Chemical Engineering
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Consider a steady flow of an incompressible, Newtonian fluid through a smooth circular pipe. Let $\alpha_{\textit{laminar}}$ and $\alpha_{\textit{turbulent}}$ denote the kinetic energy correction factors for laminar and turbulent flow through the pipe, respectively. For turbulent flow through the pipe $$\alpha_{\text{turbulent}} = \left( \dfrac{V_0}{\overline{V}} \right) ^3 \dfrac{2n^2}{(3+n)(3+2n)}$$ Here, $\overline{V}$ is the average velocity, $V_0$ is the centerline velocity, and $n$ is a parameter. The ratio of average velocity to the centerline velocity for turbulent flow through the pipe is given by $$\dfrac{\overline{V}}{V_0}=\dfrac{2n^2}{(n+1)(2n+1)}$$ For $n=7$, the value of $\dfrac{\alpha_{\text{turbulent}}}{\alpha_{\text{laminar}}}$ is ______ (round off to $2$ decimal places).
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