d[chi]/dd = [Omega]/d² + cot[beta] - 2/sin [beta] = 0,
or
[Omega]/d² = 2 cosec. [beta] - cot [beta]. (4)
or
d = [root]{[Omega] sin [beta]/(2 - cos [beta])}.
From (3) and (4),
b/d = 2(1 - cos [beta])/sin [beta] = 2 tan ½[beta].
_Proportions of Channels of Maximum Discharge for given Area and Side
Slopes. Depth of channel = d; Hydraulic mean depth = ½d; Area of
section =_ [Omega].
+-------------+-----------+--------+----------+---------+------------+
| |Inclination|Ratio of| Area of | |Top width = |
| |of Sides to| Side | Section | Bottom |twice length|
| | Horizon. | Slopes.| [Omega]. | Width. |of each Side|
| | | | | | Slope. |
+-------------+-----------+--------+----------+---------+------------+
| Semicircle | .. | .. | 1.571 d² | 0 | 2 d |
| Semi-hexagon| 60° 0´ | 3 : 5 | 1.732 d² | 1.155 d | 2.310 d |
| Semi-square | 90° 0´ | 0 : 1 | 2 d² | 2 d | 2 d |
| | 75° 58´ | 1 : 4 | 1.812 d² | 1.562 d | 2.062 d |
| | 63° 26´ | 1 : 2 | 1.736 d² | 1.236 d | 2.236 d |
| | 53° 8´ | 3 : 4 | 1.750 d² | d | 2.500 d |
| | 45° 0´ | 1 : 1 | 1.828 d² | 0.828 d | 2.828 d |
| | 38° 40´ | 1¼ : 1 | 1.952 d² | 0.702 d | 3.202 d |
| | 33° 42´ | 1½ : 1 | 2.106 d² | 0.606 d | 3.606 d |
| | 29° 44´ | 1¾ : 1 | 2.282 d² | 0.532 d | 4.032 d |
| | 26° 34´ | 2 : 1 | 2.472 d² | 0.472 d | 4.472 d |
| | 23° 58´ | 2¼ : 1 | 2.674 d² | 0.424 d | 4.924 d |
| | 21° 48´ | 2½ : 1 | 2.885 d² | 0.385 d | 5.385 d |
| | 19° 58´ | 2¾ : 1 | 3.104 d² | 0.354 d | 5.854 d |
| | 18° 26´ | 3 : 1 | 3.325 d² | 0.325 d | 6.325 d |
+-------------+-----------+--------+----------+---------+------------+
Half the top width is the length of each side slope. The wetted
perimeter is the sum of the top and bottom widths.
§ 114. _Form of Cross Section of Channel in which the Mean Velocity is
Constant with Varying Discharge._--In designing waste channels from
canals, and in some other cases, it is desirable that the mean
velocity should be restricted within narrow limits with very different
volumes of discharge. In channels of trapezoidal form the velocity
increases and diminishes with the discharge. Hence when the discharge
is large there is danger of erosion, and when it is small of silting
or obstruction by weeds. A theoretical form of section for which the
mean velocity would be constant can be found, and, although this is
not very suitable for practical purposes, it can be more or less
approximated to in actual channels.
Let fig. 117 represent the cross section of the channel. From the
symmetry of the section, only half the channel need be considered. Let
obac be any section suitable for the minimum flow, and let it be
required to find the curve beg for the upper part of the channel so
that the mean velocity shall be constant. Take o as origin of
coordinates, and let de, fg be two levels of the water above ob.
[Illustration: FIG. 117.]