Page 354 - Modelling in Transport Phenomena A Conceptual Approach

334 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

9.1.3 Flow in a Circular Tube

Consider the flow of a Newtonian fluid in a vertical circular pipe under steady con-
ditions as shown in Figure 9.3. The pressure gradient is imposed in the z-direction.

E

Figure 9.3 Flow in a circular pipe.

Simplification of the velocity components according to Figure 8.4 shows that
v, = v,(r) and v, = ve = 0. Therefore, from Table C.2 in Appendix C, the only
non-zero shear stress component is r,, and the components of the total momentum
flux are given by

dV,
Trz = rrz + (pv,) v, = r,, = - p - (9.1-65)
dr
re% = re+ + (pv,) vg (9.1-66)
A,% = 72, + (pv,) v, (9.1-67)

The pressure in the pipe depends on z. Therefore, it is necessary to consider only
the z-component of the equation of motion.
For a cylindrical differential volume element of thickness AT and length Az, as
shown in Figure 9.3, Eq. (9.1-2) is expressed as