4.4.1. Theories about silt in irrigation channels

 Kennedy's theory

This theory states that vertical eddies that rise from the bed of a channel keep silt particles suspended. It also defines a critical velocity that prevents scouring or silting. Velocities above the critical velocity cause scouring, while velocities below cause silting. Kennedy's equation relates the critical velocity to the depth of flow. 

Lacey's theory

This theory states that vertical eddies generated along the wetted perimeter of a channel keep silt in suspension. Lacey's equations and design procedure are based on factors such as discharge, silt factor, and side slopes. Lacey's theory also defines three regime conditions: true, initial, and final. However, he acknowledged that true regime can never be achieved in practice. 

Some differences between Kennedy's and Lacey's theories include:

Critical velocity ratio

Kennedy's theory uses a critical velocity ratio to make the equation valid for different silt grades. Lacey's theory uses a silt factor to make the equation valid for different silt grades.

Equation for calculating mean velocity

Kennedy's theory uses Kutter's Equation to calculate the mean velocity, while Lacey's theory developed his own equation.

Equation for calculating bed slope

Kennedy's theory does not provide an equation for calculating the bed slope, while Lacey's theory developed his own equation.

Unlined canal design

Kennedy's theory does not use trial and error techniques for unlined canal design, while Lacey's theory does. 

Please also Refer : 

https://testbook.com/civil-engineering/canal-design#:~:text=Lacey's%20Theory,-It%20states%20that&text=It%20states%20that%20the%20silt,actual%20mean%20velocity%20and%20depth.

------------------------------------------------------------Sample Canal Design --------------------------------------

Canal Design: Lined and Unlined Canals

A canal can be defined as an artificial waterway which is constructed for the passage of boats or ships inland or for the conveyance of water for the purpose of irrigation. Canal lining refers to an impermeable layer provided to resist the seepage losses from the beds and sides of the canal. Around 60-80% of the water losses can be prevented by providing canal linings.

Canal Design

When designing a canal, we first start from the design of the tail end of the canal and then proceed step by step towards the upstream side till the head regulator of the canal is reached. 

Different IS codes used for canal design are:-

IS:4745 (1968)- Used for unlined canal design.

IS:7112 (1973)- Code of practice for designing and constructing unlined canals.

IS:10430 (2000) Annex A- Used for economic justification of lining.

Unlined Canal Design

The design procedure of unlined canals for alluvial and non-alluvial soils is quite different. For alluvial soils, the unlined canals can be designed with the help of Kennedy or Lacey’s Theory. Whereas for the non-alluvial soils, the coefficient of rugosity plays a vital role. Chezy’s Equation, Manning’s Equation and Kutter’s formula are adopted for unlined canal design of non-alluvial soil.

For Alluvial Soil

• A portion of the sediments, mainly silt, is received by the channels which withdraw water directly from the rivers.
• In the case of an unlined alluvial (silty) channel (canal), the sides may be scoured if the velocity of the flow of water is very high and silting or deposition may occur if the velocity of flow is very less.
• If scouring occurs, the F.S.L. (flow surface level) of the canal reduces, and the friction on the surface increases, thereby causing reduced discharge which ultimately leads to a loss in the command area.
• If silting occurs, the capacity of the canal reduces as its cross-sectional area reduces, and hence it will irrigate lesser areas.
• Thus, alluvial channels must be designed for a velocity which is neither scouring nor silting and results in a stable (regime) channel.
• Such a velocity is known as non-silting and non-scouring velocity, and a channel in which no silting and no scouring occurs is called Regime or Stable Channel (canal).
• The notable contributions are those of R.G. Kennedy (EE, Punjab Irrigation Department) and Gerald Lacey (CE, Uttar Pradesh Public Works Department)
• The theories proposed by them are called Silt Theories. Both these theories are hypothetical, and a direct mathematical solution for the design of unlined canals is not yet established.

Kennedy’s Theory for Alluvial Unlined Canal Design

• Kennedy made observations on the Upper Bari Doab canal, which had not required any maintenance for the past 30 years and thus, they were considered as stable. Doab refers to the area between two water bodies, usually rivers. The doab region is considered to be a very fertile area for crop production.
• Some of the assumptions of Kennedy's theory for alluvial unlined canal design are:-

1. The flow of water is assumed to be steady and uniform.
2. The canal flows in infinite and incoherent alluvial soil.
3. Eddies are generated from all sides of the canal. However, eddies generated from side slopes do not have silt-supporting power.
4. The silt carried by the water is assumed to be the same material by which the canal is made.

