simulation and design models for adsorption process

Shdation and design models for adsorption processes

Walter J. Weber, Jr. Edward H. Smith

7be University of Michigan Ann Arbor, Mich. 481G9

Adsorption is a fundamental process for separating inorganic and organic contaminants from waters and waste waters. The most common environmental applications of this process are embodied in ion-exchange and acti- vatedsarbon systems. Although such processes are generally capable of highly effective separations, the design and operation of specific systems is frequently complicated by the composition of waters and wastes and by the intri- cate matrix of interactions commonly associated with heterogeneous systems. These complications are particularly severe in applications involving the removal of specifically targeted organic compounds from complex mixtures, a circumstance encountered with increasing frequency as water decontamination requirements become more stringent.

Adsorption system design tradition- ally has been based on information gained from pilot-plant test programs. The pilot-plant design approach ac- knowledges the system specificity of particular applications, but it does not lend itself easily to prediction of system responses to variables other than those specifically tested, nor to elucidation of design principles that can be e x t r a p lated to other applications.

Mathematical process models can facilitate the design of full-scale systems simply by reducing the number of pilot- scale tests required to evaluate various operating conditions and design parameters. Such models can generally be calibrated from simple and well-controlled bench-scale experiments, thus

1040 Environ. Sci. Technal.. Vai. 21. No. 11, 1987

U

Q positioning pilot test programs princi- pally for verification of design bases rather than for primary data development.

Once verified, a process model can be used to examine conditions other than those directly measured. Such a model can also project adsorber re- sponse and sensitivity to a variety of circumstances that may be anticipated but not easily reproduced or experimentally simulated in the pilot program. The logistics, time, and expense normally associated with pilot-scale test equipment and programs make these important considerations.

This paper provides an overview and analysis of certain contemporary ele-

ments and applications of adsorption process modeling. Included is a discussion that expands upon considerations initiated in a recent ES&T feature article that described processes for re- moving dissolved organic contaminants from water ( I ) . In addition, this paper lays a foundation for more detailed descriptions of research developments in adsorber modeling (2-4).

The work is not exhaustive in that it makes no attempt to present and cri- tique all models and approaches available. Rather, it focuses on the Freundlich and Ideal Adsorbed Solution Theory equilibrium models and a two-resistance, homogeneous-surface-diffusion dynamic model as examples that seem particularly suited for simulating adsorption processes and predicting their behavior in water and waste treatment applications. A discussion of modeling approaches and parameter estimation techniques that address environmental complexities is given to bridge the tbeo- retical aspects of modeling with some specific challenges encountered in field-scale design.

The bridging of theory and practice is perhaps one of the most valuable engineering functions that mathematical models can serve, particularly with respect to processes as complex as adsorption. A modeling effort can be focused either on the interpretation of basic data to facilitate an unequivocal understanding of mechanism or on the translation of empirical observations into functional design relationships.

Models developed pursuant to the first objective must by nature be mechanistic, whereas those structured primarily to serve the latter pur- pose may be pbenomenological in character. As theory merges with practice, the interpretation of observed phenomena is enlightened by a greater

-1040$01.5010 0 1987 American Chemical Smiely 001 3936W8710921

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understanding of underlying mecna- nism, and the two types of modeling efforts logically converge.

Indeed, it is often the convergence of the two different modeling approaches that both facilitates and signifies the merging of theory and practice. Re- search and development on adsorption models stands squarely at the cross- roads of theory and practice today. Spe- cific efforts may lean right or left, yet all contribute to the bridging process. An important objective of this review is to identify and document specific areas that require effective interaction and communication between differently focused modeling approaches.

Fixed-bed admrbers The most common adsorber configu-

ration used in water and waste treatment applications is the fixed-bed reactor (FBR). One critical aspect of the design of an FBR adsorber involves characterization of the effluent concentration profile as a function of through- put (Le., the volume processed or the time of operation). This profile, commonly termed the breakthrough curve, represents the specific combination of equilibrium and rate factors that control process performance in a particular application.

The aynamic behavior of an FBR ao- sorber may be visualized in terms of an active adsorption zone that moves through the bed as a function of mass loading and saturation of the adsorbent, illustrated in Figure l(a). Figure 1@) translates this behavior into the breakthrough curve, which is essentially the trace generated by movement of the adsorption zone through and out of the adsorber.

The rate of appearance and the shape of a breakthrough profile depend on several factors, including the physical and chemical properties of both the sorbate and sorbent, the particular rate- limiting mechanisms involved, the depth of the bed, and the velocity of flow. The relative effects of these factors are highly specific to each particular application, which constitutes one of the major obstacles in the design of such processes.

The first step in developing a mathematical model that describes or predicts adsorption dynamics is to provide a representation of the equilibrium behavior. Figure 2 depicts several different types of equilibrium capacity-concentration distributions, or adsorption isotherms (5). The second step is to mathematically characterize the associated rate phenomena. In general, the

rate of uptake of a contaminant by a microporous adsorbent, such as activated carbn, is controlled by a resistance to mass transport rather than by a reaction velocity.

