Identification of mass transfer resistances in microporous materials using frequency response methods

Abstract

The frequency response (FR) method, a pseudo-steady state relaxation technique employing perturbation frequency, plays an essential role in discriminating between multi-kinetic mechanisms in microporous materials for separation and catalytic processes. Experimental and theoretical principles are reviewed for three frequency response methods, including one commonly used batch system with volume perturbation and two recently developed flow-through systems with pressure or concentration oscillation. Even though these methods have different overall transfer functions, they can be linked closely through the adsorbed-phase functions, which account for individual or coupling of mass transfer resistances and heat effects. Mass transfer resistances include micropore diffusion, macropore diffusion, surface barriers, and external film resistance. By judicious application of FR methods, it is not only possible to identify dominating mass transfer resistances but also to extract reliable mass transfer coefficients based on corresponding mathematical models. Representative examples to display the ability of the FR methods in studies of zeolites, carbon molecular sieves, and other microporous materials are discussed. Mixture studies and future developments, including nonlinear frequency response and chemical reactions, have also been briefly described.

Introduction

Zeolites and other microporous materials have many applications in industrial processes for the upgrading of molecules via gas adsorption, molecular separations, and catalytic conversion. When the size of a molecule approaches the pore size of a microporous material, diffusion becomes an essential factor affecting separations and reaction processes. In commercial operations, the microporous materials are generally bound in pellets adding the complication of macropore diffusion. Knowledge of mass transfer mechanisms and rates is a prerequisite for the accurate design of adsorption separation and catalysis processes. Therefore, kinetic studies have received considerable interest from both academic and industrial researchers. Even with substantial efforts on the development of microscopic and macroscopic techniques, this area remains challenging. Among the various available methods, frequency response (FR) is unique in its ability to distinguish between the various possible rate controlling steps in microporous materials with complex pore networks.

FR is a pseudo-steady state relaxation method in which one system variable (e.g. volume or pressure) is perturbed periodically around an equilibrium state, and the resulting periodic response in another system variable (e.g. pressure or flowrate) is measured to characterize the system. In contrast to Nonlinear FR (NFR) [1, 2], the perturbation is kept small to ensure that the model equations can be appropriately linearized. At each frequency, the amplitude ratio from the perturbed and responding variables is measured and plotted against the perturbation frequency to construct a response curve. With this additional degree of freedom—the perturbation frequency—FR can decouple mass transfer resistances due to its high sensitivity to the forms of the governing transport equations. Also, it has the advantage of minimizing measurement errors because of its periodic process and non-dependence on initial conditions. Thus, FR has proven to be very useful for characterizing the transport of gases in adsorbent materials, such as zeolites [3,4,5,6,7,8,9,10,11,12,13,14], carbons [15,16,17,18,19,20], silicas [21], metal organic frameworks (MOFs) [22,23,24], and catalysts [25,26,27,28,29,30].

Most FR applications involve a closed system where the system volume oscillates, and the resulting response in system pressure is measured. Naphtali and Polinski [31] first applied the volume-swing frequency response (VSFR) technique to study chemisorption kinetics. Since then, it has been developed by Yasuda et al. [13, 32,33,34,35,36,37,38] to measure diffusion coefficients in gas-zeolite systems. Other research groups (e.g. Do [39, 40], Sun [41,42,43,44,45,46,47], and Rees [48,49,50,51]) have continued to improve FR in terms of experimental setup and theoretical analysis. The typical FR experimental apparatus allows for diffusivity and adsorption measurements using perturbation frequencies between 0.05 and 10 Hz [21, 52] while newer versions allow perturbation frequencies up to 100 Hz [24]. Bourdin et al. [41, 53, 54] introduced a thermal FR method to better distinguish thermal transfer from mass transfer limitation through the measurement of both the temperature and pressure of an adsorbent sample subjected to volume modulation. While most investigations were limited to low pressures, Hossain et al. [4, 55] recently reported a new volumetric system to expand pressures up to 1 bar.

Besides VSFR, another type of widely used FR is a flow-through technique which involves perturbations in inlet flow rates, pressure, or inlet concentrations [15, 18, 20, 56,57,58,59,60,61,62,63,64]. Compared to batch FR, the flow-through FR methods have the advantage of minimizing heat effects and flexible operation through a wide range of concentrations. For example, the pressure-swing frequency response (PSFR) technique perturbs the system pressure sinusoidally, measuring the outlet flow rate responded with the same frequency but with a different amplitude and phase lag [15, 62, 64]. The most recent development involves a concentration-swing frequency response (CSFR) method capable of measuring mixture diffusion. It incorporates a mass spectrometer to monitor outlet gas concentrations while sinusoidally varying flow rates of inlet gases, creating a concentration-perturbation [18, 20, 23, 56, 57, 63, 65, 66]. The major disadvantage of flow-through systems, however, is the limitation on the maximum frequency, which cannot exceed ~ 1 Hz for the response time of flow controllers. To take advantage of both batch and flow-through FR techniques, Giesy et al. [16] introduced a combined apparatus that incorporates pressure-swing, volume-swing, and concentration-swing FR features to characterize the broad dynamics of an adsorption system.

The ability of FR techniques to characterize kinetics relies on the availability of mathematical models to describe the FR of all transport mechanisms that could occur in the system under investigation. By comparing the experimental data of FR curves with various mathematical models depicting each of the possible transport mechanisms, only the model corresponding to the correct mass transfer mechanism will best describe the data as each model behaves differently over the entire range of frequencies. Throughout the history of the application of FR techniques to adsorption systems, mathematical models have been developed for various physical systems with different rate-limiting mechanisms [31, 33, 38,39,40, 46, 47, 56, 67,68,69]. Systems have been identified with diffusion using the Fickian diffusion model [4, 16, 23, 70], first-order reaction kinetics [71], a surface barrier with linear driving force model [3, 15, 63, 64, 72, 73], parallel diffusion processes [52], and heat effects with non-isothermal models [40, 42, 47].

This paper reviews the most critical developments in the evolution of FR methods. It presents the experimental principles for three FR techniques, including the traditional batch system (VSFR) and two recently developed flow-through systems (PSFR and CSFR), as well as the theoretical development of the adsorbed-phase transfer functions, which enable FR to discriminate among different rate-limiting mechanisms. The focus of the manuscript is to illustrate the application of these methods in various microporous materials such as zeolites, carbon molecular sieves (CMS), and MOFs to obtain reliable mass transfer rates through selected examples. Diffusion studies of mixtures, an outlook for nonlinear FR, and extensions to chemical reactions are also briefly discussed.

Experimental techniques and theoretical treatment

Depending on the setup of FR techniques using either a flow-through or a closed batch system, several types of FR have been developed, and the corresponding material balance and energy balance can be written accordingly to give different overall transfer functions. These overall transfer functions relate the phase lag and amplitude ratio of the response and perturbation variables to the adsorbed-phase transfer models, which vary widely in complexity depending on the type of adsorbent-adsorbate pair and phenomena under study. By minimizing the difference between theoretical and measured response functions, one can evaluate the parameters of equilibrium capacities and dynamic constants based on appropriate theoretical models describing FR curves of the system to the spectra. Both experimental setup and corresponding mathematical models are summarized for three FR methods, including one frequently used batch system with volume perturbations and two recently developed flow-through systems with pressure or concentration oscillations. More references for batch VSFR system can be found in several reviews by Yasuda [74], Song and Rees [75], Reyes and Iglesia [76].

Apparatus and principle for batch VSFR

VSFR usually involves a harmonic perturbation of the volume of a closed system using bellows or moving plate devices coupled mechanically to rotary motors. A typical apparatus was illustrated by Reyes et al. [21] in Fig. 1. The response variable of system pressure must be monitored with high accuracy and fast time resolution to establish the signal attenuation and the phase lag between the volume and pressure fluctuations at each perturbation frequency. The system typically allows diffusivity and adsorption measurements at input perturbation frequencies between 0.05 and 10 Hz. A newer version can allow perturbation frequencies up to 100 Hz [24].

Fig. 1
figure1

Schematic diagram of volume-swing frequency response apparatus for a batch system. (Reprinted with permission from Reyes et al. [21])

Shen and Rees [48] have improved this technique by using a square wave modulation of ± 1% instead of a sine wave for volume perturbations, as well as reducing the response time of the pressure transducer. Since the square waveform can be represented by its Fourier series expansion providing higher harmonics in pressure response, the experimental frequency range was expanded to 90 Hz when a ninth harmonic was taken with a volume perturbation up to 10 Hz. This feature provides the ability to study fast dynamics of sub-seconds despite relatively low perturbation frequencies limited by mechanical restrictions. However, the higher harmonic Fourier transform requires high quality square waves and appropriate sampling rates. As a result, the highest harmonic, which can be used, is restricted by the quality of the raw experimental data.

