ARTICLE pubs.acs.org/IECR
Prediction Strategy of Adsorption Equilibrium Time Based on Equilibrium and Kinetic Results To Isolate Taxifolin Tingting Liu, Ming Yang, Tianxin Wang, and Qipeng Yuan* State Key Laboratory of Chemical Resource Engineering, College of Life Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: We proposed a systematic strategy based on the studies about the adsorption performance of AB-8 macroporous adsorption resins for prediction of the equilibrium time in taxifolin isolation. Batch experiments were appropriately designed to obtain the best fitting model and to gain insight into the adsorption mechanisms. A modified Langmuir model was used to calculate the adsorption capacity at the adsorption equilibrium point (Qe). Subsequently, the reformed pseudo-second-order model was employed to predict the effects of the second-order rate index (k2Qe) on the adsorption time (t). Then, the effects of the initial adsorption factor (Ri) on the relative adsorption time (t/te) were predicted by the reformed intraparticle diffusion model. On the basis of mathematical deduction, the value of the adsorption equilibrium time (te) can be predicted, which may lead to reduced sampling in adsorption kinetics studies. Furthermore, the good agreement between experimental and predicted data confirmed the reliability of the proposed strategy.
can be fitted to either pseudo-first-order, pseudo-second-order, or Elovich and intraparticle diffusion models to examine whether the intraparticle diffusion is the limiting step of adsorption.22 Recently, it has been reported that characteristic curves can be plotted to yield various initial adsorption factors (Ri) using the intraparticle diffusion model. This produced four zones of initial adsorption according to Ri values from 1 to 0: 1 > Ri > 0.9 is weakly initial adsorption (zone 1), 0.9 > Ri > 0.5 is intermediately initial adsorption (zone 2), 0.5 > Ri > 0.1, is strongly initial adsorption (zone 3), and Ri < 0.1 is approaching completely initial adsorption (zone 4).23,24 When the fitting models are regarded as the best equation of a parameter that may be related to many other models, all these equations can be deduced to each other. However, there is no study about adsorption of taxifolin onto MARs using combined equilibrium and kinetic models. Batch experiments, which are helpful for designing and optimizng adsorption processes of the fixed bed, can provide information on the adsorption process, rate of adsorption, and adsorption mechanism of adsorbate onto adsorbent.2528 In adsorption kinetics studies, the adsorption equilibrium time is a key parameter which can be applied to guide the subsequent separation processes. However, determination of the adsorption equilibrium time is a time-consuming process because samples must be measured repeatedly until equilibrium, such as in the separation process of genistein and apigenin from pigeon pea roots29 and the preparative separation process of glabridin from Glycyrrhiza glabra L.30 Therefore, based on the theoretical studies of the thermodynamics and kinetics, it is necessary to propose a strategy to predict the adsorption equilibrium time during taxifolin isolation. In this study, we investigated the thermodynamics and kinetics of the AB-8 MAR adsorption performance during isolation of
1. INTRODUCTION Taxifolin (3,30 ,40 ,5,7-pentahydroxyflavanone) is the main constituent of the extracts of the rind of the Siberian larch Larix sibirica Leder and the Dahurian larch L. gmelinii Rupr. (Rupr.), syn. L. dahurica Turoz.1 Because of its flavonoid character, taxifolin can significantly dilate blood vessels, improve microcirculation, increase cerebral blood flow, and inhibit platelet aggregation.2 Taxifolin is widely used to treat cerebral infarction and sequela, cerebral thrombus, coronary heart disease, and angina pectoris.3 In recent years, it was found that taxifolin also has anti-inflammatory, antioxidant, and hepatoprotective activities.4 The conventional methods used to extract taxifolin result in a harvest of low-purity taxifolin. In order to increase its purity, further purification processes must be performed. The conventional methods to extract taxifolin with low-purity could be harvested.57 As one kind of functional polymeric material, macroporous adsorption resin (MAR) has shown its powerful separation ability and, thus, has been extensively used in chromatographic analysis,8 in medical treatment,9 and in purification of target compounds from many plant extracts.10 Except for the general advantages of the common adsorbents,11,12 MARs possess more favorable properties such as structural diversity and low cost, which makes them more promising.13 With improved synthesis technology of macroporous resins, many types of novel MARs, including AB-8,14 have been developed. Compared with classic ion exchange and adsorption resins, AB-8 resin has a larger surface area (>200 m2g1), is easily prepared (always derived from styrene), and can be easily regenerated by normal organic solvents. Therefore, AB-8 resin has been successfully used in separations of many biological compounds.1517 Although there have been many studies on the application of MARs and mechanisms of MAR adsorption, few studies about translation of theoretical results into engineering practice have been reported.1820 It is well-known that there are three classical thermodynamic equilibrium models: Langmuir isotherm, Freundlich, and Temkin isotherm.21 The kinetic data about the adsorption r 2011 American Chemical Society
Received: June 4, 2011 Accepted: November 23, 2011 Revised: November 23, 2011 Published: November 23, 2011 454
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taxifolin from the heartwood of L. gmelinii. By fitting the traditional models, a novel strategy was declared as follows: (1) A modified Langmuir model was established to express the relationship between the adsorption capacity at the adsorption equilibrium point (Qe) and the initial concentration (C0). Therefore, Qe was regarded as a known constant in the subsequent studies. (2) The reformed pseudo-second-order model was employed to predict the effects of the second-order rate index (k2Qe) on the adsorption time (t). (3) The effects of the initial adsorption factor (Ri) on the relative adsorption time (t/te) was predicted using the intraparticle diffusion model. (4) Based on our mathematical models, the value of the adsorption equilibrium time (te) can be predicted. In this way, the times of repeated sampling and measurements in the study of adsorption kinetics could be greatly reduced.