Figure 3 is a schematic representation of a porous adsorbent surrounded by a hydrodynamic boundary layer or film. The macropores of such an adsorbent are typically large enough that associated diffusion phenomena are un- hindered by the pore walls, whereas the micropores have radii comparable to the size of diffising species, so diffi- sion may be hindered by the pore walls. This conceptual view translates into the resistance rate series shown in Figure 3; the overall rate of adsorption is controlled by the step($ providing the greatest resistance to mass transport.

The third step in structuring an adsorption model is implementation of the thermodynamic principle of mass continuity. In an FBR, this results in a material balance for each component of interest in both the liquid (water) and solid (adsorbent) phases; that is, for a control volume of either phase:

MlS Mass MUS Mass rateof = flux of . flux of rite of

component component component component change in out adsorption

Environ. Sci. T&hnol.. Val. 21. NO. 11. 1987 1041

The flux of a component entering or leaving the control volume can include bulk flow, dispersive flow, and molecular diffusion.

The normal procedure for applying the continuity principle is to consider a small but finite control volume representative of the overall system. After the appropriate relationships describing the l o c a l i i mass balance are developed, the control volume is assumed to become infinitesimally small so as to generate the continuity relations on a differential scale. Boundary and initial conditions are also established to spec- ify the constraints of integration. Be- cause the partial differential equations comprising such a system are often highly nonlinear, a numerical method is generally required for their solution.

F&liirium modeling Breakthrough profiles generated

from predictive dynamic models are particularly sensitive to equilibrium parameters (610). This can be especially crucial in the very low, so-called Henrys region, of concentration as well as in very high equilibrium concentration ranges. It is thus important in the formulation of a modeling approach to select an equilibrium model that can accurately describe single-solute isotherm data over several decades of concentration and that can be trans- lated into a predictive multicomponent isotherm equation. Also, the model selected must be integrable with dynamic models in such a manner as to minimize computational efforts.

There are numerous single- and multicomponent equilibrium formulations, both theoretical and empirical in nature, that vary considerably in applica- bility and utility (11). Multicomponent models that are more predictive in nature yield values of liquid- and solid- phase concentrations for competing adsorbates based upon single-solute isotherm data or sorbate-solution characteristics. Difficulties variously associated with such models are that:

they may not be theoretically sound for the systems they purport to represent; they may contain parameters that are difficult to obtain either experimentally or analytically; they may be restricted as to the types and concentrations of compounds for which they produce accurate results; or they may not yield an explicit calculation of adsorption variables for integration into dynamic models. Interests in this research area remain

strons. oarticularlv among those at- temp&to utilize purely th&retical information and first principles as a 1

1042 Environ. Sci. Technol., Vol. 21, No. 11. I987

basis for engineering projections. The Freundlich Model. Semi-empirical models, such as the familiar Freundlich equation, generally fit single-solute experimental data well and acknowledge the surface heterogeneity of activated carbon; therefore, these semi-empirical models are frequently used in adsorber design efforts. This model has the form:

= KFC" (1)

where q and C are the equilibrium solid- and solution-phase concentrations of solute; KF and IZ are character- istic constants relating to adsorption capacity and intensity, respectively.

Extensive use of the Freundlich isotherm model has led to a significant data base of associated model coefficients for different single-adsorbate-adsorbent systems (12). A potential disad- vantage of this model is that it does not approach Henry's law at low concentrations. To accommodate this shortcom- ing, the Freundlich equation has been altered in a variety of ways to adequately describe data over wider concentration ranges (13, 14).

Expansions of the Freundlich model have also been proposed for the description of multicomponent equilibria (2, 15, 16). Due to their empirical nature, formulations such as these require multicomponent isotherm data to cali- brate model coefficients.

The Ideal Adsorbed Solution The- ory Model. The accuracy with which predictions of multicomponent adsorption equilibria can be obtained was en- hanced significantly, especially for dilute solutions, by development of the ideal adsorbed solution theory (IAST) model (17, 18). This model has re- ceived widespread use in multisolute adsorption research for a variety of reasons. Besides the fact that application of IAST necessitates only single-solute data, the model is flexible in that multicomponent calculations can be per- formed using several different single- solute isotherm relationships.

In addition, this model has a solid theoretical foundation, providing a useful understanding of the thermodynamic approach to adsorption. In this regard it is similar to the Gibbs adsorption equation upon which it is based. This is in contrast to other popularized multicomponent models such as the Langmuir competitive model, which is founded on the same limiting assumptions as the single-solute Langmuir model (Le., monolayer adsorption and a homogeneous adsorbent surface).

Comparative studies have indicated that the IAST model has greater accuracy than the Langmuir competitive model for certain multicomponent sys-

tems (14, 19). The feasibility of incorporating IAST into kinetic models to predict adsorption dynamics for multicomponent mixtures has also been demonstrated (3, 20, 21).

The IAST model assumes thermodynamic equivalence of the spreading pressure of each solute at equilibrium, where spreading pressure, T, is the dif- ference in interfacial tension between the pure solvent-solid interface and the solution-solid interface at the same temperature. Single-solute isotherm data are required to compute the spreading pressure of each solute, i, according to:

where R is the ideal gas constant, T is absolute temperature, and A is the specific surface area of the adsorbent. When single-solute isotherm data are described by an appropriate mathematical model, such as [ql =f(Ci)l, Equa- tion 2 can be expressed and spreading pressure can be calculated according to the modified relationship (22):

0

The terms Ci* and qi* in Equations 2 and 3 are, respectively, the liquid- and solid-phase concentrations of species i in single-solute systems yielding the same spreading pressure as that of the mixture. C;. and qi are the respective liquid- and solid-phase concentrations of solute i in the mixture.

