Page 357 - Modelling in Transport Phenomena A Conceptual Approach

9.1. MOMENTUM TRANSPORT 337

9.1.3.1 Macroscopic balance

Integration of the governing differential equation, Eq. (9.1-76), over the volume of
the system gives

- -
- pL) r drdedz (9.1-85)
=J0”1’“6” L
or, T,,I,=R 2nRL = TR2 (Po - PL) (9.1-86)

Drag force Pressure and gravitational
forces
The interaction of the system, i.e., the fluid in the tube, with the surroundings
manifests itself as the drag force, FD, on the wall and is given by

(9.1-87)

On the other hand, the dimensionless interaction of the system with the surround-
ings, i.e., the friction factor, is given by &. (3.1-7), Le.,

or,
nR2 (Po - 7%) = (27Frn) (9.1-89)

Expressing the average velocity in terms of the volumetric flow rate by using Eq.
(9.1-84) reduces Q. (9.1-89) to

n2D5 (Po - PL) (9.1-90)
(’) = 32 pLQ2
which is nothing more than Eq. (4.576).
Elimination of (Po - PL) between Eqs. (9.1-84) and (9.1-89) leads to

(9.1-91)

9.1.4 Axial Flow in an Annulus

Consider the flow of a Newtonian fluid in a vertical concentric annulus under steady
conditions as shown in Figure 9.4. A constant pressure gradient is imposed in the
positive z-direction while the inner rod is stationary.