4400
Ind. Eng. Chem. Res. 2000, 39, 4400-4407
Self-Diffusion of Sucrose in Molasses Martin Rampp,† Christoph Buttersack,*,‡ and Hans-Dietrich Lu 1 demann† Institut fu¨ r Biophysik und physikalische Biochemie Universita¨ t Regensburg, D-93040 Regensburg, Germany, and Institut fu¨ r Technologie der Kohlenhydrate e. V., D-38036 Braunschweig, Germany
The self-diffusion coefficient of sucrose in molasses, Dmol, has been determined via the pulsedfield-gradient 13C NMR technique at 25, 50, 80, and 100 °C and a dry matter content of up to 80% w/w. For this purpose 10% sucrose containing 99% 13C nuclei in the 1 position of the glucose moiety was added. A comparison of Dmol with Dpure of pure aqueous solutions showed Dmol < Dpure at the same sucrose water ratio. Viscosities of the molasses, ηmol, and pure sucrose ηpure solutions were measured over a wide range (1 mPa s to 5000 Pa s) and were shown to obey the empirical formula of Ge´notelle. The friction coefficient was found to be influenced by the nonsucrose substances of the molasses, the value of the product Dmolηmol being greater than that of Dpureηpure at the same sucrose/water ratio, with the deviation increasing with decreasing temperature and increasing concentration. The results have significant consequences for the modeling of the technical crystallization process. Introduction
Table 1. Composition of the Original Molasses (pH ) 8.84)
Industrial sugar production from beet or cane ends with the crystallization of the pure substance. The remaining highly viscous material (molasses) contains 50-56% sucrose and 32-40% nonsucrose.1 Energy saving requires a process control based on a mathematical model of crystallization kinetics as a function of sucrose concentration, temperature, and nonsugar substances. The overall mass-transfer coefficient depends on the volume fraction of sucrose crystals, the convective transport, and the diffusion coefficient of sucrose and water defining the transport inside the boundary layer surrounding the growing crystals. Additional steps comprise adsorption of sucrose on the sucrose crystal, surface diffusion, and reaction (insertion into the lattice).2-4 Experimentally available quantities include the concentration of sucrose in solution and, with a knowledge of sucrose solubility, the degree of oversaturation as the driving force for the whole process, the extent of which is determined by the actual mass of the crystals. The only access to transport limitation in solution is provided by the diffusion coefficient. All other parameters, such as surface diffusion, have to be calculated. The measurement of the self-diffusion of sucrose in pure aqueous solutions has been performed using tracer techniques (radioactive 14C) up to 75% w/w at 60 °C.5 New measurements via the pulse-field-gradient NMR technique based on the proton signals of the sucrose dissolved in D2O6 or the 13C signal7 yielded results in a much broader range of temperatures. However, reliable diffusion data concerning the real liquor in which the crystallization occurs is lacking. Previous measurements of sucrose diffusion in molasses by 14C tracer methods have yielded inaccurate, irreproducible results.8,9 * Corresponding author’s address: Dr. Christoph Buttersack, Institut fu¨r Technologie der Kohlenhydrate e. V., D-38036 Braunschweig, Germany. Phone: +49 531 38009 30. Fax: +49 531 38009 88. E-mail:
[email protected]. † Institut fu ¨ r Biophysik und physikalische Biochemie Universita¨t Regensburg. ‡ Institut fu ¨ r Technologie der Kohlenhydrate e. V.
substance dry mattera sucroseb nonsucrose raffinose glucose organic acids L-lactic acid citric acid malic acid oxalic acid total N betaine pyrolidone carbonic acid R-amino acids L-glutamic acid ash Na K Ca Mg nitrate sulfate
content (g/kg) 880 488 400 2.0 1.06 11.1 2.3 4.0 0.23 25.7 40.3 43.3 6.36 3.35 118 6.6 40.0 4.8 0.16 9.8 2.6
a Measured by freeze-drying and by Karl Fischer titration of the original molasses. b Measured by polarimetry12 (λ ) 589 nm, [R]20 ) 66.4).
