Generalized criteria for predicting the dynamics of continuous-flow chemical systems—I. Application to adsorber systems

Pergamon

Chrmwi

Zngmremy

S&wr,

Vol. 51. No 13. pp. 3519-3536, 1996 Coovrieht ‘c” 1996 Elsewer Sacnce Lid Prlntcd’k &eat Britam All nghts reserved 0009 2509.96 $15.00 + 0.00

0009-2509(95)00412-2

GENERALIZED CRITERIA FOR PREDICTING THE DYNAMICS OF CONTINUOUS-FLOW CHEMICAL SYSTEMS-I. APPLICATION TO ADSORBER SYSTEMS R. GUTSCHE* Department

of Process Systems Engineering, Technical University Berlin, Institute of Process Design, Object 10.5, Rudower Chaussee 5, D-12484 Berlin, Germany

and Plant

and

K. HARTMANN Department of Process Systems Engineering, Brandenburg Technical University of Cottbus, Faculty Environmental Sciences and Process Engineering, Karl-Marx-Str. 17, D-03044 Cottbus, Germany (First receined

10 January

1995; accepted

in revisedform

8 December

of

1995)

Abstract-The

theory of Markov chains is used to develop a procedure for deriving a general dimensionless parameter that allows prediction of the dynamic behaviour of continuous-flow chemical systems. The basis of this procedure is the Markovian intensities defined for all single-transport processes (as convection, diffusion, adsorption, reaction, etc.) occurring in the system under consideration. The corresponding parameter is derived by combining the Markovian intensities according to a modular construction principle which rests upon utilizing certain basic structures. With this principle the parameter may be obtained for complex systems, i.e. for systems with a large number of single physicochemical transport and conversion processes. The parameter quantitatively determines the deviation of the actual process from the corresponding equilibrium process. Linear as well as strongly non-linear systems can be described without time-consuming numerical simulation procedures. The proposed procedure is successfully used to predict the dynamics of two isothermal non-linear adsorber models. Copyright try 1996 Elsevier Science Ltd

1. INTRODUCTION

The continuous-flow fixed-bed adsorption represents a technique widely utilized in accomplishing adsorption processes. It is used for large-scale separation processes as well as for parameter estimation on the laboratory scale. In all these applications the optimum operating conditions of the chemical system are determined by the contribution of the single-transport and accumulation processes to the overall adsorber dynamics. In technical processes the dynamic equilibrium behaviour characterized by the absence of any mass and heat transport limitations is desired. The equilibrium process leads to the steepest concentration fronts giving rise to the best separation effect and to simpler models and simulation procedures as well. Evaluating single-transport parameters from fixed beds, however, requires that transport process to be rate limiting for which the characteristic physicochemical parameter is to be determined. Because of the above considerations one is forced to conclude that for the design and optimization of continuous-flow fixed-bed adsorbers a minimum number of criteria is desirable which includes the following properties.

*Corresponding 30-6392-3222.

author.

Tel.: 49-30-6392-3220.

Fax: 49-

(1) The conditions for the dynamic equilibrium behaviour and any transport limitations are quantitatively predicted. (2) They are valid for any adsorber length, i.e. for very short adsorbers on the laboratory scale and long columns as well. (3) Both the linear and the non-linear case (e.g. non-linear sorption equilibrium, concentration-dependent transport parameters) are involved. (4) They may be derived for systems of any kind of complexity, i.e. for systems including any number and combinations of single-transport processes occurring in the total system. Parameters estimating the contribution of singletransport processes to the dynamics of chemical systems were derived by a number of authors from the normalized versions of the corresponding model equations (Masamune and Smith, 1964, 1965; Garg and Ruthven, 1973; Rasmuson and Neretnieks, 1980; Raghavan and Ruthven, 1983; Do, 1985; Do and Rice, 1987; Eic and Ruthven, 1988; Mayfield and Do, 1991). However, no parameters are presented which possess all the properties characterized above. The objective of the present paper is to derive such parameters by using a generalized procedure based on the utilization of the theory of Markov chains. In Section 2 the basic mathematical algorithms of the Markov chain method are presented which allow to

3519

R. GUTSCHEand K.

3520

HARTMANN

perform dynamic simulations. Section 3 contains the general Markov chain approach used in deriving the criteria for predicting the dynamic behaviour of chemical systems. In Sections 4-6 the application of the proposed procedure to isothermal non-linear adsorber models is treated. It is important to note, however, that the isothermal models are chosen to give a simple introduction into the new procedure rather than to indicate the limitation of the proposed method to the isothermal case. The expansion to non-isothermal problems will be the objective of a following work.

