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Question 4.2: Turbulent Friction Factor Derivation for Flow in Smooth Pipe......

Turbulent Friction Factor Derivation for Flow in Smooth Pipes

\rm u^+=8.74(y^+)^{1/7}                         (4.22)

Concepts Assumptions Sketch
• Force balance for C.∀.≘(r²π) \ell • Steady fullydeveloped turbulent flow in a smooth pipe with constant fluid properties
• friction loss expression, i.e., \rm h_f \sim Δp \sim τ_{wall}
• Use of Eq. (4.22)
Step-by-Step
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• A force balance (see Sect. 2.4, Momentum RTT) on the C.∀., π r² \ell , yields:

\Sigma\,F_{x}=\Delta p(\pi\,r^{2})-\tau_{r x}(2r\pi\ \ell)=\dot{m}(v_{2}-v_{1})=0                      (E.4.3.1a, b)

which, evaluated at the wall, results in

\rm \Delta{ p}(\pi\,{ r}_{0}^{2})-\tau_{\mathrm{w}}(2{ r}_{0}~\pi\;\ell)=0

so that

\rm\tau_{\mathrm{rx}}=\tau_{\mathrm{w}}\ {\frac{r}{r_{0}}}                                    (E.4.3.2)

Note, Eq. (E.4.3.2) indicates that τ(r) is linear in pipe flow for any fully-developed flow regime. It turns out that is the case for non-Newtonian fluids as well (see Sect. 6.3).

• Expressing Δp from Eq. (E.4.3.1b) in \rm h_f of Eq. (4.5d) with \rm ΣK_L = 0 , we obtain:

h_L = \left[ f \left\lgroup  \frac{L}{D} \right\rgroup + \sum K_L \right] \frac{v^2}{2g}      (4.5d)

\rm\Delta{ p}=2\ell\tau_{\mathrm{w}}\;/\,{ r}_{0}

and hence

\rm \mathrm{h}_{\mathrm{f}}={\frac{\Delta{ p}}{\rho g}}={\frac{2\ell\tau_{\mathrm{w}}}{\rho g r_0}}=f{\frac{{ v}^{2}}{2g}}\left\lgroup{\frac{\ell}{2r_{0}}}\right\rgroup

from which

\mathrm{f}={\frac{8\mathbf{\tau}_{\mathrm{w}}}{\mathrm{\rho v}^{2}}}                          (E.4.3.3a)

or with \rm τ_w = ρu_τ^2 ,

\mathrm{f}=8\!\left\lgroup{\frac{\mathrm{u}_{\mathrm{\tau}}}{\mathrm{v}}}\right\rgroup ^{2}                        (E.4.3.3b)

• Equation (4.22) can be evaluated at the centerline, where \rm y^+ = r_0u_τ / ν, so that we can obtain an expression for the elusive friction velocity as:

\rm \mathrm{u_{\tau}}=\left\lgroup{\frac{\mathrm{u_{\max}}}{8.74}}\right\rgroup^{7/8}\left\lgroup{\frac{\mathrm{\nu}}{\mathrm{r_{0}}}}\right\rgroup^{1/8}                            (E.4.3.4)

Also,

\mathrm{u}_{\mathrm{{av}}}\equiv\mathrm{v}={\frac{1}{\mathrm{{A}}}}\int\!\mathrm{u}\,\mathrm{d}\mathrm{A}=0.817~\mathrm{u}_{\mathrm{{max}}}

so that Eq. (E.4.3.4) can now be rewritten as:

\mathrm{u_{\tau}}=\left\lgroup{\frac{\mathrm{v}}{7.1406}}\right\rgroup^{7/8}\left\lgroup{\frac{2\mathbf{\nu}}{\mathrm{D}}}\right\rgroup^{1/8}

and hence

\rm \mathrm{f}=0.3164\ \mathrm{Re}_{\mathrm{{D}}}^{-1/4}                                (E.4.3.5)

Note, Eq. (E.4.3.5), known as the Blasius correlation, differs significantly from \mathrm{f}_{\mathrm{laminar}}=64\,\mathrm{Re}_{\mathrm{D}}^{-1/2}. More extensive expressions taking pipe wall roughness into account is given as Eq. (4.5e). Alternatively, the Moody chart can be used (see App. B).

\rm f^{-1/2}\approx-1.8\ {log}\bigg[{\frac{6.9}{{Re_{D}}}}+\left\lgroup{\frac{\varepsilon /{D}}{3.7}}\right\rgroup ^{1.11}\bigg]                         (4.5e)

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