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Modelling of ultrafiltration of non-sucrose compounds in sugar beet processing
Journal of Food Engineering 65 (2004) 73–82 www.elsevier.com/locate/jfoodeng
Modelling of ultrafiltration of non-sucrose compounds in sugar beet processing s, Miodrag Tekic Mirjana Djuric *, Julianna Gyura, Zoltan Zavargo, Zita Sere Faculty of Technology, University of Novi Sad, 21000 Novi Sad, Bulevar cara Lazara 1, Serbia and Montenegro, Yugoslavia Received 29 July 2003; accepted 9 December 2003
Abstract This paper is a contribution to mathematically describing sugar syrup purification, i.e. ultrafiltration of non-sucrose compounds, based on the experimental data, acquired in a series of batch experiments on flat polyethersulfone cross-flow membranes. Experiments were performed to final concentration factors not less than 1.5 and not greater than 2.3, with flow rates 0.2 and 0.4 m3 h1 , at the temperatures 30 and 60 C and for the transmembrane pressures 1 and 4 bar. The suggested mathematical model enables prediction of separation time if the permeate flux models as well as the initial and final concentrations of undesired nonsucrose compounds are known. In this paper, the flux models are proposed as functions of concentration factor (CF ), flow rate (Q), temperature (t) and transmembrane pressure (Dp) as independent variables. So, the following one-, two- and three-variable functions are suggested: J 1 ðCF Þ, J 1 ðCF ; QÞ, J 1 ðCF ; tÞ, J 1 ðCF ; DpÞ, J 1 ðCF ; Q; tÞ and J 1 ðCF ; Q; DpÞ. By the regression analysis method, parameters in the flux models are determined and they are used for the calculation of process duration. It is concluded that the predicted times underestimate the real times by 10–40% in particular cases. One can assume that the incomplete rejection of nonsucrose compounds by polyethersulfone membranes as well as the loss of sugar due to accumulation at the membrane is responsible for process prolongation. 2003 Elsevier Ltd. All rights reserved. Keywords: Mathematical modelling; Ultrafiltration; Permeate flux; Process duration
1. Introduction Purification of the syrup––an intermediate product in sugar beet processing is an important operation, which precedes sucrose crystallization. For many years, this problem was solved by chemically induced precipitation of undesired non-sucrose compounds, after adding calcium oxide and carbon dioxide to the syrup solution. A relatively new method, applied for a reduction of nonsucrose compounds in sugar beet syrup, is a membrane separation, i.e. an ultrafiltration (UF) process (Bubnik, s, Vatai, & Hinkova, & Kadlec, 1998; Gyura, Sere Bek assy Moln ar, 2002; Rautenbach & Albrecht, 1989; Vercelloti et al., 1998; Vern, 1995; Willett, 1997). Some experiences acquired from sugar cane juice ultrafiltration (Balakrishnan, Dua, & Bhagat, 2000; Decloux, Tatoud, & Mersad, 2000; Monclin & Willett, 1996) can *
Corresponding author. Tel.: +381-21-450-277; fax: +381-21-450413. E-mail address: [email protected] (M. Djuric). 0260-8774/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2003.12.005
be applied to the purification of syrup in sugar beet processing. Despite the extensive work on this subject, many theoretical and practical aspects of sugar syrup ultrafiltration are not completely understood. Two things are especially important; the first one is the selection of adequate membrane, according to the knowledge of sizes of the non-sucrose molecules (Bubnik et al., 1998; Decloux et al., 2000; Gyura et al., 2002). The performances of the selected membrane effect the permeate fluxes which, at the other hand, influence the duration of the separation process, determining both the capacity of the equipment (related to the cost of the whole operation) and the degree of purification. The second important question is the selection of working conditions, such as: concentration factor, flow rate of the solution at the membrane, temperature of the sugar syrup, transmembrane pressure, etc. (Balakrishnan et al., 2000; Rautenbach & Albrecht, 1989). The conditions are closely related to the previous stages of sugar beet processing (e.g. temperature) as well as to the performance of the available separation equipment. It is
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
Nomenclature B parameters in flux equations C [g l1 ”kg m3 ] non-sucrose compounds concentration CF concentration factor D; E; F parameters in flux equations Dp [bar] transmembrane pressure J [l m2 h1 ], [m3 m2 h1 ] permeate flux m [kg] mass N number of experimental points Pc permeation coefficient Q [m3 h1 ] solution flow rate R rejection value
particularly important to find out how they affect permeate flux and duration of the separation process. The aim of this paper is to investigate the working conditions problem, assuming that the type of membrane has already been chosen. The experimentally determined permeate fluxes are mathematically described through a series of functions which are introduced into the mathematical model of the process, to give the separation time.
2. The mathematical model of UF process with variable permeate flux The mathematical model of UF of sugar beet syrup is relatively simple. It is based on the mass conservation law, applied to the non-sucrose substances that should be removed, or at least reduced from the initial (C0 ) to the final (Cf ) concentration, by the UF process. The differential equation for the mass balance is a relation which brings together important quantities, such as the volume of the solution, concentrations of substances present, permeate flux, duration (time) of the process, membrane surface and its other characteristics, etc.
2.1. Mass balance equation based on the non-sucrose compounds The mass conservation of the non-sucrose compounds in the sugar syrup can be expressed by a simple equation: dm ¼ 0 )
dm dðVCÞ dV dC ¼ ¼0)C þV ¼0 ds ds ds ds
ð1Þ
On the other hand, the change in the solution volume (V ) is related to the permeate flux (J ) and the membrane area (S) in a way:
S [m2 ] membrane surface T [h m2 kg1 ] normalized phase duration t [C] temperature s [h], [min] time, phase duration V [m3 ] volume of the solution in the system Subscripts 0 initial cal calculated exp experimental f final P permeate
dV ¼ JS ds
ð2Þ
If the non-sucrose compounds are completely rejected, their mass can be expressed in terms of the concentration and the solution volume: m ¼ VC or: V ¼ m=C. Substituting both Eq. (2) and the expression for volume into Eq. (1) gives: m dC dC JS 2 ¼0) ¼ C ð3Þ C ds ds m The ultrafiltration time can be determined from Eq. (3) as s in hours or as normalized duration T in hours per 1 kg of non-sucrose compounds using 1 m2 of membrane: Z Z Z m dC S dC s ¼ T ¼ ds ¼ ) ð4Þ S JC 2 m JC 2 JSC þ
The differential equation (4) can be integrated if the adequate J -function was known. It can be assumed that the flux of permeate remains constant, which is a rough approximation of the real process. However, the flux model can be defined as a function of relevant variables. 2.2. Permeate flux models This paper suggests several regression equations as mathematical models for permeate flux typical of sugar syrup purification. If the flux model has an appropriate form, the differential mass balance equation (4) can be solved analytically. As polynomials are suitable for integration, the flux equation should have a form that leads to the polynomial form of the integral. Such an approach was suggested by Jaffrin and Charrier (1994). The most important variable, which influences the permeate flux, is concentration of non-sucrose compounds or concentration factor, so that the flux function should contain these quantities. However, other independent variables can also affect the value of the flux. The most important are: flow rate of the solution through the system, temperature at the membrane and
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
transmembrane pressure (Balakrishnan et al., 2000; Gyura et al., 2002). The greater the number of relevant variables, the more realistic is the mathematical model of permeate flux. Here, three types of flux models will be introduced: (i) one-variable, (ii) two-variable and (iii) three-variable.
2.2.2. Flux as a function of two variables More complete mathematical description of permeate flux is achieved when flow rate, temperature and transmembrane pressure are taken into account together with the concentration factor, giving two-variable regressions: J 1 ðCF ; QÞ, J 1 ðCF ; tÞ and J 1 ðCF ; DpÞ. The simplest possible form of these regressions appear when assuming that the permeate flux changes with the linear change of Q, t and Dp. Consequently, two-variable regressions have a form given by Eq. (6) with D; E and F parameters presented in Table 1.
