2 = ion charge number, assumed same for positive and negative ions in electrodialysis
Greek Letters a = a safety factor, defined with regard to eq 12, taken to be 0.5 in numerical examples 0 = a dimensionless hydrodynamic performance parameter defined in eq 15 X = conventional friction factor, defined by eq 6 p = fluid viscosity coefficient, kg/m sec p = fluid density, kg/m3 C#I = potential drop across one dialysate-concentrate channel pair, V
Subscripts opt = optimum value, t h a t is, value which minimizes product cost opt, ideal = optimum value which would be attained under hydrodynamically ideal conditions
Grossman, G., Sonin, A. A,, Desalination. I O , 157 (1972). Harned, H. S., Nuttall, R. L., J. Am. Chem. Soc., 71, 1460 (1949). Hicks, R. E., Mandersloot, W. G. B., Chem. Eng. Sci., 23, 1201 (1968). isaacson, M. S., Ph.D. Thesis, Department of Mechanical Engineering, M.I.T., 1974. Kitamoto, A,, Takashima, Y., J. Chem. Eng. Jap., 3, 182 (1970). Kitamoto, A., Takashima. Y., Desalination, 9, 51 (1971). Mandersloot, W. G. B., Hicks, R. E., Ind. Eng. Chem., Process Des. Dev., 4, 304 (1967). Marriott, J., Chem. Eng., 127 (Apr 5, 1971). Mizushina, T.. Ogino. F., Oka. Y., Fukuda, H., lnt. J. Heat Mass Transfer, 14, 1705 (1971). Norris, R. H., Streid, D. D.. Trans. Am. SOC.Mech. Eng., 62, 525 (1940). Probstein, R. F., Sonin, A. A., Gur-Arie, E., Desalination, 11, 165 (1972). Process Research, Inc.. Office of Saline Water Research and Development Progress Report No. 325, 1968. Rosenberg, N. W., Tirrell. C. E., lnd. Eng. Chem., 49, 780 (1957). Schlichting, H., "Boundary-Layer Theory", 6th ed, McGraw-Hill, New York, N.Y.. 1968. Sonin, A. A,, Isaacson. M . S., lnd. Eng. Chem., Process Des. Dev., 13, 241 (1974). Sonin, A. A,, Probstein, R. F., Desalination, 5, 293 (1968). Sonin, A. A,, Probstein, I?. F.. Desalination, 6, 270 (1969). Winograd, Y., Solan, A,, Toren, M., Desalination, 13, 171 (1973).
Received for reuiew July 9, 1975 Accepted December 2,1975
Literature Cited Belfort, G., Guter, G. A., Desalination, 10, 221 (1972). Colburn. A. P.. Am. lnst. Chem. Eng. Trans., 29, 174 (1933). Davies. S. J., White, C. M., Proc. Roy. SOC.(London), A119, 92 (1928). Forgacs, C., ishibashi, N., Leibovitz, J., Sikovic, J., Spiegier, K. S.. Desalination, 10, 181 (1972).
This research was sponsored by the Office of Water Resources and Technology, U S . Department of the Interior, under Grant No. 14-30-3297.
Influence of Rheological Properties of Polymer Solutions upon Mixing and Circulation Times D. E. Ford and J. Ulbrecht' Department of Chemical Engineering, University of Salford, Salford M5 4 W ,UnitedKingdom
The influence of the shear dependent viscosity and of the viscoelasticity upon the mixing and circulation times was examined in a vessel agitated by a helical screw impeller operating in a centrally positioned draught-tube. The liquids used were aqueous polymer solutions. The mixing times were found to depend strongly upon the viscoelasticity of the solutions while no influence of the shear-thinning viscosity was detected. On the other hand, the circulation times depend on the viscoelasticity as well as on the sheardependent viscosity. Some general rules were drawn at the end of the work for scaling-up and design of agitators for polymerization reactors and mixers.
