adsorption modelling for building contaminant dispersal analysis

Indoor Air. 2.147-171 (1991) ~ ~~~~~~

0 1991 Danish Technical Press, DK-Copenhagen

Adsorption Modelling for Building Contaminant Dispersal Analysis James W. Axley* Massachusetts Institute of Technology, USA

Abstract Two families of macroscopic adsorption models are formulated, based on f undamta l principles of ads- tion science and technology, that may be used for macroscopic (e.g., whole-building) contaminant dispersal analysis. The first family of adForprion mo&ls - the Equilibrium Adsorption (EA) Models - are based upon the simple requirement of adsorptiun equilibnum between adsorbent and room air. The second family - the Boundary Layer Diffusion Controlled Adsorption (BLDC) Models - add to the equilhum requirement a boundary layer model for d i f i i on of the adsorbate from the room air to the adsorbent sutjiie. Two members of each of these families are explicitly discussed, one based on the linear adsq t ion isotherm mo&l and the other on the Lungmuir isotherm model. The linear variants of each family are applied to model the adsorptiun dynamics of f m l d e h y d e in gypsum wall board and compared to measured data. These applications and a more general cotlsideratiun of the dynamic character of adsorption provided by these modeh indicate that simple physical adsorption and &sorption transpoTt processes h e the potential to signifiantly affect the dispersal of contaminants in buildings.

KEY WORDS: Indoor air quality, Contaminant dispersal analysis, Adsorption, Multi-zone modelling.

Manuscript received: 16 November 1990 Accepted for publication: 25 January 1991

* James W. Axley, Building Technology Program, Massachusetts Institute of Technology, Cambridge, MA, USA

Introduction The quality of air in buildings is clearly dependent on both the nature of air movement into, out of, and within building systems and the character and location of contaminant sources. As a result, researchers have focused on the complementary problems of the airflow aspects of contaminant dispersal and the emission characteristics of contaminant sources. It is also well recognized that some contaminants, such as Radon-222 gas and its radioactive decay daughters, and nitrogen di- oxide undergo chemical or radiochemical de- composition, and modelling the dispersal of these contaminants demands the additional consideration of the chemical or radiochemical mass transformation processes involved.

It is also known that adsorption and desorption processes can affect the dispersal of contaminants in buildings. The evidence is pervasive. The lingering odours of winter clothes stored during the summer with naphthalene crystals, and the odour of clothes exposed to tobacco smoke or gasoline fumes that persists for days until washed are but two examples. Few would doubt that adsorption and desorption phenomena can and do play a role in the dispersal of airborne contaminants in buildings but, from the point of view of indoor air quality analysis, should we suspect these phenomena to be significant? Can adsorption and desorption significantly alter the dynamics of the dispersal of indoor air contaminants? If so, under what circumstances will the effect be significant and how may one model the dispersal of

148 Axley: Adsorption Modelling

a contaminant when these circumstances occur?

This paper demonstrates that adsorption and desorption phenomena have not only the potential to significantly alter the dynamics of contaminant dispersal but may, in some cases, completely control them. Work- ing from fundamental principles of adsorption science, two families of adsorption/desorption models are formulated that provide a means to identify the circumstances under which sorption phenomena may come to be significant.

Fundamental Considerations A considerable body of knowledge is available in existing adsorption science and technology literature. Most of this literature is directed toward industrial applications of adsorption that involve the purposeful use of specially prepared adsorbents (e.g., activated carbon, zeolites, and synthetic resins) placed in adsorption devices that are carefully de- signed and controlled to achieve high rates of adsorption (e.g., well-mixed batch reactors, well-mixed flow reactors, and fixed-bed reactors) Uaroniec and Madey, 1988; Oscik and Cooper, 1982; Ponec et al., 1974; Rodri- gues et al., 1988; Ruthven, 1984; Slejko, 1985). Our interest here, on the other hand, is directed toward modelling adsorption phenomena that are largely unintentional, involving materials that tend to act as adsorbents, but are not specially prepared to do so, that are placed in buildings for purposes other than that of adsorption separation. Conse- quently, we shall have to adapt ideas from the available adsorption literature and limit consideration to the simpler models of adsorption phenomena until additional study warrants the application and provides the re- quisite data for more complete adsorption models.

Adsorption Dynamics Adsorption involves the separation of a substance from one phase, indoor air in the

present context, and the accumulation of that substance on the surface of another phase, here, building materials. The nature of adsorption will depend on the nature of the adsorbent, the nature of the adsorbate, and the interaction that may occur between them. According to Slejko (1985), we may dis- tinguish four generic types of adsorption - exchange, physical, chemical, and specific adsorption. Exchange adsorption involves the exchange of one ionic species - the ionic adsorbate - with other ionic species attached to the surface of the adsorbent and therefore involves electrostatic attraction of the adsorbate to the adsorbent. In physical adsorption the adsorbate is bound to the adsorbent by relatively weak intermolecular Van Der Waals forces. Chemical adsorption or chemi- sorption results from chemical reactions between the adsorbate and the adsorbent, and therefore involves usually very strong binding forces associated with the sharing of elec- trons between the adsorbate and adsorbent. Finally, the term specijic adsorption is reserved for adsorption phenomena involving binding of molecular groups, without chemical transformation, to specific functional groups on the adsorbent (e.g., polar adsorbates binding to polarized sites on the surface of an adsorbent).

The nature of the adsorption process is directly linked to the strength of the forces that bind adsorbate to adsorbent with physical adsorption falling at the weak-force end of the spectrum and chemical adsorption at the strong-force end. With this in mind, it follows directly that:

(a) adsorption is typically an exothermic process with the heat of adsorption associated with physical adsorption expected to be small relative to that associated with chemical adsorption;

(b) physical adsorption may be expected to be a relatively reversible process while chemical adsorption may not; and

Axley: Adsorption Modelling 14Y

(c) chemical adsorption may be expected to be associated with the formation of a single layer of molecules of adsorbate on the surface of the adsorbent while in physical adsorption the possibility of multiple layers is more likely.

Reversibility and the heat of adsorption of a given adsorbate/adsorbent pair thus provide a direct indication of the molecular mechan- ism responsible for the adsorption.

Physical and some chemical adsorption processes may be more or less reversible and can be represented as:

Adsorbate + Adsorbent S

Adsorbate-Adsorbent Complex

where the degree of reversibility depends, in part, on the thermodynamic state of the adsorbate-adsorbent system and the nature of bonding in the adsorbate-adsorbent complex. For physical adsorption the bonding is relatively weak and thus this reversible process may be thought, instead, to be a reversible exchange between two alternative phases of the adsorbate - the free or gas phase and the adsorbed phase - where the adsorbent may be thought to activate the phase transformation, not unlike a catalyst, rather than reac- ting to form a new adsorbate-adsorbent complex. Distinguishing the concentration of adsorbate near a surface, C*, from that adsorbed on the surface, Cs, this reversible physical adsorption process may be represented as:

adsorbent c* 8 Cs + AH

where AH is the heat of adsorption and, for our purposes, concentration will be expressed in terms of mass fraction (i.e., for C* mass adsorbate per mass gas phase and for Cs mass adsorbate per mass adsorbent1).

