Page 104 - Bird R.B. Transport phenomena

§3.6 Use of the Equations of Change to Solve Flow Problems 89

Equation 3.6-13 is the same as Eq. 2.3-18. The pressure profile in Eq. 3.6-12 was not obtained
in Example 2.3-1, but was tacitly postulated; we could have done that here, too, but we chose
to work with a minimal number of postulates.
As pointed out in Example 2.3-1, Eq. 3.6-13 is valid only in the laminar-flow regime, and
at locations not too near the tube entrance and exit. For Reynolds numbers above about 2100,
a turbulent-flow regime exists downstream of the entrance region, and Eq. 3.6-13 is no longer
valid.

EXAMPLE 3.6-2 Set up the problem in Example 2.2-2 by using the equations of Appendix B. This illustrates
the use of the equation of motion in terms of т.
Falling Film with
Variable Viscosity SOLUTION

As in Example 2.2-2 we postulate a steady-state flow with constant density, but with viscosity
depending on x. We postulate, as before, that the x- and y-components of the velocity are zero
and that v z = v (x). With these postulates, the equation of continuity is identically satisfied.
z
According to Table B.I, the only nonzero components of т are r xz = T = —\x(dvjdx). The
ZX
components of the equation of motion in terms of т are, from Table B.5,
dp
O=~ + pgsmp (3.6-14)
dp
0= —f (3.6-15)
др л
(3.6-16)

where /3 is the angle shown in Fig. 2.2-2.
Integration of Eq. 3.6-14 gives
p = pgx sin j8 (3.6-17)

in which/(y, z) is an arbitrary function. Equation 3.6-15 shows that/cannot be a function of y.
We next recognize that the pressure in the gas phase is very nearly constant at the prevailing
atmospheric pressure p . Therefore, at the gas-liquid interface x = 0, the pressure is also
aim
constant at the value p . Consequently, / can be set equal to p atm and we obtain finally
atm
p = pgx sin p + p (3.6-18)
atn
Equation 3.5-16 then becomes

(3.6-19)

which is the same as Eq. 2.2-10. The remainder of the solution is the same as in §2.2.

EXAMPLE 3.6-3 We mentioned earlier that the measurement of pressure difference vs. mass flow rate through
a cylindrical tube is the basis for the determination of viscosity in commercial capillary vis-
Operation of a Couette cometers. The viscosity may also be determined by measuring the torque required to turn a
Viscometer solid object in contact with a fluid. The forerunner of all rotational viscometers is the Couette
instrument, which is sketched in Fig. 3.6-1.
The fluid is placed in the cup, and the cup is then made to rotate with a constant angular
velocity ft 0 (the subscript "o" stands for outer). The rotating viscous liquid causes the sus-
pended bob to turn until the torque produced by the momentum transfer in the fluid equals
the product of the torsion constant k, and the angular displacement в ь of the bob. The angular
displacement can be measured by observing the deflection of a light beam reflected from a
mirror mounted on the bob. The conditions of measurement are controlled so that there is a
steady, tangential, laminar flow in the annular region between the two coaxial cylinders