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PTCUL AE E&M 2017 Official Paper (Set B)

- \(\frac{{\alpha Q{T^2}}}{{g{A^3}}} = 1\)
- \(\frac{{\alpha Q^2{T^2}}}{{g{A^3}}} = 1\)
- \(\frac{{\alpha Q^2{T}}}{{g{A^3}}} = 1\)
- None of these

Option 3 : \(\frac{{\alpha Q^2{T}}}{{g{A^3}}} = 1\)

__Concept:__

Critical Flow:

It is defined as the **flow for which the specific energy is minimum** and Froude’s number is equal to unity.

Froude’s no. (Fr) = \(\sqrt {\frac{{{\rm{Inertial\;force}}}}{{{\rm{gravity\;force}}}}} \)

Where,

Inertial force = ρ × Q × v

(ρ = Density, Q = Discharge, v = velocity)

Gravity force = ρ × g × V;

(ρ = Density, g = Gravity acceleration, V = Volume)

Now,

\({{\rm{F}}_{\rm{r}}} = \sqrt {\frac{{{\rm{\rho Qv}}}}{{{\rm{\rho gV}}}}} = \sqrt {\frac{{{\rm{Qv}}}}{{{\rm{gAL}}}}} = \sqrt {\frac{{{\rm{Q}}\left( {\frac{{\rm{Q}}}{{\rm{A}}}} \right)}}{{{\rm{gA}}\left( {\frac{{\rm{A}}}{{\rm{T}}}} \right)}}} = \sqrt {\frac{{{{\rm{Q}}^2}{\rm{T}}}}{{{\rm{g}}{{\rm{A}}^3}}}} \)

(∵ A = L × T, ν = Q/A)

A = Area, L = Length, and T = Top width

**Now, for a critical flow Fr = 1**

\(\therefore {{\rm{F}}_{{{r}}}} = 1\)

\( \Rightarrow \frac{{{{\rm{Q}}^2}{\rm{T}}}}{{{\rm{g}}{{\rm{A}}^3}}} = 1\)

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