Page 349 - Modelling in Transport Phenomena A Conceptual Approach

9. I. MOMENTUM TRANSPORT 329

The volumetric flow rate can be determined by integrating the velocity distri-
bution over the cross-sectional area, i.e.,

Q = Jd" Jd" vz dzdy (9.1-25)

Substitution of EQ. (9.1-23) into EQ. (9.1-25) gives the volumetric flow rate in the
form
(9.1-26)

Dividing the volumetric flow rate by the flow area gives the average velocity as

(9.1-27)

9.1.1.1 Macroscopic balance
Integration of the governing differential equation, Eq. (9.1-19), over the volume of
the system gives the macroscopic momentum balance as

- Jd" Jd" Jd" /I 2 dxdydz = L dxdydz (9.1-28)

or -
1z=O) L W = (Po - PL) WB (9.1-29)
(~ZZ lz=B - Z Z
~
Y
Drag force Pressure and gravitational
forces
Note that EQ. (9.1-29) is nothing more than Newton's second law of motion.
The interaction of the system, i.e., the fluid between the parallel plates, with the
surroundings is the drag force, FD, on the plates and is given by

(9.1-30)

On the other hand, the friction factor is the dimensionless interaction of the
system with the surroundings and is defined by Eq. (3.1-7), Le.,

FD = AchKch(f) (9.1-31)
or,
(9.1-32)

Simplification of Eq. (9.1-32) gives

(9.1-33)