Experimental Section 1. Viscosity. Original beet sugar molasses with a dry matter content of 88.0% and a sucrose content of 48.84% (detailed analysis in Table 1) was brought to a quotient of sucrose to dry substance (purity) of q ) 0.596 by addition of sucrose (8.80 g per 100 g). This molasses was diluted to different extents by heating the mixture under shaking in a closed vessel to 40 °C. Supersaturated pure sucrose solutions were prepared by water removal in a rotary evaporator at temperatures between 25 and 35 °C. The actual water content of the samples was determined by Karl Fischer titration in methanol solvent (METROHM, Switzerland, model 758 KDF titrino). Viscosities were determined with a rotational viscosimeter (model Rotovisco RV2, HAAKE, Germany) at
10.1021/ie000266e CCC: $19.00 © 2000 American Chemical Society Published on Web 09/28/2000
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4401
several speeds. By measuring the temperature inside the split filled with the sugar solution ((0.2 K) and by controlling its concentration before and after the measurement, the viscosities obtained are judged to be reliable to (0.5%. To exclude loss of water by evaporation, the temperature was kept to ϑ < 60 °C. 2. Self-Diffusion. The original molasses (Table 1) was freeze-dried. 13C-enriched sucrose, purchased from CAMPRO (Veenendaal, The Netherlands), with the 13C content at the C1 position of the glucose moiety being 99%, was used. To 100 parts of the dried molasses was added 10 parts of 13C-enriched sucrose, yielding a dry molasses composed of 59.55% sucrose and 40.45% nonsucrose. Aqueous solutions were prepared by weighing and were degassed by freeze-pump-thaw cycles. NMR diffusion measurements were performed with a Bruker MSL 300 spectrometer operating at a magnetic field strength of 7.05 T. The self-diffusion coefficient of sucrose was determined from the 13C NMR spectra in a Hahn spin-echo experiment by application of a pulsed field gradient (PGSE technique).10 For this pulse sequence, the amplitude of the echo A after a waiting period τ between the 90° and the 180° pulses is given by
molasses investigated. Because it is impossible to separate the influences of the individual compounds on the diffusion coefficients of sucrose, the molasses is only characterized by the overall fraction of nonsucrose substances and the viscosity. An empirical set of equations describing the viscosity of molasses has been proposed by Ge´notelle.13 It is widely used in the sugar industry. In the case of pure aqueous solutions, the viscosity η can be expressed as a function of the mole fraction of sucrose xs according to
log[η/(mPa s)] ) a1 + a2xs + φϑ(b1 + b2xsn)
(3)
with the parameters
a1 ) 3.11
(4)
a2 ) 22.4
(5)
b1 ) 1.10
(6)
b2 ) 43.1
(7)
n ) 1.25
(8)
and a temperature function
2 2 2
A(2τ) ) A(0) exp(-2τ/T2) exp{-γ δ g D (∆ - δ/3)} (1)
φϑ ) (30 - ϑ)/(91 - ϑ)
with T2 the spin-spin relaxation time of the nucleus under study, γ its magnetoric ratio, and δ the length of the field gradient pulse. The pulse strength is given by
A molasses sample is typically characterized by the quotient of sucrose to dry substance (purity) q. To use these equations for the calculation of the viscosity of molasses, Ge´notelle expressed the parameter b2 as a linear function of q
g ) kI
(2)
