# Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—Pore-Surface Diffusion Control

Authors: Danica V. Brzić; Menka T. Petkovska
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the present paper, the Nonlinear Frequency Response (NFR) analysis is applied for theoretical study of kinetics of adsorption governed by pore-surface diffusion. The concept of higher-order frequency response functions (FRFs) is used. Based on a nonlinear mathematical model for adsorption of pure gas and spherical adsorbent particles, the theoretical first- and second-order FRFs, which relate the adsorbate concentration in the particle to the surrounding pressure (F1(ω) andF2(ω,ω)), have been derived. The obtained FRFs have been simulated for different steady-state pressures and different ratios (between zero and one) of surface to pore diffusion coefficients. The results show that, unlikeF1(ω),F2(ω,ω) exhibits features which unambiguously distinguish the pore-surface diffusion model from pure pore diffusion and micropore diffusion. Based on the characteristic features ofF1(ω) andF2(ω,ω), a new methodology for direct estimation of the separate values of the pore and surface diffusion coefficients has been established.


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## Body

## 1. Introduction

Although the processes in chemical engineering are mostly nonlinear in nature, it has been a common practice in chemical engineering to use linear tools for analysis of those processes, providing thus significant simplifications on the cost of limited applicability. However, modern trends of design, which include intensification and optimization methods, imply the necessity of implementation of rigorous procedures and nonlinear mathematical tools.Adsorption processes are one of the examples of nonlinear processes with complex dynamic behavior. If we consider adsorption of a gas on a porous adsorbent bead, then the overall rate of the process may be controlled by the following individual processes: mass transfer through the fluid film around the particle, macropore diffusion, surface diffusion, micropore diffusion, and adsorption/desorption [1]. Since the intrinsic rate of the adsorption/desorption on the active site is usually fast, the overall rate is usually controlled by diffusional resistances. Depending on the properties of the adsorbate/adsorbent system and operating conditions, relative contributions of individual resistances may vary. Reliable identification of relative dominancy of the individual resistances is still a challenging issue. In the literature, a number of methods for measuring diffusion coefficients in solid adsorbents have been reported: uptake rate [2, 3], piezometric method [4], chromatographic method [5], infrared spectroscopy [6], vacuum temperature-programmed desorption (TPD) [7], electrochemical impedance spectroscopy (EIS) [8], combination of TPD and EIS [9], frequency response (FR) method [10–15], and temperature FR method [16].Since the FR method uses the response of the system to sine-wave excitations, it introduces additional degree of freedom (frequency of the sine-wave) compared to the step and pulse response techniques, which makes the FR method advantageous regarding distinguishing individual time constants of multikinetic processes [17].Adsorption of gases in commercial porous adsorbents is often controlled by pore-surface diffusion, where pore and surface diffusion take place in parallel and their rates are characterized by pore diffusion coefficient (Dp) and surface diffusion coefficient (Ds). Classical FR method, which uses the small amplitudes of input oscillations in order to obtain the first-order (linear) frequency response function (FRF), fails to identify the pore-surface diffusion mechanism. Namely, the first-order FRFs for micropore diffusion, macropore diffusion, and pore-surface diffusion have the same shapes [18] making the distinguishing between them impossible. For the pore-surface diffusion mechanism, the linear FR allows estimation of only an effective diffusion coefficient,Deff [18], but not the separate values ofDp andDs.The pioneering work of Petkovska and Do [19] has made a step forward in adsorption kinetic studies by extending the linear FR to the nonlinear range. They considered the nonlinear response of the system (for larger input amplitude excitations) and used the Volterra-Weiner concept of higher-order FRFs in order to obtain the first-, second-, and third-order FRFs for nonlinear adsorption models. Their early results for single mechanism models (Langmuir, film resistance control, and micropore diffusion) [20], which showed that the second-order FRFs for different mechanisms had different shapes, indicated the potential of the nonlinear FR analysis for identification of the kinetic mechanism, based on the pattern of the second-order FRF. In the later work, the derivation of FRFs up to the second-order for more complex models (micropore-macropore diffusion with and without film resistance) confirmed that the second-order FRFs have enough specific features for distinguishing between different mechanisms [21]. The methodology was also applied for some cases of nonisothermal adsorption (nonisothermal micropore diffusion [22] and nonisothermal macropore diffusion [23]). A procedure for estimating the equilibrium and kinetic parameters was also developed [18, 22, 23].In practical applications of the nonlinear frequency response (NFR) approach, the identification of the kinetic mechanism is done by comparison of the experimental second-order FRF with the theoretical ones corresponding to different kinetic models. The procedure for obtaining the experimental FRFs up to the second-order has already been established [24, 25] and validated [26]. The broader exploiting of the NFR concept is partially limited by the fact that derivation of the theoretical second-order FRF for complex kinetic model may become rather tedious and that sometimes no analytical solution for the second-order FRF can be obtained. However, the FRFs need to be derived only once and then can be recalled from the library whenever needed. The existing library of theoretical FRFs comprises mostly the FRFs for plane geometry of the adsorbent particles, while the FRFs for the spherical geometry, which is, in many cases, much better approximation of the real adsorbent shape, are strongly needed.The aim of this work was to check the ability of the NFR concept for identification and characterization of the pore-surface diffusion control kinetics, for spherical particles. Firstly, the FRFs up to the second-order for the nonlinear pore-surface diffusion model for spherical particle geometry have been derived. Numerical simulations of the FRFs for different parameters were employed in order to recognize the characteristic features for model identification. Finally, the methodology for estimation of the separate values of the pore and surface diffusion coefficients has been established.


