J. CHEM. SOC. FARADAY TRANS., 1994, 90(2), 369-376 Frequency Response Analysis for Multicomponent Diffusion in Adsorbents L. M. Sun, G. M. Zhong, P. G. Gray and F. Meunier LIMSI-CNRS, B.P.133 91403, Orsay France Frequency response behaviour of diffusion of multiple sorbates in adsorbents is theoretically investigated. A mathematical model describing diffusion of n components with both equilibrium and diffusional interference is developed and analytical solutions for frequency response and moments are derived. It is shown that, for the model system CO, and C,H, in 4A zeolite, the fastest diffusing component can be strongly affected by other slower diffusing components and its partial-pressure frequency response curves exhibit a roll-up phenomenon induced exclusively by the diffusional interference.The frequency response curves of the total pressure gener- ally have multi-modal forms as a result of multicomponent diffusion ; the peak positions and heights within these responses, which are closely related to the diffusivities, are found to be sensitive to the extent of equilibrium and diffusional interference. Comparison of these resonance frequencies obtained from purecomponent and multicomponent experiments can therefore give a good indication of the interference between the components. The effects of film resistance and heat dissipation, which may be important for fast diffusing systems, are also discussed. The modelling of diffusion of multiple adsorbates in a micro- porous adsorbent in general requires knowledge of both the main-term and cross-term diffusivities, as the effect of the cross-term diffusivities may be important for strongly inter- acting systems.As a consequence, the number of the diffusion coefficients to be determined may increase rapidly with the number of components; n(n + 1)/2 diffusion coefficients needed for a system of n components, by application of the Onsager reciprocal relation. Simultaneous experimental mea- surement of all these coefficients is very difficult, even for a binary system. This has necessitated the development of theo- retical models capable of predicting multicomponent diffusi- vities from pure component diffusivities. The advantage of predicting multicomponent data from single component data by using theoretical predictive models has long been proved in the correlation of multicomponent adsorption equilibria.For certain systems, the diffusional interference of different components mainly arises from the coupling through multi- component equilibria. The cross-terms of the phenomenologi- cal transfer coefficients, based on the irreversible thermodynamics formulations, can therefore be neglected and the diffusion flux of a given species depends only on its own chemical potential. '-3 The resulting Fickian diffusivity matrix has non-zero cross-terms which are concentration dependent. This model can be referred to as the ideal multicomponent diffusion model. Krishna has developed a model based on a generalized Maxwell-Stefan formulation in which the adsorb- ent is treated as an additional c~mponent.~ The transfer coef- ficients of this model were composed of the pure component coefficients and those for the counter-exchange between the adsorbates which were related to the pure component diffu- sion Coefficients by using empirical correlations.Fully predic- tive models have recently been developed by Wei and co-workers using a stochastic Markovian formalism, and by Chen and Yang using a kinetic In these models, the cross-diffusivities are explicitly related to the diffusivities of pure components, and to the occupancy, through a number of parameters characterizing interactions between adsorption sites and diffusing molecules. The model of Chen and Yang has a simple form, which reduces to the ideal multicomponent diffusion model with an extended Langmuir isotherm when all the interaction parameters are equal to zero.Their model was able to predict experimental data accurately for the diffusion of C0, and C,H6 in 4A zeolite and of benzene and toluene in ZSM-5 zeolite. In order to check the applicability of predictive multi- component diffusion models, experimental multicomponent diffusion data for different representative systems are needed. However, due to the experimental difficulties in measuring the uptake of each adsorbate, very few experimental investi- gations have been reported so far in the literature and avail- able experimental techniques are quite limited : the microscopic NMR technique,' * the differential adsorption bed (DAB) technique,2*' and the constant-volume technique.' 3-' Recently, the frequency response technique has been found to be very useful for the measurement of diffusion rates of single species in adsorbents. 