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A dynamic model of physical processes in chromatographic glucose—fructose separation
Chem~rnl
Printed
Engmeering Science, in Great Bntam.
Vol. 46, No
0309~2509/91 53.00 + 0.00 0 1991 Pergamon Press plc
4, pp 959 965, 1991.
A DYNAMIC MODEL OF PHYSICAL PROCESSES IN CHROMATOGRAPHIC GLUCOSE-FRUCTOSE SEPARATION M. LUKAe’and LCMva-State
Enterprise,
Z. PEgINA
Doni Mecholupy 130, 109 02, Prague, and Research Biochemistry, Prague, Czechoslovakia
(First received
31 May
1989;
accepted
in
reuisedform
Institute
8 May
for Pharmacy
and
1990)
Abstract-The assumption of diffusion-adsorption phenomena in the chromatographic separation of does not provide, glucose and fructose on columns of Ca 2+-farm sulphone catex (styrentiivinylbenzene) according to the results, the mathematical model corresponding to the actual phenomena in the separation process. Therefore, a model based on the assumption of a stronger ion-type interaction was tested. In this model, the equilibria on the theoretical plate are mutually conditional on all components being present m both the solution and the stationary phase. However, all experiments that model glucose as one component have failed in this model. Only the model assuming that glucose is two chromatographic components corresponds to the obtained experimental results in the whole range. In determining the equilibrium the factual solution of the model leads to an (n + l)th-power equation with n components. Suitable procedures for a numerical solution were verified
INTRODUCTION
250
Considering that chromatographic separation of glucose-fructose mixtures on a resin has become an industrial process in a number of advanced countries of the world, increasing interest has been focused on the definition of the physico-chemical nature of these processes together with production of a corresponding mathematical model. The increasing concentration of glucose-fructose mixtures influences significantly the values of the distribution coefficients of the respective sugars, including mutual operation of the components of the mixture (Barker et al., 1986). A similar procedure is employed in another model (Ching et al., 1987). In another work (Ganetsos, 1987), the mutual dependence of the distribution coefficients is solved by means of nonlinear relations, with coefficients determined from the experiments. Under these conditions a very good correspondence between the model and the experiment was reached. The derived models are empirical and do not consider the mechanism of the distribution process. From the study of the distribution processes of glucose and fructose on a resin in the CaZ+-cycle two conspicuous anomalies result
(see Figs
Fig. 1. Results
of experiment
4580.
l-6):
(1) The output peak of glucose is lower and wider, i.e. its dispersion is significantly greater than the output peak of fructose. (2) In the separation of the mixture the maximum output concentration of fructose is significantly higher (in dependence on the separation column and other conditions) than the feed of the fructose. These results are documented in the experimental part.
0
to whom
correspondence
should
210
/l\
1
260 T,me
Fig. 2. Results
310
360
Inun)
of calculation
4580.
sion and physical adsorption. Therefore we tried to derive a mathematical model, based on a different mechanism of the phenomena, the outputs of which correspond to the experiments. The results of our work (LukS, 1988) are introduced.
The above-mentioned deviations indicate the distinctness of these processes from both the usual diffu+Author
/&J_
160
be addressed. 959
M. LuKAC and 2. PERINA
960
From a group of several hundred experiments the following three examples were chosen as typical. Among them are experiments on columns of different sizes and different loadings of the filling to illustrate the statement given below. The serial numbers of the experiments are given to distinguish the ‘individual experiments and have no other meaning. The conditions of the individual experiments were: No. 4580-D
= 2.3 cm, L = 93 cm, feed = 46 ml (concentration = 350 g/l), rate = 1.17 ml/min.
Tme
= 5.0 cm, L = 160 cm, feed = 675 ml = 350 g/I), rate = 5 ml/ min. No. 555&D = 16.0 cm, L = 280 cm, feed = 8570 ml (concentration = 350 g/l), rate =901 ml/min. No.
(mm)
Fig. 3. Results of experiment
5110-D
(concentration
5110
EXPERIMENTAL
The
tose
Time
coeficients
for glucose
and
fruc-
measured in direct experiments. Fifty mililiters of Zerolit SRC-14.CaZf were mixed with 1COml of glucose or fructose solutions of different definite concentrations at selected temperatures of 20, 40 or 60°C for 15 min. Then the residual concentration was assessed. The initial concentrations of the sugars in the solutions ranged from 50 to 350 g/l, totalling 14 different concentrations. The results of the measurement can be summarized by the relation
imln)
Fig. 4. Results of calculation
distribution
5110.
were
c1 = a + bc,
(1)
in Table 1. The measured results lead to the conclusion that the distribution of the sugars between the solution and the resin is linear in the whole range of measurements and is not dependent on the concentration. The dependence on the temperature is nonlinear and cannot be derived from the measurements involving three temperatures. The constant a (intercept) should in all cases be equal to zero. Its small values (in comparison 0 310
360
410
460
510
Fig. 5. Results of experiment
560
5550.
Table I.
610
Estimations of regression coefficients in the case of distribution
of glucose
between
water
and resin
300 0 Glucose x Fluctose
“C
a
b
r
s
20 40 60
0.587 -0.306 0.377
0.807 0.860 0.844
0.9990 0.9998 0.9990
4.78 i 2.209 4.467
Estimations bution “C
Trne
(mm)
Fig. 6. Results of calculalion
5550.
