Lacey’s Regime TheoryGerald Lacey -- 1930

Lacey followed Lindley’s hypothesis:

“dimensions and slope of a channel to carry a given discharge and silt load in easily erodable soil are uniquily determined by nature”.

According to Lacey:

“Silt is kept in suspension by the vertical component of eddies generated at all points of forces normal to the wetted perimeter”.

Regime Channel

“A channel is said to in regime, if there is neither silting nor scouring”.

According to Lacey there may be three regime conditions:

(i) True regime;

(ii) Initial regime; and

(iii) Final regime.CE8603 – IRRIGATION ENGINEERING/V.PRIYA/AP/CIVIL

(1)True regime

A channel shall be in 'true regime' if the following conditions are satisfied:

(i) Discharge is constant;

(ii) Flow is uniform;

(iii) Silt charge is constant; i.e. the amount of silt is constant;

(iv) Silt grade is constant; i.e., the type and size of silt is always the same; and

(v)Channel is flowing through a material which can be scoured as easily as it

can be deposited (such soil is known as incoherent alluvium), and is of the

same grade as is transported.

But in practice, all these conditions can never be satisfied. And, therefore,artificial channels can never be in 'true regime’; they can either be in initial regimeor final regime.

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(ii) Initial regime

bed slope of a channel varies

cross-section or wetted perimeter remains unaffected

(iii) Final regime

all the variables such as perimeter, depth, slope, etc. are equally free to

vary and achieve permanent stability, called Final Regime.

In such a channel,

The coarser the silt, the flatter is the semi-ellipse.

The finer the silt, the more nearly the section attains a semi-circle.

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Lacey’s Equations:

Fundamental Equations:

(Lacey’s Non-regime flow equation)

2 5 V2

5 2 RfR or f V

Af 2 140V5

2 1

V 10.8R 3 S 2

Derived Equations:

P 4.75 Q

61

Qf 2 V 140

21

23

4980R

fS

61

35

3340Q

fS

3 14 2

1

Na

V R S

D50 is average grain/particle size in mm

50where f 1.76 D ,

The equations for determination of Velocity, Slope,

etc are function of the silt factor, whereas silt factor

is function of sediment size.

For upper Indus basin,

For Sindh plain,

f = 0.8 to 1.3

f = 0.7 to 0.8

where

q = discharge intensity, and

L = actual river width at the given site

Lacey's Normal Regime Scour Depth 0.473 f

Q 1 3

The above scour depth will be applicable if river width follows the

relationship P 4.75 Q

For other values of active river width,Qq L

Lacey's Normal Scour Depth 1.35 ,f

q 1 3

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Lacey’s Channel Design Procedure

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Problem:

Design an irrigation channel in alluvial soil from following data using Lacey’s

theory:

Discharge = 15.0 cumec; Lacey’s silt factor = 1.0; Side slope = ½ : 1

Solution:

140 140

1 1Qf 2 151) 6 ( ) 6 0.689 m / secV (

15

V 0.689 21.77 m2A

Q

P 4.75 Q 4.75 15 18.4m

3.7423.742 1.36 m

(18.4)2 6.944(21.77)18.4

P P2 6.944AD

B P 5D 18.4 5(1.36) 15.36 m

1.185 m5 V 2 5 (0.689)2

2 f 2 1R

5245

(1)5 3

11 1

3340Q 6 3340 (15) 6

5

f 3

S

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Problem:

The slope of an irrigation channel is 0.2 per thousand. Lacey’s silt factor =

1.0, channel side slope = ½ : 1. Find the full supply discharge and dimensions

of the channel.

Data:

S = 0.2/1000 = (0.2 x 5) / (1000 x 5) = 1/5000

Solution:

f f 135

1

3340Q 6

35

5000

6

3340S 3340 1

Q 11.25cumecS

ff 1

4980S

23

1

4980R 2

23

1.008m

5000

2

4980 1)2 R (S

P 4.75 Q 4.75 11.25 15.93m

A PR 15.931.008 16.06m2

3.742 1.153 m

P P2 6.944AD

15.93 (15.93)2 6.944(16.06)

3.742

5(1.153) 13.35 m5D 15.93B P

Problem:

Design an earthen channel of 10 cumec capacity. The value of Lacey’s silt

factor in the neighboring canal system is 0.9. General grade of the country is

1 in 8000.