where kf and n are both Freundlich constants, kf reflects the adsorption capacity of the adsorbent, and n reflects the adsorption affinity of the adsorbent to the adsorbate.34,35 A linearized form of eq 4 can be written as ln Q e ¼ ln kf þ 1=n ln Ce
The Temkin isotherm contained a factor that explicitly took into account the adsorbentadsorbate interactions.36 The phenomenon that the heat of adsorption of all the molecules in the layer would decrease linearly with coverage was related to the interactions. The adsorption is characterized by a uniform distribution of binding energies, up to some maximum binding energy. The Temkin isotherm is expressed as RT ð6Þ ln Ce þ A Qe ¼ bT
2. MODELS
where RT/bT = B (J/mol), which is the Temkin constant related to heat of sorption, whereas A (mL/mg) is the equilibrium binding constant corresponding to the maximum binding energy. R (8.314 J/mol K) is the universal gas constant, and T (K) is the absolute solution temperature. 2.2. Adsorption Kinetics. Kinetic study is important to an adsorption process because it depicts the uptake rate of adsorbate and controls the residual time of the whole adsorption process. The pseudo-second-order model and Elovich model have the advantage that the value of the adsorption capacity at the adsorption equilibrium point (Qe) can be an unknown parameter.37 The pseudo-second-order model presented by Ho and McKay38 based on the sorption capacity of the solid phase is expressed as
2.1. Adsorption Isotherms. The Langmuir isotherm is the
best known and the most often used isotherm for the adsorption of solutes from a solution. The Langmuir equation (1) is based on a theoretical model which assumes monomolecular layer adsorption with a homogeneous distribution of adsorption energies and without mutual interaction between adsorbed molecules.31 Qe ¼
aCe , 1 þ bCe
a ¼ Q max KL ,
b ¼ KL
ð1Þ
In eq 1, Qe is the adsorption capacity at the adsorption equilibrium point (mg/g), a is the constant in the Langmuir isotherm (mL/g), Qmax is a constant related to the adsorptive capacity (mg/g), KL is the parameter which relates to the adsorption energy (mL/mg), and Ce is the equilibrium concentration of taxifolin solution (mg/ mL). A linearized form of eq 1 can be written as Ce 1 Ce ¼ þ KL Q max Qe Q max
dQ t ¼ k2 ðQ e Q t Þ2 dt
1 1 þ K L C0
ð2Þ
t 1 t ¼ þ Qt k2 Q e 2 Qe
ð8Þ
Qe and k2 can be obtained by a linear plot of t/Qt versus t. The Elovich equation39 was established through the work of Zeldowitsch, dealing with the adsorption of carbon monoxide on manganese dioxide. It has been used successfully for the description of the adsorption process.
