ARTICLE pubs.acs.org/Langmuir
Modeling the Effect of Structural Changes during Dynamic Separation Processes on MOFs Tom Remy, Gino V. Baron, and Joeri F. M. Denayer* Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
bS Supporting Information ABSTRACT: A model able to describe the effect of structural changes in the adsorbent or adsorbed phase during the dynamic (breakthrough) separation of mixtures on metalorganic frameworks (MOFs) is presented. The methodology is exemplified for a few pertinent case studies: the separation of xylene isomers and ethylbenzene on the flexible MOF MIL-53 and the rigid MOF MIL-47. At low pressures, no preferential adsorption of any component occurs on both MOFs. Contrarily, at higher pressures separation of ethylbenzene (EB) from o-xylene (oX) occurs on MIL-53 as a result of the breathing phenomenon within the MIL-53 structure. The increase in selectivity, starting from the gate-opening pressure, could be modeled by using a pressure-dependent saturation capacity for the most strongly adsorbed component oX. In the separation of m-xylene (mX) from p-xylene (pX) on the rigid MOF MIL-47, separation at higher pressures is a result of preferential stacking of pX. Here, the selectivity increases once the adsorption of pX switches from a single to a double file adsorption. By implementing a loading dependent adsorption constant for pX, the different unconventional breakthrough profiles and the observed selectivity profile on MIL-47 can be simulated. A similar methodology was used for the separation of EB from pX on MIL-47, where the separation is a result from steric constraints imposed onto the adsorption of EB.
’ INTRODUCTION The tremendous potential of metalorganic frameworks (MOFs) in molecular separation processes has recently been demonstrated extensively.16 Their wide diversity in chemical composition and large surface area make them superior to classical adsorbents in many aspects.710 Newly observed selectivity mechanisms in adsorption for MOFs include the gate-opening effect, breathing-dependent selectivity, and pore filling dependent selectivity.1114 The stated phenomena are thought to be related to structural changes in the adsorbent and/or adsorbed phase. Although such rearrangements in MOF structures have been evidenced by a variety of characterization techniques, the impact thereof on real separation processes has scarcely been studied. Hence, potential and possible advantages of MOFs in the field of separations remain largely unexplored. Changes in pore volume obviously affect the adsorption capacity, but a change in selectivity is also to be expected. Similarly, a (drastic) change in adsorption affinity is expected to have profound influence on selectivity. Hamon et al. reported breakthrough curves of CH4 with a double roll-up upon separation of a CO2/CH4 mixture on MIL-53(Al).15 They attributed this effect to the structural flexibility (breathing) of the adsorbent. Finsy et al. have shown the occurrence of an intermediate step in the o-xylene (oX) profile upon separation of ethylbenzene (EB) on MIL-53(Al) in a narrow range of conditions.16 Barcia r 2011 American Chemical Society
et al. observed a two-step breakthrough profile for n-hexane in the separation of hexane isomers on the rigid MOF Zn(BDC)(Dabco)0.5, which has been related to different interaction sites of n-hexane in the MOF.17 In the separation of p- and mdichlorobenzene on the nonflexible MOF MIL-47, the breakthrough curve of p-dichlorobenzene contained a clear step, as shown by Alaerts et al.18 This is a result of preferential packing of the p-isomer. Finsy et al. also reported such steps in the p-xylene (pX) profile during separation of pX from m-xylene (mX) on the same adsorbent starting from a certain pX pressure.14 Similar curves were obtained by Gu et al. in the separation of pX and mX on MOF-monoclinic [Zn3(terephthalate)3(H2O)3(DMF)4]n.19 It is thus clear that framework flexibility and pore-filling effects cause very unusual behavior in the dynamic separation of molecular mixtures on MOFs. Dubbeldam et al. demonstrated preferential stacking of linear and monobranched alkanes with respect to di- and tribranched alkanes on Zn(BDC)(Dabco)0.5 via Grand Canonical Monte Carlo (GCMC) simulations.20 Other researchers have focused on the development of force fields for flexible MOFs. Zhao et al. reported a new force field to describe the flexibility of Cu-BTC.21 Chmelik and co-workers used molecular simulations to predict the adsorption isotherms of n-butane, iso-butane, 2-methylbutane, and Received: August 29, 2011 Published: September 16, 2011 13064