Helical agitators (both screws and ribbons) have been widely used in the polymer and detergent industries for a number of years mainly for the mixing of viscous rheologically complex solutions and slurries. Johnson (1967) was probably the first one who published observations of anomalous flow patterns in agitated tanks in connection with polyurethane manufacturing. He found that the performance of reactors agitated by helical agitators was much better than that obtained in tanks stirred by paddles and turbines. He also identified the reason of this phenomenon as being associated with the complicated flow pattern and large unmixed zones in vessels agitated by turbines and paddles. Despite the wide appreciation of the superiority of helical agitators, most of the published research work so far has dealt with the power consumption only. I t may be concluded from the work of Chavan and Ulbrecht (1973) that while in the viscous flow regime (Re < 10) the power consumption is strongly affected by the shear dependence of
the polymer solution or slurry, the viscoelasticity has hardly any effect a t all. This, however, will not be the case a t higher Reynolds numbers (Re > 10) where the viscoelasticity leads to a torque suppression (Kale et al., 1974). Reports on the influence of the rheological properties upon the quality of mixing began to appear only recently (Coyle e t al., 1970; Carreau, 1974; Chavan e t al., 1975; U1brecht and Ford, 1974). There seems t o be general agreement that unlike the power consumption, the quality of mixing is affected both by the shear dependence of the liquid's viscosity as well as by the viscoelasticity. While the influence of the shear-dependent viscosity can vary, the viscoelasticity can reduce the mixing performance quite severely.
Analysis of the Problem The mechanism of mixing under laminar flow conditions is initially controlled by the shearing and stretching of the individual fluid elements with the subsequent recombinaInd. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
321
Table I. Geometry of the Mixer dd
0.457 m
0.457 m
0.215 m
0.4-0.5
0.5-1.0
0.055-0.110
Table 11. Rheological Properties of Liquids Used CMC, approx. 0.5% CMC, approx. 1.0% CMC, approx. 2.0% PAA, approx. 0.5% PAA, approx. 1.0% PAA, approx. 2% Glycerol Corn syrup
1.4-1.9 25-45 1000-3000 35-40 250-500 3700-4700 1.74-8.50 26-138
tion of the striations. The molecular diffusion which is ultimately responsible for the mixing on molecular scale will take an appreciable part in the mechanisms of mixing only after sufficient area between the striations has'been created. The stretching and shearing of the liquid in the agitated vessel is due t o the velocity gradients which, in turn, is brought about by the rotation of the agitator. The drag force exercised by the agitator is transported through the liquid content of the vessel due to the viscous (frictional) forces and this sets the liquid into motion which is usually termed as the primary flow. It will be thus the viscosity of the liquid, constant or shear-dependent, which together with the geometry of the system will control the pattern of the primary flow. The damping effect of the pseudoplasticity is well known since the work of Metzner and Otto (1957), and through this effect it may be expected that the shear dependent viscosity will have some effect upon the rate of mixing. The secondary flow in the vessel is usually associated with the action of the inertia, Le., the centrifugal force. The secondary flow pattern cuts across the primary one and in doing so it contributes to the recombination of the striations. In viscoelastic liquids, however, the centrifugal force has to compete with the elastic force in establishing the secondary flow pattern. In all cases where the dominant velocity gradient is [r(awlar)] the elastic force aims toward the axis of rotation thus causing the liquid to flow from the walls inward. The complex velocity pattern so created and its instabilities were recently discussed by Ulbrecht (1974) using the analysis of rotational axisymmetric flows proposed by Giesekus (1965). There is, however, a long way from an interpretation of a velocity pattern to a quantitative measure of the mixing rate. The concept of mixing time has often been found to be more useful than an illustrative sketch of the streamlines but equally often the concept of mixing times has been strongly and justly criticized as entirely arbitrary and unreliable. Because the mixing time, defined as the time required for a soluble tracer to disperse uniformly throughout the liquid batch or for a fast chemical reaction to reach the equilibrium, is not a physical quantity, its value experimentally obtained will depend upon the method chosen, upon the scale of scrutiny used, and upon the selection of the endpoint on the response curve. Coyle e t al. (1970), for example, report that depending upon the method used the mixing times can differ by a factor of 6. As long as the identity of the experimental conditions is, however, anxiously observed, there is no reason why the mixing time concept cannot be used to compare the influence of selected material properties such as the variable viscosity and viscoelasticity. The same can be said about another commonly used cri322 Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
2.0-6.0 1.7-2.5 1.15-1.30 1.3-1.5 1.20-1.35 1.05-1.10 ...
0.1-0.6 6-12 800-2500 20-22 150-400 3500-4500
...
...
...
0.35 0.6 1.4 4.7
... ...
41 7
... ...