Other equations required for the IAST calculation are:

7r j = 7r (4)

N c z j = 1 i = 1

where zi is the mole fraction of solute i in the adsorbed phase; qT is the total quantity of material adsorbed from the

mixture; and N is the total number of species in solution.

To obtain a solution, Equation 3 must be integrated over the concentration range of interest (utilizing the single- solute equilibrium model) to calculate spreading pressures, The remaining un- knowns must be evaluated by simulta- neous solution of Equations 3-8. The model thus becomes increasingly difficult to use for increasing numbers of solutes. In addition, care must be exercised to ensure that simple single-solute models provide a sufficient degree of precision over the concentration ranges of interest, whereas more sophisticated models may make evaluation of the spreading pressure (Equation 3) pro- hibitively difficult.

A number of attempts have been made to modify the IAST model to im- prove its accuracy and to minimize computational requirements to render it more amenable to complex systems. One approach to reducing the computational effort required to implement IAST is based on the assumption that the spreading pressure for multicomponent and single-solute systems can be equal only for the special case of identi- cal isotherms, or when the single-solute Freundlich isotherms for the components have equal slopes (23-26).

Average isotherm constants, which become representative of hypothetical (or fictive) solutes of the multisolute system, can be calculated to reduce the IAST equations to a single expression. The solid-phase concentration of a solute in the mixture can be calculated directly from the liquid-phase concentration and the actual and average isotherm constants.

This simplified competitive adsorption model (SCAM) allows IAST concepts to be applied to complex mixtures of many solutes. However, in this form the model does not incorporate the mass balance relationship, has limited applications over broad concentration ranges, and is not always coincident with more rigorous IAST calculations when Freundlich n values (isotherm slopes) for competing solutes are markedly different.

For some applications it is difficult to obtain accurate calculations for the spreading pressure of each solute for use in the IAST model; such calculations require accurate descriptions of single-solute data. Although the Freundlich model is often suitable for highly heterogeneous surfaces (such as activated carbon) and is simple in form, it is not always appropriate for describing equilibrium data in low concentration ranges.

In one approach (29, the log-log plot of equilibrium data is divided into several straight-line segments, each de-

Environ. Sci. Technol., Vol. 21, No. l l , 1987 1043

scribed by appropriate Freundlich constants. The method is well-suited for incorporation in multicomponent dynamic models because it calculates liquid-phase concentrations explicitly for given solid-phase concentrations. A major limitation is that the model application becomes increasingly difficult for more than two solutes. In addition, the mass balance equation is not included, rendering batch-mode verification of multicomponent concentrations less meaningful. In other words, the model is not truly predictive.

In a similar attempt a three-parameter Freundlich single-solute isotherm was proposed for which an analytical solution could be obtained for Equa- tions 3 and 4 (14, 28). The mass balance expression for each component in the mixture is incorporated:

where Co,L is the initial concentration of species i in the mixture and M/V is the carbon dose. As expected, computa- tions are more complex for the three- parameter isotherm model than for the simple Freundlich equation, and ma- nipulation of single-solute data to establish isotherm parameters is somewhat arbitrary.

The standard Freundlich equation (Equation 1) has been applied success- fully to IAST calculations for simple mixtures of similar solutes such as low- molecular-weight halocarbons (29). Different modifications of IAST have been employed for systems in which competitive interactions are significant. In one case, the simplified IAST model was modified by adding a term to account for the effects of irreversibility and unequal competition (19).

The irreversibility term was determined experimentally by sequential sorbate loading of a shallow GAC bed incorporated within a completely mixed batch reactor (CMBR) system, some- times referred to as a differential batch reactor (30). The model showed significant improvement over its predecessor for two-solute systems of mostly phenolic compounds. An alternative scheme involves empiricizing the IAST model by addition of a term, R,, to Equation 5 to more accurately describe experimental equilibria (3, 31, 32); specifically:

Approaches such as these illustrate how the interplay between scientific and engineering disciplines can be used to develop models that both maintain theoretical integrity and describe, simu-

late, and ultimately predict, process phenomena with acceptable accuracy. IAST has several inherent shortcomings that are revealed when the model is applied to nonideal organic solutions, especially mixtures containing humic substances. The concept of equivalent spreading pressure does not adequately address specific adsorption phenomena or so-called nonideal competition, nor does it account for the differential avail- ability of sorption sites on heterogeneous adsorbent surfaces induced by steric or surface energy effects (33-35).

Moreover, for many mixtures, particularly for complex mixtures containing humic materials, reliable information on the spreading pressures of solution components is not available to provide accurate input for the model. In such cases, the researcher or design engineer can either abandon the existing model in favor of a new approach or modify a given model to account for observed anomalies and preserve the ability to extract phenomenological sig- nificance from the empiricized version(s) .

