h), whereas the SE30 column was able to elute organics with retention times beyond dioctyl phthalate (2 h). The acid fraction was the most difficult fraction to chromatograph; the free acids did not elute or separate readily on SP1000, but on SE30 broad GC humps were observed. Derivatization of the acids into their corresponding methyl esters resulted in numerous identifications which were not apparent in the acid fraction alone. Fractionation, derivatization, partitioning, and glass capillary gas chromatography/mass spectrometry of the diethyl ether extract effected the identification of 460 compounds of the more than 700 detected. When examined for mutagenicity, the diethyl ether concentrate as a whole was tested. No one can predict which components caused the positive result in such a complex mixture. Was it a single compound, a class of compounds, or the synergistic effects of a mixture of compounds? These are some of the questions the toxicologists and other scientists in health related fields are trying to answer. We as research chemists do not know the answer, but hopefully we have demonstrated techniques of classifying organic contaminants into smaller groups for ease of identification, through extraction, fractionation, partitioning, and glass capillary gas chromatography. With such a separation scheme, it is conceivable that a similar scheme could be used by toxicologists to test many fractions of a mutagenic extract rather than unpartitioned concentrates to determine biologic effect. Acknouledgment
The technical assistance of Robert G. Tardiff and Donald E. Mitchell and the clerical assistance of Ms. J. Dukes are appreciated. Literature Cited (1) Kopfler, F. C., Coleman, W. E., Melton, R. G., Tardiff, R. G., Lynch, S. C., Smith, J . K., Ann. N.Y. Acad. Sci., 298.20 (1977). (2) Ames, B. N., Lee, F. D., Durston, W. E., Proc. Natl. Acad. Sci., 70,782-6 (1973). (3) Simmon, V. F., Tardiff, R. G., M u t a t . Res., 38(6), 389-90 11976). ~.-, (4) Sourirajan, S., “Reverse Osmosis”, Academic Press, New York, 1970. (5) Merton, U., “Desalination by Reverse Osmosis”, The M.I.T. Press, Cambridge, 1966. (6) Kopfler, F. C., Melton, R. G., Mullaney, J . L., Tardiff, R. G., in “Fate of Pollutants in the Air and Water Environments”, Suffet, I. H., Ed., Part 2, Wiley, New York, 1977. (7) Melton, R. G., Barone, K. A., Environmental Protection Agency, Health Effects Research Laboratory Internal Report, 1976. (8) Tardiff, R. G., Carlson, G. P., Simmon, V. F., in “Water Chlorination: Environmental Impact and Health Effects”, Vol. 1, Jolley, R. L., Ed., Ann Arbor Science, Ann Arbor, 1978.
(9) Webb, R. G., Garrison, A. W., Keith, L. H., McGuire, J. M., Environmental Protection Agency, Washington, D.C., Publication No. EPA-R2-73-277, August 1973, pp 88-9. (10) Ishiwatari, R., Hanya, T., Adu. Org. Geochem., 1051-65 (1970). (11) Law, L. M., Goerlitz, D. F., J . Assoc. Off. Anal. Chem., 53, 1276-86 (1970). (12) Leoni, V., J . Chromatogr., 62,63-71 (1971). (13) Logsdon, 0. J., Nottingham, K. E., Meiggs, T. O., paper presented a t the 91st Meeting of the Association of Official Analytical Chemists, Washington, D.C., Oct 1977, Abstr. No. 215. (14) Lin, D. C. K., Foltz, R. L., Lucas, S. V., Peterson, B. A,, Slivon, L. E., Melton, R. G., “Measurement of Organic Pollutants in Water and Wastewater”, Van Hall, C. E., Ed., ASTM S T P 686, American Society for Testing and Materials, 1979, pp 68-84. (15) Blomberg, L., J . Chromatogr., 115,365 (1975). (16) Grob, K., Grob, G., Chromatographia, 5 , 3 (1972). (17) Caironi, E., “Potential of Glass Capillary Columns and Specification of a Dedicated Instrument”, American Labroatory, Feb 1978, p 97. (18) Grob, K., Grob, K.. Jr., HRC & CC, J . High Resolut. Chromatogr. Chromatogr. Commun., 1(1),57 (1978). (19) Grob, K., Grob, G., J . Chromatogr. Sci., 7, 584 (1969). (20) Grob, K., Grob, G., J . Chromatogr. Sci., 7,587 (1969). (21) Grob, K., Grob, G., HRC & CC, J . High Resolut. Chromatogr. Chromatogr. Commun., 2(3), 109 (1979). (22) Lin, D. C. K., Lucas, S. V., Peterson, B. A., Melton, R. G., Coleman, W. E., in “High Resolution Gas Chromatography”,Cram, S. P., Ed., Academic Press, New York, in press. (23) Eichelberger, J . W., Budde, W. L., paper presented at the 22nd Annual Conference on Mass Spectrometry and Allied Topics, Philadelphia, Pa., May 1974. (24) Heller, S. R., Koniver, D. A,, Anal. Chem., 46, 947 (1974). (25) Hoyland, J. R., Neher, M. B., Environmental Protection Agency, Washington, D.C., Publication No. EPA-660/2-74-048, June 1974. (26) Hertz, H. S., Hites, R. A., Biemann, K., Anal. Chem., 43, 681 (1971). (27) Stenhagen, E., Abrahamsson, S., McLafferty, F. W., “Registry of Mass Spectral Data”, Vol. 1-4, Wiley, New York, 1974. (28) “Eight Peak Index of Mass Spectra”, Vol. 1-3, Mass Spectrometry Data Centre, AWRE, Aldermaston, Reading, RG74PR, U.K., 1974. (29) Environmental Protection Agency, Washington, D.C., Publication No. EPA-660/2-74-004, 1974. (30) Lingg, R. L., Melton, R. G., Coleman, W. E., Kopfler, F. C., in “Proceedings of the AWWA Water Quality Technology Conference”, Dallas, Tex., 1974. (31) Lingg, R. L., Melton, R. G., Kopfler, F. C., Coleman, W. E., Mitchell, D. E., J . Am. W a t e r Works Assoc., 69(11), 612 (1977). (32) “List of Organic Compounds Identified in Drinking Water in the United States”, Environmental Protection Agency, Health Effects Research Laboratory, Cincinnati, Ohio, April 1, 1977. (33) Fed. Regist., 43(21), 4108-9 (Jan 31, 1978). (34) Hites, R. A . , J . Chromatogr, Sci., 11,570-4 (1973).
Receiued for reuieu; April 12, 1979. Accepted February 7, 1980. Mention of trade names or commercial products is for identification only and does not imply endorsement by the U.S. Environmental Protection Agency.
Half-Calcination of Dolomite at High Pressures. Kinetics and Structural Changes Carol L. Steen*, Kun Li, and F. Heiskell Rogan Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pa. 15213
The cyclic use of solid sorbents for the reduction of SO2 and
H2S emissions from the combustion and gasification of coal has become an area of intensive research. One sorbent which has been found to be effective in the control of sulfur emissions is the carbonate mineral dolomite. Although dolomite may be recycled, the cyclic process is hindered by a slow and incomplete regeneration step resulting in decreased sulfur capture capacity and regenerability. In an effort to develop effective solutions to the problem of poor regenerability of spent do588
Environmental Science & Technology
lomite, the kinetics of the various reactions involved in the cyclic use of dolomite-half-calcination, sulfidation, and carbonation (regeneration)-have been studied. The half-calcination reaction:
-
+
CaMg(CO& CaCOB-MgO COz (1) is of particular interest for several reasons. The structural changes which accompany the half-calcination reaction include an increase in porosity in the size range of interest in 0013-936X/80/0914-0588$01.00/0
@ 1980 American Chemical Society
The rate of half-calcination of dolomite was studied as a function of temperature, total pressure, partial pressure of Con, pellet size, and grain size (type of dolomite). Structural data obtained from porosimetry, B E T surface area measurements, and optical and scanning electron microscopy were used to interpret the kinetic data. Over the range of conditions studied, the rate of reaction was a strong function of temper-
ature and grain size, but was independent of total pressure, partial pressure of CO2, and pellet size. Structural characteristics indicate t h a t the half-calcination reaction is a pearlite-type reaction in which the rate is controlled by the rate of product growth from active growth sites following nucleation.