ð3Þ
dQ t ¼ α expð βQ t Þ dt
C0 is the initial concentration of the adsorbate (mg/mL), and KL is a Langmuir constant (mL/mg). The value of RL indicates the shape of the isotherm to be either unfavorable (RL > 1), linear (RL = 1), favorable (0 < RL < 1), or irreversible (RL = 0). The Freundlich model33 has been extensively used in physical adsorption and chemical adsorption. It can be used to describe the adsorption behavior of monomolecular layers as well as that of multimolecular layers. It assumed that there is a heterogeneous distribution among the adsorption sites at different energies. This two-parameter model is widely employed for many different adsorbateadsorbent systems for liquid and gas phase adsorption. The experimental data were fitted to the Freundlich equation (4), describing the interaction of solutes with the resins: Q e ¼ kf Ce 1=n
ð7Þ
where k2 is the rate constant of pseudo-second-order adsorption (g 3 mg1 3 min1). Integrating eq 7 and noting that Qt = 0 at t = 0, the obtained equation can be rearranged into a linear form:
The Langmuir equation is converted to the linearized form with Ce as an independent variable and Ce/Qe as a dependent variable. The essential characteristics of the Langmuir isotherm can be expressed by a dimensionless constant called the separation factor or equilibrium parameter RL, which was first defined by Vermeulen et al.32 RL ¼
ð5Þ
ð9Þ
where α is the initial adsorption rate (mg 3 g1 3 min1) and β is desorption rate constant (g 3 mg1) during any experiment. To simplify the Elovich equation, αβt . t is assumed and by applying the boundary conditions Qt = 0 at t = 0 and Qt = Qt at t = t, the equation can be written as Qt ¼
1 1 lnðαβÞ þ lnðtÞ β β
ð10Þ
A plot of Qt versus ln(t) should yield a linear relationship with a slope of 1/β and an intercept of (1/β) ln(αβ). 2.3. Adsorption Mechanism. The intraparticle diffusion model was based on the theory proposed by Weber and Morris.40 The kinetic results can be analyzed by the intraparticle diffusion model to elucidate the diffusion mechanism. This model is
ð4Þ 455
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expressed as Q t ¼ kid t 1=2 þ C
ð11Þ
where C is the intercept and kid (mg/(g 3 min0.5)) is the intraparticle diffusion rate constant, which can be evaluated from the slope of the linear plot of Qt versus t1/2. McKay et al.41 indicated that extrapolation of the linear portion of the plot back to the axis provided intercepts which were proportional to the extent of the boundary layer thickness. That is, the boundary layer effect becomes greater with the increasing intercept. If intraparticle diffusion occurs, Qt versus t1/2 will be linear, and if the plot passes through the origin, the rate-limiting process is only due to the intraparticle diffusion. Otherwise, some other mechanism along with intraparticle diffusion is also involved.42
Figure 1. Trend of the equilibrium curves. Ce is the equilibrium concentration (mg/mL); Qe is the adsorption capacity at the adsorption equilibrium point (mg/g).
3. MATERIAL AND METHODS
3.5. Batch Experiments. 3.5.1. Adsorption Equilibrium Studies. Fifty milliliter solutions with different concentrations of
3.1. Chemicals and Reagents. The heartwood of L. gmelinii was collected from Heilongjiang Province, China. Taxifolin standards were purchased from Sigma-Aldrich Trading Co., Ltd. Ethanol and ethyl acetate, used for sample preparation and separation, were of analytical grade and were purchased from Beijing Chemical Factory (Beijing, China). Acetonitrile and trifluoroacetic acid (TFA) used for HPLC analysis were of chromatographic grade and were purchased from Dima Technology Inc. (VA, USA). All solutions prepared for HPLC were subject to filtrate through 0.45 μm nylon membranes. 3.2. Adsorbent. AB-8 macroporous resin was provided by NanKai University (Tianjin, China). Macroporous resin samples of 100 g (wet weight) were pretreated at 25 °C with 200 mL of 1 mol/L HCl and NaOH solutions successively for 4 h to remove the monomers and porogenic agents trapped inside the pores during the synthesis process, and then were subsequently washed by 400 mL of deionized water for 2 h. Pretreated adsorbents dried at 60 °C under vacuum were soaked with 200 mL of 95% ethanol for 12 h at 25 °C. Subsequently the ethanol was thoroughly replaced by filtration twice with 400 mL of deionized water. 3.3. HPLC Analysis of Taxifolin. Quantification of the taxifolin concentration was carried out on a Shimadzu LC-20AVP system with two LC-20AT solvent delivery units, an SPD-20A UVvis detector, a CTO-10ASVP column oven (Shimadzu, Kyoto, Japan), a T2000P workstation (Beijing, China), and a reversed phase C18 column (250 4.6 mm, 5 μm Diamodsil). The solvent system consisted of methanol as mobile phase A and 0.02% (v/v) TFA in water as mobile phase B. The gradient conditions were as follows: 014 min, 39% A; 1419 min, 35% A; 1931 min, 39% A. The column oven temperature was set at 30 °C. The flow rate was 1.0 mL/min, and 10 μL aliquots were injected into the column. Taxifolin was detected at 288 nm. 3.4. Preparation of L. gmelinii Crude Extracts. The heartwood of L. gmelinii was smashed into sawdust for extraction. A 100 g sample of sawdust were extracted with 80% ethanol, by reflux in a hot oil bath for 2 h under 80 °C; the insoluble residue was extracted again under the same conditions. The combined extraction solution was filtrated and then concentrated to dryness by removal of the ethanol solvent under reduced pressure in a rotary evaporator at 45 °C. The remaining solution was diluted to 400 mL by water and extracted by reflux in a hot oil bath for 2 h at 100 °C. This solution was centrifuged at a speed of 4800 rpm for 30 min to obtain sample solutions containing taxifolin in the concentration range of 0.140.784 mg/mL.