• Kennedy introduced a term called critical velocity which is defined as the mean velocity which will just keep the channel free from silting and scouring.
• The critical velocity or non-silting and non-scouring velocity is denoted as Vcr$V_{cr}$

Vcr=0.55D0.64$V_{cr}=0.55D^{0.64}$

Where, D= depth of flow (m)

Vcr$V_{cr}$= critical velocity (m/s)

• Since Kennedy’s studies were based on a given canal system only, to validate his expression of critical velocity throughout India for various types of soils, he introduced a term called critical velocity ratio denoted by ‘m’. The value of ‘m’ accounts for silt grade (type and size of silt).
• The modified form of the critical velocity formula is then written as follows:-

V′cr=0.55mD0.64$V_{cr}^{\prime }=0.55m{D}^{0.64}$

Where, V′cr$V_{cr}^{\prime }$= critical velocity (m/s).

m= critical velocity ratio.

D= depth of flow (m).

m= 1 (for upper bari doab region)

m>1 (for soil which is coarser than upper bari doab region)

m<1 ( for soil which is finer than upper bari doab region)

Alluvial Unlined Canal Design Steps by Kennedy’s Theory:-

• Given design discharge ‘Q’ and critical velocity ratio ‘m’ assume first trial depth (D)

Design Discharge Q (m3/s$m^3/s$)	First Trial depth (D)
0-20	1
20-40	2
40-80	2.5
80-100	3
>100	3.5

1.Find Critical Velocity

V′cr=0.55mD0.64$V_{cr}^{\prime }=0.55m{D}^{0.64}$

2.Find area of cross section

A=QV′cr$A=\frac{Q}{V_{cr}^{\prime }}$

Where, A= area of cross section

Q= design discharge

V′cr$V_{cr}^{\prime }$= critical velocity (m/s).

3.Find width (B) by assuming the section to be trapezoidal and side slope as 0.5H:1V

A=BD+D22$A=BD+\frac{D^2}{2}$

Where, A= cross-sectional area

D= depth (m)

4.Find Wetted Perimeter (P)

P=B+D5–√$P=B+D\sqrt{5}$

Where, B= width of channel (m)

D= depth (m)

5.Find Hydraulic Mean Radius R

R=AP$R=\frac{A}{P}$

Where, A= area of cross-section

P= wetted perimeter

6.Use Chezy’s Equation to compute actual mean velocity (V)

V=CRS−−−√$V=C\sqrt{RS}$

Where C= Chezy’s Constant

R= Hydraulic Mean Radius

S= Bed Slope

The bed slope can be assumed between 1/2000 to 1/5000 for steady and uniform flow.

The Chezy’s Constant can be calculated by following two equations:-

Bazin’s equation- C=871+KR√$C=\frac{87}{1+\frac{K}{\sqrt{R}}}$

Where C= Chezy’s Constant

R= hydraulic mean radius

K= Bazin’s Constant.

Kutter’s Formula-  C=1n+23+0.00155S1+(23+0.00155S)nR√$C=\frac{\frac{1}{n}+23+\frac{0.00155}{S}}{1+(23+\frac{0.00155}{S})\frac{n}{\sqrt{R}}}$

Where, n= coefficient of rugosity

R= hydraulic mean radius

S= bed slope

1. Compare critical velocity V′cr$V_{cr}^{\prime }$ and actual mean velocity V

If V′cr$V_{cr}^{\prime }$ = V then, design is adopted.

If V′cr$V_{cr}^{\prime }$ > V then, reduce D.

If V′cr$V_{cr}^{\prime }$ < V then, increase D.

• Limitations of Alluvial Unlined Canal Design by Kennedy’s Theory-

• According to Kennedy, eddies generated from the base of the channel have the silt supporting power but in actual practice, eddies generated from side slopes also have silt supporting power.
• The value of the critical velocity ratio had been randomly given without providing any mathematical relation.
• Kennedy’s theory does not consider the effect of the concentration of silt particles in the channel.
• Kennedy had not given his own formula for computing actual mean velocity, but instead, he recommended Chezy’s equation. The limitations of Chezy’s equation are also incorporated into Kennedy’s theory.
• Kennedy had not given his own formula for computing bed slope.

