Page 53 - Bird R.B. Transport phenomena

38 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

1A.3 Computation of the viscosities of gases at low is rotating like a rigid body. What is the angular velocity of
density. Predict the viscosities of molecular oxygen, nitro- rotation?
gen, and methane at 20°C and atmospheric pressure, and (b) For that flow pattern evaluate the symmetric and anti-
express the results in mPa • s. Compare the results with ex- symmetric combinations of velocity derivatives:
perimental data given in this chapter. (i) (dv /dx) + (dvjdy)
y
Answers: 0.0203, 0.0175, 0.0109 mPa • s (ii) (dVy/дх) - (dvjdy)
1A.4 Gas-mixture viscosities at low density. The fol- (c) Discuss the results of (b) in connection with the devel-
2
lowing data are available for the viscosities of mixtures of opment in §1.2.
hydrogen and Freon-12 (dichlorodifluoromethane) at 25°C 1B.3 Viscosity of suspensions. 3
and 1 atm: Data of Vand for sus-
pensions of small glass spheres in aqueous glycerol solu-
Mole fraction of H : 0.00 0.25 0.50 0.75 1.00 tions of Znl can be represented up to about ф = 0.5 by the
2
2
д X 10 (poise): 124.0 128.1 131.9 135.1 88.4 semiempirical expression
6
Use the viscosities of the pure components to calculate the 2
viscosities at the three intermediate compositions by = * = 1 + 2.5ф + 7Л7ф (1B.3-1)
means of Eqs. 1.4-15 and 16. Compare this result with the Mooney equation.
Sample answer: At 0.5, fi = 0.013515 cp
Answer: The Mooney equation gives a good fit of Vand's
1A.5 Viscosities of chlorine-air mixtures at low den- data if ф is assigned the very reasonable value of 0.70.
0
sity. Predict the viscosities (in cp) of chlorine-air mixtures 1C.1 Some consequences of the Maxwell-Boltzmann
at 75°F and 1 atm, for the following mole fractions of chlo- distribution. In the simplified kinetic theory in §1.4, sev-
rine: 0.00, 0.25, 0.50, 0.75, 1.00. Consider air as a single eral statements concerning the equilibrium behavior of a
component and use Eqs. 1.4-14 to 16. gas were made without proof. In this problem and the
Answers: 0.0183,0.0164,0.0150,0.0139,0.0131 cp next, some of these statements are shown to be exact
consequences of the Maxwell-Boltzmann velocity distri-
1A.6 Estimation of liquid viscosity. Estimate the viscosity
of saturated liquid water at 0°C and at 100°C by means of bution. distribution of molecular ve-
The Maxwell-Boltzmann
(a) Eq. 1.5-9, with ALZ vap = 897.5 Btu/lb w at 100°C, and (b) locities in an ideal gas at rest is
Eq. 1.5-11. Compare the results with the values in Table 1.1-1.
3/2
2
Answer: (b) 4.0 cp, 0.95 cp f{u , u , u ) = п(т/2тткТ) ехр(-ши /2кТ) (lC.1-1)
x
z
y
in which u is the molecular velocity, n is the number
1A.7 Molecular velocity and mean free path. Compute density, and f(u , u , u )du du du is the number of mole-
z
x
x
y
the mean molecular velocity п (in cm/s) and the mean free cules per unit volume z that is XJ expected to have velocities
path Л (in cm) for oxygen at 1 atm and 273.2 K. A reason- between u and u + du , u and u + du , u and u + du .
y
y
z
x
x
y
able value for d is 3 A. What is the ratio of the mean free It follows x from this equation that the distribution z of the z
path to the molecular diameter under these conditions? molecular speed и is
What would be the order of magnitude of the correspond- flu) = 4тгпи (т/2тткТ) ехр(-ши /2кТ) (lC.1-2)
2
3/2
2
ing ratio in the liquid state? (a) Verify Eq. 1.4-1 by obtaining the expression for the
6
4
Answers: и = 4.25 X 10 cm/s, Л = 9.3 X 10~ cm
mean speed п from
1B.1 Velocity profiles and the stress components T .
J; uf(u)du
For each of the following velocity distributions, draw a
meaningful sketch showing the flow pattern. Then find all и = (lC.1-3)
the components of т and p w for the Newtonian fluid. The Г f(u)du
parameter b is a constant.
of
es
(b) Obtain the mean values of the v< components u ,
the velocity
(a) v x = by, v y = 0, v z = 0 п , and u . The first of thes ;e is obtain* obtained from x
i
s
e
z
ч
(b) v x = by, v y = bx, v z = 0 р ж , + x , + x
uJbivUy^JdujujlU;
(c) v x = -by, v y = bx, v z = 0 J -o c J -o c J -о с
(d) v x = -\bx, v y = -{by, v z = bz " A = /*+OC Г +ЭС /*+OC (1C.1-4)
f(u , u , ujdujuyduz
y
x
1B.2) A fluid in a state of rigid rotation.
What can one conclude from the results?
(a) Verify that the velocity distribution (c) in Problem 1B.1
describes a fluid in a state of pure rotation; that is, the fluid
2 3
J. W. Buddenberg and C. R. Wilke, Ind. Eng. Chem. 41, V. Vand, /. Phys. Colloid Chem., 52,277-299,300-314,
1345-1347 (1949). 314-321 (1948).