I n d . Eng. Chem. Res. 1990,29, 361-366 HZSM-5 Type Catalysts. Appl. Catal. 1983, 8, 43-56. Mavrodinova, V.; Minechev, Ch.; Penchev, V.; Lechert, H. Toluene Conversion over Offretite, Omega and ZSM-5. Zeolites 1985, 5, 217-220. Meshram, N. R. Selective Toluene Disproportionation over ZSM-5 Zeolites. J . Chem. Technol. Biotechnol. 1987, 37, 111-122. Meshram, N. R.; Hegde, S. G.; Kulkarni, S. B.; Ratnasamy, P. Disproportionation of Toluene over HZSM-5 Zeolites. Appl. Catal. 1983,8, 359-367. Nayak, V. S.;Riekert, L. Catalytic Activity and Product Distribution in the Disproportionation of Toluene on Different Preparations of P e n t a d Zeolite Catalysts. Appl. Cutal. 1986, 23, 403-411.
361
Olsen, D. H.; Kokotailo, G . T.; Lawton, S. L.; Meier, W. M. Crystal Structure and Structure-Related Properties of ZSM-5. J . Phys. Chem. 1981,85, 2238-2243. Palekar, M. G.; Rajadhyaksha, R. A. Sorption Accompanied by Chemical Reaction. Catal. Rev.-Sci. Eng. 1986, 28, 371-429. Young, L. B.; Butter, S. A.; Kaeding, W. W. Shape Selective Reactions with Zeolite Catalysts. Selectivity in Xylene Isomerization, Toluene-Methanol Alkylation and Toluene Disproportionation over ZSM-5 Catalysts. J . Catal. 1982, 76, 418-432.
Received for review December 29, 1988 Accepted October 24, 1989
PROCESS ENGINEERING AND DESIGN Effect of Ice Nucleators on Snow Making and Spray Freezing James C. Liao*vt and Kam C. Ng' Bioproducts Division and Information & Computing Technologies Division, Eastman Kodak Company, Rochester, New York 14650
T h e snow-making process is a spray-freezing operation involving heat and mass transfer and ice nucleation. This latter phenomenon is of particular interest recently since some bacterial cells such as Pseudomonas syringae have been used commercially as ice nucleators in the snow-making process. This study quantifies the effect of ice nucleators on the efficiency of snow making. T h e theory of nucleation spectrum is extended to account for the variation of nucleation temperature of water droplets and is applied to this dynamic system. T h e results are then integrated with the heat- and mass-transfer calculations, using two-dimensional spline interpolant. T h e effects of nucleator concentration, ambient conditions, and droplet size distribution are discussed.
I. Introduction
to atomize the water and then spray the water droplets into
Pure water can be cooled down to -40 "C without freezing (e.g., Knight (1967)). Such a metastable state can be destroyed by either homogeneous or heterogeneous nuclei. Several organic and inorganic crystals are known to nucleate water at relatively high temperatures. For example, AgI nucleates water at -8 O C . In the past decade, some microorganisms have been found to have a similar ice-nucleation property (Phelps et al., 1986; Lindow, 1983; Maki et al., 1974) but at an even higher temperature (-5 OC). Among these organisms, P s e u d o m o n a s syringae has been used commercially as an additive in making artificial snow in ski areas. However, the effect of these ice-nucleators on the snow-making process is yet to be quantified. This study analyzes the effect of nucleators on the snowmaking process and determines the effective concentration of nucleator needed in snow-making. The snow-making process is essentially a spray-freezing operation. Water is atomized through a nozzle and sprayed into the ambient air, removing most of the latent heat generated in the freezing process. Under typical snowmaking conditions, the cooling provided by the free jet is small compared to the ambient cooling (Chen and Kevorkian, 1971). The role of the compressed air is mainly
the ambient air. Therefore, for our purpose here, the process can be modeled as a spray of water droplets cooled by a cross-flow stream of ambient air, as shown in Figure 1. Although several authors have studied artificial snow making or similar systems (e.g., Chen and Kevorkian (1971), Yao and Schrock (1976), and Chen and Trezek (1977)),most of them dealt with heat and mass transfer of the system. The effect of ice nucleators on spray freezing is yet to be quantified. The analysis of heterogeneous ice nucleation on water droplets presents additional challenges: (1)the effective nucleation temperature of each nucleator is different, and (2) the likelihood of a given nucleator to be incorporated into the water droplet has to be quantified. These two issues have to be considered along with the effects of heat and mass transfer and droplet size distribution.
*Author t o whom correspondence should be addressed a t the Department of Chemical Engineering, Texas A&M University, College Station, TX 77843. Bioproducts Division. Information & Computing Technologies Division.
11. Approach As mentioned before, the cooling capacity in snow making is mainly provided by the ambient air under normal conditions, and thus, the snow-making process is modeled as a spray of water droplets cooled by a cross-flow stream of ambient air (Figure 1). This study begins by considering a single water droplet free falling in a constant-temperature air stream. The fraction of ice in the droplets, Q(D,T,), is calculated as a function of droplet diameter, D,and nucleation temperature, T,, using literature values for the heat- and mass-transfer coefficients.
