Ind. Eng. Chem. Res. 2000, 39, 1903-1910
1903
Application of a Continuous Two Impinging Streams Reactor in Solid-Liquid Enzyme Reactions Morteza Sohrabi* and Mehdi Ahmadi Marvast Chemical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran
The isomerization of D-glucose to D-fructose, using the immobilized glucose isomerase, as a typical example of solid-liquid-catalyzed reactions has been carried out in a tangential flow two impinging streams reactor (TFTISR). The results obtained for the conversion of glucose in TFTISR were much higher than those expected in conventional reactors. This indicates the greater performance capability of TFTISR relative to those of classical reaction systems. A compartment model was devised to describe the pattern of flow within the reactor. Considering such a flow pattern, a stochastic model for the residence time distribution (RTD) of solid particles was developed, using Markov chains discrete formulation. The latter was correlated with the experimental RTD data, obtained by applying a tracer analysis method. 1. Introduction The application of two impinging streams systems as the chemical reactors was first reported in 1982.1 In such reactors, two feed streams flowing parallel or countercurrently collide with each other at a zone in which the two streams impinge. Among the multiphase reactions carried out in two impinging streams reactors, chemical absorption of CO2 gas in sodium hydroxide2 and monoethanolamine solutions3 and mononitration of toluene by mixed acids in a liquid-liquid system4 may be mentioned. Impinging streams systems have been successfully applied to other chemical processes such as dissolution, mixing, and mass- and heat-transfer operations.5-10 The tangential flow two impinging streams apparatus was first designed and applied by Bar and Tamir in 1990.11 Such a system has been employed for drying,12 dissolution of solids,13 and mixing of particles.14 In the present study, certain hydrodynamic behavior of a tangential flow two impinging streams reactor (TFTISR) has been investigated, followed by measurement and modeling of the residence time distribution (RTD) of solid particles. Finally, the isomerization of D-glucose to D-fructose using an immobilized glucose isomerase enzyme, as a typical solid-liquid bioreaction has been conducted in this reactor. A comparison has been made between the conversion of glucose determined experimentally and those predicted using a model developed from the combination of RTD formulation and kinetic parameters. This reaction has a great industrial impact, because D-glucose cannot be substituted directly for sucrose. Glucose is less sweet relative to sucrose, and also crystallization of glucose in concentrated solutions can make prior handling and processing difficult. Isomerization of glucose to fructose is a reversible reaction with an equilibrium constant of approximately unity at 50 °C. The heat of reaction is on the order of 5 kJ/mol.15 Consequently, the equilibrium product contains roughly a 1:1 ratio of glucose to fructose that does not change appreciably with temperature.
Isomerization of glucose is normally carried out in tubular reactors packed with immobilized enzyme with a space velocity of 0.2-2 h-1. One of the aims of the present study was to consider the possibility of promoting the overall performance of glucose isomerization reaction systems by employing a TFTISR. 2. Experimental System 2.1. Reactor. The reaction vessel was a tangential horizontal flow two impinging streams system, the schematic diagram of which is shown in Figure 1. The reactor setup comprised a reaction chamber made of Pyrex glass, 12 cm in diameter and 5 mm in thickness, with an inner pipe creating an annular space and directing the streams toward the impinging zone. The height of the chamber was 43 cm and the diameter of the central inner tube was 22 mm. At the lower end of the chamber a provision was made for the pressure measurement. The two streams of suspension (immobilized enzyme + substrate) were fed through symmetrically positioned acceleration pipes. The accelerated suspension feed streams impinged at the annular space and dropped instantly along the inner discharge tube down to the outlet port. The solid particles were added to the two liquid streams through a hopper type feeder (Figure 1), the exit size of which could be varied to provide the desired flow rate. 2.2. Chemicals. All chemicals used in this study were of analytical grade. The immobilized enzyme, Sweetzyme T (EC 5.3.1.5 D-xylose ketol isomerase), was provided by Novo Nordisk Co. The enzyme particles were of cylindrical shape and 0.3-1 mm in diameter having an activity of 350 IGIU/ g. D-Glucose in crystalline form was provided by Merck Co., Germany. 2.3. Analytical Technique. The determination of glucose was carried out using a glucose oxidase/peroxidase method.16 The fructose concentration was measured by a cystine-carbasols method.17 The accuracy of these methods was tested using known samples of
10.1021/ie990507m CCC: $19.00 © 2000 American Chemical Society Published on Web 04/20/2000
1904
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
Figure 1. Schematic diagram of the reactor and feeder. All dimensions are in millimeters.
