1286 WATER FLOW
Flow depth: DQ
f0 473^13
13.
./
/
(77)Blench ’ s Equations Blench (1953, 1966), using the concept
that canals possessing three degrees of freedom must, there-
fore, have three basic equations to describe their motion,
presented the stream equations for small bed load as:VD Fb(^2)
(78)
V
B
Fs
2
(79)
V
DS
VB
2
363 14
g
.( )./v /
(80)
In Eq. 78, F b is bed factor and the equation itself expresses
the statement that channels with similar water-sediment
flows tend towards the same Froude number in relation to a
suitable depth. Equation (79) describes the scouring action
on the hydraulically smooth sides and defines the side Factor,
F s. The dissipation of energy per unit mass of water per unit
time in the channel is given by Eq. (80) in which S is the
energy gradient.
For appreciable bed load, Eq. (80) becomes:
V
DS
(^2) CVB
14
363 1
g 233
.,
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
v⎠⎟
/
(81)
where C is the bed load charge in parts per hundred thousand
by weight of fluid discharge.
The bed factor, F b , for sand of subcritical flow is given
by the empirical equations:
FFbb^0 (.)^1012 C (82)
Fdbmm 0 19. (83)
in which F b 0 zero bed factor and is the value of F b when C
tends to zero, d mm median bed material size by weight in
millimeter. As a guide to the value of the side factor, F s , the
following table has been suggested by Blench (1966).
Tractive Force Method for Canals
Unit Tractive Force The stability of an erodible chan-
nel depends on (a) the resistance of the material lining the
bottom and sides against the erosive force of the stream and
(b) the ability of the stream to transport the sediment load
without giving rise to significant deposition.
The shear or drag force exerted by the water on the bed
and sides of the channel is termed the tractive force. The
average unit tractive force, t, in uniform flow is the compo-
nent of the gravity force acting on the water parallel to the
channel bottom per unit area, thus:
tg RS. (84)
For wide channels, the flow depth can replace the hydraulic
radius:
tg DS. (85)
The distribution of tractive force has been investigated by
the United States Bureau of Reclamation (Lane, 1952; 1953,
Olsen and Florey, 1951, 1952). The maximum values of the
unit tractive force for the bottom and sides of rectangular
and trapezoidal cross-sections are given in Figure 12.
Tractive Force Ratio A soil particle of effective area, A e ,
resting on the side of a channel is acted on by the tractive
TABLE 9
Values of the bed factor
Material (Fs)max Remarks
Very sandy loam banks 0.1 Erosion if (Fs)max
Silty clay loam 0.2 Erosion if (Fs)max
Very cohesive banks 0.3 Erosion if (Fs)max
Trapezoidal bottom
SS 2:1 and 1.5:1
SS 2:1, sides
SS 1.5:1, sides
SS 0:1, sides
Rectangular
SS 0:1, bottom
Width/depth ratio
Maximum unit tractive force
γDS
0
0
0.2
0.4
0.6
0.8
1.0
(^2468)
FIGURE 12 Maximum tractive force.
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