## Q. 4.2

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)

## Verified Solution

• 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)