Ind. Eng. Chem. Fundam. 1984,23,370-373
370
6 , = angle of friction between moving bed of particles and
stationary shoulders above discharge orifice t = void fraction to = void fraction of moving bed (assumed constant) 0 = polar angle for spherical coordinateswith origin at vertex of feed hopper 8, = value of 0 at wall of feed hopper (0, = a) p = coefficient of friction between particulate material and pipe wall p = air density pa = air density at atmospheric conditions ps = true density of solid material po = first term in expansion of p in powers of 0 u = rr component of particle phase stress tensor in eq 3; y y component of this tensor in eq 13 a = mean value of principal stresses in (r,O) plane uo = first term in expansion of a in powers of 6' u1 = value of usy on cross section at top of standpipe u2 = value of uys on cross section at bottom of standpipe u3 = value of urr on spherical cap spanning discharge orifice "h = value of urron spherical cap at bottom of feed hopper urr = rr component of stress tensor in spherical coordinates uro= r0 component of stress tensor in spherical coordinates asr = y r component of stress tensor in cylindrical coordinates up = y y component of stress tensor in cylindrical coordinates ayy = cross-sectional average of uys use = 00 component of stress tensor in spherical coordinates uH = $$ component of stress tensor in spherical coordinates 4 = angle of internal friction of particulate material
+
= azimuth angle of spherical coordinate system Literature Cited Brown, R. L.; Richards, J. C. "Principles of Powder Mechanics"; Pergamon Press: Oxford, 1970. Ginestra, J. C.; Rangachari, S.; Jackson, R. Powder Techno/. 1980, 2 7 , 69. Janssen, H. A. Z . Ver. Deutsch. Ing. 1895, 3 9 , 1045. Judd, M. R.; Dixon, P. D. AIChE AQM, Chicago, 1976. Judd, M. R.; Rowe, P. N. In "Proceedings of the International Fluidization Conference, Cambrldge", Davidson, J. F., Ed.; Cambridge University Press: Cambridge, 1978. Kaza, K. R. Ph.D. Thesis, University of Houston, Houston, TX, 1982. Kojabashian, C. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1958. Leung, L. S.; Wilson, L. A. Powder Technol. 1973, 7 , 343. Leung, L. S. I n "Fluidization Technology", Vol. 2; Keairns, D. L., Ed.; Hemisphere Publishing Corp.: Washington, 1976. Leung, L. S. Powder Technol. 1977, 16, 1. Leung, L. S.; Jones, P. J.; Knowlton, T. M. Powder Techno/. 1977, 19, 7. Leung, L. S.; Jones, P. J. Powder Techno/. 1978, 2 0 , 145. Matsen, J. M. Powder Technol. 1973, 7 , 93. Matsen, J. M. I n "Fluidization Technology", Vol. 2; Keairns, D. L., Ed.; Hemisphere Publishing Corp.: Washington, 1976. Nguyen, T. V.; Brennen, C.; Sabersky, R. H. J . Appl. Mech. 1979, 46, 529. Rangachari, S.;Jackson, R. Powder Techno/. 1982, 31, 185. Richardson, J. F.; Zaki, W. N. Trans. Inst. Chem. h g . 1954, 3 2 , 35. Richardson, J. F. I n "Fluidization"; Davidson, J. F.; Harrison, D., Ed.; Academic Press: London, 1971; Chapter 11. Sokolovskii, V. V. "Statics of Granular Media"; Pergamon Press: Oxford, 1965. Villadsen, J.; Micheisen, M. L. "Solution of Differential Equation Modeis by Pdynomiai Approximation"; Prentice-Hall, Englewood Cliffs, NJ, 1978. Wieghardt. K. Ann. Rev. F/u!d Mech. 1985, 7 , 89.
