6938
Ind. Eng. Chem. Res. 2003, 42, 6938-6948
Consistency of Energy and Material Balances for Bidisperse Particles in Fixed-Bed Adsorption and Related Applications Krista S. Walton and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
This paper develops material and energy balances for fixed-bed adsorption of a multicomponent nonideal gas in bidisperse particles by defining enthalpies from thermodynamic paths chosen such that the reference states are an ideal gas for the fluid-phase and adsorbate-free solid for the adsorbent. The balances are developed in detail for a three-domain system: nanoporous domains, macropore space, and interstitial fluid. It is also shown that material and energy balances may be easily developed from the same enthalpy definitions for use in batch, stirred tank, or moving-bed adsorption systems. Material balances are developed first and are subsequently used to simplify the energy balances such that terms involving reference states cancel out of the equations. Simplifying assumptions are applied to the final equation set to obtain equations that resemble, but are somewhat different from, frequently used forms. A major difference, resulting from the use of thermodynamic paths, is often found in the treatment of heat capacities and heats of adsorption. Introduction Multicomponent adsorption processes are commonly used for gas separation and purification and, with advances in adsorbent materials, are becoming viable options for gas storage applications. The design of fixedbed systems requires knowledge of adsorption equilibria and mass-transfer rates in adsorbents to determine the best mode of operation, e.g., pressure-swing adsorption (PSA) or temperature-swing adsorption (TSA). Regardless of the process, parameters such as breakthrough times, periodic (cyclic steady) states, and temperature and pressure setpoints must be determined to achieve the best system performance. Even batch systems such as gas storage cylinders containing adsorbent will reach a cyclic state at which the amount discharged equals the amount charged. Attempting to examine multicomponent adsorption processes thoroughly by experimental means is often unrealistic. However, an appropriate mathematical model that has been validated by experimental means can be an invaluable tool for exploring many different combinations of operating conditions. An adsorption process consists of simultaneous heat and mass transfer, requiring a model with partial differential equations coupled by adsorption equilibria. There are many assumptions and simplifications that can be made to facilitate the solution of such equations, but most remain too complicated to solve analytically and thus require numerical solutions. Because adsorption is accompanied by the release of energy, an adsorbent pellet will undergo a temperature change that may have an impact on the adsorption capacity and, hence, on the overall process performance. The significance of nonisothermal effects in a particle during adsorption has been examined by many authors. For example, Chihara et al.1 studied the uniform * To whom correspondence should be addressed. Tel.: (615) 322-2441. Fax: (615) 343-7951. E-mail: m.douglas.levan@ vanderbilt.edu.
temperature change of a particle during adsorption. The case of temperature gradients in a single particle has been considered by Brunovska´ et al.,2 Lee and Ruthven,3 Ilavsky´ et al.,4 and Sun and Meunier.5 The latter concluded that temperature gradients within adsorbent particles can be significant for systems in which mass diffusivity and thermal diffusivity are of the same order of magnitude. All of these studies considered only singlecomponent ideal gas adsorption on monodisperse particles. The first model taking into account both macropore and micropore diffusion was developed by Ruckenstein et al.6 Sun and Meunier7 expanded the bidisperse model for nonisothermal adsorption in which there are both mass and heat fluxes within the macropore, but the temperature within the micropore is uniform. There have been numerous studies on the mathematical description of adsorption processes. PSA and TSA have been commercialized for many years, and reviews on various operations can be found in work by Ruthven,8 Yang,9 and Suzuki.10 In other contributions, Mahle et al.11 modeled a nonisothermal PSA cycle, in which there was no temperature variation between the fluid and particles, and showed experimentally that heat effects can be significant even for a weakly adsorbed gas. Davis and LeVan12 presented results of an experimental and theoretical study of TSA cycles in which material and energy balances were developed to account for differences between bulk fluid temperature and stationaryphase temperature. Serbezov and Sotirchos13 developed a mathematical model for multicomponent, nonisothermal PSA in which monodisperse particles possessed a temperature gradient separate from that of the bulk fluid phase. Pentchev et al.14 presented a model for ideal gas adsorption in a fixed bed of bidisperse particles but did not consider multicomponent diffusion and assumed microparticles to be in thermal equilibrium with the macroparticles. Throughout the extensive literature on nonisothermal adsorption, the energy balance equations are rarely shown to be developed, in a thermodynamic sense, using
10.1021/ie0303855 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/20/2003
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6939
Figure 1. (a) Packed bed with interstitial fluid space with void fraction . (b) Particle with nanoporous domains connected by macropore space with porosity χ.
appropriate paths and reference states to define enthalpy and internal energy of the fluid and stationary phases. In this work, we develop the general material and energy balances that are applicable for multicomponent fixed-bed processes, giving special attention to the thermodynamic paths and the form of the flux terms. We expand the treatment given by LeVan et al.15 to include nonideal gas adsorption in nonisothermal bidisperse particles. The equations we develop resemble, but are different from, many of the commonly used forms, and we show what assumptions must be made to simplify these equations to the common forms. In the following sections, we show a detailed development of material and energy balances for fixed-bed processes, but the thermodynamic treatment of the fluid and adsorbed phases may be easily applied to other adsorption systems such as batch or stirred tank processes and moving-bed processes. We will discuss this further in the final section of the paper. Theory We consider a multicomponent, nonisothermal adsorption process occurring in a fixed bed of adsorbent particles with a bidisperse pore structure (see Figure 1). Each particle is assumed to consist of nanoporous domains (microparticles) interconnected by macropore space. The nanoporous domains are such that any molecule inside them is under the influence of the adsorbent walls and is, by definition, physically adsorbed. This is consistent with the IUPAC standard, which states that the entire space within the nanoporous domain (i.e., within “micropores”, defined as having a diameter or slit width of 2 nm or less) is considered to be adsorption space. Thus, adsorption is considered to occur at the outer surface of the nanoporous domains, with adsorbed molecules then being able to diffuse into the nanopores. Adsorption is also allowed to occur on macropore walls. With no other heat source, any temperature variation that occurs within the bed or the bulk fluid phase will be caused by the heat of adsorption that is released from the outer surface of the nanoporous domains and macropore walls and by additional energy released as the adsorbate diffuses into the nanoporous domains, if the heat of adsorption varies with adsorbate loading.