Following Yasuda’s treatment [74], the capacities and dynamics of underlying processes are obtained by fitting the experimental response curves, from measured quantities of volume (V), pressure (P), and their phase lag (φ), to theoretical transfer functions. Supposing the volume is perturbed harmonically around an equilibrium value V0, the gas phase pressure responds accordingly around equilibrium pressure P0; thus,

$$V(t)={V}_{0}-\Delta V{e}^{i\omega \mathrm{t}}$$
(1)
$$P(t)={P}_{0}+\Delta P{e}^{i(\omega t+\phi )}$$
(2)

where ∆V and ∆P represent the oscillation amplitudes of the system volume and pressure, respectively, ω is the angular frequency of oscillation, which equals to 2πf with frequency f in Hz. The most useful Intensity function to identify kinetic parameters from experimental data is [71]:

$$\delta =\frac{\Delta V}{\Delta P}-1$$
(3)

Real and imaginary response functions (RRF and IRF), or characteristic in-phase (δin) and out-of-phase (δout) functions for the adsorbed phase which describes the contribution of gas adsorption to the dynamic response of the system, can be outlined as

$$\frac{\Delta V}{\Delta P}\mathrm{cos}\,\phi -1={\delta }_{in}$$
(4)
$$\frac{\Delta V}{\Delta P}\mathrm{sin}\,\phi ={\delta }_{out}$$
(5)

In order to eliminate any effect associated with the physical system that are unrelated to the dynamics of diffusion in the sample, control experiments can be carried out through either employing quartz particles under identical conditions for the FR measurement or with an inert gas added to the adsorbate. These data are used to calculate the amplitude ratio PB/PZ and the phase lag \(\phi\)(Z-B) = \(\phi\)Z \(\phi\)B, where \(\phi\)B and \(\phi\)Z are the phase lag and PB and PZ are the pressure response in the absence and presence of adsorbents, respectively. Consequently, Eqs. 4 and 5 can be replaced by

$$\left(\frac{{P}_{B}}{{P}_{z}}\right)\mathrm{cos}\,{\phi }_{(Z-B)}-1={\delta }_{in}$$
(6)
$$\left(\frac{{P}_{B}}{{P}_{z}}\right)\mathrm{sin}\,{\phi }_{\left(Z-B\right)}={\delta }_{out} .$$
(7)

If the perturbation frequency is sufficiently low compared with system intrinsic rates, then the system will be in an equilibrium state. The pressure will follow the volume perturbation with no time lag so that the phase difference will be zero. On the other hand, if the perturbation is fast enough compared with intrinsic system dynamics, there will be no significant adsorption on the time scale of the cycle. Once the experimental range of frequency is wide enough to cover all the rate processes occurring in a system, the asymptotes of in-phase and out-of-phase characteristic curves should satisfy the following two relations

$$\underset{\omega \to \infty }{\mathrm{lim}}{\delta }_{in}=0; \underset{\omega \to \infty }{\mathrm{lim}}{\delta }_{out} =0$$
(8)
$$\underset{\omega \to 0}{\mathrm{lim}}\left(\frac{{P}_{B}}{{P}_{z}}\right)\mathrm{cos}\,{\phi }_{(Z-B)}-1=0;\underset{\omega \to 0}{\mathrm{lim}}{\delta }_{out}=0$$
(9)

Notably, correction of FR data to the control experiment is crucial due to the heating effect introduced by gas compression. The heat released from compression could cause substantial spurious effects for response curves at high frequencies, resulting in interference with mass transfer dynamics. The relative importance of gas compression is found to depend mainly on the gas type, pressure, and heat-exchange rate at the adsorber wall from experimental and simulation studies [53,54,55]. Figure 2a shows the simulation results of pressure in a control experiment as a sequence for gas compression—the pressure amplitude exceeds the magnitude vP0 at higher frequencies for a control experiment, leading to negative values of the in-phase component rather than to zero when the effect of gas compression heating is neglected. It also shows that gas compression effect is more severe for monatomic gases with a higher specific heat ratio k (k = 1.67) than that for linear paraffin (k = 1.04 ~ 1.32) under similar experimental conditions [54]. Hossain et al. [55] confirmed this unexpected feature at higher frequencies with systematic experimental evaluations and concluded that pressure and the type of gas are the main parameters that affect the shape or location of the high-frequency feature, shown in Fig. 2b, where the unexpected feature even started as low as 0.1 Hz with pressure close to 1 bar. Their findings from simulation and experiments agreed well to show that the gas compression effect is more severe for monatomic gases and for high pressures because of higher heat capacities of the gases [54]. This could explain why most VSFR investigations were conducted at low pressures as the heating effect from gas compression can be minimized. It is critical to use mathematical treatment in Eqs. 6, 7 to have a relative pressure response ratio because the effect of gas compression on pressure response can be considerably reduced by correcting FR data relative to the blank experiment [33, 54, 55].

Fig. 2
figure2

; Figure b reprinted with permission from Hossain et al. [55])

Pressure response curves for a blank experiment in a batch VSFR system. a Simulation results for different gases with specific heat ratio k assuming the same rate of heat exchange, b Experiments for pressure response in 3 mm glass beads with different gases at 750 torr and 30 °C. (Figure a reprinted with permission  from Bourdin  et al. [54]

The effect of gas compression was further investigated experimentally by a thermal frequency response (TFR) developed by Bourdin et al. [41, 53, 54]. This approach incorporated a rapid response infrared temperature detector to measure the sample temperature directly in addition to the pressure response of volume modulations. The temperature measurement is exceptionally accurate, having a standard error approximately 10–4 K with frequencies perturbed from 0.001 to 30 Hz [53]. The combination of temperature and pressure measurement offers more interesting features than traditional VSFR using only the pressure measurement, such as easier discrimination between heat transfer and mass transfer processes. The blank experiment is not required for the TFR because the temperature response was treated as the function of pressure, not volume in the data analysis. However, the sensing instrument for temperature was quite complex and related components were expensive. Like the VSFR, the usefulness of the TFR method was also limited to low pressure conditions, roughly to pressures less than 4 kPa [77].

Apparatus and principles for flow-through PSFR

The design of PSFR utilizes a flow-through system versus a batch system to minimize heat effects and system nonlinearity. A typical diagram of a flow-through PSFR system is shown in Fig. 3. The measured gas enters the system at a constant rate controlled by a mass flow controller (MFC) and flows through the adsorption bed, a pressure transducer (PT), mass flow meter (MFM), and pressure controller (PC) and then exits to a vent/vacuum. The system pressure is perturbed sinusoidally (± 5%) using a flow-based PC, and the response induced in the flow rate leaving the system is measured by the MFM. The adsorption bed is housed in an environmental chamber to allow kinetic studies over a range of temperatures [24]. The experimental apparatus typically allows diffusivity and adsorption measurements at input perturbation frequencies between 1 × 10–5 and 0.1 Hz. In comparison to batch VSFR, PSFR generally covers well for intermediate- and low-frequencies but has a high frequency limitation.

Fig. 3
figure3

Schematic diagram of a flow-through pressure-swing frequency response apparatus (Reprinted with permission from Wang et al. [24])

The pressure of an adsorption chamber is varied sinusoidally around a mean pressure and the flow rate follows consequently to produce a unique harmonic that has the same frequency as that of the pressure perturbation

$$P={P}_{e}-\Delta P{e}^{i\omega \mathrm{t}}$$
(10)
$$F={F}_{e}+\Delta F{e}^{i(\omega t+\phi )}$$
(11)

where ∆P and ∆F represent, respectively, the oscillation amplitudes of the system pressure and the mass flow rate leaving the system, and \(\phi\) is the phase lag between the flow rate and the pressure.

The overall transfer function G(s) and the adsorbed-phase transfer function Gn (s) are [64]:

$$G\left( s \right) = \frac{{\Delta F}}{{\Delta P}}e^{{i\Phi }} = - s\left( {M_{s} G_{n} + \frac{{V_{0} }}{{RT}}} \right)$$
(12)
$$G_{n} \left( s \right) = \frac{{\overline{n} }}{{\overline{P} }}$$
(13)

where Ms is the mass of adsorbent, n is the amount adsorbed. By using the standard change of variable, \(s=\upomega j\), the amplitude ratio and phase lag for the system are closely related to adsorption equilibrium and kinetics of the system with the real and imaginary parts of the adsorbed-phase transfer function Gn:

$$AR = \frac{\Delta F}{{\Delta P\omega }} = \left| {M_{s} G_{n} \left( s \right) + \frac{{V_{0} }}{RT}} \right| = \sqrt {\left[ {M_{s} {\text{Re}}\left( {G_{n} } \right) + \frac{{V_{0} }}{RT}} \right]^{2} + \left[ {M_{s} {\text{Im}}\left( {G_{n} } \right)} \right]^{2} }$$
(14)
$$\phi ={\mathrm{tan}}^{-1}\left[-\frac{{M}_{s}{\mathrm{Re}(G}_{n})+\frac{{V}_{0}}{RT}}{{M}_{s}{\mathrm{Im}(G}_{n})}\right]$$
(15)

Even though the mathematical treatments for pressure-swing FR and volume-swing FR are quite different, Wang and LeVan [69] developed a general model to unite these two techniques with master amplitude ratio curves from the material balances and the same form of transfer functions from the energy balances. For this purpose, the overall transfer function for the VSFR technique can be defined as:

$${G}_{v}\left(s\right)=\frac{{P}_{B}}{{P}_{Z}}\frac{{V}_{0}}{RT}=-{M}_{s}{G}_{n}(s)-\frac{{V}_{0}}{RT}$$
(16)

Therefore, the amplitude ratio (AR) curves from flow-through PSFR and batch VSFR are precisely the same if the amplitude ratio curve is defined as \({{\left| {{{\Delta F} \mathord{\left/ {\vphantom {{\Delta F} {\Delta P}}} \right. \kern-\nulldelimiterspace} {\Delta P}}} \right|} \mathord{\left/ {\vphantom {{\left| {{{\Delta F} \mathord{\left/ {\vphantom {{\Delta F} {\Delta P}}} \right. \kern-\nulldelimiterspace} {\Delta P}}} \right|} \omega }} \right. \kern-\nulldelimiterspace} \omega }\) for PSFR (Eq. 14) and \(\left|\frac{{P}_{B}}{{P}_{Z}}\right|\frac{{V}_{0}}{RT}\) = \(\left|\frac{\Delta V}{\Delta P}\right|\frac{{P}_{0}}{RT}\) for VSFR (Eq. 16). The phase lag curves are also identical for both systems.

These results demonstrate that the general response curves are not dependent on the particular FR techniques adopted, but depend on the adsorbed-phase transfer function (Gn) or equivalently so-called in-phase (\({\delta }_{\mathrm{in}})\) and out-of-phase \(\left({\delta }_{\mathrm{out}}\right)\) functions, which are related to mass-transfer resistances within adsorbents.

