Oil with kinematic viscosity ν=7.5 ×10^-4 ft^2 / s flows at 45 gpm in a 100 -fi-long horizontal

step 1 In this question we have to find the percent change in the loss if the diameter is reduced. So f

Oil with kinematic viscosity ν=7.5 ×10^-4 ft^2 / s flows at...

Oil with kinematic viscosity $\nu=7.5 \times 10^{-4} \mathrm{ft}^{2} / \mathrm{s}$ flows at 45 gpm in a 100 -fi-long horizontal drawn-tubing pipe of 1 in. diameter. By what percentage ratio will the energy loss increase if the same flow rate is maintained while the pipe diameter is reduced to 0.75 in.?

Key Concepts

Continuity and Flow Rate

In fluid dynamics the conservation of mass requires that the volumetric flow rate remains constant along a streamline. In a pipe, the flow rate (Q) is related to the cross?sectional area (A) and the average velocity (V) by the relation Q = A × V. When the diameter of the pipe changes while keeping Q constant, the velocity must adjust inversely to the cross-sectional area, having major implications on the flow characteristics and energy losses.

Pressure Drop in Viscous Flow (Hagen–Poiseuille Law)

For laminar flow, the pressure drop or energy loss due to viscous friction in a circular pipe is described by the Hagen–Poiseuille law, which shows that the pressure drop is directly proportional to the volumetric flow rate and inversely proportional to the fourth power of the pipe’s diameter. This strong dependence means that even a small reduction in the pipe diameter will notably increase the energy loss required to maintain the same flow rate.

Reynolds Number and Flow Regime

The Reynolds number is a dimensionless quantity used to predict whether the flow in the pipe is laminar or turbulent. It compares the relative effects of inertial forces to viscous forces. Understanding the flow regime is essential because the relationship between pressure drop and pipe diameter can vary significantly between laminar and turbulent flows, with the Hagen–Poiseuille relationship applicable in the laminar regime.

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