The flow rate per unit width of a wide rectangular clean-earth channel is 20 $m^3 s^{-1} m^{-1}$. The calculated critical flow depth in meter will be __________ [round off to two decimal places]
(Take $g = 9.81 m s^{-2}$)
This question asks for the calculation of the critical flow depth ($y_c$) in a wide rectangular channel, given the flow rate per unit width ($q$) and the acceleration due to gravity ($g$).
Calculating Critical Flow Depth
The specific energy ($E$) for a rectangular channel is given by $E = y + \frac{V^2}{2g}$, where $V$ is the flow velocity and $y$ is the flow depth. The flow rate per unit width ($q$) is $V \times y$, so $V = q/y$. Substituting this into the specific energy equation gives $E = y + \frac{(q/y)^2}{2g}$.
Critical flow occurs when the specific energy is minimum for a given flow rate. This condition is mathematically represented by $\frac{dE}{dy} = 0$ or, more directly for rectangular channels, by the relationship:
$ \frac{q^2}{g} = y_c^3 $
where $y_c$ is the critical flow depth.
Step-by-Step Calculation
- Identify Given Values:
- Flow rate per unit width, $q = 20 \, m^3 s^{-1} m^{-1}$
- Acceleration due to gravity, $g = 9.81 \, m s^{-2}$
- Apply Critical Flow Formula: Rearrange the formula to solve for $y_c$:
$ y_c = \sqrt[3]{\frac{q^2}{g}} $
- Substitute Values:
$ y_c = \sqrt[3]{\frac{(20 \, m^3 s^{-1} m^{-1})^2}{9.81 \, m s^{-2}}} $
- Calculate Intermediate Value:
$ y_c = \sqrt[3]{\frac{400 \, m^6 s^{-2} m^{-2}}{9.81 \, m s^{-2}}} $
$ y_c = \sqrt[3]{40.7747 \, m^2} $
- Calculate Final Depth:
$ y_c \approx 3.441 \, m $
- Round to Two Decimal Places: The calculated critical flow depth is approximately $3.44 \, m$.
The calculated value of 3.44 m falls within the provided range of 3.42 m to 3.46 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