Kinetic modeling Dual resistance models. Dynamic

models that account for both film and intraparticle diffusion appear most suitable for describing the adsorption of both target contaminants and background humic substances. Variations of such models are distinguished according to the means for expressing the intraparticle mass transport step. The simplest formulation for describing diffusion with microporous adsorbents is the linear driving force approximation (36,37j:

h i 9 = - (C, - C,,) = kSi(qsi - qi); 6t 0

i = l , 2 , 3 , . . . , N (11) Where p is the bulk density of the adsorbent and C,, and qSi are the solid and interphase concentrations. This model assumes that equilibrium exists at the fluid-particle interface and that the mass transfer coefficients for the liquid and solid phases, kL and k,, respectively, are constant and independent of the transfer rates of competing species.

A more descriptive and complex analysis of intraparticle diffusion is given by the pore transport model. Ad- sorbate molecules are assumed to dif- fuse in the pore voids and adsorb on pore surface sites. The migration of adsorbed solutes can occur only by desorption followed by pore transport to a new site, The corresponding material balance equation expresses intraparticle sorbate concentration as a function of the radial dimension of the adsorbent particle; for spherical coordinates:

with the associated boundary condi tions:

=O (14)

In the above expressions, E represents the bed void fraction; ps is the apparent particle density; r is the particle radius; and Dp is the pore diffusion coefficient.

A number of investigators have applied the pore diffusion model in conjunction with film resistance and various equilibrium expressions (38). It has been found, however, that pore diffusion coefficients calculated according to this model are frequently larger than corresponding free-liquid diffusivities. This suggests that surface (rather than pore) diffusion may be the primary intraparticle transport mechanism, thus promoting increased interest in surface diffusion models (39-43).

Surface diffusion supposes that an adsorbed molecule migrates to an adja- cent adsorption site on a carbon pore wall surface, a transport motivated by the surface concentration gradient and available sorption energy. Several different solutions to the film-surface diffusion model for liquid-solid separations have been presented (11, 38, 44, 45) *

One example of a two-resistance model that has evolved from efforts to simulate and predict adsorption processes for both CMBR and FBR systems is the homogeneous surface diffusion (HSD) version of the Michigan Ad- sorption Design and Applications Model, MADAM (4652) . Figure 4 depicts a schematic conceptualization of this model for an FBR adsorber (5). The results of wide-scale testing of the MADAM-HSD and several subsequent refinements thereof suggest that the model provides a reasonable struc- ture for simulating adsorption processes for water and waste treatment systems.

MADAM-HSD: development and application. The MADAM-HSD equations derived from mass balance equations for the fluid and solid phases in an FBR adsorber are given below in

1044 Environ. Sci. Technol., Vol. 21, No. 11, 1987

, averaged over the particle radius and included in the intraparticle diffusion coeficient; dilute solution conditions prevail, hence no bulk flow term is required in the solid-phase material balance expression; diffusion coefficients are not concentration-dependent; only longitudinal concentration gra- dients exist in the FBR (Le., no chan- neling); and longitudinal dispersion is negligible over the length of the column (53). In the earliest applications of this

model, surface diffusion coefficients were determined by a statistical search of single-solute CMBR rate data for both single-component and multicomponent systems. Film diffusion coefi- cients were estimated by empirical correlation techniques (54).

The HSD model and its variations have been applied to a rather broad range of singlecomoonent and multi- - - component solutions comprising priority pollutants and unspecified background dissolved organic matter (DOM). The implementation and important features of the HSD model have been summarized (55,56) with the goal of compiling a simplified, user-oriented guide to designing pilot-scale adsorbers for either CMBR or FBR systems.

Successful application of the model has been well established for adsorption and desorption in single-solute systems of target organics in backgrounds containing little or no DOM (57-62) and in waters containing only DOM (63, 64). These applications have involved vari-

I

ous adsorbent types and particle sizes, including cases of variable particle size

The HSD model also has been shown

=O dimensionalized spherical-coordinate form for singlecomponent systems. Liquid-phase continuity equation: r=o (I8) distributions (2, 65).

where C and q are fluid- and solid- phase concentrations of adsorbate, respectively; v, is the interstitial fluid velocity in the direction, z, of flow; f is time; Dh is the hydrodynamic dispersion coefficient; p s is the apparent particle density; and e is the void fraction of the bed.

Solid-phase continuity equation:

where r is the radial distance from the center of the carbon particle and D, is the surface diffusion coefficient. Specifi- cation of boundary and initial conditions include introduction of a film transfer coefficient, kp These conditions are then given by:

where C, is the fluid-phase concentration near the surface of a carbon part- cle of radius, a,, and C, is the influent concentration to a bed of depth L.

Assumptions associated with this model include the following:

local equilibrium OCCUIS at the exte- rior carbon surface; the labyrinth factor (a measure of carbon particle pore configuration) is

to handle transient organic loadings for several one- and two-solute systems (31, 66). Adequate prediction of breakthrough profiles for competing adsorbates, including chromatographic displacement phenomena, has been demonstrated for several applications (2, 3, 9,31, 62). The model has been less successful in predicting effluent concentrations for two situations: when a multicomponent solution contains compounds that exhibit markedly different adsorptive behaviors, or when significant interactions occur within a complex background of unknown composition (3, 6, 62). Increasing the com- plexity of the intraparticle diffusion term has resulted in improved model performance in some cases (52, 67, 68), but such modifications also involve additional parameter evaluations and analytical requirements.