Lacey’s Theory for Alluvial Unlined Canal Design​

• Lacey’s regime theory is popularly called Lacey's silt theory.
• Based on his studies, he observed that even if a channel is non-silting and non-scouring temporarily, it may not be in a state of permanent stability.
• Based upon his observations, he classified the channel into two categories- real channel and ideal channel.
• Ideal Channel- For a channel to be considered ideal, the following conditions must be established-

• • Discharge should be constant.
  • The flow in the channel should be steady and uniform.
  • Channel should be in infinite and incoherent alluvial soil.
  • It should be transporting the silt of the same characters as the material of the channel itself.
  • Silt grade (type and size of silt) and silt charge (silt concentration) should be constant.
• The conditions indicated above are called True Regime Conditions, and the channel satisfying the above conditions is called Ideal Channel.
• However, such conditions do not exist in reality. Thus, Lacey indicated that in the case of real or artificial channels, there are two types of regime-

Ideal Regime- It is the first stage of stability attained by a channel after it has been put into service. A newly constructed channel is unstable, and it first adjusts its bed slope and attains temporary stability called initial regime. This regime is temporary because the width and the depth of the channel have not been adjusted so far to attain complete stability.

Final Regime- It is the ultimate state of regime in which all the geometrical elements are suitably adjusted. This stability approaches the true regime, and the equations developed by Lacey are applicable to a channel in either true or final regime.

• The cross-section of an ideal channel, as per Lacey, is semi-elliptical. The channel is wider and shallower for coarse silt, and for finer silt, the channel is deeper and narrower for constant discharge.
• Lacey did not develop any equation for a semi-elliptical cross-section. So, for analysis purposes, the cross-section is assumed to be trapezoidal with a 0.5H:1V side slope.

• Lacey’s Equations- 

• V=25fR−−−−√$V=\sqrt{\frac{2}{5}fR}$ or 

V=10.8R2/3S1/3$V=10.8R^{2/3}{S}^{1/3}$

Where, V= mean velocity in m/s

f= silt factor

S= bed slope

f=1.76dmm−−−−√$f=1.76\sqrt{d_{mm}}$

dmm$d_{mm}$= mean particle size in mm

R= hydraulic mean radius

• V=(Qf2140)1/6$V=(\frac{Q{f}^2}{140}{)}^{1/6}$

Where, Q= discharge in m3/s$m^3/s$

• P=4.75Q−−√$P=4.75\sqrt{Q}$

Where, P= wetted perimeter

• S=f5/33340Q1/6$S=\frac{f^{5/3}}{3340Q^{1/6}}$

Where, S= bed slope

• R=52V2f$R=\frac{5}{2}\frac{V^2}{f}$
• Lacey′s Scour depth Rs=1.35(q2f)1/3$Lace{y}^{\prime }sScourdepth{R}_s=1.35(\frac{q^2}{f}{)}^{1/3}$
• Where q= discharge intensity m3/s/m$m^3/s/m$

• Alluvial Unlined Canal Design Steps by Lacey’s Theory:-

• Given Discharge (Q) and Silt factor (f) find the actual velocity (V) from the above equations
• Then find the cross-sectional area (A), which can be calculated as

A= Q/V

• Then find the wetted perimeter (P), which can be calculated from the above equations.
•  Then find the width (B) and Depth (D) by the below two equations

A=BD+D22$A=BD+\frac{D^2}{2}$

P=B+D5–√$P=B+D\sqrt{5}$

• At last, find the bed slope (S) from the above equations.
• Limitations of Alluvial Unlined Canal Design by Lacey’s Theory-
  • The true regime condition defined by Lacey is just theoretical and cannot be achieved in reality.
  • Lacey indicated that a true regime channel is semi-elliptical, but it is not reflected in any of his equations.
  • Lacey did not define the time required by a real channel to attain the final regime.
  • Lacey’s equations do not consider the silt concentration.

For Non-Alluvial Soil

The non-alluvial soils are almost impervious and stable. The coefficient of rugosity plays a vital role in the unlined canal design for non-alluvial soils. In this case, the flow velocity is very much close to the critical velocity. Chezy’s Equation, Manning’s Equation and Kutter’s formula are adopted for unlined canal design of non-alluvial soil. These equations are as follows:-

• Chezy’s Equation-

V=CRS−−−√$V=C\sqrt{RS}$

Where, V= mean velocity in m/s

C= Chezy’s Constant

R= Hydraulic Mean Radius

S= Bed Slope

The Chezy’s Constant can be calculated by following two equations:-

Bazin’s equation- 

C=871+KR√$C=\frac{87}{1+\frac{K}{\sqrt{R}}}$

Where C= Chezy’s Constant

R= hydraulic mean radius

K= Bazin’s Constant.

