Exam (16).pdf - Question The velocity profile v(r for turbulent flow in smooth pipes is sometimes estimated by the seventh-root law originally proposed

Exam (16).pdf - Question The velocity profile v(r for...

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Question: The velocity profile, v ( r ), for turbulent flow in smooth pipes is sometimes estimated by the seventh-root law, originally proposed by Blasius (1913): where V 0 is the maximum (centerline) velocity and R is the radius of the pipe. Estimate the energy and momentum correction factors corresponding to the seventhroot law. Answer : The velocity distribution in the pipe can be expressed as follows: …… (1) Here, r is the distance from the center line of the pipe, is the center line (maximum) velocity, R is the radius of the pipe. Area of the pipe can be expressed as follows: …… (2) Kinetic energy correction factor can be expressed as follows:
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Equation (4) can be written as follows: …… (3) Combine Equation (1) and (2) with Equation (3). Express the value of as follows: …… (4) To facilitate integration: …… (5) Equation (5) can be modified as follows: …… (6) Differentiate Equation (6) with respect to
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