A variety of other diffusion-controlled mass transport models have been developed to better describe the

Enviran. Sci. Technol.. Vol. 21, No. 11. 1987 1045

behavior of FBR adsorption systems in environmental applications (11). These include models that combine pore and surface diffusion (21, 69) and models that incorporate concentration-dependent diffusion coefficients (30, 70).

Modeling approaches for complex mixtures

A number of different techniques have been used for modeling the adsorption of organic contaminants in the presence of background DOM. One approach, which treats background organic matter as an important but uncharacterized local condition, experimentally evaluates model rate and equilibrium parameters for target compounds as system-specific coefficients (57, 59).

The idea of implicitly incorporating background effects in phenomenological coefficients has been applied to both single-target compound systems and more complex systems containing mixtures of several targeted organic contaminants (3, 57, 59). Although the methodology can be generalized for ad- aptation to various waters and wastes, the system-specific nature of the technique prohibits extrapolation of coefficient values from one case to another.

Several attempts at a more general approach have involved lumping all target organics and DOM into a single parameter, such as TOC, DOC, or TOX (total organic halogen), and modeling the behavior of that parameter (64, 71, 72). In a related method, which involves multicomponent modeling, each significant target compound represents a single solute; all other organics are characterized and modeled in terms of a lumped concentration parameter (57).

One rather unique approach to unspecified mixtures of adsorbing species involves hypothesizing the nonspecific matrix as a number of fictive or pseudospecies. The individual isotherm properties of these fictive species are evaluated from lumped parameter data and the IAST model (73, 74). The initial concentrations and single-solute isotherm parameters for each pseudo- component must be determined before IAST can be implemented to describe the overall isotherm and predict its competitive impact on selected target compounds.

In addition to experimental determination of the overall isotherm, a char- acteristic mixture isotherm of a tracer substance (i.e., a known compound added to the unknown mixture and de- tected by single-substance analysis) helps to establish initial qualitative esti- mates of the unknown parameters for each component, These parameters are varied systematically within a nonlinear, statistical curve-fitting routine until

deviations between experimental isotherm points and those computed by the model are minimized.

The calculated parameters can then be used for predicting breakthrough profiles and competitive adsorption equilibria for specific compounds in the presence of the mixture, thus moving toward the goal of formulating a methodology to accurately predict adsorption of target contaminants from an unknown solution containing many organic species.

This work has served as a forerunner to related research on the development of species-grouping methodologies for mixtures containing a large number of known solutes. In one development, species grouping is predicated upon Freundlich isotherm parameters for individual solutes, and a simplified expression for spreading pressure is employed in IAST model calculations (75). Multicomponent solutions are re- duced by this grouping to a system of a fewer number of pseudospecies characterized by average adsorption parameters, with adsorbing solute concentrations measured in terms of a surrogate parameter such as TOC.

This work has been extended by development of a search procedure for calculating the pseudospecies composition and adsorption equilibrium parameters from total adsorbate equilibrium concentration data (measured as TOC) obtained from CMBR experiments (76). Attempts to apply species grouping to the simplification of FBR multicomponent adsorption calculations have met with less than complete success (77), possibly because of the sim- plicity of the dynamic model employed (Le., a linear driving force relationship similar to that given by Equation 11); the lack of inclusion of competitive chromatographic displacement effects; and the fact that the criteria used for species grouping were largely based on only a sorption capacity parameter (i.e., the preexponential coefficient in the Freundlich model, Equation 1).

In this latter regard, it must be recognized that sorption energy or intensity, as reflected by the exponential coefficient of the Freundlich model, distin- guishes the equilibrium adsorption characteristics of a compound as significantly as does the capacity coefficient. Moreover, compounds that exhibit similar equilibrium characteristics may very well have vastly different mass transport properties.

In subsequent attempts to address these shortcomings, the species-grouping criteria have been modified to include requirements that adsorbates of the same pseudospecies have Freundlich exponent values that did not differ by more than 20% and that the

intraparticle diffusion coefficients must be of the same order of magnitude ( 78). Even with these tighter grouping criteria, the approach has yet to demonos- trate a capability for describing the competitive displacement of specific, weakly adsorbing compounds in FBR systems.

A further refinement of attempts to describe the adsorption behavior of complex mixtures in terms of Freundlich isotherm relationships and coefficients for pseudo- or theoretical components involves calibration of these parameters with respect to a known weakly adsorbing tracer compound. In this technique the tracer compound is either added to the mixture or is an existing component that is singled out for particular analysis. The theoretical component parameters are computed by matching the displacement of the isotherm for the tracer compound to its corresponding single-solute isotherm. The ability of the method to simulate composite isotherms for the components of background TOC and total organic chlorine (TOX) has been demonstrated (79).