It has been further demonstrated that the energy-balance transfer functions for flow-through and batch systems are equivalent, except that the parameter α in the denominators is defined differently:

$$G_{T} \left( s \right) = \frac{{\overline{T}}}{{\overline{n}}} = \frac{{M_{s} \lambda s}}{{M_{s} C_{s} s + \alpha }}$$
(17)

where λ is the heat of adsorption (taken to be negative), and α is a lumped heat transfer coefficient, which is different for flow-through systems and batch systems, defined as FinCp + hA and hA, respectively. h and A stand for the heat transfer coefficient and the area of bed for heat transfer, Cs and Cp are the heat capacities of the solid and gas, respectively. This demonstrates that the flow-through systems have an extra-contribution (FinCp) from the flowing gas that helps to dissipate heat released by adsorption. As a result, the temperature variations due to the release of the adsorption heat can be minimized for flow-through systems.

Apparatus and principles for flow-through CSFR

Another flow-through FR method involves varying the concentration around a quasi-steady-state value. The concept was initially suggested by Kramers and Alberda [78] to direct attention towards the possibilities of FR analysis of continuous processes in chemical engineering. They applied a step-wise modulation at the inlet of the flow system to get information regarding the actual distribution of residence times. Then the technique was developed by Boniface and Ruthven [56] to use a chromatography column employing a small sinusoidally fluctuating flow of adsorbate added into a much larger, invariant carrier flow. However, the unambiguous resolution of effects of mass transfer resistance and axial dispersion may not be always straightforward due to use of a long column. Harkness et al. [57] made improvements to this technique by using a short-tubular reactor and a mass spectrometer, which allowed faster detection of effluent concentrations and minimized axial dispersion. They found more extensive modulation can be applied for the flow system, based on the excellent agreement between two experiments with very different amplitude modulation, one was the usual 5% and another was much larger, approximately 80%. Based on these approaches, a more general CSFR method has been developed theoretically and experimentally for mixture diffusion [63] and pure components (either gas or vapor) mixing with an inert gas [18, 20, 22, 23, 65, 66].

In previous setups, the concentration variation was achieved by adding a small time-varying quantity of the adsorbate to a high constant flow rate of inert gas to assume total flow rate constant in model. The new version can maintain the total inlet flow rate (F) precisely steady by using individual flow rates in pairs, which are perturbed simultaneously with the same amplitude but reverse phase. Therefore, the experimental setup becomes consistent with the model assuming a constant flow rate. Figure 4 shows the diagram for this CSFR apparatus. Concentrations of the components in the feed are subjected to small sinusoidal perturbations (up to ± 10%) of frequency ω and amplitude ∆yi,in. The concentration variations cause the gases to diffuse into or out of the adsorbent particles, where they adsorb and desorb, which in turn causes the mole fractions in the bulk and the flow rate out of the adsorption bed to change. The mole fractions in the effluent of the adsorption bed respond in a periodic sinusoidal manner with an amplitude ∆yi,out. The amplitude ratio ∆yi,out/∆yi,in is used to extract mass transfer rates from mathematical models. The system was initially developed to investigate mixtures with the flexibility to operate at a relatively wide range of concentrations and frequencies (10–4 to 3 Hz), as well as with an optional carrier gas for adjusting average concentrations. By optimizing the flow rate, system volume, and adsorbent weight, the maximum frequency should not limit the kinetic measurement for moderately short-time scale systems. This apparatus was also applied to perform single-particle experiments, yielding dynamic measurements for short-time scale systems in zeolites and MOFs without the interference of heat effects and axial dispersion [22, 65, 79].

Fig. 4
figure4

Schematic diagram of a flow-through CSFR system. Helium flow is optional for adjusting adsorbate concentration. The total inlet flow rate and system pressure are held constant during experiments. (Reprinted with permission from Wang and LeVan [63])

The mathematical treatment for CSFR with mixtures is more complicated than that for the pure component because the diffusion and equilibrium interference for different molecules must be taken into account [63]. For a pure component diluted with helium gas, the model can be simplified by assuming helium does not affect the diffusion and equilibrium capacities of the pure adsorbate. Thus, the overall transfer function for a pure component is: [20]

$$G\left( s \right) = \frac{{\overline{y}_{out}}}{{\overline{y}_{in} }} = \frac{{(F/V_{1} )(F/V_{b} )}}{{\left( {s + \frac{F}{{V_{1} }}} \right)\left[ {s\left( {1 + \frac{{M_{s} G_{n} }}{{V_{b} C_{0} }}} \right) + \frac{F}{{V_{b} }}} \right]}}$$
(18)

where V1 and Vb denote inlet volume and adsorption bed void volume, and C0 is the initial molar concentration of adsorbate. The adsorbed phase function (Gn) correlates the adsorbed-phase concentration (n) with the gas phase composition (y) in the Laplace domain:

$$\overline{n} = G_{n} \overline{y}$$
(19)

CSFR is designed for mixture study and always has at least two components mixing in the system for a concentration variation. The characteristic function (based on relative amplitude ratio from outlet and inlet) has a more complex feature related to mixing and interaction coupling between different components. It is not feasible to extend the previous master curve analysis to CSFR without any simplification on interaction effects. However, for a pure component system with no interference from inert gas mixing, it should be possible to have a modified format to generate a master curve as in previous PSFR and VSFR techniques, linked closely through the adsorbed-phase functions, which account for individual or coupling of mass transfer resistances and heat effects.

Adsorbed-phase transfer functions

The associated dynamic parameters of the FR spectra can be determined by fitting the adsorbed phase functions generated by an appropriate theoretical model for description of the dynamics involved. As mentioned in the introduction section, general adsorbed-phase mathematical models have been derived to represent multiple dynamic processes, including diffusion, surface barrier, and heat effects, as well as a more complex combination of these mass transfer resistances. A summary of these general functions can be found in detail elsewhere [39, 40, 68, 69, 75]. The real and imaginary parts of the adsorbed-phase transfer function Gn in the PSFR and CSFR can be expressed in terms of the dimensionless characteristic functions (δc and δs), which is widely adopted in the traditional analysis of VSFR to have

$$\mathrm{Re}\left({G}_{n}\right)=K{\delta }_{c}$$
(20)
$${\text{Im}} \left( {G_{n} } \right) = K\delta_{s}$$
(21)

Therefore, FR data from different techniques can be analyzed with similar mathematical models developed over decades, and some commonly used models are summarized in the following sections.

Diffusion

When only a single micropore (or intracrystalline) diffusion (MD) process occurs in a microporous material, the adsorbed-phase transfer function for a spherical form with a constant microparticle or crystal radius, Rc, is:

$${G}_{n}\left(s\right)=\frac{3K}{\frac{s}{\eta }}\left\{\sqrt{\frac{s}{\eta }}\mathrm{coth}\left(\sqrt{\frac{s}{\eta }}\right)-1\right\}$$
(22)

where η = D/Rc2, K is proportional to the gradient of the adsorption isotherm. Depending on the FR technique used and isotherm units, K could be expressed differently, e.g., (dn/dP)0 for PSFR, (dn/dy)0 for CSFR, or Ms(RT/V0)(dn/dP)0 for VSFR, respectively.

The real and imaginary parts of Gn for spherical geometry are given by:

$$\mathrm{Re}\left({G}_{n}\right)=K{\delta }_{c}=\frac{3K}{\nu }\left\{\frac{\mathrm{sinh}\nu -\mathrm{sin}\nu }{\mathrm{cosh}\nu -\mathrm{cos}\nu }\right\}$$
(23)
$$\mathrm{Im}({G}_{n})=K{\delta }_{s}=\frac{6K}{{\nu }^{2}}\left\{\frac{\nu }{2}\frac{\mathrm{sinh}\nu +\mathrm{sin}\nu }{\mathrm{cosh}\nu -\mathrm{cos}\nu }-1\right\}$$
(24)

where \(\nu\) is dimensionless frequency as (2ω/η)1/2.

When multiple diffusion processes coincide, assuming they are independent of each other, the overall adsorbed-phase transfer function can be written as a summation of each process with a distribution of diffusion time constants or particle sizes, or simply a summation of parallel individual processes [33]. For example, one can describe the parallel dual diffusion process as

$$\begin{array}{*{20}c} {{\text{Re}}\left( {G_{n} } \right) = K_{1} \delta_{c,1} + K_{2} \delta_{c,2} } \\ {{\text{Im}}\left( {G_{n} } \right) = K_{1} \delta_{s,1} + K_{2} \delta_{s,2} } \\ \end{array}$$
(25)

where subscripts 1 and 2 indicate the two kinetics processes. The characteristic functions δc and δs are after Eqs. 23, 24.

For macropore (intercrystalline) diffusion dominated systems, the analytical solution is similar to the solution for the micropore diffusion model in Eq. (22) except that η is replaced by the effective macropore diffusivity \({\eta }_{\mathrm{M}}\), defined as [69]:

$${\eta }_{\mathrm{M}}=\frac{{\varepsilon }_{p}}{\rho K+{\varepsilon }_{p}}\frac{{D}_{p}}{{R}_{p}^{2}}$$
(26)

where \({\varepsilon }_{\mathrm{p}}, {D}_{\mathrm{p}}, {R}_{\mathrm{p}}\) denote macropore porosity, macropore diffusivity, and radius of macroparticles. \(\rho\) is the density of the biporous material.

The ideal shape and location of the amplitude ratio curves for diffusion models, including both single macropore and micropore diffusion, are displayed in Fig. 5a. The family curves have the same shape that is unique for the diffusion behavior, but different locations regarding to the intrinsic diffusion time constants. The slow diffusion locates in the low frequency region, while fast diffusion positions in the high frequency region. Because the single macropore diffusion and micropore diffusion have the similar mathematical expressions, the family curves overlap for the same value of η and ηM. Also for a purely diffusion controlled system, the in-phase and out-of-phase functions do not cross and instead approach zero asymptotically and concurrently. The in-phase function goes through a maximum at a dimensionless frequency ν about 4.818. [21].