16v1 'Detailed theoretical studies have revealed a high sensitivity of the technique to the nature of the governing transport equations.'8-21 For example, a surface barrier effect causes the so-called in-phase and out-of- phase functions to intersect, while intrusion of heat effects can lead to bimodal frequency response curves.The intersection and bimodal form of the frequency response curves constitute an unusual pattern of behaviour which cannot be described by a pure diffusion model.It therefore provides, in principle, a good way of distinguishing between intracrystalline diffu- sional resistance, surface resistance to mass transfer and the effect of heat dissipation. The high sensitivity of the frequency response technique may also be useful for studying multi- component diffusion in adsorbents. The first frequency response experiments for diffusion of binary systems have recently been reported in the literat~re.~~-,~ However, firm conclusions concerning the applicability of the technique cannot be drawn from these papers owing to the use of overly simplified theoretical models. In this paper a detailed theoretical analysis of frequency response for the diffusion of multiple adsorbates in mono- dispersed adsorbent particles (e.g.zeolite crystals) will be pre-sented. The model to be developed will include both equilibrium and diffusional interference and is, therefore, general enough to allow a reliable investigation of the sensi- tivity of the frequency response to main-term and cross-term diffusivities. The theoretical study, using the theory of Chen and Yang for predicting multicomponent diffusivities, will enable us to determine whether measurement of the total pressure alone is sufficient to yield a reliable estimation of both main-term and cross-term diffusivities. Moreover, effects of heat dissipation and surface barrier are also included in the model and discussed. Mathematical Model Consider an adsorption chamber in which are placed a number of well separated monodispersed adsorbent particles occupying a volume V,.The adsorbent particles are assumed to be of uniform size; a significantly non-uniform particle size distribution would lead to distortion or smoothing of the fre- quency response curves, thus resolution of characteristic peaks in the frequency spectrum would be very difficult. Hence, in any experimental study, it would be necessary to work with a rather uniform particle size distribution; an assumption which is made in this mathematical model. The chamber contains an ideal gaseous mixture of n components in a volume V which can be varied in a controlled manner. The volume variation causes the total pressure and partial pressures of the mixture to change, which in turn causes the gases to diffuse into or out of the adsorbent particles.The sorption of the adsorbates produces a change in the adsorb- ent temperature and a heat exchange occurs between the uniform-temperature adsorbent and the constant-temperature surroundings. The system can be considered as linear since volume changes are, in general, quite small (< 5%). The mass balance in the particles is given by the following multicomponent diffusion equation : with the reduced time t = t/R: and a = 0, 1, 2 for slab, cylin- drical and spherical particle geometry, respectively. q is a vector of dimension n denoting the incremental quantity adsorbed by the particles. [D] is a square matrix of diffusion coefficients in which the off-diagonal terms are generally non- zero.(See glossary for definition of other symbols). The corresponding initial and boundary conditions are : d$, 0)= 0 (2) (3) where the parameter k, is introduced to account for a surface barrier effect, which can be due to surface heterogeneity and/or film resistance. q* is related to the gas pressures p and temperature T through the linearized multicomponent adsorption equilibrium : q*=[Klp-kTT (5) The heat balance on the particles is given by the following scalar equation : where t, = R,CJ[(o + 1)h) is the time constant for heat exchange between the adsorbent and surroundings. The pressures of the adsorbates in the gas phase in the chamber are assumed to be uniform throughout the chamber J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 and are determined by the following mass conservation con- dition :20 P + diag(B,M = ; for a constant-volume condition for a volume modulation (7) i"iu exp(iot)p, ; where pi is the initial pressure in the gas phase, u and o are the relative amplitude and angular frequency of a sinusoidal volume modulation, respectively. The parameter B, is defined as KRk q/K.The zero-order moments for the above system under the constant-volume condition can be obtained from the Laplace solutions using Van der Laan's theorem.25 The solution pro- cedure is