An Equation for Modelling the Kinematic Viscosities of Binary and Ternary Solutions with Sugars and Sodium Chloride as a Function of Concentration and Temperature Experimental Data of Solutions with Lactose

Abstract

An equation to model experimental data of kinematic viscosities of aqueous solutions commonly employed in osmotic dehydration is proposed in this article. The binary and ternary systems are constituted by water as solvent and by sugars (sucrose, glucose, and lactose) and glycerol with or without sodium chloride as solutes. Experimental data of kinematic viscosity of aqueous lactose solutions (with and without sodium chloride) are determined in this work. The range of concentration of all systems studied covers the typical experimental conditions of interest (from 0 to 5.0 mol kg⁻¹ for sodium chloride and 0 up to 0.5 mol kg⁻¹ for lactose) in this type of processes and temperature varied from 20 to 50°C. The parameters of the equation proposed for each system were optimized and the resulting expression adjusted experimental data with deviations less than 3.0%.

Keywords:

• Glucose
• Glycerol
• Newtonian behavior
• Sucrose

INTRODUCTION

The dehydration operation is a common method of food materials preservation. Important contributions during the last few years have been made to improve the quality and economics of the global process, particularly, using combined methods where one step is constituted by osmotic dehydration. Osmotic dehydration consists on the partial removal of water by direct contact of a wet material with a hypertonic medium, i.e., a high concentrated sugar or salt solution. During osmotic dehydration there are complex mass transfer phenomena, which have been defined as dewatering-impregnation-soaking of a food material in concentrated solutions. The cellular membrane of food material acts as semi-permeable barrier where the mass transfer by diffusion of different components takes place. The mass fluxes depend on the driving force (osmotic pressure). These processes are commonly studied considering a global resistance to component transport in the bulk of the body of the food material (cellular membrane) neglecting the external resistance (this assumption is difficult to accept for osmotic systems). For rigorous models the mass transfer resistance in the fluid phase must be taken into account, and, in this way, the viscosity of the solutions must be known. The mass transfer rate is determined by the mass transfer coefficients values that depend inversely on kinematic viscosity. These coefficients in osmotic dehydration operation can not be known by theoretical models due to the complexity of the systems and must be evaluated experimentally or by empirical models involving the kinematic viscosity into the Reynold and Schmidt number, for example. The choice of osmotic agents depends on taste, cost, water activity lowering capacity, and viscosity values of the solutions (these values are high especially when disaccharides are employed). As the composition of the food material is changing during osmotic process one of the goals is to control the solids gain that strongly depends on desired final product characteristics and also can be related with the viscosity of the solution.[1] Additionally, osmotic dehydration is an effective way to reduce overall energy requirements operating at ambient conditions[2] (at low temperatures color and flavor retention of food material are favored[3]), but agitation power is necessary and depends directly on osmotic media viscosity, physical property that increases its value at low temperature.

Rheological properties of these solutions are function of physicochemical characteristics of the solutes, composition and temperature of solutions and must be determined experimentally. In previous papers kinematic viscosity data of binary and ternary aqueous solutions with several solutes as sodium chloride and sucrose,[4] and glucose,[5] and glycerol[6] were experimentally determined and correlated by means of several equations. Other systems of interest in osmotic dehydration are solutions of lactose with sodium chloride employed meanly for fruits.[7] Lactose solutions are mainly employed in the case that the characteristics of the final product can not be very sweet (sweet scale for relative sweetness: lactose 16; glucose 74, sucrose 100, fructose 173) and combined with sodium chloride allows the known advantages of ternary systems.[4] Viscosity of aqueous solutions of lactose or sodium chloride at 20°C is available in handbooks,[8] but little data can be found for other temperatures,[9,10] and no data on viscosities of the aqueous solutions with both solutes were found. One of the goals of this article is to report the kinematic viscosities data of ternary aqueous solutions of NaCl and lactose at various concentrations and temperatures applicable to osmotic dehydration process. These solutions are classified from rheological behavior as Newtonian fluids; in this way for experimental determination a capillary viscosimeter is appropriate and selected. Another goal is to establish a unique equation for all systems commonly employed in osmotic dehydration (cited previously ternary systems). In this way, the parameters for each system are collected and recalculated following the new more general proposed expression.

MATERIALS AND METHODS

Solutions were prepared using degassed distilled water and with solutes lactose and sodium chloride (purities > 99.5%). Both solutes were dried to constant mass by dehydration in an oven. The solutions were prepared by mass using a Mettler AJ 150 balance with a precision of ± 0.0001 g, and they were filtered before use. Solution concentrations were from 0–5.0 mol kg⁻¹ for sodium chloride and 0–0.5 mol kg⁻¹ for lactose (the most concentrated solutions are near to limit of solubility).

