Page 359 - Modelling in Transport Phenomena A Conceptual Approach

9.1. MOMENTUM TRANSPORT 339

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

Q = 1'" ll v, T drd6 (9.1-98)
(1 - .">"
Substitution of Q. (9.1-96) into Eq. (9.1-98) and integration gives

&= 1 (9.1-99)
In IC

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

(9.1-100)

9.1.4.1 Macroscopic balance

Integration of the governing differential equation, Eq. (9.1-92), over the volume of
the system gives

T,,~,=R 2nRL - T,z1r=n~27riCRL = 7rR2(1 - fC2) (PO - PL) (9.1-102)
\ * & \ Y
Drag force Pressure and gravitational
forces
Note that Eq. (9.1-102) is nothing more than Newton's second law of motion.
The interaction of the system, i.e., the fluid in the concentric annulus, with the
surroundings is the drag force, FD, on the walls and is given by
I
I FD = xR2(1 - K~) (Po - PL) (9.1-103)

On the other hand, the friction factor is defined by Eq. (3.1-7) as

or.
i
7rR2(1 - ~ (Po - PL) = [27rR(1+ n)L] (f P(V.)~) (f) (9.1-105)
)
~