Theoretical evidence has also been given that the tracer compound principle and computed equilibrium parameters for pseudocomponents may be applied to determination of effective solid-phase diffusivities for the same pseudospecies (80). However, no experimental verification of this latter aspect of the tracer compound modeling approach has been presented. More- over, work on this approach has for the most part been restricted to mixtures containing only low-molecular-weight micropollutants exhibiting similar chemical structures.

It is clear from the foregoing that our state-of-the-art ability to predictively model the adsorption characteristics of the individual components of complex mixtures reflects at least as much art as it does science. Nonetheless, work such as that described is gradually enhancing our ability to phenomenologically characterize the overall behavior of complex systems.

Model synopsis Recent developments in adsorption

models have led to more sophisticated and mechanistically more correct structures and codes. The fact remains, however, that it is still unrealistic for most water and waste applications to use models for design of prototypes directly from first principles. This is especially the case for complex waters containing mixtures of target compounds and unspecified background organic matter, such as humic substances.

Although small-scale refinements of existing models will continue to im-


prove their computational efficienc: and predictive capability, it appears tha. the mathematical description and numerical solution of dynamic models has essentially reached a sufficient level of sophistication, at least temporarily (7, 82). That is to say, the mechanistic and mathematical refinement of theoretically oriented model efforts has out- stripped our ability to adequately quan- tify all of the input parameters required by these models.

For the present, priority challenges in dynamic modeling involve improv- ing the reliability of parameter estimation techniques and evaluating and enhancing the capability of existing models to accommodate the complexities encountered in field applications. As concept and practice converge, feedback from such efforts will ultimately justify further theoretical sophistication.

Parameter estimation The majority of the models currently

used to simulate and predict adsorption processes for water and waste treatment systems are phenomenological in the sense that many of the reaction mechanisms in such systems, and the impacts of solid-solute and solute-solvent interactions on these mechanisms, are incorporated implicitly into the model coefficients. This approach provides a workable balance between simplistic pilot-scale testing and ideal predictive model development, which involves independent characterization and experimental verification of each reaction and associated mechanism. The latter a p proach is generally impractical for engineering applications that deal with the complexities of field conditions, whereas the former approach does not provide a logistically efficient means for formulating and testing hypotheses to explain how the dynamics of contaminants are controlled on a macroscopic scale.

The ability of a phenomenological model to describe and/or predict contaminant behavior in a given system is ultimately vested in the accuracy with which model input parameters are determined and how well the parameters relate to conditions anticipated at full scale. For cases of negligible dispersion, two-resistance models such as the MADAM-HSD (as given by Equations 1 6 2 2 ) require estimation of a film diffusion coefficient, a surface diffusion coefficient, and appropriate isotherm parameters for each contaminant to be modeled.

The film mass transport coefficient, kl, is often calculated from one of a number of semi-empirical correlations. Each of these correlations is distinguished by a functional relationship be-

Literature correlations for determination of external mass transfer coefficients c.rnlalID" Expr-Ion lor Ranpe(s) of VI

Williamwn et al. (54) c Re < 125 : Sc < 1300

2.4 c R e O ~ ~ W ~ ' 2

Wilson and Geankoplis 1.09 c-mR'RSc'n

Ohashi et al. (84j (83) 950 < Sc < 70.000

i ,__ masO.Wl < R e < 5.8(stokes .: .u.... $;,$2 region) 2 + 1.58 Re"' Sc'" 5.8 < Re < 500 (transition

Re > 500 (Newton 1 < Re < 10,000 0.0 < Sc < 10,OM Relr/(l - 4 < 101

region) 1.21 Wfl

2 + 0.59 Reo.e Sc'"

[*[I + 1.5(1 - e? 1.85 [(I - c)/c]'" Re'" Sc'"

Gnielinski (85).

Kataokaetal. (86)

+ (ShL2 + Sh '."I

Dwivedi-Upadhyay (87) (Ik) [0.765(~Re)~.'~ + 0.365 *Where Sh, = 0 . W Re'n Sc'" and Sh, = (0.037 Reo8 Sc) I [I + 2.443 W' ( S P -1)l.

0.01 < Re < 15,O ( C R ~ ) ~ . ~ ' ~ ] SC"~

Land Tsubsnim refer lo laminar and lurbulenl flow, rebpeclively. I =

tween the dimensionless Reynolds (Re), Schmidt (Sc), and Sherwood (Sh) numbers, the latter being related to the film diffusion coefficient by:

Sh = k@/D[

where d is the adsorbent particle diame- ter, DL is the bulk liquid diffusivity of the adsorbate and Re and Sc are defined as:

Re = v,d/v

Sc = v/DL

where v, is the interstitial flow through the bed, and vis the kinematic viscosity of the fluid. Values of DL can be estimated from the relationship (82):

where p is absolute viscosity, X and M are associated solvent parameters, and V, is the molar volume of the solute at its normal boiling point.

Table 1 contains a sample listing of hydrodynamic correlations from the literature (54, 83-87). It is important to recognize that calculations of free liquid diffusivity and of liquid diffusion from semi-empirical correlations such as these do not take into account interactions between the target contaminants and background matrices. Moreover, the equations presented in Table 1 have been developed using materials that are substantially different in chemical and physical character than typical microporous adsorbents such as activated carbon.

