Contents preface (VII) introduction 1—37

7.5.3. Total Bed-Material Load
The total bed-material load can be determined by adding together the bed load and the suspended load. There is, however, another category of methods too for the estimation of the total bed-material load. The supporters of these methods argue that the process of suspension is an advanced stage of tractive shear along the bed, and, therefore, the total load should be related to the shear parameter. One such method is proposed by Engelund and Hansen (19) who obtained a relationship for the total bed-material load q_T (expressed as weight per unit width per unit time) by relating the sediment transport to the shear stress and friction factor f. The relationship is expressed as

	fφ	T	= 0.4 τ	5/2	(7.37)
					*
where,	φT =		qT	(7.38)
ρ s g^3 / 2 d^3 /2 ∆ρs / ρ
and		f =	8 gRS	(7.39)
	U ^2

The median size d₅₀ is used for d in the above equation.
Example 7.9 Determine the total bed-material transport rate for Example 7.2.

Solution: Using Eqs. (7.37) and (7.39)

fφ

= 0.4 τ ^5/2

T

*

8 gRS

F

ρRS I ^5 /2

∴

U

2

φT

= 0.4G

J

H

∆ρsdK

L

2.5 × 1.6 × 10^−^4

O5 /2

L

0.95 × 0.95

O

or

φT

= 0.4 M

P

× M

P

1.65 × 0.3

× 10

− 3

8 × 9.81 × 2.5 × 1.6

× 10

−4

^qT

N

Q

N

Q

= 6.75

∴

∆ρs /

ρ

ρ s g3 / 2 d3 /2

∴

q = 6.75 × 2650 × (9.81 × 0.3 × 10^–3)^3/2 (1.65)^1/2

T

= 3.67 N/m/s

EXERCISES

1. 
   What is the meaning of the term ‘incipient motion of sediment’? How would you calculate criti-cal tractive stress for given sediment?
   

2. 
   Describe various bed forms that occur in alluvial channels.
   

3. 
   Describe various modes of sediment transport in an alluvial channel.
   

4. 
   Determine the depth of flow which will cause incipient motion condition in a wide channel hav-ing a mean sediment size of 5.0 mm and a channel slope of 0.0004. The specific gravity of the material is 2.65. Also determine the corresponding velocity of flow.
   

276 IRRIGATION AND WATER RESOURCES ENGINEERING

5. 
   Find the depth and velocity of flow at which the bed material of average size 5.0 mm will just move in a wide rectangular channel at a slope of 1 × 10^–3. Assume specific gravity of the material to be 2.65 and neglect viscous effects.
   

5. 
   Water flows at a depth of 0.20 m in a wide flume having a slope of 1 × 10^–4. The median diameter of the sand placed on the flume bed is 1.0 mm. Determine whether the sand grains are stationary or moving.
   

5. 
   An irrigation canal has been designed to have hydraulic radius equal to 2.50 m, depth of flow equal to 2.80 m, and bed slope equal to 1.5 × 10^–4. The sediment on the canal bed has median size of 0.25 mm. Find: (i) the bed condition that may be expected, (ii) height and length of bed forms, and (iii) the advance velocity of the bed forms.
   

5. 
   For an alluvial stream, the average width is 120.0 m and the cross-section may be considered as rectangular. The longitudinal slope of the stream is 0.0002. Prepare a stage-discharge curve up to a depth of 4.0 m. The size and specific gravity of the sediment are 0.30 mm and 2.65, respec-tively.
   

5. 
   Determine the bed load transport in a wide alluvial stream for the following conditions:
   

Depth of flow	= 2.50 m
Velocity of flow	= 1.50 m/s
Average slope of water surface	= 8 × 10^–4
Mean size of sediment	= 5.0 mm
Specific gravity of the sediment	= 2.65

10. 
    Determine the rate of bed load transport in a wide alluvial stream for the following data:
    

Depth of flow		= 4.50 m
Velocity of flow	= 1.30 m/s
Slope		= 2.0 × 10^–4
Size distribution of the sediment is as follows:
d (mm)	0.20	0.44	0.78	1.14	1.65	2.6	5.20
% finer	2.0	10.0	30.0	50.0	70.0	80.0	100.0

