shown for absorption conditions using a n ethanol-water system that the surface tension will fall from 72 dyn/cm to approximately 49 dyn/cm, a change of approximately 33%. From the photographic study, the droplet falling back toward t8heinterface either (a) was assimilated by the bulk fluid, (b) bounced back into the gas phase one or more times before coalescing, or (c) was assimilated but produced a secondary droplet which reentered the gaseous space. Data concerning this behavior are presented in Table I, and photographs illustrating (b) and (c) are shown in Figures 2, 3, and
Literature Cited
Bainbridge, G. S., Sawistowski, H., Chewa. Eng. Sci. 19, 992 (1964). Boyes, A. P., Ponter, A. B., J . Chem. Eng. Data 15,236 (1970). Hermans, J. J., "Flow Properties of Disperse Systems." p 320,, Interscience, New York, N. Y., 1953. Komasawa, I., Otake, T., J . Chem. Eng. J a p . 3,243 (1970). Ponter, A. B., Davies. G. A,. Ross, T. K.. Thornlev. P. G.. Int. J . HeaiMass'Transfe; 10, 349 (1967). ' Spielman, L. A4., Goren, S.L., Ind. Eng. Chem. 62, 10 (1970). I
"
-
I
ADRIAN P. BOYES
4.
ANTHONY B. P O S T E R
It is suggested t'hat this anomalous behavior is equivalent to the heat-transfer analog, the Leidenfrost effect. Thus the coalescence time is governed by the action of the vapor being expelled from between the two liquid interfaces. At the same time t'he droplet approaching the interface deforms the underlying liquid which on restoration imparts an added force causing the droplet to reent'er the gas phase.
Department of Chemical Engineering rniversity of N e w Brztnswick Fredericton, N.B., Canada
RECEIVED for review February 16, 1971 ACCEPTED August 20, 1971
The Maximum Number of Solid Phases in a Multiphase Reacting System The maximum number of solid phases which may exist in a multiphase reacting system a t chemical equilibrium is shown to be R - 1, where R is the rank of the atomic matrix of all the reacting species.
T h e equilibrium compositions in a complex chemically reacting mixture of ideal gaseous components may be calculated simply and directly by the method of White, et al. (1958). 13asically, this solution is iterative: one guesses a n initial set of mole numbers niawhich satisfy t,he conservation of atomic qecies requirements. The method of steepest descent is t'hen used to predict' a new set of mole numbers nil which satisfy the conservation of atomic species constraints but also correspond to a lower total Gibbs free energy for the system. The iterations are coiit'inued until the mole numbers niK+ I which corresponded to the minimum in the Gibbs free energy are found (subject to the coiiservation of atomic species). This mixture is the equilibrium mixture. .ilthough the original method of White, et al., is riot applicable to mixtures which form condensed phases, the extension to cover this case is straightforward and has been presented by several aut,hors (Balzhiser, el ai.,1971; I3oyntoii1 1959; Oliver, et al., 1962). At first glance it appears that these extensions are directly applicable to all reacting systems independent of the number of solid phases which we suspect may be present. However, it is well known that if one at'tempts t'o include in the calculation solid components which are not :xtually present in the equilibrium mixture serious problems are likely to r e d t (Balzhiser, et al., 1971; Oliver, el al., 1962). 'Thus, it' is important that extraneous solid phases not be introduced iiito the calculationr. O n the other hand, it is equally important that one retain in the calculations all solid 'phases which are actually present in the equilibrium mixture. Although it is not possible to determine the exact number (or composition) of solid phases which nil1 result from a given
reaction mix, it is possible to set a limit on the maximum number of solid phases that can occur. This information is extremely helpful in guiding the choices of which solid components to include and which to exclude from the equilibrium calculations. Theory and Discussion
Consider the reacting system FeO, Fe, C, CaC03, CaO, CO, COz, and O2 which may occur during the reduction of iron oxide to iron in a blast furnace. The system as originally envisioned contains a maximum of five solid phases, one gaseous phase, and a total of eight components if all chemical species are present a t equilibrium. To determine the maximum number of solid phases and components that are actually present at equilibrium the Gibbs phase rule must be considered. For chemically reacting systems the phase rule is given by f =
c-
P+2-*1I
(1)
where f is the degrees of freedom of the system, C is the number of chemical species present, P is the number of phases present, and M is the number of independent chemical reactions which can occur bet,ween the chemical species present. T o determine the number of independent chemical reactions, the rank, R , of the atomic coefficient matrix a3f must be determined. For the system described above, the atomic matrix is illustrated in Table I, where a j i is the number of atoms of element j in chemical species i. The rank of this matrix is easily seen to be R = 4. Ind. Eng. Chem. Fundam., Vol. 10, No.