-
i,~
+ I I t ~
1':.
-
l1
yI Figure 1. Schematic of the mixing vessel. terion of the mixing rate known as the circulation or turnover time. Here the time is measured for a tracer or a freely suspended particle to complete one cycle in a horizontal or vertical projection. An average value from a large number of cycles is usually considered t o be free from random variations. Experimental Section The mixer chosen for the assessment of the influence of rheological properties upon the rate of mixing consisted of a helical screw impeller operating in a draught-tube centrally positioned in a tank. Both the draught-tube and the vessel were made from Perspex. Although this mixer is not as widely used as the helical ribbon, mainly because of its higher manufacturing cost, it is known to be free from any dead zones so often to be found in a vessel agitated by a helical ribbon, particularly with viscoelastic solutions. The dimensions of the mixer are summarized in Table I where the symbols have the meaning given in Figure 1. The liquids used for the experiments are summarized in Table I1 where their typical rheological parameters are also given. I t will be noted that the range of the zero-shear viscosities covers three orders of magnitude and the natural time ranges from zero up to 3.65 s. The shear stress and normal stress data of all liquids used were measured using the cone-and-plate setup of the Weisenberg Rheogoniometer R18 for shear rates from 1 s-l to lo3 s-l. The formulas used to convert the torque M , the axial thrust T , and the angular velocity w into the shear
stress 0 , the primary normal stress difference N1, and the shear rate respectively, are
+,
where R is the radius of the cone-and-plate setup and u is the angle of the conical gap. Out of the wide selection of methods used for measuring of mixing times only a very few will not affect the rheological properties of the liquids used. The redox reaction between iodine and sodium thiosulfate in the presence of modified starch was used in this work after extensive preliminary tests (Ford e t al., 1972). An overall iodine concenN in the bulk liquid was used with a tration of 5 x 30% stoichiometric excess of sodium thiosulfate. The iodine was added first and the spreading of the dark tracer was observed in order to obtain information about the dispersion pattern. Then the sodium thiosulfate was added and the decolorization was monitored photooptically. The light source was a 12-V quartz-iodine spot lamp mounted upon a vibration-free heavy stand. The photocell was a silicon detector diode followed by an ac field effect transistor amplifier. The output was fed to a potentiometric recorder with back-off facilities in order to increase sensitivity. The photocell unit was mounted upon a vibration absorbent base attached to the framework of the equipment. Several positions of the photocell were tested in the preliminary experiments and it was found that this had no influence upon the terminal mixing time. Thus, a midway position between t h e bottom of the vessel and the liquid level was used in all experiments with the light entering the photocell via the annulus. The amount of stray light entering the photocell was reduced by a collimating tube. The mixing experiments were done in a constant temperature room from which all daylight was excluded. Two distinctly different oscillation patterns were superimposed on the recorded response curve: one with a regular frequency of the order of to lo-' s-l which was due to the circulation of the locally decolorized portion of the liquid and another random highfrequency oscillation due t o the system noise. The mixing was considered completed when the regular pattern of circulation was completely over-shadowed by the noise. The screw was operated either clockwise or anti-clockwise so that the liquid flowed either up or down the draught tube. The mixing times measured in this way were found to be between 30 s and 3 X 103 s and the circulation times, determined as the times between two subsequent passes of the locally decolourized liquid, were from about 7 s to 250 s. Results and T h e i r Discussion First the mixing and circulation times in Newtonian liquids were measured in order to establish a standard for comparison. From these data, see Figures 2 and 3, it can be seen that the dimensionless mixing and circulation times do not depend upon the Reynolds number up to Re = 10. From here toward higher Reynolds numbers both dimensionless times decrease as the Reynolds number increases. These findings are in good agreement with all t h e previously published reports on mixing times (Nagata e t al., 1956, 1972; Hoogendoorn and den Hartog, 1967; Zlokarnik 1967; Novak and Rieger, 1969) and circulation times (Nagata e t al., 1956; Gray, 1963; Holmes e t al., 1964; Sato and Taniyama, 1965; Sykora, 1966; Taniyama and Sato, 1966; Coyle et al., 1970; Oshima, 1970; Seichter, 1971; Chavan and U1brecht, 1973) in Newtonian fluids. T h e numerical values of both the NO, and NOMis of little relevance for the purpose of this work except for them being taken as reference value for plotting the non-Newtonian data. Thus the specific mixing time NOM of a Newtonian liquid in the viscous or
10
1
I
1
1
IO
100
I Re
Figure 2. Dimensionless mixing times for Newtonian liquidsaqueous solutions of corn syrup: 0,138 P, A, 26 P; glycerol: 0,8.50
P, D,1.74 P.
