Water undergoes viscous flow through two pipes with the same pressure difference between the ends. Pipe 1 has three times the diameter and twice the length of pipe 2. If the flow rate through pipe 1 is 0.400 cm³/s, then what is the flow rate through pipe 2? A) 6.22 × 10⁻³ cm³/s B) 7.78 × 10⁻³ cm³/s C) 8.33 × 10⁻³ cm³/s D) 9.88 × 10⁻³ cm³/s E) 10.80 × 10⁻³ cm³/s

Question

Quizplus · Accepted Answer

The flow rate $Q$ through a pipe can be described by the Hagen-Poiseuille equation, which is $Q = \frac{\pi \Delta P r^4}{8 \mu L}$, where $\Delta P$ is the pressure difference, $r$ is the radius of the pipe, $\mu$ is the dynamic viscosity, and $L$ is the length of the pipe. Given that Pipe 1 has three times the diameter (and thus three times the radius) and twice the length of Pipe 2, and using the fact that the flow rate is directly proportional to the fourth power of the radius and inversely proportional to the length, we can set up a ratio of the flow rates of Pipe 1 to Pipe 2 as follows: $\frac{Q_1}{Q_2} = \frac{(3r)^4/2L}{r^4/L} = \frac{81}{2}$. Given $Q_1 = 0.400$ cm^3/s, solving for $Q_2$ gives $Q_2 = \frac{2}{81} \times 0.400 = 0.00987654321$ cm^3/s, which is approximately $9.88 \times 10^{-3}$ cm^3/s.

Water Undergoes Viscous Flow Through Two Pipes with the Same