Page 381 - Bird R.B. Transport phenomena

Problems 363

3
2
for methane are: a = 2.322 cal/g-mole -K,b = 38.04 X 1СГ cal/g-mole • K , and с = -10.97 X
10~ cal/g-mole • K .
3
6
2
Answers: (a) pT~ exp[-(b/R)T - (c/2R)T ] = constant;
ll/R
(b) 270 atm
11B.2. Viscous heating in laminar tube flow (asymptotic solutions).
(a) Show that for fully developed laminar Newtonian flow in a circular tube of radius R, the
energy equation becomes

if the viscous dissipation terms are not neglected. Here v zmax is the maximum velocity in the
tube. What restrictions have to be placed on any solutions of Eq. 11B.2-1?
(b) For the isothermal wall problem (T = T at r = R for z > 0 and at z = 0 for all r), find the as-
o
ymptotic expression for T(r) at large z. Do this by recognizing that дТ/dz will be zero at large
z. Solve Eq. 11B.2-1 and obtain

(с) For the adiabatic wall problem (q = 0 at r = R for all z) an asymptotic expression for large z
r
may be found as follows: Omit the heat conduction term in Eq. 11B.2-1, and then average the
remaining terms over the tube cross section, by multiplying by rdr and then integrating from
r = 0 to r = R. Then integrate the resulting equation over z to get
2
(T) - T, = (4fjiV Z/max /pC R )z (11B.2-3)
p
in which T} is the inlet temperature at z = 0. Postulate now that an asymptotic temperature
profile at large z is of the form

2
T-T } = {4fjLV Z/max /pC R )z + fir)
p
Substitute this into Eq. 11B.2-1 and integrate the resulting equation for/(r) to obtain

R) 2 \R

Keep in mind that the solutions in Eqs. 11B.2-2 and 5 are valid solutions only for large z. The
complete solution for small z is discussed in Problem 11D.2.

11B.3. Velocity distribution in a nonisothermal film. Show that Eq. 11.4-20 meets the following
requirements:
(a) At x = 5, v z = 0.
(b) At x = 0, dvjdx = 0.
2
(c) Urn v (x) = ( g8 2 cos p/2fjL )[l - (x/8) ]
P
0
z
11B.4. Heat conduction in a spherical shell (Fig. 11B.4). A spherical shell has inner and outer radii
JR and R . A hole is made in the shell at the north pole by cutting out the conical segment in
2
1
the region 0 < 0 < 6 V A similar hole is made at the south pole by removing the portion (тг -
6^) < в ^ 77. The surface 9 = 6^ is kept at temperature T = T and the surface at в = тг - в is
v }
held at T = T . Find the steady-state temperature distribution, using the heat conduction
2
equation.