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Question 4.1: Derivation of the Frictional Loss Term, hf , for Steady Lami......

Derivation of the Frictional Loss Term, \rm h_f , for Steady Laminar Flow in Smooth Pipes

Concept Assumptions Sketch
• Combine Poiseuille flow solution with extended Bernoulli • As stated, with \rm L >> L_{entrance}
• Poiseuille flow
• Constant cross sectional pipes
• Constant properties and velocity average
Step-by-Step
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(A) Check entrance length and assume that \rm Re_D < 2,300

• Clearly, for any given data set \rm L_{entrance} = 0.05Re_D \,D should be much less than \rm L_{pipe} .

(B) Poiseuille flow

\vec{\bf v}=\rm [u(r),0,0]~~\text{and}~~\partial p/\partial x=-\frac{\Delta p}{L} = constant

\rm{v}={\frac{1}{\mathrm{A}}}\int\mathrm{u}(\mathrm{r})\;\mathrm{d}\mathrm{A}={\frac{1}{\pi\,\mathrm{r}_{0}^{2}}}\int\mathrm{u}_{\mathrm{max}}\left[1-\left\lgroup {\frac{\mathrm{r}}{\mathrm{r}_{0}}}\right\rgroup ^{2}\right](2\pi\,\mathrm{r}\;\mathrm{d}\mathrm{r})

={\frac{1}{2}}\mathrm{u}_{\mathrm{max}}={\frac{\mathrm{r}_{0}^{2}}{8\mu}}\left\lgroup{\frac{\Delta\mathrm{p}}{\mathrm{L}}}\right\rgroup =\mathrm{Q}/(\pi\,\mathrm{r}_{0}^{2})              (E.4.1.1a)

\tau_{\mathrm{wall}}=\mu\left.{\frac{\mathrm{du}}{\mathrm{dr}}}\right|_{\mathrm{r}=r_{0}}=4\mu\ \mathrm{v}/\ \mathrm{r}_{0}={\frac{\mathrm{r}_{0}}{2}}\left\lgroup \frac{\Delta\mathrm{p}}{ L}\right\rgroup                                   (E.4.1.1b)

(C) Extended Bernoulli equation applied to the pipe centerline from points ➀–➁

\rm \left\lgroup{\frac{\mathrm{p}}{\mathrm{\rho g}}}+{\frac{\mathrm{v}^{2}}{2\mathrm{g}}}+z\right\rgroup _{➀}=\left\lgroup{\frac{\mathrm{p}}{\mathrm{\rho g}}}+{\frac{\mathrm{v}^{2}}{2\mathrm{g}}}+z\right\rgroup _{➁}+{\mathrm{h}_{\mathrm{loss}}}                            (E.4.1.2)

\rm h_{loss}\equiv h_L=h_{\underset{<major>}{friction}}+h_{\underset{<maior>}{form} }                                  (E.4.1.3a)

•With Δz = 0, \rm v_1 = v_2

and \rm h_{form}\approx 0, h_{friction}=h_f=\frac{\Delta p}{\rho g}                          (E.4.1.3b)

(D) Combine (B) and (C) results

• From \begin{array}{c c}\rm{{\mathrm{v}=\frac{r_{0}^{2}}{8\mu}\left\lgroup\frac{\Delta{ p}}{ L}\right\rgroup ;}}&{{\mathrm{~r_{0}=D/2}}}\end{array}                                  (E.4.1.4a)

we have

Δp = 32μ v L/D²                           (E.4.1.4b)

As a result,

\mathrm{h}_{\mathrm{f}}={\frac{32\mathrm{\mu}\;{\mathrm{L}}\;\mathrm{v}}{\rho\mathrm{g}\;\mathrm{D}^{2}}}:=\frac{128\mathrm{\mu}\;{\mathrm{L}}\;\mathrm{Q}}{\pi\;\rho\mathrm{g}\;\mathrm{D}^{4}}                          (E.4.1.5a, b)

Now, defining the friction factor f (Darcy–Weisbach) as:

\mathrm{f}\equiv\frac{8\tau_{\mathrm{w}}}{\mathrm{\rho{\mathrm{{v}}}^{2}}}:=\frac{2\mathrm{D}}{\mathrm{\rho{\mathrm{{v}}}^{2}}}\left\lgroup\frac{\Delta\mathrm{p}}{\mathrm{L}}\right\rgroup                            (E.4.1.6a)

so that

\rm\Delta{ p}={f}~\frac{\rho{ v}^{2}}{2}\left\lgroup\frac{{ L}}{ D}\right\rgroup                            (E.4.1.6b)

and hence we can express the frictional loss also as:

\rm h_{\mathrm{f}}={\frac{\Delta{ p}}{\rho g}}=f{\frac{{ v}^{2}}{2{{{g}}}}}{\left\lgroup{\frac{{ L}}{{ D}}}\right\rgroup }                          (E.4.1.7)

(E) Notes

• With \tau_{ w}=4\mu\,\nu /{r}_{0}\,,\,\,\,f_{\mathrm{laminar}}=\frac{32\mu \nu }{r_{0}\,\rho\,\nu ^{2}}=\frac{64}{\mathrm{Re}_{D}}                        (E.4.1.8, 9)

• From a 1-D force balance for any fully-developed flow regime (see momentum RTT) we obtain (with \rm v_{in} = v_{out} ):

\Sigma \mathrm{F}_{\mathrm{x}}\,=\,\underbrace{\Delta{\rm p}(\pi r_{0}^{2}\,)}_{\mathrm{F_{\mathrm{net,pressure}}}}-\underbrace{\rho\mathrm{g}(\pi r_{0}^{2}\,)\mathrm{L}\,\mathrm{sin}\,\phi}_{\rm W_x}-\underbrace{\tau_{\mathrm{w}}\,(2\pi\mathrm{r}_{0}^{2}\,)}_{\mathrm{F_{viscous}}}                              (E.4.1.10)

so that with L sin \phi = − Δz :

\mathrm{h}_{\mathrm{f}}=\Delta {\rm Z}+{\frac{\Delta\mathrm{p}}{\mathrm{\rho}{\mathrm{g}}}}={\frac{2\tau_{\mathrm{w}}}{\mathrm{\rho}{\mathrm{g}}}}\,{\frac{\mathrm{L}}{\mathrm{r}_{0}}}={\frac{4\tau_{\mathrm{w}}}{\mathrm{\rho}{\mathrm{g}}}}\left\lgroup{\frac{\mathrm{L}}{\mathrm{D}}}\right\rgroup                          (E.4.1.11)

or with \rm f ≡ 8τ_w | ( ρv^2 ) we have again:

\mathrm{h}_{\mathrm{f}}=\mathrm{f}~{\frac{\mathrm{v}^{2}}{2g}}\left\lgroup{\frac{\mathrm{L}}{\mathrm{D}}}\right\rgroup                                  (E.4.1.12)

Graph:

• On a log-log graph, we have the analytical and measured f(\rm Re_D ) function:

Commen:t
• Clearly, the \rm h_f -correlation holds for both laminar and turbulent pipe flows, where

\mathrm{f}_{\mathrm{laminar}}=64\,/{\sqrt{\mathrm{Re_{D}}}}\quad\mathrm{~and~}\quad\mathrm{f}_{\mathrm{turbulent}}=\mathrm{f}\left\lgroup\,\mathrm{Re_{D}}\,,\,{\frac{{\varepsilon }}{\mathrm{D}}}\right\rgroup

Turbulent friction factor values are obtainable, for example, via Eq. (4.5e), or from the Moody chart (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)

example 4.1

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