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Journal of Food Engineering 38 (1998) 15-2.5 0 1998 Elscvier Science Limited. All rights reserved Printed in &eat Britain 0260-8774/98/$ - see front matter ELSEVIER

PII:

SO260-8774(98)00109-S

A Mathematical Model for Lactose Dissolution, Part II. Dissolution Below the Alpha Lactose Solubility Limit

E. K. Lowe” & A. H. J. Paterson’ “New Zealand ‘Department

Leather and Shoe Research Association, Palmerston North, New Zealand of Process and Environmental Technology, Massey University, Massey, New Zealand (Received

19 August 1995; accepted 27 July 1998)

ABSTRACT The rate of dissolution of lactose into water has been shown to be controlled by two diflerent mechanisms, depending on the final concentration of lactose required. If concentrations above the alpha lactose solubility limit are required then the dissolution rate is governed by the first order kinetics of the mutarotation reaction from alpha to beta lactose. At typical room temperatures this reaction is so slow that the rate of dissolution of alpha lactose into solution and the rate of mass transfer away from the surface can be considered instantaneous. At lactose concentrations below the alpha solubility limit, the initial assumption that the rate of mass transfer from the suface to the liquid controlled dissolution, has been shown to be incorrect with experimental dissolution times being much longer than those predicted using a model with the lowestpossible mass transfer co-efJicient and an instantaneous dissolution of lactose at the surface of the crystal. To account for this discrepancy, a model was proposed that assumed a first order reaction for the dissolution occuring at the surface. Appropriate empirical parameters were fitted to the experimental data at difierent temperatures and initialparticle sizes. It was concluded that the surface reaction of unbinding the bound lactose molecules from the crystal structure controls the rate of dissolution of a-lactose monohydrate crystals below the a-lactose solubility limit at temperatures below 50°C. The Arrhenius constants of E, = 37f3 kJ mole-’ and A, = 152 (43-540) m s-‘, were determined. 0 1998 Elsevier Science Limited. All rights reserved. 15

16

E. K. Lowe, A. H. J. Paterson

NOMENCLATURE a1

A

Ma Mfi k Re SC

Sh T v1 V P P AP

Crystal surface area conversion factor (crystal surface area = a,d$) Total crystal surface area available for diffusion (number of lactose crystals x aId:) (m’) Frequency factor for surface reaction (m s-‘) Concentration of a-lactose in solution (kg m-“) a-lactose solubility with a B-lactose concentration of C,{ (kg m-“) Concentration of p-lactose in solution (kg m-“) Diffusivity of lactose solution (m2 SK’) Characteristic dimension of crystal (m) Arrhenius constant for surface reaction (kJ mol-‘) Gravitational acceleration constant (m s-“) Mutarotation reaction rate constant for a-lactose to B-lactose (s-‘) Mutarotation reaction rate constant for p-lactose to cc-lactose (SK’) Liquid side mass transfer coefficient (m SK’) Surface reaction rate (m s-‘) Characteristic length (surface area of particle/maximum perimeter projected on a plane normal to the flow) (m) Mass of a-lactose in solution (kg) Mass of p-lactose in solution (kg) Number of lactose crystals in solution Best number Reynolds number Schmitt number Sherwood number Time (s) Volume conversion factor (crystal volume = v,dz) Volume of solution (m”) Viscosity of solution (Pa s-‘) Density of solution (kg me3) Difference between crystal and solution density (kg m-“)

INTRODUCTION The dissolution of a-lactose into water is of significance to many processes (e.g. tablet and milk powder dissolution). Hudson (1904) has described a-lactose dissolution process at low temperatures (0-25°C) as being a sequential process given as; a-lactose monohydrate

crystals+a-lactose

in solution+-lactose

in solution.