## 2. Nonlinear Frequency Response and Concept of Higher-Order FRFs

Since the Volterra-Weiner concept of higher-order FRFs [27] will be used for derivation of the FRFs, it will be briefly presented. This concept represents the generalization of the well-known approach of convolutional integral and definition of the FRF, used in the linear frequency domain analysis. Namely, the dynamic response of a stable linear single-input single-output system to an arbitrary input signalx(t) can be defined as a convolution integral:(1)ylint=∫-∞∞gτxt-τdτwhere g(τ) represents the impulse-response function of the system, or its kernel. By taking into account the definition of FRF [27]:(2)Gω=∫-∞∞gτe-jωτdτit is possible to relate the time domain response of the system,ylin(t), with its FRFG(ω) (which is the complex function of a single variable-frequency). For the input in the form of periodic functionx(t)=Aejωt, the response defined by Eq. (1) becomes (3)ylint=Aejωt∫-∞∞gτe-jωτdτwhich, considering definition (2), becomes (4)ylint=AejωtGωWhen a nonlinear system with polynomial nonlinearities is the subject of the same arbitrary inputx(t), the response may be represented as an indefinite sum of multidimensional convolutional integrals (Volterra series):(5)ynonlint=∑n=1∞ynt(6)ynt=∫-∞∞⋯∫-∞∞gnτ1,⋯,τn∏i=1nxt-τidτ1⋯dτnwhere gn(τ1,…,τn) is the generalized impulse response function of ordern,orn-th order Volterra kernel. The first element of the series,y1(t), has the same form asylin(t) (Eq. (1)) and represents the response of the linearized system, while each of the higher terms (n>1) represents the contribution of the nonlinearities ofn-th order.In analogy to the Fourier transform of the linear system’s kernel, which defines the FRF of a linear system (Eq. (2)), the multidimensional Fourier transform of then-th order Volterra kernel defines the generalizedn-th order FRF:(7)Gnω1,⋯,ωn=∫-∞∞⋯∫-∞∞gnτ1,⋯,τne-jω1τ1+⋯ωnτndτ1⋯dτnwhich is a complex function ofn frequencies. In order to establish the relation between the response of the nonlinear system defined by Eqs. (5) and (6) and the FRFs of the different orders, defined by Eq. (7), we will consider the input periodic functionx(t) in the form (8)xt=A1ejω1t+A2ejω2tBy expanding the Volterra series (Eqs. (5) and (6)) the response will be (9)yt=y1t+y2t+…where (10)y1t=G1ω1A1ejω1t+G1ω2A2ejω2t(11)y2t=G2ω1,ω1A12e2jω1t+2G2ω1,ω2A1A2ejω1+ω2t+G2ω2,ω2A22e2jω2tThe first elementy1(t) contains two terms, which are responses of the linearized system to the individual harmonics contained in the input (Eq. (8)). However, the second element of the response,y2(t), contains the nonlinearities of the second order, which are second order interactions of the individual harmonics ((ejω1t)2 and (ejω2t)2, the first and the third terms in Eq. (11)) and second order intermodulations (combined effect) of two input harmonics (ejω1t×ejω2t, the second term in Eq. (11)). Multiple interactions and intermodulations of input frequencies are characteristic only for the nonlinear systems.


