Note Added in Proof. After completion of this work it came to our attention that Le Goff et al. [Znd. Chirn. Belge, 11, 1141-1164 (1964)j have suggested an expression for the mass to momentum transfer efficiency in contactors, R, on the basis of economical considerations
where G is the volumetric flow rate, COand C, are the concentrations of the solute transferred, and Po and P, are the pressures at the inlet and outlet, respectively. The ratio R can be made dimensionless (P. Le Goff, personal communication) when the numerator and denominator are divided by G(C0 - C,)/NTU, which can be considered as the maximum flux to the wall at infinitely fast mass transfer in the contactor, and by the influx of kinetic energy, respectively. Multiplying the ratio NTU/Eu, thus obtained, by ( S C )3-or ~ by (Pr)2 3 for heat transfer-we get the Le Goff number. Nomenclature A = internal surface area of the tube d = tube diameter D, particle diameter Eu = Eulernumber F = liquid flowrate f / 2 = frictionfactor (f/2)p = friction factor for packed bed Lf = LeGoffnumber j = jfactor 1 = sluglength L = tubelength NTU = number of transfer units Nu = local Nusselt number Nu = average Nusselt number Nu, = viscous Nusselt number
Re = Reynoldsnumber R e = Reynolds number calculated with liquid flow rate Rep = Reynolds number for packed bed S = tube cross section Sc = Schmidtnumber St = local Stanton number 3 = average Stanton number V = average liquid velocity
Greek Letters
p = dimensionless slug length, l / d AP = pressure drop c = void fraction p = liquidviscosity p = liquid mass density Literature Cited Bird, R. B., Stewart, W. E., Lightfoot, E. N.. "Transport Phenomena," pp 197-200, Wiley, NewYork, N.Y., 1960. Colburn, A. P., Trans. A . 1.Ch.E.. 29, 174 (1933). Engasser, J. M., Horvath. C., Chem. Eng, Sci., 29, 2259 (1974). Ettre, L. S., "Open Tubular Columns in Gas Chromatography." p 2, Plenum Press, New York, N.Y.. 1965. Horvath, C., Solomon, B. A . , Engasser, J. M., lnd. Eng. Chem.. Fundam., 12, 431 (1973). Knudsen, J. G . , Katz, D . L., "Fluid Dynamics and Heat Transfer," pp 407-425, McGraw-Hill, New York, N.Y.. 1958. LeGoff, P., Chim. lnd., 103, 1805 (1970a). LeGoff, P., Chim. lnd., 103, 1959 (1970b). Opfell, J. B., Sage, B. H., in "Advances in Chemical Engineering," Vol. 1, T. H. Drew and J. W. Hoopes, Ed.. p 274, Academic Press, New York, N.Y., 1956. Pretorius, V . , Smuts, T. W.. Anal. Chem., 38, 274 (1966) Williamson, J. E., Pazaire, K. E., Geankoplis. C. J., lnd. Eng. Chem., Fundam., 2, 126 (1963).
Received for review M a y 9, 1974 Accepted J a n u a r y 30,1975 T h i s s t u d y a n d i t s p u b l i c a t i o n has b e e n m a d e possible by grants No. GM 21084-01 a n d G M 20993-01A1 f r o m t h e N a t i o n a l I n s t i t u t e s of H e a l t h , US.P u b l i c H e a l t h Service.
Mass Transfer for Inorganic Salts across Liquid Films: Systems of H3P04, NaH2P04, and KH2P04 with Water Jaygwan G. Chung and George Thodos* Northwestern University, Evanston. //l/no/s6020 1
Initial rates for the transfer of phosphates into water flowing past the surface of porous spheres in differential beds have been used to establish mass transfer coefficients and factors for the H3P04-HzO and NaHzP04-HZ0 systems for 1.3 < Re < 63. The j factors for these electrolytic systems follow essentially the same type of dependence with Re as is exhibited by nonelectrolytic systems, but are approximately six times lower. Additional data for the KH2PO4-HzO system support this behavior. This disparity cannot be explained from our current knowledge of mass transfer, but may be due to a hindrance on transfer resulting from interactions of the neutral ionic clusters diffusing through the liquid film.