The adsorption process is but one step of several that determine the nature of the adsorption dynamics. In the building context, adsorbate species are transported via diffu-

sion transport processes from the bulk air phase in building zones to near-surface locations where the adsorption processes come into play to bind the adsorbate to the adsorbent surface. Simultaneously, desorption processes release adsorbate from the adsorbent surfaces to near-surface locations where, now, diffusion transport processes act to transport the adsorbate to the bulk air phase. The rate of species transport to and from the adsorbent surface is thus determined by both the rate of diffusion transport processes and adsorption-desorption kinetics.

Diffusion Transport Diffusion transport to and within effective adsorbents invariably involves a complex variety of transport mechanisms including boundary layer molecular and turbulent diffusion from the bulk air phase and molecular, Knudsen, and surface diffusion within macroporous and microporous interstices (Ruthven, 1989). For our present purposes it will be sufficient to consider only boundary layer diffusion which may be modelled as a spatially discrete process using boundary layer theory (White, 1988); macroporous and microporous diffusion processes which are, by their nature, spatially continuous may be considered in future investigations using discrete Finite Element approximations. The pertinent details of boundary layer theory will be considered subsequently.

' The concentration of absorbate is sometimes expressed in terms of mass adsorbate per unit surface area available for adsorption or, in the indoor air quality literature, in terms of mass adsorbate per unit of projected surface area of the adsorbent (e.g., wall surface area). Effective adsorbents typically have extremely large surface areas available for adsorption due to con- voluted surfaces and high porosities. As this surface area is well correlated to mass, for effective adsorbents, it is usual to classify adsorbents by specific surface area (aredmass) and, hence, to employ mass fraction as a measure of concentration (e.g., see Vermeu- len et al., 1984).


Adsorption/Desorption Equilibria In typical situations the kinetics of the adsorption or desorption steps may be expected to be practically instantaneous relative to the diffusion transport processes and/or the rate of mass transport by advection into a given building zone; thus, it may not be necessary to explicitly model these kinetics. Although practically instantaneous, adsorption and desorption are limited by an equilibrium state. That is to say, for closed systems under steady conditions the rate at which adsorbate molecules (or ions) bind to adsorbent surfaces will eventually equal the rate at which they are released from the surface and the concentration in the free and adsorbed phases will remain constant at their respective equilibrium values, C: and Cs,. In functional notation, we may say that the adsorbed phase equilibrium concentration is related to the free phase equilibrium concentration and the thermodynamic state of the system defined by temperature, T, and gas phase pressure, P, as:

These equilibrium relations are generally unique to the adsorbate-adsorbent system being considered and, when reported for isothermal conditions at atmospheric pressure, are identified as adsorption isotherm. Experi- mentally determined adsorption isotherms, typically represented graphically, may be approximated by one of several equilibrium models including the Linear Model, Lang- muir Model, BET Model, and Freundlich Model - the first three having theoretical bases and the last being empirical.

ven, 1984; Slejko, 1985). This single parameter (i.e., K,) model may be expected to be appropriate for cases of physical adsorption of gases at very low concentrations (e.g., trace amounts) on homogeneous surfaces with ample sites available for adsorption.

Lungmuir Model: In the years 1916-1918 Lang- muir developed an adsorption isotherm model that accounted for the fact that the sites available for adsorption are limited (Ruthven, 1984; Slejko, 1985). The Langmuir model may be expressed, for our purposes, as :

(3)

where Cso is the surface concentration corresponding to complete coverage by a single monolayer (constant, for a given adsorbate- adsorbent system, and independent of temperature for the Langmuir assumption of a fixed number of sites available for adsorption) and KL is the temperature-dependent Langmuir adsorption coejjicicient. The Lang- muir model approximates the linear model for very low concentrations of adsorbate in the gas (or liquid) phase (i.e., when KL C: < < 1); thus we should expect near equality between the partition coefficient and the product of the parameters of this model as:

(4)

BET Model: Brunauer, Emmett, and Teller developed a model that accounted for the possibility of multilayer adsorption that may be expected to occur in physical adsorption (Ruthven, 1984):

Linear Model: This adsorption isotherm is defined by a simple linear relationship:

Cs, KpC, (2)

where K, is a temperature-dependent coeficient known as the partition coeficient (Ruth-

where KBET is the BET constant related to the energy of adsorption, Cso is the surface

Axley: Adsorption Modelling 151

concentration corresponding to complete coverage by a single monolayer (constant for a given adsorbate-adsorbent system), C’,,, is the air phase concentration corresponding to saturated conditions, and CX = (C’$C’,,,) is the reduced equilibrium concentration. The BET model has been found to be reliable in the range of reduced concentrations of 0.05 < (C7C’,J < 0.35 (Ruthven, 1984) for many gas adsorbate-solid adsorbent systems.

The BET model also approximates the linear model for very low concentrations of adsorbate in the gas (or liquid) phase:

thus we should expect near equality between the partition coefficient and the product of the parameters of this model as:

( Cso KBET/~* ) Kp = sat

Freundlich Model: An alternative two-parameter model that often fits experimental data well was promoted by Freundlich (Slej- ko, 1985):

where KF and n are the constant parameters associated with the Freundlich model.

From a somewhat simplified consideration of adsorption thermodynamics it may be shown that the temperature dependency of the adsorption coefficients used in the Linear, Langmuir, and BET models may be expected to be described by Arrhenius type relationships of the general form K = KgAwRT where K, is a constant, AH is an energy change associated with the adsorption process, R is the universal gas constant, and T is absolute temperature (Ruthven, 1984; Slejko, 1985). A plot of the logarithm of experiment- ally determined model constants log(K,), log(KL), or log (KBET) versus VT thus provides one indication of the correctness of the

particular model (i.e., it should be a straight line) and provides a means to evaluate the Arrhenius constants. Borrazzo recently reported this to be so for some nine hydrocarbons commonly found in indoor air adsorbed on nylon, wool, and glass fibres (An- delman, et al., 1989).

Building Zone Adsorption Dynamics Models The foregoing principles of adsorption science may be directly applied to the problem of modelling adsorption dynamics in buildings. We begin by considering the adsorption dynamics of a single contaminant species cx in a well-mixed building zone containing a volume of air with mass m, and a quantity of adsorbent material with mass ms (see Figure 1). The (average) concentration of the contaminant species in the zone air will be identified as “Cz and the average concentration in the adsorbent as CS, both expressed in terms of mass fraction (i.e.,(mass a)/(mass air) and (mass a)/(mass adsorbent) respectively).