20 40 60
of regression
of fructose a -0.404 0.663 0.044
between
coefficients water b
0.782 0.771 0.791
in the case of distri-
and resin I
0.9990 0.9990 0.9997
s
4.042 5.132 2.680
Dynamic model of physical processes
with the standard deviation s) are a consequence of experimental errors in the analytical determination. From the estimations of the constant b (slope) we can calculate the distribution coefficients between water and the resin for the individual sugars and individual temperatures by means of the equation
‘=
I-b1 ___ b
f
(the ratio of the content of sugar in the resin and in the liquid after modification multiplied by the ratio of the phases). The estimated values are introduced in Table 2. For the actual
study we had at our disposal the results of several hundred laboratory measurements for the separation of glucose-fructose mixtures on chromatographic columns of different diameters and heights of filling at different temperatures and concentrations, amount of feed and flow rate through the column. The diameters of the columns used in these experiments ranged from 2 to 30 cm with fillings of Zerolit, Ostion, Wofatit and Dowex. The selected temperatures in the experiments ranged from 20 to 70°C. The conditions of these experiments were not selected according to any rational plan. The experiments were evaluated in this way (Levenspiel, 1965): the first two distribution moments of the eluant peaks (the average and the variance) were determined by means of the equations
T xif txi) (3)
p = p, ~xff(xA
d = isf(&)
-
p2.
(4)
The dispersion function of reactors is appropriately characterized by the dimensionless criterion
the so-called reactor dispersion number (Levenspiel, 1965), which is in fact the reciprocal value of Peclet’s criterion for mass transfer. By means of this criterion we can numerically evaluate a series of experiments. It is linked with the dispersion by several relations. From these relations, in this case the relation for a closed system is relevant. 02 = 2D/uL - 2(D/UL)2(1 - .KZ”L’D).
(5)
For the reactor dispersion number this relation is not directly solvable but can be easily calculated by means of iterations. By means of this criterion we can estimate the number of “theoretical plates”: N = uL/(2D). From had at [eqs (3) [eq. (5)]
961
Table 2. Estimations of distribution coefficients P in eq. (2) Temperature
20°C
40°C
60°C
Glucose Fructose Difference (fructose glucose) Ratio (fructose/ glucose
0.648 0.752
0.441 0.803
0.500 0.712
0.104
0.362
0.212
1.160
1.821
1.424
ated the number of theoretical plates [eq. (6)]. In all cases (402 experiments), we estimated a lower number of theoretical plates for glucose than for fructose. Other authors arrived at the same conclusion (Ching and Ruthven, 1986). From the results we could empirically derive the regression equation (in the form of a criterion relation) between the HETP and optional input conditions for fructose in the performed experiments. InHETP
= - 3.854 + 0.438lnL,
+ O.l22InD,
+ 0.280 In Z + 0.763 In u, - 0.177 In m + 0.038gR r = 0.784,
n = 402,
(7)
F = 105.1
The HETP in individual experiments being dependent on the conditions of the experiment, we derived the regression equation for its determinination from the different tested linear and nonlinear equations with a different combination of input conditions. The regression equation shows the predominant influence of the length of filling, the load weight of the cross section and the flow velocity on the HETP. The influence of the diameter of the column (the influence of the walls), of the viscosity (function of the concentration) and of the kind of resin is significantly lower. This equation was selected from a number of other empirically selected equations for its best statistical parameters. On the assumption that we can explain the member (0.763 In u) in the form (0.5 x 0.763 In u’), this equation is after delogarithmization dimensionless. DERIVATION
OF
THE
MATHEMATICAL
MODEL
If the mathematical model should correspond with the real processes in all primary aspects, the relation employed must respect the physico-chemical nature of the processes in progress. The interpretation of the empirical derivation is restricted to a narrow area of the conditions in the experiment. The new mathematical model of this separation process was derived on the assumption that:
(6)
the results of all experiments whose data we our disposal their distribution moments and (4)] and reactor dispersion numbers were sequentially calculated. Then we estim-
(I) The chromatographic bed in the unit volume has a definite accurately defined number of relevant binding positions. (2) The mechanism of retention of the sugars in the
962
M.
LUKM
chromatographic bed is of the same quality for both sugars and only differs in quantity. (3) The sugars influence each other in competition for these binding positions. system glucose appears (4) In the chromatographic as a mixture of two closely related substances. To put it another way, the measured eluant plot of glucose is a superposition of two eluant plots for two chromatographically different components of glucose. (In chromatographic methods, this is not the unknown phenomenon and there are mixtures of components with similar distribution coefficients in the majority of the cases.) The first assumption is based on the structure of the bed of the separation column. The separation effect almost disappears when the sulpho groups are not occupied by Ca, Sr or Ba ions. These ions have a conclusive influence on this process. In this way a definite number of binding positions in the volume unit of the chromatographic bed is given. The next two assumptions are closely connected with the preceding one as its consequences. The latter assumption is based on the results of the performed experiments. We could not demonstrate by any modification any agreement between the experimental results and the outputs of the mathematical model when we tried to devise mathematical models on the assumption of diffusion-adsorption phenomena. The output peak of glucose was always higher and more narrow than the output peak of fructose. Big differences in the shape of the peak between the experiments and the model also occurred. Therefore we tested differences between the course of the curve with the normal distribution and the output peak of the individual sugars. Both curves included the same area and had the same width. The course of the curves was nearly coincident for fructose and considerably different for glucose. We can, however, divide the glucose peak in two mutually overlapping peaks with the half-area and distribution, which is very near to normal. A typical example is given in Fig. 7. The latter assumption calls for three substances instead of two to be incorporated into the mathematical model. Besides, the stationary phase and the pure eluant must also he incorporated into the drawn model. These are complications for the solution; nevertheless, they are not unbridgeable. The preceding assumptions are advantageous for the following reasons: The equilibrium states of any component of the mathematical mode1 are directed by the relation with the sole equilibrium constant depending only on the temperature (dependence on the pressure was not tested) The mutual influence of the components takes place exclusively through the mediation of its activities. The equation in the model contains this relation.
and Z. PE%INA
250 3
t
v “Left” glucose ! “Rqht’ glucose
Time imInI
Fig. 7.
Calculated
distribution of glucose (experiment 51 IO).