with k the coil constant and I the pulse current. D is the self-diffusion coefficient to be studied, and ∆ is the time span between two gradient pulses. The value of D was obtained from decoupled 13C NMR spectra, which demanded an accumulation of up to 4000 free induction decays. It proved most reliable to determine D from a series of 10-12 spin-echoes at increasing coil currents I with all other parameters in eq 1 kept constant. The gradient coil constant k was calibrated with water at 293 K,11 the calibration being controlled with a second substance (cyclooctane). In the cold sugar solutions, the self-diffusion coefficients become very small, and an actively shielded set of gradient coils had to be used in order to eliminate interferences from image currents. The temperature of the home-built probe head was controlled with a metal sheathed chromel/alumel thermocouple. The actual temperatures of the samples are considered reliable to (1 K. The diffusion coefficients obtained were judged to be reliable to (5%; they were reproducible to (2%. 3. Solubility. The characterization of the original molasses by its sucrose solubility was performed for four dilutions (plus 0, 10, 30, and 40 parts of water to 100 parts molasses). Sucrose was added, and the solutions were heated in closed vessels at 70 °C for 24 h. The remaining crystals were filtered off by suction, and the dissolved sucrose was measured by polarimetry (λ ) 589, [R]20 ) 66.412). Results and Discussion 1. Viscosity. Samples of sugar beet molasses vary because of the plant and the crystallization process. Table 1 shows the complex composition of the original
b2 ) (0.85-0.15q)
(9)
(10)
and replaced xs by an apparent mole fraction
xs* ) wds*/(Ms/Mw - Mwwds)
(11)
wds* ) wds[kη + (1 - kη)q]
(12)
where
is the apparent dry substance mass fraction and kη is an empirical constant expressing the specific influence of the nonsucrose components on the viscosity, which depends on the chemical composition of the nonsucrose fraction. New viscosity data for pure aqueous sucrose solutions, especially for the low-temperature region, are compiled in Table 2. As expected, also at high concentrations and low temperatures, no dependence on the shearing forces was observed, although Christoph et al.14 recently reported that highly concentrated sucrose solutions should exhibit non-Newtonian behavior. Not only did the measured viscosities fit Ge´notelle’s formula, but so did the set of data found in a handbook for sugar technologists15 (Figure 1). This is remarkable because those data and the formulas of Ge´notelle were previously experimentally determined only within a region of ϑ ) 40-80 °C. The temperature dependence of these data according to the VTF equation and the extrapolation of the absolute glass transition temperatures is discussed in another publication of the authors.7 With the accuracy of Ge´notelle’s equation having been demonstrated, viscosities of molasses, compiled in Table 3, are fit with that system of equations, yielding the factor kη, which reflects the chemical composition of the
4402
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000
Table 2. Dynamic Viscosity of Pure Aqueous Sucrose Solutions 10% w/w
30% w/w
ϑ (°C)
η (Pa s)
ϑ (°C)
-2.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0
2.75 2.59 2.19 1.88 1.63 1.45 1.32 1.18
-2.0 0.0 10.0 20.0
45% w/w
η ϑ η (Pa s) (°C) (Pa s) 7.26 6.57 4.46 3.21
0.0 5.0 10.0 15.0 20.0
22.7 16.0 13.5 10.5 8.7
50% w/w ϑ (°C)
η (Pa s)
-5.5 -2.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0
60.7 48.0 41.7 30.5 23.5 17.5 13.8 11.0 8.9
60% w/w ϑ (°C)
65% w/w
η (Pa s)
ϑ (°C)
η (Pa s)
68% w/w ϑ (°C)
η (Pa s)
76% w/w ϑ (°C)
80.1% w/w
η (Pa s)
ϑ (°C)
-10.0 551 -16.3 5 042 -17.3 18 995 -14.1 434 0.0 221 -14.4 4 024 -14.7 12 534 -9.7 189 10.0 109 -10.1 2 319 -9.8 5 478 -5.1 83.7 20.0 55.5 -4.9 1 314 -0.2 2 040 0.3 36.9 30.0 33.5 0.0 813 5.4 1 175 5.1 18.1 40.0 24.2 10.1 338 15.7 442 10.3 9.63 50.0 16.6 20.0 151 30.4 138 15.5 4.95 60.0 11.4 30.0 111 20.6 2.75 30.0 1.04 38.7 0.478
η (Pa s)
-9.8 4 506 -6.2 1 777 -4.6 1 323 -2.6 845 0.4 485 5.3 192 10.6 79.8 15.5 38.5 20.6 18.4 25.5 10.0 34.9 3.74
Table 3. Viscosities of the Molasses original molasses (q ) 0.554) wds ) 0.88
plus 10% sucrose (q ) 0.60) wds ) 30% w/w
wds ) 50% w/w
wds ) 60% w/w
wds ) 70% w/w
wds ) 76.6% w/w
wds ) 82.4% w/w