I
I
1
'Ib
'
1
'
" "100
Re
Figure 3. Dimensionless circulation times for Newtonian liquidssymbols as in Figure 2.
creeping flow regime (Re < 10) is taken as being equal to 1. The mixing times obtained for aqueous solution of CMC are shown in Figure 4 in terms of specific mixing time and Reynolds number. All the solutions were typically shearthinning and the flow curve was interpolated by an empirical formula: It = To - m+b-1
(2)
where 70,m, and b are material parameters of the flow curve. From Figure 4 it can be seen that all of the data are fitted fairly well by the solid line which is the Newtonian correlation line if the Reynolds number incorporates the shear-thinning viscosity; thus: Re"
=
ND2p 70 - mNb-l
=-ND2p
(3)
Ta
The only data points which seem to be somewhat higher than the Newtonian correlation line are those for the 2% solution. In this context a word must be said about the characteristic shear rate in the vessel which is to be used for the calculation of the apparent viscosity. Based on the work of Metzner and Otto (1957) it is believed that despite a wide spectrum of shear rates existing in a stirred tank there will be a characteristic shear rate, proportional to the rotational speed of the impeller. The proportionality constant is then evaluated using the concept of equal power both in the Newtonian and the non-Newtonian liquid. The same constant does not have to apply to other processes in a stirred tank and the choice of = N seems t o be justified for this particular Reynolds number by the coincidence of its numerical value with that for Newtonian fluids for the end of the viscous flow regime. The specific mixing times for the aqueous PAA solutions are presented in Figure 5. I t is immediately obvious that these times are up to three times higher than those for Newtonian liquids (solid line). Because the shear-thinning anomaly cannot be blamed for this deviation (it has already
+
+
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 2, 1976
323
- I N%
1
I
0
l
-
L
,
01
I
,I
.;
,
,
,
10
1
,
,
,
Re
I
IM
Figure 4. Specific mixing times for aqueous solutions of CMCapprox. concentrations 2% 0 , 1%A, A, and 0.5% 0. Full points refer to the downward pumping action of the screw, plain points to the opposite.
1
f
, , , , , , , 1
01
i0
WI
Figure 6. Specific mixing times as a function of Weissenberg number Wi: 8 , 2 % CMC;0 , 2 %PAA;A, 1% PAA;and 0,0.5% PAA.
-
(OL
i
t
01
1
IO
c rie
~
_
_
3.
Figure 7. Dimensionless mixing time correlation. Figure 5. Specific mixing times for aqueous solutions of PAAapprox. concentrations 2% 0, 1%A, and 0.5%0. been incorporated into the Reynolds number Re”) then one is left with the assumption that this deviation might be associated with the viscoelasticity of the PAA solutions manifesting itself by a nonzero shear-dependent primary normal stress coefficient
+
IC
0
cr-
&
Z L I
$1
= NdT2
(4)
If this is so then the specific mixing time of the viscoelastic solutions (both CMC and PAA) should correlate with some dimensionless measure of the viscoelasticity in terms of the primary normal stress coefficient. In the absence of any physical model of the flow the dimensional analysis will offer such a measure in the form of the Weissenberg number Wi *1 W i = + X ; X = lim -
Y -0
(5)
1
+
where X is the characteristic time of the liquid and is the characteristic shear rate in the system. Following the preceding argument about the proportionality between the characteristic shear rate and the rotational speed of the impeller, the proportionality coefficient has been taken as 1 although this choice might involve a certain degree of arbitrariness in the numerical value of the Weissenberg number. The correlation is shown in Figure 6. The shape of the correlation line seems to indicate that the influence of the elasticity upon the mixing times is far from linear. This is then revealed in the final regression of all the data points in Figure 7 where the viscoelastic term (1 0.45Wi) is raised to 0.8. The solid line in this figure again represents the Newtonian behavior. I t has often been claimed in the past that the dimensionless mixing times for both Newtonian and non-Newtonian solutions are equal and constant. I t would seem that in the light of this work this statement would need some qualification. The dimensionless mixing times are constant only in the sense that they do not depend upon the Reynolds
+
324
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
-
--
-
l
2
1
I
+
~
I
- ~ _ _----
01
130
Re