(3) The HETP is influenced only by the length and the diameter of the column, by the velocity of the eluant fluid, by the amount and concentration of the feed, and by maintaining plug flow through the column. It is not influenced by the type of substance under separation. All these conditions of the process and the temperature set the task for the mathematical model. By means of the relation derived from the experimental data C(7)] we can determine the HETP in every individual case. The basic model of the ion-exchange process in the chromatographic column can be expressed as follows: [sugar.
H,O]
+ [resin]
= [resin. sugar]
+ [H,O]. (8)
In the shortened
notation
[KY]
this is
+ [R] = [RS] + [HI
(9)
i.e. the sugar is bound either on the solvent in the liquid phase or on the binding places in the stationary phase. For the equilibrium constant the following relation holds:
(10) In this equation the individual components are expressed in molar concentrations. This equation is known as the relation for the operation of active matter. The activities of individua1 components should be applied instead of the molar concentrations. Considering that the relations for distribution coefficients are not dependent on the concentration, we can replace the activities with the concentrations, without the accuracy being lowered. In the case of nonequilibrium in the system, we can simply define the changes necessary for achieving the equilibrium by the relation K
=
s
(CRSI ([RI
+
Cxlj.(CW
-cxl).(pE]
+
Cxl)
_cxjj’
In deriving the model, an important assumption that of plug flow of the eluant through the column,
(11) is i.e.
963
Dynamic model of physical processes all the physical values (concentrations of the sugars, feed velocity, temperature and pressure) in the section perpendicular to the axis of the column are identical. Otherwise a settlement of these values through diffusion and through sharing of the warmth would occur and complicate the mathematical model. On the given assumptions we can write partial differential equations for the process in the column (Himmelblau and Bischoff, 1968):
Equation (12) refers to the phenomena in the solution; eq. (13) refers to the processes in the individual particle on the resin, on the assumption of its ballshaped form. Members on the right side of these equations express the changes in concentrations resulting from diffusion and the changes resulting from reactions. In eq. (12) the diffusion in question is lengthwise in the flow direction; eq. (13) refers to the central or opposite diffusion in the particle of the bed. Considering that the free cross section area of the column is only about 10% of its crosswise section, the used flow velocities are higher than the diffusion velocities and the particles have very small diameters (0.1-0.3 mm), the diffusion in the model can be neglected. According to the law of the conservation of mass the changes in C, are equal in both equations, except for the Signum. After this simplification of the numerical method of the solution the differential equations are replaced by the difference equations (a solution by the method of nets) AS dt+
h(u,S)=C AL
s
AS 2 = Csr = - C, At We usually split the solution into the shift between the individual plates and the subsequent calculation of the equilibrium, the length of the shift being the HETP. With the continuity of the flow, the shift will proceed in all “theoretical plates” at one time, while the next feed or eluant enters the highest plate and the same volume of liquid leaves the lowest plate. The equilibrium on the individual plates is estimated according to a modified relation (14) for the individual components. (CH&,l
- x)&r,
= (CR&J
+ x)(CHl
+ x + y + z)/([R] (Cf&l]
- Y)JG,,
= (C&J
- z)G
= (CBS,1 +
(16)
+ Y)(CHl
+ x + Y + Z)/([R] (C=,l
- x - y - z)
-x
- y - 2)
(17)
z)(CHI
+ x + y + z)/([R] - x - y - z) (18)
and then introducing
into the balance
equations:
[HI, = [H-J,+ 10.9866(x + y + z) [RI,
= CR], - (x + Y + z)-
(19)
The solution of the set (16)-(19) leads to a set of ihree equations for the unknown x, y and z, and then to one fourth-power equation. The modification of these equations is trivial even if the resulting equations have complicated coefficients. With the value 10.9866 we equalize the entry of sugar into the resin with the exit of water by the molecular mass ratio (MS/M, = 10.9866). The given set was used only in the case when all sugars were on the “theoretical plate” simultaneously. In the presence of two sugars we solved the set of two equations, which led to one third-power equation, and lastly in the presence of one sugar we solved one second-power equation. In a numerical work up, the primary question was to find a useful method which would give the nondeformed results after finishing several hundred thousand calculations on the “theoretical plate”. The difficulty consisted particularly in the choice of the roots of the fourth-power equation out of four possibilities and so on. The direct trials with individual simple methods did not lead to the goal. Therefore, we first used the Bairstow iterative method for the calculation of the roots (Ralston, 1965a), by means of which we obtained all roots of the equation solved. Thus we obtained the necessary information about the propcrtics of the roots of the polynomial, assigning the equilibrium on the “theoretical plate”. It was found that the required solution applicable for the next calculation is that real root having the smallest absolute value of all the roots. Besides, we found that, in these calculations, of which there were several tens of millions, every solved polynomial had at least one real root. That means that derivation of polynomials as models of processes is from the mathematical point of view unobjectionable. However, by using the Bairstow method the calculation time was disproportionately extended and the model would lost: practical meaning After a series of mathematical experiments the iterations were realized in accordance with Newton and Raphson (Ralston, 1965b), and so the calculation was substantially done. The distribution coefficients given in Table 2 are usable in this model after being converted into the molar concentrations in the individual phases. The concentration of the “active places” in the chromatographic bed was determined on the basis of the analytical assessment of the Ca2+ ion content. For the individual components of glucose, the empirically derived equilibrium constants increasing and decreasing towards the determined equilibrium value by 8% of this value were used. In the model, the constants
M. LuKA~ and Z. PEKINA
Y64 Table 3. Comparison
of the experimental results with results of the model: experiment no. 4580 Calculation
Experiment Time
Glucose
170 174 178 182 186 1YO 194 198 202 206 210 214 218 222 226 230 234 238 242 246 250 254 258 262 266 270 274 278 282 286 290 294 298 302 306
0.0 2.0 8.5 18.1 30.9 49.0 68.6 89.0 111.3 131.5 148.7 163.7 172.1 172.9 161.7 143.3 118.5 78.4 41.6 11.8 3.1 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0
0.0
Fructose 0.0 0.0 0.0 0.0
0.0 0.2 0.5 0.7 0.2 0.1 0.B 0.0 2.2 9.1 22.2 41.3 66.5 108.4 147.6 180.5 202.1 210.6 205.0 172.0 130.6 90.5 57.3 30.2 10.3 4.4 2.3 2.6 1.3 0.3 0.0
adjusted in this way gave results closely corresponding to the laboratory experiments. The volume of the liquid on the layer is the tota sum of the liquid volume in the free room of the resin (ca 26-30% of the whole volume of the layer) and the liquid volume contained in the resin. It can be determined by means of simple analysis. The comparison of the experimental results with the results of the model are given in Table 3.