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Table 1. Overview of the Different Types of Separations on MOFs with Their Associated Adsorbate Isotherms and Separation Mechanism
2,2-dimethylpropane on CuBTC.22 Greathouse and Allendorf developed a flexible force field to describe the interaction of MOF-5 with water, which was later validated by molecular dynamics simulations.23,24 Recently, several studies focusing on the thermodynamics of adsorption-driven structural changes in the adsorbent have emerged.2529 Coudert et al. developed a thermodynamic model in order to predict pure component isotherms on flexible MOFs and “third generation coordination polymers”.25 The thermodynamic framework was used to calculate the gate-opening and closing pressures in the third generation coordination polymers Co(BDP) and Cu(4,40 -bipy)(dhbc)2. In addition, they were able to simulate the two-step isotherms of CO2 on the breathing MOF MIL-53 with an accurate prediction of the two different transition pressures. These authors also extended their approach to predict breathing in MIL-53 upon adsorption of binary mixtures consisting of one component which induces breathing (e.g., CO2) and one which does not (e.g., CH4).26 Neimark et al. further studied breathing in MIL-53 with the adsorption of xenon on MIL-53(Al) as a case study.27 In their model, the adsorptioninduced stress causes the structure to breath, i.e., when the stress attains a critical threshold value. In a following paper, Boutin et al. applied the osmotic thermodynamic model from Coudert et al. to the adsorption of CO2 in the amino modified MIL-53(Al) in order to calculate the temperature-loading diagram.29
Although great progress has been made in the fundamental description of adsorption of pure components onto flexible and nonflexible MOFs, the analysis of the effect(s) of structural changes in the adsorbent or adsorbed phase onto real dynamic separation processes has not yet been carried out. Herein, we present a model to describe the effect of framework flexibility and molecular stacking in the dynamic (breakthrough) separation of mixtures on MOFs. We will exemplify the methodology for a pertinent separation process: the separation of xylene isomers and ethylbenzene on the flexible MIL-53(Al) and the rigid MIL-47 MOFs. A model allowing us to rationalize the effects and capable of predicting the various types of experimentally observed breakthrough profiles is presented. Several cases are discussed and analyzed: (a) EB/oX on MIL-53(Al), (b) mX/ pX on MIL-47, and (c) EB/pX on MIL-47 (see Table 1).
’ METHODOLOGY Flexible MOF MIL-53(Al). A detailed description of the experimental procedure for the isothermal vapor phase breakthrough experiments on MIL-53(Al) and the interpretation of the experimental results can be found in a former paper by Finsy et al.16 The procedure for the vapor-phase adsorption experiments is also explained therein. Adsorption isotherms of the different xylenes (o-xylene, m-xylene, and p-xylene) on the flexible MOF MIL-53(Al) show an initial 13065
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Figure 1. Experimental (Δ) and predicted (line) adsorption isotherm of o-xylene (oX) on MIL-53(Al) at 383 K showing the transition from a narrow pore form (1 xylene molecule per pore) to a large pore form (2 xylene molecules per pore). Langmuirian profile followed by a step corresponding to the breathing of the framework induced by the adsorbate (see Figure 1 for oX).16 Since the adsorption capacity strongly increases from a certain pressure (the so-called gate opening pressure Pgo16), it was decided to fit the adsorption isotherms with a Langmuir model combined with a pressure-dependent saturation capacity. A logistic function was used to model the S-shaped increase in adsorption capacity between the first and second isotherm plateaus. Logistic functions are typical model functions for S-shaped growths and find applications in biology, chemistry, statistics, medicine, and economics (for more info on logistic functions, see Supporting Information section S1). The following equation was used to fit the pure pX, mX, and oX isotherms on MIL-53(Al) at 383 K: q ¼ qlog 3
K3C 1 þ K 3C
ð1Þ
