Contents preface (VII) introduction 1—37

Fig. 11.10 Line sketch of canal trough and transitions
The flow rate Q is related to the depth of flow h and velocity of flow v in the transition as follows:

Q = (B + mh) vh

(11.7)

Applying Bernoulli’s equation between the i^th and (i + 1)^th sections (with flume bottom as the datum), one obtains

v

2

v

2

– ∆z + (∆z + h + ∆h ) + ^i

= – ∆z

i+1

+ (∆z

i + 1

+ h + ∆h ) +

i+ 1

+ hL

(11.8)

i

if

i

2g

f

i + 1

2g

i, i + 1

Here, h_f is the depth of flow in the flume and hL_{i , i + 1} is the energy loss between the i^th and

(i + 1)^th sections. Further, ∆z is considered as positive when the transition bed is lower than the flume bed, and the water surface elevation increment ∆h (with respect to the water surface in the flume) is positive when the water surface in the transition is higher than the water surface in the flume (4). Equation (11.8) can, alternatively, be written as

	^Ei + 1 ^= ^Ei ^+ ^∆^zi, i + 1 ^– ^hLi , i + 1	(11.9)
where, ∆zi, i + 1 = ∆zi + 1 – ∆zi
and specific energy	E = ∆z + h + ∆h +	v^2
		f	2g

CROSS-DRAINAGE STRUCTURES

393

Hinds (5) proposed the following equation for estimating the head loss through transitions:

v 2

− v^2

^hL

= Kh

i

i + 1

= Kh ∆ hv

(11.10)

i, i + 1

2g

On substituting Eq. (11.10) into Eq. (11.9) and simplifying, one gets,

v^2

v ^2

∆z

+ h + ∆h

+

i + 1

= ∆z + h + ∆h +

i

+ ∆z

– ∆z – K

∆h

i + 1

2g

2g

h

f

i + 1

i

f

i

i + 1

i

v

∴

∆hi + 1 – ∆hi = ∆hv (1 – Kh)

or

∆hv

=

∆hi + 1 − ∆hi

(11.11)

1

− Kh

Hence, from Eqs. (11.10) and (11.11), one obtains,

^hLi, i + 1 =

Kh (∆hi + 1 − ∆hi )

(11.12)

(1 − Kh )

A good transition design would require proper selection of the transition geometry (or determination of suitable values for B, ∆z, and m of Eqs. (11.4-11.6) which would yield minimum energy loss consistent with the convenience of design and construction.
Following methods of design of transitions have been described here: (i) Hinds’ method (5).
(ii) UPIRI method (6) which is commonly known as Mitra’s method. (iii) Vittal and Chiranjeevi’s method (4).
11.7.1. Hinds’ Method
Hinds (5) assumed a water surface profile,

∆h = f4 (x)

(11.13)

in the transition as a compound curve consisting of two reverse parabolas with an inflexion point in the middle of the transition and which join the water surface at either end of the transition tangentially. The water surface profile equation [Eq. (11.13)] is, therefore, written as

∆h = C x^2	(11.14)
1

Here, C₁ is a coefficient to be determined from the coordinates of the junction of two parabolas, and x is to be measured from the transition end for the respective parabolas. Hinds (5) also assumed a linear rise (for contraction) or drop (for expansion) in the channel bed. Thus,

∆z = C2 x

(11.15)

where, C₂ is a constant.
The transition is now divided into N sub-reaches by cross-sections 1-1, 2-2, etc., and an arbitrary set of values for m, lying between 0 and m₀ (i.e., 0 ≤ m ≤ m₀), is assigned to these sections. Here, m₀ is the side slope of the exit (or approach) channel. Using Eqs. (11.14) and (11.15), one can compute ∆h and ∆z and, hence, the depth of flow (= ∆z + h_f + ∆h) for any cross-section of the transition. Using Eqs. (11.12) and (11.8), one can determine the flow velocity at

394 IRRIGATION AND WATER RESOURCES ENGINEERING
the (i + 1)^th section for known v_i, and substitution of v_{i + 1} in Eq. (11.7) yields the bed width at the (i + 1)^th section. These computations proceed from one end of the transition to the other end. If the resulting transition is not smooth and continuous, the computations are repeated with a new set of arbitrary values for m till a smooth and continuous bed width profile is obtained.