Fig. 5
figure5

Typical theoretical amplitude ratio curves, \(\left|\Delta F/\Delta P\right|/\omega\) for PSFR and \(\left|\frac{{P}_{B}}{{P}_{Z}}\right|\frac{{V}_{0}}{RT}\) (or \(\left|\frac{\Delta V}{\Delta P}\right|\frac{{P}_{0}}{RT}\)) for VSFR, with parameters of Ms = 2 g, K = 1 mol/(kg bar), P = 1 bar, and V0/RT = 0.00172 mol/bar, a Diffusion family curves with corresponding diffusion time constants η or ηM ranging from 1E−4 to 1 s−1, b surface barrier (LDF) family curves with k ranging from 15E−4 to 15 s−1. Solid green lines represent blank response with the value of V0/RT

Surface barrier (LDF)

When surface barrier resistances, also called skin effect, or external mass transfer resistance in the gas film is dominant, and represented by a linear driving force model (LDF), the adsorbed-phase transfer function Gn is given by:

$${G}_{n}\left(s\right)=\frac{Kk}{s+k}$$
(27)

where k is the LDF mass transfer coefficient.

The real and imaginary parts of Gn can be expressed as [62, 64]

$$\mathrm{Re}\left({G}_{n}\right)=K{\delta }_{c}=\frac{K{k}^{2}}{{\omega }^{2}+{k}^{2}}$$
(28)
$$\mathrm{Im}\left({G}_{n}\right)=K{\delta }_{s}=-\frac{K\omega k}{{\omega }^{2}+{k}^{2}}$$
(29)

When multiple resistances coincide in a first-order kinetics model, a distribution function of LDF coefficients f (ki) can be introduced and the total transfer function is given by the integral summation of the individual parallel-transfer functions:

$${G}_{n}\left(s\right)=K{\int }_{0}^{\infty }\left[f\left({k}_{i}\right)\frac{{k}_{i}}{(s+{k}_{i})} \right]{\mathrm{d}k}_{i}$$
(30)

where f (ki) can be assumed to have a Gaussian distribution

$$f\left({x}_{i}\right)=\frac{1}{\sqrt{2\pi \sigma }}\mathrm{exp}\left[-\frac{{{(x}_{i}-\mu )}^{2}}{2{\sigma }^{2}}\right]$$
(31)

with µ is the mean and σ2 is the variance of the LDF coefficient.

The ideal shape and location of amplitude ratio curves for the surface barrier are presented in Fig. 5b. These family curves have the same shape that is solely determined by the LDF model, but the locations change with the values of LDF mass transfer coefficients (ki), having moderate rates to the left (slow-frequency region) and fast rates to the right (high-frequency region). Notably the shape for LDF family differs from the diffusion family even with equivalent coefficients, where k approximates 15D/r2. This demonstrates the ability of FR to distinguish between the surface barrier and diffusion resistances by the shape of amplitude ratio curves. More discussion can be found in a recent review by Ruthven et al. [80] on experimental approaches, including other macroscopic and microscopic methods, to separate between these two resistances.

Combined micropore and surface barrier resistances

If the constrictions at the microparticle surface (i.e., the pore mouth) are not negligible compared to the micropore diffusion resistance, then a model that allows for the simultaneous existence of micropore diffusion and surface barrier resistances is needed. The boundary condition imposed at the microparticle surface is expressed by:

$$D{\left.\frac{\partial n}{\partial r}\right|}_{r={R}_{c}}={\left.{k}_{B}\left({n}^{*}-n\right)\right|}_{r={R}_{c}}$$
(32)

where kB is the barrier resistance coefficient or surface permeability in m/s. For a system with two resistances from the surface barrier and intrinsic diffusion, the adsorbed-phase transfer function is [64]:

$$G_{n} \left( s \right) = \frac{{3K\beta _{{bm}} }}{{\frac{s}{\eta }}}\frac{{\sqrt {\frac{s}{\eta }} {\text{coth}}\left( {\sqrt {\frac{s}{\eta }} } \right) - 1}}{{\sqrt {\frac{s}{\eta }} {\text{coth}}\left( {\sqrt {\frac{s}{\eta }} } \right) + \beta _{{bm}} - 1}}$$
(33)
$${\beta }_{bm}=\frac{{k}_{B}}{{R}_{c}}/\frac{D}{{R}_{c}^{2}}=\frac{{k}_{B}{R}_{c}}{D}$$
(34)

where βbm is the dimensionless parameter to describe the relative contribution of surface permeation and intrinsic diffusion for a sphere microparticle. It suggests that surface barrier effects depend not only on the ratio of the rate constants but also on the size of the crystals. The smaller the crystal size, the more pronounced the contribution from the surface barrier.

Heat effect

Heat transfer resistance has been shown to have an additional resonance in the out-of-phase response in the batch volume FR system by Sun et al. [43, 46]. Even though the flow-through system mitigates heat effects compared with traditional batch experiments, the thermal effect can be an important factor in determining rate parameters [20]. With large isosteric heats and slow heat transfer, ignoring the thermal effects may lead to erroneous results [69].

Considering temperature change due to adsorption and desorption in FR systems, the adsorbed-phase concentration can be expressed by a linearized equilibrium equation [69]:

$${n}^{*}\left(P, T\right)={n}^{*}\left({P}_{0}, {T}_{0}\right)+{K}_{p}\left(P-{P}_{0}\right)+{K}_{T}\left(T-{T}_{0}\right)$$
(35)

where \({n}^{*}\) is the adsorbed amount in equilibrium, KT is the slope of the isobar and related to Kp through the Clausius–Clapeyron equation [62] :

$${K}_{T}={K}_{p}\frac{\lambda P}{R{T}^{2}}$$
(36)

With knowledge of the temperature interference on loadings, one can combine the energy balance and material balance to derive the non-isothermal LDF transfer function:

$${G}_{n}\left(s\right)=\frac{K}{s/k+1+\frac{{K}_{T}{M}_{s}\lambda }{{M}_{s}{C}_{s}+\alpha /s}}$$
(37)

Following a similar approach, the non-isothermal diffusion transfer function is obtained:

$$G_{n} \left( s \right) = \frac{{\bar{n}}}{{\bar{P}}} = \frac{{3Kl_{3} }}{{1 + 3l_{3} \frac{{K_{T} M_{s} \lambda }}{{M_{s} C_{s} + \alpha /s}}}}$$
(38)

where parameter l3 is defined as \((\eta /s)\left[\sqrt{s/\eta } \mathrm{coth}\left(\sqrt{s/\eta} \right)-1\right]\)

In summary, Fig. 6 shows characteristic response curves for these mass transfer resistances discussed here: three isothermal models of micropore diffusion, surface barrier (LDF), parallel micropore diffusion, and two non-isothermal models of micropore diffusion and surface barrier. The amplitude ratio and phase lag curves, as well as the RRF and IRF (Eqs. 4 and 5) are plotted in Fig. 6a–d. The corresponding kinetic parameters for the example cases are given in Table 1, along with other system parameters.

Fig. 6
figure6

Theoretical response curves for five different mass transfer models: micropore diffusion (MD), parallel diffusion model, surface barrier (LDF), non-isothermal micropore diffusion, and non-isothermal surface barrier. a Amplitude ratio, (ΔF/ΔP)/ω for PSFR and \(\left|\frac{{P}_{B}}{{P}_{Z}}\right|\frac{{V}_{0}}{RT}\) (or \(\left|\frac{\Delta V}{\Delta P}\right|\frac{{P}_{0}}{RT}\)) for VSFR; b Phase lag; c Real response functions, d Imaginary response functions

Table 1 Model parameters and system properties used in theoretical study

Depending on the relative time scales of the perturbation and the system intrinsic rate, the corresponding responses vary between two limiting cases: adsorption equilibrium attained (with a much larger perturbation time scale compared with the intrinsic mass transfer time scale) and essentially no adsorption occurring (with a very short perturbation time scale compared with the intrinsic mass transfer time scale). Equilibrium information K related to the gradient of isotherms can be extracted from amplitude ratio or RRF curves at slow frequencies, a plateau shown in Fig. 6a and c when the system reaches equilibrium.

Except for the two limiting situations where all response curves have the same asymptotes, the response curves from diverse mass transfer mechanisms exhibit distinctive shapes shown in Fig. 6. The micropore diffusion and surface barrier models with equivalent rates, k = 15D/r2, show different shapes in amplitude ratio and RRF curves with more spread which is characteristic of curves from MD than from LDF. Also one can visually separate the non-isothermal cases from the isothermal case, where the two non-isothermal examples—non-isothermal LDF (dashed red line) and MD (dashed blue lines)—show heat interference in the low x-axis range with an extra step compared with their corresponding isothermal LDF (solid red line) and MD (solid blue line) models. This is caused by the coupling between heat and mass transfer steps, which results in a reduction of the mass uptake by the temperature increase from the heat of adsorption. These bimodal responses are shown clearly in phase lag and IRF response curves with double peaks in Fig. 6b and d. It is noteworthy to mention that the bimodal response is also observed for the parallel MD model, where the contribution of an extra slow mass transfer step behaves like a slow heat transfer step. Further investigation of bimodal response will be discussed in the next section.