Received for review April 25, 1983 Accepted November 30, 1983
COMMUNICATIONS Absorption of Sulfur Dioxide in Calcium Hydroxide Solutions Rates of absorption of sulfur dioxide in calcium hydroxide solutions were measured by use of a laminar jet apparatus and contact times of between 10 and 25 ms. The hydroxyl ion concentration in the calcium hydroxide solutions was varied between 0.025 and 0.041 g-ion/L, and the sulfur dioxide concentration in the mixture of nitrogen and sulfur dioxide was varied between 0.96 and 2.86%. Under these conditions, the reaction is instantaneous and the results agreed well with the penetration theory.
Introduction Removal of sulfur dioxide from stack gases a t concentrations usually less than 3% is important in air pollution control. The sulfur dioxide may be removed by reaction with calcium carbonate or calcium hydroxide by either wet or dry scrubbing. In the dry scrubbing method, the solid calcium oxide particles are injected into the hot flue gas stream and, after an appropriate retention time, the sulfur dioxide is removed as calcium sulfate. In the wet scrubbing process, sulfur dioxide is absorbed in a solution or slurry, usually of sodium carbonate, sodium sulfite, ammonium sulfite, calcium oxide, calcium carbonate, or magnesium oxide. The mechanisms and rates of the reactions involved in the absorption process must be known for design of appropriate contactors. For the sulfur dioxide-calcium hydroxide solution system, the reaction mechanim may be considered to be similar to that of the sulfur dioxide-sodium hydroxide system (Onda et al., 1971) 2S02 + Ca(OH), 6 Ca(HSO& (a)
Ca(HSO&
+ Ca(OH), 6 2CaS03 + 2H20
(b)
The reaction kinetics of a gas-liquid system are usually estimated by measuring the amount of gas absorbed a t various contact times under well-defined hydrodynamic conditions. Absorption of carbon dioxide into laminar water jets has been extensively studied (Matsuyama, 1953; Scriven and Pigford, 1959; Cullen and Davidson, 1952; Toor and Raimondi, 1959). Carbon dioxide and alkaline solution systems have been investigated by Nijsing et al. (1959), Sharma and Danckwerts (1963), and Rehm et al. (1963). The absorption of sulfur dioxide into jets of sodium hydroxide solution has been studied by Onda et al. (19691, but the counterpart system of sulfur dioxide-calcium hydroxide has received little attention. Bijrle et al. (1972) attempted to measure the absorption rates of sulfur dioxide into jets of calcium carbonate slurry and calcium hydroxide slurry. With calcium hydroxide, precipitation occurred in the jet receiver, preventing proper operation. In the present study, this problem is overcome by use of a solids-free solution.
0196-4313/84/1023-0370$01.50/00 1984 American Chemical Society
U
Figure 1. Schematic representation of the experimental apparatus: (1)laminar jet; (2) glass nozzle; (3) glass receiver; (4) jet chamber; (5) gas inlet; (6) gas outlet; (7) nitrogen cylinder; (8) nitrogen manometer; (9) pyrogallol bubbler; (10)water bubblers; (11) sulfur dioxide cylinders; (12) sulfur dioxide manometer; (13) spilled liquid outlet; (14)overflow adjustment device; (15) liquid manometer; (16) overhead glass bottle; (17) monoethanolamine bubbler; (18) KOH bubbler; (19) filtration device; (20) vacuum pump.