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
ϑ (°C)
η (Pa s)
40.0 50.0 60.0
29.3 10.8 4.40
0.0 10.0 20.0 30.0
5.99 4.18 3.02 2.31
10.0 20.0 30.0 40.0 50.0
19.8 12.3 8.30 6.10 4.70
0.0 10.0 20.0 30.0 40.0 50.0
152 75.7 41.8 24.8 16.5 12.0
0.0 10.0 20.0 30.0 40.0 50.0
2 894 654 272 132 74.0 47.9
-9.7 0.0 10.0 20.0 30.0 40.0 50.0
104 26.8 7.44 2.26 0.866 0.369 0.213
0.0 10.0 20.0 30.0 40.0 50.0
667 129 29.8 8.26 2.99 1.21
Figure 1. Viscosity of pure sucrose solutions. This work (b), published data15,16 (0), pure water19 (O), calculated curve due to Ge´notelle.13
nonsucrose fraction (Figure 2). Using the data between ϑ ) 40 and 80 °C, a value of
kη ) 0.941 ( 0.002 was found. At a given sucrose/water ratio, the viscosity of the molasses sample is always greater than that of the pure solution because nearly all substances found in the molasses, especially alkaline earth salts, contribute to an increase in the flow resistance.17,18 As is evident from eq 10, the specific contribution of the nonsucrose fraction is smaller than that of sucrose itself. An analogous behavior is observed for the activation energy (Figure 3). The activation energy increases linearly with the sucrose/water ratio. Remarkable is the deviation of the
Figure 2. Viscosity of molasses (q ) 0.60). This work (O), fitted curve due to Ge´notelle.13
value of neat water. Obviously, the structure of water is altered by the addition of small amounts of sucrose. 2. Self-Diffusion. The application of the pulsed-fieldgradient NMR technique to the measurement of the diffusion of sucrose in a complex multicomponent mixture includes the difficulty of finding a well-separated signal for this molecule in the NMR spectra. A measurement based on the 1H spectrum of sucrose or water fails because of multiple overlap in the 1H signals, exchange with other hydroxyl- and amino-group-containing nonsucrose substances, and the known influence of ions, which was indicated by small relaxation times (T2 < 10 ms). Therefore, the13C nuclei were used. To enhance the sensitivity based on the natural 13C content of 1.1% of all C atoms, the original mass fraction of 55.5% sucrose of the dry matter was increased to 59.6% by the addition of 13C-enriched sucrose. The location of
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4403 Table 4. Self-Diffusion Coefficients D of Sucrose in Diluted Molasses D (10-10 m2 s-1)
wds (%w/w)
ws (%w/w)
nw/ns
25 °C
50 °C
80 °C
100 °C
Do (10-6 m2 s-1)
EA (kJ mol-1)
30 40 50 60 70 80
18 24 30 36 42 48
73.89 47.50 31.67 21.11 13.57 7.92
2.06 1.25 0.715 0.289 0.110 n.d.
4.16 2.68 1.69 0.725 0.293 0.098
7.68 5.00 3.24 1.63 0.85 0.25
14.2 10.0 6.40 3.60 1.60 0.60
3.25 3.25 3.25 3.25 3.25 3.25
24.1 25.3 26.7 28.8 31.1 34.3
Figure 3. Activation energy of the viscosity of water19 (b), of pure aqueous sucrose solution (O), and of molasses (q ) 0.60) (0) at 50 °C. Table 5. Self-Diffusion Coefficients of Sucrose in Pure Aqueous Solutions Obtained with 13C NMR and Comparison with Data from Literature (1HNMR,6 14C-tracer5)a D (10-10 m2 s-1) ws (% w/w) 0.004 2 5 10 15 20 30 40 50 60 70 a
25 °C 14C
1H
50 °C 13
C
6.14 5.08 4.32 3.75 3.24 2.26 1.28 0.75 0.34
14C
1H
11.00 6.02 5.66 4.63 3.02 1.81 1.07 0.59 0.57 0.20
9.15 7.81 7.71 6.12 4.30 2.75 1.55 0.71
80 °C 13C
14C
1H
13C
18.6 15.8 13.8 13.1 13.4 5.26 10.8 8.70 3.68 7.39 7.28 2.40 5.14 5.00 1.47 1.44 2.95 3.31 3.22 0.63 1.51 1.56 0.23 1.00
Figure 4. Self-diffusion of sucrose in pure aqueous solution. Comparison of different methods. This study: 13C method (b). Literature: 1H NMR in D2O6 (O), 14C-tracer technique5 (0). Table 6. Self-Diffusion Coefficients of Sucrose in Molasses (q ) 0.60), Comparison with Previous Measurements ϑ (°C)
a
b
30
40 60 40 60 40 60 40 60
3.10 5.41 1.14 2.12 0.51 0.99 0.21 0.43
3.13 4.64 1.03 1.85
50
7.81
Values as measured or empirically interpolated.