Time
GIucose
164.7 172.4 180.1 187.8 195.5 203.2 210.9 218.9 226.3 234.0 241.7 249.4 257.1 264.8 272.5 280.2 287.9 295.6 303.3 311.0
0.0 4.8 13.8 25.7 61.3 107.4 133.7 151.1 130.8 104.5 69.6 21.9 3.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0
Fructose 0.0 0.0 0.0 0.0
0.0 0.0
0.0 0.5 3.5 16.7 50.7 107.9 176.3 214.8 194.1 82.6 15.2 1.5 0.1 0.0
The separation is in a progress in these models in a certain ratio, while through one part of the model the solution flows faster than through the other one and both separated flows mix together at the output. In effect, this modified mode1 helps to determine the disturbances in the used apparatus. The separation of the glucose into two components was not realized. The mathematical model was derived after finishing the experimental series. From the model it cannot be concluded that glucose will really separate into two components. Our
DISCUSSION
The mathematical mode1 of the process of the separation of sugars on columns with a filling of ionexchanging resin with sulphone groups saturated with Ca2+ ions, based on ion bindings, corresponds closely with the model of the processes. In the case of some other influences on the experiment and of breaking the given conditions, a close correspondence between the experiment and the derived model can be reached by introducing the disturbance into the model, e.g. the breaking of the plug flow we can model by distributing the model of the column in two parallel models.
idea is that there can be a chain of phenomena, where one anomer binds on the resin more strongly than the other one and the equilibrium in the solution is gradualty restored. Therefore we verified the previously mentioned assumptions. There was correspondence between the experimental and model results for glucose in the model into two components with the same volume and the close distribution coefficients. A very close correspondence was also reached between the model and the experiment in the case of fructose, and between the output concentration of fructose in the
Dynamic
model of physical processes
965
u
mean axial rate
times and eluant curves were in agreement with the laboratory experiments. With the achieved results we
VI, V, xi
tops of the tangential argument of the f(xi
can assume with high probability that the derived model of the distribution of sugars on the ion-exchanging resin with SO,Ca+ groups corresponds to the factual actions in the real process.
x3 Y, 2
number of molar changes of the separated components necessary to the accessibility of the equilibrium loading of the section
Acknowledgements-The authors and Ing. M. Tadra for providing Research Institute for Pharmacy feasibility of the realization of computer. The authors thank Mr ments performed.
Greek p
model
and the experimenta
result.
Also the eluant
thank Dr M. Kulhtinek the results, as well as the and Biochemistry for the the calculations on their M. Brda for the measure-
z
zz
letrers first distribution density second
triangles
) function
moment
distribution
momentdispersion
Subscripts NOTATION
C
C D
f” fF4 gR HETP IHI
CHSI K L m M MI, Ml N P R, r
cw IW S s t
intercept distribution constant for the given sugar and selected temperature in dimensions of the experiment (slope) concentration increment (decrement) of the sugar in the solution as a result of the reaction diameter diffusion coefficient of sugar corrective factor for the ratio of phases in the experiment Fisher-Snedecor test value of function f at point x coefficient for a sort of resin height of the “theoretical plate” molar concentration of water molar concentration of sugar in the water sotution equilibrium constant length of the reactor viscosity molecular mass base lines of the tangential triangles number of “theoretical plates” distribution coefficient resolution coefficient regression coefficient molar concentration of tbe active centers in the resin molar concentration of the sugar in the resin concentration of the sugar in the solution standard time
deviation
F
column fructose
Ga
“first” glucose
Gb
“second”
H
water lengthways
C
L 0
r
s
t 1
glucose direction
original radial direction sugar time equilibrium REFERENCES
Barker, P. E., Ganetsos, G. and Thawait, S., 1986, Development of a link between batch and semicontinuous liquid systems. Chem. Engng Sci. 41, chromatographic 2595-2604 Ching, C. B., Ho, C., Hidajat, K. and Ruthven, D. M., 1987, Experimental study of simulated countercurrent adsorption system-V. Comparison of resin and zeolite adsorbents for fructose-glucose separation at high concentration. Chem. Engng Sci. 42, 2547-2555. Ganetsos, G., 1987, Prediction of the distribution coefficient (K,) variation with operating conditions in chromatographic system. J. Chromat. 411, 81-94. Himmelblau, D. M. and Bischoff, K. B., 1968., Process Analysrs crnd Simulation, Deterministic Systems, p, 9. John Wiley, New York. Levenspiel, O., 1965, Chemical Reaction Engmeering, p. 242. John Wiley, New York. LukLf, M., 1988, Dynamic models of the physical phenomena and optimalization in the production of glucose. C.Sc. thesis, eeskC VysokC Uceni Technick& (Czech Technical University), Prague. Ralston. A.. 1965a. A First Course in Numerical Analysis, p, 329. McGraw-Hill, New York. Ralston, A., 1965b, A First Course in Numerical Anulysis, p. 372. McGraw-Hill, New York.