In the present work, sulfur dioxide is absorbed into calcium hydroxide solutions in a laminar jet apparatus. The concentration of hydroxyl ion in the pure calcium hydroxide solution was varied over the range 0.0257 to 0.0417 g-ion/L. The sulfur dioxide content in the mixture of nitrogen and sulfur dioxide was varied over the range 0.96 to 2.87% and the contact time over the range 10 to 25 ms. Under the above conditions, the reaction is instantaneous. Experimental Section A schematic diagram of the apparatus is shown in Figure 1. Calcium hydroxide solution was stored in a 20-L bottle (16) under a carbon dioxide free atmosphere. A controlled amount of solution was allowed to flow through a bellmouthed glass nozzle (2) to obtain a laminar jet of diameter 0.908 mm. Spillage and gas entrainment were eliminated by carefully adjusting the liquid level in the overflow device (14). A mixture of sulfur dioxide and oxygen-free nitrogen was allowed to flow through the gas chamber (4) (i.d. = 3.7 cm) in the opposite direction to that of the solution. The jet length was controlled by adjusting the glass nozzle. Initially all the inlets to the storage bottle were closed and vacuum was applied. After distilled water was added to the filtration unit and allowed to flow into the storage bottle, the supernatant solution of the calcium hydroxide slurry was added to the filtration unit. The vacuum was then released to the atmosphere. During the experiments, air was passed through bubblers (17, 18) containing potassium hydroxide and monoethanolamine solutions to remove carbon dioxide before being admitted to the storage vessel. The sulfate ion concentration in the outlet solutions was determined by titration. A small quantity of calcium sulfite was converted to calcium sulfate as a result of the presence of dissolved oxygen. Calcium sulfate was determined by precipitation to account for all of the sulfur dioxide. The experiments were carried out at a temperature of 25 f 1 "C and a pressure of 91.1 kPa. Standardization of the Laminar Jet and Evaluation of the Gas-Phase Resistance The laminarjet may be standardized by absorbing either carbon dioxide or sulfur dioxide in water. In the present case, sulfur dioxide saturated with water vapor was absorbed in distilled water at room temperature and at-
V
'0
I
2
I
4
8
f
10
I
I
12
14
hV
Figure 2. Sulfur dioxide absorption in water.
2
L . a vc
CClSEC
10
12
Figure 3. SOz absorption in Ca(OH), solution for estimation of gas phase resistance: SOz = 1.927%; Bo = 4.13 X g-mol/cm3; 0 = 25.37 ms.
mospheric pressure. The total absorption rate, q , was determined at two jet lengths of 4.191 and 6.067 cm at various liquid flow rates. The solubility of sulfur dioxide in water, A*, at these experimental conditions may be calculated as 1.33 X g-mol/cm3 (Danckwerts, 1970). Values of (q/4A*)2,where q is the total rate of absorption in the liquid jet, were determined at various hV values and are plotted in Figure 2. The slope, DA, is 1.78 X cmz/s. This value may be compared with 1.72 X cm2/s, reported by Bijrle et al. (1972). In the chemical absorption experiments, the sulfur dioxide concentration in the gas phase was very low, suggesting the possibility of gas-phase resistance in the mass transfer process. To investigate this possibility, experiments were conducted to measure the total absorption rates at a contact time of 25.4 ms. The concentration of hydroxyl ion in the liquid was maintained at 0.0413 gion/L and the sulfur dioxide content in the gas phase at 1.93%. The gas flow rate in the jet chamber was varied from 6.12 to 12.24 cm3/s. The values of q at various gas flow rates are plotted in Figure 3. It can be concluded
372
Ind. Eng. Chem. Fundam., Vol. 23, No. 3, 1984
2'5
I 1,9
1.b
1.7
l.L L
1,i i,i 1.3
1.2 1.1
1
1.1
1.2
1.4
1.3
1.5
1.6
1.7
E,
Figure 4. SO2absorption in Ca(OH)2solutions: SO2 = 1.927%;(0) Bo = 4.173 X lo-' g-mol/cm3;(A) Bo = 2.573 X loa g-mol/cm3;(-)
theoretical.
from Figure 3 that there is no change in q with change in gas flow rate, suggesting that the gas-phase resistance is not significant (Boelter, 1943). The gas flow rate in subsequent experiments was arbitrarily maintained at 8.16 cm3/s. Results and Discussion The sulfur dioxide concentration in the gas phase was maintained at 2.88, 1.93,and 0.98%. Four different calcium hydroxide solutions having hydroxyl ion concentrations of 0.0257,0.0282, 0.0391,and 0.0417 g-ion/L were used. The contact times were varied between 10 and 25 ms. The total amount of sulfur dioxide absorbed per unit time is given by q = AV (1) The totalamount absorbed per unit area may be calculated as
Figure 5. Comparison of experimental, E, and theoretical, Ei, enhancement factors.
