Page 374 - Bird R.B. Transport phenomena

356 Chapter 11 The Equations of Change for Nonisothermal Systems

Table 11.5-3 Physical Interpretation of Dimensionless Groups
_ pvp/lp _ inertial force
/XV Q /IQ viscous force

_ pvp/lo _ inertial force
PS gravity force
Gr _ PgP(T\ ~ o> _ buoyant force
T
Re 2 pvl/l 0 inertial force
pC v (T - T )/l heat transport by convection
Pe = RePr = p 0 ] 0 0
k(T } - T )/IQ heat transport by conduction
0
fjb(v /l ) 2 heat production by viscous dissipation
B r = • o o
«T, - T )// 2 heat transport by conduction
0

A low value for the Reynolds number means that viscous forces are large in compar-
ison with inertial forces. A low value of the Brinkman number indicates that the heat
produced by viscous dissipation can be transported away quickly by heat conduction.
When Gr/Re 2 is large, the buoyant force is important in determining the flow pattern.
Since dimensional analysis is an art requiring judgment and experience, we give
three illustrative examples. In the first two we analyze forced and free convection in sim-
ple geometries. In the third we discuss scale-up problems in a relatively complex piece
of equipment.

EXAMPLE 11.5-1 It is desired to predict the temperature distribution in a gas flowing about a long, internally
cooled cylinder (system I) from experimental measurements on a one-quarter scale model
Temperature (system II). If possible the same fluid should be used in the model as in the full-scale system.
Distribution about a The system, shown in Fig. 11.5-1, is the same as that in Example 3.7-1 except that it is now
Long Cylinder nonisothermal. The fluid approaching the cylinder has a speed v x and a temperature T , and
x
the cylinder surface is maintained at T , for example, by the boiling of a refrigerant contained
o
within it.
Show by means of dimensional analysis how suitable experimental conditions can be
chosen for the model studies. Perform the dimensional analysis for the "intermediate case" in
Table 11.5-1.

SOLUTION The two systems, I and II, are geometrically similar. To ensure dynamical similarity, as
pointed out in §3.7, the dimensionless differential equations and boundary conditions must
be the same, and the dimensionless groups appearing in them must have the same numerical
values.
Here we choose the characteristic length to be the diameter D of the cylinder, the charac-
teristic velocity to be the approach velocity v x of the fluid, the characteristic pressure to be the
pressure at x = - °° and у = 0, and the characteristic temperatures to be the temperature T x of
the approaching fluid and the temperature T o of the cylinder wall. These characteristic quan-
tities will carry a label I or II corresponding to the system being described.
Both systems are described by the dimensionless differential equations given in Eqs.
11.5-7 to 9, and by boundary conditions
2
B.C.I asiP + j/ -»», v-»6. , f-»l (11.5-10)
Y
B.C. 2 at x 2 + f = i v = 0, f = 0 (11.5-11)
B.C.3 at*-» -oo and у = О, Ф -» 0 (11.5-12)