Page 379 - Bird R.B. Transport phenomena

Problems 361

In principle, we may solve Eq. 11.5-28 for Gr and obtain an expression for (T - T ). Since we
}
o
are neglecting the temperature dependence of physical properties, we may consider the
Prandtl number constant for the given fluid and write
(11.5-29)

2
2
2
Here ф is an experimentally determinable function of the group Qp gPD /k/jL . We may then
construct a plot of Eq. 11.5-29 from the experimental measurements of T u T , and D for the
o
small-scale system, and the known physical properties of the fluid. This plot may then be
used to predict the behavior of the large-scale system.
Since we have neglected the temperature dependence of the fluid properties, we may go
even further. If we maintain the ratio of the Q values in the two systems equal to the inverse
square of the ratio of the diameters, then the corresponding ratio of the values of (7^ — T )
o
will be equal to the inverse cube of the ratio of the diameters.

QUESTIONS FOR DISCUSSION
1. Define energy, potential energy, kinetic energy, and internal energy. What common units are
used for these?
2. How does one assign the physical meaning to the individual terms in Eqs. 11.1-7 and 11.2-1?
3» In getting Eq. 11.2-7 we used the relation C - C = R, which is valid for ideal gases. What is the
p v
corresponding equation for nonideal gases and liquids?
4. Summarize all the steps required in obtaining the equation of change for the temperature.
5. Compare and contrast forced convection and free convection, with regard to methods of
problem solving, dimensional analysis, and occurrence in industrial and meteorological prob-
lems.
6. If a rocket nose cone were made of a porous material and a volatile liquid were forced slowly
through the pores during reentry into the atmosphere, how would the cone surface tempera-
ture be affected and why?
7. What is Archimedes' principle, and how is it related to the term pg/3(T - T) in Eq. 11.3-2?
8. Would you expect to see Benard cells while heating a shallow pan of water on a stove?
9. When, if ever, can the equation of energy be completely and exactly solved without detailed
knowledge of the velocity profiles of the system?
10o When, if ever, can the equation of motion be completely solved for a nonisothermal system
without detailed knowledge of the temperature profiles of the system?

PROBLEMS 11A.1. Temperature in a friction bearing. Calculate the maximum temperature in the friction bear-
ing of Problem ЗАЛ, assuming the thermal conductivity of the lubricant to be 4.0 X 10" 4 cal/s
• cm • C, the metal temperature 200°C, and the rate of rotation 4000 rpm.
Answer: About 225°C

11A.2. Viscosity variation and velocity gradients in a nonisothermal film. Water is falling down a
vertical wall in a film 0.1 mm thick. The water temperature is 100°C at the free liquid surface
and 80°C at the wall surface.
(a) Show that the maximum fractional deviation between viscosities predicted by Eqs. 11.4-17
and 18 occurs when Г = л/Т Т .
0
5
(b) Calculate the maximum fractional deviation for the conditions given.
Answer: (b) 0.5%