Page 366 - Bird R.B. Transport phenomena

348 Chapter 11 The Equations of Change for Nonisothermal Systems

— — I у = dimensionless horizontal coordinate (11.4-41)
fiaHJ J
( A V / 2
ф г = ( —=z J v z = dimensionless vertical velocity (11.4-42)
ф = I -r— Vy = dimensionless horizontal velocity (11.4-43)
ч 3
\a B/
in which a = k/pC and В = pg/3(T 0 - T ).
p
}
When the equations of change, without the dashed-underlined terms, are written in
terms of these dimensionless variables, we get
дфи дф
Continuity ~ ^ + ^ F ? = 0 (11.4-44)

Motion ^ 0v T" + <k т? <fc = ^ т + ® (11.4-45)
8-j
H
Energy I ф„ ^- + ф ^- 10 = — - (11.4-46)
г
The preceding boundary conditions then become

B.C. 1: a t г) = О, ф = ф = Ъ, 0 - 1 (11.4-47)
г
у
B.C. 2: asrj-»*), ф ^ 0 , 0^0 (11.4-48)
2
B.C.3: at£ = 0, ф = О (11.4-49)
г
One can see immediately from these equations and boundary conditions that the dimension-
less velocity components ф and ф and the dimensionless temperature 0 will depend on 17
2
у
and £ and also on the Prandtl number, Pr. Since the flow is usually very slow in free convec-
tion, the terms in which Pr appears will generally be rather small; setting them equal to zero
would correspond to the "creeping flow assumption." Hence we expect that the dependence
of the solution on the Prandtl number will be weak.
The average heat flux from one side of the plate may be written as
- "
HJ 0 \ -*,)\J* (n 4 50)
The integral may now be written in terms of the dimensionless quantities
В V м Г( дв\
лг
1/4
•c

"' • 1/4

= С • £ (Г - T )(GrPr) 1/4 (11.4-51)
о 1
in which the grouping Ra = GrPr is referred to as the Rayleigh number. Because 0 is a function
of 17, £, and Pr, the derivative dS/drj is also a function of 17, £, and Pr. Then d<d/dir), evaluated
at 17 = 0, depends only on £ and Pr. The definite integral over £ is thus a function of Pr. From
the remarks made earlier, we can infer that this function, called C, will be only a weak func-
tion of the Prandtl number—that is, nearly a constant.
The preceding analysis shows that, even without solving the partial differential equa-
tions, we can predict that the average heat flux is proportional to the f-power of the tempera-
ture difference (T o - T }) and inversely proportional to the J-power of H. Both predictions
have been confirmed by experiment. The only thing we could not do was to find С as a func-
tion of Pr.