On the other hand, the Markov chain method can be also applied to process models which are continuous in space as demonstrated by Schmidt-Naake (1980) and Patwardhan (1989) with respect to the diffusion process. This becomes possible by dividing the transport space to get cells the concentration within each cell being independent of the local coordinate. For a general cell model of N volume elements (cells) the Markov chain can be formulated as (Kafarov and Klipinicer, 1969; Pippel and Philipp, 1977; Schmidt-Naake and Pippel, 1991)

2. MARKOV CHAINS FOR DESCRIBING

Equation (1) implies that the vector of dimensionless cell concentrations at the time tk+l is the product of the concentration vector at t, and the transition probability matrix P(tk). The matrix element pij(tk) of the probability matrix denotes the probability of the transition of a molecule from cell i to cell .j (i,j = 1, 2, , N) at time tk. In general, the time dependence of P(tk) arises from non-linearities of the process (e.g. non-linear sorption isotherm, concentration-dependent transport coefficients). The probability matrix is defined by (Karlin, 1968; Schmidt-Naake, 1980; Schmidt-Naake and Pippel, 1991)

TRANSPORT

AND

COlVVERSlOU PROCESSES

A number of studies dealing with the utilization of the theory of Markov chains for calculating the dynamic behaviour of mass transport processes have been reported for the past two and a half decades. The process models of Gibilaro ef al. (1967), Kafarov et al. (1968), Kafarov and Klipinicer (1968) are related only to the flow and mixing process in continuous-flow systems. The results are correlated with experimental residence time distributions. Further examples for modelling residence time distributions by means of Markov chains are the papers of Raghuraman and Varma (1974) and Fan et al. (1985). Krambeck et al. (1967), Kafarov and Klipinicer (1969), Pippel and Philipp (1977), Nassar et al. (1981), Antia and Lee (1985) and Chou et ul. (1988) combined these process models with chemical reactions. Schmidt-Naake (1980) and Patwardhan (1989) provided theoretical concepts for describing the one-dimensional diffusion process by means of Markov chains. Recently, Schmidt-Naake and Pippel (1991) demonstrated that the Markov chain method is appropriate to simulate diffusion controlled adsorption processes as well as chemical reactions. According to these studies the procedure of using the Markov chain method for calculating the dynamics of mass transport processes may be considered to fall into three fundamental parts.

(1) The mass transport

system is allocated to a convenient cell model (network, loop, cascade). The concentration within each cell is position independent. (2) Formulation of the Markov chain describing the variation in concentrations with time for the cell model as the result of interconnecting flows between the cells. (3) Combination of the Markov chain with accumulation (sorption) and/or conversion (chemical reaction) processes. It should be noted that in most cases a cascade of stirred tanks is used as the cell model because it a priori governs the condition of a spatial discretization. However, this does not mean that the cascade of tanks requires a Markov chain solution technique. For example, Deans and Lapidus (1960), Cho et al. (1983) and Do (1985) utilized the series-of-stirredtanks (SST) model without considering Markov chains.

C(tk+l) = C(t,)P(t,),

P(Tk) = exp[A(t,)

k = 0, 1,

k = 1, 2, __

At],

(1)

(2)

The elements zij(tk) (ij = 1,2, ,N) of the intensity the Markovian intensities. matrix A (tk) represent They are intensities of only those transitions that take place within a small time interval At (i.e. that take place between neighbouring cells) and may be calculated as kjCtk)

Xij&) = ~

nj(tk)

and %ittk)

=

-

c

(4)

I*ijttk)

i with

tiij(tk) ~ij(tk)

=

~

Qttk)

’

i,j = 1, 2, .

,N(i #j).

