Irrigation Engineering Test 2

Concept:

Kennedy’s Silt Theory: Kennedy’s slit theory is used for designing of unlined canal especially for the canals on alluvial soils. According to this theory, the critical velocity is given as:

\({{\rm{V}}_0} = 0.55{\rm{m}}{{\rm{y}}^{0.64}}\) where V₀ is the critical velocity, y is the depth of the water in the channel and m is critical velocity ratio.

Critical Velocity Ratio (m): The ratio of mean velocity V to the critical velocity V₀ is known as the critical velocity ratio (CVR). It is denoted by m i.e. \({\rm{CVR\;}}\left( {\rm{m}} \right) = \frac{{\rm{V}}}{{{{\rm{V}}_0}}}\).

CVR accounts for the effect of different types of silt grade on critical velocity. Generally, m ranges 1.0 to 1.2 for coarse sand and 0.7 to 1.0 for fine sand.

Note:

Silt Factor (f): Used in Lacey’s Theory to calculate mean velocity.

Rugosity Coefficient (n): Used in Kutter’s formula and Manning’s formula to calculate mean velocity.

Shape and surface factor (C): Used in Chezy’s formula to calculate mean velocity.

Concept:

Head Sluices (or Canal Head Regulator): A head sluices or canal head regulator (CHR) is provided at the head of the off-taking canal, and serves the following functions:

• It controls the entry of silt in the canal.
• It regulates the supply of water entering the canal.
• It prevents the river floods from entering the canal.

Silt Ejectors (or Silt extractors): These are the devices that extract the silt from the canal water after the silted water has traveled a certain distance in the off-take canal. These works are, therefore, constructed on the bed of the canal, and a little distance downstream from the head regulator.

Under-Sluices (or Scouring Sluices): The under sluices are the openings that are located on the same side as the off-taking canal and are fully controlled by gates, provided in the weir wall with their crest at a low level. Their main functions are:

• It helps in controlling the silt entry into the canal.
• It scours the silt deposited on the river bed above the approach channel.
• It passes the low floods without dropping the shutter of the main weir.
• It preserves a clear and defined river channel approaching the regulator.

Divide Walls: Divide wall is a masonry or concrete wall constructed at the right angle to the axis of the weir. Its main objective is to form a still and comparatively less turbulent water pocket in front of the canal head regulator so that the suspended silt can be settled down which then later be cleaned through the scouring sluices from time to time.

∴ The correct combination is 1-C, 2-D, 3-B, 4-A.

Concept:

The process by which a liquid leaks through a porous substance is known as the process of seeping.

According to darcy’s law,

Q = kiA where K = permeability of soil

i = hydraulic gradient

A = Area of percolation

So in seepage, the discharge depends upon the wetted area.

Location of the canal also affects the hydraulic gradient available for seepage. So location of canal also influences the seepage.

The time period for which the canal is having flow, during that time seepage loss will also occur. When there is no flow or the canal is dry, there will be no seepage loss in the canal. So frequency of canal usage is also a major factor in seepage loss.

Characteristics of the soil traversed by the canal system does not influence the seepage in canal.

Concept:

Lacey’s Theory: Lacey’s theory is based on the concept of regime condition of the channel. The regime condition will be satisfied if, a) the channel flows uniformly in unlimited incoherent alluvium of the same character which is transported by the channel, b) the silt grade and silt charge remains constant and c) the discharge remains constant. According to this theory, the velocity is given by,

\({\rm{V}} = {\rm{\;}}{\left( {\frac{{{\rm{Q}}{{\rm{f}}^2}}}{{140}}} \right)^{\frac{1}{6}}}\) where V is the velocity in m/ sec, f is the Lacey’s silt factor and Q is the discharge in m³/sec.

The silt factor is related with average particle size and given by, \({\rm{f}} = 1.76\sqrt {{{\rm{d}}_{{\rm{mm}}}}}\) where d_mm is the average particle size in mm.

Calculation:

Given, average particle size, d_mm = 0.016 cm = 0.16 mm and discharge, Q = 4.5 m³/sec.

∴ Silt factor, \({\rm{f}} = 1.76 \times \sqrt {0.16} = 0.704\).

