Page 335 - Bird R.B. Transport phenomena

Questions for Discussion 319

The velocity distribution in Eq. 10.9-15 may be rewritten using a dimensionless ve-
locity v = BV^Z/JL and a dimensionless coordinate у = y/B thus:
z
V) (10.9-17)
2
Here Gr is the dimensionless Grashof number, defined by
Gr = (10.9-18)
1
where Ap = p - p . The second form of the Grashof number is obtained from the first
x 2
form by using Eq. 10.9-6. The Grashof number is the characteristic group occurring in
analyses of free convection, as is shown by dimensional analysis in Chapter 11. It arises
in heat transfer coefficient correlations in Chapter 14.

QUESTIONS FOR DISCUSSION
1. Verify that the Brinkman, Biot, Prandtl, and Grashof numbers are dimensionless.
2. To what problem in electrical circuits is the addition of thermal resistances analogous?
3. What is the coefficient of volume expansion for an ideal gas? What is the corresponding ex-
pression for the Grashof number?
4. What might be some consequences of large temperature gradients produced by viscous heat-
ing in viscometry, lubrication, and plastics extrusion?
5. In §10.8 would there be any advantage to choosing the dimensionless temperature and di-
mensionless axial coordinate to be © = (T - Т )/Т and f = z/R?
л
л
6. What would happen in §9.9 if the fluid were water and T were 4°C?
7. Is there any advantage to solving Eq. 9.7-9 in terms of hyperbolic functions rather than expo-
nential functions?
8. In going from Eq. 10.8-11 to Eq. 10.8-12 the axial conduction term was neglected with respect
to the axial convection term. To justify this, put in some reasonable numerical values to esti-
mate the relative sizes of the terms.
9. How serious is it to neglect the dependence of viscosity on temperature in solving forced con-
vection problems? Viscous dissipation heating problems?
10. At steady state the temperature profiles in a laminated system appear thus:
Which material has the higher thermal conductivity?

Material I Material II
— •
^rature

6u
1

Distance -

11. Show that Eq. 10.6-4 can be obtained directly by rewriting Eq. 10.6-1 with x + Ax replaced by
r
x . Similarly, one gets Eq. 10.6-20 from Eq. 10.6-17, with r + Ar replaced by .
0
Q

2
Named for Franz Grashof (1826-1893) (pronounced "Grahss-hoff). He was professor of applied
mechanics in Karlsruhe and one of the founders of the Verein Deutscher Ingenieure in 1856.