0.7 Laminar flows with dependence on one dimension  (Page 4/6)

If the fluid is Newtonian, the equation of motion can be integrated once more to obtain the velocity profile and maximum velocity .

The volumetric rate of flow through the pipe can be determined by integration of the velocity profile across the cross-section of the pipe, i.e., $0<r<R$ and $0<\theta <2\pi$ .

This relation is the Hagen-Poiseuille law . If the flow rate is specified, then the potential gradient can be expressed as a function of the flow rate and substituted into the above expressions.

The average velocity or volumetric flux can be determined by dividing the volumetric rate by the cross-sectional area.

Before one begins to believe that the Hagen-Poiseuille law is a "law" that applies under all conditions, the following is a list of assumptions are implicit in this relation (BSL, 1960).

1. The flow is laminar-NRe less than about 2100.
2. The density is constant ("incompressible flow").
3. The flow is independent of time ("steady state").
4. The fluid is Newtonian.
5. End effects are neglected-actually an "entrance length" (beyond the tube entrance) on the order of Le = 0.035D NRe is required for build-up to the parabolic profiles; if the section of pipe of interest includes the entrance region, a correction must be applied. The fractional correction introduced in either P or Q never exceeds Le/L if L>Le.
6. The fluid behaves as a continuum-this assumption is valid except for very dilute gases or very narrow capillary tubes, in which the molecular mean free path is comparable to the tube diameter ("slip flow" regime) or much greater than the tube diameter ("Knudsen flow" or "free molecule flow" regime).
7. There is no slip at the wall-this is an excellent assumption for pure fluids under the conditions assumed in ( f ).

Friction factor and Reynolds number. Because pressure drop in pipes is commonly used in process design, correlation expressed as friction factor versus Reynolds number are available for laminar and turbulent flow. The Hagen-Poiseuille law describes the laminar flow portion of the correlation. The correlations in the literature differ when they use different definitions for the friction factor. Correlations are usually are usually expressed in terms of the Fanning friction factor and the Darcy-Weisbach friction factor.

The Reynolds number is expressed as a ratio of inertial stress and shear stress.

Both the friction factor and the Reynolds number have as a common factor, the kinetic energy per unit volume $\rho u_m^2$ . This factor may be eliminated between the two equations to express the friction factor as a function of the Reynolds number.

Recall the expressions derived earlier for the wall shear stress and the average velocity for a Newtonian fluid and substitute into the above expressions.