A Semiempirical Kinetics for Modeling and Simulation of the Crystal

Ind. Eng. Chem. Res. 2009, 48, 5105–5110

5105

RESEARCH NOTES A Semiempirical Kinetics for Modeling and Simulation of the Crystal Growth Process in Pure Solutions K. Vasanth Kumar* Departmento de Engenharia Quı´mica, Faculdade de Engenharia, UniVersidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

A kinetic model was proposed to explain the growth of crystals in pure solutions based on the solid-phase concentration, assuming that the crystal growth process follows a pseudo-second-order kinetics. The initial crystallization rate was defined based on the pseudo-second-order kinetics. The proposed model was determined to be useful in predicting the rate constant, the solid-phase concentration at equilibrium, and the initial crystallization rate. The pseudo-second-order kinetic model was applied to the experimental data of the sucrose crystal growth process for different operating temperatures and seed crystal diameters. The proposed model was accurate in modeling the experimental kinetics of sucrose crystallization process for the range of operating conditions studied. The coefficient of determination between experimental data and predicted kinetics varied from r2 ) 0.943 to r2 ) 0.982 at the studied temperatures. The calculated kinetic parameters were used to generate three-dimensional empirical correlations relating the crystal growth of sucrose for the range of temperature studied. The proposed model was very simple with only two unknown parameters, which can be easily determined by a simple linear or nonlinear regression analysis. 1. Introduction The growth of crystals in aqueous solution is a complex process controlled by diffusion, surface integration, or both. The role of diffusion and surface integration on the growth of crystals in pure solution was usually modeled using the two-step masstransfer model.1,2 According to the two-step mass-transfer model, the crystal growth process involves the diffusion of solute from the bulk liquid phase to the solid phase. The second step is the arrangement of diffused solute molecules into the crystal lattice. Several kinetic models are available and widely used to model the growth kinetics of crystals in pure solutions.1-7 Undoubtedly, these models are excellent to model any crystallization systems. However, most of the existing models involve many parameters such as interfacial tension and morphological details, which are practically difficult or which require many assumptions to determine during the crystal growth process. In the present study, a simple semiempirical kinetic model was proposed assuming the growth kinetics to be a pseudo-secondorder process. The proposed kinetics was used to model the growth of sucrose crystals in pure solutions under different operating conditions. The objective of the present study is only to propose kinetics from a different and new approach, to model the growth process without incorporating any complex parameters, which are practically difficult to determine. Another main objective of the present study is to propose a generalized semiempirical kinetic expression, which can be useful to simulate the seeded crystallization system accurately for a range of operating conditions. 2. Experimental Section The growth of sucrose crystals was performed in a 4-L batch agitated crystallizer under isothermal conditions at three different * Tel.: +351 - 91437 1937. E-mail addresses: vasanth_vit@ yahoo.com, [email protected].

temperatures (30, 40, and 50 °C). The crystallizer was connected to the online monitoring system for the continuous monitoring of brix, which is defined by the percentage of sucrose in solution, and temperature. The temperature inside the crystallizer was maintained by a crystallizer jacket, which is connected to a thermostatic water bath. The details of the crystallizer used in the present study are explained elsewhere.8 The agitation inside the crystallizer was maintained at a constant speed of 250 rpm. Sucrose solutions were prepared by dissolving the sucrose crystals at Tw + 20 °C in ultrapure water (where Tw refers to the working temperature). In all cases, the impurity was added while dissolving the sucrose at Tw + 20 °C. Supersaturation was obtained by cooling the solution to the working temperature. All of the experiments were conducted for an initial supersaturation of 20 g of sucrose/100 g of water. After the crystallizer temperature was stable, an accurately weighed amount (16 g) of sucrose seed crystals was added into the crystallizer. Crystals ranging within the sieve fractions of 0.0425-0.0500 cm and 0.0125-0.0250 cm were used as seed crystals. The average seed sizes of these two fractions were determined using the Coulter particle size analyzer and were determined to be 0.0536 and 0.01885 cm. The crystal growth experiments were performed for 24-72 h, based on the solution temperature, until the supersaturation reaches a value of ∼7 g of sucrose/100 g of water. The mass of the crystals inside the crystallizer at any time was calculated from a mass balance. 3. Results and Discussions The pseudo-second-order kinetics used to model the growth of crystals in solutions was derived by making the following assumptions. The kinetics of crystal growth in solutions is a function of supersaturation, which is dependent on the liquid-phase concentration and saturation concentration (temperature). In the present study, the change in solid-phase concentration, with