similar to that used for deriving the frequency response solution which will be presented later. The moments of the temperature T and quantity adsorbed q are: thM(T)= J':.dt = --(AH -CS R: [D]-'(a + 1)(a + 3) where [A] = [I] + [K]diag(/3,) represents the ratio of the masses contained in the adsorbent and gas phase, respec- tively.t,, = R,/[(a + l)k,] is the time constant for the surface barrier effect. The tensor product of k, and AH leads to a square matrix whose elements (i, j) are kTiAHj.q, is the amount adsorbed at the equilibrium state: 4, = [KIWI + diag(B,)[KlI -'Pi (10) The temperature moment is linearly proportional to the heat exchange time constant and the moments for the adsorbed masses are given by a linear addition of the resist- ances caused by surface barrier, diffusion and heat exchange, respectively. For one-component systems (n= l),the moment solutions [(eqn.(8) and (9)] reduce to those given in a pre- vious study.lg From a practical point of view, the zero-order moments are useful for determining both the heat transfer coefficient, h, and the total resistance to mass transfer. Analytical Frequency Response Solution The frequency response solution of the above system can be analytically obtained by using the matrix analogue.26-28 The key point is to diagonalize the matrix [D]in terms of the eigenvalues and eigenvectors : [XI -[D] [XI = diag(L,) (11) where 1, are the eigenvalues of [DJ and [XI is formed from columns of the corresponding eigenvectors. For a binary system, the eigenvalues are : and the eigenvector matrix is : 1 CXl = i D2 1 11 -D22 4,) J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 With the diagonalization of the diffusivity matrix, the n coupled diffusion equations [eqn. (l)] can be separated into n individual equations which can be easily solved as in the case of pure component systems. The periodic steady solution of eqn. (1)-(4) is: q = [Xldiag [B]([K]p -kr T)(;&::;IThe function f(+) is defined by: cosh(b,+) ; for 0 = 0 ; for 0 = 1 fk($) = I (k = 1, 2, ..., n) -+ sinh(b, +) ; for 0 = 2 where b, = (1 + i),/wR,2/(2Ak). The matrix [B] is derived from the particle surface bound- ary condition [eqn. (4)] : [BI -= [XI + diag(wtSk)[X]diag(6,,+ idck) with a+l6,k -id,, = -bk tan h(b,) ; for 0 = 0 'l(bk)/zO(bk) ; for B = 1 (k = 1, 2, ..., n) (bk coth(b,) -l)/bk; for B = 2 where d,, and 6, are in-phase and out-of-phase functions for the intrinsic diffusion mechanism only.The volumetic average for qis: 4 = CAI"Klp -k, T) (13) with [A] = [X]diag(G,, -iS,,)[B]. From the heat balance equation [eqn. (6)],the following equation is obtained: where aT= iwtd(1 + iwt,). This yields : 4 = CClCAlCKIP = CHIP (15) where [C]-' = [I] -(cl~/C,)[A][kr x hw. Eqn. (15) is introduced into eqn. (7) to obtain solutions for the gas pressures p: p = u exp(iwt)[Elp, (16) where [El-' = [I] + diag(B,)[q. This allows one to deter- mine the amplitudes and phase lags of the partial and total pressures ak,cp, and A,0,respectively. The in-phase and out-of-phase functions for the partial and total pressures are defined as: Pek ' Bk 0,6in.k -iaout,k= -eXp(-icpk) -1 = -(k = 1, 2, .. . , n)a& Yk (17) S!tot)In -i'(tot) out = APv exp(-ia) -1 = c;=1 Pk ek A C;=l Yk (18) where N 37 1 When the interference between the different components is negligible, both for the equilibrium and kinetics (Kkj = Dkj = 0, k #j), the following simple frequency response solutions are obtained: where the heat effect has been neglected and These solutions are identical to those obtained for one-component systems. ' The above frequency response functions can be normalized with respect to their asymptotes of the in-phase functions at zero frequency, which is done by replacing [HI and [El -by[K] and [I] + diag(B,)[K), respectively in eqn. (17) and (18).In the case of negligible equilibrium interference, i.e. [K] x diag(K,,), the asymptotes are given by: din,klo=O= jkKkk (20) This simplified solution has been given by Yasuda for a binary system.,, Results and Discussion The adsorption of CO, and C,H, in 4A zeolite was used as a reference system for theoretically investigating multi-component frequency response behaviour. The equilibrium and diffusion parameters of this system have been given by Chen and Yang7 and are summarized in Table 1. The main characteristic of the chosen system is that the diffusion of CO, in 4A zeolite is much faster than that of C,H, [D,(CO,)/Do(C,H,) x 501. Values of the other parameters are: V, = m3, V,= 3 x m3, 0 = 2 (spherical crystals), R, = 1.7 pm, u = 0.02, C, = 10, J m-3 K, t, = 0.1 s and t, = 0 (no surface barrier effect).With these values, the heat effect proved to be negligible owing to the very low rates of diffusion of CO, and C,H, in 4A zeolite. The equilibria of the binary mixture C0,-C,H, were cal- culated using the following LRC isotherm: where the constants b and n were assumed to be the same as for the pure components. The prediction of the binary diffusi- vities from pure-component diffusivities was made using Yang's m~del~.