Figure 2. Holdup of solid particles within the reaction system. Standard deviations of the data points are between 1.87 and 2.32 (g).
both monosaccharides. The maximum error did not exceed (4%. 3. Results and Discussion To present a suitable model for the reaction system, the behavior of the reactor should be known. 3.1. Mean Residence Time. Measurement of the mean residence time (RTD) of materials in the system, requires a knowledge of liquid and solid holdup within the reactor. To determine such parameters, vegetable grains (millet seeds) as solid particles and water as the liquid phase were employed. Millet seeds were selected as solid particles because of certain similarities between the latter and the immobilized enzyme, such as density and apparent shape. Water and vegetable grains were passed continuously through the system. After a time, the flow of materials
was suddenly ceased. The materials remaining in the reactor were collected at the outlet port. The liquidphase volume and the weight of millet seeds (after drying) were measured. In Figures 2 and 3, the solid holdup as a function of the liquid flow rate and the liquid holdup as a function of the solid flow rate are presented. The mean residence time of each phase was calculated as the ratio of the holdup of that phase to the respective mass or volumetric flow rates. The results are shown in Figures 4 and 5. The general conclusion drawn from Figure 2 may be summarized as follows: (a) At a given solid flow rate, increasing the mass flow rate increases the solid holdup. (b) Increasing the liquid flow rate decreases the solid holdup. This behavior may be explained by the increase in the collision rate between the particles when the liquid
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1905
Figure 3. Liquid holdup as a function of the solid mass flow rate. Standard deviations of the data points are between 9.37 and 18.46 (cm3).
flow rate is increased and, hence, the chance of solid particles to leave the reactor is promoted. It is apparent from Figure 3 that, at a given liquid flow rate, the holdup of the liquid phase in the reactor is almost constant and insensitive to the change in the solid flow rate. Such a behavior may be explained by noting that the flow rate of liquid in all experimental runs was 13-38 times greater than that of solid. Hence, variation in the solid flow rate may not have a profound effect on the liquid holdup. In Figure 4, it is demonstrated that, in all solid flow rates, a slight decrease in the solid residence time occurs, as the liquid flow rate is increased. This behavior is expected, because increasing the liquid flow rate should promote the chance of solid particles to be washed out of the reaction system. It is shown in Figure 5 that, at a given solid flow rate, the mean residence time ratio remains almost constant with a change in the mass flow rate ratio. This may be explained by considering that an increase in the liquid flow rate decreases both the liquid and solid residence times within the system without affecting the ratio between the two. 3.2. RTD of Solid Particles. RTD experiments were conducted to obtain an insight into the behavior of the particles in the reactor. This is an important parameter because of the particle oscillatory motion that may form internal recirculation and affect the mean residence time in the reactor. However, the determination of particle RTD is normally faced with a major difficulty. The number of particles flowing through the reaction system is on the order of 103 particles/s for a typical mass flow rate of 5 × 10-3 kg/s in comparison to 106 particles/s for continuous flow.18 Consequently, only a small number of tagged particles can be used in tracer experiments without disturbing the flow conditions. In the present study, some 200 painted millet seeds were applied as the tracer.
