Page 443 - Modelling in Transport Phenomena A Conceptual Approach

PROBLEMS 423

9.15 For laminar flow forced convection in a circular pipe with a constant wall
concentration, the governing equation for concentration of species A, Eq. (9.5-13),
is integrated over the cross-sectional area of the tube in Section 9.5.1 to obtain Eq.
(9.5-21), i.e.,
dCAb
=
Q - TD kc(CA, - CAb) (1)
dz
a) Now assume that the flow is turbulent. Over a differential volume element of
thickness Az, as shown in the figure below, write down the inventory rate equation
for the mass of species A and show that the result is identical with Eq. (1).

b) Instead of coating the inner surface of a circular pipe with species A, let us
assume that the circular pipe is packed with species A particles. Over a differential
volume element of thickness Az, mite down the inventory rate equation for mass
of species A and show that the result is

where A is the cross-sectional mea of the pipe and a,, is the packing surface area
per unit volume. Note that for a circular pipe a, = 4/0 and A = rD2/4 so that
Eq. (2) reduces to Eq. (1).

9.16 A liquid is being transported in a circular plastic tube of inner and outer
radii of R1 and R2, respectively. The dissolved 02 (species d) concentration in the
liquid is CA,. Develop an expression relating the increase in 02 concentration as a
function the tubing length as follows:
a) Over a differential volume element of thickness Az, write down the inventory
rate equation for the mass of species A and show that the governing equation is

where DAB is the diffusion coefficient of 02 in a plastic tube and CA, is the
concentration of 02 in air surrounding the tube. In the development of Eq. (l),
note that the molar rate of 02 transfer through the tubing can be represented by
Eq. (B) in Table 8.9.
b) Show that the integration of Eq. (1) leads to