(5)

According to eq. (4) Xij(tk) represents the sum of the nij(tk) of molecules to leave cell i. The intensities matrix elements xlj(tk) are the intensities of entering cell ,j. The relations (3H5) deviate slightly from the convention used by Pippel and Philipp (1977) and Schmidt-Naake and Pippel(l991). The definitions of the present work allow to omit the diagonal matrices of the cell volumina in the formulation of the Markov chain [eq.( l)]. A number of authors treated the overall system by combining a stochastic description of flow and mixing with deterministic kinetic models, e.g. according to batch kinetics within the cell i (Kafarov and Klipinicer, 1969; Pippel and Philipp, 1977) or by means of a diffusion-immobilization model to consider the chemical reaction and/or adsorption in simulating the dynamics of a diffusion-controlled process (Schmidt-Naake and Pippel, 1991). Other authors developed a unified stochastic description of

Generalized

criteria

for predicting

the dynamics

both the flow process and the chemical reaction kinetics (Nassar et al., 1981; Chou et ul., 1988). In accordance with the latter authors, we formulate Markovian intensities for the adsorption process and the chemical reaction as well. The general relation for calculating these intensities is obtained from eq. (5):

Gh)

3.f(tk) = __

n,(t!i)

i=l,2,

’

. . . . N,

k=0,1,2

,.,..

(6)

The superscript X denotes any physicochemical process which takes place within the cell i but does not represent any transition of molecules from the cell i to adjacent cells. To assess the Markov chain method from a mathematical point of view it is reasonable to express the Markov chain by a set of linear equations rather than by the matrix formalism of eqs (1)<2). Assuming any physicochemical process X (e.g. sorption, chemical reaction) to occur within the cell i of a general cell model the change in dimensionless concentration with time in the cell i may be written as

1I

C&j(tk) + CAX(tk)At Ci(tk) j + iJ = 1,2, .

x

CXji(tk)At Cj(tk)

, N,

k=0,1,2

. ... .

i#j.

The relation (7) is suitable to demonstrate that the Markov chain method is mathematically equivalent to the explicit finite difference method. For the case of a longitudinal diffusion combined with first-order chemical reaction in a finite cylinder, for example, the normalized concentration change in a volume element i of the cylinder is Ci(rk+l) = Cl - (A?,+1 + n!i- 1 + n”)At] +Xf-l,iAtC,-I(tk)

Ci(tk)

+;iP+l.iAtCi+l(tk) (8)

with

j=l3 ,‘-> ,.. 3N,

j = i - 1, i + 1.

+k,C,(f&

i=

+

chemical

systems--I

3521

procedure in solving differential equations. In association with this procedure, Moler and van Loan (1978) presented 19 different ways to compute the exponential of the matrix ;i (tk). A corresponding equivalence between the Markov chain model and the deterministic description is also found by Kumar and Pethii (1990). One has to conclude, therefore, that from the mathematical point of view the Markov chain method has no advantage over the widely used numerical methods for solving differential equations. It will be shown in the present work, however, that there is an advantage with respect to the physicochemical interpretation of complex transport processes found in chemical engineering. This advantage follows from the definition of the intensities lij(tk) and $(t,) which are the basis for deriving generalized criteria predicting the dynamics of chemical systems. 3. MARKOVIAN

INTEWITIES

CRITERIA PREDICTING

FOR THE DERIVATION

THE DYNAMICS

OF

OF CHEMICAL

SYSTEMS

A complete Markov chain algorithm as described in Section 2 is necessary only if dynamic simulations are desired. The goal of the present work, however, is to derive criteria which predict the dynamic behaviour without time-consuming dynamic simulations. There are three main hypotheses which are the basis in deriving such criteria. (1) The essential behaviour of the total system may be obtained from the intensities evaluated for one representative volume element of the cell model. The most important property of this volume element is that it involves all single transport and conversion processes occurring in the overall system. (2) Among these single processes there is a reference process which is not negligible over the whole range of operating conditions. The ratio of the intensity of the non-negligible transport process (called the wference intensity) to the overall intensity of the remaining single processes (called the churacteristic intensity) is assumed to provide the essential information about the dynamic behaviour of the total system. Hence, the dimensionless parameter