Under such conditions, the behavior of the solid phase is highly discontinuous with respect to the length of the observed phenomena. Only random fluctuations of single particles could be detected, and a reproduction of the results obtained becomes unlikely.19 An average behavior must be, therefore, calculated. Such a measurement requires a number of repetitive experiments and statistical averaging. A more applicable method is to take cumulative readings of the response of each experiment. Each reading was collected during an interval of time ∆t. When ∆t is increased from zero to some higher values, the sensitivity of reading decreases until, at a certain value of ∆t, the random fluctuations of particles are no longer observed. The only problem that remains is the determination of the correct value for ∆t to employ in experiments because ∆t should be large enough to eliminate any random oscillation but not so large that useful information is lost. To overcome this difficulty, a stochastic model for the RTD in the reaction vessel was developed based on patterns first proposed by Van de Vusse.20 When the successive experimental results were correlated with the theoretical model, the true RTD curve was determined.19 Because the collision of the particles in the impingement zone is random, a suitable mathematical technique to handle such a process could be Markov chains models. According to discrete-time Markov chains, the probability of an event at time t + 1 (t ) 0, 1, 2, ...) given only the outcome at time t is equal to the probability of the event at time t + 1 given the entire history of the system. In other words, the probability of the event at t + 1 is not dependent upon the state history prior to time t. Thus, the values of the process at the given time t determine the conditional properties for future values of the process. These values are called the state of the process, and the conditional properties are thought of as transition probabilities between the states i and j, pij. These values may be displayed in a matrix (P ) [pij]) called the one-step transition matrix. The matrix P has N rows and N columns, where N is the number of possible states for transition of the system. The rows of matrix P consist of the probabilities of all possible transitions from a given state and so sum to 1. N
pij ) 1 ∑ j)1
(1)
This matrix completely describes the Markov process. Other definitions and equations related to Markov processes are as follows: (i) si(n), the state probability, defined as the probability that the system will be in the state i after n transitions from a given starting point. (ii) S(n), the state probability vector, a line vector composed of elements si(n). S(n) ) [s1(n), s2(n), ..., sN(n)]. The equations which govern the Markov processes are as follows: N
si(n) ) 1 ∑ i)1
(2)
1906
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
Figure 4. Variation of the mean residence time of solid particles within the reaction system. Standard deviations of the data points are between 0.92 and 1.53 (s).
Figure 5. Variation of the ratio of mean residence times of solid to liquid phases within the reaction system.
This equation implies that the total probability of the system to reach one of all possible N states which can be occupied after n transitions is equal to unity. Equation 3 states that the probability that the system N
sj(n+1) )
si(n) pij ∑ i)1
n ) 0, 1, 2, ...
(3)
is in state j after n + 1 transitions is equal to the sum of its probabilities of being in any state i after n transitions, multiplied by the probability of transferring from state i to state j in one transition step. This definition is also valid for all of the states expressed by
S(n+1) ) S(n)‚P
(4)
Further details of Markov models may be found elsewhere.4,19,21,22 When the patterns of particle flow within the vessel were considered, the reaction compartment was divided into six regions with equal volumes (Figure 6). Each region was assumed to be an ideal flow reactor. A recycle stream R was also considered because of countercurrent flows in the impingement zone.
Figure 6. Model of the reactor: (a) flow regions proposed for the reaction; (b) the resulting model. 1, 2, 6: perfect plug regions. 3, 4, 5: perfect mixed regions.
Each region represents a state in the Markov process, and the transition probability pij is the probability of a solid particle leaving region i and entering region j. Now, consider region 6 in Figure 6. Whenever a particle reaches this region, it leaves the system to
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1907
state 7 where it remains. State 7 is then a “trapping state”. The sixth element of the state probability vector, s6(n) indicates the probability that a particle entering the system at n ) 0 will leave the system after n∆t (n ) 1, 2, ...). The knowledge of the numerical value of s6(n) for n ) 0, 1, 2, ..., represents the impulse response of the system after n intervals of time length ∆t. It is possible to define the various probabilities along the path of a particle from the inlet to the system down to the exit. The complete transition probabilities are arranged in matrix P.