If the mass flow rate of air into the zone is w and the species concentration flowing into the zone is “C,,, then the mass flow rates of species cx into the zone and out of the zone “W,,, and “W,,,, are simply equal to the product of the respective concentrations and the air mass flow rate as:

(9)

In an analogous manner we may describe the species mass flow rate by adsorption and desorption transport between the room and the adsorbent material in terms of discrete species mass flow rates, (IWads and UWdes, that will, in general, depend on the zone and adsorbent concentrations as well as the physical characteristics of the adsorbate species a, ad-

152 Axlev: AdsorDtion Modellina

Adsorbent Material aCs 9 ms

aWO“t= aCZW Building Zone I‘ aCz 9 mz

~

Fig. 1 Zone plus adsorbent control volume

Fig. 2 Zone control volume and adsorbent control volume.

sorbent material, and the detailed nature of the airflow in the room:

“Wads = fads (.C, “CS, ... )

“wdes = fdes (“CZ, “CS, ...

(10)

(W

Isolating the control volume surrounding the adsorbent from the control volume surrounding the zone air, the adsorption and desorption mass flow rates may be diagramma- tically represented as shown in Figure 2.

Limiting consideration for the moment to completely reversible adsorption (i.e., allow- ing no irreversible destruction or transformation of the adsorbate species a in the ad-

sorbent material or in the building zone) we may immediately write species mass balance relations for the adsorbent material, the building zone, and the systems as a whole as:

Building Zone Mass Balance

(“wout - “win) -k (awads - “Wdes) +

d“C, m, - = “Gz dt

Adsorbent Material Mass Balance

Axley: Adsorption Modelling 153 -

Building Zone and Adsorbent Material Mass Balance

daCs + m, - d%, ( a ~ o u t - “Win) + mz - dt dt

= “G, + aG, (14)

where we admit the possibility of generation of the contaminant species CI in both the building zone, “GZ and the adsorbent material “Gs.

Irreversible destruction or transformation of the contaminant species in the adsorbent or in the building zone may be introduced by inserting appropriate transformation kinetics expressions for either or both of the generation terms above, *GZ and “Gs. Thus, for example, if the analyst wishes to model linear destruction kinetics in the adsorbent, “Gs would be replaced by an expression of the form “Gs = (-msKS) uCs, where tcS is the first order rate constant associated with the destruction. To allow a focused consideration of (reversible) physical adsorption, irreversible transformation will not be considered further.

Equilibrium Adsorption To provide some insight into the dynamics of this coupled system it is useful to consider a limiting case where the adsorbed species concentration remains at all times in equilibrium with the zone concentration (i.e., diffusion, adsorption, and desorption transport rates are practically instantaneous relative to the flow transport dynamics). More specifically, if the equilibrium relationship is described by the Linear Model:

“Cs = K p W Z (154

or:

dUCs - -- - dt

the mass Equation fies to:

daC, KP - dt

balance for the system as a whole, 14 with Equations 8 and 9, simpli-

Linear Equiltbrzum Adsorption Model Wac, + (m, + Kpm,) daC, __ = CIE (16a)

(16b)

dt

a E = (aG, + aG, + WaCi,)

where all forcing terms have been collected on the right-hand side to define the system excitation “E.

Alternatively, if the equilibrium relationship is described by the Langmuir Model:

or:

the mass balance for the system as a whole simplifies to:

Langmuir Equilibrium Adsorption Model

Wac, + (m,+ ( (1 + KL~C,)’ dt

For both cases the adsorbent acts to increase the participating mass of the building zone from that of the mass of the air in the zone, mZ, to this quantity plus the mass of the adsorbent, scaled by a factor. For the Linear Model, the effective participating mass is:

For the Langmuir Model the scaling factor includes a nonlinear dependency on the zone concentration:

but for situations where the zone concentration does not vary greatly, this factor will remain reasonably constant as well. Further-


more, for those situations where the zone concentration is small enough so that the KL"CZ term in the denominator may be neglected, then this factor should be expected to equal the partition coefficient of the Line- ar Model (as discussed earlier).

Dynamic Character of the Equilihum Adsorption Model For a single well-mixed zone under conditions of constant flow the system time constant, T, (or its inverse, the nominal air change rate);

T = (mzlw) ; w = constant (21)

provides a concise and convenient measure of the system's dispersal dynamics. For a well-mixed zone under conditions of constant flow with equilibrium adsorption dynamics added it follows that the (instantaneous) system time constant, 7, becomes:

T = (meffective/w) ; w = constant (22)

Adsorption thus has the effect of increasing the system time constant in proportion to the adsorbent mass.

Example Application: Equilibrium A d s q t w n in a Room To fix these ideas and to evaluate the potential significance of the participating mass of- fered by real adsorbents in building construction it is usehl, at this point, to consider a practical example. To this end, consider a room 5 x 8 m in plan with a 3 m floor to ceiling height, 25 mm thick nylon carpeting, and 13 mm thick gypsum board walls and ceiling.

The approximate weight of the gypsum board may be estimated at 10 kg/m2 and that of the nylon fibres in the carpeting at 2.5 kg/ m'. Thus, given the room geometry, the total mass of gypsum, mgyp, board is approximately 11.8 x lo5 g and the mass of the nylon fibres in the carpet, mnylon, approximately 1.00 x 105 g.

At standard conditions of temperature and pressure the density of the air in the room is very nearly 1.2 kg/m3; thus the mass of air in the room, m,, is 1.44 x lo5 g.

Researchers at the Oak Ridge National Laboratory report a partition coefficient for the formaldehyde (HCH0)-gypsum wallboard system at 23 "C and 50% RH to be approximately 5.5 g-air/g-gypboard (Matthews et al., 1987). (Silberstein also determined the same value for the partition coefficient but did not publish this result (Silberstein, 1988; Silberstein, 1989b).) If we assume that equilibrium adsorption conditions prevail, then for this measured partition coefficient and the representative room considered here, the gypboard may be expected to contribute a participating mass of HCHo-gFK P m gyp = 5.5 (g-air/g-gypboard) x 11.8 x lo5 (g-gypboard) = 65 x lo5 (g-air) that increases the participating mass to 46 times the room air mass alone. This increase in participating mass would result in an equal increase in the time constant of the well-mixed zone, for conditions of constant airflow. Thus, for example, for a well-mixed zone without adsorption having a time constant of 1 hour (i.e., an air exchange rate of 1 air change per hour) the gypboard would alter the time constant for HCHO dispersal to 46 hours.

