Ind. Eng. Chem. Res. 1991,30, 803-804 125 O C , the adhesive vitrification in the core layer of OSB should occur during this post manufacture period and be responsible for the strength improvement. Therefore, the adhesive strength that holds the board together at press opening does not appear to involve vitrification, but probably involves the gelation, as mentioned above, that helps resist the board expansion. Our experimental method with the fast heat-up schedules did not give the details on this early curing period. Further study in this region would be fruitful. Overall the DMA method used in this study appears to be very useful for comparing the curing rates of different PF adhesive resins used in the manufacture of OSB.
Conclusion The DMA method provided a detailed observation of the rigidity increase and loss modulus changes during the curing of P F resol resins. Tan -,6 was assigned as the vitrification point, which occurred in the early part of rigidity development with higher intensities at low cure temperatures. At higher curing temperatures the tan, 6 occurred later with lower intensities. This observation was interpreted to indicate that at higher temperatures PF resins vitrify after a higher extent of polymerization than at lower temperatures. The DMA method was shown to be a useful analytical tool in that the curing process of PF resins are characterized in terms of the cure time, vitrification time, and other useful parameters. The characterization results in turn allowed the differentiation of resin compositions and the optimization of resin synthesis or curing parameters as used in the manufacture of wood composite boards.
803
Acknowledgment We appreciate the many helpful discussions with Prof. Terry Sellers, Jr., and Mr. C. R. Davis. This work was done in partial fulfillment for the requirement of the Ph.D. degree of W.L.-S.N. Financial support from the USDA Wood Utilization Research Grant Program and the Mississippi Forest Products Laboratory is gratefully acknowledged. Registry No. F/P (copolymer), 9003-35-4;(formaldehyde)(phenol)(urea) (copolymer), 25104-55-6.
Literature Cited Carswell, T. S. Phenoplasts; Interscience Publishers, Inc.: New York, 1947. Ferry, J D.Viscoehtic Properties of Polymers;John Wiley & Sone: New York, 1961. Flory, P. J. Polymer Chemistry; Cornel1 University Press: Ithaca, NY, 1953. Gillham, J. K. Formation and Properties of Network Polymeric Materials. Polym. Eng. Sci. 1979, 19, 676-82. Kim, M. G.; Amos, L. W.; Barnes, E. E. Study of the Reaction Rates and Structures of a Phenol-Formaldehyde Resol Resin by C-13 NMR and Gel Permeation Chromatography. Znd. Eng. Chem. Res. 1990, 29, 2032-7. Lofthouse, M. G.; Burroughs, P. Material Testing by Dynamic Mechanical Analysis. J. Therm. Anal. 1978, 13, 19-53. Steiner, P. R.; Warren, S. R. Rheology of Wood-Adhesive Cure by Torsion Braid Analysis. Holrforschung 1981, 35, 273-8. Young, R. H.;Kopf, P. W.; Salgado, 0. Curing Mechanism of Phenolic Resins. Tappi 1981, 64, 127-30.
Received for review April 3, 1990 Revised manuscript received October 17, 1990 Accepted November 11, 1990
CORRESPONDENCE Comments on “The Probability Distribution of Growth Rates of Anhydrous Sodium Sulfate Crystals” Sir: Recently, Klug and Pigford (1989) performed two types of crystallization experiments to study the growth behavior of anhydrous sodium sulfate crystals. In the first, many isolated single crystals were grown in a flow cell and growth rates determined from measured crystal size-time variations. In the second set of experiments, transient crystal size distributions (CSDs) measured from an isothermal batch crystallizer were used to determine the moments of probability distribution of growth rate activity. The like momenta of distributions in these two sets of experiments are different. The stochastic distribution of growth rate activities determined from a large number of single isolated crystals may not necessarily be applicable to an ensemble of growing and nucleating crystals as in an isothermal batch crystallizer. Although many other variables may contribute to this difference, it is necessary to point out that basic definitions of growth rates conventionally used in these two types of experiments are different. For a single individual crystal, the growth rate is usually determined from the gradient of crystal eize-time variation. Experimental evidence based on singlecrystal studies tends to suggest that a given
crystal grows at a constant intrinsic growth rate and different crystals can have different growth rates for the same global environmental conditions. Thus, the growth rate of a single crystal may be used as its property or be characterized by an additional independent variable as used by the authors and termed growth rate activity of an individual crystal. The average growth rate of many isolated single crystals is
where E N j is the total number of crystals used. The conslstent definition of the average growth rate used for an ensemble of crystals with changing population with time is
where fi,, is the average zeroth moment and assumed constant over a small time interval between t and t + At. From eq 1
Q888-5885/91/263Q-Q8Q3$Q2.5Q f Q 0 1991 American Chemical Society
Ind. Eng. Chem. Res. 1991,30,804
804
Gi=
moAm, - m,Amo mo2At
-
mlAmo
=G,--
mo2At
(3)
Thus, the average growth rates determined from the batch experiments are usually higher than those determined from a flow cell or single-crystal studies. Uncertainties associated with moments of population density from a seeded batch experiment may perhaps be large because of the possible discontinuity between population densities of seed and newly generated crystals and tail effects. As only moments of growth rate activity distribution for crystals grown from both a seeded and unseeded batch crystallizer can be derived from the moments of observed transient CSDs, the efficacy of this approach appears uncertain in subsequent crystallizer performance evaluation.