Printed
Engmeering Science, in Great Bntam.
Vol. 46, No
0309~2509/91 53.00 + 0.00 0 1991 Pergamon Press plc
4, pp 959 965, 1991.
A DYNAMIC MODEL OF PHYSICAL PROCESSES IN CHROMATOGRAPHIC GLUCOSE-FRUCTOSE SEPARATION M. LUKAe’and LCMva-State
Enterprise,
Z. PEgINA
Doni Mecholupy 130, 109 02, Prague, and Research Biochemistry, Prague, Czechoslovakia
(First received
31 May
1989;
accepted
in
reuisedform
Institute
8 May
for Pharmacy
and
1990)
Abstract-The assumption of diffusion-adsorption phenomena in the chromatographic separation of does not provide, glucose and fructose on columns of Ca 2+-farm sulphone catex (styrentiivinylbenzene) according to the results, the mathematical model corresponding to the actual phenomena in the separation process. Therefore, a model based on the assumption of a stronger ion-type interaction was tested. In this model, the equilibria on the theoretical plate are mutually conditional on all components being present m both the solution and the stationary phase. However, all experiments that model glucose as one component have failed in this model. Only the model assuming that glucose is two chromatographic components corresponds to the obtained experimental results in the whole range. In determining the equilibrium the factual solution of the model leads to an (n + l)th-power equation with n components. Suitable procedures for a numerical solution were verified
INTRODUCTION
250
Considering that chromatographic separation of glucose-fructose mixtures on a resin has become an industrial process in a number of advanced countries of the world, increasing interest has been focused on the definition of the physico-chemical nature of these processes together with production of a corresponding mathematical model. The increasing concentration of glucose-fructose mixtures influences significantly the values of the distribution coefficients of the respective sugars, including mutual operation of the components of the mixture (Barker et al., 1986). A similar procedure is employed in another model (Ching et al., 1987). In another work (Ganetsos, 1987), the mutual dependence of the distribution coefficients is solved by means of nonlinear relations, with coefficients determined from the experiments. Under these conditions a very good correspondence between the model and the experiment was reached. The derived models are empirical and do not consider the mechanism of the distribution process. From the study of the distribution processes of glucose and fructose on a resin in the CaZ+-cycle two conspicuous anomalies result
(see Figs
Fig. 1. Results
of experiment
4580.
l-6):
(1) The output peak of glucose is lower and wider, i.e. its dispersion is significantly greater than the output peak of fructose. (2) In the separation of the mixture the maximum output concentration of fructose is significantly higher (in dependence on the separation column and other conditions) than the feed of the fructose. These results are documented in the experimental part.
0
to whom
correspondence
should
210
/l\
1
260 T,me
Fig. 2. Results
310
360
Inun)
of calculation
4580.
sion and physical adsorption. Therefore we tried to derive a mathematical model, based on a different mechanism of the phenomena, the outputs of which correspond to the experiments. The results of our work (LukS, 1988) are introduced.
The above-mentioned deviations indicate the distinctness of these processes from both the usual diffu+Author
/&J_
160
be addressed. 959
M. LuKAC and 2. PERINA
960
From a group of several hundred experiments the following three examples were chosen as typical. Among them are experiments on columns of different sizes and different loadings of the filling to illustrate the statement given below. The serial numbers of the experiments are given to distinguish the ‘individual experiments and have no other meaning. The conditions of the individual experiments were: No. 4580-D
= 2.3 cm, L = 93 cm, feed = 46 ml (concentration = 350 g/l), rate = 1.17 ml/min.
Tme
= 5.0 cm, L = 160 cm, feed = 675 ml = 350 g/I), rate = 5 ml/ min. No. 555&D = 16.0 cm, L = 280 cm, feed = 8570 ml (concentration = 350 g/l), rate =901 ml/min. No.
(mm)
Fig. 3. Results of experiment
5110-D
(concentration
5110
EXPERIMENTAL
The
tose
Time
coeficients
for glucose
and
fruc-
measured in direct experiments. Fifty mililiters of Zerolit SRC-14.CaZf were mixed with 1COml of glucose or fructose solutions of different definite concentrations at selected temperatures of 20, 40 or 60°C for 15 min. Then the residual concentration was assessed. The initial concentrations of the sugars in the solutions ranged from 50 to 350 g/l, totalling 14 different concentrations. The results of the measurement can be summarized by the relation
imln)
Fig. 4. Results of calculation
distribution
5110.
were
c1 = a + bc,
(1)
in Table 1. The measured results lead to the conclusion that the distribution of the sugars between the solution and the resin is linear in the whole range of measurements and is not dependent on the concentration. The dependence on the temperature is nonlinear and cannot be derived from the measurements involving three temperatures. The constant a (intercept) should in all cases be equal to zero. Its small values (in comparison 0 310
360
410
460
510
Fig. 5. Results of experiment
560
5550.
Table I.
610
Estimations of regression coefficients in the case of distribution
of glucose
between
water
and resin
300 0 Glucose x Fluctose
“C
a
b
r
s
20 40 60
0.587 -0.306 0.377
0.807 0.860 0.844
0.9990 0.9998 0.9990
4.78 i 2.209 4.467
Estimations bution “C
Trne
(mm)
Fig. 6. Results of calculalion
5550.