(9)

The intensities nz and 2; are identical because the volume elements of the cylinder are equal in size. Inserting eqs (9) into eq. (8) and rearranging the terms one obtains Ci(tk+ 1) - C,(t,) D = 12 CCi&l(Q - 2Wk) At L

of continuous-flow

ci+I(tdl

1,2, ,.., N. (10)

Equation (10) is completely identical with the finite difference solution of the axial diffusion-reaction process in a cylinder [see, for example, Crank (1964)]. The equivalence of the Markov chain method and the finite difference method becomes additionally obvious if one considers the fact that the matrix algorithm of eqs (1)42) is also known from the common numerical

(11) may be the basis for the formulation of a criterion predicting the dynamic behaviour. The symbols & and 1$‘; designate the reference and the characteristic intensity, respectively. The subscript ref indicates that the intensities are related to the refirence cell (representative volume element fraction which the reference process occurs in). This subscript stands for transport processes from the reference cell to neighbouring cells as well as for conversion processes occurring within the reference cell and the representative volume element, respectively. (3) The characteristic intensity is formed by the intensities of the remaining single processes (i.e. all processes except for the reference process) on the basis

R.

3522

GUTSCHE and

of certain basic structures which are determined by kind of combination of the single processes within overall process model. The basic structure for combination of two single-transport processes ranged in series is

the the the ar-

where L;“b:is the overall intensity for the combined processes X and Y. Equation (12) is the mathematical expression for the well-known fact that for single processes in series the slower process is the rate controlling one, i.e. the overall intensity is determined by the intensity of the slower process. On the contrary, for single processes in parallel the faster process is rate controlling. In this case the intensity of the faster process must dominate the overall intensity, so that that basic structure for parallel processes is n;; = 12;, + A,‘,,.

(13)

K.

HARTMANN

model. The following single-transport processes are included. (1) The flow process (convection and axial dispersion) in the adsorber void volume described by the SST model. (2) External mass transfer, i.e mass transport from the fluid phase to the external surface area of the spherical particles forming the fixed bed (bed particles). (3) General diffusion-sorption process in the spherical adsorbent particles to which the diffusion process is related (diffusion particle). This combined process is suitable to represent one of the following special transport mechanisms: -

-

According to these assumptions the following steps are necessary to obtain the criteria for predicting the dynamics of a chemical system. (1) Construct the cell model for the system to be investigated and determine the representative volume element. (2) Identify the reference process among the singletransport processes taking place in the representative volume element. Calculate the Markovian intensities (related to the reference cell) of the corresponding single processes by means of eqs (5) and (6). The nature of the transport mechanisms determines the form of the gross mole fluxes in eqs (5) and (6). (3) Combine the intensities of all processes except for the reference process by using the basic structures to get the characteristic intensity A:,:“. (4) Calculate the general dimensionless parameter Q by using eq. (11). As only intensities rather than complete Markov chains are used, the proposed procedure will be called the Markovian intensities approach (MIA). According to eqs (5) and (6) the Markovian intensities are time and position (cell number) dependent. In the linear case these dependences are dropped so that the intensities become constant and, therefore, one parameter Q is obtained for the whole dynamic process. For strongly non-linear processes the Markovian intensities include concentration-dependent physicochemical parameters. To obtain only one parameter @ also in this case it is necessary to calculate the concentration-dependent quantities for one characteristic concentration. In this work the equilibrium concentration c, or q, (e.g. inlet concentration for a step response) is generally assumed to be this characteristic concentration. 4. THE DlMENSIONLESS ISOTHERMAL

PARAMETER

@ FOR A GENERAL

ADSORBER MODEL

The application of the MIA is first to be demonstrated in terms of a general isothermal fixed-bed adsorber

-

Combined pore-surface diffusion with local sorption equilibrium between the pore volume and the surface phase of the macrosphere (Komiyama and Smith, 1974; Do and Rice, 1987). Pore diffusion with local sorption equilibrium including both the diffusion in the pore system of a homogeneous macrosphere and the macropore diffusion in a macrosphere consisting of agglomerated microspheres (Garg and Ruthven, 1974b; Do and Rice, 1987). Solid (micropore or surface) diffusion with local equilibrium at the outer surface of the bed particles represented by microspheres (agglomerated or non-agglomerated) or monodisperse structured macrospheres (Garg and Ruthven, 1974a; Do and Rice, 1987).