taxifolin were contacted with 2 g of pretreated resin (equal to 0.58 g of dry resin) in Erlenmeyer flasks. The mixtures were continuously stirred at 120 rpm for 24 h at 25, 35, 45, or 55 °C in a thermostatic bath. The equilibrium concentration Ce was detected by the HPLC method. According to eq 12 Qe ¼
ðC0 Ce ÞV W
ð12Þ
The equilibrium adsorbed amount Qe (mg/g) adsorbed onto resins could be calculated, where V (mL) is the volume of solutions, C0 (mg/mL) is the initial concentration of the sample, Ce (mg/mL) is the equilibrium concentration, and W (g) is the dry weight of resins. Subsequently, the equilibrium adsorption isotherms of taxifolin on the AB-8 macroporous resin were obtained at different temperatures. 3.5.2. Adsorption Kinetics. On account of static adsorption isotherms, the optimal concentration of the sample was obtained under different temperatures. Subsequently, 50 mL of taxifolin solution at the optimal concentration and 2 g of pretreated resin were contacted in Erlenmeyer flasks. Samples were taken at certain intervals, but the end time of continuous sampling was less than the equilibrium time. The taxifolin concentration Ct was detected by HPLC. The amount of the target product adsorbed by AB-8 macroporous resin at time t, Qt (mg/g), was calculated by eq 13: Qt ¼
ðC0 Ct ÞV W
ð13Þ
where Ct (mg/mL) was the taxifolin concentration at time t. W (g) was the dry weight of resins. 3.6. Verification Experiment. According to traditional experimental methods, the adsorption kinetic curve was plotted again. Samples were taken every 3 min after adsorption equilibrium. The adsorption equilibrium time was determined by the method of making diagrams.
4. RESULTS AND DISCUSSION 4.1. Adsorption Equilibrium and Model Fitting. To investigate the adsorption capacity and characterize the adsorption behavior of taxifolin, equilibrium curves were drawn. As shown in Figure 1, the adsorption capacities increased with the initial concentrations. When the initial concentration of taxifolin was 456
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Table 1. Isotherm Parameters by Linear Regression Method for the Sorption of Taxifolin by AB-8 Macroporous Resina
T (°C)
KL (mL/mg)
Langmuir Isotherm Qmax (mg/g) R2
Table 2. Kinetic Model Parameters by Linear Regression Method for the Sorption of Taxifolin by AB-8 Macroporous Resina T
equation
Pseudo-Second-Order Kinetic Model k2 Qe R2
C0
equation
(°C) (mg/mL) (g 3 mg1 min1) (mg/g)
25
105.21
32.63
0.9932
Ce/Qe = 0.03065Ce + 0.0003
35
62.58
32.63
0.9893
Ce/Qe = 0.03065Ce + 0.0005
25
0.5
0.006 621
34.3643 0.9960
t/Qt = 0.0291t + 0.1279
45
45.87
32.63
0.9819
Ce/Qe = 0.03065Ce + 0.0007
35
0.5
0.009 017
33.0033 0.9990
t/Qt = 0.0308t + 0.1052
55
33.75
32.63
0.9886
Ce/Qe = 0.03065Ce + 0.0009
45
0.5
0.014 02
30.1205 0.9992
t/Qt = 0.0336t + 0.0805
55
0.5
0.019 72
26.5957 0.9983
t/Qt = 0.0375t + 0.0713
T
RL
k2
Freundlich Isotherm 1/n R2
R2
Qe
equation
T (°C)
kf
25
43.60
0.2090
0.8748
ln Qe = 0.2090 ln Ce + 3.7751
35
42.74
0.2299
0.8664
ln Qe = 0.2299 ln Ce + 3.7552
25
0.018 65
0.006 621
34.3643 0.9960
t/Qt = 0.0291t + 0.1279
0.018 65
0.008 705
33.4481 0.9978
t/Qt = 0.0299t + 0.1027
equation
(g 3 mg1 min1) (mg/g)
(°C)
45
41.40
0.2507
0.8446
ln Qe = 0.2507 ln Ce + 3.7233
35
55
41.36
0.2767
0.9054
ln Qe = 0.2767 ln Ce + 3.7223
45
0.018 65
0.011 82
33.6700 0.9967
t/Qt = 0.0297t + 0.0746
55
0.018 65
0.011 92
33.4481 0.9913
t/Qt = 0.0299t + 0.0750
Temkin Isotherm T (°C) bT (J 3 mol1) A (mL 3 mg1) R2
equation
T
C0
(1/β)
(°C)
(mg/mL)
ln(αβ)
Elovich Model 1/β R2
equation
25
536.27
38.55
0.9413 Qe = 4.6200 ln Ce + 38.55
35
514.30
37.18
0.9406 Qe = 4.9791 ln Ce + 37.18
45
507.66
36.24
0.9347 Qe = 5.2079 ln Ce + 36.24
25
0.5
15.8140
4.0884
0.9489
Qt = 4.0884 ln t + 15.8140
0.9766 Qe = 5.4014 ln Ce + 35.64
35
0.5
14.6409
4.3150
0.9967
Qt = 4.3150 ln t + 14.6409
45
0.5
15.3452
3.7257
0.9971
Qt = 3.7257 ln t + 15.3452
55
0.5
15.6223
2.8980
0.9875
Qt = 2.8980 ln t + 15.6223
T
RL
(1/β)
1/β
R2
equation
55
504.87
35.64
a
T (°C) is the absolute solution temperature; Qmax (mg/g) is a constant related to the adsorptive capacity (mg/g); KL (mL/mg) is the parameter which relates to the adsorption energy; kf is the Freundlich constant; 1/n is the Freundlich exponent; bT (J 3 mol1) is the Temkin constant related to heat of sorption (J/mol); A (mL 3 mg1) is the Temkin isotherm constant (mL/mg); R2 is the value of the linear correlations.