20 40 60
of regression
of fructose a -0.404 0.663 0.044
between
coefficients water b
0.782 0.771 0.791
in the case of distri-
and resin I
0.9990 0.9990 0.9997
s
4.042 5.132 2.680
Dynamic model of physical processes
with the standard deviation s) are a consequence of experimental errors in the analytical determination. From the estimations of the constant b (slope) we can calculate the distribution coefficients between water and the resin for the individual sugars and individual temperatures by means of the equation
‘=
I-b1 ___ b
f
(the ratio of the content of sugar in the resin and in the liquid after modification multiplied by the ratio of the phases). The estimated values are introduced in Table 2. For the actual
study we had at our disposal the results of several hundred laboratory measurements for the separation of glucose-fructose mixtures on chromatographic columns of different diameters and heights of filling at different temperatures and concentrations, amount of feed and flow rate through the column. The diameters of the columns used in these experiments ranged from 2 to 30 cm with fillings of Zerolit, Ostion, Wofatit and Dowex. The selected temperatures in the experiments ranged from 20 to 70°C. The conditions of these experiments were not selected according to any rational plan. The experiments were evaluated in this way (Levenspiel, 1965): the first two distribution moments of the eluant peaks (the average and the variance) were determined by means of the equations
T xif txi) (3)
p = p, ~xff(xA
d = isf(&)
-
p2.
(4)
The dispersion function of reactors is appropriately characterized by the dimensionless criterion
the so-called reactor dispersion number (Levenspiel, 1965), which is in fact the reciprocal value of Peclet’s criterion for mass transfer. By means of this criterion we can numerically evaluate a series of experiments. It is linked with the dispersion by several relations. From these relations, in this case the relation for a closed system is relevant. 02 = 2D/uL - 2(D/UL)2(1 - .KZ”L’D).
(5)
For the reactor dispersion number this relation is not directly solvable but can be easily calculated by means of iterations. By means of this criterion we can estimate the number of “theoretical plates”: N = uL/(2D). From had at [eqs (3) [eq. (5)]
961
Table 2. Estimations of distribution coefficients P in eq. (2) Temperature
20°C
40°C
60°C
Glucose Fructose Difference (fructose glucose) Ratio (fructose/ glucose
0.648 0.752
0.441 0.803
0.500 0.712
0.104
0.362
0.212
1.160
1.821
1.424
ated the number of theoretical plates [eq. (6)]. In all cases (402 experiments), we estimated a lower number of theoretical plates for glucose than for fructose. Other authors arrived at the same conclusion (Ching and Ruthven, 1986). From the results we could empirically derive the regression equation (in the form of a criterion relation) between the HETP and optional input conditions for fructose in the performed experiments. InHETP
= - 3.854 + 0.438lnL,
+ O.l22InD,
+ 0.280 In Z + 0.763 In u, - 0.177 In m + 0.038gR r = 0.784,
n = 402,
(7)
F = 105.1
The HETP in individual experiments being dependent on the conditions of the experiment, we derived the regression equation for its determinination from the different tested linear and nonlinear equations with a different combination of input conditions. The regression equation shows the predominant influence of the length of filling, the load weight of the cross section and the flow velocity on the HETP. The influence of the diameter of the column (the influence of the walls), of the viscosity (function of the concentration) and of the kind of resin is significantly lower. This equation was selected from a number of other empirically selected equations for its best statistical parameters. On the assumption that we can explain the member (0.763 In u) in the form (0.5 x 0.763 In u’), this equation is after delogarithmization dimensionless. DERIVATION
OF
THE
MATHEMATICAL
MODEL
If the mathematical model should correspond with the real processes in all primary aspects, the relation employed must respect the physico-chemical nature of the processes in progress. The interpretation of the empirical derivation is restricted to a narrow area of the conditions in the experiment. The new mathematical model of this separation process was derived on the assumption that:
(6)
the results of all experiments whose data we our disposal their distribution moments and (4)] and reactor dispersion numbers were sequentially calculated. Then we estim-
(I) The chromatographic bed in the unit volume has a definite accurately defined number of relevant binding positions. (2) The mechanism of retention of the sugars in the
962
M.
LUKM
chromatographic bed is of the same quality for both sugars and only differs in quantity. (3) The sugars influence each other in competition for these binding positions. system glucose appears (4) In the chromatographic as a mixture of two closely related substances. To put it another way, the measured eluant plot of glucose is a superposition of two eluant plots for two chromatographically different components of glucose. (In chromatographic methods, this is not the unknown phenomenon and there are mixtures of components with similar distribution coefficients in the majority of the cases.) The first assumption is based on the structure of the bed of the separation column. The separation effect almost disappears when the sulpho groups are not occupied by Ca, Sr or Ba ions. These ions have a conclusive influence on this process. In this way a definite number of binding positions in the volume unit of the chromatographic bed is given. The next two assumptions are closely connected with the preceding one as its consequences. The latter assumption is based on the results of the performed experiments. We could not demonstrate by any modification any agreement between the experimental results and the outputs of the mathematical model when we tried to devise mathematical models on the assumption of diffusion-adsorption phenomena. The output peak of glucose was always higher and more narrow than the output peak of fructose. Big differences in the shape of the peak between the experiments and the model also occurred. Therefore we tested differences between the course of the curve with the normal distribution and the output peak of the individual sugars. Both curves included the same area and had the same width. The course of the curves was nearly coincident for fructose and considerably different for glucose. We can, however, divide the glucose peak in two mutually overlapping peaks with the half-area and distribution, which is very near to normal. A typical example is given in Fig. 7. The latter assumption calls for three substances instead of two to be incorporated into the mathematical model. Besides, the stationary phase and the pure eluant must also he incorporated into the drawn model. These are complications for the solution; nevertheless, they are not unbridgeable. The preceding assumptions are advantageous for the following reasons: The equilibrium states of any component of the mathematical mode1 are directed by the relation with the sole equilibrium constant depending only on the temperature (dependence on the pressure was not tested) The mutual influence of the components takes place exclusively through the mediation of its activities. The equation in the model contains this relation.
and Z. PE%INA
250 3
t
v “Left” glucose ! “Rqht’ glucose
Time imInI
Fig. 7.
Calculated
distribution of glucose (experiment 51 IO).
