‫الفضاء الحر‬
‫المقطع االقتصادي او الكفوء‬
‫‪Free board‬‬
‫‪Most efficient section‬‬
EXAMPLES:
Q1: There is a steady flow at 5 m3/sec at a slope of 0.002. What is the
proper width B of a rectangular channel if we wish to minimize the
wetted perimeter for the sake of economy of construction? The channel is
made of unfinished concrete (n = 0.015).
Solution:
For a rectangular section, the best hydraulic section has a width B equal
to twice the depth
i.e. B = 2y
A = By =
P = B + 2y = B +B = 2B
⁄
R=
Q = AV
⁄
5=( )(
⁄
⁄
5=( )(
5.91589
⁄
⁄
5
B = 0.9388 m
Therefore minimum perimeter, P = 2B = 2 x 0.9388 = 1.8776 m.
Q2: A rectangular channel, 8 m wide and 1.5 m deep, has a slope of 1 in
1000 and is lined with smooth concrete plaster. It is desired to increase
the discharge to a maximum by changing the dimensions of the channel
but keeping the same quantity of lining. Find the new dimensions and the
percentage increase in discharge. Assume, Manning's coefficient, n =
0.015.
Solution:
(a) A = By = 8 x 1.5 = 12 m2
P = B + 2y = 8 + 2 x 1.5 = 11 m
R=
m
⁄
⁄
⁄
3
Q1 = Av = 12 x 2.23 = 26.79 m /sec.
(b) B = 2y for maximum discharge
P = B + 2y = 2y +2y = 4y
For the same quantity of lining, P = 11 = 4y
y = 11/4 = 2.75 m
B = 2 x 2.75 = 5.5 m
A = By = 5.5 x 2.75 = 15.125 m2
⁄
m
R=
= 1.375 m
⁄
⁄
⁄
⁄
m
Q2 = Av = 15.125 x 2.6 = 39.42 m3/sec
Therefore the percentage increase in discharge is:
Q3: A trapezoidal channel is to be designed to convey 280 m3 of
water per minute. Determine the cross sectional dimensions of the
channel if bed slope is 1 in 1600, the side slopes 1:1 and the cross
section is to be minimum. Take Chezy's constant C = 50.
Solution:
For minimum cross section
B + 2zy = 2y √
B + 2y = 2y √
B = 0.828 y
R = y/2
A = (B + zy) y = (0.828 y + y) y = 1.828 y2
P = B + 2zy √
= 0.828 y +2 √ y = 3.656 y
Q=AC√
√
y = 1.528 m
B = 0.828 y = 0.828 (1.528) = 1.265 m.
Q4: Determine the bed width of a trapezoidal channel of maximum
discharge, if the flow depth is 3.0 m and side slope 1:1. What will
be the discharge if the bed slope is 1 in 1600? Take Chezy's
constant C = 66.