10.1021/ie8018565 CCC: $40.75  2009 American Chemical Society Published on Web 04/27/2009

5106

Ind. Eng. Chem. Res., Vol. 48, No. 10, 2009

respect to time, was assumed to follow a pseudo-second-order kinetics during the growth process. In addition, it was further assumed there is no crystal breakage or nucleation. If co and c are, respectively, the concentrations of initial solute and the concentration at any time t, and Vs is the volume of solvent inside the crystallizer, then the ratio of mass crystallized at any time to the total mass of crystals inside the crystallizer (represented by the symbol qt) can be calculated by a simple mass balance expression, which is given as Vs qt ) (co - c) Mt

(1)

where Mt is the total mass inside the crystallizer at any time t and is given as Mt ) Mseeds + Mcrystallized

(2)

Similarly, if ce represents the solute concentration when the crystals and the solution inside the crystallizer are in equilibrium with each other, then the solid-phase concentration ratio at equilibrium (qe) can be defined as the ratio of crystal yield (Ycr) to the total solid phase concentration, which is given as qe ) (co - ce)

Vs Ycr ) Mt Mt

(3)

Assuming that the change in the solid-phase concentration ratio or the concentration of solute in solution, with respect to time, follows a pseudo-second-order kinetics, the crystal growth kinetics can be given by dqt ) k2(qe - qt)2 (4) dt The kinetics was the so-called pseudo-second-order kinetics, because the growth kinetics was only a function of the liquidphase concentration. Although the term qt poorly reflects the physical meaning, it refers to the change in the dimensionless solid-phase concentration ratio, as a function of the initial mass of seed crystals. Thus, the determined kinetic constant varies with the amount of seed crystals or the number of seed crystals during seeding. Thus, this effect should be considered while using a pseudo-secondorder expression, which is explained in the later sections. Previously, several kinetics have been proposed for adsorption systems based on the solid-phase concentration.9,10 In the present research, keeping them as inspiration, a semiempirical expression was proposed for crystallization systems, considering the kinetic effect, based on the change in the dimensionless solidphase concentration ratio. With respect to the initial conditions, the limiting conditions used to solve eq 4 can be written as qt ) 0; and qt ) q;

t)0 t)t

(5)

Integrating eq 4 with respect to the boundary conditions (eq 5), the growth kinetics can be given by qt )

k2qe2t 1 + k2qet

(6)

The linearized expression of eq 6 is given by t 1 t ) + qt k q 2 qe

(7)

2 e

From eq 7, the rate of change in the solid-phase concentration

ratio can be defined by qt 1 ) t [1 ⁄ (k q 2)] + (t ⁄ q ) 2 e e

(8)

When t f 0, the initial rate of reaction (Vcr) can be given by Vcr ) k2qe2

(9)

