, that is, d ¼ H a þ DZ. The next concern is be practical.
to check the channel velocity for Q(min) to ascertain 7.2.2.4.3 Spreadsheet for Parshall-Flume with Parabolic
whether v H (min) 0.75 ft=s. The selection is indicated by
the blocked out area, based upon the v H criteria being met, Section
that is, 0.23 v H 0.38 m=s (0.75 v H 1.25 ft=s). Table CD7.7 illustrates a protocol for sizing a Parshall-flume-
Tables CD7.6(a) and (b) (metric units and U.S. Custom- control with a parabolic-section-grit-chamber in terms of a
ary units, respectively) illustrate the foregoing spreadsheet spreadsheet. The spreadsheet is formatted in three parts: (1)
description. Formulae for cells are indicated at the bottom. selection of Parshall flume, (2) determination of p, and (3)
calculation of the coordinates for the parabolic section. Figure
CD7.11 is a plot of the resulting parabolic half-section, which
7.2.2.4 Parabolic Section
is linked to the spreadsheet. Example 7.6 illustrates the design
The problem of maintaining constant v H in the grit chamber is
protocol.
achieved by a parabolic section, matched with a Parshall
flume control section. The mathematical characteristics of a
parabolic section are reviewed here. Example 7.6 Design of Parabolic Grit Chamber
Section
7.2.2.4.1 Mathematics of a Parabolic Section
The coordinates for a parabola, oriented with axis in the Given
y-direction and vertex at (0,0), were given by Griffin (1936, Let the expected maximum flow for a headworks of a
3
p. 293) as wastewater treatment plant, be Q(max) ¼ 0.3067 m =s
(7 mgd).
2
x ¼ 4 py (7:14) Required
Select a Parshall flume for the flow stated and determine
the associated parabolic grit chamber section.
where
x is the x-coordinate of the parabolic shape (m) Solution
y is the y-coordinate of the parabolic shape (m) Table CD7.7 is a spreadsheet that follows the foregoing
p is the mathematical constant characteristic of the para- protocol. Part (a) shows that a 305 mm (12 in.) Parshall
3
bola shape, for example, narrow or wide (m) flume has a capacity of 0.453 m =s (10.3 mgd), which is
greater than Q(max). [The 229 mm (9 in.) flume lacks
sufficient capacity and the 457 mm (18 in.) flume has too
The ‘‘spread’’ of the parabola depends on the value of p, for
much capacity.] The coefficients C and n are given in
example, when y ¼ p, then x ¼ 2p. The area of the parabola
Part (a). Determination of p is given in Part (b) by applying
at any y is given by integration of Equation 7.14, Equation 7.17 for any Q. Knowing p permits calculation of
the coordinates of a matching parabolic section as seen in
Part (c); Figure CD7.11 shows the half-parabolic section
p y (7:15)
8 ffiffiffip 3=2
A(parabola) ¼
3 plotted by the coordinates given.