## 3. Derivation of the FRFs for Pore-Surface Diffusion Model

### 3.1. Model Equations

For the derivation of the first- and second-order FRFs, the general procedure for derivation of theoretical higher-order FRFs [28] based on Volterra-Weiner approach is followed. According to that procedure, the first step is setting up the nonlinear mathematical model of the adsorption process. For isothermal pore-surface diffusion control, the mass balance over the adsorbing porous particle (of the radius R) is given by the following equation:(12)1-ε∂qi∂t+εp∂ci∂t=1-ε1rσ∂∂rDsrσ∂qi∂r+εp1rσ∂∂rDprσ∂ci∂rwhereqi andci are the sorbate concentrations in the solid phase and in the pore, respectively, both defined as relative deviations from the corresponding steady-state values:(13)qir=Qir-QisQis(14)cir=Cir-CisCisε is porosity of the particle andεp is the modified porosity:(15)εp=εCisQisDp is pore diffusion coefficient,Ds is surface diffusion coefficient, andσ is the shape factor (σ=2 for spherical geometry). The subscripts denotes the values in the steady-state. The boundary conditions are based on the assumptions of concentration profiles symmetry (16)r=0:∂ci∂r=∂qi∂r=0and negligible film resistance (17)r=R:ci=pwhere p is the dimensionless pressure, or sorbate concentration in the gas phase, defined as (18)p=P-PsPsFurther, local equilibrium within the pore is assumed, soqi andci are related by the adsorption isotherm relation:(19)qi=fciwhich is generally nonlinear. The adsorption isotherm relation (19), as a source of nonlinearity, is expressed in the form of Taylor series around the steady state: (20)qici=aci+bci2+…wherea andb are proportional to the first and second derivatives of the adsorption isotherm, respectively (their definitions are given in Table 1). Eq. (20) is substituted into Eq. (12), and the resulting PDE has only one dependent variable,ci.Table 1
Definitions of isotherm coefficients.
a = ∂ q i ∂ c i b = 1 2 ∂ 2 q i ∂ c i 2 a e f f = ε p + 1 - ε a ε p + 1 - ε b e f f = 1 - ε b ε p + 1 - εThe overall adsorbate concentration within the particle at positionr,q(r), is defined as (21)qr=εpεp+1-εcir+1-εεp+1-εqirand the mean concentration in the adsorbent particle<q> is given as(22)q=σ+1Rσ+1∫0Rrσqrdr


### 3.2. Definitions of FRFs

In order to define the FRFs corresponding to the pore-surface diffusion model defined in the previous section, we need to define the input and output variables. Since we consider the model on the particle scale, the input is the dimensionless pressurep and the output is the dimensionless mean concentration in the adsorbent particle<q>, defined in Eq (22). Consequently, we define the main set of FRFsF1(ω),F2(ω,ω),…, which relate<q>top. However, since<q>depends onq(r) (Eq. (22)) and consequently onci(r) (Eq. (21)), we need to define two auxiliary sets of FRFs: F1∗(ω), F2∗(ω,ω),…, which relateq(r) top and H1∗(ω), H2∗(ω,ω),…, which relateci(r) top.