He showed that the first part of this reaction was virtually instantaneous when compared with the time required for a-lactose to mutarotate to /?-lactose. Hodges et al. (1993) has examined the dissolution process of a-lactose in order to model lactose dissolution, and has developed a mathematical model of the dissolution process. The model developed was based on the assumption that two main mechanistic processes occurred during dissolution; mass transfer of a-lactose from the crystal surface into solution, and mutarotation of a-lactose to /I-lactose. This lead to the development of a model with three differential equations.

A mathematical model for lactose dissolution

17

The first differential equation expressed the rate of change in the mass of cr-lactose present in solution in terms of the mass of a-lactose diffused into solution, loss by mutarotation to p-lactose, and increase by reverse mutarotation from /?-lactose.

dMa

-

dt

=kLA (Cp?S-C Y.)-k,C,V+kzC,jV

The second differential equation expresses rate of change in the mass of p-lactose present in solution in terms of the increase by mutarotation from a-lactose, and loss by reverse mutarotation to cr-lactose. dM, dt

= -k,C,V+k,C,jV

The last differential equation expressed the rate of change in crystal size in terms of the reduction of crystal volume. --=

ddc

k,a,(C” IS-C 1)

dt

3W

(3)

As the crystal size is changing throughout dissolution, the mass transfer coefficient kL cannot be assumed to be constant. Instead, k,_ was calculated from the Sherwood number using the following equation. The Sherwood number was assumed to be 2.0. ShD k,_= L’

(4)

This assumes the lactose is diffusing away from the surface into a stagnant medium, using the characteristic length L’, giving the lowest reasonable value for the mass transfer rate. This work identified three main mechanistic regions: the first was when saturated lactose solutions were required, the second was when lactose concentrations between the a-lactose solubility limit and saturated solutions are targeted, and the third is when lactose concentrations below the a-lactose solubility limit are desired. After allowing for variations in a-lactose solubility due to the presence of p-lactose, as shown by Visser (1982) the first two mechanistic regions were modeled successfully. However, a poor fit was apparent for lactose concentrations below the a-lactose solubility limit. Table 1 shows the difference between the predicted and experimental dissolution times. Here we can see that the predicted dissolution times were significantly faster than those observed. In this region, mass transfer is the controlling step, with mutarotation of a-lactose to /&lactose having minimal impact on lactose dissolution. The Sherwood number used in the model for the calculation of the mass transfer coefficient was set assuming the lactose was diffusing away from the surface into a stagnant medium, excluding turbulence effects on the mass transfer coefficient. This gave the lowest reasonable value for the mass transfer rate, taking into account the decreasing crystal size during dissolution, with the actual value likely to be higher. Even with this slowest possible mass transfer rate, the modeled dissolution was too fast.


18

Dissolution

TABLE 1 Times Below the Alpha Lactose Solubility Limit Hodges et al. (1993)

Temperature (“C)

Average size (W.4

Concentration (kg m -“)

25 25 45 45

116 233 116 233

::

Dissolution time (s) Experimental

Model

24 50 24 42

5.3 22 3.3 13

100 100

70

Difference

78% 56% 86% 69%

There are two possible explanations for the poor lack of fit of the model in this region, namely; (i) The crystal size distribution within the sample affected the results. The model developed by Hodges et_ al. (1993) is based size. Although _^ on. a single particle .__ . . . _ expertments were conducted on sieved tractions, there was still variation in particle sizes. (ii) There is a significant reaction rate step involved at the surface of the crystal. This paper will attempt to identify the cause of this lack of fit and propose an improved model. EXPERIMENTAL Alpha lactose monohydrate (‘Special dense’ grade, Lactose Company of New Zealand Ltd., Kapuni, Taranaki, New Zealand) was sieved and two size fractions collected, one between sieve sizes 150 and 210 pm, the other between sieve sizes 210 and 300 pm. The particle size distribution of each sieved fraction was determined using a Malvern particle size analyser (Lowe, 1993). Lactose dissolution experiments were conducted using the method given by Hodges et al. (1993). A defined mass of alpha lactose monohydrate was added to a stirred, temperature controlled vessel. Liquor samples were taken at 1 s intervals and the refractive index determined at 25°C. The total lactose concentration was determined from the refractive index using a calibration curve from solutions of known lactose concentrations. Replicates were conducted at four different temperatures with the particle size fraction 150-210 pm and at six different temperatures with the particle size fraction 210-300 pm. Table 2 lists the experimental conditions for each run. MODEL FORMULATION Modification of model to account for turbulence effects on mass transfer