2.2.1. Flux as a function of one variable The collection of undesired compounds at the membrane is the aim of sugar syrup purification by an UF process. Therefore, it is very appropriate to express flux in terms of the concentration of non-sucrose compounds, which have to be removed during the purification: J11 ¼ B1;1 þ B1;2 C þ B1;3 C 2
2.2.3. Flux as a function of three variables In a real separation process, more than two working conditions might change or should be adjusted. So, the flux model should contain as many relevant variables as possible. Here, three-variable models are suggested: J 1 ðCF ; Q; tÞ and J 1 ðCF ; Q; DpÞ. Basic equation in these cases is Eq. (6), with D; E and F parameters presented in Table 1. As is obvious from Table 1, two alternatives for the function J 1 ðCF ; Q; tÞ are offered; the simple one assumes a linear change of Q, and t, while the alternative suggests a slightly better but more complicated relationship. However, both relations are easy to apply.
ð5Þ
In addition to concentration, the dimensionless concentration factor ðCF ¼ C=C0 Þ is even more interesting in a UF process interpretation. Having this in mind, the flux can be expressed in the form: 2 C C 1 Ji ¼ F þ E þ D ð6Þ C0 C0 that allows a kind of generalization. If the concentration factor is the only variable, the index i takes the value: i ¼ 2, as presented in Table 1 where an overview of all suggested flux models can be seen. It also might be useful to mention that the nonsucrose compounds concentration in the retentate can be expressed in terms of other quantities, such as the initial volume in the tank V0 , the permeate volume VP , and the permeate coefficient Pc (Durante, Santos, & Durante, 2001). The following relationship is well known: ð1PcÞ ð1PcÞ V0 C V0 C ¼ C0 ) ¼ ð7Þ C0 V0 VP V0 VP
2.3. Duration of UF process Once defined, mathematical model (4) can be used for predicting the duration of the UF process. When the flux model (6), with the D; E and F parameters from Table 1, is substituted into Eq. (4) one obtains: 2 Z Cf D C þ E CC0 þ F C0 S s¼T ¼ dC ð9Þ m C2 C0 After solving the integral at the right hand side of Eq. (9) the solution can be found as the normalized duration (T ) of the UF process:
whereby, the coefficient Pc and the solute rejection R are connected by a simple expression: R ¼ 1 Pc
75
S s¼T m 1 Cf Cf C0 ¼ D 1 þ E ln F 1 C0 C0 C0 Cf
ð8Þ
Eqs. (5) and (6) are simple but do not include all relevant variables.
ð10Þ
Table 1 An overview of flux models Number of variables
Flux
Index i
Parameters D
E
F
One
J 1 ðCF Þ J 1 ðCF ; QÞ J 1 ðCF ; tÞ J 1 ðCF ; DpÞ
2 3 4 5 6 7 8
B2;3 B3;3 B4;3 B5;3 B6;3 B7;3 B8;3
B2;2 B3;2 B4;2 B5;2 B6;2 B7;2 B8;2
B2;1 B3;1 þ B3;4 Q B4;1 þ B4;4 t B5;1 þ B5;4 Dp B6;1 þ B6;4 Q þ B6;5 t B7;1 þ QðB7;4 þ B7;5 QÞ þ t1 ðB7;6 þ B7;7 t1 Þ B8;1 þ B8;4 Q þ B8;5 Dp
Two
Three
J 1 ðCF ; Q; tÞ J 1 ðCF ; Q; DpÞ
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for various values of the initial and final concentrations of the non-sucrose compounds.
7 3
3. Experimental
8
4
The syrup solution, which was exposed to ultrafiltration in the laboratory plant, was prepared for the separation process when the molasses, taken from the factory, was diluted (1:1) with distilled water. The main characteristics of the solution are given in Table 2. Ultrafiltration was performed using a flat cross-flow polyethersulfone membrane, the characteristics of which are given in Table 3. The capacity of the laboratory plant was 2 l of the initial syrup. Coloured and colloidal compounds were collected in batch experiments. A simplified schematic of the plant is presented in Fig. 1. The syrup solution from the reservoir (1) flows towards the membrane module (3) being forced by a 0.55 kW pump (2). Through the membrane module the solution flows tangentially. Permeate is taken away from the system continuously, while the retentate returns to the tank through a flow-meter (4) where its volume rate is measured. Flow rate of the solution is adjusted by the control valve (5) positioned in the pipeline. Pressure is controlled by the valve (6) and registered by the manometer (7). Temperature of the sugar syrup is kept at a constant value by thermostat (9) and it is measured by thermometer (8), which is built into the main pipeline. Permeate volume is measured after being captured in a cylinder (10). At the end of experiment, the retentate is transferred into a reservoir (11). Experiments were planned in accordance with the full factorial design presented in Table 4. After adjusting the variables from Table 4, the experiment started. The permeate volume was measured Table 2 Characteristics of the initial syrup (diluted 1:1) Statistical parameter
Dry matter (%)
pH
Invert (mass%/ d.m.)
Purity ratio Q (%)
Average value ðxÞ Standard deviation ðrn1 Þ
39.2 0.01
8.23 0.01
0.186 0.02
61.73 0.00
Table 3 Characteristics of membrane Characteristics
Membrane for UF
Manufacturer Type of membranes
Zoltek Rt MAVIBRAN, HU FS-20 Asymmetrical porous thin layer Polyethersulfone 15–20 470 300–390 6 60
Material MWCO (kDa) Active surface (cm2 ) Thickness (lm) Maximal pressure (bar) Maximal temperature (C)
6
9 1 5 10 2
11
(1) reservoir, (2) pump, (3) membrane module, (4) flow- meter, (5) and (6) valves (7) manometer, (8) thermometer, (9) thermostat, (10) cylinder for permeate volume measurements, (11) reservoir for retentate
Fig. 1. Scheme of the laboratory plant.
Table 4 Plan of experiments Number of experiment
Flow-rate (m3 h1 )
Temperature (C)
Transmembrane pressure (bar)
1 2 3 4 5 6 7 8
0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4
30 30 60 60 30 30 60 60
1 4 1 4 1 4 1 4
every 15 min so that permeate flux was defined. The concentration factor was also estimated from the flux data every 15 min. For this purpose the relation (7), between the initial volume of the syrup and the actual volume of the permeate, was applied with the assumption that the rejection of non-sucrose compounds by polyethersulfone membrane is complete (R ¼ 1). Each experiment lasted 2.5 h. After this period of time, the lowest estimated concentration factor was 1.5 while the greatest one was 2.3. The minimum value of permeate flux was 5.62 l m2 h1 while the maximum value was 9.8 l m2 h1 . In this way, a data base was acquired with the concentration factor, flow-rate, temperature and transmembrane pressure as the independent variables and the permeate flux as dependent variable. Each quantity was measured at least three times, and the mean values were used for further analysis.
4. Discussion of experimental and calculated results Parameters in the suggested flux functions (in Table 1) were determined by processing the available experimental data (J , CF , Q, t and Dp). For this purpose,
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
4.1. One-variable flux function and duration of UF process When the concentration factor is taken as the only independent variable, the mathematical model for permeate flux through the membrane is the regression equation (6). In order to illustrate such a case, 10 experimental points, with variable CF and constant Q ¼ 0:2 m3 h1 , t ¼ 30 C and Dp ¼ 4 bar, were selected. After application of the regression analysis method to the measured points, adequate B-parameters were calculated, as presented in Table 5. The analysis of the corresponding t-values shows that the linear term is more important than the quadratic term, which means that permeate flux decreases almost linearly with increase in CF -value. The comparison between measured and calculated flux values, presented in Fig. 2, leads to the conclusion that a very adequate type Table 5 Parameters in the one-variable regression for fluxes in m3 m2 h1 B-parameters
Other data
10.5
Flux [lm-2 h-1]
3
-1
Q=0.2 [m h ]
10.0
o
t= 30 [ C] ∆ p= 4 [bar]
9.5 9.0 8.5 8.0 7.5 1.2
Other data 3
-1
Q=0.2 [m h ] 120 100
o
t= 30 [ C] ∆ p= 4 [bar]
80 60 40 20 0 1.0
1.2
1.4
1.6
1.8
Concentration factor Fig. 3. Experimental duration (full line) and predicted duration (dash line) as the function of concentration factor.
