Simple Kinetic Expressions to Study the Transport Process during the

11236

Ind. Eng. Chem. Res. 2009, 48, 11236–11240

CORRELATIONS Simple Kinetic Expressions to Study the Transport Process during the Growth of Crystals in Solution K. Vasanth Kumar* Departmento de Engenharia Quı´mica, Faculdade de Engenharia, UniVersidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

First- and second-order kinetic expressions are proposed to explain the growth of crystals in solutions limited by diffusion and integration kinetics, respectively. The experimental growth kinetic data of sucrose crystals at 30 and 40 °C were fitted to the kinetic expressions by a linear regression method. A linear second-order expression proposed based on the Burton-Cabrera-Frank (BCF) model was found to be useful in identifying the transition in kinetics from the diffusion to the kinetic regime during the crystal growth process. A parameter, Atcritical (product of crystal area and time) was defined according to a second-order kinetic expression. The growth process was found to be limited by diffusion and surface reaction kinetics for At < Atcritical and At > Atcritical, respectively. An initial reaction rate was defined according to a second-order surface reaction kinetics. The mass-transfer coefficient during the growth of crystals increases with increasing temperature, whereas the reaction kinetic constant was found to decrease with increasing temperature. The total adsorption energy for the growth of sucrose crystals in the diffusion and kinetic regime during was found to be >95 kJ/mol according to the derived expressions. 1. Introduction The growth of sucrose crystals occurs mainly in two steps: diffusion of sucrose molecules onto the crystal surface, followed by incorporation of the molecules into the crystal lattice.1,2 Masstransfer processes are slow for crystals growing in viscous solutions, as in the case of the sucrose growth process. The driving force for mass transfer to occur is the concentration gradient between the bulk solution and the crystal surface. At lower supersaturation, the solution becomes less viscous, and in the presence of agitation, the diffusion process is rapid, and the kinetics can be controlled by a direct surface integration process. It is difficult to identify the transition in kinetics from diffusion to integration, as the ranges of supersaturation depend on the crystal itself and the crystal growing environment.3 The process of crystal growth from aqueous solution has usually been modeled using the classical two-step mass-transfer model.1,2 According to the two-step mass-transfer model, the crystal growth process involves the diffusion of solute from the bulk of the liquid phase to the solid phase. The second step is the arrangement of diffused solute molecules into the crystal lattice.1,2 In the present study, first-order and second-order kinetic expressions that are simple to use are proposed to explain the transport mechanism of the crystal growth process. A simple method is proposed to trace the transition in kinetic regime during the growth process. The activation energies of the mass-transfer and surfacereaction process are estimated according to the proposed expressions. 2. Experimental Section

3. Results and Discussion According to Burton-Cabrera-Frank (BCF) theory, the crystal growth process is controlled by diffusion and reaction at higher and lower supersaturations, respectively. In the present study, a simple mass-transfer kinetic expression is obtained from the model proposed by Howell et al.2 assuming diffusion as the rate-limiting step. 3.1. Diffusion Growth Kinetics. For a crystal growth process limited by diffusion, the linear growth rate, R, is described by the equation2 R)

Growth of sucrose crystals was performed in a 4-L batch agitated crystallizer at two different temperatures, 30 and 40 °C. The crystallizer was connected to an online monitoring system for continuous measuring of brix (Schmidt+Haensch, iPR2) and temperature. The details of the experimental setup are discussed elsewhere.4 The temperature within the crystallizer was maintained by a crystallizer jacket that was connected to a thermostatic water * E-mail: [email protected].

bath. All experiments were carried out at a constant agitation speed of 250 rpm. Sucrose solutions were prepared by dissolving the sucrose crystals in ultrapure water at Tw + 20 °C, where Tw refers to the working temperature. Supersaturation was obtained by cooling the solution to Tw °C. All experiments were carried out with an initial supersaturation of 20 g of sucrose/100 g of solution. Once the crystallizer temperature stabilized, an accurately weighed amount of 16 g of seed crystals with an average particle diameter of 0.0536 cm (coulter size) was added into the crystallizer. The crystal growth experiments were carried out for 24-72 h depending on the temperature. After 24 or 48 h, the solution reached a supersaturation of about 7 g of sucrose/100 g of solution. The mass of the crystals within the crystallizer at any time was calculated from a mass balance.

