## Lacey's and Kennedy's Theory for Canal Design

**Canal Design in General**

**Design Parameters**

- The design considerations naturally vary according to the type of soil.
- The velocity of flow in the canal should be critical.
- Design of canals which are known as ‘Kennedy’s theory’ and ‘Lacey’s theory’ are based on the characteristics of sediment load (i.e. silt) in canal water.

**Important Terms Related to Canal Design**

**Alluvial soil:**The soil which is formed by the continuous deposition of silt is known as alluvial soil.

**Non-alluvial soil:**The soil which is formed by the disintegration of rock formations is known as non-alluvial soil.

**Silt factor:**During the investigations works in various canals in alluvial soil, Gerald Lacey established the effect of silt on the determination of discharge and the canal section. So, Lacey introduced a factor which is known as ‘silt factor’. It depends on the mean particle size of silt. It is denoted by ‘f’. The silt factor is determined by the expression,

where, d_{mm} = mean particle size of silt in mm

**Co-efficient of rugosity:**The roughness of the canal bed affects the velocity of flow. The roughness is caused due to the ripples formed on the bed of the canal. So, a coefficient was introduced by R.G Kennedy for calculating the mean velocity of flow. This coefficient is known as the coefficient of rugosity and it is denoted by ‘n’. The value of ‘n’ depends on the type of bed materials of the canal.**Mean velocity:**It is found by observations that the velocity at a depth 0.6D from surface represents the mean velocity (V), where ‘D’ is the depth of water in the canal or river.

**Critical velocity:**When the velocity of flow is such that there is no silting or scouring action in the canal bed, then that velocity is known as critical velocity. It is denoted by ‘Vo ’. The value of Vo was given by Kennedy according to the following expression,

**V _{o} = C.D^{n}**

Where V_{o} = Critical velocity

D = Depth of channel

C & n = Constants

He found the values of C & n are __0.55__ and __0.64__

Therefore **V _{o} = 0.55 × D^{0.64}**

Later he found that critical velocity ratio has huge impact on Critical velocity and he incroporated some changes in above equation.

**Critical velocity ratio (c.v.r), m:**The ratio of**mean velocity**‘V’ to the**critical velocity**‘Vo ’ is known as critical velocity ratio (CVR). It is denoted by m i.e.

**CVR (m) = V/Vo **

When m = 1, there will be no silting or scouring.

When m > 1, **scouring** will occur

When m < 1, **silting** will occur

So, by finding the value of m, the condition of the canal can be predicted whether it will have silting or scouring

**Regime channel:**When the character of the bed and bank materials of the channel are same as that of the transported materials and when the silt charge and silt grade are constant, then the channel is said to be in its regime and the channel is called regime channel. This ideal condition is not practically possible.

**Hydraulic mean depth:**The ratio of the cross-sectional area of flow to the wetted perimeter of the channel is known as hydraulic mean depth or radius. It is generally denoted by R.

**R = A/P**

Where,

A = Cross-sectional area

P = Wetted perimeter

**Unlined Canal Design on ***Non-alluvial* soil

*Non-alluvial*soil

The non-alluvial soils are stable and nearly impervious. For the design of canal in this type of soil, the coefficient of rugosity plays an important role, but the other factor like silt factor has no role. Here, the velocity of flow is considered very close to critical velocity. So, the mean velocity given by Chezy’s expression or Manning’s expression is considered for the design of the canal in this soil. The following formulae are adopted for the design.

**Note: **

- If value of K is not given, then it may be assumed as follows,

For unlined channel, K = 1.30 to 1.75.

For line channel, K = 0.45 to 0.85

- If the value of N is not given, then it may be assumed as follows,

For unlined channel, N = 0.0225

For lined channel, N = 0.333

**Example 1: **

**Unlined Canal Design on Alluvial soil by Kennedy’s Theory **

R.G Kennedy arrived at a theory which states that the silt carried by flowing water in a channel is kept in suspension by the vertical component of eddy current which is formed over the entire bed width of the channel and the suspended silt rises up gently towards the surface.

The following **assumptions** are made in support of his theory:

- The eddy current is developed due to the roughness of the bed.
- The quality of the suspended silt is proportional to bed width.
- It is applicable to those channels which are flowing through the bed consisting of sandy silt or same grade of silt.

He established the idea of critical velocity ‘Vo ’ which will make a channel free from silting or scouring. From, long observations, he established a relationship between the critical velocity and the full supply depth as follows,

**V _{o} = C D^{n}**

The values of C and n where found out as 0.546 and 0.64 respectively, thus

V_{o} = 0.546 D^{0.64}

Again, the realized that the critical velocity was affected by the grade of silt. So, he introduced another factor (m) which is known as critical velocity ratio (C.V.R).