Thus, according to eq 7, the kinetics of growing crystals, following a pseudo-second-order kinetics, should be a perfect linear plot between t/qt versus t. The kinetic constant (k2) and the theoretical crystal yield can be determined from the slope and intercept of the plot between t/qt versus t using eq 7. The determined kinetic constant and the theoretically determined qe can be used to calculate the initial reaction rate of the growing crystals. Figure 1a shows the plot of t/qt versus time t for the growth of sucrose crystals at three different temperatures. The plot of t/qt versus time t was determined to be perfectly linear at the studied temperatures with a coefficient of determination (r2) equal to unity. The calculated kinetic constant (k) and the theoretically predicted solid-phase concentration ratio (qe) and the corresponding r2 values by linear regression analysis are given in Table 1. The higher coefficient of determination (r2 ≈ 1) suggests that the proposed model represents the solid-phase concentration very well during the growth process at the studied temperatures. Table 1 also shows the qe value calculated using the solubility data for the studied experimental conditions. Table 1 shows that the qe value determined theoretically from the pseudo-second-order kinetics was more or less near to the qe value obtained from solubility data. This shows the advantage of the proposed model in predicting the theoretical crystal yield or, in other words, the crystallization capacity under the experimental conditions studied. The effect of temperature (T) on the theoretically determined solid-phase concentration ratio at equilibrium (qe) fits the correlation qe ) -0.0003T2 + 0.204T - 30.821

(10)

Although the qe value that was determined using pseudosecond-order kinetics are more or less near to the qe value that was determined from the solubility experiments, the relationship between qe and T, as shown in eq 10, is physically insignificant. This may be due to the complexities during the fit of experimental data in the theoretical model by linear regression analysis. Because the slope of the plot of t/q versus t has a physical meaning that is related to the solubility of the solute, the kinetic constant was determined via nonlinear regression analysis, without altering the parameter qe. In the case of nonlinear regression analysis, the value of qe was determined from solubility data using a mass-balance equation. For nonlinear regression analysis, a trial-and-error method applicable to computer computation was developed to minimize the error distribution between experiments and kinetics. The error distribution was reduced by maximizing the error function coefficient of determination (r2), using a solver add-in (Microsoft spreadsheet, Microsoft Excel). Figure 1b shows the experimental data and the predicted pseudo-second-order kinetics by nonlinear regression analysis at the studied temperatures. Figure 1b shows that the growth rate of sucrose crystals increased as the operating temperature increased. The increase in solid-phase concentration was rapid at higher supersaturation and the growth rate decreased gradually at lower supersaturation. Table 1 shows the determined k, Vcr, and r2 values determined by nonlinear regression analysis. Both the kinetic constant and initial reaction rate were deter-


mined to be increasing as the operating temperature increased. The kinetic constant increases from 7.936 × 10-3 min-1 to 21.742 × 10-3 min-1 for an increase in temperature from 303 K to 323 K. The determined kinetic constant (k2) increases exponentially as the solution temperature increases and fits the Arrhenius expression, which is given by

5107

( RTE )

k2 ) k2o exp -

(11)

where E is the activation energy for the growth of crystals according to a pseudo-second-order kinetics. The pre-exponen-

Figure 1. Graphic depictions showing (a) a plot of t/q vs t for the growth of sucrose crystals in pure solutions at different temperatures and (b) experimental data and pseudo-second-order kinetics by nonlinear regression analysis for the growth of sucrose crystals at different temperatures. (For both plots, seed crystal diameter ) 0.05362 cm, agitation speed ) 250 rpm, and mass of seed crystals ) 16 g.) Table 1. Kinetic Constants Determined by Linear and Nonlinear Regression Analysis for the Growth of Sucrose Crystals in Pure Solutions at Different Temperaturesa Linear Regression

Nonlinear Regression

temperature, T (K)

qe,experimental

k2

qe,kinetics

Vcr

r2

k2

Vcr

r2

303 313 323

0.952362 0.949109 0.94722

0.00754 0.012202 0.023002

0.966923 0.992651 0.952989

0.00705 0.012023 0.02089

0.999 0.999 0.999

0.007936 0.015483 0.021742

0.007198 0.013947 0.019507

0.982284 0.943971 0.970129

a

Parameters are defined in the text.

