Page 54 - Bird R.B. Transport phenomena

Problems 39

(с) Obtain the mean kinetic energy per molecule by (b) Insert Eq. 1С. 1-1 for the Maxwell-Boltzmann equilib-
rium distribution into Eq. 1C.3-1 and perform the integra-
{mu 2 tion. Verify that this procedure leads to p = пкТ, the ideal
Jo (lC.1-5) gas law.
\ f(u)du
Jo 1D.1 Uniform rotation of a fluid.
The correct result is \mu 2 = f кТ. (a) Verify that the velocity distribution in a fluid in a state
of pure rotation (i.e., rotating as a rigid body) is v = [w x
1C.2 The wall collision frequency. It is desired to find
the frequency Z with which the molecules in an ideal gas r], where w is the angular velocity (a constant) and r is the
strike a unit area of a wall from one side only. The gas is at position vector, with components x, y, z.
+
rest and at equilibrium with a temperature T and the num- (b) What are Vv + (Vv) and (V • v) for the flow field in (a)?
ber density of the molecules is n. All molecules have a (c) Interpret Eq. 1.2-7 in terms of the results in (b).
mass m. All molecules in the region x < 0 with u x > 0 will 5
hit an area S in the yz-plane in a short time Af if they are in 1D.2 Force on a surface of arbitrary orientation. (Fig.
the volume Sw^Af. The number of wall collisions per unit 1D.2) Consider the material within an element of volume
area per unit time will be ОЛВС that is in a state of equilibrium, so that the sum of
the forces acting on the triangular faces ДОВС, AOCA,
/ • + * Г + Х Г + о с
(Su x kt)f{u x , u 4 , u AOAB, and АЛВС must be zero. Let the area of AABC be
Z = dS, and the force per unit area acting from the minus to the
plus side of dS be the vector тг„. Show that тг„ = [n • тг].
(a) Show that the area of AOBC is the same as the area of
= n the projection AABC on the yz-plane; this is (n • b )dS. Write
x
similar expressions for the areas of AOCA and AOAB.
2
ехр(-ши /2кТ) du, (b) Show that according to Table 1.2-1 the force per unit
area on AOBC is 8 тг . + 6 тг + 6 тг . Write similar force
х хд у ху 2 Г2
expressions for AOCA and AOAB.
(1C.2-1) (c) Show that the force balance for the volume element
OABC gives
Verify the above development.
«,i = 2 2 (n • 8^(8^) = [n • 2 2 8 M (1D.2-1)
1C.3 Pressure of an ideal gas. 4 It is desired to get the ' / ' /
pressure exerted by an ideal gas on a wall by accounting
for the rate of momentum transfer from the molecules to in which the indices i, j take on the values x, y, z. The dou-
the wall. ble sum in the last expression is the stress tensor IT written
(a) When a molecule traveling with a velocity v collides as a sum of products of unit dyads and components.
with a wall, its incoming velocity components are u , u u ,
XJt
z
x
and after a specular reflection at the wall, its components
are —u , u yf u . Thus the net momentum transmitted to the
z
x
wall by a molecule is 2mu . The molecules that have an x-
x
component of the velocity equal to u , and that will collide
x
with the wall during a small time interval Af, must be
within the volume SwAf. How many molecules with ve-
x
locity components in the range from u , u u to u + Au ,
x yr z x r
u y + Au (// u z + Дм will hit an area S of the wall with a ve-
2
locity u within a time interval Af? It will be f(u , x u yf u )du x
x
z
du du z times Su kt. Then the pressure exerted on the wall
y
x
by the gas will be
Fig. 1D.2 Element of volume OABC over which a force
balance is made. The vector it = [n • n] is the force per
(Su x At)(2mu x )f(u x , u yf u n
unit area exerted by the minus material (material inside
v = OABC) on the plus material (material outside OABC). The
Explain carefully how this expression is constructed. Ver- vector n is the outwardly directed unit normal vector on
ify that this relation is dimensionally correct. face ABC.
4 R. J. Silbey and R. A. Alberty, Physical Chemistry, Wiley, 5 M. Abraham and R. Becker, The Classical Theory of Electricity
New York, 3rd edition (2001), pp. 639-640. and Magnetism, Blackie and Sons, London (1952), pp. 44-45.