sulfur capture. Also, the conditions under which the halfcalcination reaction is carried out are known to affect both the rate and extent of the sulfation reaction ( I , 2):
Company. The Guelph dolomite was quarried in Gibsonburg, Ohio, and the Stockbridge dolomite in Canaan, Conn. These materials are classified by Bituminous Coal Research (BCR) as no.'s 1337 and 1341, respectively. The chemical compositions of these dolomites as determined by Pfizer Company are given in Table I. The average grain size of the Guelph dolomite is measured t o be -112 pm, while that of the Stockbridge dolomite is -236 pm. The grain size distributions of these two materials are similar. A single pellet of dolomite was placed in a small platinum wire mesh basket which was suspended by a platinum wire from the balance arm of the TGA into the reactor tube. The reactor was pressurized with nitrogen, and a continuous flow of COn and N2 was fed t o the reactor. The reactor tube was then heated to the desired temperature, and the weight loss as a function of time was continuously recorded by a Du Pont 990 thermal analyzer. The fractional conversion, x , based on weight loss data was calculated by means of the relationship:
CaCOs.Mg0
+ SOz + 0.502
-
CaS04eMgO
+ CO2
(2)
Reaction conditions are expected to have a similar effect on the sulfidation react ion: CaC03-MgO
+ H2S
-
Cas-MgO
+ COz + H20
(3)
Dolomite may undergo decomposition by either of two reaction paths. Reaction 1 occurs if there is sufficient COz present to suppress the decomposition of CaC03 (3).If there is less than the equilibrium partial pressure of COz present, full calcination occurs: CaMg((l03)z
-
CaOmMgO
+ 2C02
(4)
According to Ruth et al. ( 4 ) ,the reaction of half-calcined dolomite with H2S can he faster than the reaction of fully calcined dolomite. During the course of the half-calcination reaction the pellet volume is unchanged, although the porosity of the stone increases due to the opening of intragranular pores in the 100 to 1000 A range. These morphological changes associated with the half-calcination reaction may be reponsible for the greater activity of the half-calcined stone for H2S or SO2 absorption, and suggest that an understanding of the relationship between the kinetics and structural changes which occur is needed. T h e earliest efforts to relate the kinetics and structural changes associated with the half-calcination reaction were made by Haul and Wilsdorf ( 5 , 6). These investigators employed the techniques of X-ray diffraction and microscopy t o study single crystals of dolomite at temperatures ranging from 600 to 800 "C and a t 1 atm total pressure. More recently, the relationship between the kinetics and structural changes has been studied at atmospheric pressure by Hubble and coworkers ( I , 2, 7 ) using the techniques of X-ray diffraction, optical and scanning electron microscopy, and porosimetry. Numerous investigators have studied the decomposition of dolomite a t pressures of 1 atm or less, but little work has been done a t higher pressures. Only recently has the effect of moderate pressures on the rate of half-calcination been studied, by O'Neill e t al. (8). In the present work, the rate of half-calcinationwas studied as a function of temperature, pressure, partial pressure of COz, pellet size, and grain size (type of dolomite) using thermogravimetric analysis (TGA). In addition, structural data from optical and scanning electron microscopy, porosimetry, and B E T surface area measurements were used to interpret the kinetic data. Experimental
The experimental system has been described previously (9). I t consists of a Du Pont 951 TGA modified for operation a t high pressures and temperatures in corrosive environments along with a system for gas feeding, blending, and metering of flow rates. The system is capable of operating a t pressures u p t o 60 atm and temperatures up to 1100 "C with corrosive gases. Spherical pellets of Guelph or Stockbridge dolomite were shaped from pieces of crushed dolomite provided by Pfizer
x = weight loss/(initial pellet weight
where % MgC03 was taken from the Pfizer analysis of Table I. Some reaction (110% conversion of MgC03 to MgO) did occur during the heat-up period. However, weight loss data obtained before the reactor had reached constant temperature were not included in the data analysis. Instead, data from the constant temperature period were extrapolated from the first isothermal data point to zero conversion to approximate the initial rate under isothermal conditions. Porosity measurements were made on both unreacted and half-calcined pellets using a mercury porosimeter. These measurements covered the range from lo2 to lo6 A. Surface area was measured in a Quantasorh by the flow technique using nitrogen-helium mixtures. Samples for examination with a scanning electron microscope were prepared in the standard manner. Samples for examination with a n optical microscope were vacuum impregnated with an epoxide resin and epoxide hardener. The impregnated samples were then ground, polished, and stained with Alazanin Red-S in dilute HCl ( I O ) , which causes the half-calcined material to appear darker than the unreacted material. R e s u l t s and D i s c u s s i o n