Nomenclature G, = growth rate of the jth group of crystals, pmls Gj= average growth rate of many single isolated crystals,pm/s
a, = average growth rate of an ensemble, pm/s
mk = kth moment of population density with respect to size, number (pm)k/ g of suspension
Nj = number of crystals in the jth group, number C N j = total number of crystals t = time, s Registry No. Na2S04,7757-82-6.
Literature Cited Klug, G. L.; Pigford, R. L. The Probability Distribution of Growth Rates of Anhydrous Sodium Sulfate Crystals. Znd. Eng. Chem. Res. 1989,28, 1718-1725.
N.S . Tavare Department of Chemical Engineering University of Manchester Institute of Science & Technology (UMZST)
P.O. Box 88 Manchester, England M60 1 QD
Response to Comments on "The Probability of Growth Rates of Anhydrous Sodium Sulfate Crystals" Sir: Professor Tavare has correctly pointed out a matter that we have taken into consideration in our experimental and mathematical analysis of crystal growth, namely, the problem of determining growth kinetics from batch experiments in which both growth and nucleation are simultaneously occurring. This was accounted for in the following ways: (a) The batch crystallizer was seeded with a mass of large, monodisperse crystals that could be "tracked" and identified independent of secondary nuclei (as illustrated in Figure 4 of the paper). (b) In the case of unseeded crystallizer experiments, a fixed number, motof the largest particles were tracked for kinetic analysis. In both cases, the kinetic analysis is limited to a fixed population of crystals of constant number, m,,, As a result, the seond term in Tavares eq 3 becomes negligible, and the growth rates of fixed populations measured from eith-er the batch crystallizer or flow cell are equivalent, Gi = G,. The derivation of eq 19 from eq 18 in our paper is, in fact, based on this assumption. The accuracy of both approach a and approach b was evaluated in our study. The total number of particles, m,, from seeded experiments was calculated for each particle size distribution sample measured and found to remain essentially constant throughout each run. The experimental particle size distributions in Figure 4 show that the two distributions (seed and secondary nuclei) can be readily distinguished, and all subsequent kinetic analysis derived from this data showed remarkably little scatter, as shown by Figures 5-7. Hence, the assumption Amo = 0 for the seeded runs was satisfactorily met. The second approach, used for the analysis of the unseeded crystallizer, has somewhat less accuracy due to the fact that crystals are growing into and out of the tracked population at the lower size limit. However, if the number of particles in the tracked population is large, the "blurring" at the lower size limit has only a negligible effect on the overall measured behavior of the crystal population. As a result, the comparison of kinetic data derived from
the unseeded batch experiments with the independent kinetic measurements of single crystals in a flow cell is found to be in good agreement, as shown by the comparison in Figure 2. Hence, we find no fundamental reason why the kinetics of crystal growth measured from bdtch vs single crystal studies should differ as suggested by Tavare, unless hydrodynamics or other environmental factors come into play. It seems that with proper consideration of experimental design and resulting mathematics, such a conclusion is, on fundamental grounds, unfounded, and contradictory to our general findings. Finally, regarding the use of moment equations, it is well-known that particle size distributions can be approximated by the use of moment equations and that the approximation improves with the number of moments used. Equation 16 in the manuscript can be used to calculate as many moments of the growth rate distribution as desired. It is important, however, that the number of moments employed for kinetic measurements reflects the accuracy of the basic experimental data (in this case, particle size distributions). The accuracy of most particle size distribution data warrants consideration of only the first several moments. Greater mathematical resolution than this normally exceeds the precision of the experimental data several-fold and in no way improves the ability to obtain kinetic data or gauge subsequent crystallizer performance. Thus, the mathematical approach used does not present inherent limitations in this regard. As the accuracy of particle size measurement techniques improves, an appropriately larger number of moments may be calculated and employed for crystal growth determination. Registry No. Na2S04,7757-82-6. Diana L.Klug E. Z. du Pont de Nemours & Company Engineering Department, Experimental Station P.O. Box 80304 Wilmington, Delaware 19880-0304
0888-58~5/91/2630-0804$02.50/0 0 1991 American Chemical Society