This adsorber model called the FTDSe model (standing for Flow process, external mass Transfer, internal Diffusion and Sorption equilibrium7 is governed by the set of partial differential equations given in Appendix A. According to these equations the FTDSe model involves a concentration-dependent effective diffusivity D(cq) and any non-linear sorption isotherm expressed by the concentration dependent equilibrium parameters K(cis) and I-(&. To obtain the relations which are valid for the corresponding special transport mechanisms the general parameters of the equations in Appendix A have to be modified in the manner given in Table I. The application of the MIA to the FTDSe model requires the setting up of an appropriate cell model. For the FTDSe model there are two separate phases in which transport processes occur: the fluid phase in the adsorber void volume (flow process) and the adsorbent phase represented by the diffusion particle (general diffusion-sorption process). Unlike the Markov chain for a complete simulation run the proposed procedure requires a cell structure only for that phase for which this structure has a physical background. Hence, only the adsorber void volume is divided into cells to obtain the SST model. This model is a physically based cell model because the cell number N is related to the axial dispersion coefficient by (Turner, 1983) N=-

%&A

20,;

(14)

Generalized

criteria

Table 1. Modifications

for predicting

of the general

the dynamics

FTDSe

parameters

Mechanism

Combined

pore-surface

diffusion

of continuous-flow (Appendix

chemical

systems-1

A) for special diffusion

3523

mechanisms

d c,

NC:‘)

K(k)

G(

cf

D&P)

1

%

qi

D&J

F

AlI nd-

1

qi

D:(q,)

3

1

1

Ucf)

Pore diffusion in homogenous macrospheres or macropore diffusion in agglomerated microspheres

Surface diffusion

in homogeneous

macrospheres

Micropore

diffusion

in agglomerated

microspheres

Micropore

diffusion

in non-agglomerated

microspheres

Ab

* c*=<,a

c* (. =c,,

flow

I

nbnd

process

d@xsion particle

Fig. 1. Representative

volume element for the FTDSe model including tion processes.

The cell structure of the absorbent phase (diffusion particle) would only be a formal one necessary for a complete simulation algorithm. In the proposed procedure the whole diffusion particle is regarded as one cell. The equally volumed cells of the SST model are denoted by i running from 1 to N. The cell i is coupled by corresponding transitions with nbnd identical diffusion particles (Fig. 1). The transitions of molecules between neighbouring cells are indicated in Fig. 1 by arrows each of which represent the gross mole flux of a single-transport process. The element shown in Fig. 1 is the representative volume element of the FTDSe model because all single processes of the system are included. To identify the non-negligible single-transport process it is reasonable to interpret the limiting operating conditions of the FTDSe model. For very high flow rates in the adsorber void volume there is not enough time for the species to adsorb. Hence, the overall dynamics is determined by the flow process only. With the other extreme of very low flow rates the adsorption equilibrium is established at any adsorber position. In this case the dynamic behaviour is deter-

all single transport

and accumula-

mined by the flow process and the sorption equilibrium as well. The consequence is that for the whole range of operating conditions the contribution of the flow process cannot be neglected, i.e. the flow process and the cell i of the SST model represent the reference process and the reference cell, respectively. We are now able to calculate the Markovian intensities for the single-transport processes of the FTDSe model. This calculation requires the definition of the gross mole fluxes for the following processes: - Flow process from cell i to cell i + 1, - mass transfer from cell i to the external surface area of the bed particles and - effective diffusion-sorption process from the external surface into the bed particles. The corresponding

gross mole fluxes are rift+

ri$Jtk)

ri$(fk) =

1 (tk)

=

=

ti C;(tk)

nb kb Ab ci(tk)

2 nba D (C:‘Ah s) C&(tk). R

(15)

(16) (17)

3524

R.