Kinetic Studies. T h e rate of half-calcination was studied as a function of temperature, pressure, partial pressure of Con, pellet diameter, and grain size (type of dolomite). Guelph dolomite was the principal dolomite studied. Observations relating to Stockbridge dolomite will be so noted. The overall rate of half-calcination is strongly influenced Table 1. Typical Chemical Analysis (Weight Percent) CaC03 MSC03 Si02 A1203 Fen03
Guelph
Stockbrldge
54.6 44.7 0.40 0.35 0.05
55 43 1.25 0.2 0.3
Volume 14, Number 5, May 1980
589
I
1.01
I
I
I
I
I
I
I
1
1
1
I
0
IO
20
30
40
0
I
I
I
TIME (min)
TIME (min)
Figure 1. Effect of temperature on the rate of half-calcination ( P = 20.4 atm, Pco2 = 4.1 atm, D = 2.5 mm)
I .o
I
i
1
I
Figure 3. Effect of total pressure on the rate of half-calcination (Pco2 = 4.1 atm, D = 2.5 mm)
I
I
I .o
(765°C) ( 7 3 5 Ti
c
I
0 TIME ( m i n )
IO
1
I
20 30 TIME (min)
I
40
Flgure 2. Effect of grain size (type of dolomite) on the rate of half-calcination ( P = 20.4 atm, PcO2= 4.1 atm, D = 2.5 mm)
Figure 4. Effect of partial pressure of COPon the rate of half-calcination ( P = 13.6 atm, D = 2.5 mm)
by temperature in the region 710-800 "C as shown in Figure 1. The rate increases with increasing temperature for both Guelph and Stockbridge dolomites, although the overall rate a t any given temperature is lower for the Stockbridge dolomite as shown in Figure 2. This difference in rates may be attributed to the factor of two difference in grain sizes. Neither total pressure (6.1-27.2 atm) nor partial pressure of COz (0 6-6.8 atm) had any measurable effect on the overall rate of reaction. This point is illustrated in Figures 3 and 4. Haul and Markus ( 11 ) observed that the partial pressure of COn had no effect over the temperature range 550-685 "C. The absence of pressure effects indicates that the rate of reaction is not controlled by diffusion or flow of COz within the pellet. The rate was also found to be independent of pellet diameter in the range 2.5-5.2 mm at high temperatures (1755 "C), but a slight dependence on pellet diameter was observed a t lower temperatures as illustrated in Figure 5 . Haul and Markus ( 1 1 ) noted the absence of a size effect in the range of 0.05-2 mm. The absence of a size effect at high temperatures indicates that the external resistance to heat or mass transfer
is not important. However, this may not be the case a t lower temperatures. Structural Studies. Structural data from porosimetry and BET surface area measurements and from optical and scanning electron microscopy (SEM) studies have been used to interpret the kinetic data. Porosimetry measurements indicate that a significant increase in porosity occurs upon half-calcination as shown in Figures 6 and 7 . The porosity of Guelph dolomite increases from 12.0 to 29.2% upon half-calcination. This increase in porosity is due primarily to the opening of intragranular pores in the 100-1000-8, range. It is these pores which are of interest in sulfur capture. An increase in surface area from 0.3 to 8.9 m2/g accompanies this increase in porosity. An optical microscopy comparison of polished samples of Guelph and Stockbridge dolomites indicates that there is a factor of two difference in grain size, 112 pm for Guelph vs. 236 pm for Stockbridge. For both Guelph and Stockbridge dolomites, optical studies indicate that the half-calcination reaction occurs "homogeneously" on the pellet level but topochemically on the grain level as shown in Figures 8 and 9.
590
Environmental Science & Technology
Floure 5. Effect of Dellet diameter on the rate 01 half-calcination w =
1 Flgure 8. Topochemical behavior at the grain level. Optical photomicroaraoh of 40% half-calcined Gueloh dolomite: 160X. Light areas
0.5
0 c 0
5LL
0.4
W
a
0.0 101
io'
I0 3
PORE
I0s
IO'
DIAMETER
6)
Figure 7. Pore volume distibution of 100% half-calcinated Guelph dolomite
.
. .
..
__
.
l . l . .
1
-,.
'I'hls behavior was also ODServea ov H a u l ana w11soor1 w,
\.,
01
and Siege1e t al. (2).These observations support t h e argument t h a t neither heat nor mass transfer a t t h e pellet level is t h e controlling mechanism.
