Critical Depth Calculation for Rectangular Channel
The critical depth ($y_c$) in an open channel occurs when the specific energy is minimum for a given discharge. For a rectangular channel, the relationship is derived from the flow equation.
Formula for Critical Depth
The specific discharge ($q$) is given by $q = Q/b$, where $Q$ is the total discharge and $b$ is the channel width.
The critical depth ($y_c$) for a rectangular channel can be calculated using the formula:
$ y_c = \left( \frac{q^2}{g} \right)^{1/3} $Alternatively, substituting $q = Q/b$:
$ y_c = \left( \frac{Q^2}{b^2 g} \right)^{1/3} $Given Data
- Discharge ($Q$): $10 \text{ m}^3/\text{s}$
- Channel Width ($b$): $2 \text{ m}$
- Acceleration due to gravity ($g$): $9.81 \text{ m/s}^2$
Step-by-Step Calculation
- Calculate the specific discharge ($q$): $ q = \frac{Q}{b} = \frac{10 \text{ m}^3/\text{s}}{2 \text{ m}} = 5 \text{ m}^2/\text{s} $
- Calculate the critical depth ($y_c$) using the formula: $ y_c = \left( \frac{q^2}{g} \right)^{1/3} = \left( \frac{(5 \text{ m}^2/\text{s})^2}{9.81 \text{ m/s}^2} \right)^{1/3} $ $ y_c = \left( \frac{25 \text{ m}^4/\text{s}^2}{9.81 \text{ m/s}^2} \right)^{1/3} $ $ y_c = \left( 2.5484 \text{ m}^2 \right)^{1/3} $ $ y_c \approx 1.3658 \text{ m} $
- Round the result to two decimal places: $ y_c \approx 1.37 \text{ m} $
Conclusion
The calculated critical depth is approximately 1.37 m, which falls within the range of 1.35 m to 1.4 m.
Important Questions from Open Channel Flow
- A trapezoidal channel carries $5\text{ m}^3\text{.s}^{-1}$ of water under uniform flow condition. The channel has a bottom width of $2\text{ m}$, side slope of $2:1$ (horizontal: vertical), and a bed slope of $1\%$. If the Manning roughness coefficient is $0.03$, the conveyance of the channel, in $\text{m}^3\text{.s}^{-1}$, is
- Two rectangular channels, A and B, join to form a large rectangular channel, C. Each rectangular channel is lined with the same material and has the same bottom slope. The bottom widths of Channels A and B are $2\text{ m}$ and $1\text{ m}$, respectively. If the flow depth in each channel is $2\text{ m}$, the bottom width of Channel C (in m) is ________. (Rounded off to two decimal places)
- Water is flowing at the rate of 15 m$^3$/s through a rectangular channel of 5 m width.
If the acceleration due to gravity ($g$) is 9.81 m/s$^2$, the critical velocity for the flow is ______ m/s (rounded off to three decimal places). A Hydraulic Jump is said to be Oscillating Jump if its Froude number is ________.
The Francis formula for the discharge over cippoletti weir is