of regression has been chosen. Small value of the deviation (11), given in Table 5, proves this fact. With the parameters of the regression, defined in Table 5, Eq. (10) is solved and the duration of the UF process calculated. The results of these calculations are presented in Fig. 3 where they are compared with the measured values. It can be seen that calculated times are lower than the real times. The difference between these two increases during the process, reaching 25% at the end after 2.5 h of experiment. This is a consequence of errors in the flux estimation but it is much more the consequence of the incomplete rejection of the nonsucrose compounds by the polyethersulfone membrane. 4.2. Two-variable flux functions and duration of UF process
11.0
1.0
140
t-values
10.21926 6.80E)03 97.21664 4.51E)02 )16.43666 2.17E)02 P 2 D J =N ¼ 1.55E)08 m3 m2 h1
B21 B22 B23
160
Process duration [min]
the well-known technique––the regression analysis method––was applied. The significance of each parameter in the regressions is verified by a statistical measure known as the t-value. The adequateness of the introduced flux functions is proved by calculation of the deviation: PN 2 ðJexp Jcal Þ d¼ 1 ð11Þ N
77
1.4
1.6
1.8
Concentration factor Fig. 2. Experimental flux (full line) and predicted flux (dash line) as the function of concentration factor.
When the concentration factor is taken as the independent variable together with the flow rate, or temperature or transmembrane pressure, the regressions from Table 1 are mathematical models of permeate flux. This case is illustrated by three examples; 20 experimental points, with variable CF and Q but constant t ¼ 30 C and Dp ¼ 1 bar, were selected as the first example. Then, 20 experimental points, with variable CF and t but constant Q ¼ 0:2 m3 h1 and Dp ¼ 4 bar, were chosen as well as 20 experimental points with variable CF and Dp but constant Q ¼ 0:2 m3 h1 and t ¼ 30 C. Application of the regression analysis method to the measured points, gives adequate B-parameters, as presented in Table 6. The calculated t-values indicate that the parameter Bi4 , connected to the flow rate (i ¼ 3), temperature (i ¼ 4) and transmembrane pressure (i ¼ 5), was the most important for all two-variable regressions. This means that the mentioned working conditions are very
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
Table 6 Parameters in the two-variable regressions for fluxes in m3 m2 h1 Temperature as variable (index i ¼ 4)
Pressure as variable (index i ¼ 5)
B-parameters
B-parameters
B-parameters
t-values
2.152593 9.22E)04 248.1224 7.78E)02 )69.52132 6.46E)02 )274.2582 3.81E)01 P 2 D J =N ¼ 5.502E)08
t-values
11.3249 1.04E)02 120.7277 8.52E)02 )26.29468 5.72E)02 )0.4872019 2.21E)01 P 2 D J =N ¼ 5.082E)08
important in the real as well as in the mathematically described process. The comparison between measured and calculated flux values can be seen in Figs. 4–6 for all two-variable cases. It shows that the calculated flux values predict very well experimental fluxes. The best fit is achieved when the concentration factor–transmembrane pressure––regression is applied (Fig. 6) as it shows the smallest deviation value in the last row of Table 6. It can also be concluded that flux increases with the increase of flow rate and pressure, and especially with increase in
0.40
0.40
Flow rate [ m3 h-1 ]
10.9
8.75
10.9 8.50
10.3 9.70
0.35
9.10 -2 -1
Flux (exp) [ l m h ] 0.30
Other data ∆ p=1 [bar]
7.90
0.25
t-values
32.79082 7.79E)03 142.6764 2.37E)02 )30.94981 1.46E)02 )14.45167 2.16E)01 P 2 D J =N ¼ 4.826E)08
temperature. This is in agreement with the expectations as well as with the results of reported investigations (Balakrishnan et al., 2000; Gyura et al., 2002). As far as the concentration factor is concerned, its increase reduces permeate flux. Maximum flux is obtained at minimum concentration factor at the beginning of the process, while the minimum flux is obtained at a concentration factor around a value of 1.5. When the parameters from Table 6 are applied to Eq. (10) the duration of ultrafiltration process is predicted. The results of these calculations are presented in
Flow rate [ m3 h-1 ]
Bi1 Bi2 Bi3 Bi4
Flow rate as variable (index i ¼ 3)
o
t= 30 [ C]
10.4 9.85
0.35
-2
-1
Flux (cal) [ l m h ] 9.30
8.20
0.30
Other data ∆ p=1 [bar]
7.65
0.25
0
7.10
7.30
0.20 1.0
1.2
1.4
t= 30 [ C]
0.20
6.10
6.70
6.55
1.6
1.8
1.0
1.2
1.4
1.6
1.8
Concentration factor
Concentration factor
Fig. 4. Experimental flux and predicted flux as the function of two variables: concentration factor and flow rate.
60
60
-2
50
8.85
45 9.30
-2
10.7 10.3
-1
Flux (exp) [ l m h ]
50
Other data
9.90
3 -1
Q=0.2 [m h ] ∆ p=4 [bar]
45 40
8.40
35
8.70
o
Temperature [ C ]
Q=0.2 [m h ] ∆ p=4 [bar]
10.7 10.2
-1
Flux (cal) [ l m h ]
55
3 -1
o
Temperature [ C ]
55
40
11.1
Other data
11.1
8.30 9.50 9.10
35
7.95 9.75
30 1.2
1.4
1.6
1.8
Concentration factor
2.0
30 1.0
1.2
1.4
1.6
1.8
2.0
Concentration factor
Fig. 5. Experimental flux and predicted flux as the function of two variables: concentration factor and temperature.
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82 4.0
Transmembrane pressure [bar]
9.55
8.20
9.10
3.5
8.65
3.0
7.75 -2 -1
Flux (exp) [ l m h ] 2.5
Other data 3 -1
2.0
Q=0.2 [m h ] o
t=30 [ C]
7.30
1.5 6.40
1.0 1.0
6.85
1.2
1.4
1.6
1.8
Transmembrane pressure [bar]
4.0
79
9.15
9.60
3.5
8.70 8.25
3.0
7.80
-2
-1
Flux (cal) [ l m h ]
2.5
Other data
2.0
3 -1
Q=0.2 [m h ]
7.35
o
1.5
t=30 [ C]
6.90 6.45
1.0 1.0
1.2
Concentration factor
1.4
1.6
1.8
Concentration factor
Fig. 6. Experimental flux and predicted flux as the function of two variables: concentration factor and transmembrane pressure.
4.3. Three-variable flux functions and duration of UF process
160
0
t=60 [ C]
140
Process duration [min]
Figs. 7–9 where they are compared with the measured values. Fig. 7 presents curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 , Fig. 8 presents curves of the measured and predicted times at 30 and 60 C and Fig. 9 presents curves of the measured and predicted times for the transmembrane pressures 1 and 4 bar. In all cases, presented in Figs. 7–9, an increase in the difference between estimated and experimental times (during the process development) are noticed. At the end-points of the experiments (after 2.5 h), underestimation reaches 25–30% due to the incomplete rejection of undesired compounds by the membrane.
0
t=30 [ C]
120 100 80
Other data 60
3 -1
Q=0.2 [m h ] ∆ p= 4 [bar]
40 20 1.0
1.2
1.4
1.6
1.8
2.0
Concentration factor Fig. 8. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and temperature.
When the concentration factor, flow rate and temperature are taken as the independent variables, in the 160
160 3
Process duration [min]
Q=0.4 [m h ]
140
Process duration [min]
-1
120 3
-1
Q=0.2 [m h ]
100 80
Other data ∆ p= 1 [bar]
60
o
40
t= 30 [ C]
∆ p=4 [bar]
140 120
∆ p=1 [bar]
100 80 60
Other data
40
Q= 0.2 [m h ]
3 -1
o
20
t= 30 [ C]
20 1.2
1.4
1.6
1.8
2.0
Concentration factor Fig. 7. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and flow rate.
