GATE 2021 CH – Question 65
A viscous liquid is pumped through a pipe network in a chemical plant. The annual pumping cost per unit length of pipe is given by $C_{pump}=48.13q^2\mu/D^4$. The annual cost of the installed piping system per unit length of pipe is given by $C_{piping}=45.92D$. Here, D is the inner diameter in meter, q is the volumetric flowrate in m³/s and μ is the viscosity in Pa·s. If the viscosity is $20\times10^{-3}$ Pa·s and the volumetric flow rate is $10^{-4}$ m³/s, the economic inner diameter of the pipe is _____ meter (round off to 3 decimal places).
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Correct answer: 0.005 to 0.025
Explanation
Minimize the **sum** of annual pumping and installed-pipe costs:
$$C(D)=48.13q^2\mu D^{-4}+45.92D.$$
Differentiating gives $dC/dD=-4(48.13)q^2\mu D^{-5}+45.92$. At the optimum,
$$D^5=\frac{4(48.13)(10^{-4})^2(20\times10^{-3})}{45.92}.$$
Taking the positive fifth root gives approximately 0.0153 m, so the rounded diameter is **0.015 m**. The second derivative $20(48.13)q^2\mu D^{-6}$ is positive, confirming a minimum.