{
P) 0 0 1 0 0 0 0 0 eR 0 0 0
0 1 0
0 0 0 0 0 0 H(1 - eR) G(1 - eR) 0 H(1 - eR) G(1 - eR) 0
eR 0 0 -(1/2)(1 - e ) -(1/2)(1 - eβ) eβ 0 0 0 0 0 0 0 0 0 0 β
0 0 0
0 1 1
}
(5)
The dimensionless parameters in the above matrix are defined as follows:
R ) (-nV/2)(1 + R)∆θ
(6)
β ) -nVR∆θ
(7)
H ) (R)/(1 + R)
(8)
G ) 1/(1 + R)
(9)
∆θ ) ∆t/th
(10)
R ) r/Ws
(11)
A complete list of the symbols is given in the Nomenclature section. In this matrix, the elements located on the diagonal are the probabilities of remaining in the same region. Those on the vertical columns are the probabilities of entering region j, and the elements on the horizontal rows correspond to the probabilities of leaving region j. The initial state probability vector S(0) is also known:
S(0) ) [s1(0) s2(0) s3(0) s4(0) s5(0) s6(0) s7(0)] ) [0.5 0.5 0 0 0 0 0] (12) To perform the RTD experiments, a circular plate divided into 24 equal segments was constructed. The plate was placed under the solid discharge port of the reactor and was driven at selected speeds by means of an electric motor. Different sampling intervals were thus obtained by changing the speed of rotation. In each experimental run, water and solid particles were passed continuously through the system. At a certain time (t ) 0), 200 painted millet seeds were rapidly injected into the solid flow and the number of tagged seeds remaining in each segment of the circular plate during the time interval ∆t was measured. Four series of RTD curves were determined by varying ∆t from 0.3 to 0.6 s. The variances, σ2, and the mean residence times, ht, of the latter were calculated using the following relations and compared with those of the
theoretical curve: 24
ht )
tiCi∆ti ∑ i)1 24
(13)
Ci∆ti ∑ i)1 24
σ2 )
(ti - ht )2Ci∆ti ∑ i)1 24
(14)
Ci∆ti ∑ i)1 The “true” RTD curve was assumed to be that particular curve having a variance and a mean residence time identical with those of the theoretical model. The variance of the theoretical RTD curve was determined from the following equation: 24
σmod2 )
S6(j) (R, ∆θ)[th(j∆θ - 1)]2 ∑ j)1
(15)
The dimensionless time, ∆θ, was determined by considering the implication of equality of the two mean residence times of the experimental and theoretical RTD curves; i.e. 24
1 ) ∆θ
jS6(j)R ∑ j)1
(16)
Such a comparison revealed that the best correlation existed with ∆t equal to 0.57 s (Figure 7). All calculations were carried out by applying Mathematica version 2.2 software. 3.3. Isomerization of D-Glucose to D-Fructose. To perform such a reaction, a system was arranged according to Figure 8. The liquid feed reservoir contained 6 dm3 of a solution consisting of 0.1 M glucose and 0.01 M MgSO4 at 25 °C and pH 7.5. The temperature of the feed solution was raised to 60 °C by means of a heating element. Deaeration was performed by blowing nitrogen gas into the solution. The liquid flow rate was adjusted, and the immobilized enzyme with the desired mass flow rate was continuously added to the liquid streams. The effluent, after being passed through a membrane to separate the spent enzyme, recycled to the feed reservoir. Samples were taken at different time intervals from both the liquid effluent and the feed reservoir and analyzed for glucose and fructose. The ranges of certain pertinent parameters and operating conditions are summarized in Table 1. Some unexpected results have been obtained from the TFTISR system. These are shown in Figure 9. This figure shows that a conversion of 32% was achieved within 5 min, whereas in a laboratory scale tubular reactor, a conversion of only 5% has been reported to be obtained in 4 min.23-26 In Table 2, a comparison is made between the experimental results and those calculated as follows, using the RTD model and the apparent rate constants taken from the literature: D-glucose
k1 S D-fructose (k1 ) k1′) k1 ′
(17)
1908
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
Figure 7. Comparison between the experimental and theoretical RTDs in a TFTISR. Each increment in the x axis represents a ∆t ) 0.57 s; recycle ratio ) 0.052; variance of the experimental data points ) 7.54 (s2).
or
1
1 - Xg )
2
∑i (1 + e-2kt )Ei∆ti i
(22)
Thus, conversion of glucose after m passes, Xgm, may be found from the following relation:
1 - Xgm )
Figure 8. Experimental rig for isomerization of glucose: 1, nitrogen cylinder; 2, 9, control valve; 3, pH meter; 4, sparger; 5, mixer; 6, temperature controller; 7, heating element; 8, pump; 10, liquid flowmeter; 11, reactor.