GUTSCHE and

The subscript b stands for bed particle. With these mole fluxes the corresponding intensities of leaving cell i follow from eq. (5) so that

(18)

(19)

ny(c,) =

24,aDke) &Kke)

(20)

V,R

where Lr and 1.T denote the flow (reference) intensity and the mass transfer intensity, respectively. The sorption process is not described by a separate process intensity because it is not rate controlling. However, it is incorporated into the diffusion intensity 1:” by using the sorption equilibrium parameter K(c,). In this sense the diffusion intensity is an overall intensity representing the combination of the internal diffusion and the equilibrium sorption process. Equation (20) results from the assumption that the reservoir of moles available for the diffusion flux tifb(t,J is equal to the product ViCiS(tk) and from the convention (Section 3) that the concentration-dependent parameters are estimated for the equilibrium concentration c,. In eqs (18)-(20) the complete subscripts i, i + 1 and ib of the intensities are replaced by i because it is sufficient to indicate the reference to the cell i. The combination of the single-transport intensities to get the characteristic intensity is carried out by taking into account the basic structures defined in Section 3. Except for the flow process (as reference process it is not considered in forming the characteristic intensity), the FTDSe model includes the external mass transfer and the combined diffusion-sorption process. These two processes are obviously arranged in series. The overall intensity of the combined mass-transfer-diffusion-sorption process is then given by

representing the characteristic intensity for the FTDSe model. With this intensity we are able to determine the general dimensionless parameter CD.According to eq. (11) we get for the FTDSe model

5. APPLICABILITY PREDICTING

OF THE PARAMETER

@ FOR

THE FTDSe DYNAMICS

The objective of this section is to show that the parameter CDis suitable to obtain a criterion predicting the dynamic behaviour of the FTDSe model. For this purpose the values of the parameter @ are compared with the corresponding results obtained from complete simulations of the adsorber dynamics. The

K.

HARTMANN

dynamic simulations were carried out by solving the set of differential equations given in Appendix A according to a finite difference technique. By using the conditions (A5) and (A6) of Appendix A breakthrough curves (BCs) are obtained. The BCs are presented in the C-O diagram (Figs 2-10). Here 0 is formed by the ratio of the time t and the first moment about the mean $i. By using this kind of normalization the affect of the sorption capacity is eliminated, i.e. the course of the BC is determined only by the transport kinetics and the degree of non-linearity. Note that in all the figures the legend boxes with shadow are valid for all plots (ad) of a figure. Furthermore, in all figures the thin and thick curves represent the FTDSe and the corresponding FEQ dynamics, respectively.

5.1. Linear FTDSr model Computing BCs for the linear FTDSe model [D(cf) = D(q) = D, K(cis) = K(c,) = K] it can be shown (Fig. 2) that as the intensity ratio (22) is decreased the dynamics of the FTDSe model approach a constant breakthrough behaviour which is identical with that of the corresponding equilibrium (FEQ) model. With this model the fluid- and adsorbed-phase concentrations are in equilibrium as the result of neglecting all particle transport limitations. The FEQ model is described by the balance equation given in Appendix B. The convergence behaviour of the FTDSe model due to a decreasing @ reveals that the rate-controlling character is the less possible the more the characteristic intensity LTDS surpasses the flow intensity If. According to Fig. 2 the rate control is absent if

@=

gs<

0.1.

In this case the BCs are practically identical with the corresponding equilibrium curves. In the reversed case, i.e. with increasing @, the dynamics of the FTDSe model approach another constant breakthrough behaviour. This behaviour is characterized by the only presence of the flow process, what follows from the identity of the adsorber dynamics for @ > 10’ [Fig. 2(a)] with the dynamics of the pure flow model (SST model) governed by the differential equation of Appendix C. Hence, the criterion CDmay be regarded as the basis for a complete interpretation of the adsorber dynamics between the two limits “equilibrium behaviour” (@ < 0.1) and “pure flow behaviour” (@ > 105). For practical purposes the pure flow behaviour is not of any interest. Instead of this behaviour an asymptote should be defined which represents the limiting case for performing practical adsorption processes. From a practical point of view a breakthrough close to 0 = 0 is not acceptable. According to Fig. 3 a breakthrough later than 0 = 0 may be expected for w d 0.1 provided N > 1. It is proposed, therefore, to

Generalized

0.6

C

criteria

for predicting

the dynamics

of continuous-flow

chemical

t

0.4

thick lines: FEQ thin lines: FTDSe

-

2.0

1.5

1.0

0.8

3525

systems-1

N= 100 _upper curves: Bi = 0 Q

105+ SST

0.6 __..