with q = adsorbed amount (mol/kg), C = concentration (mol/m3), and K = adsorption constant (m3/mol). The parameter qlog contains the logistic function and is defined as 2 2 !3 3 P 6 Klog 3 Plog 3 exp4rlog 3 1 5 7 6 7 6 7 Pgo 6 7 7 2 3 1 þ qlog ¼ qsat0 3 6 ð2Þ ! 6 7 6 7 P 4 Klog 3 Plog 3 exp4rlog 3 1 5 15 Pgo with qsat0 = saturation capacity of the first plateau (mol/kg) and Pgo = the gate opening pressure. This is the pressure at which the crystal structure starts to breath and additional space becomes available for adsorption16 The other parameters are characteristic of the logistic function and are explained in Supporting Information (SI) section S1. The fit is very satisfactory and shown in Figure 1 for the pure oX isotherm (for the other xylenes, see SI Figures S3S4). The proposed isotherm model (eq 1) correctly captures the two-step shape of the experimental isotherm (see Figure 1). The difference between the predicted adsorbed amount and the observed capacity is always less than 0.3 mol/kg, except for the first experimental point (at the lowest pressure: Pvap = 4.04 105 bar), where a difference of 0.38 mol/kg prevails. This point is,
however, less reliable given the very low pressure at which it was measured. It is true that the model proposed by Coudert et al. has led to an accurate description of the isotherm of CO2 on MIL-53(Al).25 Their model considers two independent Lanmguir isotherms, one for the narrow pore form and one for the large pore form. As a result, this model is unable to fit the step in the isotherm of an adsorbate, which causes the MIL-53 structure to breath. Contrarily, our model does fit the actual step and the different experimental breakthrough profiles (vide infra). For completeness, we have also indicated a region 0 at very low pressures in Figure 1, in which the material possesses an open pore form. The characteristics of the MIL-53 material at very low pressure have been discussed in previous works.16,30 Already after introduction of the first xylene molecules, the diameter of the pores is strongly reduced giving rise to the narrow pore form of MIL-53 (see region I in Figure 1). Therefore, this region 0 is not relevant for the experiments that are being modeled. The contracted pore structure reopens at higher loadings. This reopened structure of MIL-53 is able to accommodate 2 molecules per unit cell (see region III in Figure 1). In a next step, the pressure-dependent Langmuirian isotherms of the xylenes were combined with a one-dimensional axially dispersed plug flow model in order to simulate the dynamic separation performance of MIL-53(Al) in the EB/oX separation. The relevant equations and boundary conditions are listed in the Supporting Information. The appropriate isotherm equations (q*i = f(Ci)) will be given in the following sections. The values of the parameters in the adsorption column model and the logistic function parameters are given in SI Tables S1 and S2, respectively. A more detailed description of the methodology for the EB/oX separation on MIL-53(Al) is given in SI section S3. Nonflexible MOF MIL-47. A detailed description of the experimental procedure for the isothermal breakthrough experiments on MIL47 and the interpretation of the experimental results can be found in former work by Finsy et al.14 The procedure for the vapor-phase adsorption experiments is also explained therein. The main difference is that in this case no steps are perceived in the pure component adsorption isotherms of pX, mX, oX, and EB within the range of experimental pressures. All these components show type I isotherms on the nonflexible MOF MIL-47 (see also Table 1). The adsorption column model is the same as for MIL-53(Al). A more detailed description of the methodology for the mX/pX and EB/pX separation on MIL-47 is given in SI sections S4 and S5, respectively. The mass transfer from the fluid phase to the adsorbed phase is described via a linear driving model.31 This is a good starting model for a first study, in which we want to focus on the adsorption characteristics rather than the influence of mass transfer resistance(s). Since the adsorbates were diluted in an inert carrier (helium), the changes in fluid velocity across the column are negligible.32 Therefore, the interstitial velocity was considered to be constant. In addition, the column was considered to remain at a constant temperature. The systems of equations were solved in Athena Visual Studio v 14.0 using the method of finite differences (Upwind Second Order WENO Scheme) with 50 discretization points.