A number of studies have demonstrated that the surface topography and roughness of an adsorbent can have an

impact on film-controlled mass transfer rates (58,85,88). Further, no standardized criteria have been established for determining which correlation may be best suited to a particular system, other than restrictions stipulated according to hydrodynamic conditions. Caution must therefore be exercised in applying generic correlation techniques to determination of specific mass transfer coefficients, particularly for the complex mixtures typical of water and waste treatment applications.

Traditional dynamic modeling a p proaches have utilized klvalues that are determined from correlation procedures in conjunction with intraparticle (e.g., surface) diffusion coefficients obtained from model calibrations to CMBR rate data. Such studies are relatively easy to perform and do not ne- cessitate the large volumes of solution required for column studies. If the ultimate objective is to predict the performance of FBR systems, however, it must be recognized that CMBRs do not approximate the hydrodynamic and contaminant removal patterns of column reactors.

In a CMBR, for instance, the concentration gradient decreases more rapidly with respect to the carbon surface (at least in the initial stages of operation) than it does in a column, for which the influent concentration of adsorhate@) is normally maintained at a relatively constant level. Effective intraparticle diffusion coefficients may also differ between the two types of reactor systems, particularly if an adsorbate dis- plays a hysteresis pattern with respect to the sorption-desorption process.

These considerations are particularly significant for mnlticomponent systems and systems in which interactions between target compounds and back-

Environ. Sei. Technol., MI. 21. NO. 11. 1987 1047

ground matrices are operative, especially for cases of nonideal competition among adsorbates. For these reasons, certain approaches may be suspect if they employ D, values obtained from single-solute rate studies for application in multicomponent modeling exercises,

The differential batch reactor has been utilized by several investigators as a prospective improvement on the standard CMBR apparatus (30, 89). Although this system more accurately approximates the hydrodynamics of an FBR, it suffers from the same deficiencies as the CMBR with respect to simulating the concentration and hysteresis profiles of adsorbates in flow-through column-type systems.

Conversely, the short-bed adsorber (SBA) technique (60, 61) enables determination of values for both kf and 0, in an experimental reactor that more closely approximates the configuration of practical adsorption systems. The SBA is an adsorber column sufficiently short in length to allow incipient breakthrough of the compounds being studied, yet still exhibiting hydrodynamic properties comparable to those of a deep-bed adsorber.

The design of an SBA is such that the initial stage of breakthrough is domi- nated by film transfer, regardless of the type of overall mass transport control that will eventually predominate. This enables independent determination of the film transfer coefficient followed by calculation of the surface difisivity using a search-regression routine over the entire breakthrough profile. In this way both kf and D, are calibrated from the same experimental data set using the HSD model, thus circumventing the need for correlation procedures and the potential for error compounding that is involved when CMBR systems and correlation techniques are used to deter- mine the two rate parameters.

Investigations using the SBA/ MADAM-HSD approach have demonstrated good to excellent agreement between predicted and experimental FBR breakthrough profiles in deep beds for single-solute and bisolute systems of typical organic contaminants (9, 60, 61). Descriptions of dynamic multicomponent hysteresis and displacement phenomena have been obtained for nearly every case studied. The potential of the SBA technique for mass transfer parameter determination in systems containing complex mixtures of organics has been demonstrated (3, 57, 64).

Model isotherm parameters typically have been obtained from equilibrium experiments conducted by the bottle- point technique in CMBR systems. A substantial volume of data can be generated with relative ease by this method. However, just as CMBR data

may not provide the most accurate basis for estimating rate parameters for FBR modeling, so the bottle-point method may not provide the best estimate of equilibrium relationships; this again by virtue of the potentially irreversible hysteresis patterns exhibited by certain adsorbates (34) and the different approaches to multicomponent equilibria that occur in CMBR and FBR adsorbers.

In an FBR, for example, the highest solution-phase concentration of adsorbate encountered by an adsorbent is equivalent to the equilibrium concentration at exhaustion. To achieve the same equilibrium concentration in a CMBR, the initial concentration must be higher than that applied to the column. Thus different concentration gradient patterns result for the two types of reactor systems.

As for kinetic studies, the impact of such variables on estimated values ,for the effective equilibrium capacities and model coefficients is no doubt propor- tional to the number of adsorbing species in solution. Several investigations have revealed dissimilarities between effective capacities determined by CMBR and FBR techniques (2, 90). Efforts are currently underway to develop a standardized methodology for column determinations of adsorption equilibria (2).

Conclusions and future directions The ultimate engineering objective of

process modeling is to provide a vehi- cle for efficient and cost-effective design. Although significant progress toward this objective has been made in the development of adsorption models and parameter estimation procedures, more must be accomplished before specifications for prototype systems can be generated directly by modeling efforts alone.

The role of experimental testing and empirical observation in the design of specific systems remains as important today as it has ever been, although an improved understanding of the process now enables more focused test programs and more informed interpretation of experimental information. This latter fact is attributable to advances that have been made toward a parallel objective of process modeling; to wit, the development of more rigorous mechanistic interpretations of process dynamics through continuously refined integration of experimental observations with theoretical concepts and model assumptions. Theory and practice are thus converging to benefit both objectives of process modeling.