V_{o} = 0.546 m D^{0.64}

**Drawbacks of Kennedy’s Theory**

- The theory is limited to average regime channel only.
- The design of the channel is based on the trial and error method.
- The value of m was fixed arbitrarily.
- Silt charge and silt grade are not considered.
- There is no equation for determining the bed slope and it depends on Kutter’s equation only.
- The ratio of ‘B’ to ‘D’ has no significance in his theory.

**Design Procedure**

- Critical velocity, V
_{o}=0.546 m D^{0.64}. - Mean velocity, V = C (RS)
^{1/2}

Where,

m = critical velocity ratio,

D = full supply depth in m,

R = hydraulic mean depth of radius in m,

S = bed slope as 1 in ‘n’.

The value of ‘C’ is calculated by Kutter’s formula

Where,

n = rugosity coefficient which is taken as an unlined earthen channel.

3. B/D ratio is assumed between 3.5 to 12.

4. Discharge, Q = A*V

Where,

A = Cross-section area in m^{2} ,

V = mean velocity in m/sec

5. The full supply depth is fixed by trial to satisfy the value of ‘m’. Generally, the trial depth is assumed between 1 m to 2 m. If the condition is not satisfied within this limit, then it may be assumed accordingly.

**Example 2:**

**.**

**Try Yourself**

**Unlined Canal Design on ***Alluvial soil* by Lacey’s Theory

*Alluvial soil*by Lacey’s Theory

Lacey’s theory is based on the concept of regime condition of the channel. The regime condition will be satisfied if,

- The channel flows uniformly in unlimited incoherent alluvium of the same character which is transported by the channel.
- The silt grade and silt charge remains constant.
- The discharge remains constant.

In his theory, he states that the silt carried by the flowing water is kept in suspension by the vertical component of eddies. The eddies are generated at all the points on the wetted perimeter of the channel section. Again, he assumed the hydraulic mean radius R, as the variable factor and he recognized the importance of silt grade for which in introduced a factor which is known as silt factor ‘f’.

Thus, he deduced the velocity as;

V = (2/5f R)^{0.5}

Where,

V = mean velocity in m/sec,

f = silt factor,

R = hydraulic mean radius in meter

Then he deduced the relationship between A, V, Q, P, S and f are as follows:

**Example 3:**

**Drawbacks of Lacey’s Theory **

- The concept of the true regime is theoretical and can not be achieved practically.
- The various equations are derived by considering the silt factor f which is not at all constant.
- The concentration of silt is not taken into account.
- Silt grade and silt charge is not taken into account.
- The equations are empirical and based on the available data from a particular type of channel. So, it may not be true for a different type of channel.
- The characteristics of regime channel may not be the same for all cases.

**Comparison between Kennedy’s and Lacey’s theory**

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 eddies which are 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 eddies which are generated from the entire wetted perimeter of the channel. |

It gives relation between ‘V’ and ‘D’. | It gives relation between ‘V’ and ‘R’. |

In this theory, a factor known as critical velocity ratio ‘m’ is introduced to make the equation applicable to different channels with different silt grades | In this theory, a factor known as silt factor ‘f’ is introduced to make the equation applicable to different channels with different silt grades. |

In this theory, Kutter’s equation is used for finding the mean velocity. | This theory gives an equation for finding the mean velocity. |

This theory gives no equation for bed slope. | This theory gives an equation for bed slope. |

In this theory, the design is based on trial and error method. | This theory does not involve trial and error method. |

**Design of Lined Canal**

The lined canals are not designed by the use of Lacey’s and Kennedy’s theory, because the section of the canal is rigid. Manning’s equation is used for designing. The design considerations are,

- The section should be economical (i.e. cross-sectional area should be maximum with minimum wetted perimeter).
- The velocity should be maximum so that the cross-sectional area becomes minimum.
- The capacity of the lined section is not reduced by silting.

**Section of Lined Canal:**

The following two lined sections are generally adopted:

**Circular section:**The bed is circular with its center at the full supply level and radius equal to full supply depth ‘D’. The sides are tangential to the curve. However, the side slope is generally taken as 1:1.

**Trapezoidal section:**The horizontal bed is joined to the side slope by a curve of radius equal to full supply depth D. The side slope is generally kept as 1:1.

Note: For the discharge up to 50 cumec, the circular section is suitable and for the discharge above 50 cumec trapezoidal section is suitable.

**Example 4: **

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