In another study (Andelman et al., 1989) partition coefficients for nylon, wool, and glass fibres are reported for some nine differ- ent aromatic and chlorinated hydrocarbons, designated simply as H-C-C1 here. For nylon fibres these partition coefficients range from 2.4 to 16.8 x lo4 g-air/g-nylon fibres.'

If, again, we assume that equilibrium adsorption conditions prevail, then the nylon fibres in the carpet of the representative room being considered may be expected to contri-

' The reported partition coefficient of 0.2 to 1.4 (cm3- air/g-nylon) was convened to the mass fraction units of (g-air/g-gypboard) based on an air density of 1.2 kg/m3.

Axley: Adsorption Modellins 155

bute a participating mass of HCC1-ny’onKpmnyton = (2.4 to 16.8) x (g-air/g-nylon) x 1.0 x lo5 (g-nylon) = (24 to 168) (g-air). In this case the room air mass would be effectively in- creased only slightly by 0.01 to 0.10%.

These simple calculations, based on the equilibrium adsorption assumption, indicate that adsorption can dramatically alter the dynamics of contaminant dispersal in rooms when the product of the adsorbent mass and its partition coefficient are significant. Some initial experimental studies of adsorption in rooms support this conclusion (Matthews et al., 1987; Silberstein, 1989a).

Boundary layer Diffusion Controlled Adsorption In general, adsorption mass transport rates are determined by the individual transport rates of the diffusion processes and the surface adsorption kinetics that together account for the multiple steps in the overall adsorption transport process. Throughout the adsorption literature, however, it is emphasized that adsorption dynamics, in most practical situations, are invariably controlled by diffusion transport (the adsorption kinetics step is practically instantaneous) although exceptions to this general rule exist (Ethier and Kamm, 1989). With this in mind, we will consider models for room adsorption dynamics that account for external film diffusion based on the assumptions that:

(a) equilibrium exists at the solid-gas interface at all times as defined by one of the equilibrium isotherms considered above (ie., the kinetics of adsorption is practically instantaneous); and

(b) the system remains practically isothermal (i.e., the rate of heat generation due to adsorption is negligibly small and the heat transfer processes are sufficiently large to maintain isothermal conditions).

Models for room adsorption dynamics that account for macropore diffusion and micro-

pore diffusion will be considered in subse- quent studies.

Adsorption mass transport rates from the room air to solid (ie., nonporous) adsorbents with reasonably smooth surfaces may be expected to be controlled by the rate at which a contaminant species diffuses from the bulk phase to the surface of the adsorbent. The diffusion mass transport rate will, in general, depend on the details of the contaminant concentration and velocity fields in the room, details that are both difficult to measure and to determine analytically (e.g., by computational fluid dynamics) and, conse- quently, details that normally will be neither known nor easily determined. For wall surfaces and other larger flat surfaces in rooms it is likely that any flow that exists will be directed more or less parallel to these surfaces, forming a boundary layer over the surface of interest. Therefore, to obtain esti- mates of diffusion mass transport for these surfaces, we may reasonably apply some key results from boundary layer theory.

The adsorbent surface may be thought to be separated from the room air by afilm of air (i.e., the boundary layer) over which the contaminant concentration varies from a near-surface, air-phase concentration, CZ, to the bulk air-phase concentration in the zone, ’XZ, as illustrated in Figure 3.

If mass transport to the adsorbent is due only to film diffusion (e.g., there is no bulk flow of Contaminant through the adsorbent due to pressure gradients within or across the adsorbent) and the air-phase concentrations are not varying too rapidly, then the net mass transport rate from the bulk phase to the surface, uWs, may be approximated using steady-state mass transfer relations from boundary layer theory (White, 1988) of the general form:


3 aC;

aCz

Fig. 3 Boundary layer diffusion controlled adsorption

where: - hln

P

As

a - airD

ReL < 500,000 (244 is the average film mass transfer coefficient acting over the adsorbent surface (lengthhime) - is the film density of air, the average of hm

a - airD the bulk and surface densities ReL > 500,000 ; SC = 1.0 is the projected surface area of the adsor-

bent

S ~ L E = 0.037 ReL W k Y 3 ; for

(24b)

The average film mass transfer coefficient may be measured directly (e.g., see Matthews et al., 1987) or indirectly using the naphthalene sublimation technique (White, 1988). It may also be estimated from published heat transfer coefficients or correlations (e.g., see Khalifa and Marshall, 1990) using the so- called heat and mass transjir analogy. Finally, several published mass transfer correlations are available that relate a dimensionless form of the average jilm mass transport coefficient, the average Sherwood number, S % , with the surface flow Reynolds number, ReL, and the air phase Schmidt number, Sc (White, 1988) (see also the closely related correlations for deposition velocity discussed by Nazaroff (Nazaroff and Cass, 1987; Nazaroff and Cass, 1989)). For flow parallel to a flat plate, a flow condition that may be considered to be representative of airflow past interior building surfaces, White provides the following correlations;

where:

UL ReL = __ 1,

U is the mean fluid velocity (parallel to the surface) outside of the boundary layer is the length of the surface in the direction of the flow

a-airD is the kinematic viscosity of the air phase

is the molecular difisivity of the binary a-air system being considered.

L

v s c = ~

1,

a-aim

For many gases in air the molecular diffusivity is practically constant with concentration (although it varies directly with temperature and inversely with pressure) with values, at room temperature aild atmospheric pressure,


ranging from to m2/s and the Schmidt number is typically close to 1.0 (e.g., 0.2 to 2.0).

With the expression for film diffusion transport, Equation 23, in hand we may for- mulate equations that describe the system adsorption dynamics by first demanding equilibrium at the adsorbent surface using either the Linear Model:

@Cs = KpaCi (25)

or the Langmuir Model:

‘ycs = aCs,,KLa~d (26) 1 + K L ~ c ~

and then substituting these equations and Equations 8, 9, and 23, into the basic mass balance relations for the system, Equations 12 and 13, to obtain, after simplification:

(a) for the Linear Model:

+ m Z - d a C ~ = ‘YG, (27a) dt

= aGs or, alternatively:

Boundary Layer Dffusion Controlled Ads* tion Model

where we have collected all forcing terms on the right-hand side as the system excitation vector {aE):

(b) for the Langmuir Model:

mS dorC, = a~~ dt

or, alternatively:

Langmuir Bounahry Layer Diffusion Controlled Adsorption Model -

dt = {aE} ( 2 8 ~ )

(The Langmuir Model Equations 28 are valid only up to the condition of complete coverage of the surface (i.e., (Cso - C,) > 0:) at which point Equations 28 become singular.)