Figure 8. Dimensionless circulation time correlation. number in the viscous regime of flow as long as the Reynolds number respects the shear-thinning behavior. If, however, the polymer solutions also display a noticeable viscoelasticity then a further dimensionless parameter, e.g., the Weissenberg number needs to be included in the correlation. The viscoelasticity will always tend to increase the mixing times due to the complex secondary flow pattern and a possibility of partly segregated torroidal vortices. This conclusion is in good agreement with similar findings recently presented by Carreau (1974) for a helical ribbon impeller stirring viscoelastic PAA solutions. If the mechanism, through which the viscoelasticity influences the mixing times, is entirely of a fluid mechanical nature than one would expect a similar conclusion to be found for the circulation times as well. This is, however, not the case. When the regression of all the circulation time data is carried out (Figure 8), then it can be seen that the viscoelastic term (1 0.45Wi) is raised to the power of 0.3 and that the same exponent applies also to the term representing the shear-thinning anomaly ( V ~ / T J which was entirely missing from the mixing time correlation. In other words, the mixing times depend strongly upon the elastic properties but not upon the shear-thinning viscosity. The circulation times are, however, affected both by the viscoelastic as well as by the shear-thinning properties of the liquid. Any attempt to interpret this phenomenon must take into account the particular geometry of the screw agitated mixer and the flow pattern associated with it.
+
The dispersion of a dye tracer (iodine) injected into the vessel allows to identify four different regions in the vessel. In the draught-tube the radial dispersion is very fast due to the action of the rotating screw and the concentration profile is best simulated by that of an axially dispersed plug flow. The presence of viscoelasticity in the liquid is manifested by the complexities of the secondary flow pattern superimposed over the primary flow pattern in the helix which might result into retaining of small parts of the tracer in the secondary vortices. This is particularly true in the second (radial converging flow a t the top of the draughttube) and the third (radial diverging flow a t the bottom) regions. Here the flow can be compared with the accelerating or decelerating flow in a die in which the viscoelasticity is known to play a major role. The flow in the fourth region, in the annulus, is an almost viscometric flow with the axial velocity component prevailing. In this type of flow the viscoelasticity does not come into the picture but, on the other hand, the shear-thinning viscosity exercises a significant influence upon the shape of the velocity profile and this has been confirmed by the observation of the iodine trace. We are thus left with a model in w h k h the average flow velocity and thus the circulation time are affected both by the shear-thinning anomaly (in the annulus) as well as by the viscoelastic anomaly (in the draught-tube). However, only the flow in the draught-tube contributes significantly t o the dispersion and thus to the mixing time and therefore the shear-thinning anomaly does not appear in the mixing time correlation. Conclusions a n d Scaling-up When designing a polymerization reactor or a mixer, one ought to be aware of the potential complications which the rheology of the polymer solutions may bring about. Recently, Ide and White (1974) reported that during the bulk polymerization of styrene the zero-shear viscosity TJO rose from 1 to 300 P in the region between 20% and 50% monomer conversion, and in the same region the zero-shear primary normal stress coefficient rose from a negligible value u p t o about 100. This illustrates well the type of difficulties to be expected: the polymerizing syrup a t 20% conversion will be an almost inelastic slightly shear-thinning liquid while a t 50% conversion the viscoelasticity might already be as high as to reduce the rate of mixing down by the factor of 3 or 4. However, there is some reason for moderate optimism. The normal stresses generated in the viscoelastic polymer solutions which give rise to all the peculiar secondary flow patterns are proportional to local velocity gradients. If this is kept in mind when designing the polymerizers or mixers then most of the difficulties can be avoided. In particular when scaling-up, the velocity gradients do not increase a t the same rate as the geometrical parameters. In fact, under certain circumstances the Weissenberg number which is the dimensionless measure of the viscoelasticity might even decrease with increasing dimensions of the mixer. This can be illustrated using a mixer model based on concentric cylinders (Figure 9). The liquid used for both the pilot and commercial-scale operations will be polysiloxane (silicon oil) which has a constant density and viscosity on both scales but a variable characteristic time A. If the geometrical similarity is observed then d l D 2 = d2D1. From the condition of dynamic similarity it follows that d12N1 = dz2N2. When the characteristic velocity gradient is taken as 4 = N d / h where h = D - d is the clearance between the cylinders, then $1
=Y2
(2) d
2
Figure 9. Scaling for viscoelastic liquids.