Data:

Q = 10 cumec; f = 0.9; Sn=1/8000; B = ?; D = ?; Sreq= ?.

Solution:

Which is steeper than the natural grade of the country (i.e. 1 in 8000),

therefore not feasible.

140 0.622 m/sec

12

1

Qf 2 6 100.9 6

V 140

10 16.08 m2

V 0.622A

Q

P 4.75 Q 4.75 10 15.02 m

3.742 1.25 m

P P2 6.944A 15.02 (15.02)2 6.944(16.08)D

3.742

5(1.25) 12.22 m5D 15.02B P

334010

0.953 1

16 58441

63340Q

5

f 3

Sreq

Now putting S = 1/8000 in the relationship

Hence silt factor will be reduced to 0.7454 by not allowing coarser silt to enter the

canal system by providing silt ejectors and silt excluders.

i.e. silt having mean diameter > 0.179 mm will not be allowed to enter the canal

system.

11

16 5 3

0.74548000 103340

36

1 5

3340Q 6

5

f 3

S f 3340SQ

2

ff 1.76 D50 D50 1.76 0.179 mm

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Lacey's Shock Theory

Lacey considered absolute rugosity coefficient Na as;

Constant and

Independent of channel dimensions.

the

In practice Na varies because;

V-S and y-f relationships are logarithmic,

Due to irregularities or mounds in the sides and bed of the channel (ripples), pressure on front is more than the pressure on

rear.

The resistance to flow due to this difference of pressure on the two sides of the mound is called form resistance.

Lacey termed this loss as shock loss, which is different from frictional resistance or tangential drag.

Shock loss = f (size, shape and spacing of bed forms)

Total resistance = frictional resistance + shock loss

(due to bed) (due to irregularities)

Lacey suggested:

Na should remain constant

Slope should be splited

to overcome friction and

to meet shock lossi.e.

According to Lacey

Na = 0.025 with shock loss

Na = 0.0225 without shock loss

i.e. for a channel in good condition

19 % slope for shock lossand 81 % slope for friction

Na

where, s = slope required to withstand shock losses.

V 1

R 3

4 S s 12

0.025 0.0225

Therefore, s = 0.19 S

1 1R3 4 S s1 2R3 4S1 2

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Drawbacks in Lacey’s theory:

The concept of true regime is only theoretical and cannot be

achieved practically.

The various equations are derived by considering the silt

factor of which is not at all constant.

The concentration of silt is not taken into account.

The silt grade and silt charge are not clearly defined.

The equations are empirical and based on the available data

from a particular type of channel.

The characteristics of regime of channel may not be same for

all cases.CE8603 – IRRIGATION ENGINEERING/V.PRIYA/AP/CIVIL

Kennedy theory Lacey’s theory

1.It states that the silt carried by the flowingwater is kept in suspension by the verticalcomponent of eddies which are generatedfrom the bed of the channel.

1.It states that the silt carried by the flowing water is kept in suspension by the vertical component of eddies which are generated from the entire wetted perimeter of the channel.

2. Relation between ‘V’ & ‘D’. 2. Relation between ‘V’ & ‘R’.

3. Critical velocity ratio ‘m’ is introduced tomake the equation applicable to diff.channels with diff. silt grades.

3. Silt factor ‘f’ is introduced to make theequation applicable to diff. channels withdiff. silt grades.

4. Kutter’s equation is used for finding themean velocity.

4. This theory gives an equation for finding the mean velocity.

5. This theory gives no equation for bedslope.

5. This theory gives an equation for bed slope.

6.In this theory, the design is based on trialand error method.

6. This theory does not involve trial and error method.

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