Considerable experimental work is presented in the literature on mass transfer across gaseous and liquid films for flow through packed and fluidized beds. Gamson et al. (1943) established mass transfer factors for gases from experimental information for the vaporization of water from the surfaces of spheres and cylinders to air streams flowing through the packed beds. Chu et al. (1953) extended these studies and utilized the air-naphthalene system for 110
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
both packed and fluidized beds. McCune and Wilhelm (1949) and Hobson and Thodos (1949), utilizing different experimental approaches, established relationships for mass transfer factors of liquids which have the same functional dependence with the modified Reynolds number as was previously obtained for gases. For binary aqueous systems of benzoic acid and propylene glycol, Wilson and Geankoplis (1966) obtained mass transfer factors for Reyn-
- -
+ 0
0 0
A
4
Chu, Kalil,ond Wetteroth (1953) Gomson, Thodos, ond Hougen(1943) Hobson and Thodos (1949) McConnachie and Thodos(1963) McCune and Wilhelm (1949) Wilson and Geankoplis (1966)
0346-0877 0404-0430 0475 0416-0778 0494- 0921 0401-0436
gas gas liquid gas
liquid liquid
00
Re
=F DUP
Figure 1 . Relationship of ij vs Re for the flow of gases and liquids through packed and fluidized beds.
olds numbers as low as Re = 0.0016. The combined efforts of these experimental studies and those of Gaffney and Drew (1950), DeAcetis and Thodos (1960), and Williamson et al. (1963) have demonstrated that mass transfer factors for gases and liquids possess the same dependence with the modified Reynolds number. From the experimental studies of McConnachie and Thodos (1963) for the vaporization of water from the surface of distended beds of spheres, the effect of the void fraction, c , was incorporated into the parameter c j to account for the behavior of packed and fluidized beds (Sen Gupta and Thodos, 1962). Values of tj have been plotted against the modified Reynolds number, Dup/w, in Figure 1from the data of some of these sources. A general characteristic of the systems selected for these studies is the fact that the transferable component is either organic or inorganic in nature. The selection of these systems has been quite arbitrary and was largely dictated by the convenience of the experimental procedure and analysis of the data. It seems appropriate to extend these studies to the transport, through a water film, of inorganic salts. Such salts, of necessity, must ionize, but in the absence of an electric field, neutrality must be maintained throughout the system. Transport of this type requires that both the cations and anions of the salt diffuse through the liquid film in consort and it would be expected that the mass transfer behavior of these systems would be similar to systems already investigated. T o obtain information for such systems, aqueous solutions of H3P04 and NaHzP04 were selected for study. In addition, the limited information available for the KHzP04-Hz0 system (Chung, 1972) has also been included. Experimental Procedure
The experimental technique involved the unsteady state approach reported by Hobson and Thodos (1949). Briefly, the procedure involves the upward flow of distilled water through the voids of a differential bed of porous Celite spheres, previously saturated with a standard phosphate solution. Celite was selected as the carrier,
since an experiment was performed by contacting Celite spheres with phosphate solution for 40 days without any noticeable decrease in phosphate concentration of the solution. Therefore, the phosphate solution occupies the pores of the Celite spheres without any physical bonding. The overflow from a bed of these spheres, collected at different time intervals, was analyzed for phosphate content. The average phosphate concentration of the effluent leaving the differential bed reactor was plotted against time. When the resulting relationship was extrapolated to zero time, the effluent phosphate concentration was obtained which was used to account for the initial mass transfer rate. In this study solutions of phosphoric acid (&Pod) and monobasic sodium phosphate (NaHzP04) were used in concentrations of 1000 and 1500 mg/l., respectively. The phosphate concentration in each effluent was established using the colorimetric analysis recommended by the American Public Health Association (1965). This method involves the reaction between ammonium molybdate, (NH4)3Mo03, and potassium antimonyl tartrate, K ( S b 0 ) C4H406, in the presence of orthophosphate solution and sulfuric acid to produce the phosphomolybdate ion, PO4 1 2 M 0 0 3 ~ - .The addition of ascorbic acid produces the intensely colored “molybdenum blue” which can be accounted for by spectrophotometric measurements a t 700 mp. These measurements were made using a Universal spectrophotometer (Coleman Model 14). Three different sizes of Celite spheres, of average diameter, 0.368, 0.835, and 0.955 cm, were used. These catalyst carriers were made of calcined diatomaceous earths and were highly porous. A charge of these spheres was soaked in phosphate solutions at room temperature, for a prolonged period, a t least for 24 hr before initiating an experimental run. During this soaking period, no phosphate concentration change was noted in the solution. After soaking, the spheres were rolled on cheesecloth to remove excess solution adhering to the surface without removing solution from the pores by capillary action. Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
Ill
3t
Number Celite Spheres I4 PIosttc Spheres none
3, seconds
Figure 1. Phosphoric acid concentration-time relationship for the effluent of Run 96.