The chances are then that i.2 might be 20-100 times smaller than the velocity gradient in the pilot-plant unit. The Weissenberg number will not be, however, reduced by the same factor because as the characteristic shear rate decreases the characteristic time X of the liquid (which is shear dependent) increases. In any case, however, a careful analysis of all regions in the vessel with high shear rates will help to avoid the generation of the secondary recirculating vortices and thus the worsening of the mixing performance. Nomenclature b = material parameters, eq 2 c = bottom clearance of the screw, Figure 1 d l , dz = diameter of the inner cylinder, Figure 9 di = diameter of the screw, Figure 1 d d = diameter of the draught-tube, Figure 1 D1, D2 = diameter of the outer cylinder, Figure 9 DT = diameter of the vessel, Figure 1 h = gap between cylinders, Figure 9 H1 = height of the liquid, Figure 1 1 = length of the screw, Figure 1 m = material parameter, eq 2 M = torque,eq 1 N = rotational speed N1 = primary normal stress difference, eq 1 P = pitch of the screw, Figure 1 r = radial coordinate R = radius T = axial thrust, eq 1 CY = angle of the conical gap i. = velocity gradient 7) = viscosity TJO = zero-shear viscosity, eq 2 = apparent viscosity X = characteristic time OM = mixing time 8, = circulation time o = angular velocity $1 = primary normal stress coefficient, eq 2 p = density 4 = shear stress, eq 1 Re, Re" = Reynolds number for a Newtonian and nonNewtonian liquid, respectively Wi = Weissenberg number L i t e r a t u r e Cited Carreau, P. J., Patterson, I., Yap, C. Y., paper presented at the "Chemical Engineering Rheology" Conference, Salford, 1974. Chavan. V. V., Arumugam, M., Ulbrecht. J.. AIChEJ., 21, 613 (1975). Chavan, V. V., Ulbrecht, J.. Chem. Eng. J., 6, 213 (1973). Chavan, V. V., Ulbrecht, J., hd. Eng. Chem., Process Des. Dev., 12, 472 (1973); 13, 309 (1974). Coyle, C. K., Hirschland, H. E., Michel, 6.J.. Oldshue, J. Y., AlChEJ., 16, 906 (1970). Ford, D. E., Mashelkar, R . A., Ulbrecht, J.. Proc. Techn. lnt., 17, 803 (1972). Giesekus, H., Proc. fourth Int. Congr. Rheol.. I , 249 (1965); Rheol. Acta, 4, 85 (1965). Gray, J. E., Chem. Eng. Prog., 59, 55 (1963). Holmes, 6.D.. Voncken, R. M., Dekker. J. A,. Chem. Eng. Scb, 19, 201 (1964). Hoogendoom, C. J., den Hartog, A. P., Chem. Eng. Sci., 22,689 (1967). Ide. Y., White, J. L., J. Appl. Polym. Sci., 18, 2997 (1974). Johnson, R . I., lnd. Eng. Chem., Process Des. Dev., 6, 340 (1967).
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
325
Kale, D. D., Mashelkar, R. A., Ulbrecht, J., Chem. lng. Techn., 46, 69 (1974). Nagata, S., Nishikawa, M., Katsube, T.. Takaish. K., lnt. Chem. Eng., 12, 175
(1972). Nagata. S.,Yanagimoto, M., Yokoyama, T.; Mem. Fac. Eng. Kyoto Univ., 18,
444 (1956).
Tanlyama, I., Sato, T., KagakuKogaku(€ng.Abr.), 4, 1, 95 (1966). Ulbrecht, J., Chem. Eng., 286, 347 (June 1974). Ulbrecht, J.. Ford, D. E., paper presented at the "Chemical Engineering Rha ology" Conference, Salford, 1974. Zlokarnik, M., Chem. lng. Tech., 39, 539 (1967).