Examples of FR studies in microporous materials

Employing a broad range of perturbation frequencies, FR techniques were applied to study multi-kinetics processes covering mass transfer rates over several orders of magnitude. This allows FR to cover a wide range of kinetics compared with microscopic methods, including pulse field gradient nuclear magnetic resonance (PFG NMR), quasi-elastic neutron scattering and molecular simulations for rapid self-diffusivities, as well as with other macroscopic methods (used for slower transport-diffusivities). Furthermore, the most promising feature of FR is its ability to discriminate between different rate-limiting mechanisms due to its high sensitivity to the nature of the governing equations. This feature constitutes a significant advantage over both pulse and step methods. Consequently, the high sensitivity requires accurate models to account for any limiting effects underlying the transfer processes, and to allow for precise description of observed FR behavior. As a result, it is evident in some system, where single models predicted very different behavior from experimentally measured responses, and additional transfer steps had to be included in the model Moreover, for more complex systems, it is possible that FR behavior could be described similarly by several models with combined resistances [81]. Therefore, this will require additional experimental checks that are carefully selected to further identify the dominating resistances. Available FR studies differ in scope, complexity, and their application to many specific dynamic phenomena in gases, vapors, and solids structures. Here we select representative examples to illustrate their use and demonstrate the potential of FR in the study of dynamic phenomena in gas–solid systems.

Pure, single micropore diffusion process

Diffusivities can be readily derived from FR data if only a single diffusion process occurs in a microporous material system. Depending on the shape and location of amplitude ratio curves, the intrinsic diffusion time constants can be determined, as shown in Fig. 5a. This is the simplest case commonly reported for zeolite systems.

Wang et al. [24] applied both PSFR and VSFR techniques to measure the diffusivity of ethane in ZIF-8 crystals. The PSFR can operate with elevated pressures up to 7 bar and frequencies ranging from 1 × 10–5 to 0.05 Hz, complementary to the VSFR that operates at low pressures with frequencies ranging from 0.002 to 100 Hz. The determination of the mass transfer mechanism with the PSFR technique is illustrated in Fig. 7a. The control experiments (triangular symbols), which were carried out with helium on the same adsorbent, can be represented by a horizontal line corresponding to the system volume information at a preset temperature (V0/RT). The response curve of ethane (circle symbols) approaches the plateau asymptotically when the frequency becomes very slow. The difference between the plateau of the adsorbate and the inert gas response curve provides equilibrium information about isotherm slope. Comparing the MD (Eq. 22) and the LDF (Eq. 27) models, it shows that the LDF model has a curvature that is inconsistent with the data, but the MD provides an excellent description of the data. Figure 7b shows an excellent agreement between PSFR data (symbols) and theoretical MD model descriptions (solid lines) for ethane on ZIF-8 with the same quantity of adsorbent at various pressures. The work clearly shows how the pressure dependence of diffusivity and isotherm slope can be obtained simultaneously from FR data.

Fig. 7
figure7

PSFR of ethane diffusion in ZIF-8: a amplitude ratio curves at room temperature (22 ºC) and 0.6 bar compared with isothermal LDF and MD models; b AR curves at room temperature (22 ºC) and various pressure (0.11, 0.17, 0.32, 0.61, 1.0, 1.9, and 2.8 bar) with isothermal MD model description. (Reprinted with permission from Wang et al. [24])

Further, the authors compared results to an independent measurement using batch VSFR technique on the same system. Traditional real and imaginary response functions (RRF and IRF outlined in Eqs. 6 and 7) for data analysis using VSFR, are plotted in Fig. 8a. No crossover of IRF and RRF directly demonstrates that there is no presence of a surface barrier or minimal contribution from the surface barrier. Instead, the in-phase and the out-of-phase characteristic curves merge asymptotically at a higher frequency, which indicates the diffusion dominates. If the phase lag data is too small to be accurately measured at high and low frequencies, one can use the high-quality amplitude ratio data alone to provide the same information as shown in Fig. 8b. The results from VSFR were in excellent agreement with PSFR. Diffusion time constants increase with an increase of temperature, but the related isotherm slopes decrease when temperature increases. As a result, the intensity of FR spectra decreases to give the response curve for the lowest temperature (− 25 °C) at the top and for the highest temperature (150 °C) at the bottom. Since the VSFR cannot operate lower than 0.001 Hz, the minimum perturbation frequency usually sets the lower limit of detectable diffusion time constants. As shown in Fig. 8b, AR at the temperature of − 25 and 0 °C cannot reach full equilibrium plateau.

Fig. 8
figure8

VSFR of ethane diffusion in ZIF-8: a RRF and IRF 0 ºC and 0.13 bar with isothermal MD fit; b AR curves at various temperatures (− 25, 0, 25, 50, 100, and 150 °C) with isothermal MD description. (Reprinted with permission from Wang et al. [24])

FR, and other macroscopic methods, directly report diffusion time constant D/r2, not diffusivity D. To compare diffusivity D with values from microscopic techniques, it is vital to have an accurate account of crystal size to calculate diffusivity. The corrected diffusivity, extracted as 1.0*10–11 m2/s based on a mean radius of 44 µm, was in good agreement with results obtained from IRM [82], permeation technique [83], and molecular simulation [84].

The measurable diffusivities depend on both the crystal size and frequency range covered by specific FR apparatus. It is relatively easy to follow slower diffusion processes as with most macroscopic measurements. The challenge is to extend the range of reliable measurement to faster processes that require measurements at higher frequencies. By incorporating large crystal sizes and wider perturbation frequencies, the diffusivities are measurable from 10–17 to 10–7 m2/s. Such a plot, shown in Fig. 9, provides an estimation of measurable diffusivity as a function of perturbation frequency and crystal radius [75]. The broad coverage allows direct comparison of diffusivities with other techniques, e.g., fast diffusivities from microscopic methods and slow diffusivities from other macroscopic methods as uptakes.

Fig. 9
figure9

Measurable diffusivity D (by FR) as a function of perturbation frequency f and spherical crystal radius r. (Reprinted with permission from Sun and Rees [75])

Dominance of surface barrier resistance

While many investigations demonstrate micropore diffusion in zeolites, a few examples show a surface barrier dominating or contributing to the overall mass transfer, especially in CMSs [15, 63, 64, 72, 73]. CMS is designed to separate targeted molecules kinetically by generating a restriction at the pore mouth of micropore opening. Because it has been used commercially for air separation and other applications, there have been many studies devoted to understanding transport mechanisms and adsorption rates on these materials. Applying PSFR to evaluate CMS systems has been reported by LeVan’s group on studying N2, O2, CH4, CO2, and Ar at various pressures [15, 64]

Figure 10 shows amplitude plots for O2, N2, and Ar measured at 0.5 bar in Shirasagi MSC-3R type 162 [15]. All experimental data in this figure are compared with best-fit descriptions of both the surface barrier resistance (LDF) and micropore diffusion models, and the O2 data are also compared with the combined resistance model considering both micropore diffusion and surface barrier. It is found that the transport of Ar and N2 in this system can be described by the LDF model but not by the micropore diffusion model. For the transport of O2, however, the combined resistance model offered the best description of the experimental data even though the LDF model was similar to the combined resistance model in its ability to describe the experimental data. This comparison suggested that the contribution of micropore diffusion to O2 adsorption dynamics in these materials was small. Alternatively, O2 FR data can be described well with a distribution of surface barrier resistances, which takes into account heterogeneity of the surface opening [64]. By comparing the location of the predominant change in the amplitude ratio curve for three gases in Fig. 10d, it is easily observed that O2 occurs at higher frequencies than for N2 or Ar, which indicates O2 transport is much faster than transport of both Ar and N2, with Ar being the slowest of the three. This is due to the fact that the location of FR curves corresponds to the value of mass transfer coefficients as explained in Fig. 5. The kinetic selectivity for O2 was calculated accordingly, about 45 for kO2/kAr, and 25 for kO2/kN2. Because PSFR can be performed at a wide range of pressures, it is capable to evaluate the concentration dependence of kinetics via experiments carried out at various pressures.

Fig. 10
figure10

(Adapted with permission  from Giesy and LeVan [15])

PSFR data and model descriptions for pure gases on Shirsagi MSC-3R type at 0.5 bar. a Argon; b Nitrogen; c Oxygen; d Comparison of data with the best model description.

Interference of macropore diffusion for determination of micropore diffusivity in a packed bed of crystals or biporous materials

Commercial adsorbents and catalysts generally consist of small zeolite crystals formed into macroporous pellets. Diffusion in these macropores can be a rate-controlling step in industrial processes. Errors occur year after year in the published literature reporting somewhat slow diffusivities obtained from pellets or even in assemblages of small commercial crystals because of a lack of understanding of kinetics. The most common error relates to the unrecognized intrusion of macropore (intercrystalline) resistances or heat effects. A careful study of diffusion in commercial pellets is important to establish the extent to which macropore diffusion may be in play.

Rees’s group found the micropore diffusivities extracted from their VSFR method were influenced by the depth of packed-beds of zeolite crystals [10, 85]. Figure 11 showed how the diffusion coefficients of ethane in Si-ZSM-5 varied with decreasing bed height: a sevenfold decrease in bed height (from 2.4 to 0.35 g of Si-ZSM-5) led to apparent diffusivity increases by a factor of 20 (from 2.95 × 10–12 to 5.5 × 10–11 m2 s−1). The variance of diffusivity indicated the interference of macropore diffusion because micropore diffusion time constants only correlate to the zeolite crystal length, not the bed length. Furthermore, when the 0.4 g sample of zeolite were dispersed in glass wool for separating individual crystals from each other, the diffusion coefficient increased to 3 × 10–9 m2 s−1, an 50 times increase compared to previous results with 0.35 g zeolite in the packed form [10]. This value was in good agreement with the self-diffusion coefficient of 6 × 10–9 m2 s−1 by NMR methods [86] and the transport diffusion coefficient of 2 × 10–9 m2 s−1 in the z direction by a single-crystal membrane technique [87]. Therefore, experiments using glass wool to disperse samples in a shallow bed were applied in latter VSFR measurements to avoid bed effects caused by macropore diffusion.

Fig. 11
figure11

(Adapted with permission from Van-Den-Begin and Rees [10])

IRF and RRF curves for ethane in silicalite at 50 °C and 0.01 bar with different sample weights. a 2.4 g; b 1.2 g; c 0.6 g; d 0.35 g.

