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Maximising discharge with same lining

Short answer

Keeping the same wetted perimeter (11 m), the best rectangular section (b=2y) gives y=2.75 m, b=5.5 m. Discharge increases from 26.81 to 39.43 m³/s, a 47% increase.

A rectangular channel, 8 m wide and 1.5 m deep, has a slope of 0.001 in 1 and is lined with smooth concrete plaster. It is desired to enhance the discharge to a maximum by changing the dimensions of the channel but keeping the same amount of lining. Find the new dimensions and percentage increase in discharge. For smooth concrete plaster, take Manning's constant = 0.015.

Keeping the same amount of lining means the wetted perimeter stays the same. For a given perimeter, discharge is maximum when the section is the best (most economical) rectangle: b=2yb = 2y.

Existing channel

A1=8×1.5=12 m2,P=8+2×1.5=11 mR1=12/11=1.091 mQ1=1nA1R12/3S1/2=10.015×12×1.0912/3×0.0011/2=66.67×12×1.0597×0.03162=26.81 m3/s\begin{aligned} A_1 &= 8 \times 1.5 = 12\ \text{m}^2, \quad P = 8 + 2 \times 1.5 = 11\ \text{m} \\ R_1 &= 12/11 = 1.091\ \text{m} \\ Q_1 &= \frac{1}{n}A_1R_1^{2/3}S^{1/2} = \frac{1}{0.015} \times 12 \times 1.091^{2/3} \times 0.001^{1/2} \\ &= 66.67 \times 12 \times 1.0597 \times 0.03162 = 26.81\ \text{m}^3/\text{s} \end{aligned}

New channel (same P = 11 m, b=2yb = 2y)

P=b+2y=4y=11  ⇒  y=2.75 m,b=5.5 mA2=5.5×2.75=15.125 m2,R2=15.125/11=1.375 mQ2=66.67×15.125×1.3752/3×0.03162=39.43 m3/s\begin{aligned} P &= b + 2y = 4y = 11 \;\Rightarrow\; y = 2.75\ \text{m}, \quad b = 5.5\ \text{m} \\ A_2 &= 5.5 \times 2.75 = 15.125\ \text{m}^2, \quad R_2 = 15.125/11 = 1.375\ \text{m} \\ Q_2 &= 66.67 \times 15.125 \times 1.375^{2/3} \times 0.03162 = 39.43\ \text{m}^3/\text{s} \end{aligned}

Percentage increase

Q2−Q1Q1×100=39.43−26.8126.81×100=47.1%\frac{Q_2 - Q_1}{Q_1} \times 100 = \frac{39.43 - 26.81}{26.81} \times 100 = 47.1\%
ItemExistingNew
Width (m)8.05.5
Depth (m)1.52.75
Perimeter (m)1111
R (m)1.0911.375
Q (m³/s)26.8139.43

Answer: New width = 5.5 m, depth = 2.75 m; discharge increases by about 47%.

✓ Answer checked by Claude Opus 5.5 (high effort)

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