Concise Matrix Notation The Linear Model and Langmuir Model variants of the system adsorption dynamics


equations considered so far may be represented in concise notation as:

[“W]{aC) + [“MI = {aE} (29a)

where:

(a) for Equilibrium Adsorption:

dt

{“C} = “CZ

[“W] = w

Linear Model

[aM] = mz + KpmS (294

or

Lungmuir Model

[“MI = m, + ( aCsoKL ) ms (29e) (1 + KL“CZY

(b) for Boundaty Luyer D z f i i o n Controlled Adsorptwn

Linear Model

(290

or

Lungmuir Model

Note that the Langmuir Model introduces nonlinearity in the mass matrix, [“MI, of the Equilibrium Adsorption (EA) Model (i.e., with respect to zone concentration, “C,) and in the transport matrix, [“W], of the Boundary Layer Diffusion Controlled Adsorption (BLDC) Model (i.e., with respect to zone concentration, “Cs).

Dynamic Character of the BLDC Adsorption Model The essential dynamic character of the (sca- lar) EA Model is contained in the system time constant, t, or equivalently the system eigenvalue, A = Vt which is equal to A =

pM]- ‘[“W] = (w/meffective). Analogously, the essential dynamic character of the coupled BLDC Model equations is contained in the coupled system eigenvalues that satisfy the characteristic equation of the system:

det I [“MI’ [“W] - h [I] I = 0 (30)

where, for the Langmuir Model, this eigenvalue problem would be evaluated at each specific state of the adsorbent concentration, OLCS, given the nonlinearity of the mass transport matrix in this case. In general, this eigenvalue problem has two solutions, {Al, A2); the inverses of the system time constants, {t,, t2}. For the Linear Model, by substituting Equations 29g 8z h, into Equation 30 we obtain two solutions of a quadratic relation that correspond to these eigenvalues:

I -

(29i)

Axley: AdsorDtion Modellina 159

It is interesting to consider three limiting cases of this eigenvalue solution:

No D f i s i o n : When diffusion from the building zone to the adsorbent surface drops to zero, “h,pA, = 0, the system eigenvalues correspond to the eigenvalue of the well- mixed zone without adsorption as expected:

-

when: hm p As = 0 (32)

No Infiltratwn: When the infiltration into the zone drops to zero, W = 0, the system eigenvalues become :

[E2] =

0

when: w = 0 (33)

Note: the first eigenvalue increases with increasing “h,pAs with the result that the system time constant, T~ = l/Al, approaches zero as the mass transfer rate increases.

Equilibnum Adsorption: If the rate of diffi- sion, “h,pAs, is large relative to the infiltration mass flow rate, w, then the system eigenvalues approach those corresponding to the Linear Equilibrium Adsorption Model:

[ 2 ; 2 ] -

when: h< p As > > w (34)

Again the first eigenvalue increases linearly with diffusion mass transfer rate. (The proof of Equation 34 is rather indirect. The sums (“FmpAs + w) in Equation 31 may be replaced by “&,PA, when “h,pA, > > w and the resulting equation simplified and expanded using a Taylor’s series approxima- tion of the square root term to obtain the final result.)

Example Application: Boundary Layer D i h w n Controlled Adsorption in a Room To fix these ideas and to evaluate the significance of the diffusion transport, reconsider the room studied earlier using the Adsorp- tion Equilibrium Model, now applying, instead, the BLDC Model. This time we will limit consideration to the adsorption of formaldehyde, HCHO, by gypsum board having the partition coefficient considered earlier, KP = 5.5 g-air/g-gypboard. The area of the adsorbent in this case is As = 118 m2. Assuming air flows uniformly along the length of the room, a reasonable estimate of the effective length of the adsorbent surface would be L = 8 m and an estimate of the free stream velocity of the airflow would be U = 8 m/h = 0.00222 m/s (i.e., given the air change rate of 1 air change per hour, the volumetric flow divided by the room cross- sectional area). Thus the effective Reynolds number would be (with the kinematic viscosity of the air of v = 1.5 x m2/s):

= 1,173 ReL = __ UL = (0.0022)(8)

V 1.5 x

For a Schmidt number of 1.0 we thus obtain the average Sherwood number for this problem, from Equation 24a, as:

~

ShL = 0.664 ReL y2Scfi =

0.664(1173)fi (l.O)v3 = 22.7

For a representative gas diffisivity of HCHO-

”“D = 1.4 x m2/s we obtain an estimate of


the film diffusion mass transfer coefficient from Equations 24 as:

h,=sh~( HCHO-aia ) = 2 2 . 7 ( 1.4 ) = 3.97 x m/s

With these values in hand, an infiltration rate of W = 40 g/s (i.e., based on 1.0 air change per hour and an air density of 1,200 g/m3), and the zone and adsorbent masses computed earlier - mZ = 1.44 x lo5 g and mgyp = 7.8 x lo5 g - we may apply Equation 31 to compute the eigenvalues for the gyp- boardroom system:

kl = 3.17 x

A2 = 1.15 x

s-l = 1.14 hr-’ or tl = 0.876 hr

s-l = 4.14 x h i ’ or t2

= 242 hr

revealing a bimodal response characterized by a very long time constant of 242 h (i.e., 10.1 days) and a quick response somewhat smaller

than the nominal system time constant without adsorption (i.e., (mzN) = 1.0 h). This sort of response behaviour has, in fact, been observed in experimental studies (Dunn, 1987; Dunn and Tichenor, 1987; Matthews et al., 1987; Seifert and Schmahl, 1987; Silber- stein, 1989a); the results of these studies will be considered subsequently. The response of this system, and other systems governed by the BLDC Model, is particularly sensitive to boundary layer diffusion rate as characterized by the average Sherwood number. Plot- ted below, for this example case, is the variation of the system time constants for a variation of average Sherwood number.

It is seen that the first time constant, 71 di- minishes from 1.0 h to zero as the second time constant, T~, drops from very high values for an average Sherwood number near zero to an asymtoptic value for large values. In the present case this asymtoptic value is 30.8 hours and is equal to the time constant we obtained for this case using the Equilibri- um Adsorption Model (i.e., as predicted by Equation 34).

1

71 (h)

0.8

0.6

0.4

0.2

0

0 1000 2000 3000 4000 ShL 5000

Fig. 4 Time constont variation for the exomple application.

400

320

240

160

80

0


We may conclude, then, that at one extreme, with a near-zero rate of diffusion mass transfer, the system will respond to changes of concentration in the zone as a simple well-mixed zone with a time constant nearly equal to (mZrJu). For nonzero rates of diffusion the system will rapidly respond to changes of concentration with a time constant somewhat less than the simple well- mixed zone time constant; then, if the second mode is excited (e.g., if the contaminant is generated or concentrated in the adsorbent) a very slowly changing response will be observed. Diffusion in smaller macropores and/or diffusion in micropores (e.g., having pore diameters approaching molecular sizes) may be expected to have, in essence, a similar effect to that produced by very low mass transfer rates in the BLDC Model and may be responsible for this form of observed behaviour. At the other extreme, very high rates of film diffusion mass transfer lead to the simple behaviour predicted by the EA Model as expected.

