Page 365 - Bird R.B. Transport phenomena

§11Л Use of the Equations of Change to Solve Steady-State Problems 347
Fig. 11.4-3. The temperature and velocity profiles in
the neighborhood of a vertical heated plate.

1

8
1 (Г-

ел

SOLUTION We postulate that v = b v (y, z) + 6 z;(y, z) and that T = TXy, z). We assume that the heated
y y
z
2
fluid moves almost directly upward, so that v y « v . z Then the x- and y-components of Eq.
11.3-2 give p = p(z), so that the pressure is given to a very good approximation by -dp/dz -
pg = 0, which is the hydrostatic pressure distribution. The remaining equations of change are
Continuity (11.4-33)

Motion (11.4-34)

Energy РЧ\ y^, + v ^ u (T - T ) (11.4-35)
v
Sy 2 dz 1 x
in which p and /3 are evaluated at the ambient temperature TV The dashed-underlined terms
will be omitted on the ground that momentum and energy transport by molecular processes
in the z direction is small compared with the corresponding convective terms on the left side
of the equations. These omissions should give a satisfactory description of the system except
for a small region around the bottom of the plate. With this simplification, the following
boundary conditions suffice to analyze the system up to z = H:
B.C. 1: at у = 0, v y = v z = 0 and T = T o (11.4-36)
B.C. 2: asy—> ±c and T- (11.4-37)
B.C. 3: at z = 0, v=0 (11.4-38)
Note that the temperature rise appears in the equation of motion and that the velocity distrib-
ution appears in the energy equation. Thus these equations are "coupled/ 7 Analytic solutions
of such coupled, nonlinear differential equations are very difficult, and we content ourselves
here with a dimensional analysis approach.
To do this we introduce the following dimensionless variables:

= — =J- = dimensionless temperature (11.4-39)

- — - dimensionless vertical coordinate (11.4-40)
ti