Question

how to find the friction factor of a pipe using reynolds number

Answer 1

To study the variation in friction factor, f, used in the Darcy Formula with the Reynolds number in both laminar and turbulent flow. The friction factor will be measured as a function of Reynolds number and the roughness will be calculated using the Colebrook equation.

Theory

The loss of head resulting from the flow of a fluid through a pipeline is expressed by the Darcy Formula

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where hf is the loss of head (units of length) and the average velocity is V. The friction factor, f, varies with Reynolds number and a roughness factor.

Laminar flow

The Hagen-Poiseuille equation for laminar flow indicates that the head loss is independent of surface roughness.

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Thus in laminar flow the head loss varies as V and inversely as D2. Comparing equation 1.1 and equation 1.2 it can be shown that

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indicating that the friction factor is proportional to viscosity and inversely proportional to the velocity, pipe diameter, and fluid density under laminar flow conditions. The friction factor is independent of pipe roughness in laminar flow because the disturbances caused by surface roughness are quickly damped by viscosity.

Equation 1.2 can be solved for the pressure drop as a function of total discharge to obtain

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Turbulent flow

When the flow is turbulent the relationship becomes more complex and is best shown by means of a graph since the friction factor is a function of both Reynolds number and roughness. Nikuradse showed the dependence on roughness by using pipes artificially roughened by fixing a coating of uniform sand grains to the pipe walls. The degree of roughness was designated as the ratio of the sand grain diameter to the pipe diameter (/D).

The relationship between the friction factor and Reynolds number can be determined for every relative roughness. From these relationships, it is apparent that for rough pipes the roughness is more important than the Reynolds number in determining the magnitude of the friction factor. At high Reynolds numbers (complete turbulence, rough pipes) the friction factor depends entirely on roughness and the friction factor can be obtained from the rough pipe law.

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For smooth pipes the friction factor is independent of roughness and is given by the smooth pipe law.

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The smooth and the rough pipe laws were developed by von Karman in 1930.

Many pipe flow problems are in the regime designated “transition zone” that is between the smooth and rough pipe laws. In the transition zone head loss is a function of both Reynolds number and roughness. Colebrook developed an empirical transition function for commercial pipes. The Moody diagram is based on the Colebrook equation in the turbulent regime.

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The Colebrook equation can be used to determine the absolute roughness, , by experimentally measuring the friction factor and Reynolds number.

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Alternatively the explicit equation for the friction factor derived by Swamee and Jain can be solved for the absolute roughness.

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When solving for the roughness it is important to note that the quantity in equation 1.9 that is squared is negative!

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Equations 1.8 and 1.10 are not equivalent and will yield slightly different results with the error a function of the Reynolds number.