Page 373 - Bird R.B. Transport phenomena
P. 373
§11.5 Dimensional Analysis of the Equations of Change for Nonisothermal Systems 355
Table 11.5-1 Dimensionless Groups in Equations 11.5-7, 8, and 9
Special Forced Free Free
cases —» convection Intermediate convection convection
(A) (B)
Choice
for VQ —> Щ e/'o
J± 1 Pr
Re Re
- Го) Gr 2
Neglect Gr GrPr
Re 2
[ * 1 1 1
RePr RePr Pr
Br Br Neglect Neglect
RePr RePr
- To)
Notes:
" For forced convection and forced-plus-free ("intermediate") convection, v is generally
0
taken to be the approach velocity (for flow around submerged objects) or an average
velocity in the system (for flow in conduits).
b
For free convection there are two standard choices for % labeled as A and B. In §10.9,
Case A arises naturally. Case В proves convenient if the assumption of creeping flow is
appropriate, so that Dir/Dt can be neglected (see Example 11.5-2). Then a new
dimensionless pressure difference Ф = РгФ, different from & in Eq. 3.7-4, can be
introduced, so that when the equation of motion is divided by Pr, the only dimensionless
group appearing in the equation is GrPr. Note that in Case B, no dimensionless groups
appear in the equation of energy.
It is sometimes useful to think of the dimensionless groups as ratios of various
forces or effects in the system, as shown in Table 11.5-3. For example, the inertial term in
the equation of motion is p[v • Vv] and the viscous term is /xV v. To get "typical" values
2
of these terms, replace the variables by the characteristic "yardsticks" used in construct-
2
ing dimensionless variables. Hence replace p[v • Vv] by pvl/l , and replace /iV v by
0
ILV /IO to get rough orders of magnitude. The ratio of these two terms then gives the
Q
Reynolds number, as shown in the table. The other dimensionless groups are obtained in
similar fashion.
Table 11.5-2 Dimensionless Groups Used in
Nonisothermal Systems
Re = ll v /f4 = ll o v o /p]I = Reynolds number
o oP
;
Pr = [C>/fc] = Wai = Prandtl number
Gr = lgP(T ] - T )l o/A = Grashof number
'
3
o
o r — ^JJLVQ/ K\l i -To)} = Brinkman number
Pe = RePr = Peclet number
Ra = GrPr = Rayleigh number
Ec = Br/Pr = Eckert number