the 13C nuclei at only one position (C1 of the glucose moiety) yielded an additional improvement in detection. The experimental self-diffusion coefficients of sucrose in molasses are comlied in Table 4. These values should be compared with those known for the pure aqueous sucrose solution. Table 5 shows the self-diffusion coefficients taken from three types of measurements. Schneider et al.5 monitored the concentration of radioactive 14C tracer molecules diffusing from a 14C-enriched layer into an originally unmarked layer. Compared to the alternative method based on a diaphragm separating two liquid-filled cells,20,21 the direct contact5 yields absolute self-diffusion values. The detection of sucrose via 1H NMR requires the replacement of solute water with D2O, coupled with the deuterium exchange of the eight hydroxyl protons of the sucrose itself when the PGSE method is applied.6 This leads to a somewhat
D (10-10 m2 s-1)
wds (% w/w)
60 70
c
1.45 2.78 0.14 0.36
a This study calculated with D b o and EA. Data used by Schliephake and Ekelhof.9 c Published by Emmerich et al.8
reduced molar volume of sucrose-d8-D2O compared with sucrose-H2O.22 Nevertheless, extensive studies of neat water and ionic aqueous solutions demonstrated that the dynamic isotope effect observed when H2O was replaced by D2O is quantitatively equivalent to an increase in the temperature of 5 K.23-25 This behavior has been assumed to be valid also for the transformation of sucrose-d8-D2O to sucrose-H2O.6 To verify this assumption, a 50% sucrose-H2O solution was measured via the new 13C technique. As shown in Figure 4, both methods show a good agreement. The data from the 14C tracer technique, however, are in fair accordance with those from the NMR method only at low concentrations and high temperatures. Some data obtained by the 14C technique are also available for diluted molasses. A corresponding compilation in Table 6 shows remarkable aggrement with those data reported by Schliephake and Ekelhof9 at low
4404
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000
Figure 5. Self-diffusion of sucrose in molasses (q ) 0.60) (O) compared with that in pure aqueous solution6 (- - -) for dry substances of 30-80% w/w.
Figure 7. Self-diffusion of sucrose in pure aqueous solution. This study: 13C NMR (b). Literature: 1H NMR in D2O6 (O), 14C-tracer technique5 (0). Self-diffusion of sucrose in molasses (q ) 0.60) (2) at 25 °C. Calculation of Dmol with the assumption Dmolηmol ) Dpureηpure according to Schliephake and Ekelhof9 (- - -). For the assumed course (s), the last four values of 0 were neglegted.
Figure 6. Activation energy of the self-diffusion of sucrose for pure aqueous solution (O) and molasses (q ) 0.60) (0) at 50 °C.
temperatures and low concentrations, the data of Emmerich et al.8 being out of range. The Arrhenius plot (Figure 5) shows that the diffusion of sucrose is on the same order of magnitude for equal mass fractions of both the molasses and the pure solution. A more detailed comparison using the obtained activation energies (Figure 6) reveals a basic difference compared to the behavior of the viscosities (Figure 3). At low sucrose/water ratios, the translation of sucrose molecules seems to be obstructed by nonsucrose molecules. However, at higher concentrations, the translation may by facilitated by a structure-breaking effect of the molasses constituents toward the formation of a diffusion-limiting sucrose-sucrose network. Such an explanation has been proposed for the analogous behavior of solubility. At low nonsucrose/water ratios, the solubility of sucrose passes through a minimum because of the attraction of water by these substances (salting out), whereas increased nonsucrose/water ratios enhance the solubility because of the direct nonsucrosesucrose interactions coupled with a size reduction of sucrose-sucrose clusters.26,27 The decrease in the diffusion coefficient for both the pure solution and the molasses as a function of the
Figure 8. Self-diffusion of sucrose in pure aqueous solution and molasses (q ) 0.60) at 50 °C. For symbols and calculated curve (- - -), see Figure 7.
