Page 84 - Bird R.B. Transport phenomena

Problems 69

Parabolic
velocity -
distribution

Fig. 2C.1 Particle trajectory in an electric dust collector. The particle that begins at z = 0 and
ends at x = +B may not necessarily travel the longest distance in the z direction.

(Po ~ Vdt m e particle mass m, and the gas viscosity /x. That is, for what length L will the smallest
particle present (mass m) reach the bottom just before it has a chance to be swept out of the chan-
nel? Assume that the flow between the plates is laminar so that the velocity distribution is de-
scribed by Eq. 2B.3-2. Assume also that the particle velocity in the z direction is the same as the
fluid velocity in the z direction. Assume further that the Stokes drag on the sphere as well as the
gravity force acting on the sphere as it is accelerated in the negative x direction can be neglected.
(b) Rework the problem neglecting acceleration in the x direction, but including the Stokes drag.
(c) Compare the usefulness of the solutions in id) and (b), considering that stable aerosol par-
3
ticles have effective diameters of about 1-10 microns and densities of about 1 g/cm .
2
5
Answer: (a) L min = [12(p - p ) B m/'25/A«] 1/4
L
0
2C.2 Residence time distribution in tube flow. Define the residence time function Fit) to be that
fraction of the fluid flowing in a conduit which flows completely through the conduit in a
time interval t. Also define the mean residence time t m by the relation
tdF (2C.2-1)
(a) An incompressible Newtonian liquid is flowing in a circular tube of length L and radius
R, and the average flow velocity is (v ). Show that
z
Fit) = 0 for t < (L/2(v )) (2C.2-2)
z
F(t) = 1 - iL/2(v )t) 2 for t > (L/2(v )) (2C.2-3)
z
z
(b) Show that t , = H/(v )).
n z
2C.3 Velocity distribution in a tube. You have received a manuscript to referee for a technical
journal. The paper deals with heat transfer in tube flow. The authors state that, because they
are concerned with nonisothermal flow, they must have a "general" expression for the veloc-
ity distribution, one that can be used even when the viscosity of the fluid is a function of tem-
perature (and hence position). The authors state that a "general expression for the velocity
distribution for flow in a tube" is

v
z (2C.3-1)

in which у = r/R. The authors give no derivation, nor do they give a literature citation. As the
referee you feel obliged to derive the formula and list any restrictions implied.