Lacey’s Regime Lacey’s Regime TheoryTheoryGerald 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.

(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 regime or final regime.

(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.

Lacey’s Equations:Lacey’s Equations:Fundamental Equations:

Derived Equations:

(Lacey’s Non-regime flow equation)

RVffRV

2

25or

52

52 140VAf 2

13

28.10 SRV

QP 75.4

61

2

140

QfV

21

23

4980R

fS

61

35

3340Q

fS

21

431 SR

NV

a

mmin size iclegrain/part average is D

,76.1 where

50

50Df

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, f = 0.8 to 1.3

For Sindh plain, f = 0.7 to 0.8

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

relationship

For other values of active river width,

where

q = discharge intensity, and

L = actual river width at the given site

31

473.0DepthScour Regime Normal sLacey'

fQ

QP 75.4

LQq

fq

,35.1DepthScour Normal sLacey'

31

Lacey’s Channel Design Lacey’s Channel Design ProcedureProcedure

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 : 1Solution:

sec/ 689.0)140

115()140

( 61

612

mQfV

2 77.21689.015 m

VQA

m 18.4 1575.475.4 QP

m 36.1742.3

)77.21(944.6)4.18(4.18742.3

944.6 22

APPD

m 36.15)36.1(54.185 DPB

m 185.11

)689.0(25

25 22

fVR

52451

)15(3340

)1(

3340 61

35

61

35

Q

fS

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) = 1/5000Solution:

cumecS

fQQ

fS 25.115000

133401

33403340

63

5

61

35

mS

fRR

fS 008.15000

149801)

4980(

4980

2

22

3

21

23

mQP 93.1525.1175.475.4

206.16008.193.15 mPRA

m 153.1742.3

)06.16(944.6)93.15(93.15742.3

944.6 22

APPD

m 35.13)153.1(593.155 DPB

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.

m/sec 622.0140

9.010140

61

261

2

QfV

2m 08.16622.010

VQA

m 02.151075.475.4 QP

m 25.1742.3

)08.16(944.6)02.15(02.15742.3

944.6 22

APPD

m 22.12)25.1(502.155 DPB

5844

1103340

9.0

3340 61

35

61

35

Q

fSreq

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.

7454.0108000133403340

3340

53

615

36

1

61

35

SQf

Q

fS

mm 179.076.176.12

5050

fDDf

Lacey's Shock Lacey's Shock TheoryTheoryLacey considered absolute rugosity coefficient Na as;

Constant andIndependent of channel dimensions.

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 the 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 constantSlope should be splited

to overcome friction andto meet shock loss

i.e.

where, s = slope required to withstand shock losses.

According to LaceyNa = 0.025 with shock lossNa = 0.0225 without shock loss

Therefore, s = 0.19 S

i.e. for a channel in good condition19 % slope for shock loss

and 81 % slope for friction

214

31 sSRN

Va

214321430225.01

025.01

sSRSR

Drawbacks in Lacey’s theory: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.

Kennedy theory Lacey’s theory

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 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 to make the equation applicable to diff. channels with diff. silt grades.

3. Silt factor ‘f’ is introduced to make the equation applicable to diff. channels with diff. silt grades.

4. Kutter’s equation is used for finding the mean velocity.

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

5. This theory gives no equation for bed slope.

5. This theory gives an equation for bed slope.

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

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