We saw in Section 8.7 that instead of the implicit Colebrook...

We saw in Section 8.7 that instead of the implicit Colebrook equation (Eq. 8.37 ) for computing the turbulent friction factor $f,$ an explicit expression that gives reasonable accuracy is $$\frac{1}{\sqrt{f}}=-1.8 \log \left[\left(\frac{c / D}{3.7}\right)^{1.11}+\frac{6.9}{R e}\right]$$ Compare the accuracy of this expression for $f$ with $\mathrm{F}$ g. 8.37 by computing the percentage discrepancy as a function of $R e$ and $e / D,$ for $R e=10^{4}$ to $10^{8},$ and $e / D=0,0.0001,0.001$ $0.01,$ and $0.05 .$ What is the maximum discrepancy for these Re and e/D values? Plot $f$ against $R e$ with $e j D$ as parameter.

Key Concepts

Percentage Discrepancy

Percentage discrepancy is a method used to quantify the accuracy of an approximate model relative to a reference model or data set. In this context, it measures the deviation between the friction factor calculated from an explicit expression and that obtained from the implicit Colebrook equation. This metric provides insight into the validity range and potential error associated with the approximation across different flow regimes.

Relative Roughness

Relative roughness is a non-dimensional parameter that quantifies the roughness of a pipe's surface compared to its diameter. It plays a crucial role in determining the frictional resistance in turbulent pipe flow. The higher the relative roughness, the more significant its influence on flow resistance, and thus on the computation of the friction factor.

Parametric Plotting

Parametric plotting involves graphically representing the relationship between one variable and others that act as parameters. In the study of friction factors, plotting the friction factor against the Reynolds number for different relative roughness values helps visualize the performance of various friction factor models under different flow conditions, thereby assisting in assessing the influence of each parameter on the calculated outcomes.

Explicit Friction Factor Expression

An explicit friction factor expression is designed to approximate the value obtained from the Colebrook equation without the need for iterative solution techniques. These formulas provide a direct computation of the friction factor as a function of Reynolds number and relative roughness, though they often involve trade?offs between simplicity and accuracy. They are valuable in engineering applications where computational efficiency is important.

Colebrook Equation

The Colebrook equation is an empirical formula used in fluid mechanics to determine the turbulent friction factor in pipe flow. It is an implicit relation that relates the friction factor, the Reynolds number, and the relative roughness of the pipe. Due to its implicit nature, solving the equation requires iterative methods, which can be computationally intensive in engineering calculations.

Reynolds Number

The Reynolds number is a key dimensionless parameter in fluid mechanics that characterizes the flow regime within a pipe, distinguishing between laminar and turbulent flow. It is defined as the ratio of inertial forces to viscous forces and is essential for predicting flow behavior, influencing the calculation of friction factors as well as other transport phenomena.

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