110th Anniversary: Comments on Heterogeneity of Practical Adsorbents

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Separations

110 Anniversary issue: Comments on heterogeneity of practical adsorbents Shivaji Sircar, and Timothy C. Golden Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b01025 • Publication Date (Web): 22 May 2019 Downloaded from http://pubs.acs.org on June 8, 2019

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Industrial & Engineering Chemistry Research

110th Anniversary Issue: Comments on heterogeneity of practical adsorbents S. Sircar1* and T. C. Golden2 1

Department of Chemical and Biomolecular Engineering, Lehigh University, Bethlehem, PA, 18015, U.S.A.

2

Air Products and Chemicals, Inc. Allentown, PA, 18195, U.S.A.

Abstract Most micro-mesoporous adsorbents used in practical adsorptive fluid (gas and liquid) separation processes are energetically heterogeneous. The adsorbent heterogeneity plays a significant role in establishing the shapes of the equilibrium adsorption isotherms (pure component or mixture), the corresponding heats of adsorption, and the adsorbate mass transfer rates, which in turn, determine the over-all separation performance by a process. The sources of the adsorbent heterogeneity are discussed and the pros and cons of several analytical heterogeneous models describing pure and multicomponent adsorption isotherms, heats, and adsorbate mass transfer coefficients are analyzed. Several critical effects of the adsorbent heterogeneity on the equilibrium isotherms are presented with examples to describe the complexity of the subject. The overall effect of adsorbent heterogeneity on an adsorptive process performance will, of course, depend on the specific adsorbate – adsorbent – process design combination. The results of a model simulation for the performance of a pressure swing adsorption (PSA) process designed to separate a bulk gas mixture (C2H4 + He) using a heterogeneous adsorbent (BPL activated carbon) is briefly presented as an example. The simulation shows that the separation performance deteriorated as the degree of the adsorbent heterogeneity is increased. Extensive experimental validity of the equilibrium and the kinetic models for a heterogeneous adsorbent, and the subsequent separation process performance in a pilot or a commercialscale plant must be used to confirm the actual performance by a heterogeneous adsorbent. Introduction A variety of meso-microporous adsorbents is frequently used for practical separation and purification of fluid mixtures. These include amorphous materials under the generic names of activated carbons, silica and alumina gels, and crystalline materials under the generic names of zeolites. The separation processes typically employ the principles of Pressure, Vacuum and 1 ACS Paragon Plus Environment

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Thermal Swing Adsorption (PSA, VSA and TSA) using fixed-bed adsorption columns and Concentration Swing Adsorption (CSA) using Simulated Moving Bed (SMB) adsorption columns, respectively, for gas and liquid phase applications.1 – 7 Some of the major commercial applications of these processes and the associated adsorbents are listed in Table 1. These adsorbents are generally energetically heterogeneous since they contain adsorption sites of different energies. The adsorbent heterogeneity plays a significant role in determining the overall equilibrium adsorption isotherms, the heats of adsorption, and the adsorbate mass transfer rates within the adsorbent particles for the components of a fluid mixture (adsorbates). This trinity of adsorptive properties, in turn, dictate the net separation performance by the adsorbate - adsorbent pair for a given adsorptive process design. Applications Gas Phase Separations Production of 90+% O2 from air Production of 99 + % N2 from air

Process

Adsorbent (generic)

PSA, VSA PSA, VSA

Zeolite (LiLSX) Zeolite (Na-Mordenite), Carbon molecular sieve Activated carbon, Alumina, Silica gel, Zeolite (5A, CaX,13X) CuCl/Alumina, CuCl/Zeolite Zeolite, activated carbon Alumina, Silica gel, Zeolite (3A, 4A, 13X) Activated carbon, High silica zeolites Activated carbon, Alumina, Zeolite

Production of H2 from SMR off-gas

PSA

Production of CO from SMR off- gas Production of CH4 from landfill gas Gas/Air Drying

PSA/VSA PSA PSA

Solvent vapor recovery PSA/VSA Trace impurity removal from gases TSA Liquid Phase Separations Xylene from C8 aromatics SMB Zeolite (BaX, Silicalite) Separation of normal-iso paraffin SMB Zeolite (5A) mixtures Separation of Cresol Isomers SMB Zeolite (Na13X) Separation of Glucose- Fructose SMB Resins Mixture Trace impurity removal from liquids TSA Activated carbon, Zeolites, Alumina Liquid Drying TSA Alumina, Silica gel Table 1: Major commercial applications of adsorptive separation technology. Source and Nature of Adsorbent Heterogeneity

The adsorbent heterogeneity can be caused by both the physical structure of the adsorbent and the chemical nature of the adsorbent surface. The presence of a distributed net-work of 2 ACS Paragon Plus Environment

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meso-micropores of different sizes and shapes in the adsorbent structure creates the physical heterogeneity because the adsorption energies of an adsorbate inside a pore depends on the size and geometry of the pore.8, 9 The presence of different functional groups on the adsorbent surface creates the chemical heterogeneity by producing adsorption sites exhibiting different polarizabilities and polarities, acidity and basicity towards an adsorbate molecule.10 - 14 The amorphous, disordered, activated carbons demonstrate a propensity for oxygen chemisorption owing to their unsaturated carbons at the edge of crystallites, thus creating the carbon-oxygen surface complexes.10 Various surface functional groups consisting of O, N, and S atoms may also be present on the pore walls of an amorphous adsorbent.11 These groups can be hydroxyl, acid, ketone, aldehyde, peroxide, amide, sulfide, etc. for an activated carbon and they are located at the edge of the basal plane of the carbon.10 - 12 Different sites may also exhibit different selectivity for the components of a fluid mixture giving rise to a selectivitybased heterogeneity. Figure 1a shows the variety of possible oxygen surface groups which can be present on an activated carbon surface.12

Figure 1a: Possible Oxygen Functional Groups on Activated Carbon12 The typical surface groups on an activated alumina and a silica gel are primarily surface hydroxyls in various geometries and coordination. 13, 14 These hydroxyls can be acidic (as is primarily the case in silica gel) or basic (as is primarily the case in activated alumina) depending on their coordination. Lewis acid and Lewis base sites may also exist due to a lack of full coordination of the surface species.

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Table 2 provides a list of commonly acknowledged sources of adsorbent heterogeneity showing the complex nature and causes of heterogeneity.

Physical Heterogeneity Amorphous Adsorbents Activated carbons

•

•

Activated alumina

•

•

•

Silica gel

•

Chemical Heterogeneity

Caused by a network of micro-meso-macro pores of different sizes and shapes (e.g. cylindrical, slit, inkbottle, etc.) Porosity depends on the source of the activated carbon (e.g. coal, petroleum, vegetable, polymer) and the method and conditions of activation (e.g. temperature and duration of heating using steam/air/ CO2 or other chemicals) Caused by a network of micro-meso-macro pores of different sizes and shapes. Porosity depends on the type of alumina (trasitional, gamma or alpha) and the method of activation. Crystal imperfections.

•

Caused by a network of micro-meso-macro pores of different sizes and shapes.

•

•

Caused by the incorporation of a plethora of surface functionalities (e.g. oxygen groups such as hydroxyl, acid, ketone, peroxide, etc.), which make the carbon more or less hydrophilic or polar. Surface groups containing N and H incorporates basicity. Surface heterogeneity is caused by formation a variety of surface groups due to (a) hydrogen bonding of water with the surface hydroxides and (b) reversible reaction of carbon dioxide with the hydroxides forming surface bicarbonates. Surface heterogeneity caused by a variety of surface hydroxyls


Comments

•

More mesoporous carbons exhibit less heterogeneity than more microporous carbons.

•

The surface of alumina is mostly basic due to the presence of both Lewis and Bronsted base sites.

•

The surface of silica gel is weakly acidic primarily due to Bronsted acid sites

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Crystalline Adsorbents Zeolites

•

• • •

•

Caused by a network of micro-meso-macro pores of different sizes and shapes in the zeolite binder. Binder type and amount. Zeolite crystal imperfections. Crystal diversity for surface barrier to mass transfer.

•

•

•

• •

Variation in Si/Al ratio in the crystals. Lower the ratio, higher is the heterogeneity. Degree of dehydration. Higher heterogeneity as the level of dehydration is increased. Cation type. Heterogeneity increases as the charge density of the cation increases. Cation position in the zeolite framework. Heterogeneity increases as the cation position is more exposed in the zeolite framework. Presence of mixed cations. Degree of cation hydration.

