4.4. Kennedy‘s and Lacey‘s Regime theory

1. Introduction to Canal Design

In irrigation engineering, designing unlined canals requires careful consideration of factors like flow velocity, channel dimensions, and sediment transport to prevent erosion and siltation. Two widely used theories in designing unlined canals are Kennedy’s Regime Theory and Lacey’s Regime Theory. These theories provide empirical guidelines to design canals that maintain a stable regime, ensuring efficient water flow without excessive erosion or deposition.

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2. Kennedy’s Regime Theory

A. Overview:

• Developed by R.G. Kennedy in 1895, Kennedy’s Regime Theory focuses on the critical velocity necessary to transport sediment without causing erosion or deposition.
• Based on observations of the Upper Bari Doab Canal in Punjab, India.
• Assumes that sediment transport and stability depend on the velocity of water and channel characteristics.

B. Key Concepts:

1. Critical Velocity (V₀):

   • The velocity at which the canal is just able to transport the sediment load without causing erosion or deposition.
   • Critical velocity is dependent on the depth of flow and the type of sediment.

2. Critical Velocity Ratio (m):

   • A ratio defined to consider the influence of silt grade on velocity.
   • $m = \frac{V}{V_0}$, where $V$ is the actual velocity, and $V_0$ is the critical velocity.

3. Silt Factor (f):

   • A parameter representing the size and gradation of silt particles.
   • $f = 1.76 \times \sqrt{d}$, where $d$ is the mean diameter of silt particles in mm.

C. Kennedy’s Formula:

The critical velocity (V₀) for a stable regime is given by:

$$V_0 = 0.55 \times D^{0.64}$$

Where:

• $V_0$ = Critical velocity (m/s)
• $D$ = Depth of flow (m)

D. Design Steps:

1. Determine Discharge (Q):
   Start with the known discharge requirement for the canal.

2. Assume a Trial Depth (D):
   Based on experience or initial calculations.

3. Calculate Critical Velocity (V₀):
   Use Kennedy’s formula to find the critical velocity based on the assumed depth.

4. Determine Actual Velocity (V):
   Calculate using the continuity equation:

   $$V = \frac{Q}{A}$$

   Where $A$ is the cross-sectional area of the flow.

5. Check for Stability:

   • Ensure $V$ ≈ $V_0$ for a stable regime.
   • Adjust depth (D) or width (B) if necessary.

E. Example Calculation Using Kennedy’s Theory:

Design an unlined canal with the following parameters:

• Discharge (Q): 50 m³/s
• Silt Factor (f): 1.0

Steps:

1. Assume a Trial Depth (D):
   Start with $D=2$ m.

2. Calculate Critical Velocity (V₀):

   $$V_0 = 0.55 \times D^{0.64} = 0.55 \times 2^{0.64} \approx 0.88 \, \text{m/s}$$

3. Calculate Cross-sectional Area (A):
   Assume a rectangular section for simplicity:

   $$A=B\times D$$

   Assume initial bottom width $B = 10$ m (trial).

4. Calculate Velocity (V):

   $$A = B \times D = 10 \times 2 = 20 \, \text{m}^2$$ $$V = \frac{Q}{A} = \frac{50}{20} = 2.5 \, \text{m/s}$$

5. Check for Stability:

   • Compare actual velocity (2.5 m/s) with critical velocity (0.88 m/s).
   • Actual velocity is higher; increase depth or width to reduce $V$.

6. Adjust and Recalculate:
   Increase depth to 3 m, recalculate, and repeat until $V \approx V_0$.

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3. Lacey’s Regime Theory

A. Overview:

• Developed by G.W. Lacey in 1930, based on observations of Indian canals.
• Focuses on designing stable canals that maintain their shape and size over time, under a regime condition where the canal is self-sustaining.
• Lacey’s theory considers factors like silt grade, water discharge, and channel slope.

B. Key Concepts:

1. Regime Channels:

   • Canals that achieve a balance between sediment transport and deposition, maintaining a stable geometry.
   • Channels adapt to flow and sediment conditions over time.