ln(αβ)
(°C)
about 0.5 mg/mL, the adsorption capacity was close to the maximum adsorption capacity at 25 °C. Meanwhile, the remaining concentration was about 0.13 mg/mL and the value of RL was 0.018 65 (Table 1). For the separation of natural products, some parameters obtained by batch experiments are helpful to predict the performance of the fixed bed and save operation time in the process of designing the dynamic conditions.43The final purification of taxifolin will be carried out by the fixed bed which is filled with AB-8 macroporous adsorption resins. According to the method of the separation of glabridin,30 the concentration of taxifolin used for the subsequent kinetic study was selected by the equilibrium curves. We first studied the kinetics of this adsorption system at the fixed initial concentration of 0.5 mg/mL at four different temperatures. If the initial concentration was larger than 0.5 mg/mL, the adsorption capacity more closely approached the maximum adsorption capacity. However, the remaining concentration will increase with the initial concentration, which may lead to the waste of taxifolin during the industrial production. Then, we studied the system at a fixed RL value (0.018 65) and at four temperatures which were helpful to save the adsorption medium. For interpretation of the adsorption equilibrium data, the Freundlich isotherm, Langmuir isotherm, and Temkin isotherm were used. The parameters corresponding to the Langmuir, Freundlich, and Temkin equations for other experimental conditions are shown in Table 1. Refer to the fitting method of Rodrigues et al.44 The maximum adsorption capacity was fixed at 32.63 mg/g during the fitting of taxifolin isotherms in AB-8 MARs. It was found that the Langmuir equation fitted the experimental data
25
0.018 65
15.8140
4.0884
0.9489
Qt = 4.0884 ln t + 15.8140
35
0.018 65
16.2247
4.0543
0.9748
Qt = 4.0543 ln t + 16.2247
45
0.018 65
22.0961
1.7848
0.9551
Qt = 1.7848 ln t + 22.0961
55
0.018 65
20.6370
2.1883
0.9600
Qt = 2.1883 ln t + 20.6370
a
T (°C) is the absolute solution temperature; k2 is the rate constant of pseudo-second-order adsorption (g 3 mg1 3 min1); Qe is the adsorption capacity at the adsorption equilibrium point (mg/g); 1/β is the number of sites available for adsorption; (1/β) ln(αβ) is the adsorption quantity; RL is the equilibrium parameter; R2 is the value of the linear correlations.
best (correlation coefficient R2 > 0.98) in the three isotherms, which may be due to homogeneous distribution of active sites onto the surfaces of AB-8 macroporous resins. A similar result was reported for the adsorption of Paraquat dichloride from aqueous solution by activated carbon derived from used tires.45The values of KL indicated that the adsorption process was strong. KL decreased with increased temperature, which suggests that the adsorption process was exothermic. 4.2. Kinetic Model. Two kinetic models, the pseudosecond-order equation and the Elovich equation, were applied to study the kinetics of the adsorption process. The relative parameters were calculated by Origin 8.0. If the pseudo-second-order kinetic model was applicable, the plot of t/Qt versus t should show a linear relationship. Qe and k2 can be determined from the slope and the intercept of the plot. The result showed the best agreement between the experimental Qe values and the calculated Qe values (Table 2). In addition, the correlation coefficients for the second-order kinetic model were greater than 0.99 at different temperatures, which indicated that the applicability of the pseudo-second-order kinetic 457
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Table 3. Fitting Results of Intraparticle Diffusion Modela T (°C)
C0 (mg/mL)
equation
R2
25
0.5
Qt = 1.7701t1/2 + 19.8828
0.9838
35
0.5
Qt = 2.0866t1/2 + 18.0938
0.9985
45
0.5
Qt = 2.0938t1/2 + 17.1546
0.9926
55
0.5
Qt = 1.6388t1/2 + 16.9914
0.9953
T (°C)
RL
equation
R2
25
0.018 65
Qt = 1.7701t1/2 + 19.8828
0.9838
35
0.018 65
Qt = 1.9179t1/2 + 19.9733
0.9913
45 55
0.018 65 0.018 65
Qt = 2.0170t1/2 + 20.4166 Qt = 2.4877t1/2 + 18.5320
0.9970 0.9873
a
T (°C) is the absolute solution temperature; RL is the equilibrium parameter; R2 is the value of the linear correlations.