1.0
1.2
1.4
1.6
1.8
Concentration factor Fig. 9. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and transmembrane pressure.
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
same relation, it was concluded that a seven-parameter equation proved better than the five-parameter regression (see Table 1). In the case when concentration factor, flow rate and transmembrane pressure are independent variables, the flux equation with linear terms for Q and Dp proved correct. Two examples illustrate this case; 40 experimental points, with variable CF ; Q and t but constant Dp ¼ 4 bar, are selected as well as 40 experimental points, with variable CF ; Q and Dp but constant t ¼ 60 C. After the application of the regression analysis method to the measured points, adequate B-parameters were calculated, as presented in Table 7. In accordance with the t-values from Table 7, the important terms of the seven-parameter regression are the concentration factors, followed by the flow rate to the second degree, and then all the others. In the case of five-parameter regression, the sequence of significance is as follows: pressure term, flow rate term and concentration factor terms. Three-variable regressions for permeate fluxes are impossible to present graphically. However, according to the deviations from Table 7, higher precision characterizes the concentration factor––flow rate–tempera-
ture regression than the concentration factor––flow rate–transmembrane pressure regression. When the parameters, defined in Table 7, are applied to Eq. (10), the duration of the ultrafiltration process is predicted. The results of these calculations are presented in Figs. 10 and 11 where they are compared with the measured values. Fig. 10 presents curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 and temperatures 30 and 60 C. Fig. 11 shows curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 and transmembrane pressures 1 and 4 bar. Better results are obtained in the case of the concentration factor––flow rate–temperature function (Fig. 10) than in the case of concentration factor––flow rate– transmembrane pressure function (Fig. 11). The reason lies in the fact that the function J 1 ðCF ; Q; tÞ is better for fitting the measured values than the J 1 ðCF ; Q; DpÞ. The trend of an increase in the difference between estimated and experimental times, with the process development, is typical of the three-variable examples as well. In these cases underestimation has values in an interval: 10–40% at the end-points, after 2.5 h of the experiments, which can be associated with the incomplete rejection of non-sucrose compounds.
Table 7 Parameters in the three-variable regression for fluxes in m3 m2 h1 Flow rate and temperature as variables (index i ¼ 7)
Flow rate and pressure as variables (index i ¼ 8)
B-parameters
B-parameters
t-values
10.39529 4.70E)05 53.7585 5.69E)03 )6.219534 2.18E)03 )13.71153 1.51E)05 )209.6563 1.39E)04 2502.383 8.04E)05 )17541.88 2.82E)05 P 2 D J =N ¼ 0.9143E)06 m3 m2 h1
Bi1 Bi2 Bi3 Bi4 Bi5 Bi6 Bi7
t-values
19.6139 1.88E)03 210.4558 1.64E)02 )50.56572 1.33E)02 )98.32354 2.11E)02 )25.86839 7.12E)02 – – – – P 2 D J =N ¼ 1.317E)06 m3 m2 h1
160
160 3
3 -1
Q=0.2 [m h ]
3 -1
Q=0.4 [m h ]
0
t=30 [ C]
0
t=30 [ C]
120 100 80 60
Other data
40
t= 30 [ C] ∆ p=4 [bar]
o
20 1.0
1.2
1.4
1.6
1.8
Concentration factor
2.0
Process duration [min]
Process duration [min]
140
140
Other data
120
t= 60 [ C] ∆ p=4 [bar]
o
-1
Q=0.2 [m h ] 0
t=60 [ C]
100 80 3 -1
Q=0.4 [m h ]
60
0
t=60 [ C]
40
2.2
20 1.0
1.2
1.4
1.6
1.8
2.0
2.2
Concentration factor
Fig. 10. Experimental duration (full lines) and predicted duration (dash lines) as the function of three variables: concentration factor, flow rate and temperature.
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82 160
81
160 3 -1
120
Other data ∆ p=1 [bar] 0
t=60 [ C] 100 80
3
-1
Q=0.4 [m h ] ∆p=1 [bar]
60 40 20 1.0
1.1
1.2
1.3
1.4
Q=0.2 [m h ] ∆ p=4 [bar]
140
Q=0.2 [m h ] ∆ p=1 [bar]
1.5
Process duration [min]
Process duration [min]
140
3 -1
120
Other data ∆p= 4 [bar] 0
100
t=60 [ C]
80 60
3 -1
Q=0.4 [m h ] ∆p=4 [bar]
40 20 1.0
1.2
Concentration factor
1.4
1.6
1.8
2.0
Concentration factor
Fig. 11. Experimental duration (full lines) and predicted duration (dash lines) as the function of three variables: concentration factor, flow rate and transmembrane pressure.
The incomplete rejection of non-sucrose compounds is responsible for the prolongation of the separation process because of the fact that the rejection coefficient is related to the concentration factor (according to Eq. (7)). On the other hand, it may be interesting to include sucrose rejection by the membrane as well, while considering the processing time. The data that can be used for an estimation of the loss of sugar due to the rejection are purity ratios in both permeate and retentate, at the end the experiments. They are defined as the ratios between sucrose content and the total dry matter in the solution and they might serve for an estimation of sucrose distribution between permeate and retentate. An average value of sucrose rejection coefficient, for all working conditions, was estimated at a level of approximately 0.3–0.35. This might be an additional reason for deviation of the calculated from real times.
When the duration of the sugar syrup UF process is considered, it can be concluded that the suggested model gives shorter times compared with the real times. This underestimation increases with the process development, reaching 40% at the end of the analysed experiments, and must be the consequence of partial rejection. Obviously, the polyethersulfone membrane does not reject coloured and/or colloidal compounds from the sugar syrup completely. The sugar loss due to the partial accumulation of the sucrose at the membrane surface can also be responsible for the difference between the calculated and real times. The suggested model is a useful tool for performing numerical experiments on sugar syrup purification, which can be improved and extended by taking into account other membranes, working conditions or characteristics of the purified solutions.
5. Conclusion Acknowledgements The ultrafiltration of sugar syrup was mathematically described, by a series of permeate flux functions, as well as the mass balance of the non-sucrose compounds. The permeate flux equations cover the influence of working conditions while the mass balance equation enables prediction of the duration of the whole process. Although the equations are derived based on the particular set of experiments, mathematical model provides a form of generalization. By processing the measured data, a kind of statistical judgement, concerning adequacy of the accepted types of the flux relations as well as significance of particular independent variables, is achieved. So, the general conclusion, concerning sugar syrup purification, might be that permeate flux depends on the linear change of flow rate, temperature and transmembrane pressure but on the quadratic change of the concentration factor. The flux increases with increase in Q, t and Dp values but decreases with increase in the concentration factor.
The authors acknowledge the support of the Ministry of Science, Technology and Development of Serbia (contract no. 1362) as well as assistance of colleagues from the University Szent Istvan, Budapest, Hungary.
References Balakrishnan, M., Dua, M., & Bhagat, J. J. (2000). Effect of operating parameters on sugarcane juice ultrafiltration: Results of a field experience. Separation and Purification Technology, 19, 209– 220. Bubnik, Z., Hinkova, A., & Kadlec, P. (1998). Crossflow micro- and ultrafiltration applied on ceramic membranes in impure sugar solutions. Czechoslovakian Journal of Food Science, 16(1), 29–34. Decloux, M., Tatoud, L., & Mersad, A. (2000). Removal of colorants and polysaccharides from raw cane sugar remelts by ultrafiltration. Zuckerindustrie, 125, 106–112. Durante, R. M. B. O., Santos, E. B. H., & Durante, A. C. (2001). Comparison between diafiltration and concentration operation
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modes for the determination of permeation coefficients of humic substances through ultrafiltration membranes. Analytica Chimica Acta, 442, 155–164. s, Z., Vatai, Gy., & Bekassy Molnar, E. (2002). Gyura, J., Sere Separation of non-sucrose compounds from syrup of sugar beet processing by ultra- and nanofiltration using polymer membranes. Desalination, 148, 49–56. Jaffrin, M. Y., & Charrier, J. Ph. (1994). Optimization of ultrafiltration and diafiltration processes for albumin production. Journal of Membrane Science, 97, 71–81. Monclin, J. P., & Willett, S. C. (1996). The ABC process for the direct production of refined sugar from sugar cane mixed juice. In
Proceedings workshop on separation processes in the sugar industry (pp. 16–21). Rautenbach, R., & Albrecht, R. (1989). Membrane processes. New York: Wiley. Vercelloti, J. R., Clarke, M. A., Godshall, M. A., Blanco, R. S., Patout, W. S., & Florence, R. A. (1998). Chemistry of membrane separation process in sugar industry applications. Zuckerinustrie, 123, 736–745. Vern, C. (1995). The beet sugar factory in the future. International Sugar Journal, 97, 310–314. Willett, C. C. (1997). The ABC process for the production of refined sugar from beets. International Sugar Journal, 99, 48–51.