2D dL ) (c - c*) dt δFc

(1)

Howell et al.2 obtained eq 1 based on the model proposed by Wey5 (as cited by Howell et al.2) for growth kinetics limited by diffusion. Equation 1 can also be written in terms of the mass-transfer coefficient, kd (cm/s), which is given by D δ The volume of a single crystal, V, is defined by6

10.1021/ie900823x  2009 American Chemical Society Published on Web 11/16/2009

kd )

(2)

Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009

V ) fvL

3

11237

(3)

By differentiating eq 3 with respect to L, the change in volume of crystal with respect to change in length of crystal can be defined as dV ) fv3L2 dL

(4)

The surface area of a single crystal can be related to the length dimension by6 A ) fsL2

(5)

Combining eqs 1-5, we obtain 6DA dV ) (c - c*) dt Fcδ(fs /fv)

(6)

Equation 6 can be written in terms of the change in mass with respect to time as 6DA dm ) (c - c*) dt δ(fs /fv)

(7)

If c and ce represent the liquid-phase concentration at any time and at equilibrium, respectively, the corresponding amounts of the liquid-phase concentration crystallized at any time and at equilibrium are given by q ) (co - c)

(8)

qe ) (co - ce)

(9)

If Vs is the volume of solvent inside the crystallizer, then the mass crystallized at any time, m, is given by m ) Vs(co - c) ) Vsq

(11)

Substituting eqs 2 and 8-11 into eq 7, we obtain 6kdA dq ) (q - q) dt Vs(fs /fv) e

(12)

From now on, eq 12 is referred to as a first-order kinetic expression, as the reaction rate is proportional to supersaturation raised to the power of unity. The surface area of the crystal, A, at any time, t can be calculated from eq 13 as7 A ) N0.33fv

( ) M Fc fs

2/3

(13)

The number of growing crystals, N, was assumed to be constant and was predicted using the expression7 N)

mo

(14)

fvFcLo3

With respect to the initial conditions, the limiting conditions to solve eq 12 are given by q ) 0: t ) 0 (15) q ) q: t ) t Integrating eq 12 with respect to the boundary conditions in eq 15, the linear first-order kinetic expression representing the growth of crystals limited by a diffusion process can be given by ln(qe - q) ) ln(qe) -

6kdA t Vs(fs /fv)

Thus, the mass-transfer coefficient, kd, can be calculated from the plot of ln(qe - q) versus area times time, At, using eq 16. A nonlinear expression for eq 16 can be written as

(16)

(

[

q ) qe 1 - exp -

)]

6kdA t Vs(fs /fv)

(17)

In the literature, experimental growth kinetics have been modeled using the values of D and δ obtained from an Arrhenius-type expression and a mass-transfer correlation, respectively.2 The parameter δ is important because it determines the maximum growth rate and strongly affects the shape of growth kinetic curve; it can be determined using the expression8

(10)

From eqs 8 and 9, the supersaturation, ∆c, can be defined as ∆c ) (qe - q) ) (c - ce)

Figure 1. Experimental data and first-order mass-transfer kinetics for the growth of 0.0536-cm sucrose crystals in pure solutions at 30 and 40 °C.

δ)

( )( )

3 FfD 2 µ

1/3

µL Ffυ

1/2

(18)

The boundary layer can be calculated from the local Reynolds number, Re, as δ(L) ) L × Re1/2.9 Kwon et al.10 suggested different expressions for calculating δ values in the diffusion and kinetic regimes. The boundary layer thickness δ is a measure of the resistance offered by the diffusion process, and this parameter accounts for the combined surface and volume diffusion. In this study, no attempts were made to obtain the kinetic curves directly from eq 1 with the D and δ values obtained from empirical or mass-transfer correlations. Instead, the mass-transfer coefficient, kd, that represents the combined effects of diffusion and advection, was obtained theoretically from the slope of a linear plot of ln(qe - q) versus At using eq 16. Figure 1 shows a plot of ln(qe - q) versus At for the growth of sucrose crystals in pure solutions at 30 and 40 °C. The masstransfer coefficient, kd, was calculated from the slope of Figure 1. From Figure 1, it can be observed that, at both studied temperatures, eq 16 well represents the experimental kinetics for At values less than a critical value, Atcritical (in units of cm2 · min). For At > Atcritical, the fitted kinetics deviates from linearity, suggesting another mechanism, more probably that the overall process is controlled by surface integration. This is an expected observation, as growth processes are limited by diffusion and integration kinetics at higher and at lower supersaturations, respectively.11 For the studied experimental conditions, the Atcritical value was estimated as ∼29.27 × 103 and ∼15.81 × 104 cm2 · min at 30 and 40 °C, respectively. An easier and more reliable way to determine Atcritical is explained in a later section of this article. 3.2. Surface Integration Growth Kinetics. To understand the complete mechanism, the experimental data were fitted to

