A horizontal pipe carries fluid in fully developed turbulent...

A horizontal pipe carries fluid in fully developed turbulent flow. The static pressure difference measured between two sections is 750 psi. The distance between the sections is $15 \mathrm{ft}$, and the pipe diameter is 3 in. Calculate the shear stress, $\tau_{w},$ that acts on the walls.

Key Concepts

Fully Developed Flow

In fluid mechanics, fully developed flow refers to a flow regime where the velocity profile does not change along the length of the pipe. This means that the effects of the entrance region or developing region have decayed, and the shear stress and velocity profile remain constant along the flow direction. This concept is crucial as it allows for the use of simplified relationships between pressure gradients and wall shear stress.

Turbulent Flow

Turbulent flow is a regime characterized by chaotic and irregular fluid motion, often occurring at high Reynolds numbers. In turbulent flow, the mixing and momentum transfer due to eddies are significant, and the velocity distribution is more uniform compared to laminar flow. Understanding turbulent flow is key because it influences how friction and energy losses are calculated in pipe flows.

Pressure Gradient

The pressure gradient is the rate at which pressure decreases along the flow direction. In the context of pipe flow, a constant pressure gradient drives the movement of fluid and directly relates to the frictional forces acting on the pipe wall. This concept is important since the wall shear stress is computed from the pressure gradient in fully developed flows.

Wall Shear Stress

Wall shear stress is the force per unit area exerted by the flowing fluid on the walls of the pipe due to viscous effects. It is directly linked to the friction factor and is derived from the momentum balance in the flow. In fully developed pipe flow, the relationship between the pressure gradient and wall shear stress simplifies calculations, making it a key concept in determining frictional losses.

Pipe Geometry

The geometry of the pipe, particularly its diameter, plays a critical role in determining the characteristics of the flow. The diameter influences the relationship between the pressure gradient and the wall shear stress. In many analytical relationships, such as the one for fully developed flow in a circular pipe, the pipe diameter is used to scale the pressure gradient to yield the wall shear stress.

Please give Ace some feedback