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Chapter 8: Problem 108

For a given head loss per unit length, what effect on the flowrate does doubling the pipe diameter have if the flow is (a) laminar, or (b) completely turbulent?

Short Answer

Expert verified
Doubling the pipe diameter increases the flowrate 16 times in laminar flow and approximately doubles the flowrate in completely turbulent flow.

Step by step solution

01

Laminar Flow Case

In a laminar flow, the flowrate Q, pipe diameter D, and head loss hL are related by the Hagen-Poiseuille equation: \( Q = \frac{πD^4hL}{128μL} \), where μ is the dynamic viscosity and L is the length of the pipe. According to the problem, the head loss per unit length is constant meaning \( hL/L \) is constant. So if the diameter is doubled, the flowrate increases by a factor of 16, since \( Q \propto D^4 \). Thus, the new flow rate \( Q' = 16Q \).
02

Turbulent Flow Case

For completely turbulent flow, the Darcy-Weisbach equation is often used to relate flowrate Q, diameter D, and head loss hL: \( hL = \frac{fLQ^2}{gD^5} \), where f is the friction factor, L is the length of the pipe, and g is the gravitational acceleration. Since the head loss per unit length, \( hL/L \), is constant and f is assumed constant in fully developed turbulent flow for a given roughness, if we double the diameter, the flowrate will approximately double, as the relationship would be practically \( Q \propto D \). Hence, the new flow rate \( Q'≈2Q \).

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