Research Article  Open Access
Danica V. Brzić, Menka T. Petkovska, "Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—PoreSurface Diffusion Control", Mathematical Problems in Engineering, vol. 2019, Article ID 7932967, 11 pages, 2019. https://doi.org/10.1155/2019/7932967
Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—PoreSurface Diffusion Control
Abstract
In the present paper, the Nonlinear Frequency Response (NFR) analysis is applied for theoretical study of kinetics of adsorption governed by poresurface diffusion. The concept of higherorder 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 secondorder FRFs, which relate the adsorbate concentration in the particle to the surrounding pressure (F_{1}(ω) and F_{2}(ω,ω)), have been derived. The obtained FRFs have been simulated for different steadystate pressures and different ratios (between zero and one) of surface to pore diffusion coefficients. The results show that, unlike F_{1}(ω), F_{2}(ω,ω) exhibits features which unambiguously distinguish the poresurface diffusion model from pure pore diffusion and micropore diffusion. Based on the characteristic features of F_{1}(ω) and F_{2}(ω,ω), a new methodology for direct estimation of the separate values of the pore and surface diffusion coefficients has been established.
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 temperatureprogrammed 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 sinewave excitations, it introduces additional degree of freedom (frequency of the sinewave) 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 poresurface diffusion, where pore and surface diffusion take place in parallel and their rates are characterized by pore diffusion coefficient (D_{p}) and surface diffusion coefficient (D_{s}). Classical FR method, which uses the small amplitudes of input oscillations in order to obtain the firstorder (linear) frequency response function (FRF), fails to identify the poresurface diffusion mechanism. Namely, the firstorder FRFs for micropore diffusion, macropore diffusion, and poresurface diffusion have the same shapes [18] making the distinguishing between them impossible. For the poresurface diffusion mechanism, the linear FR allows estimation of only an effective diffusion coefficient, D_{eff} [18], but not the separate values of D_{p} and D_{s}.
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 VolterraWeiner concept of higherorder FRFs in order to obtain the first, second, and thirdorder FRFs for nonlinear adsorption models. Their early results for single mechanism models (Langmuir, film resistance control, and micropore diffusion) [20], which showed that the secondorder 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 secondorder FRF. In the later work, the derivation of FRFs up to the secondorder for more complex models (microporemacropore diffusion with and without film resistance) confirmed that the secondorder 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 secondorder FRF with the theoretical ones corresponding to different kinetic models. The procedure for obtaining the experimental FRFs up to the secondorder 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 secondorder FRF for complex kinetic model may become rather tedious and that sometimes no analytical solution for the secondorder 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 poresurface diffusion control kinetics, for spherical particles. Firstly, the FRFs up to the secondorder for the nonlinear poresurface 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 HigherOrder FRFs
Since the VolterraWeiner concept of higherorder FRFs [27] will be used for derivation of the FRFs, it will be briefly presented. This concept represents the generalization of the wellknown approach of convolutional integral and definition of the FRF, used in the linear frequency domain analysis. Namely, the dynamic response of a stable linear singleinput singleoutput system to an arbitrary input signal x(t) can be defined as a convolution integral:where (τ) represents the impulseresponse function of the system, or its kernel. By taking into account the definition of FRF [27]:it is possible to relate the time domain response of the system, y^{lin}(t), with its FRF G(ω) (which is the complex function of a single variablefrequency). For the input in the form of periodic function x(t)=A, the response defined by Eq. (1) becomes which, considering definition (2), becomes When a nonlinear system with polynomial nonlinearities is the subject of the same arbitrary input x(t), the response may be represented as an indefinite sum of multidimensional convolutional integrals (Volterra series):where (τ_{1},…,) is the generalized impulse response function of order n, or nth order Volterra kernel. The first element of the series, y_{1}(t), has the same form as y^{lin}(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 of nth 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 the nth order Volterra kernel defines the generalized nth order FRF:which is a complex function of n 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 function x(t) in the form By expanding the Volterra series (Eqs. (5) and (6)) the response will be where The first element y_{1}(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, y_{2}(t), contains the nonlinearities of the second order, which are second order interactions of the individual harmonics ( and , the first and the third terms in Eq. (11)) and second order intermodulations (combined effect) of two input harmonics (, 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 PoreSurface Diffusion Model
