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6.6 Office Studies of Pipe Networks 211

Column 5 lists the assumed flows Q 0 in MGD or MLD. They are preceded by positive signs if
they are clockwise and by negative signs if they are counterclockwise. The distribution of flows has
been purposely misjudged in order to highlight the balancing operation. At each junction the total
flow remaining in the system must be accounted for.
Column 6 gives the hydraulic gradients in ft per 1,000 ft (‰) or in m per 1,000 m when the
pipe is carrying the quantities Q 0 shown in Col. 5. The values of s 0 can be read directly from tables
or diagrams of the Hazen-Williams formula.
Column 7 is obtained by multiplying the hydraulic gradients (s 0 ) by the length of the pipe in
1,000 ft; that is, Col. 7 Col. 6 (Col. 3/1,000). The head losses H 0 obtained are preceded by a
positive sign if the flow is clockwise and by a negative sign if counterclockwise. The values in Col.
7 are totaled for each circuit, with due regard to signs, to obtain ∑H.
Column 8 is found by dividing Col. 7 by Col. 5. Division makes all signs of H 0 >Q 0 positive.
This column is totaled for each circuit to obtain ∑(H 0 >Q 0 ) in the flow correction formula.
Column 9 contains the calculated flow correction q 0 ∑H 0 >(1.85 ∑H 0 >Q 0 ). For example,
in circuit I, ∑H 0 16.5, ∑(H 0 >Q 0 ) 43.1; and ( 1 6.5) (1.85 43.1) 0.21; or q 0 0.21.
Because pipe 4 operates in both circuits, it draws a correction from each circuit. However, the second
correction is of opposite sign. As a part of circuit I, for example, pipe 4 receives a correction of
q 0.07 from circuit II in addition to its basic correction of q 0.21 from circuit I.
Columns 10 to 14 cover the once-corrected flows. Therefore, the hydraulic elements (Q, s, H,
and q) are given the subscript 1. Column 10 is obtained by adding, with due regard to sign, Cols. 5
and 9; Cols. 11, 12, 13, and 14 are then found in the same manner as Cols. 6, 7, 8, and 9.
Columns 15 to 19 record the twice-corrected flows, and the hydraulic elements (Q, s, H, and q)
carry the subscript 2. These columns are otherwise like Cols. 10 to 14.
Columns 20 to 23 present the final result, Cols. 20 to 22 corresponding to Cols. 15 to 18 or 10
to 12. No further flow corrections are developed because the second flow corrections are of the
order of 10,000 gpd (37,850 L/d) for a minimum flow of 200,000 gpd (757,000 L/d), or at most
5%. To test the balance obtained, the losses of head between points A and D in Fig. 6.12 via the
three possible routes are given in Col. 23. The losses vary from 25.0 to 25.5 ft (7.62 to 7.77 m). The
average loss is 25.3 ft (7.71 m) and the variation about 1%.

EXAMPLE 6.4 ANALYSIS OF A WATER NETWORK USING THE RELAXATION METHOD
OF BALANCING FLOWS
Balance the network of Fig. 6.13 using the balancing flows method. Necessary calculations are
given in Table 6.5.

Solution:
The schedule of calculations includes the following:
Columns 1 to 5 identify the pipes at the three free junctions.
Columns 6 and 7 give the assumed head loss and the derived hydraulic gradient that
determines the rate of flow shown in Col. 8 and the flow-head ratio recorded in Col. 9 (Col. 8
divided by Col. 6).
Column 10 contains the head correction h 0 as the negative value of 1.85 times the sum of
Col. 8 divided by the sum of Col. 9, for each junction in accordance with Eq. 6.4:
ngQ
h =- (6.4)
g(Q>H)
A subsidiary head correction is made for shared pipes as in Example 6.2.
Column 11 gives the corrected head H 1 H 0 h 0 and provides the basis for the second flow
correction by determining s, Q, and Q 1 H 1 in that order.