The concentration of glucose after the first pass of reactants from the reactor may be calculated as follows:
∫0∞Cg(t)E dt
Cg )
(18)
where Cg is the glucose concentration in the effluent, Cg(t) is the glucose concentration remaining in an element of age t within the reactor, E is the RTD function, and t is time. Isomerization of glucose is a first-order reversible reaction (eq 5). The rate constants for the forward and reverse reactions, k1 and k1′, have identical values.27
k1 ) k1′ ) k
(19)
Assuming that the overall rate is controlled solely by the intrinsic rate of reaction, then
Cg(t) 1 ) 1 - Xg ) (1 + e-2kt) Cg0 2
(20)
where Cg0 is the initial concentration of glucose and Xg is the conversion of the latter. Then,
1 - Xg )
∫0∞(1 + e-2kt)E dt
1 2
(21)
1
∑∑i (1 + e-2kt )Eim∆ti
2m
i
(23)
The number of passes, m, was found by dividing the total reaction times by the mean residence times of the system. Thus, the duration of each pass is approximately equal to the mean residence time in the reactor. Table 2 clearly shows that the measured and estimated values of the conversion differ significantly. One explanation for the much higher values of the measured conversion is that the overall rate of isomerization of glucose carried out under conventional conditions is not governed only by the intrinsic rate of reaction. In other words, the overall rate may be affected by both internal and external mass-transfer resistances. In this case, the rate constants, k1 and k1′, account for mass-transfer and kinetic effects. In the present two impinging streams system the diffusion resistance of the liquid film surrounding the solid particles is low because of the high velocity of the particles. Moreover, the energy released and dissipated as the result of strong collision of particles may also contribute to total energy of the reacting system and thus enhances the rate of reaction. 4. Conclusion A closed tangential flow two impinging streams system with recycle was employed as a chemical reactor to conduct the isomerization of glucose to fructose. It was observed that the rates of glucose conversion in such a system were much higher than those obtained in conventional reactors. This may be attributed to the elimination of both internal and external mass-transfer resistances and to the energy released as the result of particle collision in the impingement zone. Some aspects of hydrodynamic behavior of TFTISR were studied under a range of operating conditions. The RTD of solid particles within the system was determined using tracer analysis. A compartment model was pro-
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1909 Table 1. Range of Certain Operating Conditions and Pertinent Parameters
experiment
range of solid flow rates (g/s)
range of liquid flow rates (dm3/s)
range of solid mean residence time (s)
range of liquid mean residence time (s)
range of solid holdup (g)
range of liquid holdup (cm3)
holdup RTD isomerization reaction
4.10-7.00 5.97 0.95-1.44
0.097-0.147 0.122 0.111
1.47-2.01 2.28-3.02
2.14-2.60
6.64-14.05
224-346
Figure 9. Isomerization of D-glucose to D-fructose in a closed TFTISR with recycle. Standard deviations of the data points are between 1.55 and 1.81. Table 2. Experimental and Estimated Values for Conversion of Glucose in a TFTISR
reaction conditions solid flow rate ) 1.44 g/s liquid flow rate ) 400 dm3/h mean residence time ≈ 2.0 s T ) 60 °C, pH ) 7.5; substrate initial concentration in feed reservoir ) 0.1 M
no. of conversion (%) passes (m) (obtained)a (estimated)b 1 10 30 54 84 90 105
0.005 0.04 11.1 16.7 24.2 26.0 31.94
0.0 0.0 0.4 0.8 1.2 2.1 3.1
a Calculated from eq 23. b Average of at least five runs. Standard deviations of the measured data points are between 1.55 and 1.81.
