Page 247 - Fair, Geyer, and Okun's Water and wastewater engineering : water supply and wastewater removal

JWCL344_ch06_194-229.qxd 8/2/10 9:51 PM Page 208

208 Chapter 6 Water Distribution Systems: Components, Design, and Operation
Take-off
Q i Q 1 Q i Q 1 T
Inflow
Inflow
Assumed flow Q i
incorrect by q

Q 3

Assumed flow Q 2 Figure 6.11 Simple Network
incorrect by q
Illustrating (a) the Derivation
of the Hardy-Cross Method
Q 2 Q o Q 2 Q o and (b) the Effect of
Outflow
Outflow
(a) (b) Changing Flows

H (clockwise) and -H 2 (counterclockwise) or gH = H - H = 0 . If the assumed split
1
1
2
flows Q and -Q 2 are each in error by the same small amount q, then
1
n
gH = gk(Q + q) = 0
Expanding this binomial and neglecting all but its first two terms, because higher
powers of q are presumably very small, we get
n
n
gH = gk(Q + q) = gkQ + gnkqQ n-1 = 0, whence
gkQ n gH
q =- n-1 = (6.3)
ngkQ ngH>Q
If a takeoff is added to the system as in Fig. 6.11b, both head losses and flows are affected.
In balancing flows by correcting assumed heads, necessary formulations become alge-
braically consistent when positive signs are arbitrarily assigned to flows toward junctions other
than inlet and outlet junctions (for which water table elevations are known) and negative signs
to flows away from these intermediate junctions, the sum of the balanced flows at the junctions
being zero. If the assumed water table elevation at a junction, such as the takeoff junction in
Fig. 6.8b, is in error by a height h, different small errors q are created in the individual flows Q
n
leading to and leaving from the junction. For any one pipe, therefore, H h k(Q q)
n
kQ h, where H is the loss of head associated with the flow Q. Moreover, as before,
h nkqQ n–1 nq(H Q) and
q (h n)(Q H)
Because ∑(Q q) 0 at each junction,

∑Q ∑q and
∑q (h n)∑(Q H), or
∑Q (h n)∑(Q H), therefore,
ngQ
h - (6.4)
g(Q>H)
The corrections q and h are only approximate. After they have been applied once to
the assumed flows, the network is more nearly in balance than it was at the beginning, but
the process of correction must be repeated until the balancing operations are perfected. The
work involved is straightforward, but it is greatly facilitated by a satisfactory scheme of
bookkeeping such as that outlined for the method of balancing heads in Example 6.3 for
the network sketched in Fig. 6.12.