+ e, -
c
Figure 2. Schematic diagram of differential bed reactor.
7
1
of a differential bed, it became necessary to dilute the active spheres in a matrix of inactive solid plastic spheres. A run consisted by introducing distilled water into the reactor. In Figure 3, a schematic diagram of the experimental unit is presented which includes a distilled water recirculating system provided with a constant head facilit y to maintain a steady flow rate into the reactor. The time of the run commenced when the water front reached the lower Teflon ring and was continued after the appearance of effluent. The first effluent sample was most important and was followed by six to seven continuous samples, all of which were analyzed for phosphate content. Interpretation of D a t a A typical relationship of phosphate concentration vs. time for the effluent is presented in Figure 4 for Run 96. For this run, 7 sec was required for the liquid front to travel from the lower Teflon ring to the effluent port and consequently, for this run, the initial concentration is co = 2.18 mg of H3P04/1. This value enables the calculation of the initial rate of mass transfer and corresponding driving force. This information, along with the available mass transfer area, allows the calculation of the mass transfer coefficient from the relationship Y = k,aV(c,
Figure 3. Schematic diagram of experimental unit.
Three glass reactors of different inside diameters accommodated the differential beds of these studies. A schematic diagram of a typical differential bed reactor is presented in Figure 2 . This reactor includes inlet and effluent sections, which can be disengaged in order to introduce the differential bed between them. Each differential bed was confined between two stainless steel screens which were separated by a Teflon cylindrical ring. The lower screen was supported in place by a Teflon ring fitted tightly against the inside glass surface while another Teflon ring held in place the top screen in the effluent section which was filled with glass spheres in order to reduce the void volume in this section. The inside diameter of the Teflon cylindrical rings were 1.89, 2.54, and 3.81 cm and represent the diameters of the differential bed reactors of this study. In order to reduce the bed height of some runs and a t the same time retain the requirements 112
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
L
-
c),
(1)
Mass transfer coefficients, h ~ calculated , for the runs of the H ~ P 0 4 - H z 0and NaH2P04-HzO systems are presented in Table I. Diffusion coefficients available in the literature (International Critical Tables, 1928; Parsons, 1959) enabled the calculation of the Schmidt number, Sc, correspondirig to these systems. For the H3P04-H20 system, D1 = 0.89 x 10-5 cm2/sec a t 20°C while for the NaH2P 0 4 - H ~ oL)1 = 0.874 x 10-5 cmZ/sec at 25°C. The experimental and literature information were utilized to calculate the mass transfer factor j =&a(”) en (2) Ll.11 PD, f corresponding to each run. These values are presented in Table I and have been plotted against the modified Reynolds number to produce for both systems the single relationship presented in Figure 5 . The earlier studies of Chung (1972) with the KHzP04-H20 system cannot be included in Figure 5 because the diffusion coefficient for this system is not available. However, since the Schmidt number represents a system parameter, a plot of k l / L vs.
O5I 04
03
-
020
0 0 0 0 0 0 0 0 0 0
H,P0,-H20
0 NoH,PO,-H,O
J
006 005 004 010 -
~ w ~ - ~ m r n m m m t 3 w m w w ~ r n r n t - m
008
109?909"90
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
w ~ w o w w w ~ o m mw m m m w o m w m m w f - w w m r-cooma+ 3 m o o o o o o o o 0 0 3 0 0 3 3
003-
3 3 3 3 3 3 3 3
wwwcornw
WOE-wwrn
"41"4c?