The macroporous structure will provide the pathways for pore fluid to diffuse to the surface of the nanoporous domains. Therefore, there will be a concentration gradient in the fluid phase within the macropore space. The development of relations for three domains allows more control over property variations. The nanoporous domains can have a different temperature from the macropore space, which can also differ from the interstitial fluid. The interstitial fluid, which is the bulk fluid flowing through the fixed bed, will undergo material loss and energy gain (or material gain and energy loss for desorption) as molecules diffuse into the macropore space and adsorb on the pore walls, on the outer surface, and in the interior of the nanoporous domain. Accordingly, the material and energy balances for the interstitial fluid will include contributions from the macroparticle. The use of reference states in defining enthalpy and internal energy from thermodynamic paths will be described in detail in a later section. The reference state of the fluid is considered to be an ideal gas at temperature Tref and pressure Pref. The reference state for the particles is adsorbate-free solid at Tref. We develop material balances first and use them to simplify the energy balances. Material Balances The material balances here will be similar to those of Ruckenstein et al.,6 but to maintain generality, we will not assume any particular geometry for the particles. For nanoporous domains, all molecules will be adsorbed, by definition. Therefore, the material balance will be made up of adsorbed-phase accumulation only, a with a molar flux of adsorbed molecules, Jni . The material balance on the nanoporous domains for component i is
Fs
∂nni a ) -Fs∇Jni ∂t
(1)
where Fs is the skeletal particle density with units of kg/m3 of nanoporous domain. The skeletal particle density is of no real consequence to eq 1, but it does serve to manipulate the units into a particular per volume basis. Additionally, Fs is related to the particle density, Fp, by
(1 - χ)Fs ) Fp
(2)
which will appear in the material balance for the macropore space. The macropore space will undergo both fluid-phase and adsorbed-phase accumulation if adsorption is allowed on macropore walls. There will also be a diffusional flux at the surface of the nanoporous domain that is defined by averaging eq 1 over the nanoporous domain volume. Taking into account all of these effects, the material balance for the macropore space is
χ
∂nmi ∂n j ni ∂cmi g a - Fp∇Jmi (3) + Fp + Fp ) -χ∇Jmi ∂t ∂t ∂t
where χ is the porosity of the particle and Fp is the particle density with units of kg/m3 of particle volume. The first two terms on the left-hand side of the equation
6940 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
are properties of the macropore space, and the third term is eq 1 averaged over the volume of the nanoporous domain. The material balance for the interstitial fluid must account for fluid-phase accumulation and flow through the fixed bed. Because all adsorption will take place within the particles, the molar flux contribution is from the fluid gradient in the interstitial space only. The material balance for component i in the interstitial fluid space can then be written as
∂cfi ∂cˆ mi ∂nˆ mi + ∇(vcfi) + (1 - )χ + Fb + ∂t ∂t ∂t ∂nˆ ni ) -∇Jfig (4) Fb ∂t where is the bed volume voidage and Fb is the bulk density with units of kg/bed volume. The first two terms on the left-hand side of the equation are properties of the interstitial fluid, and the last three terms are simply the macropore space material balance from eq 3 averaged over the particle volume. Note that eq 4 allows for macropore adsorption in addition to adsorption in the nanoporous domains. Keeping the two adsorption terms separate makes it possible to use different adsorption isotherms for the macropore space and nanoporous domains. Additionally, because there is a significant bulk flow of the interstitial fluid, the second term in eq 4 allows for velocity gradients and concentration gradients. Often the velocity can be considered constant along the length of the bed, but velocity variations may be seen for systems containing strongly adsorbed components or deviations from plug flow. Molar Fluxes. In general, multicomponent diffusion effects should be accounted for in systems containing three or more components if any of the following are true: high concentration of components, strong thermodynamic interactions between components, greatly differing molecular weights, or one solute gradient much larger than the others.16 For true multicomponent diffusion, there are four contributions to the molar flux: ordinary (concentration) diffusion, pressure diffusion, thermal diffusion (Soret effect), and forced diffusion.17 Jaumann’s entropy-balance equation is given by18
F
DS ) -(∇‚s) + gs Dt
(5)
where s is the entropy-flux vector and gs is the rate of entropy production per unit volume. This equation provides the basis for the development of multicomponent fluxes of bulk fluids from nonequilibrium thermodynamics. This equation, along with the equations of change for mass, momentum, and energy, has been used to develop an equation for the production of entropy in a vanishingly small fixed-volume element of a multicomponent mixture. This fixed-volume element is assumed to be a system that is approaching equilibrium; thus, the differential expression for internal energy for this system is valid17 and can be written as
dU ) T dS - P dV +
Gi
∑i M
dωi
(6)
i
Using this definition, eq 5 can then be expressed in terms of the substantial derivatives of U, V, and ωi. The
equations of change for energy, momentum, and mass are used to define the substantial derivatives, and the resulting equation for entropy production can then be written as a sum of products of fluxes and forces containing the diffusional driving force, di. It has been shown that concentration, pressure, and forced diffusion contributions are contained within this driving force19 given by N-1
cRTdi ) ciRT
∑ j)1
( ) ∂ ln ai ∂yj
∇yj + (ciV h i - ωi)∇P -
T,p,y N
Fjgj ∑ j)1
Figi + ωi
(7)
The thermal diffusion term is expressed separately in the flux equation with its own diffusion coefficient. This expression shows that the true driving force for ordinary diffusion is actually the gradient of chemical potential instead of concentration, a fact that was proven experimentally by Haase and Siry,20 who studied diffusion in binary liquid mixtures near the consolute point. The two most commonly used formulations for molar flux in multicomponent diffusion are the generalized Fickian equations and the generalized Maxwell-Stefan (GMS) equations. It is outside the scope of this paper to discuss the detailed development of both of these models, but excellent reviews are available.17,19,21,23 The generalized Fickian equations are expressed in terms of the driving forces as N
Ji ) -DTi ∇ ln T + Fi
D ˜ ijdj ∑ j)1
i ) 1, 2, 3, ..., N (8)
where DTi is the multicomponent thermal diffusion coefficient and the multicomponent Fickian diffusivity, D ˜ ij, is defined by
D ˜ ij ) -
cRTRij FiFj
(9)
where Rij is a phenomenological coefficient. The diffusional driving force, di, was developed for a multicomponent fluid phase, in which it was shown that diffusion is driven by gradients in the chemical potential. Because molecules in the nanoporous domain never escape the influence of the pore walls, diffusion in this domain will be driven by the chemical potential gradient of the adsorbed species. Thus, for the adsorbed phase, we can write the Fickian equations for the adsorbedphase molar flux as21 N
Daij∇ni ∑ j)1
Jai ) -
i ) 1, 2, 3, ..., N
(10)
where, using the adsorption isotherm, the diffusion coefficient can be defined as
Daij ) LiRTni
∂ ln fi ∂nj
(11)
which is similar to the Darken22 relation, where Li is a temperature-dependent mobility coefficient. Diffusion of this type is known as micropore or surface diffusion.
