## General Relations For Pipe Flows

For steady, uniform, fully developed flow in a pipe (or any conduit), the conservation of mass, energy, and momentum equations can be arranged in specific forms that are most useful for the analysis of such problems. These general expressions are valid for both Newtonian and non-Newtonian fluids in either laminar or turbulent flow.

### A. Energy Balance

Consider a section of uniform cylindrical pipe of length L and radius R, inclined upward at an angle 0 to the horizontal, as shown in Fig. 6-2. The steady-state energy balance (or Bernoulli equation) applied to an incompressible fluid flowing in a uniform pipe can be written

where O = P + pgz, Kf = 4/L/D, and f is the Fanning friction factor.

B. Momentum Balance

We can write a momentum balance on a cylindrical volume of fluid of radius r, length L, centered on the pipe centerline (see Fig. 6-2) as follows:

Fx = (Pi - P2)^r2 - nr2Lpg sin 6 + 2nrLxrx = 0 (6-3)

where xrx is the force in the x direction acting on the r surface of the fluid system. Solving Eq. (6-3) for xrx gives

AOr r

where AO = AP + pgL sin 6 = AP + pg Az, and rw is the stress exerted by the fluid on the tube wall [i.e., rw = (—trx)r=R]. Note that Eq. (6-4) also follows directly from integrating the axial component of the microscopic momentum equation of motion in cylindrical coordinates (i.e., the z-com-ponent equation in Appendix E).

Equation (6-4) is equivalent to Eq. (6-2), because f — Tw — Kf — ef 6

Note that from Eq. (6-4) the shear stress is negative (i.e., the fluid outside the cylindrical system of radius r is moving more slowly than that inside the system and hence exerts a force in the -x direction on the fluid in the system, which is bounded by the r surface). However, the stress at the wall (rw) is defined as the force exerted in the +x direction by the fluid on the wall (which is positive).

### C. Continuity

Continuity provides a relationship between the volumetric flow rate (Q) passing through a given cross section in the pipe and the local velocity (vx), i-e-, xQ = «CT^1

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