### 3.3. First-Order FRF

In order to derive the first-order FRF,F1(ω), the input variable (p) is expressed as a single harmonic periodic function:(23)p=Aejωtand the output variableci(r,t) is expressed as a response according to Eq. (4): (24)cir,t=AejωtH1∗ωThe input and output relations (Eqs. (23) and (24)) are substituted into the model equations (12) to (22). By equalizing the coefficients of Aejωt on each side of the model equations the following ODE in which H1∗(ω) is the dependent variable is obtained:(25)d2H1∗ω,rdr2+2rdH1∗ω,rdr-jωDeffH1∗ω,r=0where (26)Deff=1-εaDs+εpDp1-εa+εpwith boundary conditions(27)r=0dH1∗ω,rdr=0(28)r=RH1∗ω,r=1Equation (25) is a second-order homogeneous ODE and has the following analytical solution:(29)H1∗ω,r=Rrsinh⁡rjω/Deffsinh⁡Rjω/DeffUsing Eq. (21) the function F1∗(ω,r) is obtained:(30)F1∗ω,r=aeffH1∗ω,rwhereaeff is the effective first-order concentration coefficient of the adsorption isotherm, defined in Table 1. Finally, the first-order FRF with respect to the mean concentration in the particle <q> is obtained using Eq. (22):(31)F1ω=aeff3R2Rjω/Deff·ctghRjω/Deff-1jω/Deff


### 3.4. Second-Order FRF

In order to derive the second-order FRF, the input variable (p) is expressed as a sum of two harmonics of different frequencies (ω1,ω2):(32)p=A1ejω1t+A2ejω2tand the output variableci(r,t) is expressed according to Eqs. (9), (10), and (11):(33)cir,t=H1∗ω1A1ejω1t+H1∗ω2A2ejω2t+H2∗ω1,ω1A12e2jω1t+2H2∗ω1,ω2A1A2ejω1+ω2t+H2∗ω2,ω2A22e2jω2tThe input and output relations (Eqs. (32) and (33)) are substituted into the model equations (12) to (22). By equalizing the coefficients of A1A2ejω1+ω2t on each side of the model equations, the following expression for H2∗(ω1,ω2), for the caseω1=ω2, is obtained: (34)d2H2∗ω,ω,rdr2+2rdH2∗ω,ω,rdr-2jωDeffH2∗ω,ω,r=2fbjωH1∗2ω,r-4rfbDsH1∗ω,rdH1∗ω,rdr-2fbDs1+H1∗ω,rd2H1∗ω,rdr2where parameterf is defined:(35)f=1-ε1-εaDs+εpDpwith boundary conditions (36)r=0dH2∗ω,ω,rdr=0(37)r=RH2∗ω,ω,r=0By incorporating Eq. (29) for H1∗ω,r into Eq. (34), as well as the expressions for first and second derivatives of H1∗ω,r (given in the Appendix), the final equation defining the H2∗ω,ω,r function is obtained. This nonhomogeneous ODE cannot be solved analytically, so it was solved numerically, by using thebvp4c solver in MATLAB. By using Eq. (21), the function F2∗ω,ω,r is obtained:(38)F2∗ω,ω,r=aeffH2∗ω,ω,r+beffH1∗2ω,rwherebeff is the effective second-order coefficient of the adsorption isotherm (Table 1). Finally, the second-order FRF with respect to the mean concentration in the particle <q> is obtained by using Eq. (22):(39)F2ω,ω=3R3∫0Rr2F2∗ω,ω,rdr