The model, as expressed by Hodges et al. (1993), assumes for the calculation of the mass transfer rate that the lactose was diffusing away from the surface of the crystal into a stagnant medium. This excluded turbulence effects on the mass transfer


19

TABLE 2 Experimental Conditions Temperature (“C)

4 :: 37 50 70

Sieved fractions (pm)

210-300 150-210,210-300 150-210,210-300 210-300 150-210,210-300 150-210,210-300

Target concentration (kg m -“)

25.0 24.9 50.0 100.0 123.9 201.3

coefficient. This gave the lowest reasonable value for the mass transfer rate, while taking into account the decreasing crystal size during dissolution. However, the actual mass transfer coefficient will be higher. A more applicable assumption is that the mass transfer of the crystal can be approximated by a sphere in viscous creeping flow, as suggested by Clift et al. (1978). Under these conditions, the Sherwood number can be calculated from the Reynolds and Schmitt numbers using eqn (5). Sh= 1+(1+Re.Sc)“3 The Reynolds number can be calculated using correlations from the ‘Best number’ ND, as determined in eqn (6). N,,=

(5) given in Clift et al. (1978)

4pApgL’3 3P2

Using this calculation methodology, at 25°C the initial Sherwood number for the model with an initial crystal size of 284.5 pm will be 11.8, which will trend towards 2 as the particle size decreases during dissolution. This initial value is much higher than the previously assumed value for the Sherwood number, and it makes the difference between the predicted and experimental results even larger. Surface reaction model development A potential reason for the lack of fit is that there is a significant rate limiting disassociation reaction occurring at the crystal surface. To model this step, a first order reaction rate constant k, was incorporated into the model. This alters the differential equations for crystal size and the alpha lactose concentration, giving the following differential equations;


20

dM,

-

dt

=

A(@ rs -C 1

1

)

cx

-k, C,V+k,CpV

(8)

The mass transfer coefficient k,_ is not constant and was calculated as in the modified procedure given above. One thing to note from these equations is that if k, is much greater than k,_ (i.e. if the surface reaction is sufficiently fast) then the equations above revert to the equations developed by Hodges et al. (1993). Values of k, could not be found in the literature nor determined by independent experiment. Instead, values of k, were fitted to the experimental data by using an iterative process using a minimized least squares fit criteria of the residual sum of squares.

RESULTS

AND DISCUSSION

Table 3 shows the experimental dissolution times achieved for each temperature and particle size compared to predicted dissolution times using the model from Hodges et al. (1993) including the constant Sherwood number assumption. These results confirm the work by Hodges et al. (1993), showing that for dissolution below the a-lactose solubility limit the model over-predicts the dissolution rate. An interesting point to note from these results, is that as the temperature of dissolution increases, the lack of fit decreases. This indicates that the error in the model is not caused by a systematic error. Table 4 shows the results of the particle size distribution within each sieved fraction, showing the mean value within the fraction along with percentile particle

TABLE 3

Predicted and Experimental Dissolution times for Different Temperatures and Crystal Size Fractions Temperature (“C)

1: 25 Z 70

Size fraction (W)

210-300 150-210 210-300 150-210 210-300 210-300 150-210 210-300 150-210 210-300

Dissolution time (s) fiperimental

Predicted

%

D#erence

325

109.4

66%

1:; 29

49.8 28.7 23.8

45% 56% 18%

5:2 Zi5

41.2 34.0 14.7 8.5

34% 35% 30% 11%

1:

2:;

13% 40%


21

TABLE 4 Particle Size Analysis of Sieved Fractions (Malvern Instruments MASTER Particle Sizer) Sieved fraction

150-210 210-300

Particle size (pm)

202.9 266.9

268.7 411.5

141.4 186.4

201.2 281.3

189.4 258.8

sizes. It can be seen that although the lactose has been sieved, there is still some variation in particle size within each fraction. To assess whether this variation in particle size can explain the lack of fit of the model, the effect of the 10% and 90% percentile particle sizes on the predicted dissolution time was examined. Figure 1 shows the impact of this at the lowest temperature of 4” for an average crystal size of 284.5 pm. Only by assuming that all crystals are initially at the 90% percentile value does the predicted dissolution curve approximate the experimental results. However, an examination of the predicted dissolution times (Table 5) shows that even this does not slow dissolution enough to match the experimental dissolution time. Table 5 shows that it is not until the

30

g 25 Y 0'

0

'i; a 20 b c 8 5 15 0 ii 5 10 9 7 ij c

5

0 0

100

200

300

400

500

600

Time (seconds)

Fig. 1. Lactose dissolution to a final concentration below the cr-lactose solubility limit. (0.1976 kg of anhydrate lactose (DC = 233 pm) added to 2 1 of water at 45°C.)

22

E. K. Lowe, A. H. J. Paterson TABLE 5 Effect of Varying Particle Size on Predicted Size fraction (W)

Temperature (“C)

4 15

;:

325 52 112 29

210-300 150-210 210-300 150-210 210-300

70

Time

Predicted dissolution time

Experimental

210-300 150-210 210-300 HO-210 210-300

25

Lactose Dissolution

;;9.5 21 4 10

~~~,0.S,

D&W?,

D&R,)

109.4 28.7 49.8 23.8 41.2

257.6 50.5 118.2 41.7 97.7

52.9 14.0 24.3 11.6 20.1

34.0 8.5 14.7 3.5 6.0

79.1 14.9 34.3 6.1 13.9

16.7 4.2 8.4 1.7 3.0

temperature is above 15°C that increasing the modeled particle size to the 90% percentile value decreases the predicted dissolution time sufficiently to match the experimental results. These results lead to the conclusion that systematic error in the crystal size does not explain the lack of fit of the model for dissolution of lactose below the a-lactose

1

0

2

3

4

5

6

n

0

7

8

/.

0

I

9

10

Time (seconds) 1

q

Exnerimental

-

Hodoes et al. (I 9931 1 Variable Sh ................

Fig. 2. Lactose dissolution to a final concentration below the a-lactose solubility concentration of 205 kg m-‘, D, = 210-300 pm, 70°C.)

limit. (Final


23

solubility limit, and that even with the lowest reasonable mass transfer coefficients chosen, variation in crystal size cannot explain the lack of fit in the model. It must be acknowledged that the assumption of a single crystal size limits the applicability of the model for real lactose samples with wide particle size distributions. A later paper will address this issue. The experiments conducted at 70°C highlight the effect of changing the method of calculating the mass transfer coefficient on the predicted dissolution rate. Figure 2 shows the predicted dissolution curve using the two different mass transfer calculation methodologies without the inclusion of any surface reaction for an average crystal size of 284.5 pm. This shows that at high temperatures, the method for estimating mass transfer rates proposed by Hodges et al. (1993) under-predicts the rate of dissolution, where as the modified procedure more closely follows the experimental results, giving greater dissolution rates. This suggests that the modified procedure for estimating mass transfer is appropriate and that at high temperatures the limiting factor is the rate of mass transfer of a-lactose from the crystal surface to the bulk of the liquid. At lower temperatures, fittin a surface reaction rate gave a good fit to the experimental data (Fig. 3) with R values in the order of 0.95-0.99. Figure 4 shows an Arrhenius plot of the fitted surface reaction rates calculated using the modified mass transfer calculation procedure. Some scatter is evident, but the typical relationship for a chemical reaction is apparent. A regression analysis of this data results in the Arrhenius constants of Es = 3713 kJ mole-’ and As = 152 (43-540) m SK’. The only chemical reaction that can be taking place is the actual

0 0

100

200

300

400

500

600

Time (seconds) -Surface

Fig. 3.

Lactose dissolution to a final concentration concentration

Reaction Model

below the c.+lactose solubility of 25 kg me3, D, = 210-300 pm, 4°C.)

limit. (Final


24

lE-03

IE-04

1 E-05

lE-06

I

-F 0.003

0.0032

0.0033

0.0034

0.0035

0.0036

0.0037

Inverse Temperature (l/K) Large Crystals

Fig. 4. An Arrhenius

Small Crystals

-----Regression

plot of the fitted surface reaction rates.

disassociation of the lactose molecules from the crystal structure into solution, so the activation energy measured here could represent the energy binding the lactose molecules into the crystal. Visser and Bennema (1983) have examined the bonding strengths of lactose molecules being held in a lactose crystal. By using dimensionless bonding strengths, calculated for each plane of the growing crystal, they have shown that the presence of the j-lactose is the probable cause of the shape of a lactose monohydrate crystal. Their calculations show that the total dimensionless bonding strengths for a molecule inside the crystal is given as 78.8. Using this to estimate the bonding strength of a lactose molecule on the surface, where only a half of the hydrogen bonds are active, gives an estimate of 98 kJ mole-‘. This can be compared with the heat of dissolution of lactose of - 15.5 kJ mole-’ (Perry & Green, 1984) and the value of the activation energy required to release a molecule from the lactose crystals in this work of 37 kJ mole-‘. The differences between the data experimental and theoretical values can possibly be explained by the levels of hydrogen bonding that will exist between the water molecules and the lactose molecules in solution, meaning that not all of the bound energy of a lactose molecule is released when it dissolves into solution. When a molecule of lactose first dissolves away from the surface, not all the final hydrogen bonds with the surrounding water will be formed immediately and thus might explain why the activation energy as measured by the Arrhenius pilot for the lactose dissolution process is greater than the heat of solution.


25

It is concluded that the lactose dissolution at lower temperatures and below the cr-lactose solubility limit is controlled by the surface reaction rate at which the lactose molecules are disassociated from the crystal surface. CONCLUSIONS The rate at which a-lactose monohydrate crystals are dissolved into water at lower temperatures at concentrations below the a-lactose solubility limit is controlled by the surface reaction of disassociation of cr-lactose molecules from the crystal structure. The bond energy that must be overcome has been estimated at 37+3 kJ mole-‘. REFERENCES Clift, R., Grace, J. R. & Weber, M. E. (1978) Bubbles, Drops and Particles, Academic Press, p. 114. Hodges, G. E., Lowe, E. K. & Paterson, A. H. J. (1993). A mathematical model for lactose dissolution. The Chemical Engineering Journal, 53, B25-B33. Hudson, C. S. (1904). The hydration of milk-sugar in solution. Journal of the American Chemical Society, 26, 1065-1082. Lowe, E. K. (1993) The dissolution of alpha lactose monohydrate. A mathematical model for predicting dissolution times. M.Tech. Thesis. Massey University, New Zealand. Perry, R. H. & Green, D. (1984) Perry’s Chemical Engineers Handbook. McGraw Hill. Visser, P. A. (1982). Supersaturation of alpha lactose in aqueous solutions on maturation equilibrium. Neth. Milk Dairy J., 36, 89-101. Visser, R. A. & Bennema, P. (1983). Interpretation of the morphology of cr-lactose hydrate. Neth. Milk Dairy J., 37, 109-137.