Table 2: Sources of adsorbent heterogeneity on various practical adsorbents. Amorphous adsorbents are heterogeneous by nature. A zeolite, on the other hand, has a welldefined, repetitive, crystalline structure. Thus, it may be expected to be homogeneous, but heterogeneity can be introduced by one or more of the factors listed in Table 2. The most unobvious reason is the crystal diversity whereby different zeolite crystals of similar size and shape from the same batch exhibit different surface barriers to adsorbate mass transfer.15 The subject is further complicated by the fact that the adsorbent heterogeneity of practical adsorbents cannot be experimentally quantified by today’s technology. 5 ACS Paragon Plus Environment


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Location and access of adsorption sites A practical adsorption process is generally carried out using a fixed bed adiabatic column (called an adsorber) which is packed with the porous adsorbent particles. An adsorbate molecule diffuses from the bulk fluid phase (gas or liquid) to an adsorption site, which is located inside the particle, and then gets adsorbed on a site. The desorption process follows a reverse path. The probability of adsorption of a pure gas molecule at any given pressure and temperature is the same for all sites of a homogeneous adsorbent, while higher energy sites are predominantly occupied at the lower gas pressures and the lower energy sites are predominantly occupied at the higher gas pressures by the adsorbate molecules on a heterogeneous adsorbent. All sites are occupied at all times for adsorption from a liquid mixture since the adsorbent is immersed in the liquid mixture. The exact locations of the adsorption sites inside an amorphous adsorbent particle is generally not known. A random distribution of sites of different energy is generally presumed. The adsorption sites in a crystalline adsorbent like a zeolite are provided by the cations. Figure 1b schematically shows the locations of Li+ cations inside the framework of a low silica X zeolite framework.16 LiLSX is the preferred adsorbent for commercial production of oxygen enriched gas from ambient air using a PSA process.4

Figure 1b: Cation sites in LiLSX16


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The site I in the X zeolite cage depicted by Figure 1b is the preferred location for the Li cation since it is solvated by 12 oxygen atoms in the hexagonal double prism. 17 However, Li ions in that site do not affect adsorption since they are invisible to the adsorbate gas molecules. Site II cations are accessible to gas molecules, but the cations are solvated by 6 oxygen atoms and they are recessed into the framework. Thus, the cations in that site, although accessible by the adsorbate gas, are relatively ineffective for gas adsorption. The Li ions located in the site III of the zeolite framework are exposed and solvated by only 4 oxygen atoms and they are most effective for adsorption.16 - 18 This shows that the adsorption properties of the zeolite depend greatly on cation positions which gives rise to adsorbent heterogeneity. Adsorbent heterogeneity is also exhibited by a zeolite crystal when there is more than one type of cation within the framework which also influences the cation positions. Furthermore, different cations have different charge densities which gives rise to adsorption sites having different electrostatic field strengths in the zeolite framework. This manifests as site to site variation of adsorption heat, hence heterogeneity. It is this variation in electrostatic field strength that creates large heats of adsorption of polar molecules. The values of charge densities (esu/nm3) of common ions found in practical zeolites are Li = 727; Na = 227; Ca = 478; Ba = 190. Different degrees of hydration of the cations inside a zeolite framework may also introduce a heterogeneity which is exhibited by drastic reductions in the Henry’s Law constants and intracrystalline self- diffusivity of gases even with an extremely small amount of pre-adsorbed water in the zeolite framework.19 - 21 Heterogeneity in a zeolite pellet used in a practical adsorber can also be introduced by the presence of an amorphous binder and its type (clay, alumina, zeolite, etc.).22 Most adsorptive separation processes of practical interest are based on physisorption of the adsorbate molecules on the adsorption site, which is generally an extremely fast process.23 Consequently, the finite rate of transport (slow or fast) measured during a practical adsorption process is caused by a slow Fickian diffusion of the adsorbate molecules through a labyrinthic, heterogeneous, porous network of meso- micropores inside the adsorbent particle. The effective mass transfer coefficient for transport of an adsorbate is thus, governed by the structural heterogeneity of the adsorbent particle. In contrast, the rate of chemisorption of an adsorbate on the site can be relatively fast or very slow.23



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Qualitative proof of existence of adsorbent heterogeneity Qualitative proofs of the existence of adsorbent heterogeneity can be demonstrated by the following measurements: (a) Pore size distribution (PSD) using mercury porosimetry as shown by Figure 1a for various activated carbons.5 (b) Scanning electron micrograph (SEM) of adsorbent cross section as shown by Figure 2b (coconut-shell charcoal)24 and 2c (bound Na-Mordenite pellet).25 (c) Thermogravimetric analysis (TGA) of chemical functionalities on an activated carbon surface by monitoring the evolution of CO and CO2 by heating the adsorbent in helium as shown by Figure 2d for a Norit carbon.14 (d) Zero point of charge (ZPC) to estimate acidity or basicity of an adsorbent as shown by Figure 2e.26 (e) X-ray photoelectron spectroscopy (XPS) for surface group analysis as shown by Figure 2f for a carbon fiber made from wood.27

(a)


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(b)

(c)

Activated Carbon Silica Gel

A

Activated Alumina 0

5

10 ZPC in PH

(d)

(e)

(f)



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Figure 2: Qualitative evidence of adsorbent heterogeneity by various techniques. The data shown by Figures 2(a – f) directly or indirectly support the existence of physical and chemical heterogeneity in practical adsorbents. However, a quantitative measurement of the adsorbent heterogeneity cannot be derived from these data. One complication is that the degree of adsorbent heterogeneity exhibited by an adsorbent towards an adsorbate molecule also depends on the physical and chemical properties of the adsorbate molecule such as polarizability, polarity, acidity, basicity etc. Thus, adsorbent heterogeneity must be discussed under the confines of an adsorbate-adsorbent pair. The same adsorbent can be more or less heterogeneous depending on the nature of the adsorbate molecule. Conversely, different adsorbents can exhibit different degrees of adsorbent heterogeneity towards the same gas. A further complication is that the heterogeneity exhibited by an adsorbent towards an adsorbate is also influenced by the presence of other components in the bulk phase.20 Heterogeneity in commercial adsorbent samples There may be an inherent variability in adsorbent properties in a commercial sample of an adsorbent which is introduced during their large-scale production processes. This is usually indicated by a variance in the properties like bulk density, surface area, nominal pore size, etc. reported by the manufacturer in the property data sheet. Commercial producers do their best to produce a consistent final product from batch to batch but the variation caused by precursor variability (particularly for activated carbons) is often beyond their control. This factor raises a very important question. How reliable are the basic adsorption data (e.g. equilibrium, heats, mass transfer rates) for an adsorbate-adsorbent pair measured in a laboratory using a very small sample of the adsorbent (often as small as a few milligrams to a few grams) for scale up of adsorptive separation process concepts? Consequently, extensive testing of adsorptive processes in pilot or commercial scale units cannot be avoided for commercially acceptable process design. By the same token, reliable adsorptive process design using published adsorptive properties without verification may not be justified. Heat of adsorption as a tool for quantitative measurement of adsorbent heterogeneity The differential iso-excess heat of adsorption (IEHA) and the differential heat of immersion (DHI) constitute two basic thermodynamic properties of gas-solid and liquid-solid adsorption systems, respectively 28, 29. They can be experimentally measured unambiguously. The IEHA of a pure gas on an adsorbent and its variation with the amount adsorbed provides a quantitative measure of the adsorbent heterogeneity. In fact, it is the only experimental thermodynamic property of an adsorbate-adsorbent system which can be used to judge whether an adsorbent 10 ACS Paragon Plus Environment

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is energetically heterogeneous or homogeneous for a specific adsorbate. In contrast, the DHI of an adsorbent into a pure liquid or a liquid mixture cannot practically be used to identify adsorbent homogeneity or heterogeneity. The IEHA of a pure gas i (𝑞𝑖0 ) or that of the component i of a gas mixture (𝑞𝑖 ) can be estimated from the experimental pure gas Gibbsian Surface Excess (GSE) isotherms [ 𝑛𝑖𝑚0 ] or the mixed gas GSE isotherms of component i [𝑛𝑖𝑚 ] at different temperatures by using the following thermodynamic relationships 28 : 𝑞𝑖0 𝑅𝑇 2

=[

𝜕𝑙𝑛𝑃 𝜕𝑇

]

𝑛𝑖𝑚0

;

𝑞𝑖 𝑅𝑇 2

=[

𝜕ln(𝑃𝑦𝑖 ) 𝜕𝑇

]

(1)