11238


the BCF expression representing crystal growth kinetics following a surface integration process11 -

()

3fvAFc c1 dc ) (c - ce)2 dt fsWs σ1

(19)

Because the solvent used in this study was water, the concentration of sucrose in the solvent expressed in terms of g/cm3 of solvent and g of sucrose/g of water are nearly the same. However, when applying eq 19, care should be taken with units of the terms c and cs. Combining eqs 8, 9, and 19, we obtain

()

3fvAFc c1 dq ) (q - q)2 dt fsWs σ1 e

(20)

Henceforth, eq 20 will be called the second-order kinetic expression. The constants c1 and σ1 are complex temperature-dependent constants given by11

(

c1 )

)

βΛnoΩ β1 a

(21)

9.5γa skTλs

(22)

σ1 )

where β and Λ are dimensionless factors that are less than unity and that describe the effects of kinks in steps and of steps, respectively. no is the concentration of growth units on the surface (particles/cm2), Ω is the specific molecular volume of a molecule or atom (cm3), a is the dimension of the growth unit normal to the advancing step (cm), λs is the average diffusion distance of the growth units on the surface (cm), k is the Boltzmann constant, T is the temperature (in K), and s is a measure of strength of the source of cooperating spirals. Bennema and Gilmer12 suggested that s is in the range of 1-10. Integrating eq 20 with respect to the limiting conditions in eq 15, one obtains the following expression for the growth kinetics kv1qe2t q) 1 + kqet

(23)

where the constant kv is given by kv1 )

()

3fvAFc c1 fsWs σ1

(24)

The linear expression of eq 23 is given by At 1 At ) + q qe kv2qe2

(25)

where the constant kv2 in eq 25 is given by kv2 )

()

3fvFc c1 fsWs σ1

(26)

Equation 25 can be rewritten as 1 q ) At At 1 + qe kv2qe2

(

)

(27)

When the growth process is limited solely by a surface reaction process, eq 27 can be used to calculate the initial reaction rate, Vk, for t f 0 as Vk ) kv2Aqe2

(28)

Figure 2. Second-order surface integration kinetics for the growth of sucrose crystals in solutions at 30 and 40 °C.

According to eq 25, a plot of At/q versus At should be perfectly linear with r2 almost equal to unity if the crystal growth process is limited by surface integration kinetics. This property of this plot can be used to trace the transition in kinetic regimes during a growth process easily. Figure 2 shows a plot of At/q versus At for the growth of sucrose crystals at two different temperatures. The fitted kinetics for each temperature is divided into two sections by a dotted line; these dotted lines represent the values ofAtcritical at 30 and 40 °C. Figure 2 shows that, at the studied temperatures, for At > Atcritical, the experimental growth kinetics follows a surface integration kinetics according to eq 25 with r2 ≈ 1. This shows that the proposed expression based on the BCF model successfully represents the experimental kinetic data at the studied temperatures. In addition, the proposed expression was also found to be helpful in identifying the kinetic regimes limited by diffusion and integration. This suggests that the second-order kinetic expression (eq 25) can be useful for other crystallization systems in identifying the rate-controlling step, which is a function of At and the supersaturation of solution at any time t. The calculated results for kv2 and c1 and the corresponding r2 values are given in Table 2 below. The calculated kinetic constant, c1, was found to decrease with increasing temperature for the sucrose crystal growth process. The magnitude of the c1 value obtained at 40 °C from the proposed expression was found to be in good agreement with the values reported by Bennema13 for sucrose systems. At 30 °C, the c1 values obtained in this study are higher than the values reported by Bennema13 by a factor of 10. From Figures 1 and 2, it can be observed that the growth of sucrose crystals is a function of the area of the growing crystals inside the solution. The growth of sucrose crystals was found to be limited by diffusion and integration processes for At < Atcritical cm2 · min and At > Atcritical cm2 · min, respectively. The values of the supersaturation inside the crystallizer, ∆c, for At ) Atcritical, for the studied experimental conditions are given in Table 2 for reference. In this study, the initial reaction rate was not estimated, as a transition from diffusion to surface reaction kinetics occurs at the studied experimental conditions. 3.3. Growth Mechanism. In this study, from Tables 1 and 2, it can be observed that the mass-transfer coefficient, kd increases with increasing temperature and the surface reaction kinetic constant, c1 (obtained from eq 25) decreases with increasing temperature. Sahin et al.14 reported similar observations, with increasing kd and decreasing reaction kinetic constant during the growth of ammonium pentaborate crystals in the presence of NaCl in solution. The increase in mass-transfer coefficient can be attributed to a decrease in film thickness at