5108


tial factor (k2o) and E can be determined from the plot of ln(k2) versus 1/T, using the linearized Arhenius expression E (12) RT Figure 2 shows the plot of ln(k2) versus 1/T for the growth of sucrose crystals in pure solution. The activation energy and the pre-exponential factor according to the second-order kinetic model for the sucrose system were determined to be 41.131 kJ/mol and 102 615 min-1, respectively. Similarly, the relationship between the initial crystallization rate and temperature fits the Arrhenius expression, which is given by ln(k2) ) ln(k2o) -

40688.72 (13) RT From eq 13, it can be observed that the activation energy (E), based on the initial reaction rate and the kinetic constant, are more or less the same for the growth of sucrose crystals. Thus, the growth rate was determined to be a constant for the entire range of studied supersaturations at the three studied temperatures. The relationship between qe (determined from the solubility data11) and temperature is given by

(

Vcr ) 7807 exp -

qe ) -0.0003T + 1.03

)

(14)

Substituting eqs 11 and 14 into eq 7, a generalized pseudosecond-order expression valid for any operating temperature within the studied ranges can be obtained: qt )

102615 exp[-41131 ⁄ (RT)](-0.0003T2 + 1.03)2t 1 + 102615 exp[-41131 ⁄ (RT)](-0.0003T2 + 1.03)t (15)

Equation 15 can be used to predict the growth kinetics of sucrose crystals, with respect to time, for temperatures ranging from 30 °C to 50 °C. The graphical representation of eq 15 is shown in Figure 3 for the growth of sucrose crystals. In the present study, the pseudo-second-order kinetics was determined to be excellent in regard to predicting the

Figure 2. Plot of ln(K) versus 1/T for the growth of sucrose crystals.

crystallization kinetics as a function of the dimensionless solid-phase concentration. However, from a design point of view, it is necessary to obtain results that predict the change in mass of crystals or the solute concentration inside the crystallizer at any time. The relation between M and qt can be directly obtained from the plot of qt vs 1/M, using the mass-balance expression as in eq 1. The relationship between M and qt for the sucrose system at the studied temperatures is given by 1 ) -0.0625qt + 0.0625 (16) M Equation 16 is valid for the range of temperatures studied, and it also is valid for the range of seed crystal diameters studied. Figure 4 shows the change in the mass of crystals inside the crystallizer determined experimentally and the M value predicted using eq 16 and pseudo-second-order kinetics for the range of solution temperatures studied. From Figure 4, it can be observed that the pseudo-second-order kinetics better predicts the solid-phase concentration inside the crystallizer at the studied temperatures. Equation 16 can also be used to predict the solubility of solute molecules in the solvent from the kinetics using theoretically predicted qe values. To study the effect of the number of crystals inside the crystallizer, experiments were conducted with different-sized seed crystals. The effect of seed crystals was studied by performing separate crystal growth experiments by seeding 16 g of sucrose crystals of different sizes (0.01885 and 0.0536 cm). The number of crystals inside the crystallizer was determined using eq 17:12 N )

Mseed FcRL1⁄3

(17)

where Mseed is the mass of seed crystal diameter (in grams), L the length of crystal (in centimeters), and Fc the density of crystal (expressed in units of g/cm3). The calculated kinetic constant (k2) was determined to increase from 0.00854 min-1 to 0.0105 min-1 for increases in the average seed crystal diameter from


5109

Figure 3. Simulated pseudo-second-order kinetics for the growth of sucrose crystals at different temperatures (seed crystal diameter ) 0.05362 cm, agitation speed ) 250 rpm, mass of seed crystals ) 16 g).

Figure 4. Length of crystals predicted from the mass-balance equation and from the pseudo-second-order kinetics for different seed crystal diameters (temperature ) 40 °C; agitation speed ) 250 rpm; mass of seed crystals ) 16 g).

0.01855 cm-1 to 0.0536 cm-1, respectively. This is expected because the larger crystals grow faster than the small-sized crystals. The length of the crystals growing inside the crystallizer at any time was calculated using eqs 16 and eq 17 L)

[

1 ⁄ (-0.0625qt + 0.0625) FcRN

]

ment with the length of crystals determined from the massbalance equation. Thus, the pseudo-second-order kinetics, in combination with eq 16, can be very useful in predicting the linear growth rate of growing crystals.