## 3.1. Model Equations

For the derivation of the first- and second-order FRFs, the general procedure for derivation of theoretical higher-order FRFs [28] based on Volterra-Weiner approach is followed. According to that procedure, the first step is setting up the nonlinear mathematical model of the adsorption process. For isothermal pore-surface diffusion control, the mass balance over the adsorbing porous particle (of the radius R) is given by the following equation:(12)1-ε∂qi∂t+εp∂ci∂t=1-ε1rσ∂∂rDsrσ∂qi∂r+εp1rσ∂∂rDprσ∂ci∂rwhereqi andci are the sorbate concentrations in the solid phase and in the pore, respectively, both defined as relative deviations from the corresponding steady-state values:(13)qir=Qir-QisQis(14)cir=Cir-CisCisε is porosity of the particle andεp is the modified porosity:(15)εp=εCisQisDp is pore diffusion coefficient,Ds is surface diffusion coefficient, andσ is the shape factor (σ=2 for spherical geometry). The subscripts denotes the values in the steady-state. The boundary conditions are based on the assumptions of concentration profiles symmetry (16)r=0:∂ci∂r=∂qi∂r=0and negligible film resistance (17)r=R:ci=pwhere p is the dimensionless pressure, or sorbate concentration in the gas phase, defined as (18)p=P-PsPsFurther, local equilibrium within the pore is assumed, soqi andci are related by the adsorption isotherm relation:(19)qi=fciwhich is generally nonlinear. The adsorption isotherm relation (19), as a source of nonlinearity, is expressed in the form of Taylor series around the steady state: (20)qici=aci+bci2+…wherea andb are proportional to the first and second derivatives of the adsorption isotherm, respectively (their definitions are given in Table 1). Eq. (20) is substituted into Eq. (12), and the resulting PDE has only one dependent variable,ci.Table 1
Definitions of isotherm coefficients.
a = ∂ q i ∂ c i b = 1 2 ∂ 2 q i ∂ c i 2 a e f f = ε p + 1 - ε a ε p + 1 - ε b e f f = 1 - ε b ε p + 1 - εThe overall adsorbate concentration within the particle at positionr,q(r), is defined as (21)qr=εpεp+1-εcir+1-εεp+1-εqirand the mean concentration in the adsorbent particle<q> is given as(22)q=σ+1Rσ+1∫0Rrσqrdr


## 3.2. Definitions of FRFs

In order to define the FRFs corresponding to the pore-surface diffusion model defined in the previous section, we need to define the input and output variables. Since we consider the model on the particle scale, the input is the dimensionless pressurep and the output is the dimensionless mean concentration in the adsorbent particle<q>, defined in Eq (22). Consequently, we define the main set of FRFsF1(ω),F2(ω,ω),…, which relate<q>top. However, since<q>depends onq(r) (Eq. (22)) and consequently onci(r) (Eq. (21)), we need to define two auxiliary sets of FRFs: F1∗(ω), F2∗(ω,ω),…, which relateq(r) top and H1∗(ω), H2∗(ω,ω),…, which relateci(r) top.


## 3.3. First-Order FRF

In order to derive the first-order FRF,F1(ω), the input variable (p) is expressed as a single harmonic periodic function:(23)p=Aejωtand the output variableci(r,t) is expressed as a response according to Eq. (4): (24)cir,t=AejωtH1∗ωThe input and output relations (Eqs. (23) and (24)) are substituted into the model equations (12) to (22). By equalizing the coefficients of Aejωt on each side of the model equations the following ODE in which H1∗(ω) is the dependent variable is obtained:(25)d2H1∗ω,rdr2+2rdH1∗ω,rdr-jωDeffH1∗ω,r=0where (26)Deff=1-εaDs+εpDp1-εa+εpwith boundary conditions(27)r=0dH1∗ω,rdr=0(28)r=RH1∗ω,r=1Equation (25) is a second-order homogeneous ODE and has the following analytical solution:(29)H1∗ω,r=Rrsinh⁡rjω/Deffsinh⁡Rjω/DeffUsing Eq. (21) the function F1∗(ω,r) is obtained:(30)F1∗ω,r=aeffH1∗ω,rwhereaeff is the effective first-order concentration coefficient of the adsorption isotherm, defined in Table 1. Finally, the first-order FRF with respect to the mean concentration in the particle <q> is obtained using Eq. (22):(31)F1ω=aeff3R2Rjω/Deff·ctghRjω/Deff-1jω/Deff