-
~i~~~~ 9. Detail of grain boundary: Arrow indicates forming siab of half-calcined dolomite: 12OOX ( T = 715 C, P = 6.8 atm, Pcoz= 3.4 atm, D = 3.0 mm) I.._L^_ ., I\.u,,,un
c L""., . I , ,w,ay rnon I>Y"
EO,
.,-I
Fiaure 10. Lamellar structure of half-calcined dolomite SEM Dhotomicrograph of 40% half-calcined Guelph dolomite; 48000X ( T = 775 O C . P = 13.6 atm, Pco, = 6.8atm, D = 2.9 mm) SEM studies provided a great deal of useful information. It was observed that little change occurs in the grain size and shape during half-calcination under the conditions employed in this study. SEM examination also indicated the presence of a uniform distribution of magnesium in both unreacted and half-calcined samples, an observation also made by Siege1 e t al. ( 2 ) . By far, the most important observation is that the product of half-calcination appears to be formed in alternating layers, possibly of MgO and CaC03, as suggested by Huhble et al. (7), or of solid layers and void spaces. Evidence of this alternating structure is shown in a typical sample in Figure 10. This structure is quite similar in character to that of pearlite, the decomposition product of austenite (an ironcarbon alloy). Details of the pearlite structure are shown in Figure 11 (12). Model. A number of models have been proposed to represent the decomposition of dolomite. The,list of postulated reaction mechanisms has been reviewed by other authors (13), and includes primary dissociation of dolomite into CaC03 and MgC03 followed by independent decomposition of the carbonates, primary dissociation of dolomite into CaO and MgO followed by recombination of CaO with gas-phase COS,and the formation of solid solutions of MgO in CaC03. However, the rate-limiting step is not adequately represented by any of these mechanisms when applied to the data presented in this work. Kinetic data indicate that no classical gas-solid reaction model suitably describes the half-calcination reaction, hut the structural data provide the key to the reaction mechanism. The structural observations set forth in this work provide strong evidence that the half-calcination reaction is a pearlite-type reaction in which the rate-controlling step is the rate of growth of product from active growth sites following nucleation. The essential features of the pearlite reaction are as follows (14). 592
Environmental Science & Technology
Figure 11. Lamellar structure of pearlite. Transmission electron micrograph: 54400X. Reprinted with permission from ref 12.Copyright 1962. American Institute of Mining, Metallurgical,and Petroleum Engineers, New York.
* Nucleation sites are not randomly distributed; rather, they are found only a t grain boundaries, corners, or edges. Upon nucleation, the pearlite grows radially into the untransformed material. * The nucleation sites become saturated very early in the course of the reaction, saturation being indicated by the formation of slabs of pearlite hetween grains (15). Since site saturation occurs so rapidly, the nucleation rate vanishes from the kinetic law, and the growth of pearlite becomes the ratecontrolling step. The product is characterized by the lamellar morphology shown in Figure 11. In drawing analogies between the pearlite reaction and dolomite decomposition, the following observations should he emphasized. * As shown in Figure 8, the half-calcined dolomite nucleates initially a t gram boundaries, indicating that nucleation sites are not randomly located, and grows radially into the unreacted dolomite. The formation of slabs of half-calcined dolomite between grains is apparent in Figure 9, indicating that site saturation has occurred in those regions a t moderate overall conversion levels. * The lamellar morphology of half-calcined dolomite is evident in Figure 10. It should be noted here that the exact shape of the layers is not relevant to the development of the mechanism. Cahn (16)has shown that reactions involving site saturation are best represented by the equation: = 1 - e-kt" (6) where k and n are functions of temperature only. When conversion-time data are plotted as log In [l/(l- x ) ] vs. log t , the result is a straight line with slope, n , and intercept, k . The value of n may range from 1to 3 depending on whether the
6.0
the range 710-800 “C, and decreases with increasing grain size. The rate of reaction was found to be independent of both total pressure (6.1-27.2 atm) and partial pressure of COS (0.6-6.8 atm). A t high temperatures (755-800 “C),the rate of reaction was found to be independent of pellet size, although a slight dependence on pellet size was observed a t lower temperatures. The reaction was characterized as “homogeneous” on the pellet level and topochemical on the grain level. Both structural and kinetic data indicate that the rate of reaction is controlled by the rate of growth of half-calcined dolomite from active growth sites a t the grain boundaries or edges as in the pearlite reaction.
