Page 280 - Modelling in Transport Phenomena A Conceptual Approach

260 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

Left Side Right Side
Part (a) 5.9 9.7
Part (b) 0.2 0.3

Note that the physical significance of the Biot number was given by Eq. (7.1.141,
2.e.,
(Difference in driving force),,lid
Bi =
(Difference in driving
Therefore, when Bi is large, the temperature drop between the surface of the wall
and the bulk temperature is small and the physical properties can be calculated at the
bulk fluid temperature rather than the film temperature in engineering calculations.
On the other hand, when Bi is small, the temperature drop between the surface of
the wall and the bulk fluid temperature is large and the physical properties must be
evaluated at the film temperature. Evaluation of the physical properties at the bulk
fluid temperature for small values of Bi may lead to erroneous results especially if
the physical properties of the fluid are strongly dependent on temperature.

8.2.2 Conduction in Cylindrical Coordinates
Consider a onedimensional transfer of energy in the r-direction in a hollow cylin-
drical pipe with inner and outer radii of R1 and Rz, respectively, as shown in Figure
8.14. Since T = T(r), Table C.5 in Appendix C indicates that the only non-zero
energy flux component is e, and it is given by
dT
e,. = q,. = - k - (8.2-20)
dr
For a cylindrical differential volume element of thickness AT, as shown in Figure
8.14, Eq. (8.2-1) is expressed in the form

(8.2-21)

Dividing Eq. (8.2-21) by AT and taking the limit as AT -+ 0 gives

lim (Aqr)lT - (Aqr)lr+ar =O (8.2-22)
Ar+a Ar
or,
(8.2-23)

Since flux times area gives the heat transfer rate, Q, it is possible to conclude that

A qr = constant = Q (8.2-24)