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6941
The GMS formulation for mass transfer has been shown to provide a more appropriate description of multicomponent diffusion in both bulk fluids and nanoporous domains.23,24 It has been shown that the GMS equation for the fluid-phase molar flux in terms of the diffusional driving force is expressed by
cRT
( ) Jk
Cik ∑ F k*i
k
-
Ji Fi
( )
Cik ∑ k*i
) di - cRT
DTk Fk
-
DTi Fi
(∇ ln T) (12) The GMS equations can be written for the surface diffusion in an (n + 1)-component system of n sorbed species and the vacancy, n + 1, as25
θi
∇T,Πµi ) RT
n+1θ
∑ j)1
a a iJj - θjJi
(13)
where i ) 1, 2, ... n + 1 and the chemical potential gradient is at constant temperature and spreading pressure. Now that molar fluxes have been defined, it should be noted that, in addition to the material balances, additional equations connecting the balances from each domain are needed for solving the system of equations. These are of two forms: continuity of flux (or a relation between flux into a phase and accumulation in that phase) and continuity of concentration (or adsorption equilibrium). The volume averages appearing in eqs 3 and 4 can be expressed as gradients at the boundary of the respective domains. For example, if we consider a nanoporous domain in which gradients occur only in the radial direction, then the adsorbed-phase accumulation averaged over the volume of the nanoporous domain can be written as
|
(
dh )
∂h ∫TT (∂T ′)P F
ref
dT ′ +
ref
∂h ∫PP (∂P′ )T F
ref
dP′
(16)
F
The last integral in eq 16 is the enthalpy departure function and will be exactly zero for an ideal gas. This function is also known as the residual enthalpy, HR, and is defined as the difference between the actual and ideal gas enthalpy at the same temperature and pressure.26 Evaluation of the departure function for a real gas requires an equation of state or correlation such as the virial equation, Peng-Robinson equation, or LeeKessler correlation and the use of pseudocritical properties calculated from appropriate mixing rules.27 Thus, for a mixture eq 16 becomes
hyF ) hyref +
∫TT C°py(T ′) dT ′ + HTR F
F
ref
(17)
where
hyref )
∑i yihgiref
(18)
C°py )
∑i yiC°pgi
(19)
The internal energy of the gas phase is defined by
∂cˆ mi ∂nˆ mi ∂nˆ ni + Fp + Fp ) ∂t ∂t ∂t ∂nmi ∂ymi G a g FpDmi + χDmi cm Rm ∂rm rm)Rm ∂rm
[ ( )
yF
(14)
rn)Rn
in which G ) 2 and 3 respectively for cylindrical and spherical coordinates, and Rn is the radius of the nanoporous domain. Similarly, adsorbed-phase and fluid-phase accumulation averaged over the volume of the macropore space can be expressed as
χ
∫hh
yref
nsi Dij
a Dni ∂n j ni ∂nni ) FsG Fs ∂t Rn ∂rn
internal energy of the gas and stationary phases. These properties are generally not absolute measurable quantities, and their values are determined relative to a reference state that is normally considered to be an ideal gas at temperature Tref and pressure Pref. It will be shown that numerical values used for the reference state should have no effect on the energy balance if it is consistent with the material balance equations. Because enthalpy and internal energy are state variables and therefore independent of the thermodynamic path that is used to evaluate them, we choose a path that first heats the gas mixture isobarically from Tref to TF and then isothermally changes the pressure from Pref to PF. This is given by
) ]
(15)
rm)Rm
in which Rm is the radius of the macroporous domain, which is the boundary between the macropore space and interstitial fluid. When appropriate boundary conditions are applied, the equation set is now complete for modeling an isothermal adsorption process. The following sections provide a detailed development of energy balances for nonisothermal adsorption. The expressions for the volume averages of the accumulation of energy contained therein will be analogous to those written above for material accumulation. Thermodynamic Paths Before energy balances for fixed-bed processes are developed, it is necessary to define the enthalpy and
hF uF ≡ hF - PV
(20)
The reference state for the adsorbent is adsorbatefree solid at Tref. Along all paths the temperature and pressure of the solid adsorbent will be changed, with the pressure change contributing negligibly to the change in the internal energy of the solid. Thus, for any path we can write for the internal energy of the stationary phase (solid plus adsorbate)
us ) usol + uAn
(21)
where n is the total loading and
usol ) usol ref +
∫TT Csol(T ′) dT ′ s
ref
hA uA ≡ hA - PV
(22) (23)
Furthermore, when it is recognized that the PV h A term in eq 23 can be neglected because of the insignificant molar volume of the adsorbate, eq 21 can be written as
us ) usol + hAn
(24)
The enthalpy of the adsorbed phase, hA, can be evaluated using different thermodynamic paths. Two
6942 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
Sircar28 has developed the thermodynamics of multicomponent adsorption using excess properties. This method defines thermodynamic properties of the entire adsorption system, resulting in an overall heat capacity defined as
CA )
( ) ∂h° ∂T
)
nm
n°Cg + Csol -
∂ ( ∂T
∫0n qST dnm)P + qST m
( ) ∂nm ∂T
P
(32)
We may obtain a similar expression, in terms of adsorbed amounts instead of excess amounts, by using path 1 to evaluate the adsorbed phase and defining an overall enthalpy, h°, as
h° ) hs + hg
Figure 2. Thermodynamic path for defining stationary-phase enthalpy change. Path 1: solid line. Path 2: dashed line.