sucrose-water ratio is plotted in Figure 7 (25 °C), Figure 8 (50 °C), and Figure 9 (80 °C). The concentration dependence of the sucrose self-diffusion coefficients in the concentration region where the crystallization occurs is important for the mathematical modeling of the process. The sucrose/water ratio of saturated molasses is usually calculated by multiplying the corresponding known value of pure solutions28 with a saturation coefficient29,30
ysat ) a(wns/ww)+ b
(13)
which is independent of temperature. The parameters for the molasses investigated in this study were found to be
a ) 0.183 and
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4405
Figure 9. Self-diffusion of sucrose in pure aqueous solution and molasses (q ) 0.60) at 80 °C. For symbols and calculated curve (- - -), see Figure 7. Table 7. Sucrose/Water Ratio (ns/nw) at Saturation Conditions in Pure Aqueous Solution and in the Molasses Investigated (q ) 0.60)' ϑ (°C) purea molasses a
ns/nw (mol/mol) D (10-10m2s-1) ns/nw (mol/mol) D (10-10m2s-1)
0
25
50
80
100
0.09
0.11 0.08 0.12 0.04
0.14 0.2 0.16 0.05
0.19
0.25
0.25
0.36
0.1
Values taken from Bubnik et al.28
b ) 0.859 Calculated solubilities are compiled in Table 7. A model established by Ekelhof and Schliephake3,9 estimated the diffusion coefficient for highly concentrated molasses solutions by using the assumption that the Stokes-Einstein law is only influenced by the content of water. Thus
ηpureDpure ) ηmolDmol
(14)
has been assumed, and the diffusion of sucrose in molasses, Dmol, has been derived from the known data for the viscosities and the self-diffusion coefficients available from the 14C experiments with pure solutions.5 However, at 25 °C, this assumption fails considerably. A rough extrapolation to the sugar/water ratio of saturated molasses (Table 7) yields a deviation of about 1 order of magnitude compared with the new experimentally founded course. Diffusion coefficients extrapolated from the experimental data to saturation conditions of the pure solution and molasses are given in Table 7. On one hand, the difference between the diffusion in pure and impure solutions is reduced with increasing temperature at a given water content; on the other hand, this behavior is accompanied by a simultaneous increase in the sucrose solubilities. After all, this leads to an unacceptably long range of extrapolation for both the pure and impure solutions, especially for higher temperatures. The experimental equipment and the efforts undertaken in this study are, therefore, insufficient for a definitive determination of self-diffusion coefficients for concentrations of saturated molasses at technical relevant crystallization temperatures, not to mention typical super-
Figure 10. Friction coefficient of sucrose for pure aqueous sucrose solution (s) and molasses (q ) 0.60) (- -) at 25, 50, and 80 °C. For better representation the coefficient for 80 °C is multiplied by 0.5.
saturation coefficients in the range of 1.1-1.3.2 The detection of lower diffusion coefficients requires shorter and larger magnetic field gradients. Another approach would be supported by a theoretically founded extrapolation, which will be presented by the authors in a separate publication. Nevertheless, the analysis of the experimental data can be summarized in the fundamental finding that the Stokes-Einstein law depends on the presence of (mainly ionic) nonsucrose substances present in molasses. The Stokes-Einstein law
D ) kT/(6πηr)
(15)
is theoretically derived for a hard-sphere molecule moving inside a continuous phase, k being the Boltzmann constant and r the radius of the diffusing solute molecule. The continuous phase is approximated by solvent molecules significantly smaller than the molecules of the solute. At higher concentrations, this condition does not hold, and k is efficiently interpreted as a friction coefficient between the spherical molecule with radius r and the surrounding medium.31 According to Figure 10, the friction coefficient is found to be influenced by the nonsucrose substances in the molasses, with the value of Dmolηmol