Nonisothermal flow of gases through packed beds
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150(1 -- 6) 2 Ix ~3(~dp)2 p
1.75(1 G
+
-- e ) G 2
~(6d~)
p
[1]
The extension of the Ergun equation to nonisothermal packed beds has received much treatment in the literature. Ergun 2 applied Eq. [1] to the iron blast furnace by evaluating property values at an integral mean temperature for the process: = -~
[21
T(x) dx
The discrepancies between the predicted and observed pressure drops, however, were as high as 40 pct. In another method proposed by Mitchell, 3 the bed was subdivided into several zones each of which was assigned average property values. This method predicted pressure drops to within 30 pct of the observed values. Szekely and Carr 4 proposed a more rigorous formulation for the extension of Eq. [1] to nonisothermal packed beds. The basis was the differential mechanical energy balance: + gdx +-- + Fdx = 0 P
d
[3]
where the above terms represented, respectively, the kinetic energy, potential energy, pressure energy, and work done against friction. Next the potential energy term was neglected and the friction term was defined in terms of the Ergun equation, as follows: 1
[4]
F dx = --s[CIIxG + CzGZ]dx p"
where C,-
150(1 - e) 2 e3(qbdp) 2
C: -
1.75(1 - e) ~34~dp
and
After substitution of Eq. [4] into [3], and multiplying by p2, Szekely and Carr integrated Eq. [3] to give the relation: In
+
CIIxG + C2G2)dx = -
fox
pdP
[51 E.P. WONCHALA, Doctoral Student, and J. R. WYNNYCKYJ, Professor, are with the Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Manuscript submitted July 7, 1986.
METALLURGICAL TRANSACTIONS B
Szekely and Carr solved Eq. [5] numerically for P(x) over a given temperature path T -- T(x), via the temperature dependence of IX and p. Since 19 = p(T,P), the integration procedure was iterative. The value of Eq. [5] is that it can deal with a very wide range of conditions. It provides a general means to solve problems of flow through beds where both temperature as well as pressure vary over a wide range. In particular the dependency of density on pressure may become complex at very high pressures. The Szekely and Carr analysis may, however, give a somewhat misleading impression concerning the need for Eq. [5] in its full complexity for packed-bed problems common in extractive metallurgy. Table I compares the sizes of the three terms of Eq. [5] for three packed-bed systems. Case 1 is the system studied by Szekely and Carr, and the other two examples are packed beds with large pressure drops. As Table I shows, the good agreement between the experimental and calculated profile in Figures 4 and 5 of their paper is not due to the fact that the kinetic energy term was retained upon deriving Eq. [5] from Eq. [3]. Rather, as is evident from the following, the agreement was predominantly due to the fact that Eq. [5] was integrated over the true temperature distribution. By neglecting the kinetic energy term of Eq. [3] and expressing the viscosity in terms of temperature 5 (i.e., tx = Ix0T") and density in terms of the ideal gas law, p =
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