The bed effect was also examined with CSFR technique to evaluate axial dispersion effects on diffusion study [20, 57]. Harkness et al. [57] found characteristic functions for propane were unaffected by the dispersal of the silicalite-1 crystals in glass wool, and the position and shape of the peak in the imaginary function were in good agreement with a short bed experiment with no glass wool, suggesting that macropore resistance is minor for a short bed setup. Instead, the presence of a small amount of axial dispersion was detectable in the short bed results but did not interfere in the determination of both the micropore diffusion coefficients and Henry’s law constants. Wang and LeVan [20] investigated the diffusion of chloroethane in activated carbon pellets with a shallow bed and a packed bed configuration. Figure 12a shows results for a shallow bed experiment with two equal-sized granules weighing a total of 13.03 mg. The data are described well by both the shallow bed (assuming no axial dispersion) and axial dispersion models with micropore diffusion, as well as the non-isothermal shallow bed model. This was expected because the bed effect could be neglected for the shallow bed with almost zero length. In addition, the heat effect can be negligible for the CSFR apparatus due to the advantage of flowing a large amount of carrier gas through an adsorption bed with a small amount of adsorbent.

Fig. 12
figure12

CSFR curves for experiments and models for chloroethane balanced with helium in activated carbon at room temperature. a 500 ppmv chloroethane using a shallow bed of 0.013 g; b 10% chloroethane using packed bed with different lengths, 86 mg and 30 mg samples, respectively. (Reprinted with permission from Wang and LeVan [20])

In contrast, Fig. 12b shows results for fixed-bed operation with longer bed lengths using sample weights of 30.3 and 86.1 mg, respectively. The shallow bed model (dashed lines) can no longer capture the curve shape accurately for the packed bed operation; instead, the model that considers dispersed flow described all the data well in solid lines. Table 2 summarizes the fitted parameters from these two models for experiments with different lengths. The agreement between the diffusion time constants D/R2 indicates that the bed length and axial dispersion did not preclude the accurate extraction of system parameters in the CSFR. This is consistent with the previous conclusion from Harkness et al. [57]. Both studies showed that the CSFR technique could well capture the interference of axial dispersion with unambiguous resolution of these two resistances. The longer bed led to a large axial dispersion coefficient (Dz) but with the same micropore diffusion time constant and isotherm slope, shown in Table 2.

Table 2 Parameters for 10 mol% Chloroethane on BPL Activated Carbon. (Reprinted with permission from Wang and LeVan [20])

It is challenging to obtain accurate micropore diffusion coefficients of small molecules in adsorbent pellets because macropore diffusion can interfere with the process. Since both micropore and macropore diffusion have a similar mathematical form, the discrimination between these two models cannot depend only on the shape of the response curve. Additional experiments with varying pellet or crystal size are needed to further distinguish between these two mechanisms. LeVan’s group [23, 79] applied the CSFR method to study one of the model systems, CO2 diffusion in both Cu-BTC crystals and pellets. They found the dominating resistances changed with the size and form of the adsorbents. Figure 13a compares the CSFR curves of CO2 for samples of large Cu-BTC single crystals and powder crystals. The authors found the isothermal micropore diffusion model provides a good fit for the large millimeter-scale Cu-BTC crystals, but not for a powder sample with much smaller crystal sizes (~ 10 μm). This indicates that other mechanisms, such as surface barriers or external mass transfer, may also be involved in addition to micropore diffusion. By increasing the diffusion length, i.e., of the radius of the crystal, micropore diffusion becomes the dominant factor limiting the mass transfer rate, ensuring that the measured diffusion coefficients truly represent diffusion inside the Cu-BTC micropore. The dominance of micropore diffusion was further validated by conducting additional experiments with varying crystal sizes. Figure 13b shows the entire data sets described well with the diffusion model, and extracted values of \(D/{r}_{\mathrm{c}}^{2}\) decreasing with increasing crystal size [79].

Fig. 13
figure13

Figure c adapted with permission from Liu et al. [23])

Comparison of mass transfer studies of CO2 in Cu-BTC crystals and pellets by a concentration-swing FR. a Amplitude ratio curves on large Cu-BTC crystals (mm size) and small crystals (~ 10 um); b CSFR curves and micropore diffusion fitting parameters for different Cu-BTC crystal sizes at 0.5% CO2 concentration; c FR data and a macropore diffusion model fit for Cu-BTC pellets with an equivalent radius of 0.67 mm. Fitted diffusion time constants η are summarized in the right table with different pellet thickness. (Figure a and b reprinted with permission from Tovar et al. [79].

As a comparison, the mass transfer study was evaluated on a pellet form for the same system of CO2 in Cu-BTC pellets consisting of small crystals, shown in Fig. 13c [23]. In this case, the extracted diffusion time constants varied with the equivalent pellet radii from different pellets (Cu-1, Cu-2, and Cu-3), indicating that macropore diffusion was controlling. These examples illustrate the importance of differentiating between the micropore and macropore diffusion controlled regimes. Failure to make this distinction could easily lead to an erroneous report of micropore diffusivity if macropore diffusion dominates instead. Experimental checks with different adsorbent sizes help to differentiate the controlling resistance between the macropore or micropore regime.

Bimodal response caused by additional mass transfer or heat transfer step

From the shape and the pattern of the response curves, information about kinetic mechanisms taking place in the system can be obtained as demonstrated in previous sections. However, it is not always straightforward when complexity increases. As an example, the bimodal response of the FR spectra was not uniquely defined as several theoretical models could produce similar FR spectra [14, 46]. Yasuda and Yamamoto [14] observed the bimodal response behavior of linear ethane and propane in zeolite 5A and contributed it to two independent diffusive processes associated with tightly and loosely bound species within the zeolite channel. The large diffusivity agreed with the value obtained from NMR experiments, but the small diffusivity corresponded to none of the diffusivities by standard methods. Later the same data sets were reanalyzed by Sun et al. [46] to find that the bimodal response was equally well represented by a non-isothermal model using several combinations of mass transfer resistances. Therefore, they concluded that it is not possible to extract reliable values for the diffusion coefficient with the reported data and the nature of the controlling mass transfer resistances cannot be established with certainty.

Rees’ group reported bimodal response curves for several hydrocarbons with silicalite-1, such as C1-C6 n-alkanes, p-xylene, 2-butyne, and cyclic hydrocarbons [6, 8, 88,89,90]. Here p-xylene in silicalite-1 is selected as an example to illustrate how to derive the dominating mechanisms for the bimodal behavior. Figure 14a, b shows the bimodal behavior of p-xylene diffusion in silicalite-1 at low temperatures. The data was described by two independent diffusion processes in the straight and sinusoidal channels of silicalite-1, respectively, each process having its own equilibrium constant and diffusion rate. Sun and Bourdin [43] proposed alternative explanations for this bimodal behavior by a non-isothermal single diffusion model and a diffusion-rearrangement model, which considered the straight and sinusoidal channels of silicalite-1 as the transport and storage channels. The diffusion-rearrangement model only qualitatively described the bimodal form. However, the non-isothermal model provided a good description but having heat transfer coefficients varied 6 times over gas pressures from 0.2 to 3 torr. The authors deemed it impossible to discriminate unequivocally between the three possible explanations based on the presently available experimental data; they suggested to obtain more experimental data with different shapes and size fractions of the adsorbent crystals to resolve the controversy around interpretation.

Fig. 14
figure14

FR spectra of p-xylene in silicalite-1 samples using VSFR. Continuous lines are fitted by parallel diffusion model, dash and dash-dot lines represent the two different diffusion processes in the sinusoidal and the straight channels, respectively. Symbols present experimental in-phase and out-of-phase characteristic function data. (Reprinted with permission from Song and Rees [75])

Later more experiments for this p-xylene diffusion in silicalite-1 system were carried out at various pressures and temperatures [9, 49, 90]. The reasoning from heat dissipation for bimodal behavior was discarded after analyzing these additional data. Shen and Rees [49] found the non-isothermality parameter obtained from data fitting disagreed with the value calculated from the isothermal slope obtained from the same fit at 373 K. Moreover, by analyzing data at different pressures at 398 K, Song et al. [90] showed the ratio of the intensity of the lower frequency peak to the intensity of the higher frequency peak decreases slightly with increasing pressure, which is inconsistent compared to the trend caused by heat effects, leading to increase the ratio as the pressure increases. Therefore, with these additional experimental evaluations, the bimodal FR curve of p-xylene in silicalite-1, indeed, can be ascribed to the two diffusion processes down the two channels rather than the adsorption heat effect. FR data show a single diffusion process at high temperatures, as displayed in Fig. 12c–f, which was explained by the rotation of p-xylene molecules at intersections from the sinusoidal channel to the preferred straight channel direction at high temperature, resulting in diffusion down the straight channel direction being the dominating resistance.

The difficulty of discriminating between an additional mass transfer or heat transfer step for a bimodal behavior can be tackled by investigating the systems carefully over a broad range of reasonable or possible parameter values, e.g., the variation of pressures or temperatures, and then testing the validity of theoretical models by analyzing the physical parameters derived from the best description, as illustrated in previous examples. However, this approach requires a great deal of experimental effort. Recently we evaluate a simple experimental check to identify cause of the bimodal behavior by an additional heat or diffusion step [91]. It simply requires carrying out an additional experiment by adding stainless steel (SS) beads in adsorbents operated at the identical condition. If the FR experiments with and without stainless steel balls behave the same, then this suggests that additional mass transfer step causes the bimodal behavior because the inert beads would not change any properties of adsorbents. However, if heat effects cannot be neglected, then experimental results would differ in the presence of stainless steel beads due to change of heat dissipation.