~ where Do, is the zero coverage Fickian diffusivity of com- ponent k, and wkjrepresents the ratio of the sticking prob- ability of molecule k on adsorbed molecule j to that on a Table 1 Equilibrium and diffusion parameters of CO, and C,H, in 4A zeolite at 25 "C taken from Chen and Yang.' no.sorbate qJkgm-' bfPa-' n -AH/J kg-' D,/m's-' UJ~ 1 CO, 290 3.7 x lo-' 0.530 0.75 x lo6 4.2 x 10 Is 0.0936 2 C,H, 121 7.0 x 0.982 0.93 x lo6 8.2 x lo-" 0.0117 vacant site, which can be obtained from pure-component data (AHk and okk).When all o are equal to zero, the above model leads to the same expressions obtained from irrevers- ible thermodynamics for ideal multicomponent diffusion with an extended Langmuir i~otherm."~ When all o are equal to unity, the model predicts independent diffusion behaviour with constant main-term diffusivities and zero off-diagonal diffusivities(Dkk= Do,and Dkj= 0). All the simulations were made with P, = 15 kPa and x(C0,) = 0.33, which gave the following values for [K] and CDl : 0.0067 -0.0023 w1=( -0.0010 0.0012 1.4 10-l~ 1.1 x 10-l~) ['I = (3.7 10-17 1.2 x 10-14 Effect of Equilibrium and Diffusional Interference on the Partial-pressure Response The frequency response of the binary mixture, with both dif- fusional and equilibrium interference, was calculated using the reference equilibrium and diffusion values.The normal- ized in-phase and out-of-phase functions for both partial and total pressures are shown in Fig. 1. It can be seen that the frequency response curves of the slower diffusing component (C,H4)behave monotonically (as in a single component case) and are little affected by the presence of CO,. On the other hand, the faster diffusing component (CO,) is strongly influ- enced by C,H4 and has a negatively valued out-of-phase function near the resonance frequency of the slower com- ponent, while its in-phase function becomes greater than unity.This much stronger influence of the slower component on the faster component had previously been pointed out by Chen and Yang.' Effects of the diffusional interference can be better under- stood by comparing Fig. 1 with Fig. 2, wherein are shown frequency response curves obtained by cancelling the cross- term diffusivities (ill,= D,,= 0). In this case, the behaviour of the frequency response of CO, becomes 'normal', in the sense that the in-phase function does not exceed unity and the out-of-phase function is always positive. Owing to the equilibrium interference, however, the frequency response of CO, exhibits an additional maximum of smaller magnitude at lower frequency, which disappears completely when the cross-terms of the equilibrium are also set to zero (K12 = K,, = 0, Fig.3). The comparison of these figures suggests that the 'roll-up' (in-phase greater than unity and out-of- ...1.2 -f in-phase \\ I\ total0.6 0.4 0.2 0.0 --.-.--I out-of-p hase -0.21 '"'~1'1 111111111 ' 1I''I111 1 IIIII 1 TIIlITT 10-~ lo4 10-3 lo9 lo-' 1 freq uency/H z Fig. 1 Normalized in-phase and out-of-phase functions of the partial and total pressures for the reference case. Short dashed line: CO, ,long dashed line: C,H, and solid line: total pressure. J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 co2 f req uency/Hz Fig. 2 Same as in Fig. 1 except that the cross-term diffusivities (D,, and D,,)have been set to zero frequency/Hz Fig. 3 Frequency response of two independent components with zero equilibrium and diffusion cross terms (KI2= K,, = D12= D,,= 0), i.e. the independent single component case. Reference values for the other parameters. phase less than zero) of the CO, partial-pressure frequency response curves is caused by diffusional interference. This can be further confirmed by Fig. 4, wherein the frequency response is obtained by swapping the zero-coverage diffusi- vities of CO, and C,H, . The roll-up now occurs for the light component C,H, which has a larger diffusivity, but the bimodal pattern of the total-pressure frequency response -.1.2-1 , 10" lo4 10" lo-* lo-' freq u ency/H z Fig.4 Frequency response obtained by exchanging the zero-coverage diffusivities of CO, and C,H,. Reference values for the other parameters. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 curves has largely disappeared owing to the smaller adsorp- tion capacity of the fast-diffusing component. The change in behaviour of the CO, frequency response curves in Fig. 1-3 can be explained by investigating the two components, phase shift (qco2)and amplitude (aco2),of the in-phase (din) and out-of-phase (dout)functions, where (25) The phase shift and amplitude behaviour of CO, corre-sponding to Fig. 1-3 is shown in Fig. 5-6. Curves (a)in Fig. 5 and 6 show the behaviour of the amplitude (aco2)and phase shift (qco,)representative of a pure component with neither equilibrium nor diffusional interference (Kl2 = K,, = D,, = D,,= 0 corresponding to Fig.3). When equilibrium inter- ference is introduced uia the non-zero cross-terms, K,, and KZ1(curves (b)in Fig. 5 and 6, corresponding to Fig. 2), one observes that the amplitude increases in two stages with increasing frequency, while there are now two peaks in the phase shift spectrum. The additional phenomena in the amplitude and phase shift spectra at low frequency are induced by the behaviour of C,H, in the region of its own resonant frequency (ca. lop4Hz). As the frequency increases and enters the resonant frequency region of C2H6, the resist- I -____-----__---.____----250-200 $ 150 lo4 io4 lo-' 1 10 frequency/ Hz Fig. 5 Gas-phase CO, partial pressure amplitude us.frequency. Curve (a):no equilibrium or diffusional interference (K,,= KZ1= D1,= D,, = 0),corresponding to Fig. 3. Curve (b):no diffusional interference (D,,= D,,= 0, K,,, K,,# 0),corresponding to Fig. 2. Curve (c): both diffusional and equilibrium interference, reference case (K,,, K,, ,D,,,D,, # 0),corresponding to Fig. 1. 0.20 0.1 5 v) m.-x 0.10 s-'c r 0.05 v) c 0 0.00 -0.05 lo4 lo-' I 10 frequency/ Hz Fig. 6 Phase shift as a function of frequency. Parameters for curves (a),(b)and (c)as in Fig. 5. ance to diffusion inhibits the movement of C& into and out of the adsorbent, resulting in a concentration profile; conse- quently the average adsorbed-phase amplitude decreases and the gas-phase amplitude increases.Expanding the multi- component equilibrium [eqn. (5)], for constant temperature and no surface barrier, the particle surface concentration of CO, is given by: qco2I*= 1 = Kl lPCO2 + K12PC2H6 (26) As K,, is negative, the increasing C,H6 gas-phase amplitude causes a decrease in the (amplitude of) CO, surface concen- tration oia eqn. (26),and consequently a quasi-instantaneous decrease in the amplitude of the CO, concentration within the adsorbent and therefore an increase in the gas-phase CO, amplitude [the first stage increase of curve (b)in Fig. 51. This CO, response is entirely induced by the equilibrium inter- ference of C,H6 [oia eqn.(26)],giving a positive phase shift peak at the same frequency as that for C,H6 (ca. Hz, Fig. 6).The second stage increase of the CO, amplitude, and the second CO, phase shift peak (at CQ. lo-, Hz) are caused by the retarded transfer of CO, resulting from the adsorbent diffusional resistance (at a timescale characterized by D,l). Curves (c) in Fig. 5 and 6 (corresponding to Fig. 1) show the CO, behaviour with both equilibrium and diffusional interference. The CO, in-phase function greater than unity in Fig. 1 is mainly due to the decrease in amplitude (in the region 10-5-10-2 Hz) in Fig. 5, while the negative CO, out-of-phase function (region 10-5-10-3 Hz in Fig. 1) is mainly a result of the negative phase shift in Fig.6. This unusual behaviour is due to a combination of C,H6 diffu- sional and equilibrium interference and can be explained by inspecting the concentration profiles within the adsorbent particle. The particle concentration profiles of CO, and C2H6, for the three cases of equilibrium and diffusional inter- ference (presented in Fig. 1-3), were generated by numerical solution (finite difference) of model eqn. (1)-(4). These profiles (in Fig. 7) are given for a frequency of lop3Hz (diffusional resistance is very large for CzH6, but negligible for CO,), at 1/8 of a period (i.e. volume decreasing, total pressure increas- ing, adsorption in progress). It is evident from the similarity of all three ethane curves in Fig.7 that the C2H6 profiles are little affected by either equi- librium or diffusional interference with CO,; the C2H6 behaviour is dominated by diffusional resistance within the particle at a timescale characterized by D,,.Unlike the pro- files of C,H6, Fig. 7 shows that the CO, profiles are strongly influenced by equilbrium and diffusional interference. The CO, profile with no interference (corresponding to Fig. 3, i.e. 0.25 I P" 0.154 -0.05 !0.0 I0.2 I0.4 10.6 I0.8 1.o radial coordinate (+) Fig. 7 Adsorbed phase concentration profiles after 1/8 period (volume decreasing) at Hz. Parameters for curves (a), (b)and (c) as in Fig. 5. independent single component behaviour) is given by curve (a),which has a very weak gradient at the surface as a result of negligible diffusional resistance (at 10-Hz).As explained previously, the addition of equilibrium interference via the cross-terms, K,, and K,, (corresponding to Fig. 2), results in a decrease in