Kutter’s Formula-  

C=1n+23+0.00155S1+(23+0.00155S)nR√$C=\frac{\frac{1}{n}+23+\frac{0.00155}{S}}{1+(23+\frac{0.00155}{S})\frac{n}{\sqrt{R}}}$

Where n= coefficient of rugosity

R= hydraulic mean radius

S= bed slope

• Manning’s Equation-

V=1nR2/3S1/2$V=\frac{1}{n}{R}^{2/3}{S}^{1/2}$

• Discharge (Q) can be calculated as Q=A×V$Q=A\times V$

Where A= cross sectional area

• If the value of ‘K’ is not provided, then it can be assumed as:-

K= 1.3 to 1.75 for unlined canals

K= 0.45 to 0.85 for lined canals

• If the value of ‘n’ is not provided, then it can be assumed as:-

n= 0.0225 for unlined canals

n= 0.333 for lined canals

Difference between Kennedy’s and Lacey’s Theory

Some main differences between Kennedy’s theory and Lacey’s theory are mentioned below. 

Kennedy’s Theory	Lacey’s Theory
It states that the silt carried by the flowing water is kept in suspension by the vertical component of the eddies generated from the bed of the channel.	It states that the silt carried by the flowing water is kept in suspension by the vertical component of the eddies generated from the entire wetted perimeter of the channel.
It gives the relation between the actual mean velocity and depth.	It gives the relation between actual mean velocity and hydraulic mean radius.
Critical velocity ratio ‘m’ is used in Kennedy's theory to make the equation valid for different channels having different silt grades.	Silt factor ‘f’ is used in Lacey's theory to make the equation valid for different channels having different silt grades.
Kutter’s Equation is used in Kennedy’s theory to calculate the mean velocity.	Lacey developed his own equation to calculate the mean velocity.
Kennedy gave no equation for calculating the bed slope.	Lacey developed his own equation for calculating the bed slope.
The unlined canal design, according to this theory, can be done by trial and error method.	This theory does not use trial and error techniques for unlined canal design.

Lined Canal Design

Lined canals have rigid beds and sides. So the lined canal design is not done by Kennedy or Lacey’s theory. Manning’s Equation is adopted for lined canal design. Some of the design considerations are as follows:-

1. The section of the channel should be economical.
2. The velocity of the flowing water should be maximum so that the cross-sectional area can be minimised
3. The capacity of the lined section is not reduced by silting.

The following sections are adopted for the lined canal design:-

Circular Section- The bed is circular with full supply depth as its radius. The side slope is taken as 1:1

Fig.1: Circular section

Design parameters for circular section:-

Design Parameter	Side Slope
	1:1	1.5:1	1.25:1
Sectional Area (A)	1.785×D2$1.785\times D^2$	2.088×D2$2.088\times D^2$	1.925×D2$1.925\times D^2$
Wetted Perimeter (P)	3.57×D$3.57\times D$	4.176×D$4.176\times D$	3.85×D$3.85\times D$
Hydraulic mean radius (R)	0.5×D$0.5\times D$	0.5×D$0.5\times D$	0.5×D$0.5\times D$
Velocity (V)	V=1nR2/3S1/2$V=\frac{1}{n}{R}^{2/3}{S}^{1/2}$	-	-
Discharge (Q)	A×V$A\times V$	-	-

Trapezoidal Section- The side slope is kept as 1:1. The horizontal bed is connected to the side slope by a curve of radius equal to the full supply depth D.

Fig.2: Trapezoidal Section

Design parameters for trapezoidal section:-

Design Parameter	Side Slope
	1:1	1.5:1	1.25:1
Sectional Area (A)	BD+1.785×D2$BD+1.785\times D^2$	BD+2.088×D2$BD+2.088\times D^2$	BD+1.925×D2$BD+1.925\times D^2$
Wetted Perimeter (P)	B+3.57×D$B+3.57\times D$	B+4.176×D$B+4.176\times D$	B+3.85×D$B+3.85\times D$
Hydraulic mean radius (R)	A/P	A/P	A/P
Velocity (V)	V=1nR2/3S1/2$V=\frac{1}{n}{R}^{2/3}{S}^{1/2}$	-	-
Discharge (Q)	A×V$A\times V$	-	-

Note:- Circular section is adopted for discharge up to 50 cumecs and trapezoidal section is preferred for discharge above 50 cumecs.

Difference between Unlined and Lined Canals

Some important differences between unlined and lined canals are mentioned below. 

Unlined Canal	Lined canal
Erosion of the bed and sides of the canal is more.	Erosion of the bed and sides of the canal is less due to lining the canal.
Low construction cost.	High construction cost.
High maintenance cost.	Low maintenance cost.
Seepage losses are more in the case of unlined canals.	Seepage losses are minimal in the case of lined canals.
Friction to the flowing water is more as the bed and sides of the unlined canals are rough.	Friction to the flowing water is less as the bed and sides of the lined canals are smooth.
Head loss of the flowing water is more.	Head loss of the flowing water is less.
Discharge is less for constant area	Discharge is more for constant area.