3.1. Model Equations
For the derivation of the first and secondorder FRFs, the general procedure for derivation of theoretical higherorder FRFs [28] based on VolterraWeiner approach is followed. According to that procedure, the first step is setting up the nonlinear mathematical model of the adsorption process. For isothermal poresurface diffusion control, the mass balance over the adsorbing porous particle (of the radius R) is given by the following equation:where q_{i} and c_{i} are the sorbate concentrations in the solid phase and in the pore, respectively, both defined as relative deviations from the corresponding steadystate values:ε is porosity of the particle and ε_{p} is the modified porosity:D_{p} is pore diffusion coefficient, D_{s} is surface diffusion coefficient, and σ is the shape factor (σ=2 for spherical geometry). The subscript s denotes the values in the steadystate. The boundary conditions are based on the assumptions of concentration profiles symmetry and negligible film resistance where p is the dimensionless pressure, or sorbate concentration in the gas phase, defined as Further, local equilibrium within the pore is assumed, so q_{i} and c_{i} are related by the adsorption isotherm relation:which 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: where a and b 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, c_{i}.
The overall adsorbate concentration within the particle at position r, q(r), is defined as and the mean concentration in the adsorbent particle <q> is given as
3.2. Definitions of FRFs
In order to define the FRFs corresponding to the poresurface 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 pressure p and the output is the dimensionless mean concentration in the adsorbent particle <q>, defined in Eq (22). Consequently, we define the main set of FRFs F_{1}(ω), F_{2}(ω,ω), which relate <q> to p. However, since <q> depends on q(r) (Eq. (22)) and consequently on c_{i}(r) (Eq. (21)), we need to define two auxiliary sets of FRFs: (ω), (ω,ω), which relate q(r) to p and (ω), (ω,ω), which relate c_{i}(r) to p.
3.3. FirstOrder FRF
In order to derive the firstorder FRF, F_{1}(ω), the input variable (p) is expressed as a single harmonic periodic function:and the output variable c_{i}(r,t) is expressed as a response according to Eq. (4): The input and output relations (Eqs. (23) and (24)) are substituted into the model equations (12) to (22). By equalizing the coefficients of on each side of the model equations the following ODE in which is the dependent variable is obtained:where with boundary conditionsEquation (25) is a secondorder homogeneous ODE and has the following analytical solution:Using Eq. (21) the function (ω,r) is obtained:where a_{eff} is the effective firstorder concentration coefficient of the adsorption isotherm, defined in Table 1. Finally, the firstorder FRF with respect to the mean concentration in the particle <q> is obtained using Eq. (22):
3.4. SecondOrder FRF
In order to derive the secondorder FRF, the input variable (p) is expressed as a sum of two harmonics of different frequencies (ω_{1},ω_{2}):and the output variable c_{i}(r,t) is expressed according to Eqs. (9), (10), and (11):The input and output relations (Eqs. (32) and (33)) are substituted into the model equations (12) to (22). By equalizing the coefficients of on each side of the model equations, the following expression for , for the case ω_{1}=ω_{2}, is obtained: where parameter f is defined:with boundary conditions By incorporating Eq. (29) for into Eq. (34), as well as the expressions for first and second derivatives of (given in the Appendix), the final equation defining the function is obtained. This nonhomogeneous ODE cannot be solved analytically, so it was solved numerically, by using the bvp4c solver in MATLAB. By using Eq. (21), the function is obtained:where b_{eff} is the effective secondorder coefficient of the adsorption isotherm (Table 1). Finally, the secondorder FRF with respect to the mean concentration in the particle <q> is obtained by using Eq. (22):
4. Simulations of the FRFs and Characteristic Properties
The first and secondorder FRFs derived in the previous section were simulated using MATLAB software for the system CO_{2}/zeolite 5A, by using an adsorption isotherm from literature [29]. In order to identify the characteristic properties of the FRFs, simulations for different steadystate pressures (20 mbar, 40 mbar, and 75 mbar) and different ratios of D_{s}/D_{p} were performed. The pore diffusion coefficient was kept constant D_{p}=10^{−7} m^{2}/s, and the surface diffusion coefficient was varied D_{s}=(10^{−9}  10^{−7}) m^{2}/s. The case D_{s}=0 (D_{s}/D_{p}=0), which corresponds to pure pore diffusion mechanism, was also considered for comparison. The values of the adsorption isotherm coefficients a, b, a_{eff} and b_{eff} for all three considered pressures at 298 K are given in Table 2. Simulations were done for a particle radius R=8.5e04 m and particle porosity ε=0.35. The simulated FRFs are presented in the form of Bode plots (amplitude vs. frequency and phase vs. frequency).

