Page 212 - Fair, Geyer, and Okun's Water and wastewater engineering : water supply and wastewater removal

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174 Chapter 5 Water Hydraulics, Transmission, and Appurtenances

12
1. Low-pressure pipe
11
2. Pressure tunnel
10 3. River crossing
9 8
Cost, 10 4 dollars (c) 7 6 5

3 4
Tunnel
2
1
Figure 5.8 Lagrangian Optimization of
0 10 20 30 40 50 60 Conduit Sections by Parallel Tangents.
Head loss, ft (h) Conversion factor: 1 ft 0.3048 m.

c for a given change in head h is the same for each kind. The proof for this statement is
provided by Lagrange’s method of undetermined multipliers. As shown in Fig. 5.8 for
three components of a conduit with an allowable, or constrained, head loss H, the
Lagrangian requirement of c > h c > h c > h is met when parallel tangents
1
1
2
3
3
2
to the three c:h curves identify, by trial, three heads h , h , and h that satisfy the constraint
2
3
1
h h h H.
2
3
1
EXAMPLE 5.8 DETERMINATION OF THE MOST ECONOMICAL DISTRIBUTION OF HEADS FOR THREE
CONDUIT SECTIONS
Given the costs and losses of head shown in Fig. 5.8 for three sections of a conduit, find the most
economical distribution of the available head H 60 ft (18.3 m) between the three sections.
Solution
By trial, the three heads h 1 , h 2 , and h 3 that satisfy the constraint h 1 h 2 h 3 H are as follows:
ft m
h 1 13.5 4.1 c l 2.0 l0 4
h 2 19.0 5.8 c 2 2.1 l0 4
h 3 27.5 8.4 c 3 2.2 l0 4
H 60.0 18.3 C 6.3 10 4

EXAMPLE 5.9 UPGRADING CAPACITY AND SIZE OF PIPES

A gravity water supply system consists of a water pipe (C 100) 150 mm in diameter and 3,000 m
long that joins two reservoirs that have a difference of water surface elevation of 13 m.
Expecting a higher water demand in the near future, the flow needs to be increased between
the reservoirs to three times what would be produced by the present system. Two alternatives are