A rectangular channel is 4.0 m wide and carries a discharge of $2.0{\text{ m}}^3/\text{s}$ with a depth of 0.4 m. The channel transitions to a maximum width contraction at a downstream location, without influencing the upstream flow conditions. The width (in meters) at the maximum contraction is ____________ (rounded off to 2 decimal places).

In the range: 3.3 - 3.7

Rectangular Channel Contraction Width Calculation

This solution calculates the maximum contraction width in a rectangular channel using specific energy principles, focusing on the conditions at the throat.

Upstream Flow Conditions Analysis

Determine the specific energy (E) of the upstream flow. Using the acceleration due to gravity $g=9.81{\text{ m/s}}^2$.

- Given: Upstream Width $b_1=4.0$ m, Discharge $Q=2.0{\text{ m}}^3/\text{s}$, Upstream Depth $y_1=0.4$ m.
- Upstream Area: $A_1=b_1\times y_1=4.0\times 0.4=1.6{\text{ m}}^2$.
- Upstream Velocity: $V_1=\frac{Q}{A_1}=\frac{2.0}{1.6}=1.25\text{ m/s}$.
- Specific Energy: $E_1=y_1+\frac{V_1^2}{2g}$.
- $E_1=0.4+\frac{(1.25{)}^2}{2\times 9.81}\approx 0.4+\frac{1.5625}{19.62}\approx 0.4+0.0796\approx 0.4796$ m.

Throat (Maximum Contraction) Analysis

At maximum contraction, the channel width ($b_2$) is minimized. This typically occurs when the flow becomes critical at the throat (choked flow). The condition "without influencing the upstream flow conditions" means the upstream depth ($y_1$) and specific energy ($E_1$) remain unchanged. Thus, the specific energy available at the throat is $E_1$.

- At critical flow (throat), the depth is the critical depth $y_c$, and the specific energy is the minimum specific energy $E_{\text{min}}=1.5\times y_c$.
- Equating available energy to minimum required energy: $E_1=1.5\times y_c$.
- $0.4796=1.5\times y_c$.
- Calculate critical depth at the throat: $y_c=\frac{0.4796}{1.5}\approx 0.3197$ m.

Throat Width Calculation

Use the critical flow relationship $Q=b_2\times y_c\times V_c$, where $V_c=\sqrt{g\times y_c}$, to find the throat width ($b_2$).

- The formula simplifies to: $Q=b_2\times y_c\times \sqrt{g\times y_c}$.
- Rearrange to solve for $b_2$: $b_2=\frac{Q}{y_c\sqrt{g\times y_c}}$.
- Substitute the known values: $b_2=\frac{2.0}{0.3197\times \sqrt{9.81\times 0.3197}}$.
- $b_2=\frac{2.0}{0.3197\times \sqrt{3.136}}$.
- $b_2=\frac{2.0}{0.3197\times 1.771}\approx \frac{2.0}{0.5661}$.
- $b_2\approx 3.533$ m.

Rounded to 2 decimal places, the width at maximum contraction is 3.53 m.