4, 1971
643
Table 1. Atomic Matrix for the Reacting System i 1 j
Atomic species
1 Fe 2 0 3 c 4 C a
2
3
FeO
Fe
C
1 1 0 0
1 0 0 0
0 0 1 0
4
5
6
Chemical species CaCOa C a O C O
0 3 1 1
0 1 0 1
0 1 1 0
7
8
COz
0 2
0 2 1 0
0 2 0 0
Amundson (1966) has shown that the number of independent chemical reactions is given by
(2)
M = C - R so that the phase rule becomes
(3)
f = R - P + 2
If the system pressure and temperature are to be arbitrarily chosen, thenf = 2 and the total number of phases present is given by P = R
(4)
However, one of these phases is the gaseous phase and so the number of solid phases, P,, is given by
P,=R-1
(5)
(This is actually the maximum number of solids that may be present during the reaction between a given set of species. It is often possible to choose the feed composition in such a way that fewer solid phases than this are present. This reduction comes about because the inlet feed conditions, as well as P and T,have been arbitrarily fixed. Thus more than two degrees of freedom are required, and the number of phases present drops accordingly.)
For the system considered above, i t was found that R = 4, so that a maximum of three solid phases can occur at equilibrium. Thus, a t least two solid phases must vanish as the reaction proceeds. Unfortunately, the phase rule does not tell us which solid phases must vanish. This must be determined as the equilibrium calculation proceeds. However, the choice of which solid phases to remove from the calculation is not completely arbitrary. For example, in the reaction system considered above one would not attempt to remove both Fe and FeO. This unfortunate choice would drop the rank of the atomic matrix to 3 and would obviously wreak havoc with the requirement that atomic iron be conserved. Similarly, CaC03 and CaO would not be removed simultaneously, although FeO and CaC03 might well be. Oliver, et al. (1962), suggest a procedure for determining whether a species which has been removed from the calculation should be reinstated or not. However, for every species reinstated in this fashion it is necessary to remove another so that the total number of solid phases is always less than or equal to R - 1. Literature Cited
Amundson, N. R., “hlathematical Methods in Chemical Engineering,” pp ,5044, Prentice-Hall, Englewood Cliffs, N . J., 1966. Balzhiser, R. E., Samuels, AI. R., Eliassen, J. D., “Chemical Engineering Thermodynamics,” Prentice-Hall, Englewood
,
R. C., Stephanson, S;E., I
2 ,.)
l Y l -
White, W. B., Johnson, S. RI., Dantzig, G. B., J . Chem. Phys. 2 8 , 751 (1958). MICHAEL R. SAMUELS
Department of Engineering University of Delaware Newark, Del. 19711 RECEIVED for review May 21, 1971 ACCEPTEDAugust 25, 1971
An Envelope Characteristic of the Inversion Curve in the Z, P, Plane It is shown that the Joule-Thompson inversion curve is the envelope of the isotherms in the compressibilityreduced pressure plane. From this property it is proved that the curve cannot cross itself in this plane and that it cannot have a smooth vertical tangent a t its point of maximum pressure if it i s to be stable a t this point. Instead, a cusp characteristic of the inversion curve emerges a t the maximum pressure point.
M i l l e r (1970) shows the superposition of a three-constant inversion curve on a generalized compressibility chart based on the 2, = 0.27 tables of Lydersen, et al. (1955). H e indicates that the locus loops and crosses over itself and that it closely follows the envelope of the constant reduced temperature lines in the pressure region of practical interest ( P , 5 lo), thus lying mostly outside the temperature region of interest. I n the small loop the isotherms near the maximum pressure point almost intersect the inversion curve loop 644
Ind. Eng. Chem. Fundam., Vol. 10, No.
4, 1971
a t a right angle. However, as will be shown, the inversion curve is the envelope of the isotherms in this plane and as such does not permit a primary contact between itself and an isotherm to he other than tangential. The distinction between primary and secondary isothermal contacts with the inversion curve is seen in hliller’s Z , P, plane where the T,= 1 curve makes a primary tangential contact with the lower branch of the inversion curve and a t higher 2 makes a secondary nontangential contact with the upper branch. I n this