(3) The HETP is influenced only by the length and the diameter of the column, by the velocity of the eluant fluid, by the amount and concentration of the feed, and by maintaining plug flow through the column. It is not influenced by the type of substance under separation. All these conditions of the process and the temperature set the task for the mathematical model. By means of the relation derived from the experimental data C(7)] we can determine the HETP in every individual case. The basic model of the ion-exchange process in the chromatographic column can be expressed as follows: [sugar.
H,O]
+ [resin]
= [resin. sugar]
+ [H,O]. (8)
In the shortened
notation
[KY]
this is
+ [R] = [RS] + [HI
(9)
i.e. the sugar is bound either on the solvent in the liquid phase or on the binding places in the stationary phase. For the equilibrium constant the following relation holds:
(10) In this equation the individual components are expressed in molar concentrations. This equation is known as the relation for the operation of active matter. The activities of individua1 components should be applied instead of the molar concentrations. Considering that the relations for distribution coefficients are not dependent on the concentration, we can replace the activities with the concentrations, without the accuracy being lowered. In the case of nonequilibrium in the system, we can simply define the changes necessary for achieving the equilibrium by the relation K
=
s
(CRSI ([RI
+
Cxlj.(CW
-cxl).(pE]
+
Cxl)
_cxjj’
In deriving the model, an important assumption that of plug flow of the eluant through the column,
(11) is i.e.
963
Dynamic model of physical processes all the physical values (concentrations of the sugars, feed velocity, temperature and pressure) in the section perpendicular to the axis of the column are identical. Otherwise a settlement of these values through diffusion and through sharing of the warmth would occur and complicate the mathematical model. On the given assumptions we can write partial differential equations for the process in the column (Himmelblau and Bischoff, 1968):
Equation (12) refers to the phenomena in the solution; eq. (13) refers to the processes in the individual particle on the resin, on the assumption of its ballshaped form. Members on the right side of these equations express the changes in concentrations resulting from diffusion and the changes resulting from reactions. In eq. (12) the diffusion in question is lengthwise in the flow direction; eq. (13) refers to the central or opposite diffusion in the particle of the bed. Considering that the free cross section area of the column is only about 10% of its crosswise section, the used flow velocities are higher than the diffusion velocities and the particles have very small diameters (0.1-0.3 mm), the diffusion in the model can be neglected. According to the law of the conservation of mass the changes in C, are equal in both equations, except for the Signum. After this simplification of the numerical method of the solution the differential equations are replaced by the difference equations (a solution by the method of nets) AS dt+
h(u,S)=C AL
s
AS 2 = Csr = - C, At We usually split the solution into the shift between the individual plates and the subsequent calculation of the equilibrium, the length of the shift being the HETP. With the continuity of the flow, the shift will proceed in all “theoretical plates” at one time, while the next feed or eluant enters the highest plate and the same volume of liquid leaves the lowest plate. The equilibrium on the individual plates is estimated according to a modified relation (14) for the individual components. (CH&,l
- x)&r,
= (CR&J
+ x)(CHl
+ x + y + z)/([R] (Cf&l]
- Y)JG,,
= (C&J
- z)G
= (CBS,1 +
(16)
+ Y)(CHl
+ x + Y + Z)/([R] (C=,l
- x - y - z)
-x
- y - 2)
(17)
z)(CHI
+ x + y + z)/([R] - x - y - z) (18)
and then introducing
into the balance
equations:
[HI, = [H-J,+ 10.9866(x + y + z) [RI,
= CR], - (x + Y + z)-
(19)
The solution of the set (16)-(19) leads to a set of ihree equations for the unknown x, y and z, and then to one fourth-power equation. The modification of these equations is trivial even if the resulting equations have complicated coefficients. With the value 10.9866 we equalize the entry of sugar into the resin with the exit of water by the molecular mass ratio (MS/M, = 10.9866). The given set was used only in the case when all sugars were on the “theoretical plate” simultaneously. In the presence of two sugars we solved the set of two equations, which led to one third-power equation, and lastly in the presence of one sugar we solved one second-power equation. In a numerical work up, the primary question was to find a useful method which would give the nondeformed results after finishing several hundred thousand calculations on the “theoretical plate”. The difficulty consisted particularly in the choice of the roots of the fourth-power equation out of four possibilities and so on. The direct trials with individual simple methods did not lead to the goal. Therefore, we first used the Bairstow iterative method for the calculation of the roots (Ralston, 1965a), by means of which we obtained all roots of the equation solved. Thus we obtained the necessary information about the propcrtics of the roots of the polynomial, assigning the equilibrium on the “theoretical plate”. It was found that the required solution applicable for the next calculation is that real root having the smallest absolute value of all the roots. Besides, we found that, in these calculations, of which there were several tens of millions, every solved polynomial had at least one real root. That means that derivation of polynomials as models of processes is from the mathematical point of view unobjectionable. However, by using the Bairstow method the calculation time was disproportionately extended and the model would lost: practical meaning After a series of mathematical experiments the iterations were realized in accordance with Newton and Raphson (Ralston, 1965b), and so the calculation was substantially done. The distribution coefficients given in Table 2 are usable in this model after being converted into the molar concentrations in the individual phases. The concentration of the “active places” in the chromatographic bed was determined on the basis of the analytical assessment of the Ca2+ ion content. For the individual components of glucose, the empirically derived equilibrium constants increasing and decreasing towards the determined equilibrium value by 8% of this value were used. In the model, the constants
M. LuKA~ and Z. PEKINA
Y64 Table 3. Comparison
of the experimental results with results of the model: experiment no. 4580 Calculation
Experiment Time
Glucose
170 174 178 182 186 1YO 194 198 202 206 210 214 218 222 226 230 234 238 242 246 250 254 258 262 266 270 274 278 282 286 290 294 298 302 306
0.0 2.0 8.5 18.1 30.9 49.0 68.6 89.0 111.3 131.5 148.7 163.7 172.1 172.9 161.7 143.3 118.5 78.4 41.6 11.8 3.1 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0
0.0
Fructose 0.0 0.0 0.0 0.0
0.0 0.2 0.5 0.7 0.2 0.1 0.B 0.0 2.2 9.1 22.2 41.3 66.5 108.4 147.6 180.5 202.1 210.6 205.0 172.0 130.6 90.5 57.3 30.2 10.3 4.4 2.3 2.6 1.3 0.3 0.0
adjusted in this way gave results closely corresponding to the laboratory experiments. The volume of the liquid on the layer is the tota sum of the liquid volume in the free room of the resin (ca 26-30% of the whole volume of the layer) and the liquid volume contained in the resin. It can be determined by means of simple analysis. The comparison of the experimental results with the results of the model are given in Table 3.
