Page 25 - Modelling in Transport Phenomena A Conceptual Approach

6 CHAPTER 1. INTRODUCTION

surroundings through the boundaries of the system. In either case, the rate of
input and/or output of a quantity is expressed by using the flux of that particular
quantity. The flux of a quantity may be either constant or dependent on position.
Thus, the rate of a quantity can be determined as

[ (Flux)(Area) if flux is constant

Inlet/Outlet rate = (1.3-1)
Flux dA if flux is position dependent

where A is the area perpendicular to the direction of the flux. The differential
areas in cylindrical and spherical coordinate systems are given in Section A.l in
Appendix A.

Example 1.3 Note that the velocity can be interpreted as the volumetric flux
( m3/m2. s). Therefore, volumetric flow rate can be calculated by the integration
of velocity distribution over the cross-sectional area that is perpendicular to the
flow direction. Consider the flow of a very viscous fluid in the space between two
concentric spheres as shown in Figure 1.1. The velocity distribution is given by
Bird et al. (1960) as

Ue =
2 pCLE(c) sin 0
where
E(€) = ln ( )
1 +cos€
1-cose

Use the velocity profile to find the volumetric flow rate, &.

Solution

Since the velocity is in the 0-direction, the differential area that is perpendicular
to the flow direction is given by Eq. (A.l-9) in Appendix A as
dA = r sin 0 drdq5 (1)

Therefore, the volumetric flow rate is