(33)
The overall heat capacity is then found to be convenient paths are shown in Figure 2. Along path 1, the temperature and pressure of the gas phase are changed from the ideal gas reference state to the conditions of the stationary phase and are then adsorbed at constant pressure and temperature. Fluid-phase heat capacities are used to define step AB in Figure 2, and the integrated isosteric heat of adsorption, Λ, released during step CD is a function of the stationary-phase temperature, Ts, and is also a function of the loading of each component. The enthalpy of the adsorbed phase based on path 1 can then be expressed as
CA ) n°C°pg + Csol -
(25)
where hxF is the enthalpy of the adsorbable components while still in the gas phase but at the desired temperature and pressure. Thus, hxF is defined in a way similar to that of eq 16 by
hxF ) hxref +
∫TT C°px(T ′) dT ′ + HTR s
ref
S
(26)
hxref )
∑i xihgiref
(27)
C°px )
∑i xiC°pgi
(28)
where xi is the mole fraction of component i in the adsorbed phase. It is important to note that the adsorbed-phase heat capacity, C°px, is the fluid-phase heat capacity based on adsorbed-phase concentrations because of the thermodynamic path. To simplify the notation, we let the integrals of the heat capacity from eqs 17 and 26 respectively be defined by
∆hy ≡
∫TT C°py(T ′) dT ′
(29)
∆hx ≡
∫TT C°px(T ′) dT ′
(30)
F
ref
s
ref
and for component i at temperature T
∆hi ≡
∫TT C°pgi(T ′) dT ′ ref
(31)
∫0n qST T dna)n a
(34)
a
It can then be shown from the total differential of f(na,T), where f is the integral appearing in the last term of eq 34, that
-
∂ ( ∂T
∫0n qST T dna)n
)
a
1 hA ) hxF - Λs n
∂ ( ∂T
a
∂ ( ∂T
ST ∫0n qST T dna)P + qT a
( ) ∂na ∂T
P
(35)
For a pure-component or multicomponent system, the isosteric heat of adsorption pertaining to adsorption at constant pressure and temperature is given by
λi ) ZRT 2
(
)
∂ ln(Pyi) ∂T
ni
(36)
where λi is the isosteric heat of adsorption of component i at component loading ni and Z is the compressibility factor for the mixture. In many adsorption calculations, the isosteric heat of adsorption is often considered to be independent of loading; this assumption is often valid for the adsorption of pure gases on energetically homogeneous or weakly heterogeneous adsorbents over limited temperature ranges. It has been shown29 that the isosteric heat of adsorption can increase with loading for a nonpolar gas on a polar adsorbent, such as NaX zeolite, because of lateral interactions between the adsorbed molecules. However, it is well-known that the adsorption heat usually decreases with increasing loading and temperature for an energetically heterogeneous adsorbent. The form of Λ used here is a function of temperature and loading and thus accounts for the variation in the isosteric heat released as adsorbed molecules diffuse to sites of higher energy. The calculation of multicomponent heats of adsorption that depend on loadings is often difficult, and heat of adsorption data for mixtures are almost nonexistent, prompting most researchers to use pure-component heats of adsorption at the conditions of the mixture. There are several paths that may be used for calculating Λ of a multicomponent mixture. For example, the integrated heat of adsorption for a ternary
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6943
mixture can be calculated by
Λ)
∫0n λ1(n′1,0,0,T) dn′1 + ∫0n λ2(n1,n′2,0,T) dn′2 + ∫0n λ2(n1,n2,n′3,T) dn′3 (37) 1
2
heat capacities to be equal to fluid-phase heat capacities to maintain thermodynamic consistency and is strictly an approximation.
3
in which the first component is adsorbed first and held constant while the second is adsorbed and then both are held constant while the third component is added. This equation can be easily extended to multicomponent mixtures and can be expressed more generally as
Λ)
∑i ∫0 λi(nj ′j,Ts) dn′i ni
∫TT
s
ref
( ) ∂hA ∂T ′
dT ′
n
(39)
in which the isosteric heat will only vary with loading because it is evaluated at the reference temperature. The term within the integral on the right-hand side of eq 39 can be written as
|
∂hA ∂T ′
) n
∂hA ∂T ′
|
+
P′
|
∂hA ∂P′
T′
∂P′ ∂T ′
|
n
(40)
Because this path changes the temperature of the adsorbed phase, the first term on the right-hand side of eq 40 is the heat capacity of the adsorbed phase
∂hA ∂T ′
|
P′
) Cpa )
∑i xiCpai
(41)
and will generally not be equal to the fluid-phase heat capacity. The last term in eq 40 can be neglected because the adsorbed-phase volume is very small, i.e.
