3.1: Pipe Wall Shear Stress

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In general objects in a fluid flow experience a resistance proportional to the dynamic pressure of the fluid:

\[\ \frac{1}{2} \cdot \rho_{1} \cdot \mathrm{v_{1 s}^{2}}\]

For an object in a fluid flow (like settling particles) the drag force on the object is the dynamic pressure times a characteristic cross section times a drag coefficient, giving:

\[\ \mathrm{F}_{\mathrm{drag}}=\mathrm{C}_{\mathrm{D}} \cdot \frac{\mathrm{1}}{\mathrm{2}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{\mathrm{2}} \cdot \mathrm{A}_{\mathrm{o} \mathrm{b} \mathrm{j}}\]

The drag coefficient normally depends on the Reynolds number of the flow. Now with pipe flow, there is no flow around an object, but there is flow inside the pipe. The basic principles however remain the same, giving for the wall shear stress:

\[\ \tau_{w}=\mathrm{f} \cdot \frac{1}{2} \cdot \rho_{l} \cdot \mathrm{v_{ls}^{2}}\]

The proportionality coefficient f is the so called Fanning friction factor, named after John Thomas Fanning (1837- 1911). The friction force or drag force on a pipe with diameter D_p and length ΔL is now:

\[\ \mathrm{F}_{\mathrm{drag}}=\tau_{\mathrm{w}} \cdot \mathrm{A}_{\mathrm{p w}}=\mathrm{f} \cdot \frac{\mathrm{1}}{\mathrm{2}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2} \cdot \pi \cdot \mathrm{D}_{\mathrm{p}} \cdot \Delta \mathrm{L}\]

The pressure difference over the pipe with diameter D_p and length L is the drag force divided by the pipe cross section A_p:

\[\ \Delta \mathrm{p}_{\mathrm{l}}=\frac{\mathrm{F}_{\mathrm{d r a g}}}{\mathrm{A}_{\mathrm{p}}}=\frac{\mathrm{f} \cdot \frac{\mathrm{1}}{\mathrm{2}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{\mathrm{2}} \cdot \pi \cdot \mathrm{D}_{\mathrm{p}} \cdot \Delta \mathrm{L}}{\frac{\pi}{\mathrm{4}} \cdot \mathrm{D}_{\mathrm{p}}^{\mathrm{2}}}=\mathrm{2} \cdot \mathrm{f} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{\mathrm{2}}\]

The notation using the Darcy friction factor also called the Darcy Weisbach friction factor or the Moody friction factor is more convenient here for using the dynamic pressure, giving:

\[\ \Delta \mathrm{p}_{\mathrm{l}}=\lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{\mathrm{2}} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{\mathrm{2}}\]

Note that the Darcy Weisbach friction factor is 4 times the Fanning friction factor. In terms of the shear stress this gives:

\[\ \tau_{w}=\frac{\lambda_{1}}{4} \cdot \frac{1}{2} \cdot \rho_{1} \cdot \mathrm{v_{1 s}^{2}=\frac{\lambda_{1}}{8} \cdot \rho_{1} \cdot v_{1 s}^{2}}\]

The hydraulic gradient i_w (for water) or i_l (for a liquid in general) is:

\[\ \Delta \mathrm{p}_{\mathrm{l}}=\lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2} \quad\text{ and }\quad \mathrm{i}_{\mathrm{l}}=\mathrm{i}_{\mathrm{w}}=\frac{\Delta \mathrm{p}_{\mathrm{l}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}\]

In this book the Darcy Weisbach friction factor is used.