Kinematic viscosities were determined in a Schott Geräte AVS 350 automatic Ubbelohde viscosimeter, using the experimental protocol described elsewhere.[11] All measurements were made in five replicates. The accuracy of the viscosity measurements was 0.2% (greater than usually required in an osmotic dehydration process). The precision of the temperature control in all measurements was ± 0.1°C. All the solutions were studied at temperatures ranging from 20 to 50°C in 5°C steps. Statistical analysis of data and non-linear fittings were carried out using the program Table Curve 3D (Jandel Scientific).

RESULTS

Kinematic viscosities of solutions with lactose[1] and sodium chloride[2] at several concentrations and at several temperatures are collected in Table 1.

Table 1 Kinematic Viscosities, ν·10⁶ m²·s^{− 1}, of Lactose[1] + Sodium Chloride[2] Aqueous Solutions at Various Molalities m and Temperatures T.

	m2
m1 = 0.1 T (K)	0.0	0.5	1.0	2.0	3.0	4.0	5.0
293.1	1.101	1.144	1.150	1.224	1.329	1.439	1.583
298.1	0.972	0.997	1.024	1.093	1.187	1.287	1.411
303.1	0.871	0.891	0.919	0.986	1.066	1.157	1.269
308.1	0.785	0.805	0.831	0.892	0.965	1.051	1.142
313.1	0.712	0.735	0.757	0.816	0.881	0.957	1.042
318.1	0.651	0.671	0.692	0.743	0.807	0.880	0.956
323.1	0.595	0.617	0.637	0.685	0.744	0.812	0.880
	m2
m1 = 0.2 T (K)	0.0	0.5	1.0	2.0	3.0	4.0	5.0
293.1	1.200	1.214	1.246	1.358	1.438	1.576	1.747
298.1	1.052	1.081	1.107	1.212	1.281	1.408	1.551
303.1	0.940	0.965	0.993	1.063	1.150	1.258	1.382
308.1	0.841	0.873	0.899	0.962	1.039	1.133	1.247
313.1	0.764	0.790	0.817	0.881	0.949	1.033	1.136
318.1	0.694	0.723	0.748	0.804	0.870	0.944	1.037
323.1	0.639	0.659	0.685	0.737	0.800	0.869	0.950
	m2
m1 = 0.3 T (K)	0.0	0.5	1.0	2.0	3.0	4.0	5.0
293.1	1.297	1.333	1.368	1.454	1.567	1.716	1.909
298.1	1.145	1.160	1.209	1.291	1.393	1.541	1.682
303.1	1.023	1.040	1.078	1.150	1.246	1.374	1.496
308.1	0.917	0.937	0.973	1.034	1.125	1.237	1.348
313.1	0.827	0.848	0.882	0.940	1.022	1.125	1.220
318.1	0.753	0.779	0.806	0.863	0.935	1.025	1.112
323.1	0.687	0.711	0.739	0.793	0.860	0.939	1.018
	m2
m1 = 0.4 T (K)	0.0	0.5	1.0	2.0	3.0	4.0	5.0
293.1	1.404	1.442	1.479	1.616	1.725	1.883	2.088
298.1	1.238	1.277	1.309	1.406	1.527	1.658	1.851
303.1	1.100	1.136	1.168	1.261	1.360	1.476	1.643
308.1	0.990	1.022	1.049	1.134	1.225	1.329	1.470
313.1	0.891	0.925	0.951	1.029	1.110	1.201	1.329
318.1	0.810	0.838	0.865	0.932	1.013	1.093	1.206
323.1	0.741	0.770	0.792	0.856	0.931	1.024	1.104
	m2
m1 = 0.5 T (K)	0.0	0.5	1.0	2.0	3.0	4.0	5.0
293.1	1.563	1.587	1.613	1.737	1.885	2.080	2.303
298.1	1.368	1.380	1.429	1.545	1.669	1.840	2.026
303.1	1.207	1.225	1.269	1.376	1.486	1.632	1.794
308.1	1.087	1.098	1.140	1.230	1.333	1.461	1.602
313.1	0.974	0.993	1.030	1.111	1.206	1.317	1.447
318.1	0.880	0.900	0.935	1.009	1.095	1.195	1.308
323.1	0.801	0.825	0.855	0.922	1.000	1.098	1.195

For modelling, first the binary solutions viscosity was adjusted, ν_i (m²s⁻¹), with salt and lactose, separately. In this way, the obtained expression was:

where i = 1, lactose and 2, sodium chloride, ν_w (m² s⁻¹) is the kinematic viscosity of water given by:[4]

and T_R the reduced temperature:

where T (K) is the absolute temperature, and F_i is the term that involves the effect of the concentration m_i (molality) of the solute as:

where a, b, and c are parameters. Particularly, for lactose solutions the following values of the parameters showed in the Table 2 were found, after the corresponding non-linear fit at each working temperature. It can be observed that the parameters a and c don't appreciably change with temperature. Optimizing the values of both parameters (with the values 0.720 and 1.00) as fixed variables, the following simple mathematical dependence of the parameter b with the temperature was found:

where d and e are parameters for optimizing.