Surjbce Smoothness The BLDC Model presumes diffusion from the zone air to a smooth surface. This begs the question: what is meant by smooth? In the present context, it would seem reasonable to define an adsorbent surface as being “smooth” (i.e., sufficiently smooth to apply the BLDC Model) if:

(a) the surface roughness is small relative to the thickness of the boundary layer; and

(b) the structure of the surface roughness is coarser than that considered to be macroporous (ie., the width of surface irre- gularities or indentations are much larger than the mean free path of the dif- fusing contaminant species) so that the diffusion obeys Fick’s Law.

The thickness of the boundary layer, 6, may be estimated as (Beek and Muttzall, 1975):

L 6 == ShL

(35)

For the example considered above we obtain 6 = 8 m / 27.9 = 0.29 m which is certainly large relative to the (likely) physical roughness of the gypboard surface, thus justifying the application of the BLDC Model in this case.

Adsorption Element Equations for General Multi-Zone Analysis Contaminant dispersal analysis theory may be formulated using an element assembly approach wherein equations approximating the behaviour of the building airflow system or subsystem, the system equations, are assem- bled from equations that describe contaminant mass transport in discrete portions or elements of the system model. The details of this approach are presented elsewhere (Axley, 1989). In this section we shall simply describe element equations that correspond to the adsorption models used above in the single zone case that may be used for multizone system models of arbitrary complexity.

In general, for the dispersal of species a, contaminant dispersal element equations have the following form:

where {aCe) is a vector of contaminant species a discrete concentration variables associated with the element “eyy, a subset of system discrete variables {%I, and {”$) is a vector of element derived contaminant species generation rates. The element matrices [“W] and [“me] are square transformation matrices known as the element species transport matrix and species mass matrices respectively.

The element equations corresponding to the models presented above may be derived directly from Equations 29 by simply sub- tracting the single zone mass and airflow transport terms from the system equations:


Equilihum A d s q t w n Element: a single node element, say element “e”, with an adsorbent mass of m: associated with a well-mixed zone “i” with :

{ w e } = {ac;} (374

= [OI (37b)

Linear Model

[amel = KpmeS[l] (374

or

Boundary hyer D i m w n Controlled Ads+ tion: a two node element, say element “e”, associated with a well-mixed zone “i” and an adsorbent “j’’ with an adsorbent mass of mes:

Linear Model

[a*] = (%,A3

or

Langmuir Model

[a*] = ( E p AeS)

-1 1 -

KP

KP

1 -1 -

where all variables are defined as earlier (those variables with a following superscript “e” are specific to the adsorption element “e”). Again, note that the Langmuir Models introduce nonlinearities, now at the element level, and, for the boundary layer diffusion controlled case, are defined for adsorbent concentrations less than

Implementation Using Existing Multi-Zone Program The Linear Models for both cases above may be implemented using existing multi-zone contaminant dispersal analysis programs that provide only well-mixed zones and discrete flow paths. For equilibrium adsorption modelling the analyst need only increase the associated well-mixed zone mass by K,m: to account for mass transport to the adsorbent.

The boundary layer diffusion controlled case requires a slightly indirect tactic. By re- defining the element concentration variables as:

the element arrays may be rewritten as:

Linear Model

[aw‘] = ( g p AeS) 1 -1 -1 1

These element equations have the form of a subassembly of a well-mixed zone linked to inflow and outflow flow elements; thus, these element equations may be added to existing multizone contaminant dispersal models by linking a pseudo-well-mixed zone


to the actual well-mixed zone containing the adsorbent by two pseudo-flow paths (i.e., flow elements) having equal and opposite flows. By setting the pseudo zone mass to K,m: and the pseudo-mass flows to (he, p ks), the behaviour of the adsorbent will be modelled although the computed concentration response in the pseudo-well-mixed zone must be scaled by K, to obtain the species concentration in the adsorbent, q.

Application and Validation In this section we will compare results from the EA and BLDC models with the results of adsorption/desorption studies of formaldehyde on gypsum wall board reported by researchers at Oak Ridge National Lab (ORNL) and the National Institute of Stan- dards and Technology (NIST). Each of these groups developed a semi-empirical model. Similar semi-empirical sorption models and closely related sourcehink models have been reported by others (Clausen et al., 1990; Dunn, 1987; Dunn and Tichenor, 1987; Ti- chenor er al., 1990; Tucker, 1988). In contrast, the EA and BLDC models examined here are based on the theory presented above. Be- fore considering the details of these studies two other studies deserve mention.

Cleary and Sonderegger, at Lawrence Berkeley Lab, and Thomas and Burch, at

NIST, developed models to predict the adsorption and desorption of water in building materials (Cleary and Sherman, 1984; Cleary and Sonderegger, 1984, Thomas and Burch, 1990). Both groups considered models similar to the BLDC model developed above, although the NIST researchers presented this approach as a limiting case of a more general model that included diffusion within the porous adsorbent. They combined boundary layer solutions with equilibrium adsorption isotherms to form an adsorption dynamics model. In comparisons of modelled response to measured response these models proved accurate.

ORNL Study Researchers at the Oak Ridge National Lab investigated the adsorption and desorption characteristics of gypsum wall board specimens in a small test chamber and demonstrated that an empirical three-parameter ex- ponential response model could be fitted to measured response data with reasonable accuracy. A diagram of the test set up is shown in Figure 5.

In this investigation six specimens of gypsum board (0.25m x 0.24m) were placed in one of a pair of dual test chambers (volumes of 0.20 m3) and air, at 23 "C and 50% RH, containing a controlled concentration of formaldehyde, "C,,, was passed through this

GvDboard Chamber Vol = 0.20 23OC. 50%

m3 RH

I Reference Chamber

Vol = 0.20 m3 23"C, 50% RH

Fig. 5 ORNL Dual chamber test facility.

