Page 341 - Bird R.B. Transport phenomena

Problems 325

- T a )
Answer: P = •
(1 - K )R (KR) 2 1 - Л 1 - 2 In к
2
2h 4k к
10B.9. Plug flow with forced-convection heat transfer. Very thick slurries and pastes sometimes
move in channels almost as a solid plug. Thus, one can approximate the velocity by a con-
stant value v over the conduit cross section.
0
(a) Rework the problem of §10.8 for plug flow in a circular tube of radius R. Show that the
temperature distribution analogous to Eq. 10.8-31 is
Щ,& = 2£ + 1?-\ (10В.9-1)

in which f = kz/pC^R}, and @ and £ are defined as in §10.8.
(b) Show that for plug flow in a plane slit of width 2B the temperature distribution analogous
to Eq. 10B.7-4 is
(10B.9-2)
2
in which £ = kz/pC pv QB , and в and a are defined as in Problem 10B.7.
10B.10. Free convection in an annulus of finite height (Fig. 10B.10). A fluid is contained in a vertical
annulus closed at the top and bottom. The inner wall of radius KR is maintained at the tem-
perature T , and the outer wall of radius R is kept at temperature T v Using the assumptions
K
and approach of §10.9, obtain the velocity distribution produced by free convection.
(a) First derive the temperature distribution
Ing (10B.10-1)
In к
in which f = r/R.
(b) Then show that the equation of motion is

(10B.10-2)

in which A = (R /fi)(dpIdz + g) and В = ((p^AT)R /fi In к) where AT = Г, - T .
2
2
P}
K
(c) Integrate the equation of motion (see Eq. C.l-11) and apply the boundary conditions to
evaluate the constants of integration. Then show that A can be evaluated by the requirement
of no net mass flow through any plane z = constant, with the final result that
2
2
(1 - к )(1 - Зк ) - 4K 4 In к
= •
v 7 2 2 4
16 fi (1 - к ) + (1 - к ) In к
(10B.10-3)

)rfff
- T = T K at r = KR

11
- T = Ti at r = R

Fig. 10B.10. Free convection pattern in an annular space
with 7 ! > T .
1
K