In this way, Eq. (4) incorporating Eq. (5) can be rewritten:

Table 2 Parameters corresponding to Eq. (4).

T(K)	TR			c
293.1	1.073	0.720	0.843	1.01
298.1	1.091	0.721	0.769	1.00
303.1	1.110	0.719	0.705	0.99
308.1	1.128	0.719	0.650	1.01
313.1	1.146	0.720	0.601	1.01
318.1	1.165	0.721	0.557	0.98
323.1	1.183	0.720	0.519	1.00

Fig. 1 shows the experimental data of the kinematic viscosity of aqueous lactose solutions together with the calculated values with Eqs. (1), (2), and (6). A good agreement between them was found with deviations less than ± 1.9%. In all cases, at fixed temperature, when the solute concentration is increased the kinematic viscosity increases and this property decreases with temperature (the same behavior is shown by the systems studied). With respect to aqueous solutions of sodium chloride, analyzing data in the same way as it was explained for lactose solutions, this expression was obtained:[6]

Figure 1 Experimental and calculated values [Eqs. (1), (2), and (6)] of kinematic viscosity of lactose aqueous solutions at several temperatures and concentrations.

The Eqs. (6) and (7) have the same mathematical form, and for this reason, this type of equation is employed to fit the kinematic viscosity of aqueous solutions with sucrose, glucose and glycerol, cited previously. The parameters for each system are shown in Table 3. The deviations are in all cases less than ± 1.9%.

Table 3 Parameters of the model for binary systems [Eqs. (4) and (5)].

		c	d (m1 )^b	e(m1 )^b
Glycerol	0.125	0.22	2.291	−1.000
Glucose	0.325	0.89	3.730	−1.689
Sucrose	0.762	1.12	2.430	−1.001
Lactose	0.720	1.00	1.768	−1.000
NaCl	0.014	0.20	−0.127	0.909

The experimental kinematic viscosity data for ternary solutions of lactose and sodium chloride were adjusted using an equation that involves the previous expressions for binary systems and an additional term where the interaction of both solutes is included. The equation has the following form:

For these systems it is necessary to obtain 3 new parameters, f, g, and h, (the last two are practically constant with temperature and a mean value is used) and f shows a dependence with temperature that can be modelled as:

Taking into account the previous considerations, the next equation is obtained after optimization of the values of parameters:

This equation allows reproducing the experimental kinematic viscosity data of ternary solutions of lactose and sodium chloride with high accuracy (deviations less than ± 2.5%) using parameters shown in Tables 3 and 4. Figures 2 and 3 show experimental and calculated data with Eq. (10). The same treatment of experimental data for the other osmotic solutions was carried out and the parameters for each studied ternary system are collected in the Table 4 obtaining satisfactory fittings for all of them (maximum deviations of 3.0%).

Table 4 Parameters of the model for ternary systems [Eqs. (8) and (9)].

	j(mi)^−2	k	g 1(m1)^−1
Glycerol	0.028	−0.866	0.076
Glucose	0.261	−1.644	0.241
Sucrose	0.851	−2.130	0.600
Lactose	0.117	−1.000	0.256
NaCl			g 2 (m2)^−1
Glycerol			0.136
Glucose			0.072
Sucrose			0.073
Lactose			0.235

Figure 2 Experimental data and calculated values [Eqs. (2), (6), (7), and (10)] of kinematic viscosity of sodium chloride solutions 4.0 molal at different concentrations of lactose and temperatures.

Figure 3 Experimental and calculated [Eqs. (2), (6), (7), and (10)] values of kinematic viscosity of aqueous solutions with different concentrations of sodium chloride and lactose at 25°C.

CONCLUSIONS

Experimental data of kinematic viscosity of binary and ternary solutions with lactose and sodium chloride were determined showing a typical behavior, as it is an increase with the solute concentration and a decrease with the temperature. Proposed equations for binary and ternary systems were tested with several solutions of interest in osmotic dehydration that allow to estimate the kinematic viscosity in the working typical range as a function of concentration and temperature of solution. Equations for ternary systems involve the parameters of the corresponding binary systems and with an additional term are successfully able to fit experimental data (with maximum deviations of 3.0%). In this way, a mere equation form can predict the kinematic values of typical osmotic media.

ACKNOWLEDGMENTS

The authors acknowledge to the MCYT and the FEDER the partial support with the project (AGL2002–00255).