Internal flow

In internal flow, a pipe is completely filled with a fluid. Laminar flow is characterized by smooth streamlines and highly ordered motion, and turbulent flow is characterized by unsteady disorderly velocity fluctuations and highly disordered motion. The Reynolds number is defined as$L_h.$

$Re=\frac{Intertial forces}{Viscous forces}=\frac{V_{avg}D}{v}=\frac{\rho V_{avg}D}{\mu }$

Under most practical conditions, the flow in a pipe is laminar at Re < 2300, turbulent at Re > 4000, and transitional in between.

The region of the flow in which the effects of the viscous shearing forces are felt is called the velocity boundary layer. The region from the pipe inlet to the point at which the flow becomes fully developed is called the hydrodynamic entrance region, and the length of this region is called the hydrodynamic entry length $L_h.$ It is given by:

$\frac{L_{h,lamninar}}{D}\cong 0.05 Re and \frac{L_{h,turbulent}}{D}\cong 10$

The friction coefficient in the fully developed flow region is constant. The maximum and average velocities in fully developed laminar flow in a circular pipe are:

$u_{max}=2V_{avg} and V_{avg}=\frac{∆P{D}^2}{32\mu L}$

The volume flow rate and the pressure drop for lamninar flow in a horizontal pipe are

$$\begin{gathered} \dot{V}=V_{avg}{A}_c=\frac{∆P{\mathrm{\pi D}}^4}{128\mu L} \\ and \\ ∆P=\frac{32\mu L{V}_{avg}}{D^2} \end{gathered}$$

The pressure loss and head loss for all types of internal flows (laminar or turbulent, in circular or noncircular pipes, smooth or rough surfaces) are expressed as

$$\begin{gathered} ∆P_L=f\frac{L}{D}\frac{\rho V^2}{2} \\ and \\ {h}_L=\frac{∆P_L}{\rho g}=f\frac{L}{D}\frac{V^2}{2g} \end{gathered}$$

where $\frac{\rho V^2}{2}$is the dynamic pressure and the dimensionless quantity f is the friction factor. For fully developed laminar flow in a round pipe, the friction factor is $f=64/Re.$

For noncircular pipes, the diameter in the previous relations is replaced by the hydraulic diameter defined as $$\begin{gathered} D_h=4A_c/p, where A_c is the cross-sectional area of the pipe and \\ p is its wetted perimeter. \end{gathered}$$