Modelling of ultrafiltration of non-sucrose compounds in sugar beet processing s, Miodrag Tekic Mirjana Djuric *, Julianna Gyura, Zoltan Zavargo, Zita Sere Faculty of Technology, University of Novi Sad, 21000 Novi Sad, Bulevar cara Lazara 1, Serbia and Montenegro, Yugoslavia Received 29 July 2003; accepted 9 December 2003
Abstract This paper is a contribution to mathematically describing sugar syrup purification, i.e. ultrafiltration of non-sucrose compounds, based on the experimental data, acquired in a series of batch experiments on flat polyethersulfone cross-flow membranes. Experiments were performed to final concentration factors not less than 1.5 and not greater than 2.3, with flow rates 0.2 and 0.4 m3 h1 , at the temperatures 30 and 60 C and for the transmembrane pressures 1 and 4 bar. The suggested mathematical model enables prediction of separation time if the permeate flux models as well as the initial and final concentrations of undesired nonsucrose compounds are known. In this paper, the flux models are proposed as functions of concentration factor (CF ), flow rate (Q), temperature (t) and transmembrane pressure (Dp) as independent variables. So, the following one-, two- and three-variable functions are suggested: J 1 ðCF Þ, J 1 ðCF ; QÞ, J 1 ðCF ; tÞ, J 1 ðCF ; DpÞ, J 1 ðCF ; Q; tÞ and J 1 ðCF ; Q; DpÞ. By the regression analysis method, parameters in the flux models are determined and they are used for the calculation of process duration. It is concluded that the predicted times underestimate the real times by 10–40% in particular cases. One can assume that the incomplete rejection of nonsucrose compounds by polyethersulfone membranes as well as the loss of sugar due to accumulation at the membrane is responsible for process prolongation. 2003 Elsevier Ltd. All rights reserved. Keywords: Mathematical modelling; Ultrafiltration; Permeate flux; Process duration
1. Introduction Purification of the syrup––an intermediate product in sugar beet processing is an important operation, which precedes sucrose crystallization. For many years, this problem was solved by chemically induced precipitation of undesired non-sucrose compounds, after adding calcium oxide and carbon dioxide to the syrup solution. A relatively new method, applied for a reduction of nonsucrose compounds in sugar beet syrup, is a membrane separation, i.e. an ultrafiltration (UF) process (Bubnik, s, Vatai, & Hinkova, & Kadlec, 1998; Gyura, Sere Bek assy Moln ar, 2002; Rautenbach & Albrecht, 1989; Vercelloti et al., 1998; Vern, 1995; Willett, 1997). Some experiences acquired from sugar cane juice ultrafiltration (Balakrishnan, Dua, & Bhagat, 2000; Decloux, Tatoud, & Mersad, 2000; Monclin & Willett, 1996) can *
Corresponding author. Tel.: +381-21-450-277; fax: +381-21-450413. E-mail address: [email protected] (M. Djuric). 0260-8774/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2003.12.005
be applied to the purification of syrup in sugar beet processing. Despite the extensive work on this subject, many theoretical and practical aspects of sugar syrup ultrafiltration are not completely understood. Two things are especially important; the first one is the selection of adequate membrane, according to the knowledge of sizes of the non-sucrose molecules (Bubnik et al., 1998; Decloux et al., 2000; Gyura et al., 2002). The performances of the selected membrane effect the permeate fluxes which, at the other hand, influence the duration of the separation process, determining both the capacity of the equipment (related to the cost of the whole operation) and the degree of purification. The second important question is the selection of working conditions, such as: concentration factor, flow rate of the solution at the membrane, temperature of the sugar syrup, transmembrane pressure, etc. (Balakrishnan et al., 2000; Rautenbach & Albrecht, 1989). The conditions are closely related to the previous stages of sugar beet processing (e.g. temperature) as well as to the performance of the available separation equipment. It is
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
Nomenclature B parameters in flux equations C [g l1 ”kg m3 ] non-sucrose compounds concentration CF concentration factor D; E; F parameters in flux equations Dp [bar] transmembrane pressure J [l m2 h1 ], [m3 m2 h1 ] permeate flux m [kg] mass N number of experimental points Pc permeation coefficient Q [m3 h1 ] solution flow rate R rejection value
particularly important to find out how they affect permeate flux and duration of the separation process. The aim of this paper is to investigate the working conditions problem, assuming that the type of membrane has already been chosen. The experimentally determined permeate fluxes are mathematically described through a series of functions which are introduced into the mathematical model of the process, to give the separation time.
2. The mathematical model of UF process with variable permeate flux The mathematical model of UF of sugar beet syrup is relatively simple. It is based on the mass conservation law, applied to the non-sucrose substances that should be removed, or at least reduced from the initial (C0 ) to the final (Cf ) concentration, by the UF process. The differential equation for the mass balance is a relation which brings together important quantities, such as the volume of the solution, concentrations of substances present, permeate flux, duration (time) of the process, membrane surface and its other characteristics, etc.
2.1. Mass balance equation based on the non-sucrose compounds The mass conservation of the non-sucrose compounds in the sugar syrup can be expressed by a simple equation: dm ¼ 0 )
dm dðVCÞ dV dC ¼ ¼0)C þV ¼0 ds ds ds ds
ð1Þ
On the other hand, the change in the solution volume (V ) is related to the permeate flux (J ) and the membrane area (S) in a way:
S [m2 ] membrane surface T [h m2 kg1 ] normalized phase duration t [C] temperature s [h], [min] time, phase duration V [m3 ] volume of the solution in the system Subscripts 0 initial cal calculated exp experimental f final P permeate
dV ¼ JS ds
ð2Þ
If the non-sucrose compounds are completely rejected, their mass can be expressed in terms of the concentration and the solution volume: m ¼ VC or: V ¼ m=C. Substituting both Eq. (2) and the expression for volume into Eq. (1) gives: m dC dC JS 2 ¼0) ¼ C ð3Þ C ds ds m The ultrafiltration time can be determined from Eq. (3) as s in hours or as normalized duration T in hours per 1 kg of non-sucrose compounds using 1 m2 of membrane: Z Z Z m dC S dC s ¼ T ¼ ds ¼ ) ð4Þ S JC 2 m JC 2 JSC þ
The differential equation (4) can be integrated if the adequate J -function was known. It can be assumed that the flux of permeate remains constant, which is a rough approximation of the real process. However, the flux model can be defined as a function of relevant variables. 2.2. Permeate flux models This paper suggests several regression equations as mathematical models for permeate flux typical of sugar syrup purification. If the flux model has an appropriate form, the differential mass balance equation (4) can be solved analytically. As polynomials are suitable for integration, the flux equation should have a form that leads to the polynomial form of the integral. Such an approach was suggested by Jaffrin and Charrier (1994). The most important variable, which influences the permeate flux, is concentration of non-sucrose compounds or concentration factor, so that the flux function should contain these quantities. However, other independent variables can also affect the value of the flux. The most important are: flow rate of the solution through the system, temperature at the membrane and
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
transmembrane pressure (Balakrishnan et al., 2000; Gyura et al., 2002). The greater the number of relevant variables, the more realistic is the mathematical model of permeate flux. Here, three types of flux models will be introduced: (i) one-variable, (ii) two-variable and (iii) three-variable.