## 3.4. Second-Order FRF

In order to derive the second-order FRF, the input variable (p) is expressed as a sum of two harmonics of different frequencies (ω1,ω2):(32)p=A1ejω1t+A2ejω2tand the output variableci(r,t) is expressed according to Eqs. (9), (10), and (11):(33)cir,t=H1∗ω1A1ejω1t+H1∗ω2A2ejω2t+H2∗ω1,ω1A12e2jω1t+2H2∗ω1,ω2A1A2ejω1+ω2t+H2∗ω2,ω2A22e2jω2tThe input and output relations (Eqs. (32) and (33)) are substituted into the model equations (12) to (22). By equalizing the coefficients of A1A2ejω1+ω2t on each side of the model equations, the following expression for H2∗(ω1,ω2), for the caseω1=ω2, is obtained: (34)d2H2∗ω,ω,rdr2+2rdH2∗ω,ω,rdr-2jωDeffH2∗ω,ω,r=2fbjωH1∗2ω,r-4rfbDsH1∗ω,rdH1∗ω,rdr-2fbDs1+H1∗ω,rd2H1∗ω,rdr2where parameterf is defined:(35)f=1-ε1-εaDs+εpDpwith boundary conditions (36)r=0dH2∗ω,ω,rdr=0(37)r=RH2∗ω,ω,r=0By incorporating Eq. (29) for H1∗ω,r into Eq. (34), as well as the expressions for first and second derivatives of H1∗ω,r (given in the Appendix), the final equation defining the H2∗ω,ω,r function is obtained. This nonhomogeneous ODE cannot be solved analytically, so it was solved numerically, by using thebvp4c solver in MATLAB. By using Eq. (21), the function F2∗ω,ω,r is obtained:(38)F2∗ω,ω,r=aeffH2∗ω,ω,r+beffH1∗2ω,rwherebeff is the effective second-order coefficient of the adsorption isotherm (Table 1). Finally, the second-order FRF with respect to the mean concentration in the particle <q> is obtained by using Eq. (22):(39)F2ω,ω=3R3∫0Rr2F2∗ω,ω,rdr