Novak, V.. Rieger, F.. Trans. lnst. Chem. Eng., 47, 335 (1969). Oshima. E., Yuge, K., KagakuKogaku, 34 (4),439;(l),115 (1970). Sato T., Taniyama, I., Kagaku Koiaku, 29 (3),153 (1965). Seichter P., Trans. lnst. Chem. Eng., 49, 117 (1971). Sykora, S..Coll. Czech. Chem. Commun., 31, 2664 (1966).
Received for reuiew July 17,1975 Accepted October 30,1975
Correlation of the Equilibrium Moisture Content of Air and of Mixtures of Oxygen and Nitrogen for Temperatures in the Range of 230 to 600 K at Pressures up to 200 Atm Kraemer D. Luks,* Patrick D. Fitzgibbon, and Julius T. Banchero Department of Chemical Engineering, University of Notre Dame, Notre Dame, lndiana 46556
The correlation of the equilibrium moisture content of air for temperatures in the range of 230 to 600 K at pressures up to 200 atm is presented, based on existing gas and liquid phase data in the literature. The scheme uses a third virial expansion to describe the gas phase and Henry's law to describe the liquid phase. The correlational scheme is generalized, employing a pseudo-single gas component description, to arbitrary mixtures of nitrogen and oxygen. The significance of this work is that a correlational scheme in one body is made available for the above-mentioned systems, which would be useful in the design of wet oxidation processes, which use air or oxygen-enriched air.
Introduction Previous studies by Landsbaum et al. (1955) and Zimmerman (1958) have attended to the prediction of the equilibrium moisture content of the gas phase in air-water vapor-liquid systems. Limited experimental studies of airwater vapor-liquid equilibria have been performed by Politzer and Strebel (1924), Webster (1950), and Goff et al. (1941, 1943, 1945, 1949, 1965): the nitrogen-water vaporliquid equilibria have been examined by Bartlett (1927), Saddington and Krase (1934), and Rigby and Prausnitz (1968). The recent studies of Hyland and Wexler (1973a) can be considered to be most exacting as well as valuable in light of the formally rigorous way in which they correlated their results. Their work covered the temperature range of -35 to +60 "C up to pressures of 100 bars, with their experimental studies focusing on the temperature range of 30 to 50 "C up to pressures of 100 bars. The objectives of this present study are to employ their correlational approach and extend such a correlation to high temperatures, up to 600 K, for the gas component air, and then to generalize the scheme to arbitrary mixtures of nitrogen and oxygen. Such a scheme would be valuable to designers of wet oxidation processes which use air or oxygen-enriched air. Thermodynamics In this section, the thermodynamics of a multicomponent gas-water vapor-liquid system shall be developed. Although arbitrary mixtures of nitrogen and oxygen are treated as a "pseudo-single" component, the validity of such an approximation was examined in light of a three-component 326
Ind. Eng. Chern., Process Des. Dev., Vol. 15, No. 2, 1976
(Nz-Oz-HzO) prediction. Furthermore, the authors intend
to extend this work to multicomponent gases (other than 0 2 and Nz) in the future. The thermodynamics detailed here is similar in spirit but differs from the development of Haar and Levelt-Sengers (1970) in that the states are depicted by (P, T ) rather than ( u , T ) variables. The derivation of Hyland and Wexler (1973b) purports to do the (P, T ) problem; however, as presented, it suffers from discrepancies, although their final result (their eq 29a) is correct. Let i = 1, . . . , n denote the gas components (excluding water). For a vapor-liquid system one has n 1 chemical P) equilibria a t some (T,
+
/JW~(T, P,Nw', {Nig))= /JW'(T,P, Nw', {Nil))
(1)
~ i g ( TP,, N w ~W, i g { ) = fii1(T> P, Nw', (Nil))
(i = 1,. . . , n) (2)
If one fixes T , P, and the water-free gas composition ( { x i / x n { , i = 1,.. . , n - 1)and replaces eq 2 with a set of empirical relationships in the form of Henry's law
Pi H ( P , T)xi
+
(i = 1,.. . , n )
(3)
the (n 1)equations can be reduced to a single equation in terms of a single mole fraction, such as xwg or x,g, e.g. In the spirit of the derivations of Hyland and Wexler (1973b) and Haar and Levelt-Sengers (1970), one can define as the change in the chemical potential of water in the a-phase (a = g, 1) from its pure saturated state to a state characterized by P > P,,,, that state being brought