𝑛𝑖𝑚

where the variables P, T and 𝑦𝑖 are, respectively, the pressure, the temperature and the mole fraction of component i in the bulk gas phase. Eq. 1 is derived for an ideal gas by using the GSE framework of adsorption thermodynamics. The variable P in Eq. 1 should be replaced by fugacity (f) for a non-ideal gas. The IEHA is equal to the traditional isosteric heat of adsorption (𝑞𝑖𝑠𝑡 ) under certain conditions where the GSE is approximately equal to the actual amount adsorbed (𝑛𝑖𝑎 ).ǂ The estimation of 𝑞𝑖0 for a pure gas using Eq. 1 is straight forward, while the estimation of 𝑞𝑖 for a gas mixture using Eq. 1 is rather complex, requiring extensive multicomponent isotherm data which may not be easy to gather.30, 31.Alternatively, 𝑞𝑖0 or 𝑞𝑖 can be directly measured as a function of 𝑛𝑖𝑚0 or 𝑛𝑖𝑚 , respectively, at a constant T by using differential calorimetry. A Tian Calvet type calorimeter can be conveniently used for this purpose. A detailed description of the calorimeter design, the experimental protocol, and the method of data processing can be found elsewhere.28 ________________________________________________________________________ ⱡ The GSE of component i of a gas mixture (𝑛𝑖𝑚 ) is defined by 𝑛𝑖𝑚 = 𝑛𝑖𝑎 − 𝑉 𝑎 ρ𝑦𝑖 . The GSE of component i of a liquid mixture (𝑛𝑖𝑒 ) is defined by 𝑛𝑖𝑒 = 𝑛𝑖𝑎 − 𝑥𝑖𝑎 ∑𝑖 𝑛𝑖𝑎 . The variable 𝑛𝑖𝑎 is the actual amount of component i adsorbed (AAA) and 𝑉 𝑎 is the volume of the adsorbed phase. The variable ρ is the density of the bulk gas phase, and 𝑦𝑖 and 𝑥𝑖 are, respectively, the mole fractions of component i in the bulk gas and the bulk liquid phases. The GSE of a pure gas i (𝑛𝑖𝑚0 ) is defined by 𝑛𝑖𝑚0 = 𝑛𝑖𝑎0 − 𝑉 𝑎0 𝜌. The superscript 0 symbolizes pure gas i (𝑦𝑖 = 1). The GSE (𝑛𝑖𝑒0 ) of a pure liquid i (𝑥𝑖 = 1) is zero by definition. 𝑛𝑖𝑚 is a function of the gas phase pressure (P), the system temperature (T), and 𝑦𝑖 . 𝑛𝑖𝑒 is a function of only T and 𝑥𝑖 . 𝑛𝑖𝑚 ~ 𝑛𝑖𝑎 for an adsorbate at a low pressure when it is strongly and very selectively adsorbed. 𝑛𝑖𝑒 ~ 𝑛𝑖𝑎 when component i is dilute (𝑥𝑖 1) at different T rapidly decreases with increasing total amount adsorbed,4 while Figure 7 (d) shows that (i) Hmordenite selectively adsorbs CO2 (1) over C3H8 (2) at P = 40.8 kPa and T = 303.1 K, (S12 > 1), when 𝑦1 < 0.3, but then it reverses its selectivity in favor of C3H8 (S12 < 1) at higher values of 𝑦1 , and (ii) 13 X zeolite selectively adsorbs i-C4H10 (1) over C2H4 (2) at P = 137. 8 kPa and T = 298.1 17 ACS Paragon Plus Environment


K, (S12 >1), when 𝑦1 < 0.7 and then changes selectivity towards C2H4 (S12 < 1) when 𝑦1 is large.35 The last two cases are extreme examples of adsorption azeotropy.35 All of these behaviors of an adsorption system can be attributed to the adsorbent heterogeneity. Thus, the adsorbent heterogeneity can introduce a complex functional dependence of selectivity on P, T and 𝑦𝑖 (or 𝑛𝑖𝑚 and T), thereby influencing the separation performance of an adsorptive process where thermodynamic selectivity plays an important role.

Iso Excess heat of Adsorption of pure Carbon dioxide 6

IEHA, Kcal/mple

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c

5.5 5

b

4.5

a

4 3.5 3 0

2

4

6

8

GSE, mole/kg

Figure 7 (b): IHEA of pure CO2 on various carbons of Figure 7(a): a- BPL; b- RB; c-PCB.

Figure 7 (c): Selectivity of adsorption of C2H4 (1) over CH4 (2) on BPL Carbon from a binary gas mixture (𝑦1 = 0.235) as functions of total amount adsorbed at different temperatures: □ = 212.7 K; ∆ = 260.2 K; ◊ =301.4 K. 18 ACS Paragon Plus Environment

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Figure 7 (d): Examples of adsorption azeotrope.33 The role of adsorbent heterogeneity in separation of a liquid mixture is manifested primarily through (a) the shape of the GSE isotherms of the adsorbates in a dilute system and those for the binary pairs of the components of a bulk system, and (b) the mass transfer coefficients of the adsorbates and the rates of transport into a porous adsorbent. Figure 8 shows model calculations of the GSE isotherms for component 1 ( [𝑛1𝑒 vs 𝑥1 ] of an ideal bulk liquid mixture consisting of equal sized molecules.36 It also shows the variations in the corresponding dimensionless heat of immersion [F* =

[𝑄 ∗ − 𝑄2∗ ] [𝑄1∗ − 𝑄2∗ ]

] as a function of 𝑥1 . The variable 𝑄∗ is the heat

of immersion of the adsorbent into a liquid mixture (mole fraction = 𝑥1 ) and 𝑄𝑖∗ is the heat of immersion of the adsorbent into pure liquid i. The model is based on the pore-filling mechanism of liquid phase adsorption and a uniform distribution of limiting selectivity of component 1 at 𝑥1 → 0 to describe the heterogeneity. The variable 𝑚1 is the pore filling capacity of pure liquid 1.



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Figure 8: Effect of adsorbent heterogeneity on liquid phase binary GSE isotherm (𝑛1𝑒 ) and the corresponding dimension-less heat of immersion (F*). The degree of adsorbent heterogeneity increases in the order [c > b > a]35 It may be seen from Figure 8 that the adsorbent heterogeneity has a strong impact on the shape of the binary liquid phase GSE isotherm. The U- shaped isotherm depicted by case (a) is for a homogeneous adsorbent. Moderate adsorbent heterogeneity (case b) lowers the maximum value of the GSE in the isotherm, but still retains the U-shape. High adsorbent heterogeneity (case c) significantly lowers the maximum value of the GSE in the isotherm and turns it into an S-shaped isotherm indicating that the adsorbent switches selectivity from component 1 to component 2 of the liquid mixture (azeotropy) at some intermediate composition of the bulk liquid phase. Consequently, adsorbent heterogeneity can have a large impact on the performance of an adsorptive processes for separation of a bulk liquid mixture. In contrast, Figure 8 shows that adsorbent heterogeneity has a small effect on the heat of immersion of a liquid mixture. Dynamic properties: The popular Linear Driving Force (LDF) model has proved to be a practically adequate model to describe the net rate of mass transfer (𝐽𝑖 moles/g/s) of an adsorbate i from a fluid mixture into an adsorbent particle.23, 37 The instantaneous rate is given by 𝑗𝑖 = 𝑘𝑖 (𝑛𝑖𝑚∗ − 𝑛𝑖𝑚 ) for a gas phase, or 𝑗𝑖 = 𝑘𝑖 (𝑛𝑖𝑒∗ − 𝑛𝑖𝑒 ) for a liquid phase adsorption system when the instantaneous GSE of component i in the adsorbent is 𝑛𝑖𝑚 or 𝑛𝑖𝑒 (moles/g) and the GSE of component i that will be 20 ACS Paragon Plus Environment

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in instantaneous equilibrium with the super incumbent fluid mixture are 𝑛𝑖𝑚∗ or 𝑛𝑖𝑒∗ (moles/g). The variable 𝑘𝑖 (s-1) is an effective mass transfer coefficient for component i, which is a function of 𝑛𝑖𝑚 and T (or 𝑛𝑖𝑒 and T). The effective mass transfer coefficient (𝑘𝑖 ) is predominantly determined by the heterogeneous pore structure of an adsorbent, while the driving force for adsorbate mass transport is dependent on the equilibrium GSE of that component, which is affected by the adsorbent heterogeneity. Consequently, adsorbent heterogeneity plays a critical role in this area too. Models for describing adsorbent heterogeneity A mathematical model describing the mass and heat conservation equations inside an adsorber vessel is generally used to numerically simulate the performance of an adiabatic adsorptive separation process.39, 62 The models may also include other details like fluid pressure drop in the adsorber, axial mass and heat dispersions, heat loss from the adsorber, etc. The key input variables for these models are the adsorption equilibria, the heats of adsorption, and the adsorbate mass transfer characteristics for the adsorbate-adsorbent system of interest. It is desirable that these inputs are analytically described as functions of the intensive variables (P, T and 𝑦𝑖 or 𝑥𝑖 ) or (T, 𝑛𝑖𝑚 or 𝑛𝑖𝑒 ) in order to facilitate the solution of the process model using a numerical algorithm. Furthermore, the functions describing pure and multicomponent equilibrium GSE isotherms and the heats of adsorption should exhibit the following characteristics: (i) Exhibit a Henry’s law region for a pure gas and a multicomponent gas or liquid adsorption isotherm. (ii) Show thermodynamic consistency between the pure and the binary gas isotherm models. (iii) Provide analytical expressions for IEHA for the components of a gas mixture as functions of (P, T and 𝑦𝑖 or 𝑥𝑖 ) or (T, 𝑛𝑖𝑚 or 𝑛𝑖𝑒 ) using the isotherms and Eq. 1. (iv) Demonstrate validity of the models to describe experimental GSE, IEHA and mass transport data covering the entire range of conditions of P, T, 𝑦𝑖 (or 𝑥𝑖 ) encountered by a specific adsorptive separation process of interest. (v) Use a limited number of adjustable parameters. Item (i) is important for data extrapolation in the very low adsorbate concentration region needed for modeling a process to produce a very high purity product. Item (ii) is necessary for reliable estimation of a binary or a multicomponent adsorption isotherm using the corresponding pure component data. Item (iii) is essential to numerically account for the variation in IEHA with changes in GSE, which governs the local temperature changes in a 21 ACS Paragon Plus Environment