As an example, Fig. 15 shows PSFR results for propylene on commercial ZIF-8 crystals with submicron size at 30 °C and 0.5 bar—a bimodal response curve clearly observed. Therefore, neither the single diffusion nor surface barrier model could describe data well. Instead, the bimodal response is described reasonably well by the non-isothermal MD and LDF model, as well as the isothermal parallel diffusion model shown in Fig. 15. Since the description from the non-isothermal micropore diffusion control is quite close to that from the parallel micropore diffusion control, it is practically impossible to determine which resistance is dominating. This observation is similar to the previous studies for propane in Linde 5A system, where the bimodal behavior was well described by either a diffusion model with heat effects or a parallel diffusion model having different mobilities [14, 38, 46].

Fig. 15
figure15

Amplitude ratio by PSFR for propylene in commercial ZIF-8 crystals at 30 °C and 0.5 bar and reasonable descriptions from parallel diffusion, non-isothermal micropore diffusion, and non-isothermal LDF model

To understand the nature of this additional step, we carry out an experimental check by dispersing inert stainless steel beads into ZIF-8 samples. Figure 16 shows the comparison of amplitude ratios obtained from experiments performed with (green circle symbols) and without (blue diamond symbols) stainless steel beads under identical conditions. It is clear that the response curve moved up (quite obviously) after introducing these inert metal balls, resulting from a better heat dissipation. This indicates that the bimodal behavior is caused by the heat effect, not by an additional mass transfer step. Therefore, the parallel diffusion model is not the correct mass transfer mechanism for this case. Without recognizing the intrusion of heat effect, data analyzed by the isothermal MD model (commonly assumed in kinetics studies) can easily lead to error, especially when curve-fitting results cannot differentiate between the rate-controlling mechanisms. Further discrimination between non-isothermal LDF and non-isothermal MD models for this case requires higher frequency data above 0.1 Hz as indicated in Fig. 15, whereas the bimodal behavior is dominated by the heat transfer at low frequencies. This example clearly illustrates effectiveness of using additional experiment with inert metal beads to differentiate the contribution from an additional heat or mass transfer step for the bimodal behavior shown in Fig. 16b and d , a challenging case reported in previous literature [46, 49].

Fig. 16
figure16

Comparison of amplitude ratio curves of propylene on ZIF-8 at 0.5 bar and 30 °C with and without stainless steel balls

Multi-component mixtures in microporous materials

The behavior of mixture transport in microporous materials is essential with respect to both its theoretical foundations and its possibilities of practical application in separation and catalytic processes. Although various methods for experimental measurements of diffusivities such as uptake, chromatography, and NMR methods have been proposed, quantitative application of them to a binary or multi-component diffusion has been limited. The FR method is powerful in measurements of pure component mass transfer rates, but has very few applications for multi-component systems [81, 88, 92, 93]. When molecules interact, response functions no longer correspond to the sum of the individual contributions from each component in the mixture. The interaction of different molecules also adds on cross-term diffusivities Dij in addition to the change in the main-term diffusivities Dii.

The VSFR was first applied to a binary gas mixture, CH4-He or CH4-Kr in Linde 5A at 195 K by Yasuda et al. [93]. The experimental results were reproduced well according to a theoretic procedure based on a more general form of Fick's law, which contains two coupled differential equations. The measurements revealed an interesting result that each component never diffused independently despite of low equilibrium partial pressures. D11 and D22 varied considerably when the mole ratio of the mixture was changed. Similar trends were observed for the N2-O2 mixture on 4A at 273 K [92]. Later, Wang and LeVan [94] applied the flow-through PSFR system to investigate the N2-O2 mixture in CMS system at several concentrations and found the cross-term diffusivities can be essential to describe mass transfer rates for mixture separation of kinetic origin. Their correlation results showed that the main-term diffusivities are weak functions of the concentration, while the diffusive cross-terms are strong functions of the composition.

Adsorption data of Chen and Yang [95] for binary CO2 and C2H6 in 4A zeolite was used to theoretically investigate mixture FR behavior by Sun et al. [81] for a closed FR apparatus, by Wang and LeVan [96] for a flow-through PSFR apparatus, and by Park et al. [61] for a continuous flow CSFR system. All simulation results showed that the response curves of the faster-diffusing component (CO2) was strongly influenced by the slower component (C2H6). Chen and Yang [95] had previously pointed out this much stronger influence of the slower component on the faster component. Figure 17 shows the simulated FR curves of the binary mixture, with both diffusional and equilibrium interference, calculated using the reference equilibrium and diffusion values. The normalized in-phase and out-of-phase functions for both partial and total pressures for a VSFR are shown in Fig. 17a, compared to the amplitude ratio and phase lag for both partial and total flow rates calculated for a PSFR system in Fig. 17b. A good agreement was obtained between these two studies. Both depict that the slower diffusing component C2H6 (long dashed curves) behaves monotonically (as in a single component case) and is little affected by the presence of the faster-diffusing component CO2 (short dashed curves). On the other hand, the FR curves for CO2 are strongly influenced by C2H6 and show a roll-up phenomenon in both in-phase function (Fig. 17a) and amplitude ratio (Fig. 17b). By examining cases with negligible equilibrium or diffusional interferences, this roll-up phenomenon for the fast-diffusing component was found to be due to diffusional interference, not equilibrium interference.

Fig. 17
figure17

Simulated FR spectra for binary CO2 and C2H6 in zeolite 4A. a Normalized in-phase and out-of-phase functions of partial and total pressures using a VSFR apparatus. b Amplitude ratio and phase lag of partial and total flow rates using a PSFR apparatus. Short dashed curve: CO2. Long dashed curve: C2H6. Solid curve: total. (Figure a reprinted with permission from Sun et al. [81] Figure b reprinted with permission from Wang and LeVan [96])

However, both VSFR and PSFR techniques can only measure data based on the total pressure or the total flow rate response, not the response from each component. Therefore, the interference for each component cannot be observed to discriminate between the different types of interference. If the total pressure out-of-phase peaks cannot be separated, then it is very challenging to identify the type of interference. Based on theoretical studies, Sun et al. [81] found the separation also depends on the magnitude of the diffusivities of each component and the thermodynamic properties of the system, as well as the influence of total pressure and composition of gas mixtures. Additionally, an inappropriate choice of the experimental conditions can also lead to inseparable out-of-phase peaks or cause one peak to disappear. All of these restrictions led to limited applications to mixture studies.

Compared to VSFR and PSFR techniques for mixture study, CSFR has apparent advantages due to its capabilities to measure the concentration responses for each individual component simultaneously. Wang and LeVan [63] applied the CSFR method theoretically and experimentally to perform measurements of adsorption rate parameters for mixtures and examined transport behavior of binary mixtures of CH4 and CO2 in Takeda CMS-3 K. Figure 18 shows FR curves for CO2 and CH4 mixtures at 1 atm and 298 K with three different compositions: 80% CO2/20% CH4, 50% CO2/50% CH4 and 20% CO2/80% CH4. The feed composition was perturbed sinusoidally by 10%, and the outlet compositions were measured using the mass spectrometer. When in equilibrium at low frequencies, the outlet modulations followed the inlet composition perturbation with a 10% amplitude. With an increase of frequency, the FR curves responded with less amplitude depending on mass transfer rates of the mixture.

Fig. 18
figure18

Frequency response curves for binary CO2 and CH4 in CMS at 1 atm and 298 K with different compositions. Dashed lines represent prediction from pure components with negligible cross-term diffusivities. Solid lines are fit from a non-constant Fickian diffusion model. (Reprinted with permission from Wang and LeVan [63])

The authors derived the analytical solutions for a binary system considering micropore diffusion, a surface barrier resistance, external film resistance, and axial dispersion. The thermodynamic factor derived from the multisite-Langmuir isotherm was introduced to consider the concentration dependence of the mixture diffusivities defined generally for an m-component mixture as

$${D}_{ij}=\sum_{k=1}^{m}{D}_{ij}^{0}{\Gamma }_{ijk}$$
(39)

where \({D}_{ii}^{0}\) is corrected main-term diffusivity, and \({D}_{ij}^{0}\) (i ≠ j) is corrected cross-term diffusivity, and Γ is a thermodynamic factor. This non-constant Fickian diffusivity model described mixture diffusion well over the entire range of the experimental data with results shown in solid lines in Fig. 18. For this system, the corrected main-term diffusivities were in excellent agreement with the corrected diffusivities calculated from pure component results, which were carried out by the pressure-swing FR method at 298 K and 1 atm. The transport mechanism of pure CO2 on CMS was found to be controlled by micropore diffusion, but pure CH4 is mainly controlled by a surface barrier resistance. The ratio of the diffusivities of CO2 and CH4 was approximately 100 at atmospheric pressure for the Takeda CMS-3 K type 161 under the experimental conditions.

To understand the effect of cross-term diffusivities, we compared previous results with mixture behavior predicted with pure component diffusivities only without cross-term diffusivities, as shown in Fig. 18. Without consideration of cross-term diffusivities, this prediction overestimates the amount of CO2 diffusing into the adsorbent. Similar to previous observation for CO2 and C2H6 in 4A zeolite [95], the cross-term diffusivities are very important to accurately characterize the rate behavior, with slow diffusing component CH4 retarding the fast-diffusing component CO2. However, the effect of cross-coefficients introduced by multicomponent interaction is system-dependent. As a result of a binary study of O2/Ar mixture in CMS using CSFR [72], the experimental data was described well using the pure component transport models and coefficients with no additional cross-coefficients. The data suggested that the adsorbed gas could be treated as noninteracting, contrary to previous CO2 and CH4 in CMS system. Therefore, transport of gas mixtures is an important and nontrivial subject, worthy of a thorough investigation to gain more knowledge of mixture diffusion.