’ RESULTS AND DISCUSSION Case 1: Breathing on MIL-53(Al). All the xylenes (oX, mX, pX) show large breathing upon adsorption on MIL-53(Al), with capacities of the second isotherm plateau being about twice that of the first plateau. In contrast to the xylenes, EB does not show a large step in adsorption capacity from a certain pressure onward, but only a small kink at a pressure of 0.003 bar.16 Since EB does not have a steep jump in its single component isotherm on MIL-53(Al) (see Table 1) and 13066
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breakthrough experiments did not reveal any step in the binary adsorption isotherm of EB on MIL-53(Al), it is reasonable to implement the logistic function only for oX in the simulation of binary EB/oX mixtures. The corresponding isotherm equations are listed in eqs 35 qoX ¼ qsat1oX 3
KoX 3 CoX 1 þ KoX 3 CoX þ KEB 3 CEB
þ qlogoX 3
qlogoX ¼
8 > > > >
Ptot > > Klog 3 Plog 3 exp rlog 3 1 > = Pgo " " !# # > Ptot > þ Plog 3 exp rlog 3 1 1 > > ; Pgo "
> > > > :Klog
ð4Þ qEB ¼ qsatEB 3
KEB 3 CEB 1 þ KEB 3 CEB þ KoX 3 CoX
ð5Þ
with Ptot = total pressure (bar), Pgo = gate-opening pressure (Pgo = 0.005 bar), qsat1oX = adsorption capacity for oX at the first plateau (qsat1oX = 1.4 mol/kg), qsatEB = saturation capacity for EB (qsatEB = 1.4 mol/kg). Equation 3 is used in an application of the multiregion extended Langmuir model put forward by Bai and Yang.33 This model ranks the saturation capacities of the different N components in the mixture from small to large in order to obtain n different adsorption sites. The adsorbed amount of component i in region j is then qj1 qj 3 bi 3 pi qi, j ¼ i ¼ 1, 2, :::, N j ¼ 1, 2, :::, n ð6Þ N 1 þ
qA, 1 ¼
qsatB 3 bA 3 pA 1 þ bA 3 pA þ bB 3 pB
ð7Þ
qB, 1 ¼
qsatB 3 bB 3 pB 1 þ bA 3 pA þ bB 3 pB
ð8Þ
ðqsatA qsatB Þ 3 bA 3 pA 1 þ bA 3 pA
ð9Þ
Site 2
qB, 2 ¼ 0
Total adsorbed amounts
j
∑ bk 3 pk k¼1
with q0 = 0 and qi,j = 0 when qs,i < qj. The multiregion extended Langmuir approach is typically used for a binary mixture if one type of site is accessible for both components whereas the second type can only be occupied by the strongest-adsorbing component due to size exclusion or lack of competition. For the case of a binary mixture of components A and B, eq 6 leads to the following equations for the two types of sites and the total adsorbed amounts: Site 1
qA, 2 ¼
Figure 2. Experimental (left) and simulated (right) breakthrough profiles for ethylbenzene (EB) and o-xylene (oX) on MIL-53(Al) at 383 K, showing the three types of elution profiles that can be obtained depending on the partial oX pressure. The insets show the corresponding zone in the oX isotherm.
ð10Þ
qA ¼ qA1 þ qA2
ð11Þ
qB ¼ qB1 þ qB2 ¼ qB1
ð12Þ
In the eqs 712, component A is the component with the largest saturation capacity (qsatA > qsatB). The model is thermodynamically consistent and considers two types of sites for the most strongly adsorbed component, which are available over the whole range of pressures. This does not reflect the physical reality in our case, since the large pore form of MIL-53 will only exist at pressures above the gate opening pressure. The proposed isotherm equation (eq 3) for oX (the most strongly adsorbed component) is of the same form as the one for the most strongly adsorbing component in the multiregion extended Langmuir model (see eq 11). The difference between eqs 3 and 11 lies in the fact that the second term of eq 3 will only become nonzero at pressures above the gate-opening pressure. Therefore, it is a better representation of the physical reality since the second type of site (the large pore form of MIL-53(Al)) will only be present above the gate opening pressure. As such, it becomes an improved version of the multiregion extended Langmuir model for the studied situation. In this case, we have attributed to the parameter Klog of the logistic function the difference in saturation capacities for oX 13067
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Figure 3. Predicted binary isotherms of EB (O) and oX (0) compared to the experimentally obtained capacities during breakthrough separation of equimolar EB (b)/oX (9) mixtures on MIL-53(Al) at 383 K. The secondary axis shows the simulated selectivities (Δ).
