Final answer:
The minimum diameter of a pipe for laminar flow can be calculated using the Reynolds number formula, adjusted for the flow rate, while the pressure difference required to maintain the flow rate is derived from the Hagen-Poiseuille equation.
Step-by-step explanation:
The student's question pertains to the principles of fluid dynamics, specifically the application of the continuity equation and the concept of Reynolds number in relation to laminar and turbulent flow. The information given states that gasoline is piped underground with a flow rate, viscosity, and density provided.
To answer part (a), we should calculate the minimum diameter of the pipe that would result in a Reynolds number less than 2000, indicating laminar flow. The Reynolds number (Re) can be calculated using the equation Re = (Density x Velocity x Diameter) / Viscosity. Reorganizing to solve for diameter gives us Diameter = Re x Viscosity / (Density x Velocity). However, since the question includes the flow rate but not the velocity, we can find the velocity by dividing the flow rate by the cross-sectional area of the pipe (Velocity = Flow rate / Area). By plugging this into the equation for Reynolds number, we can solve for the diameter.
For part (b), in order to maintain the flow rate of gasoline, a certain pressure difference must be maintained per kilometer of pipe. This can be determined using the Hagen-Poiseuille equation, which relates the flow rate (Q) to the pressure difference (∆P), the viscosity (η), the length of the pipe (L), and the radius of the pipe (r): ∆P = (8 x Viscosity x Flow rate x Length) / (π x r⁴). Thus, for a given flow rate, viscosity, and pipe radius, we can calculate the necessary pressure difference per kilometer.