( ) ∂hA ∂P′
T′
) va - T
( ) ∂va ∂T
P
≈0
(42)
Thus, eq 39 becomes
1 hA ) hxref - Λref + n
∫T
Ts ref
Having defined the enthalpy and internal energy of the fluid and stationary phases, we can now write the general energy balances for the nanoporous domain, macropore space, and interstitial fluid respectively in the forms
(38)
where n j j is the vector of loadings, n′j ) 0 for j > i, and n′j ) nj for j < i. Evaluating hA along path 2 in Figure 2 requires adsorbing the components at reference state conditions and then heating and pressurizing to the final conditions at constant loading according to the adsorption isostere. Therefore, hA is defined as
1 hA ) hxref - Λref + n
Energy Balances
Cpa dT ′
Fs
χ
∂un ) -Fs∇qan ∂t
(44)
∂ums ∂u jn ∂(cmumg) + Fp + Fp ) -χ∇qgm - Fp∇qam (45) ∂t ∂t ∂t ∂(cˆ) ∂(cfuf) ∂uˆ ms mumg + ∇(vcfuf) + (1 - )χ + Fb + ∂t ∂t ∂t ∂uˆ n Fb ) -∇qgf (46) ∂t
In writing these equations, contributions of kinetic and potential energies, viscous dissipation of energy, and radiation have been ignored because these terms are usually of minor importance. Additionally, note that it is not necessary to include a source term for the heat of adsorption in the general energy equations. The form of the equation used here will account for the heat source implicitly from the internal energy change.17 As shown in the material balances for the three domains, it can also be seen here that the energy balance for the macropore space contains the adsorbed-phase energy accumulation for the nanoporous domain averaged over that volume. Likewise, the interstitial fluid energy balance contains the macropore space energy accumulation averaged over the particle volume. Energy Fluxes. Because we are considering multicomponent systems, there will be two contributions to the energy fluxes (qan, qgm, qam, and qam) appearing on the right-hand sides of eqs 44-46: conduction and energy flux due to the interdiffusion of various species in the mixture. Thus, we have
q ) qc + qd
(47)
This is similar to the development of fluxes in terms of transport properties17 in that
(43)
Path 1 is usually preferred over path 2 because calculations can be carried out more accurately since changes in fluid-phase properties during heating and pressurization are generally known. However, it is also necessary to accurately estimate adsorption equilibria over a wide range of temperatures to evaluate the isosteric heat of adsorption. Path 2 requires the evaluation of the isosteric heat of adsorption at only the reference temperature but also requires knowledge of adsorbed-phase heat capacities and their temperature dependencies, which are not well-known. On the basis of thermodynamic paths, making the assumption for an ideal gas (HR ) 0) that the isosteric heat of adsorption is independent of temperature forces adsorbed-phase
n
qd )
∑i hh iJi
(48)
where Ji is the molar flux of component i and h h i is the partial molar enthalpy of component i. Partial molar enthalpies of the fluid and adsorbed phases respectively are defined by
h h gi ≡ h h ai ≡
( ) ( ) ∂(chF) ∂ci
P,TF,cj
∂(nhA) ∂ni
P,TS,nj
(49)
(50)
6944 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
Using the definitions for enthalpy in eqs 16 and 25, we obtain
h h gi ) hgiref + ∆hFi + HTRF + c
h h ai ) hgiref + ∆hsi + HTRs + n
χ
( ) ∂HTRF ∂ci
(
∂(cm∆hym)
∂ni
-
P,Ts,nj
∂Λs ∂ni
P,Ts,nj
(52)
φFi ≡
θsi ≡
( ) ( )
ζsi ≡
∇h h ai
) C°pgi∇Ts +
(54)
TS,P,nj
( ) ∂Λs ∂ni
(55)
(
TS,P,nj
∇HTRF
∇HTRs
+ ∇(cφFi)
+ ∇(nθsi) - ∇ζsi
(57)
Fb Csoln
(58)
After the substitutions are made and the reference terms are canceled, the energy balances for the nanoporous domain, macropore space, and interstitial column fluid become respectively
∂t
[Csoln + nnC°pxn] + nn
∂t
-
)
∂Λn
∂t
+
∂t
∂t
∂T ˆm
(
∂t
-
∂t
∂t
)
+
m
∂(cfHTRF)
+
(
ˆ) ∂(n m∆hxm
∂t
∂t
+
∂(n ˆ) n∆hxn
∑i
+
∂t
R ˆ ∂(n mHTm)
n
Tn
)
R ∂(cˆ mHTm)
-
∂t ˆ ∂(n HR ) +
(60)
+ ∇(vcfHTRf) -
∂t
∂t +
a ∇hˆ mia) ∑i Jmi
∂(cˆ) m∆hym
)
∂t -
)
∂Λ ˆm
∂Λ ˆn
∂t ∂t ∂t (∆hfi + HRTf + cfφfi)∇Jfig -
+
+
)
∑i Jfig ∇hh fig (61)
Alternative Forms. The equations in the preceding section were developed using enthalpy definitions from path 1 in Figure 2. Because enthalpy is a state function, an equivalent equation set can be derived using path 2. If adsorbed-phase enthalpy, hA, from eq 39 is substituted into the energy balance equations in eqs 44-46, then the resulting general equations will contain both fluid- and adsorbed-phase heat capacities. In the equation development for either path, the reference temperature always appears in terms that also include heat capacities. This is not a problem for path 1 equations because only fluid-phase heat capacities appear in the final energy balances, allowing cancellation of all reference state terms. However, path 2 equations contain both fluid- and adsorbed-phase heat capacities, and because Cpai * C°pgi, the reference temperature cannot cancel out of the equations. This is an inconvenience because the reference state then must be defined numerically if the energy balances are to be evaluated. If the expression for hA developed from path 1 is equated to that obtained from path 2, this gives
1
∑∫ λi(n′j,Ts) dn′j + ∫T n i 0
Ts
ni
ref
(59)
)
∂Λ hn
+
m
+
∂T ˆn
)
∂Λm
g ∇hˆ mig) - Fp(∑[∆hmi + HTR ∑i Jmi i
∂t kf∇2Tf -
-
)
∑i Jnia ∇hh nia - Fs∑i [nnθni - ζni]∇Jnia
Fskn∇2Tn - Fs
j nHTRn) ∂(n
)
(56)
Final General Forms
(
∂(n j n∆hxn)
- ∇(vP) + (1 - )χ
(
∂t
Fb Csolm
By substituting the definitions for enthalpy and internal energy developed from path 1 into the energy balances, we recognize that all reference states appearing in the energy balances multiply terms that sum to zero in the material balances. Therefore, all reference states will cancel out of the energy balances, and it is not necessary to assign them numerical values.
∂HTRn
∂t
+ ∇(vcf∆hyf) +
∂t
∂P
qc ) -k∇T
Fs
+
∂t
-
∑i [∆hmi + HTR
∂(cf∆hyf)
The conduction term can be represented simply as Fourier’s law
∂Tn
(
∂T hn
+
∂(nmHTRm)
a + nmθmi - ζmi]∇Jmi
∂HTRs
) C°pgi∇TF +
∂t
TF,P,cj
∂ni
∂(nm∆hxm)
+
g cmφmi]∇Jmi +
(53)
then the gradients of the partial molar enthalpies can be expressed as
∇h h gi
∂Tm
+
(Fpkm + χkmg)∇2Tm - χ(
∂HTRF ∂ci
∂t
Fp Csoln
If we let
+
∂t
Fp Csolm
( ) ( ) ∂HTRs
(
(51)
P,TF,cj
)
∂(cmHTRm)
-
1
C°px(T ′) dT ′ + HTRs )
n T λ (n′,T ) dn′i + ∫T Cpa(T ′) dT ′ ∫ ∑ 0 i j ref n i i
s
ref
(62)
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6945
This relation can then be used to replace the adsorbedphase heat capacity terms and isosteric heat of adsorption at the reference state in the energy balance. This allows all of the reference states to cancel, resulting in equations that are identical with those shown in the final general forms. There are a few important points to note about eq 62. As mentioned in the section on thermodynamic paths, if the isosteric heat is assumed to be independent of temperature for an ideal gas (HR ) 0), then the heat capacity of the adsorbed phase must be equal to that of the fluid phase. However, for an ideal gas system in which λ is temperature-dependent, the adsorbed-phase heat capacity cannot be the same as the fluid-phase heat capacity to maintain thermodynamic consistency. Because fluid-phase heat capacities are generally easier to measure relative to adsorbed-phase heat capacities, eq 62 provides a convenient method for calculating Cpa if isosteric heats are known.