model to describe the adsorption process of taxifolin on AB-8 macroporous resin was rational. 4.3. Adsorption Mechanism. The adsorption process was usually governed by either the external mass transport rate or the intraparticle mass transport rate. Therefore, diffusive mass transfer was incorporated into the adsorption process. The intraparticle diffusion model based on the theory proposed by Weber and Morris41 was applied to identify the diffusion mechanism of taxifolin onto the AB-8 macroporous resin. The results showed that the values of the linear correlations of Qt and t1/2 were good (R2 g 0.9838) at different temperatures, but not all plots passed through the origin (Table 3). This indicated that, although external mass transfer was involved in this absorption process, the intraparticle diffusion was the main rate-limiting step. Maybe due to the wide distribution of the pore size of AB-8 macroporous adsorption resins, all straight lines have significant intercepts. Wu et al. observed similar phenomena in their study on the initial behavior of the intraparticle diffusion model.24
Figure 2. Prediction strategy of equilibrium time. C is the intercept of the fitted straight line about the intraparticle diffusion model; C0 is the initial concentration of the sample (mg/mL); Ce is the equilibrium concentration (mg/mL); d, m, and n are nonlinear fitting constants about the modified Langmuir model; k2 is the pseudo-second-order rate constant (g 3 mg1 3 min1); k2Qe (min1) is the second-order rate index; kid is the intraparticle diffusion rate constant (mg/(g 3 min0.5)); Qe is the adsorption capacity at adsorption equilibrium point (mg/g); Qt is the amount of target product adsorbed per unit of adsorbent at time t; Ri is the initial adsorption factor of the intraparticle diffusion model; t/te is the relative adsorption time; t is the adsorption time.
5. PREDICTION STRATEGY In the above studies, some best fitting expressions regarding a parameter were obtained. Subsequently, the theoretical model and the actual prediction were linked by a novel mathematical idea. The research process of the study is shown in Figure 2. This strategy was mainly based on sections 5.1, 5.2, 5.3, and 5.4. 5.1. Modified Langmuir Model. According to the previous thermodynamic study, the relation of Ce and Qe obeyed the Langmuir equation well. The Langmuir equation expressed the relationship between Ce and Qe. When Ce is an unknown quantity, Qe cannot be calculated by the Langmuir model. In order to predict the value of Qe, a modified Langmuir model was established where C0 was chosen as argument. Qe ¼
aCe , 1 þ bCe
VC0 WSe Ce ¼ V
a ¼ Q max KL ,
b ¼ KL
ð14Þ
d þ mC0 1 þ nC0
KL V V KL WSe
ð17Þ
d ¼
Q max KL WSe V KL WSe
ð18Þ
m¼
Q max KL V V KL WSe
ð19Þ
where Qe is the adsorption capacity at the adsorption equilibrium point (mg/g); C0 is the initial concentration (mg/mL); Qmax is a constant related to the adsorptive capacity (mg/g); KL is a Langmuir parameter which relates to the adsorption energy (mL/mg); Se is the solid surface equilibrium concentration (mg/g); W is the weight of the solid (g); V is the volum of solvent (mL). 5.2. Prediction of t. In the kinetic study, the pseudosecond-order kinetic model fitted the experimental data best,
ð15Þ
Combining eqs 14 and 15, the following equations were obtained: Qe ¼
n¼
ð16Þ 458
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Figure 3. Effects of variation of k2Qe on t. k2Qe is the second-order rate index (min1); t is the adsorption time (min); Qt/Qe is the degree of adsorption process.
Figure 4. Effects of variation of Ri on t/te. Ri is the initial adsorption factor of the intraparticle diffusion model; t/te is the relative adsorption time; Qt/Qe is the degree of adsorption process.