∴ Velocity, \({\rm{V}} = {\rm{\;}}{\left( {\frac{{4.5\; \times \;{{0.704}^2}}}{{140}}} \right)^{\frac{1}{6}}} = 0.502{\rm{\;m}}/{\rm{sec}}\)

Mistake Point:

Concept:

Manning’s Formula: This formula is widely used for finding out mean stream/ canal velocity which is dependent upon hydraulic mean radius of flow and bed slope of the stream/canal. According to Manning’s formula,

\({\rm{V}} = \frac{1}{{\rm{n}}}{{\rm{R}}^{\frac{2}{3}}}{{\rm{S}}^{\frac{1}{2}}}\) where V is the mean velocity of the canal, R is the hydraulic mean radius of flow, S is the bed slope of the canal and n is the actual rugosity coefficient.

Strickler’s Formula: This formula gives theoretical rugosity coefficient depending upon the grain size of the sediment particles and is given by,

\(n' = \frac{1}{{24}}d_{mean}^{\frac{1}{6}}\) where n’ is the rugosity coefficient pertaining to Strickler’s formula and d_mean is the mean size of the bed sediments in meter.

Calculation:

Given, Flow depth, D = 3.8 m; bed slope, S = 1.5 × 10^-4 and velocity, V = 1.9 m/sec.

Now, for a wide rectangular channel, it is known that hydraulic mean radius (R) is approximately equal to the depth of flow (D).

∴ Hydraulic mean radius, \({\rm{R}} \approx {\rm{D}} = 3.8{\rm{\;m}}\) .

So, from Manning’s formula, \(1.9 = \frac{1}{{\rm{n}}} \times {3.8^{\frac{2}{3}}} \times {\left( {1.5 \times {{10}^{ - 4}}} \right)^{\frac{1}{2}}}\;\;\;\;\therefore n = 0.01569\)  

Also, it is given, mean sediment particle size, d_mean = 1.6 mm = 1.6 × 10^-3 m

So, from Strickler’s formula, \(n' = \frac{1}{{24}} \times {\left( {1.6 \times {{10}^{ - 3}}} \right)^{\frac{1}{6}}} = 0.01425\)

∴ The ratio of rugosity coefficient given by Strickler’s formula to the actual rugosity coefficient = \(\frac{{n'}}{n}\)

∴ \(\frac{{n'}}{n} = \frac{{0.01425}}{{0.01569}} = 0.908\)

Mistake Point:

Strickler’s formula is empirical in nature. So, the value for mean particle size must be substituted in m, otherwise, the constant given in the formula is invalid. So, if in the formula the mean particle size is substituted in mm, the answer will be wrong.

Concept:

Exit Gradient: The exit gradient is the hydraulic gradient at the downstream end of the flow line where percolating water leaves the soil mass and emerges into the free water at the downstream.

Khosla’s Exit Gradient: As per Khosla’s theory the exit gradient G_E is dependent upon floor length and downstream vertical cut-off length and is given by,

\({{\rm{G}}_{\rm{E}}} = \frac{{\rm{H}}}{{\rm{d}}}\frac{1}{{{\rm{\pi }}\sqrt {\rm{\lambda }} }}{\rm{\;\;\;where\;\lambda }} = \frac{{1 + \sqrt {1 + {{\rm{\alpha }}^2}} }}{2}{\rm{\;and\;\alpha }} = \frac{{\rm{b}}}{{\rm{d}}}{\rm{\;}}\)

Where, G_E is the Khosla’s Exit Gradient; H is the upstream water depth; d is the depth of vertical cut-off (or depth of downstream sheet pile) at downstream; b is the weir floor-length between upstream and downstream. 

Calculation:

Given, safe exit gradient, G_E = 1 in 7 = 0.1428.

downstream sheet pile depth = 8 m ⇒ d = 8m.

Total weir floor-length, b = 14 m.

\(\therefore {\rm{\;\alpha \;}} = \frac{{\rm{b}}}{{\rm{d}}}{\rm{\;}} = \frac{{14}}{8}{\rm{\;}} = 1.75{\rm{\;\;\;\;}}\therefore {\rm{\;\lambda }} = {\rm{\;}}\frac{{1 + \sqrt {1 + {{1.75}^2}} }}{2} = 1.5077\)

\(\therefore 0.1428 = \frac{{\rm{H}}}{8} \times \frac{1}{{{\rm{\pi }}\; \times \;\sqrt {1.5077} }}{\rm{\;}} ⇒ {\rm{H}} = 4.41{\rm{\;m}}\)