Page 344 - Bird R.B. Transport phenomena

328 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

(a) Derive an equation for the steady-state temperature distribution in the wire, assuming
that Г depends on z alone; that is, the radial temperature variation in the wire is neglected.
Further, assume uniform thermal and electrical conductivities in the wire, and a uniform heat
transfer coefficient from the wire to the air stream.
(b) Sketch the temperature profile obtained in (a).
(c) Compute the current, in amperes, required to heat a platinum wire to a midpoint temper-
ature of 50°C under the following conditions:
T = 20°C h = 100 Btu/hr • ft 2 • F
L
D = 0.127 mm k= 40.2 Btu/hr • ft • F
1
L = 0.5 cm k = 1.00 X 10 ohm" cm" 1
5
e
Answers: (a) T - T = Щ-1 1 - w o " v - ^ " j . ) 1.01 amp
L ( b
4/*/c V coshy/W/DU
e
10B.17. Non-Newtonian flow with forced-convection heat transfer. 1 For estimating the effect of
non-Newtonian viscosity on heat transfer in ducts, the power law model of Chapter 8 gives
velocity profiles that show rather well the deviation from parabolic shape.
(a) Rework the problem of §10.8 (heat transfer in a circular tube) for the power law model
given in Eqs. 8.3-2,3. Show that the final temperature profile is
3
3
2(s + 3) (s + 3) 2 2 . +3 ( s + > ~ 8 (ЮБ17-1)
s
(s + 1) * Us + 1) g (s + l)(s + 3) g 4(s + l)(s + 3)(s + 5)
in which s = \/n.
(b) Rework Problem 10B.7 (heat transfer in a plane slit) for the power law model, Obtain the
dimensionless temperature profile:
1 • | s + 3 (s + 2)(s + 3)(2s + 5) - б]

Note that these results contain the Newtonian results (s = 1) and the plug flow results (s = oo).
See Problem 10D.2 for a generalization of this approach.
2
10B.18. Reactor temperature profiles with axial heat flux (Fig. 10B.18).
(a) Show that for a heat source that depends linearly on the temperature, Eqs. 10.5-6 to 14
have the solutions (for m + Ф m_)
m m_(exp m — exp m_)
& = 1 + — + + — exp [(m + 4- m_)Z] (10B.18-1)
exp tn
exp m_
m m+ exp tn+ - mi exp m
mi
-m (exp nQ(exp
0» =
m + exp m + - m_ exp m_
2 УИ
ш =
~-
exp (m
+ + m_) (10B.18-3)
+ exp m - mi exp m_
+
Here Ш, = ^B(l ± Vl - (4N/B), in which В = pv C L/K ei{tZZ . Some profiles calculated from
p
0
these equations are shown in Fig. 10B.18.
1 R. B. Bird, Chem.-Ing. Technik, 31,569-572 (1959).
2
Taken from the corresponding results of G. Damkohler, Z. Elektrochem., 43,1-8, 9-13 (1937), and
J. F. Wehner and R. H. Wilhelm, Chem. Engr. Sci. 6, 89-93 (1956); 8, 309 (1958), for isothermal flow reactors
r
with longitudinal diffusion and first-order reaction. Gerhard Damkohler (1908-1944) achieved fame for
his work on chemical reactions in flowing, diffusing systems; a key publication was in Der Chemie-
Ingenieur, Leipzig (1937), pp. 359-485. Richard Herman Wilhelm (1909-1968), chairman of the Chemical
Engineering Department at Princeton University, was well known for his work on fixed-bed catalytic
reactors, fluidized transport, and the "parametric pumping'' separation process.