Question

Pipes of length l₁, l₂ etc and diameters d₁, d₂, etc. are connected in series. A single pipe of length l₁ + l₂ + l₃ + …. having same pressure loss as all pipes in series will have diameter equal to
1. \({\left( {\frac{{{l_1} + {l_2} + {l_3} + \ldots }}{{\frac{{{l_1}}}{{d_1^4}} + \frac{{{l_2}}}{{d_2^4}} + \ldots }}} \right)^{\frac{1}{4}}}\)
2. \({\left( {\frac{{{l_1} + {l_2} + {l_3} + \ldots }}{{\frac{{{l_1}}}{{d_1^5}} + \frac{{{l_2}}}{{d_2^5}} + \ldots }}} \right)^{\frac{1}{5}}}\)
3. \({\left( {\frac{{{l_1} + {l_2} + {l_3} + \ldots }}{{\frac{{{l_1}}}{{d_1^3}} + \frac{{{l_2}}}{{d_2^3}} + \ldots }}} \right)^{\frac{1}{3}}}\)
4. \({\left( {\frac{{{l_1} + {l_2} + {l_3} + \ldots }}{{\frac{{{l_1}}}{{d_1^2}} + \frac{{{l_2}}}{{d_2^2}} + \ldots }}} \right)^{\frac{1}{2}}}\)

Answer 1

Correct Answer - Option 2 : \({\left( {\frac{{{l_1} + {l_2} + {l_3} + \ldots }}{{\frac{{{l_1}}}{{d_1^5}} + \frac{{{l_2}}}{{d_2^5}} + \ldots }}} \right)^{\frac{1}{5}}}\)

According to Dupuit’s equation of equivalent pipe.

\(\frac{{{l_{eq}}}}{{{{\left( {{D_{eq}}} \right)}^5}}} = \frac{{{L_1}}}{{D_1^5}} + \frac{{{L_2}}}{{D_2^5}} + \frac{{{L_3}}}{{D_3^5}} + \ldots \)

L_eg = L1 + L2 + L3 ….. for series connection

\(\therefore {D_{eq}} = {\left( {\frac{{{L_1} + {L_2} + {L_3} + \ldots }}{{\frac{{{L_1}}}{{D_1^5}} + \frac{{{L_2}}}{{D_2^5}} + \frac{{{L_3}}}{{D_3^5}} \ldots }}} \right)^{\frac{1}{5}}}\)