Status and outlook

FR techniques with a broad range of perturbation frequencies can be applied to characterize multiple kinetic processes occurring simultaneously in gas–solid systems. The range of diffusion time constants measured by FR depends on both the size of the adsorbent crystals and the range of perturbation frequencies covered by the particular FR apparatus. With developments of multiple FR techniques to cover frequency ranges from 1 × 10–5 to 100 Hz, and syntheses of adsorbents through a range of crystal sizes, measurable diffusivities can now cover over 10 orders of magnitude. Therefore, FR methods have been widely used to measure the mass transfer of many gases and vapors within microporous materials, allowing direct comparison with self-diffusivities from microscopic methods and transport diffusivities from other macroscopic methods. These microscopic techniques are usually limited to processes involving fast diffusion while the other macroscopic methods are typically reliable for slow diffusing systems.

The accurate interpretation of FR data requires models that account for all dynamic processes responding to the applied perturbations. One or more of a variety of possible mechanisms can occur, including micropore diffusion, macropore diffusion (ordinary diffusion, Knudsen diffusion, or Poiseuille flow), transport across a surface barrier, and external mass transfer, as well as heat and particle shape/size effects. The shape of FR curves identifies the dominating mass transfer resistances, while the location of the curve determines the value of the corresponding rate constant. Indeed, for some cases, additional experimental checks are needed to further differentiate between dominating resistances because seemingly similar response functions for partial frequency spectrum may reflect different dynamic mechanisms. For instance, it may be necessary to vary the adsorbent size or bed length to distinguish between a macropore or micropore diffusion controlled regime. Additionally, a bimodal response behavior can be equally well explained by both the parallel diffusion and non-isothermal diffusion models. This requires conducting more experiments through a range of pressures or temperatures to test the validity of theoretical models, and ultimately deduce the dominating mechanism. Alternatively, a simple test can be performed to compare FR curves with and without inert metal balls, under the same operating conditions, to distinguish between the additional heat or mass transfer step for the bimodal behavior.

With increasing levels of complexity, fundamental modeling developments are expected to continue to grow, making it impractical to derive analytic solutions. Therefore, mathematical equations will need to be solved numerically, as is the case for molecular mixtures. The CSFR method can also be extended to ternary or higher-order multi-component systems with no difficulty, as the technique permits simultaneous detection of all the components by the mass spectrometer and theoretical developments already have taken multi-components into consideration. However, deriving an analytical solution for multi-components is challenging owing to the increasing complexity of multiple component systems. It should be simpler to use numerical solutions for mixtures when the number of components is over two, especially when a more complicated model involving heat and mass transfer resistances is needed, in addition to diffusion models. Adopting numerical solutions as an alternative to analytical solutions may also be beneficial for exploring the FR systems with large amplitude perturbations, which no longer need to be restricted to simple harmonic functions.

A NFR approach is an extension of the linear FR method using relatively large perturbations for the volume to result in a complex periodic response, which contains both the first harmonic, as well as the second and higher harmonics, as pioneered by Petkovska and Do [2]. Petkovska et al. have also carried out comprehensive theoretical work to show the potential of NFR to identify kinetic mechanisms based on the pattern of second-order frequency response functions (FRF) [97,98,99,100]. However, the practical application of the NFR approach runs into the mathematical complexity of the derivation of the theoretical second-order FRFs, along with substantial technical challenges. The proper choice of the input amplitude is critical, as it needs to be a large enough input amplitude to allow detection of the second harmonic from the system output, but can adversely affect the data quality due to the interference of higher-order FRFs if the input amplitude is too large. The accurate measurement and determination of the experimental first- and second-order FRFs are vital to implementing the NFR technique, but remains challenging. Recently Brizic and Petkovska [101] reported the first experimental verification for NFR on a CO2/zeolite 5A system with input amplitudes of 3% and 7% of the volume, compared to 1 or 2% in traditional VSFR. They used NFR data by comparing the characteristic features of the NFR curves, while not fitting the entire dataset; non-isothermal macropore diffusion was recognized as the governing mechanism. In their later theoretical study of adsorption governed by parallel macropore and micropore diffusion, they found the second-order FRFs exhibit bimodal characteristics that reflect the dynamics of the parallel macropore and micropore diffusion processes. In the combination of second-order FRFs, NFR offers the potential to estimate the values of the macropore and micropore diffusion coefficients, both of practical importance [102]. This suggests the potential of NFR to tackle some of the difficulties of interpreting FR data, while allowing for the simultaneous determination of multiple dynamic rates with shorter experimental times, although the complexity of this analysis is significantly increased. As a result, this area still will require significant developments to overcome both the experimental and theoretical hurdles required for practical application.

The potential of FR methods in the analysis of the dynamics of complex catalytic sequences remains mostly unexploited. Naphtali and Polinski [31] reported the first application for chemisorption kinetics for a hydrogen-on-nickel system using a closed system. Later, adsorption and desorption rate constants were reported for the chemisorption of ethylene on zinc oxide and carbon monoxide and hydrogen on a rhodium catalyst [103] using a similar approach. Typically, determining the kinetics of surface-catalyzed reactions is commonly conducted under steady-state conditions in flow systems, and the application of FR methods to surface reactions in these systems could be a natural extension of these investigations. Schrieffer and Sinfelt [104] carried out an FR analysis for the surface processes occurring in flow systems, and illustrated the utility of FR data in yielding more basic kinetic information than the conventional steady-state mode of operation. Other application could also involve the measurement of reaction rates of a heterogeneous catalytic reaction by periodically varying gas space of a steady-flow reactor [35, 36, 105]: in this instance, a mass spectrometer would be used to monitor the pressure variation of each component in the system. Yasuda et al. [36] applied this approach to study CO oxidation over a Ru/Al2O3 catalyst in order to confirm its efficacy in determining thirteen rate coefficients at five elementary steps. Since all elementary reactions are more or less coupled to each other, the range of frequency played an essential role in measuring "reaction rate spectroscopy". Therefore, compared to using transient methods to investigate heterogeneous catalyzed reactions, one can use small perturbations in the FR technique to linearize any rate equation, and simultaneously determine all the adjustable parameters by scanning through a wide range of frequencies. However, this technique requires, in theory, specifically well-defined reaction mechanism to deduce the characteristic functions. Application to other kinds of reactions still needs to be evaluated; likewise, in-depth studies will be needed for more accurate investigations into reaction kinetics. One possibility might be to extend CSFR, using a periodically varying concentration instead of the volume oscillation, to a continuous flow reactor system in order to derive the concentration dependence of rate constants for specific catalytic reactions. An FR apparatus with an experimental high-frequency range extended beyond current limits, will also be desirable owing to the faster kinetic processes associated with the small crystals typically needed for catalytic reactions.

Abbreviations

A :

Heat transfer area, m2

C 0 :

Initial molar concentration of adsorbate, mol/m3

C s :

Heat capacities of the adsorbent, J/(g K)

C p :

Heat capacities of the gas, J/(mol K)

D :

Micropore diffusivity, m2/s

D ii :

Main-term diffusivities in a mixture, m2/s

D ij :

Cross-term diffusivities in a mixture, m2/s

D p :

Macropore diffusivity, m2/s

f :

Frequency, Hz

G :

Overall transfer function

G n :

Adsorbed-phase transfer function

k :

Specific heat ratio of gas, or the LDF mass transfer coefficient

K :

Equilibrium constant related to the gradient of the adsorption isotherms

k B :

Barrier resistance coefficient or surface permeability, m/s

K V :

Dimensionless equilibrium constant used in VSFR, defined as Ms (RT/V0) (dn/dP)0

K p :

Equilibrium constant defined as dn/dP, mol/(kg bar)

M s :

Adsorbent mass, kg

n :

Adsorbed amount, mol/kg

n * :

Adsorbed amount in equilibration, mol/kg

h :

Heat transfer coefficient, W/(m2 K)

l 3 :

Lumped parameter for a non-isothermal model as \((\eta /s)\left[\sqrt{s/\eta } \mathrm{coth}\left(\sqrt{s/\eta} \right)-1\right]\)

P :

Pressure, bar

P B :

Pressure response in the absence of adsorbents

P Z :

Pressure response in the presence of adsorbents

P 0 :

Initial equilibrium and reference pressure for a system with adsorbent, bar

R:

Universal gas constant, Pa m3 mol−1 K−1

R c :

Radius of micropore crystal or microparticle, m

R p :

Radius of adsorbent macroparticle, m

t :

Time, s

T :

Temperature, K

T o :

Initial temperature, K

V :

Working volume, m3

V 0 :

Equilibrium volume, m3

V 1 :

Inlet volume for a CSFR system, m3

V b :

Adsorption bed void volume in a CSFR, m3

α:

Lumped heat-transfer rate, J/(K s)

β bm :

Dimensionless parameter to describe the relative contribution of surface permeation and intrinsic diffusion for a sphere microparticle

δ :

Intensity function in Eq. 3

δin :

In-phase function

δout :

Out-of-phase function

δ c :

Dimensionless in-phase function

δ s :

Dimensionless out-of-phase function

ε p :

Macropore or pellet porosity

λ:

Heat of adsorption (taken to be negative), kJ/mol

µ :

Mean value of Gaussian distribution

η :

Parameter for D/r2, 1/s

η M :

Effective macropore diffusion time constants defined in Eq. 26, 1/s

ω:

Angular frequency of oscillation, rad/s

ν:

Dimensionless frequency as (2ω/η)1/2

ρ :

Density of the adsorbent, kg/m3

σ 2 :

Variance of the Gaussian distribution

φ:

Phase lag

ΔP :

Amplitude of pressure change in a system

ΔF :

Amplitude of flow rate change in system

V :

The oscillation amplitudes of the system volume, m3

Γ :

Thermodynamic factor

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The author thanks Ned Corcoran and Ben McCool for providing valuable suggestions and support for the work.

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Wang, Y. Identification of mass transfer resistances in microporous materials using frequency response methods. Adsorption 27, 369–395 (2021). https://doi.org/10.1007/s10450-021-00305-z

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Keywords

  • Frequency response
  • Mass transfer rates
  • Micropore diffusion
  • Macropore diffusion
  • Surface barrier
  • Mixture diffusion
  • Heat effect