11239

Table 1. Mass-Transfer Coefficients, Kinetic Constants, and Activation Energies Obtained by a First-Order Kinetic Expression temperature (K)

kd (cm/s)

c1 (cm/s)

β1 (cm/s)

W (kJ/mol)

r2

303 313

9.25 × 10-5 1.54 × 10-4

1.17 × 10-4 1.95 × 10-4

9.82 × 10-11 1.64 × 10-10

95.28482 97.18468

0.9945 0.9922

Table 2. Kinetic Constants and Activation Energies Obtained by a Second-Order Kinetic Expression r2

temperature (K) 303 313

qe

0.9978 0.9961

kv2

0.178619 0.209144

1.57 × 10 2.34 × 10-6

higher temperatures. The decrease in surface reaction constant can be related to the surface roughness of the crystal. At higher temperature, the growth of crystals is rapid, and thus, the rate of healing of crystals from a rough to smooth surface is comparably rapid at higher temperatures. Sahin et al.14 experimentally confirmed that the surface reaction constant is related to the activity of the surface as a function of roughness of the crystal. However, studies at the microscopic level are required to confirm these observations. The total energies of adsorption at the studied temperatures were obtained using the concepts of the BCF model for crystal growth. For higher supersaturation, the BCF surface diffusion model is given by11 R ) c1(c - cs)

(29)

where R is the linear growth rate of crystals in solution. Equation 29 can be rewritten in terms of the change in solute concentration as -

3fvAFc dc ) c (c - cs) dt fsWs 1

(30)

From eqs 16 and 30, the values of kd and c1 can be related as c1 )

2kdWs FcVs

(31)

From eqs 31 and 21, the mass-transfer coefficient can be used to calculate the activation energy for a growth process limited by diffusion kinetics as β1 )

2kda FcβΛnoΩ

(32)

The parameter no refers to the number of molecular positions available for adsorption on the crystal surface and can be obtained from the ratio of the total surface area of the sucrose crystals available for the growth of crystals in supersaturated solution to the area occupied by one molecule.15 The value of no for the present system was determined as 6.9232 × 1024 positions/cm2. Assuming a ) Ω1/3 and using Ω ) 715.04 × 10-22 cm3, the kinetic coefficient, β1, can be obtained from the determined kd values. From the determined kinetic coefficient, β1, the total adsorption energy for the growth of crystal under the diffusion regime can be calculated as

(

β1 ) aν exp -

∆Gads kBT

)

c1 (cm/s)

γ1 -5

(33)

where ∆Gads is the total adsorption energy, which is the sum of adsorption energy factors from the solution to the surface and from the surface to the kink where the growth unit is incorporated into the crystal surface. ν is a frequency factor on the order of the atomic vibration frequency. Assuming that ν ) kBT/hP, the kinetic coefficient, β1 and the total adsorption energy can be determined using eq 32 and 33 respectively. The

-2

6.00 × 10 3.00 × 10-2

-5

3.67 × 10 2.60 × 10-6

b1 (cm/s)