posed to describe the flow pattern in this reactor. On the basis of such a flow pattern and application of Markov chains discrete formulation, a theoretical twoparameter model was derived for the RTD of the solid phase. The two parameters were evaluated, and the model was adjusted using experimental RTD data. The RTD expression was used in conjunction with the glucose isomerization rate equation, available in the literature, to predict the behavior of the impinging system as a chemical reactor. The estimated values for the conversion of glucose, however, were much lower than the experimental data. This may be related to the effect of diffusional resistance on the overall rate of reaction and, hence, on the apparent rate constants, determined under conventional conditions. Notations C ) concentration of tagged particles (number of tagged particles in each segment/total number of tagged particles) Cg ) glucose concentration in the effluent (M) Cg(t) ) glucose concentration with age t (M) Cg0 ) initial concentration of glucose (M) E ) RTD function
G ) constant, eq 9 H ) constant, eq 8 k ) rate constant (time-1) k1 ) rate constant for the forward reaction (time-1) k1′ ) rate constant for the reverse reaction (time-1) nV ) number of regions in the flow model pij ) transition probability P ) transition matrix r ) solid recycle stream R ) solid recycle stream ratio RTD ) residence time distribution si(n) ) state probability S(n) ) state probability vector Si(j) ) ith element of the S(j) vector t ) time (s) ht ) mean residence time (s) T ) temperature (K) Wl ) liquid mass flow rate (kg/s) Ws ) solid mass flow rate (g/s) Xg ) glucose conversion Xgm ) glucose conversion after m passes Greek Symbols R ) constant, eq 6 β ) constant, eq 7 ∆t ) time increment (s) ∆θ ) dimensionless time ) ∆t/th σ2 ) variance (s2) νl ) liquid holdup (cm3) νs ) solid holdup (g) Subscripts exp ) experimental l ) liquid mod ) model s ) solid
1910
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
Literature Cited (1) Elperin, I.; Tamir, A. Israeli Pat. Appl. 66162; 29-6, 1982. (2) Tamir, A.; Herskovitz, D. Chem. Eng. Sci. 1985, 40, 2149. (3) Sohrabi, M.; Jamshidi, A. M. J. Chem. Technol. Biotechnol. 1997, 69, 415. (4) Sohrabi, M.; Kaghazchi, T.; Yazdani, F. J. Chem. Technol. Biotechnol. 1993, 58, 363. (5) Tamir, A.; Kitron, A. Chem. Eng. Commun. 1987, 50, 241. (6) Tamir, A. Chem. Eng. Sci. 1984, 39, 139. (7) Tamir, A. Chem. Eng. Sci. 1989, 85, 53. (8) Kitron, A.; Tamir, A. Ind. Eng. Chem. Res. 1987, 26, 2454. (9) Kitron, A.; Tamir, A. Ind. Eng. Chem. Res. 1988, 27, 1760. (10) Goddis, E. S.; Vogelpohl, A. Chem. Eng. Sci. 1992, 74, 2877. (11) Bar, T.; Tamir, A. Can. J. Chem. Eng. 1990, 68, 541. (12) Tamir, A.; Elperin, I.; Luzzatto, K. Chem. Eng. Sci. 1984, 39, 139. (13) Tamir, A.; Grinholtz, M. Ind. Eng. Chem. Res. 1987, 26, 726. (14) Luzzatto, K.; Tamir, A.; Sartana, D.; Salomon, S. AIChE. J. 1985, 30, 1744. (15) Tewari, Y. B.; Goldberg, R. N. J. Solution Chem. 1984, 13, 523. (16) Coburn, H. J. Clin. Chem. 1973, 19, 127.
(17) Verhoff, F. H. Biotech. Bioeng. 1982, 24, 703. (18) Luzzatto, K.; Tamir, A.; Elperin, I. AIChE. J. 1984, 30, 600. (19) Tamir, A. Impinging Streams Reactors, Fundamentals and Applications; Elsevier Science B.V.: Amsterdam, The Netherlands, 1994. (20) Van de Vusse, J. G. Chem. Eng. Sci. 1962, 17, 507. (21) Sohrabi, M.; Jamshidi, A. M. Paper presented at the International Symposium on Reaction Kinetics and the Development of Catalytic Processes, Brugge, Belgium, April 19-21, 1999. (22) Papoulis, A. Probability, Random Variables, and Stochastic Processes, 3rd ed.; McGraw-Hill Inc.: New York, 1991. (23) Vlaev, S. D. Acta Biotechnol. 1991, 11, 49. (24) Vostic-Raki, D. Bioprocess. Eng. 1991, 7, 183. (25) Donno, G. Agric. Biol. Chem. 1970, 34, 1795. (26) Hemmingsen, S. H. Appl. Biochem. Bioeng. 1979, 2, 157. (27) Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2nd ed.; McGraw-Hill Book Co.: New York, 1986.
Received for review July 13, 1999 Revised manuscript received December 3, 1999 Accepted February 16, 2000 IE990507M
