Water Conveyance Loss and Designing Conveyance System - Practices of Irrigation and On-farm Water Management - page 16

Fig. 1.5 Schematic
presentation of trapezoidal
channel section

Freeboard

d

1

z

b

zd

Let us consider a channel of trapezoidal cross-section as shown in Fig. 1.5
Assume that

b = width of the channel at the bottom,

d = depth of the flow, and

1

z =

side slope (i.e., 1 vertical to z horizontal)

2 2 × zd zd 2

Thus, Area of flow, A = ( d × b ) + d × = bd + = d ( b + zd )

A
d =

+

or b zd

A
d −

b = zd (1.11)

AC √ RS

and discharge, Q = A × V =

AC A

=

P S

[since R = A / P ]

Keeping A , C , and S constant in the above equation, the discharge will be maximum,
when A / P is maximum, or the perimeter P is minimum. From the theory of maxima
and minima, P will be minimum when,

d P
d d =

0

2 √ z 2 d 2 2 d √ z 2 d 2 We know that P = b + + = b + + 1.

Substituting the value of b from Eq. ( 1.11 ) ,

2 d z 2

A
d −

P = zd + + 1

Differentiating the above equation with respect to d and equating the same to zero,