Page 210 - Bird R.B. Transport phenomena

194 Chapter 6 Interphase Transport in Isothermal Systems

in which the Reynolds number is defined by 6A.10 Determination of pipe diameter. What size of cir-
cular pipe is needed to produce a flow rate of 250 firkins
(6A.7-3) per fortnight when there is a pressure drop of 3 X 10 scru-
5
ples per square barleycorn? The pipe is horizontal. (The
The values of G, H, and К are given as: 1
authors are indebted to Professor R. S. Kirk of the Univer-
sity of Massachusetts, who introduced them to these
H К
units.)
0.00 4.000 0.400 1.000 6B.1 Effect of error in friction factor calculations. In a
0.05 3.747 0.293 0.7419 calculation using the Blasius formula for turbulent flow in
0.10 3.736 0.239 0.7161 pipes, the Reynolds number used was too low by 4%. Cal-
0.15 3.738 0.208 0.7021 culate the resulting error in the friction factor.
0.20 3.746 0.186 0.6930 Answer: Too high by 1 %
0.30 3.771 0.154 0.6820
0.40 3.801 0.131 0.6759 6B.2 Friction factor for flow along a flat plate. 2
0.50 3.833 0.111 0.6719 (a) An expression for the drag force on a flat plate, wetted
0.60 3.866 0.093 0.6695 on both sides, is given in Eq. 4.4-30. This equation was de-
0.70 3.900 0.076 0.6681 rived by using laminar boundary layer theory and is
0.80 3.933 0.060 0.6672 known to be in good agreement with experimental data.
0.90 3.967 0.046 0.6668 Define a friction factor and Reynolds number, and obtain
the / versus
Re relation.
1.00 4.000 0.031 0.6667
(b) For turbulent flow, an approximate boundary layer treat-
ment based on the 1/7 power velocity distribution gives
Equation 6A.7-2 is based on Problem 5C.2 and reproduces y5
the experimental data within about 3% up to Reynolds F = 0.072pviWL(Lv p/fjir (6B.2-1)
x
k
numbers of 20,000. When 0.072 is replaced by 0.074, this relation describes the
5
(a) Verify that Eqs. 6A.7-1 and 2 are equivalent to the re- drag force within experimental error for 5 X 10 < Lv^pZ/x
7
sults given in §2.4. < 2 X 10 . Express the corresponding friction factor as a
(b) An annular duct is formed from cylindrical surfaces of function of the Reynolds number.
diameters 6 in. and 15 in. It is desired to pump water at 6B.3 Friction factor for laminar flow in a slit. Use the
60°F at a rate of 1500 cu ft per second. How much pressure results of Problem 2B.3 to show that for the laminar flow in
drop is required per unit length of conduit, if the annulus a thin slit of thickness IB the friction factor is / = 12/Re, if
is horizontal? Use Eq. 6A.7-2. the Reynolds number is defined as Re = 2B(v )p//A. Com-
z
(c) Repeat (b) using the "mean hydraulic radius" empiri- pare this result for / with what one would get from the
cism. mean hydraulic radius empiricism.
6A.8 Force on a water tower in a gale. A water tower 6B.4 Friction factor for a rotating disk. 3 A thin circular
has a spherical storage tank 40 ft in diameter. In a 100-mph disk of radius R is immersed in a large body of fluid with
gale what is the force of the wind on the spherical tank at density p and viscosity /x. If a torque T is required to make
z
0°C? Take the density of air to be 1.29 g/liter or 0.08 lb/ft 3 the disk rotate at an angular velocity ft, then a friction fac-
and the viscosity to be 0.017 cp. tor/may be defined analogously to Eq. 6.1-1 as follows,
Answer: 17,000 Ц T /R = AKf (6B.4-1)
z
6A.9 Flow of gas through a packed column. A horizon- where reasonable definitions for К and A are К = ^p(£lR) 2
2
tal tube with diameter 4 in. and length 5.5 ft is packed with and A = 2(TTR ). An appropriate choice for the Reynolds
2
glass spheres of diameter 1/16 in., and the void fraction is number for the system is Re = R ftp//x.
0.41. Carbon dioxide is to be pumped through the tube at For laminar flow, an exact boundary layer develop-
300K, at which temperature its viscosity is known to be ment gives
1.495 X 10~ 4 g/cm • s. What will be the mass flow rate T z = 0.6167rpR V/xft7p (6B.4-2)
4
through the column when the inlet and outlet pressures
are 25 atm and 3 atm, respectively?
Answer: 480 g/s 2 H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New
York, 7th edition (1979), Chapter XXI.
3
T. von Karman, Zeits. fur angew. Math. u. Mech., 1, 233-252
D. M. Meter and R. B. Bird, AIChE Journal, 7, 41-45 (1961). (1921).
1