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heterogeneous adsorbent, during the operation of a separation process. Item (iv) is a must before a model can be reliably chosen for use in a process separation simulation. Item (v) is necessary for meaningful estimation of model parameters from the pure gas GSE isotherms. It should be pointed out that sufficient experimental data are often not available for a system of interest to cover the entire range of conditions of operation of a process and a large amount of data interpolation and extrapolation are required to run the process simulations. This emphasizes the usefulness of items (i – v) listed above. Analytical Models for GSE, IHEA and DHI Most practical adsorptive separation process employ heterogeneous micro-meso porous adsorbents. These adsorbents typically exhibit a type I GSE isotherm for the components of a feed mixture for (i) a bulk gas phase separation process (e.g. a low to medium pressure PSA or a TSA process), (ii) a liquid phase process for separation of dilute solutes from a bulk liquid mixture (e.g. a SMB process) and (iii) a gas or liquid phase process for removal of trace impurities from a contaminated stream (e.g. a TSA process). The adsorption isotherms, on the other hand, are U or S shaped for separation of a bulk binary liquid mixture (e.g. a CSA process). Numerous analytical models for describing Type I gas phase pure and multi-component GSE isotherms and Type U or S liquid phase binary GSE isotherms on heterogeneous adsorbents can be found in the literature. These isotherm models are empirically used in process simulation. Tables 3 and 4 list a few popular GSE isotherm models for bulk gas-solid and liquid -solid adsorption systems, respectively. Tables 3 and 4 list several extensively used GSE isotherm models for adsorption of trace or dilute solutes, respectively, from a bulk gas or a bulk liquid mixture on a heterogeneous adsorbent. Models (a) Modified Langmuir39, 40 (b) Dual Site Langmuir (DSL)41, 42 (c) Sips43,44

Pure Gas

Henry’s law region yes

Consistent* no

Analytic IHEA **

+ 1+∑2

yes

yes

**

𝑚𝑖 (𝑏𝑖 𝑝𝑖 )𝑘𝑖 [1+ ∑𝑖(𝑏𝑖 𝑃)𝑘𝑖 ]

No

no

**

Mixed Gas ( i = 1,2….)

𝑚𝑏𝑃

𝑛𝑖𝑚0 = [1+𝑖 𝑏𝑖 𝑃 ]

𝑚 𝑏𝑖 𝑝𝑖 𝑖 𝑏𝑖 𝑝𝑖 ]

𝑛𝑖𝑚 = [1+ ∑𝑖

𝑖

𝑖0 𝑏𝑖 = 𝑏𝑖0 exp [𝑞𝑠𝑡 /𝑅𝑇] 𝑚𝑖 = 𝑚𝑖 (T) 𝑚 𝑏 𝑃

𝑚 𝑏 𝑃

𝑛𝑖𝑚0 = 1+1 𝑏1𝑖 𝑃 + 1+2 𝑏2𝑖 𝑃 1𝑖

2𝑖

𝑖0 𝑏𝑗𝑖 = 𝑏𝑗𝑖0 exp [𝑞𝑠𝑡𝑗 ]

𝑚𝑖 (𝑏𝑖 𝑃)𝑘𝑖 [1+ (𝑏𝑖 𝑃)𝑘𝑖 ] 0 𝑖0 𝑏𝑖 exp [𝑞𝑠𝑡 /𝑅𝑇];

𝑛𝑖𝑚0 =

𝑚 𝑏 𝑝𝑖 𝑗 1𝑗 𝑝𝑗

𝑛𝑖𝑚 = 1+∑1 𝑏𝑖1

𝑛𝑖𝑚 =

𝑚 𝑏2𝑖 𝑝𝑖 𝑗 𝑏1𝑗 𝑝𝑗

𝑏𝑖 = 𝑘𝑖 = 𝑘𝑖 (T); 𝑚𝑖 = 𝑚𝑖 (𝑇)


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𝑚𝑏 𝑃

(d) Toth45, 46 (e) Sircar (HeteroLangmuir)47, 48

𝑖 𝑛𝑖𝑚0 = [1+ (𝑏 𝑃) 𝑘 ]1/𝑘

𝑛𝑖𝑚 =

𝑖

𝑖0 𝑏𝑖 = 𝑏𝑖0 exp [𝑞𝑠𝑡 /𝑅𝑇] k = k (T) 0 𝜃𝑖0= 𝜃𝑖𝐻 [1 −

𝜃𝑖0 =

𝑛𝑖0 𝑚

0 (1− 𝜃𝑖𝐻

𝑓(𝑧𝑖0 )]

0 𝜃𝑖𝐻

𝜃𝑖 = 𝜃𝑖𝐻 [1

𝑉 ( 𝑚 )𝑐𝑥

Yes 𝑚𝑖 = m

yes

z = ∑𝑖 𝜓𝑖 𝜃𝑖𝐻 1

1+𝑧

−1

𝑉 ( 𝑀 )𝑏𝑥

𝑉 ( 𝑀 )𝑏𝑥

𝑣 𝑣 n(x) = 1+(𝑐−1)𝑥 + [1+(𝑏−1)𝑥 ][ 𝑉 𝑣

yes

𝜇𝑝

yes

𝑣 + 1+(𝑏−1)𝑏𝑥 ; 0≤ x ≤ 𝑥𝑚 1+(𝑐−1)𝑥 𝑣

yes

𝑓(𝑧) = 2𝑧ln[1−𝑧]- 1

1+ 𝑧𝑖0 1 ] 0 ln[ 2𝑧𝑖 1− 𝑧𝑖0 𝑉 ( 𝑚 )𝑐𝑥

yes 𝑚𝑖 = m 𝑘𝑖 = k

𝑖 𝑖 𝑖

𝑖

𝑖0 𝜇𝑖 = 𝜇𝑖0 exp 𝑞𝑠𝑡 /𝑅𝑇 0 0 𝜎𝑖 = 𝜎𝑖 exp 𝜆𝑖 /𝑅𝑇

n(x) =

𝑓(𝑧)]

𝑛

𝜎

(f) Sircar (Model based on PSD) (pure gas only)50

(𝜓𝑖 −𝑧) 𝑧

yes

𝜃𝑖 = 𝑚𝑖 ; 𝜃𝑖𝐻 = 1+ ∑𝑖 𝜇𝑖 𝑝

𝜇𝑖 𝑃 1+ 𝜇𝑖 𝑃

0 ; 𝜃𝑖𝐻 =

0 𝑧𝑖0 = 𝜓𝑖 𝜃𝑖𝐻 ; 𝜓𝑖 = √3𝜇𝑖

𝑓(𝑧𝑖0 ) =

𝑚𝑏𝑖 𝑝𝑖 𝑘 [1+ (∑𝑖 𝑏𝑖 𝑝𝑖 ) ]1/𝑘

𝛤{𝑝+1,,𝜀} ] 𝛤(𝑝+1)

**

+

( )[𝛤 (𝑝+1,𝜀𝑚 )− 𝛤(𝑝+1,𝜀) ]

[

λ(r) =

𝛤 (𝑝+1) 𝛼 (𝑝=1) 𝑟 𝑝 exp{𝛤(𝑝+1)

] ; 𝑥𝑚 ≤ x ≤ 1

αr}, ε = αr;

*Integral and differential thermodynamic consistency between pure and binary gas GSE isotherms.; ** Underived

Table 3: Examples of analytical bulk Gas phase GSE Isotherm models for heterogeneous adsorbents. Comments on heterogeneous isotherm models for bulk gas adsorption described in Table 3: (a) A physical constraint requires that the total number of adsorption sites in an adsorbent, and hence the saturation adsorption capacity (𝑚𝑖 ) of an adsorbate i, be independent of temperature (T) as in the classic homogeneous Langmuir isotherm model. However, an empirical modification is often made by assuming that 𝑚𝑖 is a function of T, which introduces an artificial heterogeneity to the model.38 - 40 (b) The popular DSL model assumes that the adsorbent consists of two sites (j = 1, 2) of different energy. The adsorption isotherm on each site is described by the Langmuir model and the over-all isotherm is obtained by simply adding the contributions from each site.41, 42 𝑖0 Thus, the model has six adjustable Langmuir parameters for each adsorbate i (𝑚𝑗 , 𝑏𝑗𝑖0 , and 𝑞𝑠𝑡𝑗 ; J = 1, 2], which may require a large amount of isotherm data for reliable evaluation of the model parameters. 23 ACS Paragon Plus Environment