fluid at all times. Therefore, the contribution to the energy flux due to interdiffusion of various species in this domain comes only from changes in the fluid-phase enthalpy. If the particles were monodisperse, then molecules within the particle domain would exist in both the fluid and adsorbed phases. For this case, the righthand sides of the equations would contain flux terms for the fluid and adsorbed phases. The material and energy balances for the interstitial fluid are
′
(
C°py
∂Tm ∂t
(
∂ci ∂n j ni g + Fp ) -χ∇Jmi ∂t ∂t
+ Fp Csoln
∂T hm ∂t
+ C°pxn
∂Tm
(63)
∑i λi
-
∂t
∂n ji ∂t
)
g C°pgi ∑i Jmi
χkmg∇2Tm - χ∇Tm
) (64)
Because these equations are written for a bidisperse particle with no adsorption in the macropore space, the actual adsorption is a property of the nanoporous domains. Therefore, the rate of adsorption in the equation is also a property of that domain and will be a volume-averaged value. As a result, the flux term appearing on the right-hand side of eq 64 is for the fluid phase because adsorption does not occur on the macropore walls; the term is simply the means by which adsorbable components reach the surface of the nanoporous domain. Because there is no stationary phase present within the macropore space, the particle is assumed to be in thermal equilibrium with the pore
∂P ∂t
∂T ˆm ∂t
(65)
)
- ∇(vP) +
(1 - )χC°pyc
Fb Csoln
The general equations presented in the previous section provide the framework for modeling fixed-bed adsorption of nonideal gas mixtures. The resulting model is a set of three energy balances coupled by adsorption equilibria with the corresponding material balances. Many different forms of these equations have been used by previous researchers. In this section, we show what simplifying assumptions must be made to the general energy balance equations to recover the forms resembling those used in the literature. Form 1. This example considers multicomponent adsorption in bidisperse particles. The following assumptions are applied: (1) it is an ideal gas mixture; (2) adsorption occurs only in the nanoporous domain (nni ) ni and nmi ) 0); (3) interstitial fluid concentrations are in local equilibrium with the macropore space fluid (cfi ) cmi ) ci); (4) Tn is uniform; (5) Tm is nonuniform; (6) Tn ) Tm * Tf; (7) C°pgi and λi are constants. For the macropore space, the material and energy balances become respectively
χC°pyc
∂t
+ ∇(vcTf) -
(
Simplified Forms
χ
∂(cTf)
∂ci ∂nˆ i + ∇(vci) + Fb ) -∇Jfig ∂t ∂t
∂Tm
∂t ∂(nT ˆ) m
+
)
∂nˆ i
∑i λi ∂t ) ∂t (kf∇2Tf - ∑C°pgi∇(TfJfig )) i
+ C°px
-
(66)
As a consequence of assumption (7) and the use of thermodynamic paths, the adsorbed-phase heat capacity for component i would be equal to the fluid-phase heat capacity of that component (note eq 62). However, it is important to note that C°px * C°py because they are based on their respective concentrations in the fluid or adsorbed phase (see eqs 19 and 28). This case would apply for a system in which uptake is controlled by adsorbate diffusion into the interior of the nanoporous domain; therefore, adsorption in the macropore space is considered to be negligible compared to adsorption in the nanoporous domain. The assumption of uniform temperature in the nanoporous domain is valid for systems in which the Lewis number, the ratio of thermal to mass diffusivity, is larger than 10.5 The difference between the particle temperature and bulk fluid temperature must be accounted for if the heat of adsorption is high and if the adsorption isotherms are steep.2 Form 2. For this case, we will further simplify form 1 by making the following additional assumptions: (8) there are no temperature gradients within the particle; (9) the rate equation for energy transfer at the interface of the particle and bulk fluid can be written using a convective heat-transfer coefficient. This leads to
χ
χC°pyc
∂Tm ∂t
[
∂n ji ∂ci g + Fp ) -χ∇Jmi ∂t ∂t
+ Fp Csolm
∂T hm ∂t
+ C°pxn
∂Tm ∂t
-
(67)
∑i λi
∂n ji ∂t
]
)
-hmam(Tm - Tf) (68) for the particles and
′
∂nˆ i ∂ci + ∇(vci) + Fb ) -∇Jfig ∂t ∂t
(69)
6946 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
(
C°py
∂(cTf)
+ ∇(vcTf) -
∂P
)
- ∇(vP) -
∂t ∂t (1 - )hmam(Tm - Tf) ) (kf∇2Tf -
′C°pyc
∑i C°pgi∇(TfJfig ))
∂T
+
∂t
χ
χC°pyc
∂Tm ∂t
[
∂ci ∂n ji g + Fp ) -χ∇Jmi ∂t ∂t
+ Fp Csolm
∂T hm
-
∂t
∑i λi
∂n ji
(71)
]
) ∂t -hmam(Tm - Tf) (72)
and for the interstitial fluid
′
(
C°py
∂(vcT)
-
∂z
∂P
(
(73)
)
∂(cTf) ∂(vcTf) ∂P ∂(vP) + ∂t ∂z ∂t ∂z (1 - )hmam(Tm - Tf) ) -hwaw(Tf - Tw) (74)
Equations 72 and 74 are very similar to the forms frequently found in the literature for modeling fixed-bed adsorption processes with differences between particle and bulk fluid temperature. The significant difference is the addition of the fluid-phase contribution to the temperature change in the particle found in the first term of eq 74. This term is usually small. Assuming plug flow through the bed with no mass dispersion is often valid if the system is operating at a Reynolds number that is greater than 100.8 For larger Re, there will be adequate mixing such that each particle is surrounded by a uniform boundary concentration. Making this assumption for a system that is actually operating in the low Peclet number regime will result in predicted breakthrough times that are slower than experimental results. Form 4. Finally, when the assumptions that (13) the particle is in thermal equilibrium with the interstitial fluid and (14) the adsorption process takes place under adiabatic conditions are applied to form 3, the energy balance for the particle is no longer needed, and the energy balance for the interstitial fluid becomes
)
∂(vP)
∂t
∂z ∂T ∂t
-
+
∑i λi
)
∂nˆ i ∂t
) 0 (75)