1⁄3

(18)

Figure 4 shows the length of crystals determined from the mass-balance expression and the length determined using eq 18, as a function of seed crystal diameter. Figure 4 shows that the length of growing crystals at any time, as determined from the pseudo-second-order kinetics, was in good agree-

3. Conclusions A pseudo-second-order kinetics based on the solid-phase concentration was proposed and validated with sucrose crystallization kinetic data. The proposed model was determined to be very accurate in predicting the growth kinetics of sucrose crystals at three different temperatures. The

5110


proposed model successfully predicts the crystallization capacity, the initial reaction rate, and the kinetic constant from the experimental growth kinetics. The kinetic constant was determined to increase as the temperature and seed crystal diameter each increased. The pseudo-second-order kinetics also successfully predicts the linear growth rate of sucrose crystals. The proposed model is very simple without incorporating any parameters, which are practically difficult to determine. The proposed kinetics can be used to model other crystallization systems; however, experimental data are required to validate the kinetics. Notations qt ) dimensionless solid phase concentration at any time t qe ) dimensionless solid phase concentration at equilibrium t ) contact time (min) Co ) initial solute concentration (g of sucrose/L of water) Ce ) concentration of sucrose in solution at equilibrium (g of sucrose/L of water) C ) concentration of sucrose at any time t (g of sucrose/L of water) k2 ) pseudo-second-order kinetic constant (min-1) T ) temperature (°C, K) E ) energy of activation (kJ/mol) Vcr ) initial crystallization rate (min-1) L ) length of growing crystals at any time t (cm) Fc ) density of sucrose crystals (g/cm3) N ) number of crystals inside the crystallizer M ) mass of crystals inside the crystallizer at any time t (g) Mseed ) mass of seed crystals (g) r2 ) coefficient of determination

Literature Cited (1) Mullin, J. W.; Gaska, C. The growth and dissolution of potassium sulphate crystals in a fluidized bed crystallizer. Can. J. Chem. Eng. 1969, 47, 483–489. (2) Garside, J. The concept of effectiveness factors in crystal growth. Chem. Eng. Sci. 1971, 26, 1425–1431. (3) Bennema, P. Analysis of crystal growth models for slightly supersaturated solutions. J. Cryst. Growth 1967, 1 (5), 278–286. (4) Chernov, A. A.; Rashkovich, L. N.; Mkrtchan, A. A. Solution growth kinetics and mechanism: Prismatic face of ADP. J. Cryst. Growth 1986, 74 (1), 101–112. (5) Burton, W. K.; Cabrera, N.; Frank, F. C. The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. London, A 1951, 243, 299–358. (6) Sangwal, K. Growth kinetics and surface morphology of crystals grown from solutions: Recent observations and their interpretations. Prog. Cryst. Growth. Charact. 1998, 36, 163–248. (7) Martins, P. M.; Rocha, F. Characterization of crystal growth using a spiral nucleation model. Surf. Sci. 2007, 601 (16), 3400–3408. (8) Kumar, K. V.; Martins, P.; Rocha, F. Modelling of the batch sucrose crystallization kinetics using artificial neural networks: comparison with conventional regression analysis. Ind. Eng. Chem. Res. 2008, 47 (14), 4917– 4923. (9) Lagergren, S. Zur theorie der sogenannten adsorption gelo¨ster stoffe. Kungliga SVenska Vetenskapsakademiens. Handlingar 1898, 24 (4), 1–39. (10) Ho, Y. S. Pseudo-isotherms using a second order kinetic expression constant. Adsorption 2004, 10 (2), 1572–8757. (11) Bubnik, Z.; Kadlec, P.; Urban, D.; Bruhns, M. Sugar Technologists Manual; Bartens: Berlin, Germany, 1986. (12) Sa, S.; Bento, L. S. M.; Rocha, F.; Guimaraes, L. Investigation of crystal growth in a laboratory fluidized bed. Int. Sugar J. 1995, 97 (115), 199–204.

ReceiVed for reView October 9, 2008 Accepted April 16, 2009 IE8018565

A Semiempirical Kinetics for Modeling and Simulation of the Crystal

Recommend Documents