In considering avenues for future progress, several deficiencies in our current Understanding of the process

and in our abilities to represent it in precise mathematical form emerge from the foregoing discussion. With few exceptions, the approaches described have not sufficiently empha- sized interactions and impacts effected by natural background organic matter on the adsorption of specifically targeted organic contaminants in complex multicomponent systems. Investiga- tions to discern potential relationships between experimentally determined coefficients and critical system properties could prove particularly useful in ad- dressing this deficiency.

Variations in background solution characteristics can alter the adsorption equilibrium and mass transfer behavior of target species. Modeling approaches can be structured to reflect these altera- tions in isotherm and kinetic model coefficients. As lumped-parameter techniques and phenomenological approaches for characterizing the physical and chemical properties of unspecified backgrounds are developed and refined, it may be possible to correlate these parameters with model coefficients to predict adsorber performance relative to perturbations in background water composition.

Measureable solution parameters to be examined in this regard would logically include pH, ionic strength, the presence of multivalent cations, and the character and concentration of background dissolved organic matter. In the latter regard, background organic species characterizations may have to be done phenomenologically, in terms of such measures as apparent molecular weight distributions and polarity- hydrophobicity.

Column methods for determination of isotherm parameters to be used in FBR adsorber modeling should be more extensively tested and analyzed. Small, bench-scale column methodologies have been developed for single- and dual-component systems of target organics in relatively simple background waters. Evaluation of both equilibrium and rate parameters from the same experimental design may eliminate potential problems created when CMBRs are used to estimate coefficients for subsequent column predictions.

Bench-scale column techniques could enable such complex features of fullscale FBR systems as reactor hydrodynamics, influent conditions, time- vari- ant concentration patterns, and hysteresis phenomena to be more closely simulated and their effects cap- tured in evaluation of effective adsorption capacities and rates. It is likely that these factors exert impacts on such measurements in direct proportion to the number of relevant components


present, and certainly the difficulty of otherwise explicitly accounting for such impacts so increases. It is anticipated that the use of smaller experimental designs will also permit greater flex- ibility in expanding ranges of system conditions that can be examined easily (e:g., system flow rate, bed depth, and initial conditions) and thus effect sav- ings of time and material in the design process by extending model calibration and verification capabilities at bench scale.

In all of this must be appreciated the possibility that no single model, regardless of how modified or manipulated, may be universally applicable. Perhaps a more reasonable objective for future modeling efforts, at least in the near term, is the structuring of a general modeling methodology that incorpo- rates decision-making capabilities based upon a set of well-defined pre- liminary observations and feedback refinement ability.

Modeling approaches that are system-specific in nature appear to be war- ranted for most practical applications in view of the variable solution and adsorption characteristics of target compounds when associated with complex background waters. Such approaches can accommodate both the immediate need for establishing economical and accurate scale-up specifications for field-scale design as well as the longer term scientific objective of investigat- ing fundamental relationships between model and system variables.

Acknowledgment This article has been reviewed for suitabil- ity as a critical review by Francis A. Di- Giano, University of Nor th Carol ina, Chapel Hill, N.C. 27514 and by Gordon McKay, The Queens University of Belfast, Northern Ireland. U.K.

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1983; pp. 247-68.

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(78) Ramaswami. S.: Tien, C. Ind. ond C h m . Proe. Des. and De". 1986. 25, 133- 39.

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(81) Weber, W. J.. Jr. In Proceedings. First Imemnrional Conference on the Fundomen- I d s of Adsorprim; Myers. A. L.: Belfort, G., Eds.; Engineering Foundation and AIChE: New York. 1984; pp. 745-48.

(82) Wilke, C. R.; Chang. P. J. AIChE 1955. I 7M-7n .._".

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(84) Oharhi, H. e l al. I . Chon. Eng. Japan 1981, 14.433-38.

( 8 5 ) Roberls. P. V; Cornel. F?; Summers. R. S. J . Environ. Eng. 1985. 111. 891-905.

(86) Kataoka. T.: Yoshida, H.; Ueyama, K. J. Chem. Eng. Japon 1972.5, 132-36.

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(88) Young, B. D.; van Vliel. B. M.. submitted far publication in In:. J. H m Moss

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TRZ"SfW.

I .-.-, I 4 : , \ " I I

Walter J . Weber, Jr., (1) is the Eornesr Bo>?(T Disrinruished Professor of Engi- nerring and direrror of Environmental and Worrr Resources Engineering ar the Uni- versify of Michigan. He holds engineering degrees from Brown. Rurgers. and Har- vard universities. His research and teach- ing activiries focus on rhe characrerizarion and modeling of physicochemical processes in narural and engineered sysrems. He is widelv recog;rized for his work on research and development of adsorption processes in environmenral engineering applicarions.

L?dd H. S d h (r) is a research associ- ate in the Universiry of Michigan :T Envi- ronmental and Water Resources Engineer- ing Program. He holds bachelnr:~ and master? degrees in civil en8ineering from the Universiry of Delaware, and a Ph.D. in environmenral engineering from rhe Uni- versify of Michigan. His principal areas of research include rhe adsoplion of organic and inorganic conraminants by activated carbon, and rhe applicarion of adsorprion processes ro deconraminarion of surface and subsurface wafer supplies.

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