2.2.2. Flux as a function of two variables More complete mathematical description of permeate flux is achieved when flow rate, temperature and transmembrane pressure are taken into account together with the concentration factor, giving two-variable regressions: J 1 ðCF ; QÞ, J 1 ðCF ; tÞ and J 1 ðCF ; DpÞ. The simplest possible form of these regressions appear when assuming that the permeate flux changes with the linear change of Q, t and Dp. Consequently, two-variable regressions have a form given by Eq. (6) with D; E and F parameters presented in Table 1.
2.2.1. Flux as a function of one variable The collection of undesired compounds at the membrane is the aim of sugar syrup purification by an UF process. Therefore, it is very appropriate to express flux in terms of the concentration of non-sucrose compounds, which have to be removed during the purification: J11 ¼ B1;1 þ B1;2 C þ B1;3 C 2
2.2.3. Flux as a function of three variables In a real separation process, more than two working conditions might change or should be adjusted. So, the flux model should contain as many relevant variables as possible. Here, three-variable models are suggested: J 1 ðCF ; Q; tÞ and J 1 ðCF ; Q; DpÞ. Basic equation in these cases is Eq. (6), with D; E and F parameters presented in Table 1. As is obvious from Table 1, two alternatives for the function J 1 ðCF ; Q; tÞ are offered; the simple one assumes a linear change of Q, and t, while the alternative suggests a slightly better but more complicated relationship. However, both relations are easy to apply.
ð5Þ
In addition to concentration, the dimensionless concentration factor ðCF ¼ C=C0 Þ is even more interesting in a UF process interpretation. Having this in mind, the flux can be expressed in the form: 2 C C 1 Ji ¼ F þ E þ D ð6Þ C0 C0 that allows a kind of generalization. If the concentration factor is the only variable, the index i takes the value: i ¼ 2, as presented in Table 1 where an overview of all suggested flux models can be seen. It also might be useful to mention that the nonsucrose compounds concentration in the retentate can be expressed in terms of other quantities, such as the initial volume in the tank V0 , the permeate volume VP , and the permeate coefficient Pc (Durante, Santos, & Durante, 2001). The following relationship is well known: ð1PcÞ ð1PcÞ V0 C V0 C ¼ C0 ) ¼ ð7Þ C0 V0 VP V0 VP
2.3. Duration of UF process Once defined, mathematical model (4) can be used for predicting the duration of the UF process. When the flux model (6), with the D; E and F parameters from Table 1, is substituted into Eq. (4) one obtains: 2 Z Cf D C þ E CC0 þ F C0 S s¼T ¼ dC ð9Þ m C2 C0 After solving the integral at the right hand side of Eq. (9) the solution can be found as the normalized duration (T ) of the UF process:
whereby, the coefficient Pc and the solute rejection R are connected by a simple expression: R ¼ 1 Pc
75
S s¼T m 1 Cf Cf C0 ¼ D 1 þ E ln F 1 C0 C0 C0 Cf
ð8Þ
Eqs. (5) and (6) are simple but do not include all relevant variables.
ð10Þ
Table 1 An overview of flux models Number of variables
Flux
Index i
Parameters D
E
F
One
J 1 ðCF Þ J 1 ðCF ; QÞ J 1 ðCF ; tÞ J 1 ðCF ; DpÞ
2 3 4 5 6 7 8
B2;3 B3;3 B4;3 B5;3 B6;3 B7;3 B8;3
B2;2 B3;2 B4;2 B5;2 B6;2 B7;2 B8;2
B2;1 B3;1 þ B3;4 Q B4;1 þ B4;4 t B5;1 þ B5;4 Dp B6;1 þ B6;4 Q þ B6;5 t B7;1 þ QðB7;4 þ B7;5 QÞ þ t1 ðB7;6 þ B7;7 t1 Þ B8;1 þ B8;4 Q þ B8;5 Dp
Two
Three
J 1 ðCF ; Q; tÞ J 1 ðCF ; Q; DpÞ
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
for various values of the initial and final concentrations of the non-sucrose compounds.
7 3
3. Experimental
8
4
The syrup solution, which was exposed to ultrafiltration in the laboratory plant, was prepared for the separation process when the molasses, taken from the factory, was diluted (1:1) with distilled water. The main characteristics of the solution are given in Table 2. Ultrafiltration was performed using a flat cross-flow polyethersulfone membrane, the characteristics of which are given in Table 3. The capacity of the laboratory plant was 2 l of the initial syrup. Coloured and colloidal compounds were collected in batch experiments. A simplified schematic of the plant is presented in Fig. 1. The syrup solution from the reservoir (1) flows towards the membrane module (3) being forced by a 0.55 kW pump (2). Through the membrane module the solution flows tangentially. Permeate is taken away from the system continuously, while the retentate returns to the tank through a flow-meter (4) where its volume rate is measured. Flow rate of the solution is adjusted by the control valve (5) positioned in the pipeline. Pressure is controlled by the valve (6) and registered by the manometer (7). Temperature of the sugar syrup is kept at a constant value by thermostat (9) and it is measured by thermometer (8), which is built into the main pipeline. Permeate volume is measured after being captured in a cylinder (10). At the end of experiment, the retentate is transferred into a reservoir (11). Experiments were planned in accordance with the full factorial design presented in Table 4. After adjusting the variables from Table 4, the experiment started. The permeate volume was measured Table 2 Characteristics of the initial syrup (diluted 1:1) Statistical parameter
Dry matter (%)
pH
Invert (mass%/ d.m.)
Purity ratio Q (%)
Average value ðxÞ Standard deviation ðrn1 Þ
39.2 0.01
8.23 0.01
0.186 0.02
61.73 0.00
Table 3 Characteristics of membrane Characteristics
Membrane for UF
Manufacturer Type of membranes
Zoltek Rt MAVIBRAN, HU FS-20 Asymmetrical porous thin layer Polyethersulfone 15–20 470 300–390 6 60
Material MWCO (kDa) Active surface (cm2 ) Thickness (lm) Maximal pressure (bar) Maximal temperature (C)
6
9 1 5 10 2
11
(1) reservoir, (2) pump, (3) membrane module, (4) flow- meter, (5) and (6) valves (7) manometer, (8) thermometer, (9) thermostat, (10) cylinder for permeate volume measurements, (11) reservoir for retentate
Fig. 1. Scheme of the laboratory plant.
Table 4 Plan of experiments Number of experiment
Flow-rate (m3 h1 )
Temperature (C)
Transmembrane pressure (bar)
1 2 3 4 5 6 7 8
0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4
30 30 60 60 30 30 60 60
1 4 1 4 1 4 1 4
every 15 min so that permeate flux was defined. The concentration factor was also estimated from the flux data every 15 min. For this purpose the relation (7), between the initial volume of the syrup and the actual volume of the permeate, was applied with the assumption that the rejection of non-sucrose compounds by polyethersulfone membrane is complete (R ¼ 1). Each experiment lasted 2.5 h. After this period of time, the lowest estimated concentration factor was 1.5 while the greatest one was 2.3. The minimum value of permeate flux was 5.62 l m2 h1 while the maximum value was 9.8 l m2 h1 . In this way, a data base was acquired with the concentration factor, flow-rate, temperature and transmembrane pressure as the independent variables and the permeate flux as dependent variable. Each quantity was measured at least three times, and the mean values were used for further analysis.