Page 24 of 45

(c) The empirical Sips model also assumes that 𝑚𝑖 is a function of temperature. Furthermore, it uses a temperature dependent parameter ( 𝑘𝑖 ) to represent the degree of heterogeneity exhibited by an adsorbent for adsorption of adsorbate i. Thus, the model can account for different degrees of adsorbent heterogeneity for different adsorbates by the same adsorbent.43, 44 The Sips model, however, does not exhibit Henry’s law limits and does not pass the thermodynamic consistency tests. (d) The empirical Toth isotherm model assumes that both the temperature dependent heterogeneity parameter (𝑘𝑖 = k) and the temperature independent saturation capacity (𝑚𝑖 = m) are same for all adsorbates, which is required to satisfy the thermodynamic consistency tests. The model is valid for the case where the adsorbent exhibits the same degree of heterogeneity for each adsorbate.45, 46 The degree of heterogeneity of an adsorbent is increased as the parameter k (0 ≤ k ≤ 1 ) decreases. The model collapses to the Langmuir model when k = 1. The model yields analytical expressions for isosteric heat of adsorption of a gas (pure or mixture).4, 31 The pure gas heat, however, approaches negative infinity when the adsorbate loading approaches its saturation value. Thus, the model cannot be used for very high gas pressure. (e) The heterogeneous Sircar model is based on a ‘patch-wise homogeneous concept’ of the adsorbent heterogeneity.47,48 It assumes that the Langmuir model describes the pure and multicomponent gas isotherms on a homogeneous patch, and a uniform distribution of Langmuir Henry’s law constants with superimposition of cumulative distribution functions of different gases account for adsorbent heterogeneity and site matching relationships, respectively. The parameters 𝜇𝑖 and 𝜎𝑖 are, respectively, the mean and the dispersion of the 𝑖0 uniform distribution function for pure gas i. The variables 𝑞𝑠𝑡 is the isosteric heat of adsorption in the Henry’s law region for pure component i. The parameter 𝜇𝑖 is also the Henry’s law constant for a pure gas i on the heterogeneous adsorbent. The variables 𝜇𝑖0 𝜎𝑖0 and 𝜆0𝑖 are constants. The degree of heterogeneity for adsorption of component i by an adsorbent 𝜎 is accounted for by the parameter 𝜓𝑖 (= √3 𝑖 ). The adsorbent is more heterogeneous as ψ 𝜇𝑖

increases (0 ≤ ψ ≤1). The model is capable of describing a case where the adsorbent exhibits different degrees of heterogeneity for different gases which can lead to adsorption azeotropy. The model collapses to the Langmuir model for a homogeneous adsorbent (ψ = 0). The model, however, requires that the saturation capacities of each gas be same (𝑚𝑖 = m). The parameter 𝑏𝑖 in models a, c and d of Table 3 is the gas-solid interaction parameter for component i on the heterogeneous adsorbent which is an exponential function of T ([= 𝑏𝑖0exp 𝑖0 (𝑞𝑠𝑡 /𝑅𝑇)]. The corresponding Henry’s law constant at temperature T is given by (𝐾𝑖 = 𝑚𝑖 𝑏𝑖 ). The variable 𝑝𝑖 (= P𝑦𝑖 ) is the partial pressure of component i in the equilibrium gas phase. 24 ACS Paragon Plus Environment

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(f) Model based on structural heterogeneity:50 A versatile analytical model to describe the adsorption of a pure condensable vapor on a structurally heterogeneous adsorbent containing micro and meso pores has been developed. Table 3 gives the isotherm equation relating the amount adsorbed n(x) as a function of x (= P/Ps) at temperature T, where P is the equilibrium gas phase pressure of the adsorbate and Ps is its vapor pressure at T. The model assumes (i) a gamma -type distribution (𝑝=1) 𝑝

[λ(r) = 𝛼𝛤(𝑝+1)𝑟

exp{- αr}, ε = αr]

of the adsorbent pore radius (r), (ii) Langmuir type adsorption

isotherms on the micro and mesopore walls, and (iii) condensation of vapor in the mesopores governed by the Kelvin equation. The variables of the model include the specific volumes of the micropores (𝑉𝑚 ), the mesopores (𝑉𝑀 ), the total pore volume (V = 𝑉𝑚 + 𝑉𝑀 ); the gas-solid Langmuir interaction parameters in the micropore (c) and the mesopores (b); and the parameters (p, 𝛼) of the Γ distribution function. The micropores are bound by 0 ≤ r ≤ 𝑟𝑚 and the mesopores are bound by 𝑟𝑚 ≤ r ≤ ∞. The parameter 𝑥𝑚 represents the value of x at which condensation of vapor starts in pores defined by r ≥ 𝑟𝑚 at T.



Page 26 of 45

(a)

O Experiment ---- Theory

(b) O Experiment ---- Theory

(c) O Experiment ---- Theory

Figure 9: Condensable Vapor adsorption on various carbons. All three types of isotherm shapes (I, IV and V) by the Brunauer classification, which are exhibited by adsorption of a condensable vapor on a porous adsorbent, can be described by the model based on structural heterogeneity. Figure 9 shows a few examples: (a) adsorption of condensable hydrocarbons on a microporous molecular sieve carbon at different 26 ACS Paragon Plus Environment

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temperatures (Type I), (b) adsorption of C2H5Cl vapor on mesoporous Darco Charcoal at 0 C (Type IV), (c) adsorption of water vapor on mesoporous sugar charcoal at 100 C (Type V). More details about this model can be found elsewhere.50 Ideal Adsorbed Solution Theory (IAST) and adsorbent heterogeneity: The adsorbent heterogeneity plays a critical role in prediction of mixed gas adsorption isotherms from the corresponding pure gas isotherms using the Ideal Adsorbed Solution Theory (IAST). 35, 51 The prediction can be fair to poor when the adsorbent heterogeneity is moderate to large.35,51 The concept of a non-ideal adsorbed phase is often invoked to explain the failure of the IAST predictions. However, it has been shown that the adsorbent heterogeneity may be the true reason for the apparent non-ideal behavior of the adsorbed gas phase.50 The apparent non-ideal behavior of the adsorbed phase of a binary liquid mixture can also be explained by the adsorbent heterogeneity.52 On the other hand, it is a common practice to curve fit the pure gas adsorption equilibrium data using an analytical heterogeneous isotherm model and subsequently use the model in conjunction with the IAST to predict multi-component gas adsorption equilibria.35 The model is used to interpolate and extrapolate the pure gas isotherm data.

(a) Pore- filling mechanism on a heterogeneous adsorbent.53

𝑚

𝑆𝐻

𝑥

1/(𝛽−1)

1 2 𝑛1𝑒 (𝑥1 ) = (𝑆𝐻 −𝑆 𝐿 ) {𝑎 [(𝑆 𝐻 ) 0

𝑎2 𝑎1 (𝛽−1)

[

𝑆0𝐻 𝑆𝐻

1

0

−

𝑆0𝐿 𝑆𝐿

0

𝑆𝐿

1/(𝛽−1)

− (𝑆𝐿 )

]+

Yes

0

]}

𝑚

β = 𝑚1 2

𝑆 𝐻 = 𝑆𝑜𝐻 (𝑆 𝐻 𝑎1 + 𝑎2 )(𝛽−1)/𝛽 𝑆 𝐿 = 𝑆𝑜𝐿 (𝑆 𝐿 𝑎1 + 𝑎2 )(𝛽−1)/𝛽 (b) Amagat’s law describes molar volumes in bulk liquid and adsorbed phases.54

𝑉𝑎

underived

𝑛1𝑒 (𝑥1 ) = [ 𝑣0 ] 𝑥2 [1 − 𝐹(𝜓, 𝑥1 ) ]; 1

𝛽𝑥 + 𝑥

{(1+ 𝜓)𝜇𝛽−1}𝑥 +1

𝑣0

1 2 F (ψ, 𝑥1 ) = [2𝜇𝛽𝜓𝑥 ]ln[{(1− 𝜓)𝜇𝛽−1}𝑥1 +1]; β = 𝑣10 ; 1

ψ=

1

2

√3𝜎 𝜇

Table 4: Examples of analytical bulk liquid phase GSE isotherm models for heterogeneous adsorbents.



Page 28 of 45

Comments on heterogeneous isotherm models for bulk liquid adsorption described in Table 4: (a) The popular pore-filling mechanism (PFM) for adsorption of a bulk binary liquid mixture on a homogeneous adsorbent, where all the pores of the adsorbent are filled with the liquid mixture, has been extended by Sircar to analytically describe the GSE of component 1 of a binary liquid mixture on a heterogeneous adsorbent as shown by Table 4. The model utilizes a ‘patch-wise homogeneous’ concept to describe adsorbent heterogeneity and a uniform distribution of the limiting selectivity of component 1 at infinite dilution [ 𝑆0 at 𝑥1 → 0] on a homogeneous patch is used to describe the heterogeneity. The selectivity of adsorption of component 1 over component 2 is defined by [S =

𝑥1𝑎 𝑎2 𝑥2𝑎 𝑎1

] where 𝑥𝑖𝑎 and 𝑎𝑖 are respectively, the

mole fraction and the activity of component i in the adsorbed and the bulk liquid phases, respectively. The lowest and the highest values of the distributed parameter 𝑆0 on the 𝑚 adsorbent are given by 𝑆0𝐿 and 𝑆0𝐻 , respectively. The parameter β is defined by 1 , where 𝑚𝑖 is 𝑚2

the pore filling capacity of pure adsorbate i. Thus, the model is capable of describing equilibrium GSE of a non-ideal binary liquid mixture of unequal adsorbate sizes on a heterogeneous adsorbent. (b) The model based on the validity of the Amagat’s law to describe the molar volumes of adsorbate mixtures in both the adsorbed and the bulk liquid phases assumes that the entire pore volume (𝑉 𝑃 ) of the adsorbent does not constitute the adsorbed phase like the PFM model. The selectivity of adsorption is defined by [S =

𝑥1𝑎 𝑥2 𝑥2𝑎 𝑥1

] and the adsorbent heterogeneity is

described by a uniform distribution of S ranging between 𝑆𝐿 and 𝑆𝐻 . The parameters of the analytical model describing 𝑛1𝑒 are the volume of the adsorbed phase [ 𝑉 𝑎 < 𝑉 𝑃 ], the ratio of the molar volumes (𝑣𝑖0 ) of pure adsorbates i [β (=

𝑣10 𝑣20

)], the mean (𝜇) and the dispersion (σ) of

the uniform distribution. The degree of heterogeneity of the adsorbent is defined by ψ (=

√3𝜎 𝜇

).