and the reformed equation can be calculated through eq 20: 1 1 t ¼ 1 ð20Þ k2 Q e 1 Q t =Q e
capacity at the adsorption equilibrium point; te is the adsorption equilibrium time; Ri is the initial adsorption factor of the intraparticle diffusion model. The effects of varying Ri on the value of the relative adsorption time (t/te) as predicted by the reformed model equation are shown in Figure 4. As shown in Figure 4, the value of t/te increased with increasing Ri. 5.4. Prediction of te. In the separation process of natural products by macroporous resins, there are two problems in identifying the adsorption equilibrium time. One is the problem of frequent sampling, and the other is that the HPLC detection time of one sample is too long. For example, the whole detection time of taxifolin is 31 min (Figure 5). Hence, time can be saved from these two processes. In industry, effective utilization of the adsorption medium will be helpful to reducing costs. In the batch adsorption experiment, we guarantee the adsorption degree reaches equilibrium for the purpose that the separation medium will be fully used. Therefore, the operating time should be longer than or equal to the equilibrium time. However, when the operation time is shorter than the equilibrium time, the adsorption process is not mostly effective and the subsequent static elution experiment will not be accurate. Contrasting Figures 3 and 4, it was found that the value of Qt/ Qe equals 1 Ri when t/te equals 0 (Figure 4). In the actual adsorption process, te was a fixed value and t/te = 0 means t = 0. However, when Qt/Qe was equal to 1 Ri, the value of t was equal to t1 (Figure 3). Hence, we proposed the following mathematical guess: (1) All experimental models were seen as approximate expressions for the AB-8 adsorption system. (2) For accurate quantification, we suppose that eqs 20 and 21 are not consistent. The cause of the inconsistency may be that the start time used for fitting was not equal to zero. (3) For the AB-8 adsorption system, the adsorption degree (Qt/Qe) was equal to zero at beginning. Therefore, the time starting point difference (t1) must be considered. The vale of t1 can be calculated by the following equation: 1 1 t1 ¼ 1 ð25Þ k2 Q e Ri
where Qt is the amount of target product adsorbed per unit of adsorbent (mg 3 g1) at time t; Qe (mg/g) is the adsorption capacity at the adsorption equilibrium point; t is the adsorption time (min); k2Qe (min1) is the second-order rate index. The effect of varying k2Qe predicted by the reformed model equation is shown in Figure 3. When Qt/Qe was fixed, an increase of the second-order rate index (k2Qe) decreased the time. 5.3. Prediction of t/te. For the adsorption system, Wu et al. have mentioned that the initial adsorption behavior often occurs for synthetic resin adsorbents with large particle sizes and uniform pores. They also have listed the analysis results of 86 adsorption systems by the intraparticle diffusion model such as XAD-4 synthetic resin adsorption systems. In our study, when we applied the intraparticle diffusion model to describe the kinetics result, a good linear relation between Qt and t1/2 was obtained under four different temperatures. In addition, it was also pointed out that this equation can be used to determine the adsorption conditions in engineering practice in the published paper of Wu et al. Hence, eq 21 was used in the AB-8 resin adsorption system to predict t/te. 2 t 1 Qt ¼ 1 1 ð21Þ tref Ri Q ref Ri ¼
Q ref C C ¼ 1 Q ref Q ref
ð22Þ
When the replacement method where the value of te equals the value of tref and the value of Qe equals the value of Qref was used in our study, the following equation can be obtained. t 1 Qt 2 ¼ 1 1 ð23Þ te Ri Qe Ri ¼ 1
C Qe
In the subsequent derivation process, the following equation was used to simplify the substitution:
ð24Þ
1
where tref is the longest time in the adsorption process and Qref is the adsorptive capacity (mg/g); Qe (mg/g) is the adsorption 459
Qt ¼η Qe
ð26Þ dx.doi.org/10.1021/ie201197r |Ind. Eng. Chem. Res. 2012, 51, 454–463
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Figure 5. HPLC curve of taxifolin.
Combining eqs 26 and 20, we have 1 1 1 t ¼ k2 Q e η Combining eqs 26 and 23, we have t η 2 ¼ 1 te Ri
Table 4. Prediction Results and Relative Parametersa ð27Þ
ð28Þ
Considering the time starting point difference (t1), eqs 29 and 30 were obtained: t t1 ð29Þ te ¼ þ t1 η 2 1 Ri
Qe (mg/g)
Ri
k2Qe
te (min)
25
0.5
32.0362
0.3794
0.2121
57.42
35 45
0.5 0.5
30.4893 27.5914
0.4066 0.3783
0.2749 0.3868
41.10 31.58
55
0.5
25.5316
0.3345
0.5035
27.70
T (°C)
RL
Qe (mg/g)
Ri
k2Qe
te (min)
25
0.018 65
32.0362
0.3794
0.2121
57.42
35
0.018 65
31.8701
0.3733
0.2774
44.68
45
0.018 65
31.7000
0.3559
0.3747
34.82
55
0.018 65
31.6439
0.4144
0.3772
29.34
T (°C) is the absolute solution temperature; Qe (mg/g) is the adsorption capacity at the adsorption equilibrium point; Ri is the initial adsorption factor of the intraparticle diffusion model; RL is the equilibrium parameter; k2Qe (min1) is the second-order rate index; te is the adsorption equilibrium time (min).
