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) |
• 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)=0so 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\rgroupfrom 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)