Comments on heterogeneous isotherm models for trace or dilute solutes: The Freundlich and the Langmuir- Freundlich models are traditionally used to represent equilibrium adsorption isotherms of single and multicomponent trace (or dilute) solutes from an inert gas or liquid on a heterogeneous adsorbent.2, 55, 56 Tables 5 and 6 show the analytical isotherms. The variable 𝑞𝑖 is the amount of component i adsorbed from a bulk fluid phase where the concentration of that component is 𝐶𝑖 . The parameters, 𝐾𝑖 m, 𝑛𝑖 (> 1), 𝑎𝑖 , and 𝑏𝑖 are empirical constants.


Page 29 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The Freundlich model has been empirically used to describe single solute isotherms of many trace organic chemicals from water in view of water pollution control application. 2, 56 It can also be derived theoretically by the ‘patch-wise homogeneous concept’ of adsorbent heterogeneity in conjunction with the previously described homogeneous PFM model and a gamma distribution of the property 𝑆0 .57

Single solute (a) Freundlich2, 55 (b) Langmuir Freundlich2, 55

𝑞𝑖 = 𝐾𝑖 𝑐𝑖 1/𝑛𝑖

Mixed solutes 1/𝑛𝑖

𝑞𝑖 =

1/𝑛𝑖

𝑞𝑖 =

𝑞𝑖 =

1/𝑛 𝑏𝑖 𝐶𝑖 𝑖

Thermodynamic Analytic consistency IHI no **

no

no

1/𝑛𝑗

∑𝑗 𝑏𝑗 𝐶𝑗

1/𝑛𝑖

𝑚𝑏𝑖 𝐶𝑖 1+

𝑎𝑖 𝐶𝑖

Henry’s law region no

𝑎𝑖 𝐶𝑖

**

1/𝑛𝑗 1+ ∑𝑗 𝑏𝑗 𝐶𝑗

Table 5: Examples of analytical GSE Isotherm models for adsorption of trace solutes from a bulk gas on heterogeneous adsorbents.

Models (a) Freundlich2, 56

Single solute 𝑞𝑖 = 𝐾𝑖 𝑐𝑖 1/𝑛𝑖

1/𝑛𝑖

𝑞𝑖 =

1/𝑛𝑖

(b) LangmuirFreundlich2, 56

𝑞𝑖 =

(c) Bi- Langmuir7

𝑞𝐴 = 1+ 𝑎

Henry’s law region no no

Multi-solute 𝑎𝑖 𝐶𝑖

1/𝑛𝑗

∑𝑗 𝑏𝑗 𝐶𝑗

1/𝑛𝑖

𝑚𝑏𝑖 𝐶𝑖

𝑞𝑖 =

1/𝑛𝑖

1+ 𝑏𝑖 𝐶𝑖

𝑎0𝐶𝐴

𝑞𝐴 = 1+ 𝑎

𝐴 𝐶𝐴 +𝑎𝐵 𝐶𝐵

𝑎𝑖 𝐶𝑖

1/𝑛𝑗

1+ ∑𝑗 𝑏𝑗 𝐶𝑗

𝑎0𝐶𝐴 𝐴 𝐶𝐴 +𝑎𝐵 𝐶𝐵

+

𝑏0 𝐶𝐵 1+ 𝑏𝐴 𝐶𝐴 +𝑏𝐵 𝐶𝐵

N

no y yes

Table 6: Examples of analytical GSE Isotherm models for adsorption of trace solutes from a bulk liquid on heterogeneous adsorbents.

The bi-Langmuir model empirically describes equilibrium adsorption of dilute binary solutes (A and B) from a bulk liquid. The specific equilibrium amounts of the solutes adsorbed are 𝑞𝐴 and 𝑞𝐵 from a solution having concentrations of 𝑐𝐴 and 𝑐𝐵 for components A and B, respectively. The model parameters are 𝑎0, 𝑏0, 𝑎𝐴 , 𝑏𝐴 , 𝑎𝐵 and 𝑏𝐵 . This model is popular for design of SMB processes.7 Model for adsorbate mass transfer into mesopores of a heterogeneous adsorbent particle: The complex nature of the micro-meso porous structure of a practical amorphous adsorbent (or the binder in a pelletized crystalline adsorbent) cannot be experimentally measured by 29 ACS Paragon Plus Environment


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today’s technology. It may be impossible to formulate a realistic model for such a structure even if it was known. Consequently, a simple empirical approach is popularly used to estimate the effective mass transfer coefficient ( 𝑘𝑖 ) of an adsorbate i into the mesopores of a heterogeneous adsorbent particle (diameter= 𝑑𝑝 ) for practical process design: 23 𝑘𝑖 =

60 𝐷𝑖𝑒 2 𝐾𝑖 𝑑𝑝

; 𝐷𝑖𝑒 =

𝑒𝑝

[

1

𝜏𝑝 𝐷𝐾𝑖

+

1 𝐷𝑀𝑖

]

(2)

where, 𝐷𝐾𝑖 𝐷𝑀𝑖 and 𝐷𝑖𝑒 are, respectively, the Knudsen diffusivity through the mesopore, the molecular diffusivity, and the effective overall diffusivity of the adsorbate i through the mesopore. The variables 𝜀𝑝 and 𝜏𝑝 are, respectively, the meso pore void fraction and tortuosity. The variable 𝐾𝑖 is the Henry’s law constant for the adsorbate i. The tortuosity is a fudge factor to account for the unknown labyrinthic pore structure of a practical adsorbent. The value of τ depends on the adsorbate – adsorbent pair. A range of 2 – 65 for τ has been reported in the literature.55 Eq. 2 is strictly valid in the Henrys law region only, but it is freely used beyond that. It should be noted that there is no Knudsen flow for liquid adsorbates where the pores remain filled with the liquid. Eq. 2 is valid for an adsorbent having a mono-disperse pore structure. Analytical models are also available for estimating 𝐷𝑖𝑒 for a bi-disperse pore structure, such as the random pore distribution model for diffusion of a binary gas: 61 𝐷𝑖𝑒 = 𝐷𝑀𝑖 [(1−

𝜀𝑙2 𝑙 𝛼𝑦1 ) + 𝐷𝑀𝑖 /𝐷𝐾𝑖

+

𝜀𝑠2 𝑠 (1− 𝛼𝑦1 ) + 𝐷𝑀𝑖 /𝐷𝐾𝑖

4𝜀𝑙 (1− 𝜀𝑙 )

+ (1− 𝛼𝑦1 ){1+

]

(1− 𝜀2 (1− 𝜀2 ) 𝐷𝑠 𝐷 𝑙) } + 𝑀𝑖 { 2 𝑙 + 𝐾𝑖 } 𝑠 2 𝐷𝐾𝑖 𝜀𝑠 𝜀𝑠 𝐷𝑙𝐾𝑖

(3)

where the superscripts l and s, respectively, stand for the large and the small pores of a bidisperse pore structure.

Other parallel or series resistances for adsorbate transport such as an external film resistance, a barrier at the adsorbent crystal or pellet surface, a surface flow in the mesopores, a micropore resistance in series, etc. can also be important in a practical adsorber.23 Simple Linear Driving Force (LDF) model formulations are available to include these resistances into 𝑘𝑖. 23 The diffusivity of an adsorbate into a micropore of an adsorbent is usually measured experimentally. It has been found to be a very complex function of (a) adsorbate-adsorbent pair, (b) the adsorbate loading, and (c) the presence of other components. The present state of the art does not permit a priori estimation of this property. 30 ACS Paragon Plus Environment

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It should be obvious from the above discussions that macroscopic modeling of the thermodynamic properties of a heterogeneous adsorbent is further developed than that for the dynamic properties. In reality, there may not be any incentive to develop a more complicated model for describing the pore structure of an amorphous adsorbent if an empirical tortuosity factor is used to cover up for the model’s deficiency. Experimental measurement of 𝑘𝑖 for the adsorbate-adsorbent pair of interest remains the best choice for practical use. Critical examples of adsorbent heterogeneity – A case history for gas adsorption Figure 10 shows that the heterogeneous equilibrium models proposed by Toth and by Sircar, described in Table 3, can very well fit the pure N2 adsorption isotherm data on a commercial sample of pelletized LiLSX zeolite at different T over a pressure range of 0 – 6 atm.48 Even the homogeneous Langmuir model provides a fair fit of the same data. Figure 10, on the other hand, shows that the Sircar model fits the pure O2 adsorption isotherm data very well, while the Langmuir and the Toth models underestimate the amount of O2 adsorbed at any given P at all T, the Toth model being worse than the Langmuir model.