In Figure 1 the firstorder FRF (Figures 1(a) and 1(b)) and the secondorder FRF (Figures 1(c) and 1(d)) for three different steadystate pressures and constant ratio D_{s}/D_{p}=0.1 are presented. From Figures 1(a) and 1(b) it can be seen that the amplitude of F_{1}(ω) has horizontal low frequency asymptote and a high frequency asymptote with a slope 0.5 and the phase of F_{1}(ω) has the low frequency asymptote zero, single inflection point, and the high frequency asymptote of π/4. As expected, the steadystate pressure influences only the amplitude curves. Although the poresurface 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 of F_{1}(ω) and a single inflection point of the phase might lead to misinterpretation of the poresurface diffusion as a single rate process.
(a)
(b)
(c)
(d)
In contrast to the frequency spectrum of the firstorder FRF, where only one characteristic frequency (which corresponds to the inflection point of phase) is recognized, in the frequency spectrum of the secondorder FRF (Figures 1(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 of F_{2}(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 of F_{2}(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 of F_{1}(ω), the steadystate pressure influences only the amplitude but has no influence on the shape of the amplitude curve nor on the phase characteristics. The observed pattern of F_{2}(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 of F_{2}(ω,ω).
In Figure 2 the firstorder FRF (Figures 2(a) and 2(b)) and the secondorder FRF (Figures 2(c) and 2(d)) for a constant steadystate pressure of 20 mbar and different ratios of / are given. The case =0 (/=0, dashed line) corresponds to the pure pore diffusion. It can be seen from Figures 2(a) and 2(b) that F_{1}(ω) has the same shape for pure pore diffusion (/=0) and poresurface diffusion, which shows that no distinction between those two mechanisms can be made based on F_{1}(ω). However, Figures 2(c) and 2(d) show that F_{2}(ω,ω) has qualitatively different shapes for pure pore and poresurface diffusion. The amplitude of F_{2}(ω,ω) for pure pore diffusion has one inflection while for poresurface diffusion it has two inflections. The phase of F_{2}(ω,ω) for =0 has a single minimum, while for all other ratios / 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 of F_{2}(ω,ω) can be used for reliable discrimination between poresurface and pure pore mechanisms. Moreover, comparison of the characteristic features of the amplitude and phase of F_{2}(ω,ω) for poresurface 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.

(a)
(b)
(c)
(d)
5. Estimation of and
For the adsorption process governed by poresurface diffusion, two time constants can be defined:
(i) Pore diffusion time constant ()
(ii) Surface diffusion time constant ()By taking into account the effective diffusion coefficient , defined by Eq. (26), it is also common to define the effective time constant ():It is already known [18] that can be estimated from the firstorder FRF, actually from the locus of the maximum of negative imaginary part of F_{1}(ω) (Figure 3), by using the equation (which is valid for spherical particles). can then be calculated from Eq. (42).