164 Axlev: Adsorotion Modellina

chamber and an identical reference chamber at constant and identical rates. The exit formaldehyde concentrations from each of the chambers, “c,, and “creb were measured at periodical intervals and the ratio, R, of these concentrations, adjusted for initial conditions and normalized, was reported. Finally, three-parameter (i.e., A, Z, and 7) exponen- tial relationships of the following forms:

R = Z + (A-Z)(l-e-t/T) for adsorption (40)

R = (A-Z)e-dT for desorption (41)

were fitted to measured response data. It must be emphasized, to use the words of these researchers, that “Although exponen- tial sorption and desorption processes may represent the physical behavior of the experimental system, the precise form of the models is empirically derived”. The gypsum wall board chamber was modelled using the Line- ar variants of the Equilibrium Adsorption (EA) and Boundary Layer Diffusion Con- trolled (BLDC) models, Equations 16 and 27 proposed above, and the reference chamber was modelled as a well-mixed zone, as shown in Figure 6. (For these and the following

~ ~ ~ ~ ~ ~ ~~~~

computational studies the density of the gypsum wall board was estimated to be 769,000 &m3, the molecular diffusivity of formaldehyde in air at 23 “C was estimated to be 1.39 x m2/s using the Chapman-Enskog for- mula (Bird et al., 1960; Satterfield, 1970), and the kinematic viscosity of air was estimated to be 1.53 x m2/s.)

The response of these modelled chambers was computed, using CONTAM87 (Axley, 1988), for a twenty-eight-day adsorption test followed by a four-day desorption test. The results of these analyses are compared in Fig- ures 7 and 8 to the ORNL empirical model along with an indication of its uncertainty (i-e., the + err and - err curves) based on the reported standard errors of the model parameters obtained from nonlinear regression analysis.

It is seen that the initial response appears to be well-captured by the EA model and the longer term response is better captured by the BLDC Model. An examination of the ORNL raw data (see Figure 3 of Matthews et al., 1987) reveals greater scatter, especially in the bend of these curves, than indicated by the error bounds shown above; thus a response falling between the EA and BLDC

c ref

llllll Illill

mz = 240 g meff = 20,106 g

C in Cin Cin Reference Chamber Gypboard Chamber Gypboard Chamber Well-Mixed Model EA Model BLDC Model

Fig. 6 Idealizations of ORNL test Chambers.


1.20

R

1 .oo

0.80

0.60

0.40

0.20

0.00

- - ORNL+err

Fig. 7

1 .oo

R'

0.80

0.60

0.40

0.20

0.00

28 24 Days 0 4 8 12 16 20

Comparison of modelled adsorption response and ORNL empirial model.

4 Days 0 1 2

Fig. 8 Comparison of modelled desorptian response and ORNL empirical model.


Models would be considered practically accurate. The BLDC model is sensitive to the value used for the average film mass transfer coefficient L. In the present case, this coefficient was estimated using the correlations defined by Equations 24 which require an estimate of the flow Reynolds number. The flow Reynolds number was based on an estimate of the mean flow velocity in the chamber equal to the volumetric flow rate divided by the square of the cube root of the chamber volume (i.e., an estimate of the chamber cross-sectional area). More detailed infor- mation of the chamber geometry could not be obtained. This resulted in an estimate of Re = 4.25, an extremely low value; it is cer- tain that the obstruction of the test specimens would have resulted in a larger estimate. The effect of higher Reynolds numbers (i.e., greater diffusion rates) would bring the BLDC Model curve closer to the EA Model curve in the early response and thus improve the accuracy of the modelling.

The desorption response falls between the EA and BLDC model predictions. Again, the accuracy of the estimate of the flow Rey- nolds number is critical; a larger and, pre- sumably more reasonable, Reynolds number would have improved the BLDC modelled results by “pulling” the BLDC curve closer to the EA curve.

The adsorption dynamics models presented in this paper depend critically on the equilibrium adsorption isotherm model employed, in the present case, on the partition coefficient K,. A partition coefficient of 5.5 (g-air/g-gypboard) was used for these studies and the NIST studies below. This is a value that was determined by both ORNL and NIST investigators using a rather intuitive method based on integrating the desorption response data that corresponds, in essence, to an integral form of the EA Model equations, Equations 16, or:

where one integrates response and excitation values over an arbitrary time interval, t1...t2, and evaluates the change of concentration over the same interval:

Solving this equation for the partition coefficient:

K, = msAaCz mS

defines the method used by these investigators to determine K, (although both groups neglected to consider the second term on the right-hand side, (mz/ms), which was practically insignificant in both studies).

This determination of the partition coefficient implicitly assumes linear equilibrium adsorption behaviour and therefore stacks the deck in favour of the Linear Equilibrium Adsorption Model. Silberstein, in the NIST study, also, however, used an alternate strat- egy to determine the partition coefficient and obtained practically identical results, thereby providing some validation of the value used here. It would be best to determine this equilibrium adsorption parameter or, better, the equilibrium adsorption characteristics using one of several available independent means to do so (Ruthven, 1984).

NIST Study In tests to study formaldehyde levels in a test house due to emissions from a variety of pressed-wood products, researchers at the National Institute of Standards and Tech-


............................................... ........................................................... \ , , , 23OC \ , \

\ \

5 23°C 50%RH CZ

\

\

llllll I llllll m,u = 161,056 \

\ \

, \ \ , ,

: w=2201 g/ \

\ \ \ c=o .......................................................... c=o ..............................................

NIST Chamber NIST Chamber EA Model BLDC Model

Fig. 9 Idealization of the NIST test chamber.

nology found that it took days to reach equilibrium concentration levels and suspected that these long delays were due to adsorption and desorption (Grot et al., 1985). To invest- igate this phenomena more directly these researchers conducted a separate investigation in a “medium-sized” dynamic measuring chamber, formulated a semi-empirical model of the dynamics of the sorption processes, and numerically extracted values for the model parameters from measured data (Sil- berstein, 1989a). Unlike the ORNL study, the environmental chamber employed was nearly room size, the gypsum board was attached to the walls of the chamber, as it would be in building construction, exposing a single side, and the proposed model was based, qualita- tively, on the likely physical mechanisms responsible for adsorption. This model was not, however, directly formulated from fundamental principles of adsorption science and therefore must be considered a semi-empirical model.

In the NIST study a 1.2 m by 2.4 m speci- men of 0.013 m thick gypsum wall board was secured to a wall of a “medium size” test chamber with inside dimensions of 1.22 m x 2.44 m x 0.61 m. This test chamber was placed in a closed environment maintained

at 23 “C and 50% RH, a constant fresh airflow rate through the chamber of 1.01 air changes per hour was maintained, and, for adsorption studies, formaldehyde gas was generated within the chamber at a constant rate.

The NIST chamber was modelled using the Linear variants of the EA and BLDC models estimating the density of the gypsum wall board, the molecular diffusivity of the formaldehyde, and the kinematic viscosity of the air, as above, and employing the partition coefficient of 5.5 (g-air/g-gypboard). Dia- grams of these models are shown in Figure 9. The response of these models to an 11-day adsorption test, with formaldehyde generated within the chamber at a constant rate of 0.24 mgh, followed by a 10-day desorption test, without the generation of formaldehyde, were computed using CONTAM87. The results of these computational analyses are compared to the measured data in Figure 10.