being greater than Dpureηpure at the same sucrose water ratio and the deviation increasing with decreasing temperature and increasing concentration. According to the hypothesis already mentioned, direct nonsucrose-sucrose interactions cause a reduction in size of sucrose-sucrose clusters.26 The overall effect of increasing the friction forces by the nonsucrose material should, therefore, be the result of a superposition of an enhancement of friction by the nonsucrose-sucrose interaction and a reduction of friction by the loosening of the sucrose network. Conclusion The modeling of the industrial crystallization process of sucrose requires the knowledge of the diffusion coefficient of sucrose and water in the multicomponent liquid from which the crystallization occurs. The NMRPGSE method has been shown to be a versatile tool for
4406
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000
the measurement of the diffusion of a single component in such a complex mixture. A measurement based on the 1H spectrum of sucrose or water fails because of multiple overlap in the 1H signals and the exchange with other hydroxyl- and amino-group-containing nonsucrose substances. However, in the case of sucrose, the 13C nuclei can be used, provided that some 13C-enriched sucrose is added to the multicomponent mixture. The lowest diffusion coefficients that could be detected by this method were situated around 10-11 m2 s-1; lower values are assumed to be detectable by the application of shorter and larger magnetic field gradients. The self-diffusion coefficient, Dmol, of sucrose in molasses, the dry mass of which contained 60% sucrose and 40% nonsucrose, was determined for a dry matter content ranging from 30-80% and temperatures between 25 and 100 °C. A comparison of these data with the self-diffusion coefficient, Dpure, of aqueous solutions taken from literature yields Dmol < Dpure at the same sucrose/water ratio, the difference increasing with increasing sucrose/water ratio. Obviously, the nonsucrose material exerts an obstruction effect. Known data for the viscosity values ηpure and ηmol, respectively, for the solutions mentioned above were extended to lower temperatures and higher concentrations by new measurements up to 5000 Pa s and found to obey the empirical formula of Ge´notelle. The result that the product Dpureηpure increases with increasing concentration and falling temperature indicates an increased friction toward the diffusing sucrose molecules, obviously by the formation of a sucrosesucrose network. The increase in Dmolηmol with increasing sucrose/water ratio was found to be greater than that in Dpureηpure, which is probably due to a strong sucrose-nonsucrose interaction. Acknowledgment Dr. M. Rampp was supported within a cooperation AFO project of the Zuckerverbund Magdeburg, Germany. Symbols D ) self-diffusion coefficient Do ) self-diffusion coefficient extrapolated for infinite temperature (Arrhenius or VTF equation) EA ) activation energy T ) absolute temperature M ) molecular weight w ) mass fraction n ) mole number q ) purity (mass of sucrose per dry substance) y ) saturation coefficient Greek Letters η ) dynamic viscosity ϑ ) temperature °C Indices ds ) dry substance s ) sucrose ns ) nonsucrose w ) water pure ) pure aqueous solution mol ) molasses sat ) saturation
Literature Cited (1) Higginbotham, J. D.; McCarthy, J. Quality and storage of molasses. In Sugar Technology; van der Poel, P. W., Schiweck, H., Schwartz, T., Eds.; Bartens: Berlin, 1998, 973-992. (2) Mantovani, M.; Vaccari, G. CrystallizationsTheoretical principles. In Sugar Technology; van der Poel, P. W., Schiweck, H., Schwartz, T., Eds.; Bartens: Berlin, 1998, 649-675. (3) Ekelhof, B.; Schliephake, D. Beschreibung der Kristallwachstumsgeschwindigkeit in reinen und unreinen Saccharoselo¨sungen durch einen gemeinsamen kinetischen Ansatz. Proceedings of the 20th General Assembly of C.I.T.S. (Commision Internationals Technique de Sucrerie); Mu¨nchen, Germany, 1995; pp 216-235. (4) Gilmer, G. H.; Ghez, R.; Cabrera, N. An analysis of combined surface and volume diffusion processes in crystal growth. J. Cryst. Growth 1981, 8, 79-93. (5) Schneider, F.; Emmerich, A.; Finke, D.; Panitz, N. Untersuchungen u¨ber die Diffusion in reinen und technischen Saccharoselo¨sungen. I. Die Diffusion von Saccharose in reinen wa¨ssrigen Lo¨sungen. Zucker 1976, 29, 222-229. (6) Girlich, D.; Lu¨demann, H.