## 4. Simulations of the FRFs and Characteristic Properties

The first- and second-order FRFs derived in the previous section were simulated using MATLAB software for the system CO2/zeolite 5A, by using an adsorption isotherm from literature [29]. In order to identify the characteristic properties of the FRFs, simulations for different steady-state pressures (20 mbar, 40 mbar, and 75 mbar) and different ratios ofDs/Dp were performed. The pore diffusion coefficient was kept constantDp=10−7 m2/s, and the surface diffusion coefficient was variedDs=(10−9 - 10−7) m2/s. The caseDs=0 (Ds/Dp=0), which corresponds to pure pore diffusion mechanism, was also considered for comparison. The values of the adsorption isotherm coefficientsa,b, aeff andbeff for all three considered pressures at 298 K are given in Table 2. Simulations were done for a particle radiusR=8.5e-04 m and particle porosityε=0.35. The simulated FRFs are presented in the form of Bode plots (amplitude vs. frequency and phase vs. frequency).Table 2
Adsorption isotherm coefficients at 298K [29].
P s (mbar) 20 40 75  a 0.4369 0.3684 0.3105 b -0.1736 -0.1642 -0.1511 a eff 0.4373 0.3691 0.3147 b eff -0.1735 -0.1640 -0.1508In Figure1 the first-order FRF (Figures 1(a) and 1(b)) and the second-order FRF (Figures 1(c) and 1(d)) for three different steady-state pressures and constant ratioDs/Dp=0.1 are presented. From Figures 1(a) and 1(b) it can be seen that the amplitude ofF1(ω) has horizontal low frequency asymptote and a high frequency asymptote with a slope -0.5 and the phase ofF1(ω) has the low frequency asymptote zero, single inflection point, and the high frequency asymptote of -π/4. As expected, the steady-state pressure influences only the amplitude curves. Although the pore-surface diffusion model has two rate constants (one corresponds to pore diffusion and the other to surface diffusion), they cannot be distinguished in the frequency window of the phase ofF1(ω) and a single inflection point of the phase might lead to misinterpretation of the pore-surface diffusion as a single rate process.Figure 1
(a) AmplitudeF1(ω) vs. frequency; (b) phaseF1(ω) vs. frequency; (c) amplitudeF2(ω,ω) vs. frequency; (d) phaseF2(ω,ω) vs. frequency (Dp=10−7 m2/s;Ds= 10−8 m2/s).
(a) (b) (c) (d)In contrast to the frequency spectrum of the first-order FRF, where only one characteristic frequency (which corresponds to the inflection point of phase) is recognized, in the frequency spectrum of the second-order FRF (Figures1(c) and 1(d)), two characteristic frequencies, which correspond to the extreme values (one minimum and one maximum) of the phase, can be observed. The amplitude ofF2(ω,ω) has a horizontal low frequency asymptote, two inflection points, and a high frequency asymptote with a slope -0.5 (Figure 1(c)). The phase ofF2(ω,ω) has a low frequency asymptote of π, a distinct minimum followed by a distinct maximum, and a high frequency asymptote of 3π/4 (Figure 1(d)). As in the case ofF1(ω), the steady-state pressure influences only the amplitude but has no influence on the shape of the amplitude curve nor on the phase characteristics. The observed pattern ofF2(ω,ω) with two extreme values of the phase indicates that two time constants (pore and surface diffusion) can be separated in the frequency window ofF2(ω,ω).In Figure2 the first-order FRF (Figures 2(a) and 2(b)) and the second-order FRF (Figures 2(c) and 2(d)) for a constant steady-state pressure of 20 mbar and different ratios of Ds/Dp are given. The case Ds=0 (Ds/Dp=0, dashed line) corresponds to the pure pore diffusion. It can be seen from Figures 2(a) and 2(b) thatF1(ω) has the same shape for pure pore diffusion (Ds/Dp=0) and pore-surface diffusion, which shows that no distinction between those two mechanisms can be made based onF1(ω). However, Figures 2(c) and 2(d) show thatF2(ω,ω) has qualitatively different shapes for pure pore and pore-surface diffusion. The amplitude ofF2(ω,ω) for pure pore diffusion has one inflection while for pore-surface diffusion it has two inflections. The phase ofF2(ω,ω) for Ds=0 has a single minimum, while for all other ratios Ds/Dp a minimum followed by a maximum is observed. Both are shifted towards higher frequencies with increase of the contribution of surface diffusion. This result clearly shows that the shapes of the amplitude and phase ofF2(ω,ω) can be used for reliable discrimination between pore-surface and pure pore mechanisms. Moreover, comparison of the characteristic features of the amplitude and phase ofF2(ω,ω) for pore-surface diffusion, pure pore diffusion (this work), and micropore diffusion model [18] in Table 3 proves that these characteristic features are distinct for each of these three mechanisms and can, therefore, be used for their discrimination.Table 3
Characteristic features ofF2(ω,ω) for different mechanisms (spherical geometry).
AmplitudeF2(ω,ω) PhaseF2(ω,ω)  Pore-surface diffusion Two inflections Minimum followed by maximum Pure pore diffusion One inflection Single minimum Micropore diffusion [18] No inflections Sigmoidal (single inflection)Figure 2
(a) Amplitude ofF1(ω) vs. frequency; (b) phase ofF1(ω) vs. frequency; (c) amplitude ofF2(ω,ω) vs. frequency; (d) phase ofF2(ω,ω) vs. frequency (steady-state pressure 20 mbar).
(a) (b) (c) (d)


