A channel, apex at the top carries a critical flow with a depth o
A channel, apex at the top carries a critical flow with a depth of 1.6 m as shown in the figure below. (Take g = 10 m/sec2)
Choose the correct statements among the following:
A. The discharge passing through the channel is 17.47 m<sup>3</sup>/sec
B. The specific energy of the flow is minimum for this particular discharge at which the flow is happening.
C. The discharge passing through the channel is 17.65 m<sup>3</sup>/sec
D. The discharge passing through the channel is minimum for this particular specific energy at which flow is happening.
Please scroll down to see the correct answer and solution guide.
Right Answer is:
SOLUTION
Concept:
It is mentioned in the question that the flow is taking place at critical condition. So the depth of flow is the critical depth for that particular discharge.
We have this relation for critical flow: \({{\rm{Q}}^2}{\rm{T}} = {\rm{g}}{{\rm{A}}^3}\)
(This equation is valid for all type of cross section of flow, but the flow should be critical)
T= top width of the channel
g = 10 m/sec2
A= area of the channel
It can be seen that for a particular discharge, the lowest energy of the flow is happening at the critical flow, i.e., for a given discharge, the energy of the flow is minimum at critical flow.
∴ option b) is correct.
The figure below shows the variation of depth vs discharge for a given value of specific energy-
It can be seen from the graph below that for a particular value of specific energy, the discharge is maximum at the critical flow. i.e., the discharge is maximum at the critical flow for a given value of specific energy.
∴ option d) is incorrect.
Calculation:
Given,
T= top width of the channel = 1.4 m
g = 10 m/sec2
A= area of the channel = 0.5 × 1.6 × ( 1.4+3 ) = 3.52 m2
\({\rm{Q}} = {\rm{\;}}\sqrt {\frac{{{\rm{g}}{{\rm{A}}^3}}}{{\rm{T}}}} = \sqrt {\frac{{10{\rm{\;}} \times {\rm{\;}}{{3.52}^3}}}{{1.4}}{\rm{\;}}} \) = 17.65 m3/sec
∴ option c) is correct.
Mistake point
In the question, it is mentioned to take value of ‘g’ as 10 m/sec2. If you take g as 9.8 m/sec2, then you will get option a) as the correct option, which is actually wrong.