reaction, this independence can occur only if the reaction is instantaneous. (Instantaneous reaction models are discussed by Onda et al. (1971).)Enhancement factors are computed for the following reaction scheme SOz + Ca(OH)2 CaS03 + 2H20 (C)
-
or SO2 + 20H-
S032- + H2O
(d)
The enhancement factor for this type of instantaneous reaction is independent of time and may be given by 1 E; = (6) P erf (Dd1I2 where P is to be determined from the equation (Danckwerts, 1970) e(P2/DB) erfc(P/(Dg)'/2)
where the contact time 8 is given by e=----had2 4v The enhancement factor E is given by
-+
=
(Bo/2A*)(DB/DA)1/2e(Pz/DA) erf(P/(DA)1/2)(7) (3)
E = - Q(8)chem
(4) 8(0),hp Q(8), is the amount absorbed per unit area and may be calcuhed from
(5) The diffusivity, DA,may be taken to be the same as that in distilled water (1.78X cm2/s) since the solutions are very dilute. A* values are calculated with the method of Van Krevelen and Hoftijzer (Danckwerts, 1970) at various hydroxide concentrations and sulfur dioxide partial pressures (Hobbler, 1966). Enhancement factors for two different hydroxide solutions are plotted in Figure 4 with the square root of the contact time as the other parameter. The enhancement factors are fairly independent of contact time for all gas and liquid phase concentrations studied. For a bimolecular
Dg, the effective diffusivity of hydroxyl ion in the liquid, may be estimated as 3.654 X lo+ cm2/s (Onda et al., 1971). Equation 7 was solved for P with various Bo and A* values by Muller's iteration technique. The results are plotted in Figures 4 and 5. It is evident from Figure 5 that the sulfur dioxide-calcium hydroxide system falls into the instantaneous reaction category.
Nomenclature A = concentration of sulfur dioxide in the liquid phase, gmol/L B = concentration of hydroxyl ion in the liquid phase, g-ions/L d = diameter of the liquid jet, or jet nozzle, cm Di= diffusivity of species i in the liquid phase, cm2/s E = enhancement factor as defined in eq 4 Ei= enhancement factor in the case of instantaneous reaction as defined in eq 6 h = length of the liquid jet, cm q = total rate of absorption in the liquid jet, g-mol/s &(e) = total amount absorbed per unit area, g-mol/cm2 R(0) = rate of absorption per unit area of gas-liquid interface, g-mol/cm2s V , V , = flow rates of liquid and gas, respectively, cm3/s
373
Ind. Eng. Chem. Fundam. 1964, 23,373-374
Greek Symbols
j3 = a parameter determined from eq 7 or a measure of reaction plane movement, cm/s1I2 0 = time of contact, s Subscript 0 = initial Superscript * = at the interface Registry No. SOz, 7446-09-5; Ca(OH)z,1305-62-0.
Literature Cited B i b , I.; Bengetsson, S.; Farnkvist, K. Chem. Eng. Scl. 1972, 27, 1853. Boelter. L. M. K. Trans. Am. Inst. Chem. Eng. 1943, 39, 557. Cullen, E. J.; Davldson, J. F. Trans. Faraday Soc. 1962, 53, 113. Danckwerts. P. V. “Gas-Lbuid Reactbns”; Mcoraw-HiII: New York, 1970; p 81. Hobbler, T. “MassTransfer and Absorbers”; Pergamon Press: Oxford, 1968; p 481.
Matsuyama, Y. Mem. Fac. Eng. Kyoto Unlv. 1953, 15, 142. NiJsing, R. A. T. 0.; Hendriksz, R. H.; Kramers, H. Chem. Eng. Sci. 1959, 10, 88. Onda, K.; Kobayashl, T.; Fugine, M.; Takehashi, M. Chem. Eng. Scl. 1971, 26, 2009. Onda, K.; Sada, E.; Kobayashi, T.; Odaka, M. Kagaku Kogaku 1969, 33, 886. Rehm, T. R.; Moll, A. J.; Babb, A. L. AIChEJ. 1963, 9 , 780. Scriven, L. E.; Plgford, P. L. AIChEJ. 1959, 5 , 397. Sharma, M. M.; Danckwerts, P. V. Chem. Eng. Sci. 1963, 18, 729. Toor, H. L.; Raimondi, P. AIChE J. 1959, 5 , 86.