E-m-lowm 3
3 N 3
Figure 5. Fklationship between J and N I { ~ resulting for the packed bed mass transfer studies of the H3P04-Hz0 and NaHz-
P04-HzO systems.
0004
0
H,P4 - H 2 0
0 NoHpPOa-H20
-
0 KHpP04-Hz0
' 1
s- 00010 00008 00006
5 ;Nir
00004 00003 0 00021
i
2
3
4
6 8 1 0
20
30
40
60 80 100
0
?""""""?"E-.
m w E - 3 w w r r m w m N r n 3 c o 0 3 N m
Figure 6. Relationship between k l / L and N I ~for , three different
3 3 3 N 3
phosphate-water systems.
r ( ~ w m ~ w w 3 w m
N N E - . S ~ ~ S N E - . ~ O O O N O O N O O N
m m m w m m w m m w
against Re in Figure 6. These three systems produce essentially a single relationship, indicating that their diffu4 4 4 9.4 4 4 4 4 4 4 4 4 4 4 4 o o o o o o p o o o 0 0 0 0 0 0 sion coefficients are of the same order of magnitude. In order to compare the results obtained for the transooornoolnoorn oolnorno 3 3 3 3 3 port of these inorganic salts with those obtained for other nonelectrolytic systems, values of t j for the H3P04-H20 3 3 w m 3 v r n 3 w m 3 v r n 3 r n w t - E - 3 3 e 3 3 E - 3 3 E - 3 3 - 3 3 and NaHzP04-HzO systems were included in Figure 1. It can be seen that significant disagreement exists between 04c9'9eU." the relationships for these electrolytic and nonelectrolytic m N 3 N N 3 NNNNCUN systems. To reaffirm the validity of the results obtained for the W O E - c o t - 0 1993.99 H3P04-HzO and NaH2P04-HzO systems, additional ex3 3 3 3 3 3 perimental work was conducted to verify the capability of the equipment used in the present study to reproduce mass transfer data on a nonelectrolytic system reported in the literature. In this regard, Run 111 was conducted cowlnrncornlnwrnrn cornlnmrnrn wwmrnwmrnwmln wmrnwrnm using the isobutyl alcohol-water system and was commmcommcommmm c?=?~m.903. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 pared with Run 2A of Hobson and Thodos which corresponds to the same flow conditions. A comparison of the w r n w ~ - w m o r ( ~ m wrnwt-mm m m m m m m o o o o 0 0 0 0 0 0 results of these two runs is listed in Table 11. The close 3 3 3 3 3 3 3 3 3 3 agreement of j factors for essentially the same Reynolds number of these two runs indicates that the equipment used in this study is reliable and capable of reproducing data obtained in 1949 on a different facility. This agreeRe should produce a relationship characteristic of each ment indicates that the experimental results reported in system. For the two systems investigated in this study this study, although unusual, are reliable from an experiand the KHzP04-HzO system values of k l / L are plotted mental consideration. wwE-wwt-wwE-w
wE-wwwE-
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
113
Table I1
D I = liquid diffusion coefficient of system, cm2/sec Dp = diameter of particle, cm i
Run
Investigator
System
2A Hobson-Thodos Isobutyl alcohol-water 11 1 C hung-Thodos Isobutyl alcohol-water
Fac Year tor Re 1949 1.82 4.40
1972 1.76 4.36
Conclusions
j = mass transfer factor, dimensionless kl = mass transfer coefficient for liquid film, g-mol/sec cm2 (g-mol/cm3) L = superficial mass velocity, g/sec cm2 M = molecular weight of flowing liquid Re = modified Reynolds number, D,L/l Sc = Schmidt number, k / p D ~ u = superficial velocity, cm/sec V = volume of packed bed, cm3
Greek Letters = void fraction d = time, sec = absolute viscosity, g/sec cm p = density, g/cm3 t