which is a common expression used for modeling a nonisothermal, adiabatic PSA process. This form will be valid for the case of a weakly adsorbed species11 in large adsorption columns. An adiabatic model typically will not work well for laboratory-scale columns because small columns usually have large heat transfer through the wall. If not valid, the adiabatic assumption may incorrectly predict breakthrough times, as well as the shape of the breakthrough curve. Further Applications The thermodynamics of the adsorbed and fluid phases developed in the section on thermodynamic paths may be easily applied to adsorption processes other than fixed-bed systems. For batch or stirred tank systems with bidisperse particles, the material and energy balances for the nanoporous domain and macropore space will be the same as eqs 1, 3, 44, and 45. The material balance for the fluid phase in the tank is
Msol
[
Msol
]
∂(Vfci) ∂nˆ ni ∂nˆ mi (1 - )χ ∂cˆ mi + + + ) ∂t ∂t Fb ∂t ∂t Fincini - Foutci (76)
and the energy balance is
∂ci ∂(vci) ∂nˆ i + + Fb )0 ∂t ∂z ∂t
-
Fb Csolm
(70) for the interstitial fluid. This form would be valid for cases in which the thermal conductivity of the pellet is high, and the heattransfer resistance is due to that of the stagnant film surrounding the pellet.21 Considering this type of system to be isothermal would result in predictions of breakthrough times that are typically somewhat later than experimental results. Form 3. To simplify further, we assume the following: (10) adsorbed-phase heat capacities are negligible; (11) there is plug flow through the bed and no mass dispersion in the interstitial fluid (∇Jfig ) 0); (12) the interstitial fluid has a uniform radial temperature Tf, and the heat loss at the wall is given by hwaw(Tf - Tw). Now the form 2 equations become for the particle
(
[
]
∂(Vfcuf) ∂uˆ n ∂uˆ ms (1 - )χ ∂(cˆ miumg) + + + ) ∂t ∂t Fb ∂t ∂t Fincinuf - Foutcuf (77)
Material and energy balances may also be developed for moving-bed systems by including appropriate velocity terms in the particle and fluid equations and for applications with chemical reactions such as adsorptive reactors. Conclusions We have developed general material and energy balances for nonideal multicomponent adsorption processes for three domains: nanoporous domains, macropore space, and interstitial fluid. The energy balances were derived by defining the enthalpy and internal energy of the fluid and stationary phases using thermodynamic paths. The energy balance equations allow for temperature differences among the domains, as well as temperature gradients within the domains. While developed in detail for fixed-bed processes, the equations may also be extended for use in adsorption systems such as batch or stirred tank systems and moving-bed systems. Two important features to note about the definition for the adsorbed-phase enthalpy are the inclusion of the enthalpy departure function to account for nonideality and the form of the isosteric heat of adsorption that can vary with temperature and loading. Heat capacities can also be a function of temperature and are defined as fluidphase heat capacities for path 1 in Figure 2, even for the adsorbed phase. All reference state terms cancel out
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6947
of the equations when developed using path 1. Path 2 uses adsorbed-phase heat capacities, and because Cpai * C°pgi, the reference temperature will not cancel out of the equations. Thus, path 1 is used to develop energy balances that do not require actual values for the reference state. These equations resemble, but are different from, the forms that have been used previously for modeling PSA and TSA systems; the principal difference is in the treatment of heat capacities and heats of adsorption, which arise from the use of thermodynamic paths. The equations provide a useful starting point for describing fixed-bed adsorption processes.
′ ) total bed voidage, (1 - )χ θi ) fractional surface occupancy of species i λi ) isosteric heat of adsorption, J/mol Λ ) integrated multicomponent isosteric heat of adsorption, J/kg µi ) chemical potential of species i, J/mol Fb ) bulk density of packing, kg/m3 Fi ) mass of species i per unit volume of mixture, kg/m3 Fp ) particle density, kg/m3, of a macroparticle Fs ) skeletal particle density, kg/m3, of a nanoporous domain χ ) particle porosity (intraparticle void fraction) ωi ) mass fraction
Notation
Subscripts
ai ) activity of species i am ) heat-transfer area per unit particle volume c ) fluid-phase concentration, mol/m3 C ˜ ik ) multicomponent inverse diffusivity19 Cpa ) heat capacity of the adsorbed phase, J/(mol K) C°pg ) heat capacity of an ideal gas, J/(mol K) Csol ) heat capacity of a solid, J/(kg K) di ) diffusional driving force, m-1 DTR ) generalized thermal diffusion coefficient, kg/(m s) D ˜ ij ) multicomponent diffusion coefficient, m2/s Dij ) GMS diffusivity for countersorption of species i and j, m2/s fi ) fugacity of component i F ) volumetric flow rate, m3/s gi ) force per unit mass acting on species i, N/kg gs ) rate of entropy production per unit volume hA ) enthalpy of the adsorbed phase at the final state, J/kg hF ) enthalpy of the fluid phase at the final state, J/mol hm ) heat-transfer coefficient for a macroparticle, J/(m2 s) h h i ) partial molar enthalpy, J/mol HR ) residual enthalpy, J/mol Ji ) molar flux of component i, mol/(m3 s) k ) thermal conductivity, W/(m K) Li ) mobility coefficient M ) molecular weight, g/mol Msol ) mass of the adsorbent, kg n ) loading, mol/kg nsi ) surface concentration of species25 i, mol/m2 nm ) surface excess, mol/kg N ) total number of components in the mixture P ) pressure, Pa q ) energy flux, W/m2 R ) ideal gas constant, J/(mol K) R ) radius of the nanoporous domain or macropore space respectively in eqs 12 and 13 s ) entropy-flux vector S ˆ ) entropy per unit mass T ) temperature, K t ) time, s U ˆ ) internal energy per unit mass u ) internal energy, J/mol for the fluid phase, J/kg for the stationary phase V h ) molar volume, m3/mol V ˆ ) volume per unit mass Vf ) extraparticle volume, m3 v ) interstitial velocity, m/s xi ) adsorbed-phase mole fraction yi ) fluid-phase mole fraction Z ) compressibilty factor