between the second and first isotherm plateau on MIL-53(Al, PVA) (Klog = 1.0 mol/kg). Simulations are able to catch the different observed experimental breakthrough profiles (Figure 2). At low total pressures (Ptot < Pgo; zone I of the oX isotherm in Figure 1), no separation at all is observed and predicted (Figure 2a), since both components have the same adsorption constants and saturation capacities. For total hydrocarbon pressures near the gate opening pressure (Ptot ≈ Pgo; zone II of the oX isotherm), the two-step breakthrough profile is correctly predicted (Figure 2b). The transition region in which this type of elution profile occurs is very narrow (about 2 mbar in total hydrocarbon pressure), which is in concordance with the steep transition from the first to the second plateau in adsorption capacity for the xylenes (see also Figure 1). At higher pressures (Ptot > Pgo; zone III of the oX isotherm), classical breakthrough profiles are obtained with a high selectivity (Figure 2c). Here, the loading of the pores corresponds to the second isotherm plateau from the start of the experiment. Observed adsorbed amounts are also accurately predicted, with a clear step in the binary oX adsorption isotherm at 0.005 bar (Figure 3). Consequently, the selectivity starts to increase from this point on and reaches a maximum value of 3.0 (Figure 3). In the future, it will be examined whether the above approach could be used to describe hysteresis in breathing MOFs. Only few publications have already dealt with the influence of hysteresis on the adsorption characteristics. Ritter and Yang investigated the effect of hysteresis on fixed bed adsorption based on the local equilibrium theory. Their results showed that hysteresis can increase the time needed for desorption by 1030%.34 Stepanek et al. demonstrated that hysteresis can have a detrimental impact on the performance of a pressure swing adsorption (PSA) unit.35 For the adsorption of water vapor onto silica gel, significant differences in product purity (as high as 1 order of magnitude) occur within certain parameter ranges between an adsorption column considering classical isotherms and one with hysteresis-dependent isotherms. Therefore, the impact of hysteresis on the performance of flexible materials in cyclic adsorptiondesorption processes such as
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Figure 4. Experimental (9) and simulated (dotted line) selectivity profiles for the separation of p-xylene (pX) from m-xylene (mX) on MIL-47 at 343 K. The secondary axis shows the experimental adsorbed amount of pX (2) in molecules per unit cell and the predicted one (dotted line).
temperature or pressure swing adsorption is an important topic for future studies. Case 2: Preferential Stacking on MIL-47. It is remarkable that a similar approach can be applied to molecular stacking induced selectivity on rigid MOFs. Several experimental and fundamental modeling studies have already been carried out on the nonflexible MOF MIL-47.14,3639 However, none of them has ever tried to describe the unusual breakthrough profiles in the mX/pX separation, as a result of the structural changes in the adsorbed phase. Since this MOF is not breathing, it was decided to implement a loading-dependent adsorption constant K instead of the pressure-dependent saturation capacity we used for the MIL-53(Al) case (vide supra). The corresponding isotherm equations are given below qpX ¼ qsatpX 3
klogpX
klogpX 3 CpX 1 þ klogpX 3 CoX þ KmX 3 CmX
ð13Þ
8 2 0 ! 13 9 > > qpX qpX > > > 4 @ A5 > > > P exp r K log log log > > 3 3 3 > > qsatpX qsatpX = < crit 2 0 1 3 ¼ K0, pX 3 1 þ ! > > > > qpX qpX > > > A5 1> Klog 3 Plog 3 exp4rlog 3 @ > > > > ; : qsatpX qsatpX crit
ð14Þ qmX ¼ qsatmX 3
KmX 3 CmX 1 þ klogpX 3 CpX þ KmX 3 CmX
ð15Þ
The values of the parameters in the adsorption column model and the logistic function parameters for the mX/pX case on MIL47 are given in SI Tables S3 and S4, respectively. A more detailed description of the methodology for the mX/pX separation on MIL-47 is given in SI section S4. The adsorption constants of mX and pX are similar in the simulation which is in line with previous literature on the subject.14,37,39 The fact that only the Henry constant of pX is changed is based on previous results by Alaerts et al.36 They have shown by Rietveld refinement that mX stacks less efficiently than pX within the MIL-47 pores. The small value of the lattice constant in the a-direction forces the aromatic rings within the 13068
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Figure 5. Predicted binary isotherms for mX and pX on MIL-47 at 343 K.