The significant disparity between cj values obtained in this study for the aqueous phosphate systems and those for other systems reported in the literature is difficult to explain at this time. The possible existence in the transport process of a species containing a number of water molecules would produce a constituent of a higher molecular weight than that for the molecular species of each phosphate and therefore should produce higher cj values than normally expected. The fact that the c j values for phosphates are lower may be attributed to the electrolytic nature of the species being transported, Further experimental studies with other inorganic salt-water systems must be carried out in order to establish the characteristic behavior of this type of species. Nomenclature a = transfer area, cm2/cm3 c = phosphate concentration in solution, g-mol/cm3 clf = concentration of nontransferable component in film, g-mol/cm3 co = initial phosphate concentration, g-mol/cm3 cs = phosphate concentration at solid surface, g-mol/cm3
Literature Cited American Public Health Association, "Standard Methods for the Examination of Water and Wastewater," 12th ed, New York, N.Y., 1965. Chu, J. C., Kalil, J.. Wetteroth, W. A,, Chem. Eng. Progr., 49, 141 (1953). Chung, J. G., Ph.D. Dissertation, Northwestern University, Evanston. Ill., June, 1972. DeAcetis, J., Thodos, G., Ind. Eng. Chem., 52, 1003 (1960). Gaffney, B. J., Drew, T. E., Ind. Eng. Chem., 42, 1120 (1950). Gamson, B. W., Thodos, G.. Hougen, 0. A,, Trans. Am. Inst. Chem. Eflg., 39, 1 (1943). Hobson, M.,Thodos, G., Chem. Eng. Progr., 45, 517 (1949). "International Critical Tables." Vol. 5. p 65, McGraw-Hill Book Co., New York, N.Y., 1928. McConnachie, J. T. L., Thodos. G., AIChE J., 9, 60 (1963). McCune, L. K . , Wilhelm, R. H., Ind. Eng. Chem., 41,1124 (1949). Parsons, R.. "Handbook of Electrochemical Constants," p 80, Butterworths, London, 1959. SenGupta, A,, Thodos, G.. Chem. Eng. Progr., 58, 58 (1962). Williamson, J. E., Bazaire, K. E., Geankoplis, C. J.. Ind. Eng. Chem., Fundam., 2,126 (1963). Wilson, E. J., Geankoplis, C. J.. Ind. Eng. Chem., Fundam., 5, 9 (1966).
Received for review January 10, 1974 Accepted D e c e m b e r 5, 1974
Experimental Test of a Transition State Theory for Diffusion Coefficients in Multicomponent Liquids Robert G. Mortimer,* William R. Marlow,' and Jerry L. Shenep Department of Chemistry. Southwestern at Memphis, Memphis, Tennessee 381 72
Multicomponent diffusion coefficients for the system 2,2-dichloropropane-l,l,l-trichloroethane-carbon tetrachloride were measured at 25°C to test a previously published transition state model theory of Mortimer and Clark, which includes correlated motions of pairs of particles as well as motions of individual molecules. Empirical Onsager diffusion coefficients in the center-of-mass coordinate frame were determined and compared with theoretical predictions. The Onsager coefficients can be represented as a function of mole fraction by L i i = axj bxj2 and Lj, = bxjx, with a = (24.65 f 1.71) X l o - ' ' mol* c m - ' tal-' s e c - ' and b = (24.92 f 1.22) X l o - " mol' c m - ' c a l - ' sec-'. Theory and experiment agreed within experimental error. The results are interpreted to indicate that the principal molecular diffusion process in this system is the exchange of position of two adjacent molecules, with a free energy of activation of 4.9 kcal mol-'.
+
Isothermal diffusion in a number of ternary systems has been studied by use of the Guoy diffusiometer and various diaphragm cells (Wendt and Shamim, 1970; Shuck and
' National Science Foundation Undergraduate Research Participant 114
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
Toor, 1963; Kett and Anderson, 1969; Cussler and Dunlop, 1966; Cullinan and Toor, 1965; Burchard and Toor, 1962; Kirkaldy, 1970). These measurements are not only of inherent interest, but are useful in evaluating model theories of multicomponent diffusion, such as those of