Figure 10: Pure gas adsorption isotherms of N2 and O2 on Low silica LiLSX zeolite at different T. Experiments: o, ∆; Models: dotted line - Langmuir; solid line – Toth; dashed line- Sircar.48 Figures 11(a) and 11 (b), respectively, show the variations in the iso excess (or isosteric) heats for adsorption of pure N2 and O2 on the LiLSX zeolite as functions of the corresponding GSE (circles). They were estimated from the isotherms of Figure 10 using the thermodynamic relationship given by Eq.1. The solid and the dashed lines in Figure 11 (a and b) show 31 ACS Paragon Plus Environment


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equivalent plots estimated by the analytical heterogeneous models developed by Toth and by Sircar, respectively. It may be seen that the Sircar model traces the thermodynamic heat vs GSE plots more closely than the Toth model for both N2 and O2. Figure 11c compares the variations in heats vs GSE for N2 and O2 on the zeolite. The heat decreases with increasing GSE for both gases indicating that the adsorbent is heterogeneous for both gases. However, the decrease in heat is more pronounced for N2 than that for O2, which indicates that the degree of heterogeneity exhibited by the zeolite is larger for N2 than O2. The binary selectivity of adsorption (S12) of N2 (1) over O2 (2) on the zeolite was experimentally measured under various conditions of P, T and 𝑦1 and compared to the S12 values estimated by the IAST, the Langmuir, the Toth, and the Sircar models.48 The difference in the experimental and the model estimated S12 values was largest for the Langmuir/Toth models, followed by the IAST, followed by the Sircar model. This clearly demonstrated the importance of the difference in the degree of heterogeneity exhibited by the adsorbent for N2 and O2. The over-all LDF mass transfer coefficients exhibited by the pelletized LiLSX sample for adsorption of N2 and O2 from inert helium were also reported under various conditions of P and T.49 It was found that a resistance at the surface of the adsorbent pellet contributed to ~ 60 + % of overall resistance. The surface resistance was introduced by a structural heterogeneity of the adsorbent, which could not be predicted a priori.

(a) (b)


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(c)

Figure 11: Isosteric heat of adsorption on LILSX zeolite as function of GSE: (a) pure N2, (b) pure O2. (c) comparative heats: Circles and triangles – heat calculated using pure gas isotherms of Figure 10 and thermodynamic Eq.1; Toth model (solid line); Sircar model (dashed line).48 The above example demonstrates the complexity of adsorbent heterogeneity and its role on establishing the key thermodynamic and dynamic properties like isotherm shapes, the selectivity, the heats of adsorption., and the adsorbate mass transfer coefficient for gas phase adsorption. These effects of adsorbent heterogeneity cannot be predicted. They are exposed through experimental measurement only. Critical examples of adsorbent heterogeneity – Two cases of liquid adsorption Figures 12 (a) and 12 (b) show the GSE isotherms for adsorption of benzene (1) + I, 2 dichloroethane (2) binary liquid mixtures on micro-mesoporous Calgon BPL carbon at 30o C 58 and non-porous Cabot Graphon at 0o C 59, respectively. The molar volumes of component 1 and 2 are, respectively, 89.3 and 79.3 cm3/ mole at 298 K and the liquid mixture is nearly ideal. It may be seen that the isotherm is U shaped for adsorption on the BPL carbon. Thus, benzene is selectively adsorbed over 1, 2 dichloroethane at all compositions. The data can be described 33 ACS Paragon Plus Environment


very well [solid line in Figure 12 (a)] by a homogeneous GSE model [model (b) in Table 4] which suggests that the carbon is homogeneous for adsorption of the binary liquid mixture of interest.54 Figure 12 (b), on the other hand, shows that the GSE isotherm of the same liquid mixture on Graphon is S shaped indicating that benzene is selectively adsorbed over I, 2 dichloroethane until 𝑥1 < 0.75, and then the selectively is switched towards 1, 2 dichloroethane. Such behavior is caused by adsorbent heterogeneity. The data can be described very well by a heterogeneous GSE model [Model (a) in Table 4] which provides further evidence of adsorbent heterogeneity.53 These results may be surprising since the micro-meso porous BPL carbon is expected to be more heterogeneous than the non-porous Graphon.

Surface Excess of Benzene, mmoles/g.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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GSE isotherm: Benzene (1) + 1,2 Dichloroethane (2) on Calgon BPL Carbon at 30 C 0.7 0.6 0.5

(a)

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

Mole fraction pf Benzene


0.8

1

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Figure 12: GSE isotherms of benzene (1) + 1, 2 di-chloroethane (2) binary liquid mixture on (a) BPL carbon at 30 o C and (b) Graphon at 0 o C. The above examples reveal the complexity and the nuances of adsorbent heterogeneity in determining the GSE isotherm for adsorption of a liquid mixture. They cannot be predicted. They are exposed through experimental measurement only. On the other hand, physically and thermodynamically consistent mathematical models of GSE isotherms can help to understand the nature of the heterogeneity and provide a tool for data interpolation and extrapolation, adsorbent modification, as well as for numerical simulation of the separation performance of an adsorptive process. Effect of adsorbent heterogeneity on the separation performance of a PSA process A few publications on modeling bulk gas separation by an adiabatic PSA process have used a heterogeneous adsorption isotherm model in the process simulation.39, 40, 44, 62 - 65 However, it is often assumed that the isosteric heat of adsorption for the adsorbate is not a function of the adsorbate loading, – instead an average heat is used for all loadings in the heat balance equation of the process model.38, 39, 43, 61, 62 This suppresses a key effect of the adsorbent heterogeneity on the process performance. To our best knowledge only two PSA process model studies avoid that assumption.60, 63 One of them carried out a parametric evaluation of the effect of the degree of adsorbent heterogeneity on the performance a PSA Process.63 The process was designed to produce high purity (99.999 +%) helium from a bulk C2H4 (1) + He (2) gas mixture containing 20 % C2H4. The feed gas pressure was 405.2 kPa and the temperature was 298.1 K. The final desorption pressure was 101.3 kPa. The Calgon BPL carbon was used as the model adsorbent in conjunction with a four-step PSA process consisting of (i) pressurization of the column with the feed gas, (ii) adsorption to produce a helium rich effluent gas, (iii) counter-current depressurization of the column, and (iv) counter-current back purging using a part of the effluent gas from step (ii). The effluents from steps (iii) and (iv) were wasted. A part of the effluent gas from step (ii) was withdrawn as the product gas. Figure 13 (a) shows that the Toth model describes the GSE isotherms of C2H4 on the BPL carbon at different temperatures very well. Figures 13 (b) and 13 (c), respectively, show the virtual variations in the shapes of the C2H4 isotherms at 298.1 K and the corresponding isosteric heats as functions of loadings for different values of Toth heterogeneity parameter k, 0 while maintaining the same Henry’s law region characteristics (m, 𝑏 0 and 𝑞𝑠𝑡 ).



(a )

(b)

(c)


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Figure 13: (a) C2H4 adsorption isotherms on BPL Carbon at different T: Experiment (symbols), Toth Model (solid lines); (b) Virtual C2H4 isotherms at 298 K for different values of k; (c) Virtual isosteric heat of adsorption of C2H4 vs GSE at 298 K for different values of k. An analytic expression describing the variation of the isosteric heat of C2H4 with loading for the Toth model was used to generate Figure 13 (c).4, 31 It may be seen from Figures 13 (a - c) that (i) the equilibrium adsorption capacity of C2H4 at any given pressure decreases, (ii) the isosteric heat of adsorption of C2H4 at any given loading decreases, and (iii) the slope of the heat vs loading plot at any given loading increases as the adsorbent becomes more heterogeneous (lower k). Apparently, item (i) is unfavorable for a process because it lowers the capacity of C2H4 adsorption at feed conditions, while items (ii) and (iii) are favorable because they lower the adsorbent temperature rise during the adsorption process. The over-all effect of the adsorbent heterogeneity on the process performance, however, is complex, and it can be analyzed only by carrying out a numerical simulation of the process. Figures 14 shows one process simulation result for the PSA process described earlier which was not previously published. More detailed results can be found elsewhere.65 The effects of adsorbent heterogeneity on two key parameters measuring the efficiency of the separation process are reported by Figure 14. These include the ‘Bed Size Factor’ (BSF, kg of adsorbent/kg of helium product/hour) and the ‘Helium Product Recovery’ from the fresh feed gas (R, %). The process design goals are to lower the BSF (smaller adsorbent inventory) and to increase R (smaller loss of valuable product).



