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The velocity of a liquid flowing in a circular pipe of
Chapter 5, Problem 94P(choose chapter or problem)
Problem 94P
The velocity of a liquid flowing in a circular pipe of radiusR varies from zero at the wall to a maximum at the pipe center. The velocity distribution in the pipe can be represented asV(r), wherer is the radial distance from the pipe center. Based on the definition of mass flow rate m, obtain a relation for the average velocity in terms ofV(r), R, and r.
Questions & Answers
QUESTION:
Problem 94P
The velocity of a liquid flowing in a circular pipe of radiusR varies from zero at the wall to a maximum at the pipe center. The velocity distribution in the pipe can be represented asV(r), wherer is the radial distance from the pipe center. Based on the definition of mass flow rate m, obtain a relation for the average velocity in terms ofV(r), R, and r.
ANSWER:
Step 1 of 4:
Consider a circular pipe of radius R. Let us assume that the flow is steady and incompressible. The density of the liquid is . The velocity of the flow near the wall is zero and increasing radially towards the center of the pipe. The velocity is maximum at the center. The velocity gradient perpendicular to the flow of the liquid is
. The cross-sectional area of the pipe is AC =
. r is the instantaneous radius of the layer of liquid. r varies from 0 to R.