If the flood discharge flowing in a river is 3600 m3/s, its perimeter as per Lacey’s theory is likely to be
285 m
Calculating River Perimeter using Lacey's Theory
The question asks us to determine the likely wetted perimeter of a river based on a given flood discharge, according to Lacey's theory. Lacey's theory is a well-known method used in irrigation engineering and river morphology to design stable channels and predict their dimensions based on discharge and sediment properties.
Lacey's Formula for Wetted Perimeter
According to Lacey's regime theory, for a channel in regime (stable conditions), the wetted perimeter ($P$) is related to the discharge ($Q$). The formula for the wetted perimeter in metres, when the discharge is in cubic metres per second, is given by:
\(P = 4.75 \sqrt{Q}\)
Where:
- \(P\) is the wetted perimeter in metres (m).
- \(Q\) is the flood discharge in cubic metres per second (m<sup>3</sup>/s).
Applying the Formula
We are given the flood discharge, \(Q = 3600 \text{ m<sup>3</sup>/s}\). We can substitute this value into Lacey's formula to find the wetted perimeter.
Given: \(Q = 3600 \text{ m<sup>3</sup>/s}\)
The formula is:
\(P = 4.75 \sqrt{Q}\)
Substitute the value of \(Q\):
\(P = 4.75 \sqrt{3600}\)
First, calculate the square root of 3600:
\(\sqrt{3600} = 60\)
Now, multiply this by 4.75:
\(P = 4.75 \times 60\)
Performing the multiplication:
\(P = 285 \text{ m}\)
Comparing with Options
The calculated wetted perimeter is 285 m. Let's compare this with the given options:
- Option 1: 360 m
- Option 2: 300 m
- Option 3: 285 m
- Option 4: 270 m
Our calculated value of 285 m matches Option 3 exactly.
| Parameter | Value |
|---|---|
| Flood Discharge (Q) | 3600 m<sup>3</sup>/s |
| Lacey's Formula | \(P = 4.75 \sqrt{Q}\) |
| Calculation | \(P = 4.75 \sqrt{3600} = 4.75 \times 60\) |
| Calculated Perimeter (P) | 285 m |
Therefore, as per Lacey's theory, the likely wetted perimeter for a flood discharge of 3600 m<sup>3</sup>/s is 285 m.
Revision Table: Lacey's Regime Theory
| Parameter | Formula | Units |
|---|---|---|
| Wetted Perimeter (P) | \(P = 4.75 \sqrt{Q}\) | metres (m) |
| Silt Factor (f) | \(f = 1.76 \sqrt{d_{avg}}\) | Dimensionless (where \(d_{avg}\) is in mm) |
| Velocity (V) | \(V = \left(\frac{Q f^2}{140}\right)^{1/6}\) | m/s |
| Hydraulic Radius (R) | \(R = \frac{5}{2} \frac{V^2}{f}\) or \(R = \frac{2}{3} \frac{V^2}{f}\) (Note: Different derivations exist, P=4.75sqrt(Q) often uses R=2/3 V^2/f or derived from P,V,R relationships) | metres (m) |
| Bed Slope (S) | \(S = \frac{f^{5/3}}{3340 Q^{1/6}}\) | Dimensionless |
Additional Information on Lacey's Theory
Lacey's regime theory provides a set of empirical relationships used for the design of stable alluvial channels. A channel is said to be in "regime" when there is neither silting nor scouring, and the flow velocity is just sufficient to transport the sediment load introduced into the channel.
Key assumptions and concepts of Lacey's theory:
- The channel is flowing uniformly in unlimited incoherent alluvium of the same character as the bed load.
- The sediment grade and sediment concentration are constant.
- The discharge is constant.
In reality, achieving a perfect regime channel is often difficult due to variations in discharge, sediment load, and channel material. However, Lacey's theory provides a valuable basis for initial channel design and analysis, particularly for unlined canals carrying silt.
The formula \(P = 4.75 \sqrt{Q}\) relates the wetted perimeter directly to the discharge, assuming the channel is in a regime state. This relationship is fundamental in determining the plan form dimensions of the channel cross-section.
Important Questions from Design of Canals
Upon a detailed topographical investigation, an engineer wants to align a canal. Along which of the following should be align the canal?
Which of the following components of the weir divides the river width into weir portion and under sluices pocket?
Which canal irrigates only on one side because the area on the other side is higher?
In which type of canal escapes is the crest of the weir wall kept at R.L. equal to the canal FSL?
If the discharge in canal equals to 70 m3/s with it silt factor √2, the velocity of flow in canal as per Lacey’s theory is