3 3 2 1 1 η 1 7 7 1 6 1 6 Ri 7 6 η Ri 7 6 þ t te ¼ ¼ 7 7 6 6 1 k2 Q e 4 k2 Q e 4 η 25 η 25 1 η 1 Ri Ri 3 2 7 1 6 1 7 6 7 þ t1 6 η 5 k2 Q e 4 η 1 Ri
C0 (mg/mL)
a
2
þ t1 ¼
T (°C)
concentration of 0.5 mg/mL, we can predict that the adsorption equilibrium time was 57.42 min (Table 4); the actual value was 57 min (Figure 6). In this prediction model, the last data used for fitting was at 45 min. Comparing this predicted strategy and the traditional experimental method, we can calculate an equation which will be as a standard: 45/57 = 0.79. Also, 0.79 is less than 0.9695, which means that although the adsorption progress reached 0.9695, the remaining adsorption time ratio is not 3% but 21%. Furthermore, using the traditional method, we must sample until the equilibrium is identified. As shown in Figure 6, the last sampling time was 120 min. Obviously, our predicted strategy will be helpful to saving labor. For the separation of natural products, AB-8 resin is a commonly used adsorption medium especially in the separation process of flavonoid substances. Using the results of verified experiments at the temperature of 25 °C with an initial concentration of 0.5 mg/mL, we studied the stability range of this model (relative error e5%) and the relative error (Table 5). First, the effect of the sample range on the error was studied when the sample interval was fixed. Four stability ranges of this model were listed: from 9 to 45 min, from 12 to 42 min, from 15 to 39 min, and from 18 to 36 min. Then, the effect of the sample range on error was studied when the sample number was fixed. It was found that the stability range remained the same. Therefore, the
ð30Þ
When η equals Ri/2, te has a minimum value (this value is seen as the prediction results): 1 5 1 ð31Þ te ¼ k2 Q e Ri where k2Qe (min1) is the second-order rate index. te is the adsorption equilibrium time. Ri is the initial adsorption factor of the intraparticle diffusion model, Qt is the amount of target product adsorbed per unit of adsorbent (mg 3 g1) at time t, and t is the adsorption time (min). In the adsorption study, sampling must be carried out at a certain interval after equilibrium which will consume a lot of experimental time, especially when intraparticle diffusion plays a major role in the adsorption process. A simple strategy about the prediction of the equilibrium time was also shown in Figure 2. 5.5. Method Performance and the Comments on Error. From calculation of the fitting results at 25 °C with the initial 460
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Figure 7. Adsorption kinetics curves of taxifolin on AB-8 at different temperatures at the same RL value. t is the adsorption time (min); Qt is the amount of target product adsorbed per unit of adsorbent at time t.
Figure 6. Adsorption kinetics curves of taxifolin on AB-8 at different temperatures at fixed sample concentration. t is the adsorption time (min); Qt is the amount of target product adsorbed per unit of adsorbent at time t.
Table 5. Effects of Different Fitting Conditions on the Errora time range
saturation
sampling interval
(min)
range (%)
time (min)
Ri
k2Qe
error (%)
648 945
70.6298.57 77.7796.95
3 3
0.3986 0.3794
0.2254 0.2121
10.15 0.74
1242
79.7795.92
3
0.4059
0.1971
0.74
1539
81.2594.95
3
0.4050
0.1900
4.76
1836
82.6693.82
3
0.4109
0.1913
2.42
2133
86.9090.42
3
0.3733
0.1981
9.76 error
time range
saturation
(min)
range (%)
sample no.
Ri
k2Qe
(%)
648 945
70.6298.57 77.7796.95
7 7
0.4112 0.3579
0.2293 0.2228
14.62 2.13
1242
79.7795.92
7
0.3949
0.1949
4.97
1539
81.2594.95
7
0.3892
0.1982
4.86
1836
82.6693.82
7
0.4109
0.1913
2.42
2133
86.9090.42
7
0.4037
0.1840
8.56
Figure 8. Correlation plot of predicted data with experiment data.
6. CONCLUSIONS In this study, we proposed a prediction strategy of adsorption equilibrium time based on the equilibrium and kinetic performance of AB-8 macroporous resin to isolate taxifolin. On the basis of mathematical deduction, some equations were established to guide the calculation of several important parameters. Using the modified Langmuir equation, the adsorption capacity at the adsorption equilibrium point (Qe) can be calculated. The relationship between the adsorption time (t) and the degree of the adsorption process (Q t/Qe) could be revealed by the reformed pseudo-second-order kinetic equation. Also, the reformed intraparticle diffusion model could reflect the relationship of the relative adsorption time (t/te) and the degree of adsorption process (Q t/Qe). By variable substitution, a prediction equation of the adsorption equilibrium time (te) was proposed. The good agreement between experimental and predicted results confirmed the reliability of the proposed strategy.
error
time range
saturation
sample
(min)
range (%)
no.
Ri
k2Qe
(%)
945 945
77.7796.95 77.7796.95
13 7
0.3794 0.3579
0.2121 0.2228
0.74 2.13
945
77.7796.95
5
0.3757
0.2315
6.72
a
Ri is the initial adsorption factor of the intraparticle diffusion model; k2Qe (min1) is the second-order rate index.
best sample range was from 9 to 45 min and the corresponding saturation range was from 77.77 to 96.95%. The effect of sample number on error was also studied at fixed sample range, and the appropriate sample number was 7. The corresponding saturation range will guide us to the predicted subsequent six results. All prediction results are shown in Table 4. The experimental results are shown in Figures 6 and 7. The correlation coefficient of six results is shown in Figure 8 (R2 = 0.9858). The high consistency indicates the good stability and predictive capability of the strategy. This prediction strategy was helpful to selecting suitable adsorption equipment and simplifying the batch experiment, through saving operating time.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Tel: +86-10-64437610. Fax: +86-10-64437610. 461
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Author Contributions
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Tingting Liu and Ming Yang contributed equally to this work.
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