Contents preface (VII) introduction 1—37

Table 7.2 Relationship between φ_B and Ψ′ (17)

ψ′	27.0	24.0	22.4	18.4	16.4	11.5	9.5	5.5	4.08	1.4	0.70
^φB	10^–4	5 × 10^–4	10^–3	5 × 10^–3	10^–2	5 × 10^–2	10^–1	5 × 10^–1	1.0	5.0	10.0

Example 7.7 Determine the amount of bed load in Example 7.2 using Einstein’s method. Solution: From the solution of Example 7.2,
R′ = 0.651 m

∴	ψ′ =	∆ρ sd
ρ R ′ S

165. 
     × 0.3 × 10^–3
     
• 
  0.651 × 16. × 10^–4
  

• 
  4.752
  

∴	φB = 0.763	(from Fig. 7.10)
∴	q = 0.763	× 2650 × (9.81 × 0.3 × 10^–3)^1.5 (1.65)^1/2
	B

= 0.415 N/m/s
7.5.2. Suspended Load
At the advanced stage of bed load movement the average shear stress is relatively high and finer particles may move into suspension. With the increase in the shear stress, coarser fractions of the bed material will also move into suspension. The particles in suspension move with a velocity almost equal to the flow velocity. It is also evident that the concentration of sediment

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273

particles will be maximum at or near the bed and that it would decrease as the distance from the bed increases. The concentration of suspended sediment is generally expressed as follows:

(i) Volume concentration: The ratio of absolute volume of solids and the volume of sedi-ment-water mixture is termed the volume concentration and can be expressed as percentage by volume. 1 % of volume concentration equals 10,000 ppm by volume.
(ii) Weight concentration: The ratio of weight of solids and the weight of sediment-water mixture is termed the weight concentration and is usually expressed in parts per million (ppm).
Variation of Concentration of Suspended Load
Starting from the differential equation for the distribution of suspended material in the vertical and using an appropriate diffusion equation, Rouse (18) obtained the following equation for sediment distribution (i.e., variation of sediment concentration along a vertical):

C	F h − y		a	I ^Zo
	= G		×		J	(7.34)
Ca	y
		H		h − aK

where, C = the sediment concentration at a distance y from the bed, C_a = the reference concentration at y = a,
h = the depth of flow,

Z	o	=	wo	and is the exponent in the sediment distribution equation,
U *k

w_o = the fall velocity of the sediment particles, and k = Karman’s constant.
Rouse’s equation, Eq. (7.34), assumes two-dimensional steady flow, constant fall velocity and fixed Karman’s constant. However, it is known that the fall velocity as well as Karman’s constant vary with concentration and turbulence. Further, a knowledge of some reference concentration C_a at y = a is required for the use of Eq. (7.34).
Knowledge of the velocity distribution and the concentration variation (Fig. 7.11) would enable one to compute the rate of transport of suspended load q_s. Consider a strip of unit width and thickness dy at an elevation y. The volume of suspended load transported past this strip in
a unit time is equal to ₁₀₀¹ Cudy.

Velocity		Concentration
profile		profile
u		c
h
y	y	a
		ca

Fig. 7.11 Variation of velocity of flow and sediment concentration in a vertical

274 IRRIGATION AND WATER RESOURCES ENGINEERING
Here, C is the volume concentration (expressed as percentage) at an elevation y where the velocity of flow is u. Thus,

qs =	ρs g	za^h Cudy	(7.35)
100

where, q_s is the weight of suspended load transported per unit width per unit time. Since the suspended sediment moves only on top of the bed layer, the lower limit of integration, a, can be considered equal to the thickness of the bed layer, i.e., 2d.
Instead of using the curves of the type shown in Fig. 7.11, one may use a suitable velocity distribution law and the sediment distribution equation, Eq. (7.34). For the estimation of the reference concentration C_a appearing in Eq. (7.34), Einstein (17) assumed that the average concentration of bed load in the bed layer equals the concentration of suspended load at y = 2d. This assumption is based on the fact that there will be continuity in the distribution of suspended load and bed load. Making use of suitable velocity distribution laws, the velocity of the bed

layer was determined as 11.6 u ′ and as such the concentration in the bed layer was obtained as

*

(qB /ρs g)	. Hence, the reference concentration Ca	(in per cent) at y = 2d is given as
(11.6U ^′ )(2d)
	*
		Ca = C2d =	(qB /ρs g)		× 100	(7.36)
	(23.2U^′ )(d)
		*

Equation (7.35) can now be integrated in a suitable manner.
Example 7.8 Prepare a table for the distribution of sediment concentration in the vertical for Example 7.2. Assume fall velocity of the particles as 0.01 m/s.
Solution: From the solution of Example 7.2 and 7.3, q_B = 0.286 N/m/s
and R′ = 0.651 m Using Eq. (7.36)
_{Ca =} ^(qB ^{/ ρ}s ^g) _{× 100} 23.2U_*^′ d

0.286 × 100 ⁼ (2650 × 9.81) × 23.2 × (9.81 × 0.651 × 16. × 10^{− 4} )^{1/ 2} (0.3 × 10⁻³ )

= 5%

C

L h − y

a

O ^Zo

Now

= M

×

P

Ca

y

N

h − a Q

a = 2d = 2 × 0.3 × 10^–3 m

Z =

wo

=

0.01

= 0.4

o

U * k ( 9.81 ×

2.5 × 1.6 × 10 ^−^4 ) ^1/ 2 × 0.4

C

L 2.8 − y

2 × 0.3 × 10^−^3

O0.4

∴

= M

×

P

5.0

y

2.8

− (2

× 0.3 × 10

−3

N

) Q

_0.17F 2.8 − yI ^0.4
∴ C = G J H y K

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275

The variation of C with y can now be computed as shown in the following table:

y (m)	0.1	0.2	0.5	1.0	1.5	2.0	2.5	2.7	2.8
C (%)	0.635	0.474	0.313	0.215	0.161	0.118	0.0728	0.0455	0
C (ppm)	6350	4740	3130	2150	1610	1180	728	455	0