Regrettably, there was some confusion re- garding the actual formaldehyde generation rate and infiltration rate used during the test. Test data were obtained from two draft reports of the tests (Silberstein, 1988; Silber- stein, 1989b). In one of these reports an infiltration rate of 1.01 ACH was reported but the


0.15

[HCHO] rng/rn3

0.10

0.05

0.00

I I I I I

20.00 l6.O0 Days 0.00 4.00 8.00 12.00

Fig. 10 Comparison of modelled response and NlST measured data. Formaldehyde was generated within the chamber from day 0 through day 10.

uncertainty associated with this value was great. The formaldehyde generation rate was reported to be 0.313 mg/h for the adsorption tests yet hypothetical steady-state concentration levels for an empty chamber were reported that indicated the actual generation rate to be 0.24 mg/h. Computations were based on an air exchange rate of 1.01 ACH and a generation rate of 0.24 mg/h. (Silberstein suggested, in a personal communication, that an air exchange rate closer to 1.3 ACH and a generation rate of 0.313 mg/h might be more appropriate. Since the ratios of these values are nearly identical to the values used, the computed results would be very similar.)

As in the ORNL case, the measured response very nearly falls between the EA Model and the BLDC Model responses. Again the BLDC response was especially sensitive to the estimation of the average mass transfer coefficient that was, in this case, estimated using the correlations with the flow Reynolds number reported above. In this case the Reynolds number was estimated to be 110, a relatively low value. In

Figure 10 the BLDC Model response is plot- ted for a ten-fold increase in Reynolds number (i.e., a fl increase in diffusion mass transfer rate) and it may be seen that the modelled response in this case more closely follows the measured response. Although a somewhat higher Reynolds number would have led to excellent agreement, no reliable means are available to estimate the flow Rey- nolds number accurately, clearly a limitation of the use of these correlations.

Conclusion Two families of macroscopic adsorption models have been formulated, based on fundamental principles of adsorption science and technology, that may be used for the purposes of indoor air quality analysis. The first family of adsorption models - the Equi- l i h u m Adsorption (EA) Models - are based upon assuming that zone air concentrations and adsorbent concentrations remain in equilibrium at all times, where the equilibri-


um condition may be defined by any number of equilibrium isotherms. Two members of this family are explicitly discussed, the first based on the linear isotherm model and the second on the Langmuir isotherm model. The second family of adsorption models - th Boundmy Lyer Diffusion Con- trolled Adsorption (BLDC) Models - are based on the use of boundary layer theory to model the rate of mass transfer from the bulk phase through the boundary layer to the solid adsorbent phase and the assumption that equilibrium exists at the solid-gas interface. Again this equilibrium may be modelled using any one of several adsorption isotherm models. Two members of this model are explicitly discussed, one based on the linear adsorption isotherm model and the other on the Langmuir model, as before.

An analysis of the linear variants of the EA and BLDC models, presented in this paper, demonstrated that the EA model and the simple well-mixed zone model (ie., without adsorbent) are simply limiting cases of the BLDC model. This analysis provides for- mal proof of the physical consistency of the BLDC approach.

It is shown that the linear variants of both the EA and BLDC models lead directly to contaminant dispersal sorption elements that may be used directly with existing contaminant dispersal analysis programs such as the CONTAM family of programs. These models have the attractive feature that they depend on a single parameter, the partition coefficient, that may be measured using one of several well-documented methods (Ruth- ven, 1984). They must be expected, however, to be limited to the modelling of physical adsorption of trace contaminants because the linear adsorption isotherm upon which they are based is generally only valid for very low adsorbate concentrations. The BLDC cap- tures the biphasic or bimodal nature that has been observed in adsorption and desorption tests (Dunn, 1987; Dunn and Tichenor, 1987; Matthews et al., 1987; Seifert and Schmahl,

1987; Silberstein, 1989a) but it relies on an estimation of an average film mass transfer coefficient, an uncertain problem that demands knowledge of the nature of airflow in the zone which may not be available. DiffL sion through a boundary layer must be expected, however, to be an important step in practically all cases of adsorption and, short of complete solution of the microscopic equations governing the flow in a zone, the boundary layer solutions to this problem are the best available; the analogous solutions for boundary layer heat transfer have served the building energy simulation community well for some fifty years (Khalifa and Mar- shall, 1990).

Applications of the linear variants of the EA and BLDC models to formaldehyde adsorption and desorption tests were presented to provide some first validations of these simple models. In these studies measured response data, reported by NIST and ORNL researchers, were compared with modelled response and the agreement was good. These practical applications and the example applications considered to clarify the develop- ment of the models clearly indicate the significant, in fact, overwhelming effect that simple physical adsorption and desorption transport processes may have on the dispersal of contaminants in buildings. Further- more, physical adsorption must be expected to be pervasive depending not so much on special characteristics of adsorbate-adsorbent systems but simply on the availability of sufficient adsorbent surface. To date, sorption dynamics have seldom been considered in practical indoor air quality analysis and, im- portantly, in the closely related field of tracer gas analysis of building airflows. Both areas of analysis should be re-evaluated in light of the implications of the results reported here.

The success of the work reported by Cleary (Cleary and Sherman, 1984; Cleary and Sanderegger, 1984) and Thomas (Tho- mas and Burch, 1990) in modelling moisture sorption in building materials supports the


hope that the nonlinear variants (e.g., Lang- muir and BET adsorption isotherms) of the EA and BLDC models will prove effective in modelling physical sorption mass transport when adsorbate concentrations vary over wide ranges and may not be at trace levels (e.g., moisture variations in buildings in general), but this will have to be investigated.

Not presented in this paper are physical adsorption and desorption models based on existing knowledge of macroporous and microporous diffusion processes. It is clear, however, that macroscopic models could be readily developed based on finite element approximations to available governing differen- tial equations for these classes of diffusion (e.g., the so-called Glueckauf Model for the diffusive transport of an adsorbate down a cylindrical pore in a macroporous adsorbent (Holland and Liapis, 1983)). The numerical studies reported clearly indicate the potential importance of diffusion rate limitations -

diffusion resistance can lead to extremely long time constants in adsorption dynamics. It is entirely possible that the exponentially- diminishing “source” models reported by some investigators and the apparent irreversible, but small, conversion of formaldehyde in gypsum wall board are both due to more deeply adsorbed (e.g., within macroporous or microporous structures) constituents. It seems likely that a consideration of these more rate-limiting diffusion steps in the adsorption process may lead to some rationally based physical “source” models.

Acknowledgments The research reported was supported by the U.S. National Institute of Standards and Technology (NIST) and the U.S. Department of Energy (DOE) and this paper was abstrac- ted from a report prepared for these agencies (Axley, 1990).

Axlev: Adsorption Modellinq 171

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