Department of Chemical Engineering and Applied Chemistry University of Toronto Toronto, Ontario, Canada M5S l A 4
Dasari Ram Babu’ G . Narsimhan’ Colin R. Phillips*
Received for review April 20, 1982 Accepted January 25, 1984 Research Laboratory, ylderabad, India. * Regional Bendei State University, Nigeria.
CORRESPONDENCE Comments on “Materlai Stabiilty of Multicomponent Mixtures and the Multiplicity of Solutions to Phase-Equllibrlum Equations. 1. Nonreacting Mixtures” Sir: A paper by Van Dongen et al. on “Material Stability of Multicomponent Mixtures and the Multiplicity of Solutions to Phase Equilibrium Equations” appeared in the November 1983 issue of this journal. The acceptance for publication of this work deserves some comments, since in my opinion the paper lacks relevance, uses substantial amounts of space for repeating known material, and is in error. The bulk of the paper is concerned with stability, and according to basic definitions mixtures of fixed overall composition at given temperature and pressure are classified as either stable, metastable, or intrinsically unstable. A stable mixture cannot decrease its Gibbs free energy. Metastable mixtures are unstable and form separate phases in virtually all situations of practical interest. The distinction between metastable mixtures and intrinsically unstable mixtures is that for the former the matrix of second composition derivatives of the Gibbs free energy has no negative eigenvalues (i.e., is positive semidefinite), whereas for intrinsically unstable mixtures at least one eigenvalue is negative. For metastable mixtures a split into phases with a macroscopic difference in composition is required to decrease the Gibbs energy, and for intrinsically unstable mixtures the decrease is obtainable with infinitesimal differences. In the paper by Van Dongen et al., only intrinsically unstable mixtures are considered unstable, whereas metastable mixtures are treated as stable. The inadequacy of this treatment is well demonstrated by a liquid mixture of water and benzene at 298 K. Using the UNIQUAC model with parameters from Serrensen and Arlt (1979, p 341) the mixture will form two liquid phases when the overall water concentration is in the range 0.003 < XHl0 < 0.9996. However, intrinsic instability is only observed in the range 0.43 < X H I O < 0.91. This result is of little value, since any algorithm for practical industrial application must be able to correctly handle metastable mixtures. Recent examples of such algorithms are those 01964373/04/1023-0373$01.50/0
of Gautam and Seider (1979), Fournier and Boston (1981), and Michelsen (1982). The authors of the present paper do not refer to these attempts but only state that a more complete stability analysis requires a tremendous effort in trial and error search. Admittedly, the relevance of any subject is open to discussion. A more detailed treatment of the criterion of intrinsic instability might be justified if “the extension of the theories to multicomponent mixtures is nontrivial and lacks careful development”,and if, as claimed, “several new facts about material stability and the spinodal curve have been found”. However, the authors’ “new facts” are often incorrect. Section 5 contains a 3-page “proof“ that a two-phase equilibrium will be stable if and only if the phases are individually stable with respect to phase splitting, and it is stated that “this is the first satisfactory proof“ of an often quoted result. The basic equation used by the authors is eq 58 H = G’ GI1 where H is the matrix of second derivatives of the total Gibbs energy with respect to the amount of material transferred between phases, and the G matrices represent matrices of second derivatives of the Gibbs energy with respect to mole numbers for the individual phases. The authors claim that H is positive definite if and only if the G matrices are both positive semidefinite. The if is trivially correct since the individual terms in the sum of quadratic forms, eq 59
+
1
+1
G = -eTG’c -eTG”c 2 2 are both nonnegative when the G matrices are both positive semidefinite. The sum can only be zero provided t is a common eigenvector with zero eigenvalue for both matrices. By virtue of the Gibbs-Duhem equation, both matrices have a zero eigenvalue eigenvector with elements 0 1984 American Chemical Society