4. Discussion of experimental and calculated results Parameters in the suggested flux functions (in Table 1) were determined by processing the available experimental data (J , CF , Q, t and Dp). For this purpose,
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
4.1. One-variable flux function and duration of UF process When the concentration factor is taken as the only independent variable, the mathematical model for permeate flux through the membrane is the regression equation (6). In order to illustrate such a case, 10 experimental points, with variable CF and constant Q ¼ 0:2 m3 h1 , t ¼ 30 C and Dp ¼ 4 bar, were selected. After application of the regression analysis method to the measured points, adequate B-parameters were calculated, as presented in Table 5. The analysis of the corresponding t-values shows that the linear term is more important than the quadratic term, which means that permeate flux decreases almost linearly with increase in CF -value. The comparison between measured and calculated flux values, presented in Fig. 2, leads to the conclusion that a very adequate type Table 5 Parameters in the one-variable regression for fluxes in m3 m2 h1 B-parameters
Other data
10.5
Flux [lm-2 h-1]
3
-1
Q=0.2 [m h ]
10.0
o
t= 30 [ C] ∆ p= 4 [bar]
9.5 9.0 8.5 8.0 7.5 1.2
Other data 3
-1
Q=0.2 [m h ] 120 100
o
t= 30 [ C] ∆ p= 4 [bar]
80 60 40 20 0 1.0
1.2
1.4
1.6
1.8
Concentration factor Fig. 3. Experimental duration (full line) and predicted duration (dash line) as the function of concentration factor.
of regression has been chosen. Small value of the deviation (11), given in Table 5, proves this fact. With the parameters of the regression, defined in Table 5, Eq. (10) is solved and the duration of the UF process calculated. The results of these calculations are presented in Fig. 3 where they are compared with the measured values. It can be seen that calculated times are lower than the real times. The difference between these two increases during the process, reaching 25% at the end after 2.5 h of experiment. This is a consequence of errors in the flux estimation but it is much more the consequence of the incomplete rejection of the nonsucrose compounds by the polyethersulfone membrane. 4.2. Two-variable flux functions and duration of UF process
11.0
1.0
140
t-values
10.21926 6.80E)03 97.21664 4.51E)02 )16.43666 2.17E)02 P 2 D J =N ¼ 1.55E)08 m3 m2 h1
B21 B22 B23
160
Process duration [min]
the well-known technique––the regression analysis method––was applied. The significance of each parameter in the regressions is verified by a statistical measure known as the t-value. The adequateness of the introduced flux functions is proved by calculation of the deviation: PN 2 ðJexp Jcal Þ d¼ 1 ð11Þ N
77
1.4
1.6
1.8
Concentration factor Fig. 2. Experimental flux (full line) and predicted flux (dash line) as the function of concentration factor.
When the concentration factor is taken as the independent variable together with the flow rate, or temperature or transmembrane pressure, the regressions from Table 1 are mathematical models of permeate flux. This case is illustrated by three examples; 20 experimental points, with variable CF and Q but constant t ¼ 30 C and Dp ¼ 1 bar, were selected as the first example. Then, 20 experimental points, with variable CF and t but constant Q ¼ 0:2 m3 h1 and Dp ¼ 4 bar, were chosen as well as 20 experimental points with variable CF and Dp but constant Q ¼ 0:2 m3 h1 and t ¼ 30 C. Application of the regression analysis method to the measured points, gives adequate B-parameters, as presented in Table 6. The calculated t-values indicate that the parameter Bi4 , connected to the flow rate (i ¼ 3), temperature (i ¼ 4) and transmembrane pressure (i ¼ 5), was the most important for all two-variable regressions. This means that the mentioned working conditions are very
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
Table 6 Parameters in the two-variable regressions for fluxes in m3 m2 h1 Temperature as variable (index i ¼ 4)
Pressure as variable (index i ¼ 5)
B-parameters
B-parameters
B-parameters
t-values
2.152593 9.22E)04 248.1224 7.78E)02 )69.52132 6.46E)02 )274.2582 3.81E)01 P 2 D J =N ¼ 5.502E)08
t-values
11.3249 1.04E)02 120.7277 8.52E)02 )26.29468 5.72E)02 )0.4872019 2.21E)01 P 2 D J =N ¼ 5.082E)08
important in the real as well as in the mathematically described process. The comparison between measured and calculated flux values can be seen in Figs. 4–6 for all two-variable cases. It shows that the calculated flux values predict very well experimental fluxes. The best fit is achieved when the concentration factor–transmembrane pressure––regression is applied (Fig. 6) as it shows the smallest deviation value in the last row of Table 6. It can also be concluded that flux increases with the increase of flow rate and pressure, and especially with increase in
0.40
0.40
Flow rate [ m3 h-1 ]
10.9
8.75
10.9 8.50
10.3 9.70
0.35
9.10 -2 -1
Flux (exp) [ l m h ] 0.30
Other data ∆ p=1 [bar]
7.90
0.25
t-values
32.79082 7.79E)03 142.6764 2.37E)02 )30.94981 1.46E)02 )14.45167 2.16E)01 P 2 D J =N ¼ 4.826E)08
temperature. This is in agreement with the expectations as well as with the results of reported investigations (Balakrishnan et al., 2000; Gyura et al., 2002). As far as the concentration factor is concerned, its increase reduces permeate flux. Maximum flux is obtained at minimum concentration factor at the beginning of the process, while the minimum flux is obtained at a concentration factor around a value of 1.5. When the parameters from Table 6 are applied to Eq. (10) the duration of ultrafiltration process is predicted. The results of these calculations are presented in
Flow rate [ m3 h-1 ]
Bi1 Bi2 Bi3 Bi4
Flow rate as variable (index i ¼ 3)
o
t= 30 [ C]
10.4 9.85
0.35
-2
-1
Flux (cal) [ l m h ] 9.30
8.20
0.30
Other data ∆ p=1 [bar]
7.65
0.25
0
7.10
7.30
0.20 1.0
1.2
1.4
t= 30 [ C]
0.20
6.10
6.70
6.55
1.6
1.8
1.0
1.2
1.4
1.6
1.8
Concentration factor
Concentration factor
Fig. 4. Experimental flux and predicted flux as the function of two variables: concentration factor and flow rate.
60
60
-2
50
8.85
45 9.30
-2
10.7 10.3
-1
Flux (exp) [ l m h ]
50
Other data
9.90
3 -1
Q=0.2 [m h ] ∆ p=4 [bar]
45 40
8.40
35
8.70
o
Temperature [ C ]
Q=0.2 [m h ] ∆ p=4 [bar]
10.7 10.2
-1
Flux (cal) [ l m h ]
55
3 -1
o
Temperature [ C ]
55
40
11.1
Other data
11.1
8.30 9.50 9.10
35
7.95 9.75
30 1.2
1.4
1.6
1.8
Concentration factor
2.0
30 1.0
1.2
1.4
1.6
1.8
2.0
Concentration factor
Fig. 5. Experimental flux and predicted flux as the function of two variables: concentration factor and temperature.
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82 4.0
Transmembrane pressure [bar]
9.55
8.20
9.10
3.5
8.65
3.0
7.75 -2 -1
Flux (exp) [ l m h ] 2.5
Other data 3 -1
2.0
Q=0.2 [m h ] o
t=30 [ C]
7.30
1.5 6.40
1.0 1.0
6.85
1.2
1.4
1.6
1.8
Transmembrane pressure [bar]
4.0
79
9.15
9.60
3.5
8.70 8.25
3.0
7.80
-2
-1
Flux (cal) [ l m h ]
2.5
Other data
2.0
3 -1
Q=0.2 [m h ]
7.35
o
1.5
t=30 [ C]
6.90 6.45
1.0 1.0
1.2
Concentration factor
1.4
1.6
1.8
Concentration factor
Fig. 6. Experimental flux and predicted flux as the function of two variables: concentration factor and transmembrane pressure.
4.3. Three-variable flux functions and duration of UF process
160
0
t=60 [ C]
140
Process duration [min]
Figs. 7–9 where they are compared with the measured values. Fig. 7 presents curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 , Fig. 8 presents curves of the measured and predicted times at 30 and 60 C and Fig. 9 presents curves of the measured and predicted times for the transmembrane pressures 1 and 4 bar. In all cases, presented in Figs. 7–9, an increase in the difference between estimated and experimental times (during the process development) are noticed. At the end-points of the experiments (after 2.5 h), underestimation reaches 25–30% due to the incomplete rejection of undesired compounds by the membrane.