I
765 “C 2.0
I .o
0.8
t
0.6 0.4
Acknowledgment
The authors wish to thank Dr. B. R. Hubble of Argonne National Laboratory for many helpful discussions. Mr. Jeffrey H.-G. Yen and Mr. C. S. Huang contributed to the construction of the experimental system and the collection of data.
0.2 /
0
0.I
8
200
1
1
500
1000
Literature Cited
I
2000
5000
TI ME (seconds 1 Figure 12. Model for grain boundary nucleated reaction; n = 1.77 (Guelph); n = 1.42 (Stockbridge)
active growth sites are the grain boundaries (l),grain edges (2), or grain corners (3). All kinetic data obtained in the course of this study were found to fit Equation 6. For Guelph dolomite, the values of n were 0.98 a t 710 “C, 1.24 a t 735 “C, 1.51 a t 755 “C, and 1.77 a t 765 “C. These specific values seem to indicate a shift in active growth sites from the grain boundaries (0.98) to the grain edges (1.77). The data for Guelph and Stockbridge dolomite a t 765 “Care shown in Figure 12, and suggest that the value of n decreases with increasing grain size: 1.77 for Guelph vs. 1.42 for Stockbridge. The temperature dependence of n reflects the effect of temperature on nucleation and/or growth rates. For the pearlite reaction (decomposition of austenite), for example, the nucleation rate decreases while the growth rate increases with increasing temperature (17). Conclusions
The half-calcination reaction has been studied from both the kinetic and structural points of view leading to the following conclusions. The overall rate of half-calcination was found to be a strong function of temperature and grain size (type of dolomite). The rate increases with increasing temperature over
(1) Hubble, B. R., Siegel, S.,Fuchs, L. H., Hoekstra, H. R., Nielsen, E. L., Tani, B. S., “Chemical Engineering Division Environmental Chemistry Annual Report”, Argonne National Laboratory Report ANL-76-107. Julv 1975-June 1976. DV 35-71. (2) Siegel, S.,Fuchs, L. H., Hubble, B . k . , Nielsen, E. L., Enuiron. Sci. Technol., 12,1411-6 (1978). (3) Harker, R. I., Tuttle, 0. F., A m . J . Sci., 253, 209-24 (1955). (4) Ruth, L. A,, Squires, A. M., Graff, R. A., Enuiron. Sci. Technol., 6, 1009-14 (1972). (5) Haul, R. A. W., Wilsdorf, H. G. F., Nature ( L o n d o n ) , 167(9), 945-6 (1951). (6) Haul, R. A. W., Wilsdorf, H. G. F., Acta Crystallogr., 5 , 250-5 (1952). (7) Hubble, B. R., Siegel, S., Fuchs, L. H., Cunningham, P. T., “Chemical, Structural, and Morphological Studies of Dolomite in Sulfation and Regeneration Reactions”, Proceedings of the Fourth International Conference on Fluidized-Bed Combustion, the MITRE Corporation, McLean, Va., May 1976, pp 367-91. (8) , _ O’Neill. E. P.. Keairns. D. L.. Kittle. W. F.. Thermochirn. Acta. 14,209-20 (1916). (9) Li, K.. Roaan, F. H., Thermochim. Acta, 26.185-90 (1978). (10) Davies, P. J., Till, R., J . Sediment. Petrol.,’ 38(1), 235 (1968). (11) Haul, R. A. W., Markus, J., J . A p p l . Chem., 2, 298-306 (1952). (12) Darken, L. S., Fisher, R. M., in “Decomposition of Austenite by Diffusional Processes”, Zackay, V. F., Aaronson, H. I., Eds., Interscience, New York, 1962, p 286. (13) Haul, R. A. W.: Heystek, H., A m . Mineral., 37,166-79 (1952). (14) Cahn, J . W., in ref 12, p p 131-96. (15) Cahn, J. W., Trans. A m . Inst. Min. Eng., 209,140-4 (1957). (16) Cahn, J . W., Acta Metall., 4,449-59 (1956). (17) Christian, J. W., “The Theory of Transformation in Metals and Alloys”, Pergamon Press, Oxford, England, 1965, pp 677-96.
Received for reuieu: J u l y 25, 1979. Accepted February 7, 1980. Work performed under Contract No. EX-76-S-01-2408 for the lJ.S. Department of Energy.
Volume 14, Number 5, May 1980
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