W (kJ/mol) -11

3.07736 × 10 2.17874 × 10-12

98.20 108.42

determined BCF constant, c1, β1 and ∆Gads at the studied temperatures are given in Table 1 for the growth of sucrose crystals in pure solutions. Table 1 shows that ∆Gads values for the growth process in the diffusion regime are very high and in the range of 95-97 kJ/mol at the studied conditions. Because no works interpreting kinetic data with a modified BCF expression as in the present study are available in the literature, no comparisons can be made based on ∆Gads values obtained from this study. The total adsorption energy, ∆Gads, for the growth process in kinetic regime, i.e., At > Atcritical cm2 · min, was calculated from the determined c1 values that were obtained from second-order linear expression (eq 27). Table 2 shows that the growth process in the kinetic regime is associated with an increase in ∆Gads with increasing temperature. In the kinetic regime, the ∆Gads value increased from 98 to 108 kJ/mol as the temperature was increased from 30 to 40 °C. The ∆Gads values obtained in this study under the diffusion and kinetic regimes cannot be compared with those reported previous works, as the mathematical structure of the proposed expression is different from that of the original BCF expression. However, it is worth pointing out that the theoretical concepts behind the linear second-order expression obtained in this study and the BCF expression for a surface integration process are the same. As reported in Tables 1 and 2 the total adsorption energy was found to be >90 kJ/mol for the growth process in the diffusion and kinetic regimes according to the expressions proposed in this study. Considering sucrose crystallization systems, the ∆Gads values were evaluated using the c1 values reported by Bennema13 for sucrose systems and found to be 103.43 and 97.18 kJ/mol at 30.7 and 40.5 °C, respectively. This is in good agreement with the values obtained using a secondorder kinetic expression in this study. In this study, the transport process during the growth of sucrose crystals at the studied temperatures was also analyzed using a two-step mass-transfer model assuming a second-order surface integration process (not shown). However, the two-step mass-transfer model poorly fit the experimental data at the studied conditions. Further, it produced kinetic constants (for the surface integration step) without physical significance (negative values). A similar observation was reported for the growth of calcite crystals at higher supersaturations.16 Further attempts to determine the kd value directly from eq 1 produced a very poor fit of the experimental kinetic data. In this study, for the similar case of the growth of sucrose crystals, the proposed expression successfully represented the crystal growth data and was also found to be useful in predicting the masstransfer coefficient. 4. Conclusions Two simple kinetic models are proposed to model the crystal growth process in the diffusion and kinetic regimes. The proposed models were used to determine the mass-transfer

11240


coefficient and reaction kinetic constant at two different temperatures. The total adsorption energies for the growth of crystals in both the diffusion and kinetic regimes were determined using the proposed expressions. The proposed expressions are simple to use and resemble first- and second-order kinetics, and they successfully represent the diffusion and surface integration processes, respectively. A second-order expression based on the BCF model was found to be helpful in predicting the transition in crystal growth kinetics as a function of the area of crystal that changes with time and supersaturation. An initial reaction rate was proposed according to a pseudo-second-order expression. The present study is limited only to the growth of sucrose crystals at two different temperatures. Many more studies are required to validate the proposed expressions for different crystallization systems, including sucrose systems under different experimental conditions.

Vs ) volume of solvent, cm3 Ws ) mass of solvent, g Greek Variables β ) constant in BCF expression β1 ) kinetic coefficient, cm/s γ ) interfacial tension (in eq 24), J/cm2 δ ) film thickness, cm ∆Gads ) total adsorption energy, kJ/mol λs ) diffusion length Λ ) thickness of adsorption layer, cm ν ) frequency factor, s-1 Fc ) density of sucrose crystal, g/cm3 σ1 ) constant in BCF equation Ω ) molecular volume, cm3