CO, concentration within the particle due to the surface CO, concentration being reduced by C,H6 inter- ference. This is shown by curve (b) in Fig. 7 where, again, the profile is relatively flat due to minimal diffusional resistance. Profile (c) in Fig. 7 results from both equilibrium and diffu- sional interference (Kl,, K21, D,,, D,, non-zero), corre-sponding to the frequency response curves presented in Fig. I. Expanding eqn. (1) for CO, gives (27) showing that the C2H6 gradient within the particle can also contribute to the accumulation of CO, (as D,,is positive and the same order of magnitude as Ill,).Hence, the strong posi- tive C,H6 gradient near the surface induces an additional flux of CO, into the particle which increase the mass of CO, adsorbed in the particle, raising profile (c) above profile (b); the adsorbed mass amplitude is therefore greater, and the gas-phase CO, amplitude is smaller, than for case (b) (see Fig. 5). The additional CO, flux induced by the C,H6 gradient is, however, not the only phenomenon affecting the shape of the CO, profile. As in case (b),C,H6 equilibrium interference (via the K,, term), and also reduced CO, in the gas phase, reduces the CO, surface concentration much below that inside the particle (independently induced by the C,H6 gradient).This results in a steep negative CO, gradient near the adsorbent surface [curve (c), Fig. 71 and, consequently, desorption from the particle in the region of the surface, Although it appears paradoxical that both co-diffusion and counter-diffusion can occur simultaneously, this is in fact per- missible as the two CO, fluxes, due to the CO, and C,H6 gradients [eqn. (27)], are completely independent of each other; the result of these two processes being the negative CO, phase shift (see Fig. 6) in the region of lo-’ to Hz. Conversely, if the sign of the diffusion cross-terms (D,,, D2J is made negative, the CO, out-of-phase peak (and hence the phase shift) at lop4Hz becomes positive and the CO, in-phase is always less than unity (Fig.8), and the ‘roll-up’ 0.0 , 10-5 lo4 lo3 lo-’ 1 frequency/Hz Fig. 8 Normalized in-phase and out-of-phase functions of the CO, (short dashed line) and C,H, (long dashed line) partial pressures and the total pressure (solid line). Parameters same as reference case except with D,, and D,, negative. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 behaviour of Fig. 1 completely disappears. The very large C,H, gradient at the particle surface (starting to appear before ca. Hz), combined with the negative value of largely blocks the net movement of CO, into and out of the adsorbent [via eqn. (27)]. Thus the CO, out-of-phase peak at ca. Hz is more pronounced than that at the CO, reson- ant frequency (ca.Hz); the opposite behaviour to that observed for diffusional interference with positive cross-terms (Fig. 1). Comparing Fig. 1, 2, 3 and 8, one can clearly see that the roll-up of the CO, partial pressure frequency response curves is uniquely a consequence of diffusional interference with positive cross-terms ; no other type of interference can cause this behaviour. Sensitivity of the Total-pressure Response The out-of-phase curves of the total pressure shown in Fig. I, 2, 3 and 8 exhibit a bimodal form due to the difference in diffusion rates of the two components: the peak at the lower frequency is induced by diffusion of C,H6 and the peak at the higher frequency by diffusion of CO,. While the frequency response curves of the partial pressures have qualitatively different forms due to equilibrium and diffusional interference between the components (as discussed previously), it is evident from Fig.1, 2, 3 and 8 that the corresponding responses of the total pressure have similar forms and there- fore are only quantitatively different. This is summarized in Fig. 9, wherein the total in-phase and out-of-phase curves from Fig. 1,2, 3 and 8 are assembled. It is clear from Fig. 1, 2, 3 and 8 that one could easily discriminate between the different types of interference if the CO, and C,H6 partial-pressure responses were experimen- tally measured. However, current frequency response tech- niques can only give the total-pressure response, so it is important to know whether this is sufficient to identify accu- rately the type of interference involved.The difference between the curves in Fig. 9 indicates that, in principle, it should be possible to identify the type of interference based on the experimental total-pressure response alone; noting that, although curves (a) and (b)(no interference and equi- librium interference, respectively) are virtually identical when normalized, they are quite different in the unnormalized form. In practice, however, one would require very ‘clean’ data (i.e. with very little experimental scatter); which has not always been the case, i.e. some frequency response data appearing in the literature has quite a lot of scatter of the experimental 1.o 0.8 0.6 0.4 0.2 0.010-5 10-4 10-3 10-2 lo-’ frequency/Hz Fig.9 Comparison of total-pressure responses (from Fig. 1, 2, 3 and 8) for different types of interference. Curves (a),(b) and (c) same as in Fig. 5. Curve (4:both diffusional and equilibrium interference, but with negative cross-term diffusion coefficients (K ,,, K,, # 0; D,,, D,,< 0),corresponding to Fig. 8. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 points. Without 'clean' data, one could not discriminate between the different theoretical curves on a statistically sig- nificant basis. The separation between the total-pressure out-of-phase peaks is dependent not only on the type of interference, but also on the magnitude of the diffusivities of each component and the thermodynamic properties of the system. Thus, for an experimental study, one should choose a binary system in which the diffusivities of each component differ by more than an order of magnitude, otherwise, as shown in Fig.10, the out-of-phase peaks will not be sufficiently distinct. Further- more, the influence of total pressure and composition of the gas mixture should not be ignored as, particularly when the isotherms for each component are quite different (as in the case of CO,/C,H, on 4A zeolite, see ref. 7), an inappropriate choice of these experimental conditions may render the out-of-phase peaks inseparable or cause one peak to disap- pear. Effect of Surface Resistance For multicomponent diffusion in zeolite crystals, the surface barrier effect can be either of an intrinsic nature and/or caused by film resistance in the gas phase.The former can result, for example, from obstructions of the pore entrances or the inte- rior windows by hydrothermal pretreatment3 and can be quantified by pure-component experiments. The latter is due to the concentration gradient in the gas phase created by the difference in diffusion rates of the components inside the crys- tals and does not exist in a pure-component experiment. As can be seen from Fig. 11 (reference case with the addition of a surface barrier of time constant is), the existence of any surface resistance may cause the in-phase and out-of-phase functions of the total pressure to intersect, as in the case of pure systems.In principle, the presence of a gas-phase con- centration gradient induced surface resistance may be detected by a quasi-single-component experiment in which the adsorbable component is mixed with a second, non-adsorbable, component (e.g., helium), hence artificially cre- ating a surface barrier. Intrusion of film resistance in multicomponent frequency response experiments has been observed by Shen and Rees when studying diffusion of p-xylene and benzene in silicalite-I:24 the experimental data of the binary mixture revealed the presence of a surface resist- ance, while the frequency response curves obtained from pure-component experiments excluded the existence of any structural surface barrier. 375 I in-phase4 0.8 % 1 out-of-Dhase I 0-5 lo4 10" lo-* 10-1 frequency/Hz Fig. 11 Effect of surface resistance on the frequency response of the total pressure. Solid line: t, = 0 (no surface barrier effect); long dashed line: t, = 6 s and short dashed line: t, = 60 s.Reference values for the other parameters. Effect of Heat Dissipation For diffusion of the mixture CO, and C2H, in 4A zeolite, the rate of diffusion is so slow that the effect of heat dissipation is negligible under usual experimental conditions. However, for fast diffusing systems, the heat effect may become important and significantly alter the frequency response behaviour. This is illustrated by the curves in Fig. 12 where, by artificially increasing the diffusivities more than lOO-fold, a thermal resistance is induced ; the frequency response curves were obtained using Do, = 5 x lo-'' m2 s-', Do,2= m2 s-', R, = 10 pm, t, = 7 s for the non-isothermal case, P, = 10 kPa and x(C0,) = 0.5 (reference values for the other parameters).It can be seen that the out-of-phase curves, which are bimodal in the isothermal case, exhibit an addi- tional peak in the non-isothermal case, as a result of a rate- limiting heat exchange. The three peaks remain, however, sufficiently distinct that the corresponding mechanisms could be discriminated with good experimental data. The fact that the resonance frequencies of each of the different transport mechanisms are little affected by the other mechanisms con- stitutes one of the most attractive advantages of the fre- quency response technique.1.o 0.8 -\'vphasea-, 10 f req uency/H z frequency/Hz Fig. 10 Out-of-phase curves of the total pressure