Page 471 - Bird R.B. Transport phenomena

Problems 451

0.7632 cp, C = 0.9986 cal/g • С and к = 0.363 Btu/hr • ft • F. in which 7 is the current required to maintain the desired
p
Also, the density of water in the neighborhood of 90°F is temperature, v^ is the velocity of the approaching fluid,
and С is a constant. How well does this equation agree
T(C) 30.3 31.3 32.3 33.3 34.3 with the predictions of Eq. 14.4-7 or Eq. 14.4-8 for the fluid
3
p(g/cm ) 0.99558 0.99528 0.99496 0.99463 0.99430
and wire of (a) over a fluid velocity range of 100 to 300
Answer: Q/L = 32 Btu/hr • ft ft/s? What is the significance of the constant С in Eq.
14A.9. The ice-fisherman on Lake Mendota. Compare 14B.3-1?
the rates of heat loss of an ice-fisherman, when he is fish- 14B.4. Dimensional analysis. Consider the flow system
ing in calm weather (wind velocity zero) and when the described in the first paragraph of §14.3, for which dimen-
wind velocity is 20 mph out of the north. The ambient air sional analysis has already given the dimensionless veloc-
temperature is -10°F. Assume that a bundled-up ice-fish- ity profile (Eq. 6.2-7) and temperature profile (Eq. 14.3-9).
erman can be approximated as a sphere 3 ft in diameter.
(a) Use Eqs. 6.2-7 and 14.3-9 and the definition of cup-
14B.1. Limiting local Nusselt number for plug flow mixing temperature to get the time-averaged expression.
with constant heat flux.
T 2 iU = a Unction of Re, Pr, L/D (14B.4-1)
j 2j
(a) Equation 10B.9-1 gives the asymptotic temperature
distribution for cooling a fluid of constant physical proper- (b) Use the result just obtained and the definitions of the
ties in plug flow in a long tube with constant heat flux at heat transfer coefficients to derive Eqs. 14.3-12,13, and 14.
the wall. Use this temperature profile to show that the lim-
iting Nusselt number for these conditions is Nu = 8. 14B.5. Relation between h Xoc and /t . In many industrial
ln
tubular heat exchangers (see Example 15.4-2) the tube-
(b) The asymptotic temperature distribution for the analo-
gous problem for plug flow in a plane slit is given in Eq. surface temperature T varies linearly with the bulk fluid
o
10B.9-2. Use this to show that the limiting Nusselt number temperature T . For this common situation /t and /z may
b
ln
Ioc
isNu = 12. be simply interrelated.
(a) Starting with Eq. 14.1-5, show that
14B.2. Local overall heat transfer coefficient. In Prob- 2
h ]oc (7TDdz)(T h - T o ) = -(lrrD )( P C p (v)dT)
lem 14A.1 the thermal resistances of the condensed steam
film and wall were neglected. Justify this neglect by calcu- and therefore that
lating the actual inner-surface temperature of the tubes at \ ,_ i А „,_л W) ~ 40) (14B.5-2)
that cross section in the exchanger at which the oil temper- Jo 'о l bhn
ature is 150°F. You may assume that for the oil /z, is con-
oc (b) Combine the result in (a) with Eq. 14.1-4 to show that
stant throughout the exchanger at 190 Btu/hr • ft 2 • F. The
tubes are horizontal. К = T \ K4z (14B.5-3)
L Jo
14B.3. The hot-wire anemometer. 1 A hot-wire anemome- in which L is the total tube length, and therefore that (if
ter is essentially a fine wire, usually made of platinum, (dh /dL) = 0, which is equivalent to the statement that
loc
z
which is heated electrically and inserted into a flowing axial heat conduction is neglected)
fluid. The wire temperature, which is a function of the fluid
temperature, fluid velocity, and the rate of heating, may be fcioclz=L = fc + L ^ t (14B.5-4)
determined by measuring its electrical resistance. ln
(a) A straight cylindrical wire 0.5 in. long and 0.01 in. in 14B.6o Heat loss by free convection from a pipe. In Ex-
diameter is exposed to a stream of air at 70°F flowing past ample 14.6-1, would the heat loss be higher or lower if the
the wire at 100 ft/s. What must the rate of energy input be pipe-surface temperature were 200°F and the air tempera-
in watts to maintain the wire surface at 600°F? Neglect ra- ture were 180°F?
diation as well as heat conduction along the wire.
14C.1. The Nusselt expression for film condensation
(b) It has been reported that for a given fluid and wire at heat transfer coefficients (Fig. 14.7-1). Consider a laminar
2
given fluid and wire temperatures (hence a given wire film of condensate flowing down a vertical wall, and as-
resistance)
sume that this liquid film constitutes the sole heat transfer
2
I = BVz^ + С (14В.З-1) resistance on the vapor side of the wall. Further assume
that (i) the shear stress between liquid and vapor may be
neglected; (ii) the physical properties in the film may be
See, for example, G. Comte-Bellot, Chapter 34 in The
1
Handbook of Fluid Dynamics (R. W. Johnson, ed.), CRC Press, Boca evaluated at the arithmetic mean of vapor and cooling-
Raton, Fla. (1999). surface temperatures and that the cooling-surface temper-
L. V. King, Phil. Trans. Roy. Soc. (London), A214, 373-432
2 ature may be assumed constant; (iii) acceleration of fluid
(1914). elements in the film may be neglected compared to the