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362 Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990
where CD is expressed as a function of Reynolds number, Re (Chen and Trezek, 1977)
Water Droplets /
'
I
l i i i 8
,
,
CD = 24/Re
< 1000
CD = 0.6649 - 0.2712 X lOW3Re+ 1.22 X 10-7Re210.919 X 10-12Re3 1000 < Re < 3600 (8b)
T,
Figure 1. Cross-flow representation of the snow-making process.
Although this section is a direct synthesis of literature correlation, it provides a basis for analyzing the effect of ice nucleators. We then extend the theory of nucleation spectrum (Vali, 1971) to analyze the distribution of nucleation temperature of each droplet. The probability that the nucleation temperature is between T , and T, + AT, is determined by G(T,,D,C), which is the probability density for a droplet to initiate freezing at temperature T,, given the droplet diameter (D ) and nucleator concentration ( C ) . This treatment is necessary since the nucleation temperature depends on the probability that a droplet contains at least one nucleator that is effective at temperature T,. The total fraction of frozen water in a plume, or snow quality, SQ(C 1, is determined by the following integration:
SQ(C) =
I ,U, = components of droplet velocity in the direction of I. y, and z , respectively
Yn‘ = humidity at droplet-air interface Yg’ = humidity of ambient air Greek Symbols pa = density of air pIl = density of the p = viscosity
L i t e r a t u r e Cited Chen, J.; Kevorkian, V. Heat and Mass Transfer in Making Artificial Snow. Ind. Eng. Chem. Process Des. Deu. 1971, 10, 75-78. Chen, K. H.; Trezek, G. J. The Effect of Heat Transfer Coefficient, Local Wet Bulb Temperature and Drop Size Distribution Function on The Thermal Performance of Sprays. J. Heat Transfer 1977, 99, 381-385. Dickinson, D. R.; Marshall, W. R. The Rates of Evaporation of Sprays. AIChE J . 1968, 14, 541-552. IMSL. I M S L MathlLibrary User’s Manual, Version 1.0; IMSL, Inc.: Houston, TX, 1987; Vol. 11, pp 446, 465, 604. Knight, C. A. The Freezing of Supercooled Liquids; D. Van Nostrand Co., Inc.: Princeton, NJ, 1967; p 23. Lindow, S. E. The Role of Bacterial Ice Nucleation in Frost Injury to Plants. Annu. Reu. Phytopathol. 1983,21, 363-84. Maki, L. R.; Galyan, E. L.; Chang-Chien, M. M.; Caldwell, D. R. Ice Nucleation Induced by Pseudomonas syringae. Appl. Microbiol. 1974, 28, 456-59. McLay, B. Development of a Model of the Air/Water Snow-making Process. Bachelor of Engineering Project Report, Dartmouth College, Hanover, NH, 1986. Phelps, P.; Giddings, T. H.; Prochoda, M.; Fall, R. Release of CellFree Ice Xuclei by Erwinia herbicola. J . Bacteriol. 1986, 167, 192-502. Ranz, W. E.; Marshall, W. R. Evaporation from Drops. Chem. Eng. Prog. 1952, 48, 141-180. Vali, G. Quantitative Evaluation of Experimental Results on the Heterogeneous Freezing Nucleation of Supercooled Liquids. J . Atomos. Sei. 1971, 28, 402-409. Yao, S. C.; Schrock, V. E. Heat and Mass Transfer from Freely Falling Drops. Trans. A S M E 1976, Feb, 120-126. Zarling, J. P. Heat and Mass Transfer from Freely Falling Drops at Idow Temperatures. CRREL Report 80-18, ADA090522, 1980.
W,= weight of ice in the droplet Wn
=
droplet
Receiued for review July 13, 1989 Accepted November 16, 1989
total weight of the droplet
A New Method for the Derivation of Steady-State Gains for Multivariable Processes Heleni S. Papastathopoulou and William L. Luyben* Department of Chemical Engineering, 11 1 Research Drive, Lehigh University, Bethlehem, Pennsylvania 18015
A new method for the derivation of steady-state gains for a multivariable process is presented. It is based on small perturbations in t h e controlled variables instead of changes in t h e manipulated variables, as is t h e case in t h e traditional method. The availability of accurate steady-state gains for a multivariable process facilitates significantly the control system design procedure. The steady-state gains provide the zero frequency characteristics of the system. This piece of information enables the initial screening and selection of proper manipulated and/ or controlled variables, variable pairing, and initial evaluation of candidate control structures (Grosdidier et al., 1985: Yu and Luyben, 1986; Shinskey, 1988). The steady-state gains can be determined by using either plant tests (although it has been shown (Luyben, 1987a) that the results might be seriously different from those of a linearized model of the process) or some kind of a rating
* Author to whom correspondenceconcerning this paper should be addressed. 0888-5885/90/2629-0366$02.50/0
program (Buckley et al., 1985). A third and more complex alternative is to get the steady-state gains through a transfer function identification procedure, if dynamic plant data or data from a dynamic model of the process are available. In some cases, like the autotuning variation method (Luyben, 1987b), the steady-state gains are required for the identification. In some others, like least-squares or instrumental variable methods (Ljung and Soderstrom, 1987), the gains are not required. Still, obtaining the steady-state gains independently (e.g., use of a rating program) enables validation of the transfer function derivation procedure. A rating program suitable for the determination of steady-state gains for a distillation process requires as input, apart from the design variables of the column
Z’1990 American Chemical Society