Time
GIucose
164.7 172.4 180.1 187.8 195.5 203.2 210.9 218.9 226.3 234.0 241.7 249.4 257.1 264.8 272.5 280.2 287.9 295.6 303.3 311.0
0.0 4.8 13.8 25.7 61.3 107.4 133.7 151.1 130.8 104.5 69.6 21.9 3.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0
Fructose 0.0 0.0 0.0 0.0
0.0 0.0
0.0 0.5 3.5 16.7 50.7 107.9 176.3 214.8 194.1 82.6 15.2 1.5 0.1 0.0
The separation is in a progress in these models in a certain ratio, while through one part of the model the solution flows faster than through the other one and both separated flows mix together at the output. In effect, this modified mode1 helps to determine the disturbances in the used apparatus. The separation of the glucose into two components was not realized. The mathematical model was derived after finishing the experimental series. From the model it cannot be concluded that glucose will really separate into two components. Our
DISCUSSION
The mathematical mode1 of the process of the separation of sugars on columns with a filling of ionexchanging resin with sulphone groups saturated with Ca2+ ions, based on ion bindings, corresponds closely with the model of the processes. In the case of some other influences on the experiment and of breaking the given conditions, a close correspondence between the experiment and the derived model can be reached by introducing the disturbance into the model, e.g. the breaking of the plug flow we can model by distributing the model of the column in two parallel models.
idea is that there can be a chain of phenomena, where one anomer binds on the resin more strongly than the other one and the equilibrium in the solution is gradualty restored. Therefore we verified the previously mentioned assumptions. There was correspondence between the experimental and model results for glucose in the model into two components with the same volume and the close distribution coefficients. A very close correspondence was also reached between the model and the experiment in the case of fructose, and between the output concentration of fructose in the
Dynamic
model of physical processes
965
u
mean axial rate
times and eluant curves were in agreement with the laboratory experiments. With the achieved results we
VI, V, xi
tops of the tangential argument of the f(xi
can assume with high probability that the derived model of the distribution of sugars on the ion-exchanging resin with SO,Ca+ groups corresponds to the factual actions in the real process.
x3 Y, 2
number of molar changes of the separated components necessary to the accessibility of the equilibrium loading of the section
Acknowledgements-The authors and Ing. M. Tadra for providing Research Institute for Pharmacy feasibility of the realization of computer. The authors thank Mr ments performed.
Greek p
model
and the experimenta
result.
Also the eluant
thank Dr M. Kulhtinek the results, as well as the and Biochemistry for the the calculations on their M. Brda for the measure-
z
zz
letrers first distribution density second
triangles
) function
moment
distribution
momentdispersion
Subscripts NOTATION
C
C D
f” fF4 gR HETP IHI
CHSI K L m M MI, Ml N P R, r
cw IW S s t
intercept distribution constant for the given sugar and selected temperature in dimensions of the experiment (slope) concentration increment (decrement) of the sugar in the solution as a result of the reaction diameter diffusion coefficient of sugar corrective factor for the ratio of phases in the experiment Fisher-Snedecor test value of function f at point x coefficient for a sort of resin height of the “theoretical plate” molar concentration of water molar concentration of sugar in the water sotution equilibrium constant length of the reactor viscosity molecular mass base lines of the tangential triangles number of “theoretical plates” distribution coefficient resolution coefficient regression coefficient molar concentration of tbe active centers in the resin molar concentration of the sugar in the resin concentration of the sugar in the solution standard time
deviation
F
column fructose
Ga
“first” glucose
Gb
“second”
H
water lengthways
C
L 0
r
s
t 1
glucose direction
original radial direction sugar time equilibrium REFERENCES
Barker, P. E., Ganetsos, G. and Thawait, S., 1986, Development of a link between batch and semicontinuous liquid systems. Chem. Engng Sci. 41, chromatographic 2595-2604 Ching, C. B., Ho, C., Hidajat, K. and Ruthven, D. M., 1987, Experimental study of simulated countercurrent adsorption system-V. Comparison of resin and zeolite adsorbents for fructose-glucose separation at high concentration. Chem. Engng Sci. 42, 2547-2555. Ganetsos, G., 1987, Prediction of the distribution coefficient (K,) variation with operating conditions in chromatographic system. J. Chromat. 411, 81-94. Himmelblau, D. M. and Bischoff, K. B., 1968., Process Analysrs crnd Simulation, Deterministic Systems, p, 9. John Wiley, New York. Levenspiel, O., 1965, Chemical Reaction Engmeering, p. 242. John Wiley, New York. LukLf, M., 1988, Dynamic models of the physical phenomena and optimalization in the production of glucose. C.Sc. thesis, eeskC VysokC Uceni Technick& (Czech Technical University), Prague. Ralston. A.. 1965a. A First Course in Numerical Analysis, p, 329. McGraw-Hill, New York. Ralston, A., 1965b, A First Course in Numerical Anulysis, p. 372. McGraw-Hill, New York.