Fully Developed Pipe Flow and Head Losses

Fully developed internal flow wall shear stress

In fully developed internal flow, the velocity profile no longer changes in the flow direction, so the streamwise pressure gradient is constant and is balanced by the shear stress at the wall. For a control volume of length L with cross-sectional area A and wetted perimeter P, the pressure force (p_1-p_2)A balances the wall shear force \tau_w P L, so (p_1-p_2)A=\tau_w P L. For flow between two parallel walls separated by height h, the balance per unit width gives \Delta p\,h = 2\tau_w L, so \tau_w = \dfrac{\Delta p\,h}{2L}. For a circular pipe of diameter D, the same balance gives \tau_w = \dfrac{\Delta p\,D}{4L}. Use the absolute value when only the magnitude of the average wall stress is asked, since the pressure drop is positive in the flow direction. For example, in a pipe of diameter 0.04 m with a pressure drop of 80 Pa over 2 m, \tau_w = \dfrac{80\times0.04}{4\times2}=0.4 Pa. Doubling the length at fixed velocity, diameter and fluid properties doubles the head loss but does not change the friction factor, because the friction factor depends only on Reynolds number and relative roughness.

In the entrance (developing) region, the boundary layer grows from the inlet and the velocity gradient at the wall is largest at the entrance. The wall shear stress therefore decreases in the flow direction. Once the boundary layer fills the pipe and the flow becomes fully developed, the wall shear stress becomes constant along the pipe. Thus a section near the entrance has a larger wall shear stress than a section further downstream in the developing zone, while two sections in the fully developed zone have equal wall shear stress. In a laminar pipe flow with a parabolic profile, the shear stress varies linearly with radius, being zero at the centreline and maximum at the wall. The wall value can be found from the velocity gradient at r=R as \tau_w=\mu\left|\dfrac{du}{dr}\right|_{r=R}.

Darcy and Fanning friction factors

The Darcy friction factor f is defined through the Darcy–Weisbach head loss relation h_f = f\dfrac{L}{D}\dfrac{V^2}{2g}, so that the head loss per unit length is \dfrac{h_f}{L}=\dfrac{fV^2}{2gD}. The Fanning friction factor C_f is defined through the wall shear stress, \tau_w = C_f\dfrac{\rho V^2}{2}. Combining these definitions gives \tau_w = \dfrac{f\rho V^2}{8}, so C_f = f/4, or equivalently f = 4C_f. Many textbooks and exam questions use these two factors interchangeably, so it is essential to check which one is being asked for.

For fully developed laminar flow in a circular pipe, f = 64/Re_D with Re_D = \rho V D/\mu, and the friction factor depends only on the Reynolds number. The laminar major head loss can also be written directly as h_f = \dfrac{32\mu L V}{\rho g D^2}, which is useful when the pressure drop and viscosity are known. The total head loss in a pipe system is the sum of the major loss h_f and the minor losses from fittings, valves and entries, h_m = K\dfrac{V^2}{2g}. When a valve is inserted in a pipe, the minor loss equals the measured total loss minus the computed major loss. Given a friction factor, the head loss per metre or the pipe diameter can be found by rearranging \dfrac{h_f}{L}=\dfrac{fV^2}{2gD}. For example, with f=0.03, V=2 m/s and D=0.2 m, the head loss per metre is \dfrac{0.03\times4}{2\times9.81\times0.2}\approx0.031 m.

Entrance head loss coefficient

When a fluid enters a pipe from a reservoir, the flow contracts and separates near the edge, creating an entrance loss. This loss is written as h_e = K_e\dfrac{V^2}{2g}, where the entrance loss coefficient K_e depends on the geometry of the entry. A sharp-edged flush entry has K_e \approx 0.5. A projecting (re-entrant) entry, where the pipe sticks into the reservoir, causes stronger separation and contraction, and K_e is close to 1.0. A well-rounded (bell-mouth) entry keeps the flow attached, so K_e is very small, about 0.05. The ranking of the coefficients is therefore projecting entry > sharp-edged entry > rounded entry. For example, a rounded entry with K_e=0.05 at V=2 m/s gives h_e = \dfrac{0.05\times4}{2\times9.81}\approx0.010 m. The entrance loss is a local loss and is added to the friction loss when computing the total head loss in the pipe.