0
t=30 [ C]
120 100 80
Other data 60
3 -1
Q=0.2 [m h ] ∆ p= 4 [bar]
40 20 1.0
1.2
1.4
1.6
1.8
2.0
Concentration factor Fig. 8. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and temperature.
When the concentration factor, flow rate and temperature are taken as the independent variables, in the 160
160 3
Process duration [min]
Q=0.4 [m h ]
140
Process duration [min]
-1
120 3
-1
Q=0.2 [m h ]
100 80
Other data ∆ p= 1 [bar]
60
o
40
t= 30 [ C]
∆ p=4 [bar]
140 120
∆ p=1 [bar]
100 80 60
Other data
40
Q= 0.2 [m h ]
3 -1
o
20
t= 30 [ C]
20 1.2
1.4
1.6
1.8
2.0
Concentration factor Fig. 7. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and flow rate.
1.0
1.2
1.4
1.6
1.8
Concentration factor Fig. 9. Experimental duration (full lines) and predicted duration (dash lines) as the function of two variables: concentration factor and transmembrane pressure.
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M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82
same relation, it was concluded that a seven-parameter equation proved better than the five-parameter regression (see Table 1). In the case when concentration factor, flow rate and transmembrane pressure are independent variables, the flux equation with linear terms for Q and Dp proved correct. Two examples illustrate this case; 40 experimental points, with variable CF ; Q and t but constant Dp ¼ 4 bar, are selected as well as 40 experimental points, with variable CF ; Q and Dp but constant t ¼ 60 C. After the application of the regression analysis method to the measured points, adequate B-parameters were calculated, as presented in Table 7. In accordance with the t-values from Table 7, the important terms of the seven-parameter regression are the concentration factors, followed by the flow rate to the second degree, and then all the others. In the case of five-parameter regression, the sequence of significance is as follows: pressure term, flow rate term and concentration factor terms. Three-variable regressions for permeate fluxes are impossible to present graphically. However, according to the deviations from Table 7, higher precision characterizes the concentration factor––flow rate–tempera-
ture regression than the concentration factor––flow rate–transmembrane pressure regression. When the parameters, defined in Table 7, are applied to Eq. (10), the duration of the ultrafiltration process is predicted. The results of these calculations are presented in Figs. 10 and 11 where they are compared with the measured values. Fig. 10 presents curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 and temperatures 30 and 60 C. Fig. 11 shows curves of the measured and predicted times for the flow rates 0.2 and 0.4 m3 h1 and transmembrane pressures 1 and 4 bar. Better results are obtained in the case of the concentration factor––flow rate–temperature function (Fig. 10) than in the case of concentration factor––flow rate– transmembrane pressure function (Fig. 11). The reason lies in the fact that the function J 1 ðCF ; Q; tÞ is better for fitting the measured values than the J 1 ðCF ; Q; DpÞ. The trend of an increase in the difference between estimated and experimental times, with the process development, is typical of the three-variable examples as well. In these cases underestimation has values in an interval: 10–40% at the end-points, after 2.5 h of the experiments, which can be associated with the incomplete rejection of non-sucrose compounds.
Table 7 Parameters in the three-variable regression for fluxes in m3 m2 h1 Flow rate and temperature as variables (index i ¼ 7)
Flow rate and pressure as variables (index i ¼ 8)
B-parameters
B-parameters
t-values
10.39529 4.70E)05 53.7585 5.69E)03 )6.219534 2.18E)03 )13.71153 1.51E)05 )209.6563 1.39E)04 2502.383 8.04E)05 )17541.88 2.82E)05 P 2 D J =N ¼ 0.9143E)06 m3 m2 h1
Bi1 Bi2 Bi3 Bi4 Bi5 Bi6 Bi7
t-values
19.6139 1.88E)03 210.4558 1.64E)02 )50.56572 1.33E)02 )98.32354 2.11E)02 )25.86839 7.12E)02 – – – – P 2 D J =N ¼ 1.317E)06 m3 m2 h1
160
160 3
3 -1
Q=0.2 [m h ]
3 -1
Q=0.4 [m h ]
0
t=30 [ C]
0
t=30 [ C]
120 100 80 60
Other data
40
t= 30 [ C] ∆ p=4 [bar]
o
20 1.0
1.2
1.4
1.6
1.8
Concentration factor
2.0
Process duration [min]
Process duration [min]
140
140
Other data
120
t= 60 [ C] ∆ p=4 [bar]
o
-1
Q=0.2 [m h ] 0
t=60 [ C]
100 80 3 -1
Q=0.4 [m h ]
60
0
t=60 [ C]
40
2.2
20 1.0
1.2
1.4
1.6
1.8
2.0
2.2
Concentration factor
Fig. 10. Experimental duration (full lines) and predicted duration (dash lines) as the function of three variables: concentration factor, flow rate and temperature.
M. Djuric et al. / Journal of Food Engineering 65 (2004) 73–82 160
81
160 3 -1
120
Other data ∆ p=1 [bar] 0
t=60 [ C] 100 80
3
-1
Q=0.4 [m h ] ∆p=1 [bar]
60 40 20 1.0
1.1
1.2
1.3
1.4
Q=0.2 [m h ] ∆ p=4 [bar]
140
Q=0.2 [m h ] ∆ p=1 [bar]
1.5
Process duration [min]
Process duration [min]
140
3 -1
120
Other data ∆p= 4 [bar] 0
100
t=60 [ C]
80 60
3 -1
Q=0.4 [m h ] ∆p=4 [bar]
40 20 1.0
1.2
Concentration factor
1.4
1.6
1.8
2.0
Concentration factor
Fig. 11. Experimental duration (full lines) and predicted duration (dash lines) as the function of three variables: concentration factor, flow rate and transmembrane pressure.
The incomplete rejection of non-sucrose compounds is responsible for the prolongation of the separation process because of the fact that the rejection coefficient is related to the concentration factor (according to Eq. (7)). On the other hand, it may be interesting to include sucrose rejection by the membrane as well, while considering the processing time. The data that can be used for an estimation of the loss of sugar due to the rejection are purity ratios in both permeate and retentate, at the end the experiments. They are defined as the ratios between sucrose content and the total dry matter in the solution and they might serve for an estimation of sucrose distribution between permeate and retentate. An average value of sucrose rejection coefficient, for all working conditions, was estimated at a level of approximately 0.3–0.35. This might be an additional reason for deviation of the calculated from real times.
When the duration of the sugar syrup UF process is considered, it can be concluded that the suggested model gives shorter times compared with the real times. This underestimation increases with the process development, reaching 40% at the end of the analysed experiments, and must be the consequence of partial rejection. Obviously, the polyethersulfone membrane does not reject coloured and/or colloidal compounds from the sugar syrup completely. The sugar loss due to the partial accumulation of the sucrose at the membrane surface can also be responsible for the difference between the calculated and real times. The suggested model is a useful tool for performing numerical experiments on sugar syrup purification, which can be improved and extended by taking into account other membranes, working conditions or characteristics of the purified solutions.
5. Conclusion Acknowledgements The ultrafiltration of sugar syrup was mathematically described, by a series of permeate flux functions, as well as the mass balance of the non-sucrose compounds. The permeate flux equations cover the influence of working conditions while the mass balance equation enables prediction of the duration of the whole process. Although the equations are derived based on the particular set of experiments, mathematical model provides a form of generalization. By processing the measured data, a kind of statistical judgement, concerning adequacy of the accepted types of the flux relations as well as significance of particular independent variables, is achieved. So, the general conclusion, concerning sugar syrup purification, might be that permeate flux depends on the linear change of flow rate, temperature and transmembrane pressure but on the quadratic change of the concentration factor. The flux increases with increase in Q, t and Dp values but decreases with increase in the concentration factor.
The authors acknowledge the support of the Ministry of Science, Technology and Development of Serbia (contract no. 1362) as well as assistance of colleagues from the University Szent Istvan, Budapest, Hungary.
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