-D.; Buttersack, C.; Buchholz, K. c,T-Dependence of the self-diffusion in concentrated aqueous sucrose solutions. Z. Naturforsch. 1994, 49c, 258-264. (7) Rampp, M.; Buttersack, C.; Lu¨demann, H.-D. c,T-dependence of the viscosity and the self-diffusion of some aqueous carbohydrate solutions. Carbohydr. Res. 2000, 328, 561-572. (8) Emmerich, A.; Finke, D.; Panitz, N.; Rieck, H. Untersuchungen u¨ber die Diffusion in reinen und technischen Saccharoselo¨sungen. II. Die Diffusion von Saccharose und Nichtsaccharosestoffen in Mehrstoffsystemen. Zucker 1976, 29, 302-307. (9) Schliephake, D.; Ekelhof, B. Beitrag zur vollsta¨ndigen Berechnung der Kristallisationsgeschwindigkeit der Saccharose. Zuckerindustrie 1983, 108, 1127-1137. (10) Stejskal, E. O.; Tanner, J. E. Spin diffusion measurements: spin-echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 1965, 42, 288-292. (11) Weinga¨rtner, H. Self-diffusion in liquid water. A reassessment. Z. Phys. Chem. (Frankfurt) 1982, 132, 129-149. (12) Bubnik, Z.; Kadlec, P.; Urban, D.; Bruhns, M. Sugar technologists manual; Verlag Bartens: Berlin, 1995; p 166. (13) Ge´notelle, J. Expression de la viscosite´ des solutions sucre´es. Ind. Aliment. Agric. 1978, 95, 747-755 (14) Christoph, D.; Schmidt, T.; Senge, B. Fliesseigenschaften von reinen und technischen Saccharoselo¨sungen im Temperaturbereich von 30 bis 130 °C. Zuckerindustrie 1998, 123, 876-882. (15) Reference 12, p 146. (16) Schneider, F.; Schliephake, D.; Klimmek, A. U ¨ ber die Viskosita¨t von reinen Saccharoselo¨sungen. Zucker 1963, 16, 465473. (17) Mathlouthi, M.; Ge´notelle, J., Rheological properties of sucrose solutions and suspensions. In SucrosesProperties and Applications; Mathlouthi, M., Reiser, P., Eds.; Blackie Academic: London, 1995; pp 126-154. (18) Bohuon, P.; Le Maguer, M.; Raoult-Wack, A. L. Densities and viscosities of ternary systems of NaCl-sucrose-water from 283 to 303 K. J. Chem. Eng. Data 1997, 42, 266-269. (19) Blanke, W., Ed. Thermophysikalische Stoffgro¨ ssen; Springer: Berlin, 1989. (20) Irani, R. R.; Adamson, A. W. Transport processes in binary liquid systems. Diffusion in the sucrose-water system at 25 °C. J. Phys. Chem. 1958, 62, 1517-1521. (21) Tilley, J. F.; Mills, R. Relations between mutual and intradiffusion coefficients in aqueous sucrose solutions. J. Phys. Chem. 1967, 71, 2756-2757. (22) Bernal, P. J.; van Hook, W. A. Apparent molar volumes, isobaric expansion coefficients and isentropic compressibilities, and the hydrogen/deuterium isotope effects for some aqueous carbohydrate solutions. J. Chem. Thermodyn. 1986, 18, 955-968. (23) Mills, R. J. Self-diffusion in normal and heavy water in the range 1-45 °C. J. Phys. Chem. 1973, 77, 685-688. (24) Lang, E. W.; Lu¨demann, H.-D. NMR Basic Princ. Prog. 1990, 24, 129-187. (25) Prielmeyer, F. X.; Lang, E. W.; Speedy, R. J.; Lu¨demann, H. D. The pressure dependence of self-diffusion in supercooled light and heavy water. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1111. (26) Schliephake, D. U ¨ ber die Strukturwa¨ssriger Saccharoselo¨sungen. Zucker 1963, 16, 523-528.
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4407 (27) Schneider, F.; Reinefeld, E.; Amding, F. U ¨ ber den Einfluss der Nichtzuckerfraktion auf die Lo¨slichkeit der Saccharose. Zucker, Zucker-Beihefte 1962, 4, 55-78. (28) Reference 12, p 124. (29) Wagnerowski, K.; Dabrowska, D.; Dabrowski, C. Melasseerscho¨pfung, Z. Zuckerindustrie 1962, 12, 662-671. (30) Wicklund, O. Moleku¨lverbindungen zwischen Saccharose und Salzen. Zucker 1955, 8, 266-277.
(31) Heyes, D. M. The liquid state, Applications of molecular simulations; Wiley: Chichester, U.K., 1998; p 116.
Received for review February 25, 2000 Revised manuscript received June 16, 2000 Accepted August 8, 2000 IE000266E