for three ratios of Fig. 12 Effect of heat dissipation on the frequency response of the diffusion rates: short dashed line (Do, = 4.2 x lo-"), long dashed total pressure. Solid line: isothermal case (th= 0 s) and dashed line: line (Do, = 4.2 x 10-16) and solid line (Do,,= 4.2 x 10-15) non-isothermal case (th= 7 s). Curves obtained with Do, = 5 (reference values for the other Parameters). The ratios of Do, to Do,, x mz s-l, Do,z = m2 s-l, R, = 10 pm,P, = 10 kPa are given by the numbers on the curves. and x(C0,) + = 0.5 and reference values for the other parameters. J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 Conclusions Owing to its high sensitivity to the nature of rate-limiting transport processes, the frequency response technique offers the possibility of studying multicomponent adsorption systems. The present work has theoretically investigated the sensitivity of the partial-and total-pressure frequency responses to the nature of the interference affecting the multi- component diffusion. It has been shown, in the case of a binary system, that partial-pressure frequency response curves of the fast diffusing component can be qualitatively different, depending on the presence or absence of equi-librium and diffusional interference, while the resulting total- pressure frequency response curves are only quantitatively different. Nevertheless, with good experimental data, it should be possible to identify the type of interference from the total-pressure frequency response, particularly if pure and multicomponent experimental results are compared. Because of the complexity of multicomponent diffusion, however, it is preferable to make experiments over a range of conditions (compositions, pressures and temperatures), guided by a pre- dictive diffusivity model, in order to have a precise and unique determination of the multicomponent diffusivities. Furthermore, it has been shown that, for the case of equi- librium plus diffusional interference with positive cross-diffusivities, both counter-diffusion and co-diffusion of the faster diffusing component can coexist.Frequency response experiments can be disturbed by the presence of surface barrier and heat effects. The former can result from surface heterogeneity and/or film resistance in the gas phase (not present in pure-component experiments, but may arise in multicomponent experiments) and lead to inter- sections of the in-phase and out-of-phase functions. The effect of the intrinsic surface barrier can, in principle, be determined from pure-component experiments while the effect of the film resistance has to be determined from multicomponent experi- ments. In principle, the latter effect could be estimated using a binary mixture composed of an adsorbate and an inert gas which does not diffuse in adsorbents. For fast diffusing systems, effect of heat dissipation can be important and lead to an additional peak for the frequency response of the total pressure.In order to avoid attributing this thermal peak to a diffusion process, it is highly necessary to quantify the heat effect, which once again can be estimated from pure-component experiments. Glossary cs volumetric heat capacity of crystals/J K-' Do diffusion coefficient at zero coverage/m2 s-' D Fickian diffusion coeficient/m2 s-' h heat transfer coefficient/W m-* K-AH heat of adsorption/J kg- ' mass transfer coefficient for surface resistance/m s-' kS derivative of isotherm with respect to the temperature/ kT kg m-3 K-' K adsorption isotherm equilibrium constant/kg m-Pa-' n number of components P incremental pressure of adsorbates/Pa Pi initial gas pressures in the chamber/Pa absolute partial pressures at equilibrium/Pa Pe P incremental total pressure/Pa Pe absolute total pressures at equilibrium/Pa incremental amount adsorbed by crystals/kg m- 4s saturated amount adsorbed by crystals/kg m-3 r spatial coordinate/m R gas constant/J kg- ' K-' RC radius of crystals/m time/s time constant of heat exchange/s time constant of surface resistance to mass transfers/s incremental temperature/K amplitude of volume modulation volume of the chamber excluding the adsorbent sample/m3 volume occupied by the adsorbent sample/m3 molar fraction surface coverage shape parameter of particles reduced time (= t/R:)/s m2 normalized spatial coordinate ( =r/Rc) angular frequency of the volume modulation/s -',ratio of sticking probabilities in Yang's model [eqn.(23) and (2411 Subscripts m equilibrium values after a pressure or volume step e reference state j,k component index Superscripts -volume average values variables in the Laplace domain Matrix and Vector Notation boldface letters vectors of dimension n [ 3 square matrix of dimension n x n References 1 J. 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