Figure 6. (left) Concentration profiles of pX within the adsorption column at different times showing three different zones: favorable zone I (blue), unfavorable zone II (red), and favorable zone III (green). The black arrow shows the sense of increasing time: (right) Elution profiles for m-xylene (black) and p-xylene on MIL-47 at 343 K (PpX = PmX = 0.03 bar). Full lines correspond to the experimental values, dotted lines for simulated values.
pairwise adsorbed xylene molecules away from each other, thereby reducing the ππ interactions. In contrast, pX pairs adsorb in an almost parallel configuration, which leads to strong ππ interactions between the pX molecules. The use of similar adsorption constants (and saturation capacities) results in a selectivity of about one at low pore occupancy (Figure 4). Adsorption of pX is favored with increasing degree of pore filling (see Table 1). Concurrently, a three-staged breakthrough profile is obtained experimentally for pX over quite a large region of total hydrocarbon pressures (Ptot = 550 mbar) due to a more efficient stacking of pX once the available pore space becomes critical. The onset of the increase in selectivity coincides with the transition from a single to a double file adsorption of pX molecules within the MIL-47 pores (Figure 4). Implementation of a loading-dependent adsorption constant enabled us to catch the effect of preferential pX stacking, yielding the correct shape of the breakthrough curves (see Figure 6 and SI Figure S5). For the criterion on loading (q) that is used within the logistic function, one could argue whether to use the total adsorbed
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Figure 7. Experimental (9) and simulated (dotted line) selectivity profiles for the separation of pX from EB on MIL-47 at 383 K. The secondary axis shows the experimental total adsorbed amount (2) in molecules per unit cell and the simulated one (dotted line).
amount qtot or the adsorbed amount of a given component i (qi). The simulations have shown that, in order to create a broad transition region, in which two-step breakthrough profiles are obtained for the strongest adsorbed component pX, a criterion on the adsorbed amount of the most strongly adsorbed pX needs to be used. If one would choose the total adsorbed amount as a criterion, a much narrower transition region (only a few mbars) would be obtained. This means that a certain amount of pX in the adsorbent’s pores is necessary to unlock the preferential stacking of pX in the mX/pX separation. One could therefore refer to the pX stacking process as “autocatalytic”; a phenomenon which prevails in many other fields of chemistry. The preferential pX stacking, starting from a certain concentration of pX in the MIL-47 pores, also results in an S-shaped selectivity profile that is well matched by the simulations (Figure 4). Adsorbed amounts also correspond well (Figure 4). Within this simple model, the mathematical origin of the intermediate step in the pX profiles results from the creation of two inflection points in the binary pX/mX isotherm (Figure 5 and SI Figure S6). Therefore, the binary pX equilibrium isotherm shifts from a concave shape (also called “favorable equilibrium”) to a convex shape (called “unfavorable equilibrium”) and back to a concave one giving rise to three different parts in the pX concentration front within the adsorption column (Figure 6). Since the isotherm contains an unfavorable zone (convex part), the pX concentration front will broaden within this pressure region (zone II in Figure 6).40 However, as the binary pX isotherm switches back to a favorable equilibrium an intermediate step in the breakthrough profile arises (Figure 6). Similarly, adsorbates showing a BET isotherm, which has only one inflection point, yield a breakthrough curve consisting of a shock wave followed by a dispersive front (SI Figure S7).41 Another important aspect of the mX/pX separation is the fact that both components have similar high adsorption constants. Simulations show that, if the differences in Henry constants would be larger (say a factor 2 difference or more), intermediate steps in the breakthrough profiles would never occur (see also SI Figures S9S15). The presented model also accurately simulates the shape of the pure component elution profiles where no intermediate steps 13069
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were observed (SI Figure S16). The occurrence of intermediate steps in the experimental pX profiles during mixture separations is a result of the preferential stacking of pX during mixture adsorption and is predicted well by the simulations. Case 3: Steric Constraints on MIL-47. For the final case, separation of EB/pX on MIL-47, the separation is the result of steric constraints imposed on the adsorption of EB as shown in earlier work (see also Table 1).14 Therefore, a loading-dependent adsorption constant is implemented, but now, the total loading (qtot) constitutes the criterion within the logistic function (SI section S5). The corresponding isotherm equations are given in eqs 1619 qpX
klogpX
klogpX 3 CpX ¼ qsatpX 3 1 þ klogpX 3 CoX þ KEB 3 CEB
ð16Þ
0 ! 13 9 > qtot qtot > 4 @ A5 > > Klog 3 Plog 3 exp rlog 3 > > qsatpX qsatpX = crit 2 0 1 3 ¼ K0, pX 3 1 þ ! > > > > qtot qtot > > > A5 1> Klog 3 Plog 3 exp4rlog 3 @ > > > > ; : qsatpX qsatpX 8 > > > > > >
