Laceys Regime TheoryGerald Lacey -- 1930

Lacey followed Lindleys hypothesis:

dimensions and slope of a channel to carry a given discharge and silt load ineasily erodible soil are uniquely determined by nature.

According to Lacey:

Silt is kept in suspension by the vertical component of eddies generated atall 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 regimeor final regime.

(ii) Initial regime

cross-section or wetted perimeter remains unaffected

bed slope of a channel varies

(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.

Laceys Equations:

Fundamental Equations:

Derived Equations:

(Laceys Non-regime flow equation)

R

VffRV

2

2

5or

5

2

52 140VAf

21

32

8.10 SRV

QP 75.4

61

2

140

QfV

21

23

4980R

fS

61

35

3340Q

fS

21

431

SRN

Va

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'

f

Q

QP 75.4

L

Qq

f

q

,35.1DepthScour Normal sLacey'

31

Laceys Channel Design Procedure

Determine D and B using these

relationships for trapezoidal

section with side slope 1/2:1

Problem:

Design an irrigation channel in alluvial soil from following data using Laceys

theory:

Discharge = 15.0 cumec; Laceys silt factor = 1.0; Side slope = : 1

Solution:

sec/ 689.0)140

115()

140( 6

16

12

mQf

V

2 77.21689.0

15m

V

QA

m 18.4 1575.475.4 QP

m 36.1742.3

)77.21(944.6)4.18(4.18

742.3

944.622

APP

D

m 36.15)36.1(54.185 DPB

m 185.11

)689.0(

2

5

2

5 22

f

VR

52451

)15(3340

)1(

3340 61

35

61

35

Q

fS

Problem:

The slope of an irrigation channel is 0.2 per thousand. Laceys silt factor = 1.0,

channel side slope = : 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:

cumecS

fQ

Q

fS 25.11

500013340

1

33403340

6

35

61

35

mS

fR

R

fS 008.1

500014980

1)

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(93.15

742.3

944.622

APP

D

m 35.13)153.1(593.155 DPB

Problem:

Design an earthen channel of 10 cumec capacity. The value of Laceys 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 622.0

140

9.010

140

61

261

2

QfV

2m 08.16622.0

10

V

QA

m 02.151075.475.4 QP

m 25.1742.3

)08.16(944.6)02.15(02.15

742.3

944.622

APP

D

m 22.12)25.1(502.155 DPB

58441

103340

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.

7454.0108000

1334033403340

53

615

3

61

61

35

SQf

Q

fS

mm 179.076.1

76.12

5050

fDDf

Lacey's Shock Theory

Lacey considered absolute rugosity coefficient Na as;

Constant and

Independent of channel dimensions.

In practice Na varies because of irregularities or mounds in the sides and bed of the channel (i.e. 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 + form resistance (i.e. shock loss)

(due to bed) (due to irregularities)

Lacey suggested:Na should remain constantSlope should be splitted

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

21

431

sSRN

Va

214321430225.0

1

025.0

1sSRSR

Drawbacks in Laceys theory:

The concept of true regime is only theoretical and cannot be

achieved practically.

The various equations are derived by considering the silt

factor only, that is not appropriate for various phases of flow

along different bed and side conditions.

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 Laceys 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. Kutters 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.