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Figure 14: The effect of the Toth heterogeneity parameter (k) on the BSF and the R of a PSA process producing pure helium from a 20 % C2H4 + He feed gas mixture using the BPL carbon and a four-step process. Total PSA process cycle time: 16 seconds (squares) and 60 seconds (circles). It may be seen from Figure 14 that the BSF remains practically unaffected until the adsorbent becomes sufficiently heterogeneous (low k) and then it increases rapidly. The R, on the other hand, slowly decreases with decreasing k in the larger k region, and then it decreases very rapidly when the k is further reduced. The same behavior was observed for total PSA cycle times of 16 and 60 seconds. More results for other cycle times can be found elsewhere.65 The general conclusion from this study is that the adsorbent heterogeneity is detrimental to the process performance of this PSA process. The extent of the effect of adsorbent heterogeneity on a separation process will, however, depend on the adsorbate – adsorbent – process design combination. Furthermore, practically reliable process performance data can only be obtained by continuous operation of the process in a pilot or commercial scale unit. Summary Practical micro-mesoporous adsorbents used in the separation of fluid (gas or liquid) mixtures by an adsorptive process are generally energetically heterogeneous. The cause of adsorbent 38 ACS Paragon Plus Environment

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heterogeneity (physical or chemical) and the qualitative tests to identify its existence are discussed. The variation of the iso-excess (or isosteric) heat of adsorption of a pure gas with the adsorbate Gibbsian surface excess (GSE) on an adsorbent can be used to quantify the degree of adsorbent heterogeneity. In contrast, there is no easy method to establish adsorbent heterogeneity by measuring binary liquid phase GSE isotherms or the heat of immersion of the adsorbent into a pure liquid. The effect of adsorbent heterogeneity on the shape of the equilibrium gas phase (pure or mixture) and liquid phase (binary or multi-component) GSE isotherms can be significant and complex, and they must be experimentally measured. Numerous analytical models to describe heterogeneous adsorption are available to describe these isotherms. Some of these models yield analytical expressions for pure and multicomponent isosteric heat of adsorption or heat of immersion for heterogeneous adsorbents. They are discussed and their pros and cons are emphasized using many examples. The complexity of the pore structure of an amorphous adsorbent prevents realistic quantification and modeling of the same, and empirical models are used to describe the adsorbate mass transfer coefficient into a heterogeneous adsorbent particle. An empirical tortuosity factor is used to cover - up the ignorance of the structure. Numerical process studies of adiabatic gas separation processes using a heterogeneous adsorbent are few. Many of them ignore the variation in the heat of adsorption with adsorbate loading. Thus, they may be unrealistic. Processes for liquid phase separation are generally operated isothermally. The simulation results of a four-step PSA process for production of pure helium from a bulk C2H4 + He binary gas mixture using a heterogeneous BPL carbon as the adsorbent, where the Toth model is used to describe the isotherms and the variation of isosteric heats, in conjunction with a four-step PSA process, showed that the over-all performance of the process deteriorated with the increase in the degree of adsorbent heterogeneity. The extent of the effect of adsorbent heterogeneity on a separation process will, however, depend on the adsorbate – adsorbent – process design combination. Practically reliable process performance data can only be obtained by continuous operation of the process in a pilot or commercial scale unit. Process models, however, may be useful for adsorbent and process screening, process optimization, scale up, etc. Author Information Corresponding Author 39 ACS Paragon Plus Environment


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*Email: [email protected] ORCID Shivaji Sircar: 0000-0002.9648-4083 Notes The authors declare no competing financial interest. References 1. Yang, R. T. Adsorbents: Fundamentals and Applications; John Wiley and Sons: Hoboken, NJ, 2003. 2. Noll, K. E.; Gounaris, V.; Hou, W-S. Adsorption Technology for air and water pollution control; Lewis Publishers: Chelsea, MI, 1992. 3. Sircar, S: Drying processes. In Handbook of Porous Solids; Sch𝑢̈ th, F; Sing, K. S. W; Weitkamp, J. Eds.: Wiley-VCH: Weinheim, Germany. 2002; Chapter 6.3, pp 2533 - 2565. 4. Sircar, S; Myers, A. L: Gas separation by zeolites. In Handbook of Zeolite Science and Technology; Aurbach, S. M.; Carrado, K. A.; Dutta, P. K. Eds.: Marcell Dekker, New York, NY. 2003; Chapter 22, pp 1063 – 1164. 5. Sircar, S.; Golden, T. C.; Rao, M. B. Activated carbon for Gas separation and storage. Carbon. 1996, 32, 1. 6. Sircar, S.; Golden, T. C. Pressure swing adsorption technology for hydrogen production. In Hydrogen and syngas production and purification technologies; Liu, K.; Song, C.; Subramani, V. Eds.: John Wiley and Sons, Hoboken, NJ. 2010; Chapter 10, pp 414 – 450. 7. Rodrigues, A. E.; Pereira, C.; Minceva, M.; Pais, L. S.; Ribeiro, A.; Silva, M.; Graca, N., Santos, J. C. Simulated moving bed technology; Elsevier, New York, NY. 2015. 8. Gregg, S. J.; Sing, K. S. W: Adsorption surface area and porosity; Academic press, London, 1967. 9. Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by powders and porous solids: Principles, Methodology and Applications. Academic Press, London, 1999. 10. Golden, T. C.; Jenkins, R. G.; Otake, Y.; Scaroni, A. W. Oxygen complexes on carbon surfaces. Presented at DOE workshop on the electrochemistry of carbon, Case Western Reserve University Library. 1983. 40 ACS Paragon Plus Environment

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11. Capelle, A.; de Vooys, F. Eds., Activated carbon – a fascinating material. Norit N. V., Amersfoort, The Netherlands. 1983. 12. Shafeeyan, M. S.; Daud, W. M. A. W.; Houshmand, A.; Shamiri, A: A review on surface modification of activated carbon for carbon dioxide adsorption. J. Analytical and Applied Pyrolysis. 2010, 89, 143. 13. Peri, J. B. Infrared and gravimetric study of the surface hydration of γ- alumina. J. Phts. Chem. 1965, 69, 211. 14. van Roosmalen, A. J.; Moi, J. C. An infrared study of the silica gel surface. J. Phys. Chem. 1978, 82, 2748. 15. Saint Remi, J. C.; Lauerer, A.; Chmelik, C.; Vandendael, I.; Terryn, H.; Baron. G. V.; Denayer, F. M.; Karger, J. The role of crystal diversity in understanding mass transfer in nano-porous. materials. Nature Materials. 2016, 15, 401. 16. Gaffney, T. R. Porous solids for gas separation. Current opinion in solid state and materials science. 1996, 1, 69. 17. Kirner, J. F. Nitrogen adsorption with highly Li exchanged X- zeolites with low Si/Al ratio. U. S. Patent, 5, 268, 023. 1993. 18. Kazansky, V. B.; Bulow, M.; Tichomirova, E. Specific adsorption sites for nitrogen in NaLSX and LiLSX. Adsorption, 2001, 7, 291. 19. Sircar, S.; Rao, M. B.; Golden, T. C. Fractionation of air by zeolites. In Studies in surface science and catalysis. V. 120. Elsevier, Amsterdam. 1998, pp 395-423. 20. Karger, J.; Pfeifer, H. NMR self-diffusion studies on zeolite science and technology. Zeolites. 1987, 7, 90. 21. Brandini, F.; Ruthven, D. M. The effect of water on the adsorption of CO2 and C3H8 on X zeolites. I & EC Res. 2004, 43, 8339. 22. Cao, D. V.; Sircar, S. Isosteric heats of adsorption of pure SF6 and CO2 on silicalite pellets with alumina binder. I & EC Res., 2001, 40, 156. 23. Sircar, S. Adsorbate mass transfer rates into porous adsorbents - A practical viewpoint. Sep. Purif. Tech. 2018, 193, 283.



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24. Ansuzzaman, S. M.; Joseph, C. G.; Taufiq-Yap, Y. H.; Krishnaiah, D; Tay, V. V. J, King Saud.Univ.-Sci. Modification of commercial activated carbon for the removal of 2, 4 dichlorophenol from simulated waste water. 2015, 27, 318. 25. Kumar, R.; Sircar, S. Skin resistance for adsorbate mass transfer into extruded adsorbent pellets. Chem. Eng. Sci. 1986, 42, 2215. 26. Kosmulski, M. pH dependent surface charging and points of zero charge. IV. update and new approach. J. Colloid. Interface Sci. 2009, 337, 439. 27. Liu. H.; Zhao, G. Effect of temperature and time on microstructure and surface functional groups on activated carbon fibers made from liquified wood. Bio Resources. 2012,7, 5552. 28. Sircar, S.; Mohr, R. J.; Ristic, C.; Rao, M. B. Isosteric heat of adsorption – Theory and experiment. J. Phys. Chem. 1999, 103, 6539. 29. Sircar, S. Gibbsian thermodynamics and column dynamics for adsorption of liquid mixtures. Ind Eng. Chem. Res. 1993, 32, 2430. 30. Sircar, S. Excess properties and thermodynamics of multicomponent gas adsorption. J. Chem. Soc. Faraday Trans. I. 1985, 81, 1527. 31. Sircar, S. Isosteric heats of multicomponent gas adsorption on heterogeneous adsorbents. Langmuir, 1991, 7, 3065. 32. Golden, T. C.; Sircar, S. Synthetic heterogeneity in X zeolite for gas adsorption. J. Colloid and Interface Sci. 1991, 147, 274 33. Sircar, S.; Gupta, R. A semi-empirical adsorption equation for single component gas-solid equilibria. AIChE J. 1981, 27, 806. 34. Dunne, J.; Rao, M. B.; Sircar, S.; Gorte, R. J.; Myers, A. l. Calorimetric heats of adsorption and adsorption isotherms – Mixtures of CH4 and C2H6 on Silicalite and CO2 and C2H6 on NaX zeolite. Langmuir. 1997, 13, 4333. 35. Wu, C-W.; Sircar, S. Comments on binary and ternary gas adsorption selectivity. Sep. Purif. Tech. 2016, 170, 453. 36. Sircar, S. Thermodynamics of adsorption from binary liquid mixtures on heterogeneous adsorbents. J. Chem. Soc. Faraday Trans. I. 1986, 82, 831. 37. Sircar, S.; Hufton, J. R. Why does linear driving force model for adsorption kinetics work? Adsorption, 2000, 6, 13. 42 ACS Paragon Plus Environment

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110th Anniversary: Comments on Heterogeneity of Practical Adsorbents

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