2. Lacey’s Silt Factor (f):

   • A measure of the sediment size and gradation, similar to Kennedy’s theory.
   • $f = 1.76 \times \sqrt{d}$, where $d$ is the silt size in mm.

3. Lacey’s Equations:

   • Provide relationships between discharge, velocity, channel dimensions, and slope for a stable regime.

C. Lacey’s Equations:

1. Regime Velocity (V):

   $$V = 0.64 \times (Qf)^{1/6}$$

   Where:

   • $V$ = Velocity in m/s
   • $Q$ = Discharge in m³/s
   • $f$ = Silt factor

2. Regime Width (B):

   $$B = 4.8 \times (Q^2/f)^{1/3}$$

3. Regime Depth (D):

   $$D = 0.3 \times (Q/f)^{1/3}$$

4. Longitudinal Slope (S):

   $$S = \frac{Q^{1/2}}{f \times P}$$

   Where $P$ is the wetted perimeter.

D. Design Steps:

1. Determine Discharge (Q):
   Known requirement for the canal.

2. Calculate Regime Velocity (V):
   Use Lacey’s velocity formula to find the stable velocity.

3. Calculate Regime Width (B) and Depth (D):
   Use Lacey’s equations for width and depth based on the discharge and silt factor.

4. Check Slope (S):
   Ensure the calculated slope meets practical and stability considerations.

E. Example Calculation Using Lacey’s Theory:

Design an unlined canal with the following parameters:

• Discharge (Q): 50 m³/s
• Silt Factor (f): 1.0

Steps:

1. Calculate Regime Velocity (V):

   $$V = 0.64 \times (Qf)^{1/6} = 0.64 \times (50 \times 1)^{1/6} \approx 0.97 \, \text{m/s}$$

2. Calculate Regime Width (B):

   $$B = 4.8 \times (Q^2/f)^{1/3} = 4.8 \times (50^2/1)^{1/3} \approx 28 \, \text{m}$$

3. Calculate Regime Depth (D):

   $$D = 0.3 \times (Q/f)^{1/3} = 0.3 \times (50/1)^{1/3} \approx 1.8 \, \text{m}$$

4. Check Longitudinal Slope (S):
   Using Lacey’s slope formula, ensure that the slope is practical for construction and stable for canal flow.

4. Comparison of Kennedy’s and Lacey’s Regime Theory

A. Similarities:

• Both theories aim to design stable unlined canals that maintain a balance between erosion and sediment deposition.
• Both use empirical relationships derived from observations of actual canals.
• Emphasize the role of flow velocity and sediment characteristics in maintaining canal stability.

B. Differences:

Aspect	Kennedy’s Theory	Lacey’s Theory
Basis	Critical velocity concept	Regime conditions (width, depth, slope)
Velocity	Depends on depth	Function of discharge and silt factor
Design Approach	Empirical, based on critical velocity	Analytical, based on regime equations
Application	Simpler, easier for preliminary design	More comprehensive, suitable for detailed design

C. Real-Life Applications:

1. Kennedy’s Theory:
   Applied in designing smaller irrigation canals where simplicity and ease of calculation are required. Suitable for regions with similar silt characteristics to the Upper Bari Doab Canal.

2. Lacey’s Theory:
   Widely used in the design of larger irrigation projects and river training works. Example: Design of the Sutlej-Yamuna Link Canal in India, which required a detailed understanding of sediment transport and channel stability.

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5. Conclusion

• Kennedy’s and Lacey’s Regime Theories provide valuable tools for designing unlined canals in irrigation engineering.
• Understanding the principles of sediment transport and channel stability is crucial for efficient water distribution.
• Proper application of these theories ensures long-term sustainability, reduces maintenance costs, and optimizes water use in irrigation systems.

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These lecture notes provide an in-depth understanding of Kennedy’s and Lacey’s Regime Theories for the design of unlined canals, highlighting their key concepts, design steps, and real-life applications with numerical examples. This knowledge is essential for civil engineering students specializing in irrigation and water resources management.