## 5. Estimation ofDs and Dp

For the adsorption process governed by pore-surface diffusion, two time constants can be defined:(i) Pore diffusion time constant (τp)(40)τp=R2Dp1-εa+εpεp(ii) Surface diffusion time constant (τs)(41)τs=R2Ds1-εa+εp1-εaBy taking into account the effective diffusion coefficient Deff, defined by Eq. (26), it is also common to define the effective time constant (τeff):(42)τeff=R2DeffIt is already known [18] that τeff can be estimated from the first-order FRF, actually from the locus of the maximum of negative imaginary part ofF1(ω) (Figure 3), by using the equation ωτeff=11.56 (which is valid for spherical particles). Deff can then be calculated from Eq. (42).Figure 3
Negative imaginary part ofF1(ω) vs.ωτeff [18].In the previous section it was shown that the most remarkable feature ofF2(ω,ω) is maximum right after the minimum of the phase ofF2(ω,ω). In Figure 4 the phase ofF2(ω,ω) (for different ratios Ds/Dp) is plotted vs. frequency (ω) (Figure 4(a)) and vs. dimensionless frequency, defined as frequency multiplied by surface diffusion time constant (ωτs) (Figure 4(b)). It turns out that, when plotted againstωτs, all curves overlap and the maximum corresponds toωτs=25. As a consequence, τs can be estimated from the frequency at which the phase ofF2(ω,ω) has a maximum, and Ds can be further calculated from Eq. (41).Figure 4
(a) Phase ofF2(ω,ω) vs. frequency; (b) phaseF2(ω,ω) vs.ωτs.
(a) (b)The procedure for estimation ofDs and Dp from the experimentalF1(ω) andF2(ω,ω) can be summarized in three steps as follows:(1) Identify the frequency at which the negative imaginary part ofF1(ω) has a maximum, ωcrit,1. Calculate τeff from the equation τeff=11.56/ωcrit,1. Calculate Deff using Eq. (42) as Deff=R2/τeff.(2) Identify the frequency at which the phase ofF2(ω,ω) has a maximum, ωcrit,2. Calculate τs from equation τs=25/ωcrit,2. Calculate Ds using Eq. (41) as Ds=R2/τs(1-ε)a+εp/(1-ε)a.(3) CalculateDp using Eq. (26) as Dp=Deff(1-ε)a+εp-Ds(1-ε)a/εp.


## 6. Conclusions

In this work the Nonlinear Frequency Response analysis is applied for theoretical treatment of adsorption governed by parallel pore and surface diffusion, which is commonly encountered in commercial adsorbents. The derivation of the theoretical first- and second-order FRFs (F1(ω) andF2(ω,ω)) for the pore-surface diffusion model and spherical particle geometry is given step by step. The derived FRFs relate the adsorbate concentration in the particle to the pressure of the surrounding gas (in the dimensionless form). For the first-order FRF the analytical solution was obtained, while the second-order FRF was obtained numerically. The derived FRFs were simulated by using the literature equilibrium data for CO2/zeolite 5A. The varied parameters were the steady-state pressure and the ratio of surface to pore diffusion coefficient (Ds/Dp). Unlike the first-order FRF, the second-order FRF exhibits bimodal characteristics which reflect the dynamics of the parallel pore and surface diffusion processes. For all tested combinations of parameters, the amplitude and phase characteristics ofF2(ω,ω) show the same patterns: two inflections of the amplitude and two extrema (a minimum followed by a maximum) of the phase. These features enable clear distinction of the pore-surface diffusion from pure pore diffusion and micropore diffusion mechanisms. Besides reliable mechanism identification, the second-order FRF, considered together with the first-order FRF, gives the possibility to estimate the separate values of the pore diffusion coefficient (Dp) and surface diffusion coefficient (Ds).This work has both theoretical and practical importance. It gives a detailed procedure for the derivation of the first- and second-order FRFs which can be used for derivation of FRFs for other mechanisms. Further, the existing library of theoretical FRFs is extended. The potential of the nonlinear frequency response analysis in identification of the adsorption kinetic mechanism is proven. The practical importance lies in the fact that we offer a procedure for estimation of the separate values of the pore and surface diffusion coefficients which is very often needed in engineering praxis.



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