Page 207 - Bird R.B. Transport phenomena

§6.4 Friction Factors for Packed Columns 191

which is the Blake-Kozeny equation. 3 Equations 6.4-8 and 9 are generally good for
(D G /JJL(1 - s)) < 10 and for void fractions less than s = 0.5.
P
0
(b) For highly turbulent flow a treatment similar to the above can be given. We begin
again with the expression for the friction factor definition for flow in a circular tube. This
time, however, we note that, for highly turbulent flow in tubes with any appreciable
roughness, the friction factor is a function of the roughness only, and is independent of
the Reynolds number. If we assume that the tubes in all packed columns have similar
roughness characteristics, then the value of / tube may be taken to be the same constant for
all systems. Taking/ tube = 7/12 proves to be an acceptable choice. When this is inserted
into Eq. 6.4-7, we get
H(^) (6 4 10)
--

When this is substituted into Eq. 6.4-1, we get

which is the Burke-Plummer^ equation, valid for (D G //x(l — s)) > 1000. Note that the
p 0
dependence on the void fraction is different from that for laminar flow.
(c) For the transition region, we may superpose the pressure drop expressions for (a)
and (b) above to get

7 | " " e (6.4-12)
,Dj/ s 3 4VD,
For very small v , this simplifies to the Blake-Kozeny equation, and for very large v , to
0
0
the Burke-Plummer equation. Such empirical superpositions of asymptotes often lead to
satisfactory results. Equation 6.4-12 may be rearranged to form dimensionless groups:
• т (6-4-13)
5
This is the Ergun equation, which is shown in Fig. 6.4-2 along with the Blake-Kozeny and
Burke-Plummer equations and experimental data. It has been applied with success to
gas flow through packed columns by using the density p of the gas at the arithmetic av-
erage of the end pressures. Note that G o is constant through the column, whereas v 0
changes through the column for a compressible fluid. For large pressure drops, however,
it seems more appropriate to apply Eq. 6.4-12 locally by expressing the pressure gradient
in differential form.
The Ergun equation is but one of many 6 that have been proposed for describing
packed columns. For example, the Tallmadge equation 7

2
G 0 J\L JV e) \D P G 0 //JLJ \D p G 0 /
is reported to give good agreement with experimental data over the range 0.1 <
5

3
F. C. Blake, Trans. Amer. Inst. Chem. Engrs., 14,415-421 (1922); J. Kozeny, Sitzungsber. Akad. Wiss. Wien,
Abt. Ibi, 136,271-306 (1927).
4
S. P. Burke and W. B. Plummer, lnd. Eng. Chem., 20,1196-1200 (1928).
S. Ergun, Chem. Engr. Prog., 48, 89-94 (1952).
5
6
1. F. Macdonald, M. S. El-Sayed, K. Mow, and F. A. Dullien, lnd. Eng. Chem. Fundam., 18,199-208
(1979).
J. A. Tallmadge, AIChE Journal, 16,1092-1093 (1970).
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