Question

Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of
1. 4
2. 16
3. 32
4. 64

Answer 1

Correct Answer - Option 4 : 64

Concept:

Head loss due to friction is given by \({h_f} = \frac{{fL{V^2}}}{{2gD}}\)

\({h_f} = {\frac{{fLv^2}}{{2gD}}}= {\frac{{fLQ^2}}{{2gDA^2}}}\)

Where,

F = friction factor \( = \frac{{64}}{{{R_e}}}\)

Re = Reynold’s number, L = length of pipe, V = velocity of fluid flowing through the pipe, D = Diameter of the pipe

Calculation:

Given: Mass flow rate, density and friction factor to be constant. L' = 2L, d' = 0.5 d

\({h_f}= {\frac{{fLQ^2}}{{2gDA^2}}}\)

Or \(h∝ \frac{L}{{d.A^2 }}∝\frac{L}{{d^5 }}\)

But we know that \(A = \frac{\pi }{d}{d^2}\) or A ∝ d²

\( \frac{{h'}}{h} = \frac{{L'}}{L} \times {\left( {\frac{d}{{d'}}} \right)^5} = 2 \times {\left( 2 \right)^5} = 64\)

h' = 64 h