11. 
    For an alluvial stream having a slope of 0.00015 and depth of flow equal to 2.40 m, the following velocity profile was observed:
    

y (m)	0.215	0.30	0.425	0.670	0.885	1.035	1.28	1.77	2.07	2.35
Velocity (m/s)	1.31	1.37	1.45	1.56	1.65	1.66	1.68	1.69	1.70	1.65

If the fall velocity for the average size of the suspended load is 8.00 mm/s, plot the distribution of suspended load in a vertical section. Assume Karman’s constant k equal to 0.4 and the concen-tration of sediment at y = 0.215 m as equal to 4 N/litre.

REFERENCES

1. 
   Garde, RJ and KG Ranga Raju, Mechanics of Sediment Transportation and Alluvial Stream Problems, 2nd Ed., New Age International Publishers, New Delhi, 1985.
   

2. 
   Shields, A, Anwendung der Aehnlichkeitsmechanic und der turbulenzforschung auf die Geschiebebewegung, Mitteilungen der Pruessischen Versuchsanstalt für Wasserbau and Schiffbau, Berlin, 1936.
   

3. 
   Swamee, PK and MK Mittal, An Explicit Equation for Critical Shear Stress in Alluvial Streams, CBIP J. of Irrigation and Power, New Delhi, April, 1976.
   

4. 
   Yalin, MS and E Karahan, Inception of Sediment Transport, J. of Hyd. Div., Proc. ASCE, Vol. 105, Nov., 1979.
   

5. 
   Garde, RJ and ML Albertson, Sand Waves and Regimes of Flow in Alluvial Channels, Proc. 8th IAHR Congress, Vol. 4, Montreal, 1959.
   

HYDRAULICS OF ALLUVIAL CHANNELS

277

6. 
   Kondap, DM and RJ Garde, Velocity of Bed-forms in Alluvial Channels, Proc. 15th IAHR Con-gress, Vol. 5. Istanbul, 1973.
   

6. 
   Ranga Raju, KG and JP Soni, Geometry of Ripples and Dunes in Alluvial Channels, J. of Hyd. Research, Vol. 14, 1976.
   

6. 
   Garde, RJ, and KG Ranga Raju, Regime Criteria for Alluvial Streams, J. of Hyd. Div., Proc. ASCE, Nov. 1963.
   

6. 
   Keulegan, GH, Laws of Turbulent Flow in Open Channels, U.S. Deptt. of Commerce, NBS, Vol. 21, Dec. 1938.
   

6. 
   Lacey, G, Regime Flow in Incoherent Alluvium, CBIP Publication No. 20, 1939.
   

6. 
   Garde, RJ and KG Ranga Raju, Resistance Relationship for Alluvial Channel Flow, J. of Hyd. Div., Proc. ASCE, Vol. 92, 1966.
   

6. 
   Ranga Raju, KG, Resistance Relation for Alluvial Streams, La Houille Blanch, No. 1, 1970.
   

6. 
   Ranga Raju, KG, Flow through Open Channels, Tata McGraw-Hill Publishing Company Limited, New Delhi, 1986.
   

6. 
   Einstein, HA, Formulas for the Transportation of Bed Load, Trans. ASCE, Vol. 107, 1942.
   

6. 
   Einstein, HA, and NL Barbarossa, River Channel Roughness, Trans. ASCE, Vol. 117, 1952.
   

6. 
   Meyer-Peter, E and R Müller, Formulas for Bed Load Transport, Proc. 2nd IAHR Congress, Stockholm, 1948.
   

6. 
   Einstein, HA, The Bed Load Function for Sediment Transportation in Open Channel Flows, USDA, Tech. Bull. No. 1026, Sep., 1950.
   

6. 
   Rouse, H, Modern Conceptions of Mechanics of Fluid Turbulence, Trans. ASCE, Vol. 102, 1937.
   

6. 
   Engelund, F and E Hansen, A Monograph on Sediment Transport in Alluvial Streams, Teknisk Forlag, Denmark, 1967.