Page 120 - Bird R.B. Transport phenomena

Problems 105

concentric cylinder 4.500 cm in diameter. The length L is tank and extending vertically upward 1 ft from the tank
4.00 cm. The viscosity of a 60% sucrose solution at 20°C is bottom. It is known from experience that, as molasses is
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about 57 cp, and its density is about 1.29 g/cm . withdrawn from the tank, a vortex will form, and, as the
On the basis of past experience it seems possible that liquid level drops, this vortex will ultimately reach the
end effects will be important, and it is therefore decided to draw-off pipe, allowing air to be sucked into the molasses.
calibrate the viscometer by measurements on some known This is to be avoided.
solutions of approximately the same viscosity as those of It is proposed to predict the minimum liquid level at
the unknown sucrose solutions. which this entrainment can be avoided, at a draw-off rate
Determine a reasonable value for the applied torque of 800 gal/min, by a model study using a smaller tank. For
to be used in calibration if the torque measurements are re- convenience, water at 68°F is to be used for the fluid in the
liable within 100 dyne/cm and the angular velocity can be model study.
measured within 0.5%. What will be the resultant angular Determine the proper tank dimensions and operating
velocity? conditions for the model if the density of the molasses is
1.286 g/cm 3 and its viscosity is 56.7 cp. It may be assumed
3A.5 Fabrication of a parabolic mirror. It is proposed to that, in either the full-size tank or the model, the vortex
make a backing for a parabolic mirror, by rotating a pan of shape is dependent only on the amount of the liquid in the
slow-hardening plastic resin at constant speed until it tank and the draw-off rate; that is, the vortex establishes it-
hardens. Calculate the rotational speed required to pro- self very rapidly.
duce a mirror of focal length/ = 100 cm. The focal length is
one-half the radius of curvature at the axis, which in turn 3B.1 Flow between coaxial cylinders and concentric
is given by spheres.
(a) The space between two coaxial cylinders is filled with
' < = ! + X (3A.5-1) an incompressible fluid at constant temperature. The radii
dr 2 of the inner and outer wetted surfaces are KR and R, re-
spectively. The angular velocities of rotation of the inner
Answer: 21.1 rpm
and outer cylinders are Д and Д. Determine the velocity
3A.6 Scale-up of an agitated tank. Experiments with a distribution in the fluid and the torques on the two cylin-
small-scale agitated tank are to be used to design a geo- ders needed to maintain the motion.
metrically similar installation with linear dimensions 10 (b) Repeat part (a) for two concentric spheres.
times as large. The fluid in the large tank will be a heavy
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oil with /JL = 13.5 cp and p = 0.9 g/cm . The large tank is to Answers:
have an impeller speed of 120 rpm.
(a) Determine the impeller speed for the small-scale (a) v e =
model, in accordance with the criteria for scale-up given in
Example 3.7-2.
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(b) Determine the operating temperature for the model if (b) v, = ^ - [(Д, - Д * )(^) + (Д - Д ) ( ^ sin в
water is to be used as the stirred fluid.
Answers: (a) 380 rpm, (b) T = 60° 3B.2 Laminar flow in a triangular duct (Fig. 3B.2). 2
One type of compact heat exchanger is shown in Fig.
3A.7 Air entrainment in a draining tank (Fig. ЗА.7). A ЗВ.2(я). In order to analyze the performance of such an
molasses storage tank 60 ft in diameter is to be built with a apparatus, it is necessary to understand the flow in a duct
draw-off line 1 ft in diameter, 4 ft from the sidewall of the whose cross section is an equilateral triangle. This is done
most easily by installing a coordinate system as shown in
Fig. 3B.2(W.
(a) Verify that the velocity distribution for the laminar
flow of a Newtonian fluid in a duct of this type is given
by
( ^ o - 1 2 2
v = - (y - H)0x - у ) (ЗВ.2-1)
2 4/xLH
= 1.00 f

2 An alternative formulation of the velocity profile is given
by L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon,
Fig 3A.7. Draining of a molasses tank. Oxford, 2nd edition (1987), p. 54.
o