Acknowledgment

(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) Howell, T. A., Jr.; Yoseph, E. B.; Rao, C.; Hartel, R. W. Sucrose crystallization kinetics in thin films at elevated temperatures and supersaturations. Cryst. Growth Des. 2002, 2 (1), 67–72. (3) Shiau, L.-D. The distribution of dislocation activities among crystals in sucrose crystallization. Chem. Eng. Sci. 2003, 58, 5299–5304. (4) 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. (5) Wey, J. S. Basic crystallization process in silver halide precipitation. In Preparation and Properties of Solid State Materials; Wilcox, W. R., Ed.; Marcel Dekker: New York, 1981; Vol. 6, Chapter 2, pp 67-117. (6) Garside, J.; Mullin, J. W.; Das, S. N. Importance of Crystal Shape in Crystal Growth Rate Determinations. Ind. Eng. Chem. Process Des. DeV. 1973, 12, 369–371. (7) Guimaraes, L.; Sa, S.; Bento, L. S. M.; Rocha, F. Investigation of crystal growth in a laboratory fluidized bed. Int. Sugar J. 1995, 97, 199– 204. (8) Gilmer, G. H.; Ghez, R.; Cabrera, N. An analysis of combined and volume diffusion processes in crystal growth. J. Cryst. Growth 1971, 8, 79–83. (9) Sizaret, S.; Fedioun, I.; Barbanson, L.; Chen, Y. Crystallization in flowsII. Modelling crystal growth kinetics controlled by boundary layer thickness. Geophys. J. Int. 2006, 167, 1027–1034. (10) Kwon, Y. I.; Dai, B.; Derby, J. J. Assessing the dynamics of liquidphase solution growth via step growth models: From BCF to FEM. Prog. Cryst. Growth Charact. Mater. 2007, 53, 167–206. (11) Burton, W. K.; Cabrera, N.; Frank, F. C. The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. R. Soc. A 1951, 1934, 299–358. (12) Bennema, P.; Gilmer, G. H. Kinetics of Crystal Growth. In Crystal Growth: An Introduction; Hartman, P., Ed.; North Holland: Amsterdam, 1973; pp 263-327. (13) Bennema, P. Surface diffusion and the growth of sucrose crystals. J. Cryst. Growth 1968, 3-4, 331–334. (14) Sahin, O.; Ozdemir, M.; Genli, N. Effect of impurities on crystal growth rate of ammonium pentaborate. J. Cryst. Growth 2004, 260, 223– 231. (15) Koutsopoulos, S. Kinetic study on the crystal growth of hydroxyapatite. Langmuir 2001, 17, 8092–8097. (16) Tai, C. Y.; Chen, P. C.; Tsao, T. M. Growth kinetics of CaF2 in a pH-stat fluidized-bed crystallizer. J. Cryst. Growth 2006, 290 (2), 576– 584.

The experimental work in this study was carried out under the direction of Prof. F. Rocha, FEUP, Portugal. We thank the anonymous reviewers for their valuable suggestions, which helped to improve this work. Notations A ) area of growing crystals, cm2 a ) dimension of the growth unit normal to the advancing step (in eq 24), cm Am ) area occupied by one molecule, cm2 Atot ) total surface area of the sucrose crystals available for the growth, cm2 BCF ) Burton-Cabrera-Frank c ) sucrose concentration at any time t, g/cm3 c*, ce ) sucrose concentration at equilibrium, g/cm3 c1 ) kinetic constant, cm/s D ) diffusivity of sucrose in solution, cm2/s fv ) volume shape factor fs ) area shape factor R ) linear growth rate, cm/s kB ) Boltzmann constant, J/K kd ) mass-transfer coefficient, cm/s kv1 ) constant in eq 25, g of solvent/[(g of crystal) s] kv2 ) constant in eq 25, g of solvent/[(g of crystal) cm2 s] L ) length of growing crystals at any time t, cm Lo ) average diameter (coulter size) of seed crystals, cm m ) mass crystallized at any time, g mo ) mass of seed crystals (g) M ) total mass of crystals inside crystallizer, g no ) concentration of growth units on the surface, particles/cm2 N ) number of crystals q ) difference in concentration, g/cm3 (refer to eq 11) qe ) difference in concentration, g/cm3 (refer to eq 12) R ) gas constant, 8.314 J/(K mol) s ) strength of the source of cooperating spirals SSA ) specific surface area of the sucrose crystals T ) temperature, K t ) reaction time, s Tw ) working temperature, °C or K V ) volume of crystal, cm3

Literature Cited

ReceiVed for reView May 19, 2009 ReVised manuscript receiVed October 11, 2009 Accepted November 2, 2009 IE900823X

Simple Kinetic Expressions to Study the Transport Process during the

Recommend Documents