A ) final state of the adsorbed phase in a thermodynamic path a ) adsorbed-phase property f ) interstitial fluid domain F ) final state of the fluid phase in a thermodynamic path g ) fluid-phase property m ) macropore space (macroparticle) domain n ) nanoporous domain ref ) reference state property s ) final state of the stationary phase in a thermodynamic path sol ) solid property x ) fluid-phase property based on adsorbed-phase composition y ) fluid-phase property based on fluid-phase composition
Greek Letters Rij ) phenomenological coefficient β ) volume expansivity, K-1 ∆h ) integral of heat capacity (see eqs 27-29) ) void fraction of packing
Superscripts a ) adsorbed-phase property c ) energy flux due to conduction d ) energy flux due to diffusion g ) fluid-phase property ° ) total property ′ ) dummy variable X h ) average over the nanoporous domain (except eqs 5 and 6) X ˆ ) average over the particle volume (except eqs 5 and 6)
Literature Cited (1) Chihara, K.; Suzuki, M.; Kawazoe, K. Effect of Heat Generation on Measurement of Adsorption Rate by Gravimetric Method. Chem. Eng. Sci. 1976, 31, 505-507. (2) Brunovska´, A.; Hlava´cek, V.; Ilavsky´, J.; Valty´ni, J. An Analysis of a Nonisothermal One-Component Sorption in a single Adsorbent Particle. Chem. Eng. Sci. 1978, 33, 1385-1391. (3) Lee, L.-K.; Ruthven, D. M. Analysis of Thermal Effects in Adsorption Rate Measurements. J. Chem. Soc., Faraday Trans. 1 1979, 75, 2407-2422. (4) Ilavsky´, J.; Brunovska´, A.; Hlava´cek, V. Experimental Observation of Temperature Gradients Occurring in a Single Zeolite Pellet. Chem. Eng. Sci. 1980, 35, 2475-2479. (5) Sun, L. M.; Meunier, F. A Detailed Model for Nonisothermal Sorption in Porous Adsorbents. Chem. Eng. Sci. 1987, 42, 15851593. (6) Ruckenstein, E.; Vaidyanathan, A. S.; Youngquist, G. R. Sorption by Solids with Bidisperse Pore Structures. Chem. Eng. Sci. 1971, 26, 1305-1318. (7) Sun, L. M.; Meunier, F. Nonisothermal Adsorption in a Bidisperse Adsorbent Pellet. Chem. Eng. Sci. 1987, 42, 2899-2907. (8) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. (9) Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Boston, 1987. (10) Suzuki, M. Adsorption Engineering; Elsevier Science: New York, 1990. (11) Mahle, J. J.; Friday, D. K.; LeVan, M. D. Pressure Swing adsorption for Air Purification. 1. Temperature Cycling and Role of Weakly Adsorbed Carrier Gas. Ind. Eng. Chem. Res. 1996, 35, 2342-2354.
6948 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 (12) Davis, M. M.; LeVan, M. D. Experiments on Optimization of Thermal Swing Adsorption. Ind. Eng. Chem. Res. 1989, 28, 778785. (13) Serbezov, A.; Sotirchos, S. V. Mathematical Modeling of Multicomponent Nonisothermal Adsorption in Sorbent Particles Under Pressure Swing Conditions. Adsorption 1998, 4, 93-111. (14) Pentchev, I.; Paev, K.; Seikova, I. Dynamics of nonisothermal adsorption in packed bed of biporous zeolites. Chem. Eng. J. 2002, 85, 245-257. (15) LeVan, M. D.; Carta, G.; Yon, C. M. Section 16: Adsorption and Ion Exchange. Perry’s Chemical Engineers’ Handbook, 7th ed.; McGraw-Hill: New York, 1997. (16) Cussler, E. L. Multicomponent Diffusion; Elsevier Scientific Publishing Co.: Amsterdam, The Netherlands, 1976. (17) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd ed.; Wiley: New York, 2002. (18) Jaumann, G. A. J. Closed system of physical and chemical differential laws. Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A 1911, 120, 385-530. (19) Curtiss, C. F.; Bird, R. B. Multicomponent Diffusion. Ind. Eng. Chem. Res. 1999, 38, 2515-2522. (20) Haase, R.; Siry, M. Diffusion in Critical Mixing Region of Binary Fluid Systems. Z. Phys. Chem. F 1968, 57, 56. (21) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (22) Darken, L. S. Diffusion, mobility and their interrelation through free energy in binary metallic systems. Trans. Am. Inst. Min., Metall. Pet. Eng. 1948, 175, 184-201. (23) Krishna, R.; Wesselingh, J. A. The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 1997, 52, 861-911.
(24) Broeke, L. J. P.; Krishna, R. Experimental Verfication of the Maxwell-Stefan Theory for Micropore Diffusion. Chem. Eng. Sci. 1995, 2507-2522. (25) Krishna, R. Multicomponent Surface Diffusion of Adsorbed Species: A Description Based on the Generalized Maxwell-Stefan Equations. Chem. Eng. Sci. 1990, 1779-1791. (26) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 5th ed.; McGrawHill: New York, 1996; pp 190-196. (27) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (28) Sircar, S. Excess Properties and Thermodynamics of Multicomponent Gas Adsorption. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527-1540. (29) Sircar, S.; Cao, D. V. Review: Heat of Adsorption. Chem. Eng. Technol. 2002, 25, 945-948. (30) Sircar, S.; Rao, M. B. Heat of Adsorption of Pure Gas and Multicomponent Gas Mixtures on Microporous Adsorbents. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J. A., Contescu, C. I., Eds.; Marcel Dekker: New York, 1999; Chapter 19,
Received for review May 5, 2003 Revised manuscript received September 26, 2003 Accepted October 1, 2003 IE0303855
