Calculating Canal Velocity using Lacey's Theory

This question asks us to determine the velocity of flow in a canal based on Lacey's theory, given the discharge and the silt factor. Lacey's theory is a prominent method for designing stable alluvial channels in regime conditions.

Understanding Lacey's Theory Parameters

Lacey's theory provides empirical formulas for designing irrigation canals flowing through alluvial soil. It defines a "regime" condition where the channel dimensions are stable and the sediment transport is in equilibrium. Key parameters in Lacey's theory include:

• Discharge (Q): The volume of water flowing per unit time ($m^3/s$).
• Silt factor (f): A dimensionless parameter representing the characteristics of the silt or sediment in the channel. It is related to the average particle size ($d_{avg}$ in mm) by the formula $f = 1.76 \sqrt{d_{avg}}$.
• Velocity (v): The average speed of water flow ($m/s$).
• Wetted Perimeter (P): The length of the channel boundary in contact with the flowing water ($m$).
• Hydraulic Radius (R): The ratio of the cross-sectional area to the wetted perimeter ($m$).

Lacey's Formula for Velocity

Lacey's theory provides several formulas relating these parameters. The formula relevant to finding the velocity (v) when discharge (Q) and silt factor (f) are known is:

$\qquad v = \left(\frac{Qf^2}{140}\right)^{1/6}$

where:

• $v$ is the velocity of flow in m/s
• $Q$ is the discharge in m$^3$/s
• $f$ is the silt factor

Step-by-Step Calculation

We are given:

• Discharge, $Q = 70 \, m^3/s$
• Silt factor, $f = \sqrt{2}$

Now, we substitute these values into Lacey's velocity formula:

$\qquad v = \left(\frac{Qf^2}{140}\right)^{1/6}$

First, calculate $f^2$:

$\qquad f^2 = (\sqrt{2})^2 = 2$

Now, substitute Q and $f^2$ into the velocity formula:

$\qquad v = \left(\frac{70 \times 2}{140}\right)^{1/6}$

Simplify the expression inside the parentheses:

$\qquad v = \left(\frac{140}{140}\right)^{1/6}$

$\qquad v = (1)^{1/6}$

Any root of 1 is 1. Therefore:

$\qquad v = 1 \, m/s$

The calculated velocity of flow in the canal, according to Lacey's theory, is 1.0 m/s.

Comparing with Options

Let's compare our calculated velocity with the given options:

Our calculated velocity is 1.0 m/s, which matches Option 3.

Revision Table: Key Lacey's Theory Formulas

Additional Information on Lacey's Theory

Lacey's theory assumes that the channel is flowing through uniform, incoherent, alluvial material and has achieved a state of equilibrium (regime). This theory is widely used in irrigation engineering for the design of stable channels, particularly in the Indian subcontinent.

• Regime Conditions: Lacey defined three regime states: initial, permanent, and final. The final regime is when the channel dimensions, slope, and sediment transport reach a stable state for a given discharge and sediment load.
• Limitations: Lacey's theory is empirical and based on observations from specific canal systems. It may not be directly applicable to channels with significantly different sediment characteristics, cohesive soils, or non-alluvial beds.
• Importance: Despite its limitations, Lacey's theory provides a foundational approach to understanding and designing alluvial channels and remains a valuable tool for engineers.