In the previous section it was shown that the most remarkable feature of F_{2}(ω,ω) is maximum right after the minimum of the phase of F_{2}(ω,ω). In Figure 4 the phase of F_{2}(ω,ω) (for different ratios /) is plotted vs. frequency (ω) (Figure 4(a)) and vs. dimensionless frequency, defined as frequency multiplied by surface diffusion time constant (ω) (Figure 4(b)). It turns out that, when plotted against ω, all curves overlap and the maximum corresponds to ω=25. As a consequence, can be estimated from the frequency at which the phase of F_{2}(ω,ω) has a maximum, and can be further calculated from Eq. (41).
(a)
(b)
The procedure for estimation of and from the experimental F_{1}(ω) and F_{2}(ω,ω) can be summarized in three steps as follows:
(1) Identify the frequency at which the negative imaginary part of F_{1}(ω) has a maximum, . Calculate from the equation . Calculate using Eq. (42) as .
(2) Identify the frequency at which the phase of F_{2}(ω,ω) has a maximum, Calculate from equation . Calculate using Eq. (41) as .
(3) Calculate using Eq. (26) as .
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 secondorder FRFs (F_{1}(ω) and F_{2}(ω,ω)) for the poresurface 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 firstorder FRF the analytical solution was obtained, while the secondorder FRF was obtained numerically. The derived FRFs were simulated by using the literature equilibrium data for CO_{2}/zeolite 5A. The varied parameters were the steadystate pressure and the ratio of surface to pore diffusion coefficient (/). Unlike the firstorder FRF, the secondorder 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 of F_{2}(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 poresurface diffusion from pure pore diffusion and micropore diffusion mechanisms. Besides reliable mechanism identification, the secondorder FRF, considered together with the firstorder FRF, gives the possibility to estimate the separate values of the pore diffusion coefficient () and surface diffusion coefficient ().
This work has both theoretical and practical importance. It gives a detailed procedure for the derivation of the first and secondorder 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.
Appendix
The first and the second derivatives of the function defined by Eq. (29) are the following:
(i) The first derivative
(ii) The second derivative
Nomenclature
:  Amplitude of the input 
:  Firstorder coefficient of the adsorption isotherm (Table 1) 
b:  Secondorder coefficient of the adsorption isotherm (Table 1) 
:  Dimensionless sorbate concentration in the pore 
:  Sorbate concentration in the pore (mol/m^{3}) 
:  Pore diffusion coefficient (m^{2}/s) 
:  Surface diffusion coefficient (m^{2}/s) 
:  Effective diffusion coefficient (m^{2}/s) 
:  Parameter (Eq. (35)) 
(ω_{1}, ω_{2}, … ):  nth order FRF relating <q> to 
(ω_{1}, ω_{2}, … ):  nth order FRF relating () to 
(ω_{1}, ω_{2}, … ):  nth order FRF relating to 
(ω_{1}, ω_{2}, … ):  nth order FRF relating to 
:  Dimensionless pressure 
:  Pressure (bar) 
:  Dimensionless sorbate concentration in the solid phase 
:  Sorbate concentration in the solid phase (mol/m^{3}) 
:  Radius of the adsorbent particle (m) 
:  Spatial coordinate 
:  Time (s) 
:  Input 
:  Output. 
ε:  Porosity of the adsorbent particle 
:  Modified porosity of the adsorbent particle 
σ:  Shape factor 
:  Pore diffusion time constant (Eq. (40)) 
:  Surface diffusion time constant (Eq. (41)) 
:  Effective diffusion time constant (Eq. (42)) 
ω:  Frequency (rad/s). 
Data Availability
The simulation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The financial support from the Ministry of Education and Science, Republic of Serbia, project number 172022, is gratefully acknowledged.
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Copyright
Copyright © 2019 Danica V. Brzić and Menka T. Petkovska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.