If the flow is critical in a channel then : (Symbols have normal meanings)

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

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

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

Q2T = gA3