C

104

.__...

0.4

-

103 100

--

nn 0.0

10 I

0.2

-

0.1

“.”

0.5

1.0

1.5

0.0

0.5

2.0

1.5

0

0 Fig. 2. Convergence of the breakthrough behaviour behaviour due to decreasing @,. In the plots (a)(d)

1.0

of the linear FTDSe model to the corresponding FEQ the behaviour is shown for different N (LA = const.).

1.0

0.8

0.6

C

0.4

0.2

0.0

0.8

0.6

C

0.4

0.2

0.0 C

Fig. 3. Effect of N (LA = const.) and axial dispersion, respectively, on the linear effect is shown in the plots (a)(d) for different a’.

FTDSe

dynamics.

The

R.

3526

GUTSCHE

and K. HARTMANN for nb, ti, N and A,/Vb one obtains

choose the BCs for @’= 0.1 [Fig. 3(c)] as such asymptotes. Hence, the condition for practically reasonable adsorption processes is proposed to be

The relation or

@
N

(N>

1).

Note that @’is obtained from Q, if nb (the number of bed particles exchanging mass with the cell i) is replaced in all intensities forming Q, by the number nb of particles within the total adsorber. As nb = Nnh the two quantities @ and W are correlated by the lefthand equation of the inequality (24a). Replacing nb by nb is equivalent to a transformation of the local quantity @ into the integral quantity @‘. Hence, the criteria (23) and (24) represent local and integral properties, respectively, of the adsorber system. The local character and some further properties of @ and the criterion (23), respectively, become obvious from a further modification of eq. (22). Rearranging the terms in this equation yields

The first conclusion is corroborated by Fig. 2 because the identity between the FTDSe curves for 0 = 0.1 and the corresponding equilibrium curves is found for any N. Note that for N = 1 the difference between the FTDSe and the FEQ model is noticeable but negligible from a practical point of view. A confirmation for the second conclusion is given in Fig. 4. Here BCs of the FDSe and the FEQ models are presented for increasing adsorber length. The FDSe (instead of the FTDSe) model was chosen to render possible a direct comparison of @ and the corresponding literature parameters (Table 2) with respect to predicting mass transport limitations. In the FDSe model mass transfer is neglected so that according to

It should be noted that according to this equation the total parameter @ is uniquely determined by relating the intensity of each of the remaining transport processes to the flow (reference) intensity. If the intensities in eq. (25) are replaced according to the expressions (18)<20) modified by the corresponding definitions

./‘

1.0 / 0.8 0.6 -

N = 1,000

li’ ,./

= 0.02

I’

PRl4

!;

/’

conclusions:

(1) the axial dispersion parameters D,, is completely involved in the quantity @ so that the criterion (23) is independent of the effect of axial dispersion; (2) for a given sorption system, particle size and velocity the quantity @ is independent of the adsorber length and N, respectively; and (3) the parameter @ depends on N if the variation in N is only due to a change in D,,. This behaviour may be important in laboratory-scale experiments where molecular diffusion is responsible for axial dispersion (Gutsche, 1993).

(24b)

-’

(26) entails the following

1,0000.02

-’

_x

-

;’

N = 10,000

i

i .’ i./ :/

P RN = 0.002

f/

’

/-

/“’

; C 0.4 -

,/j

; !

0.2 -

0.0

f

(cj,,,/’

_.-.’ / 0.8

thick lines: thin lines:

FEQ FDSe

_ _

,/j 1.2

f

/i ’ i (dj ,;’ ;’ /. / _ / ’ *.’

:’ 1.0

/j

0.90

0.95

1.oo

1.05

Fig. 4. Effect of N (at increasing LA) on the difference between the linear models FDSe and FEQ.

1.10

Generalized

criteria

Table 2. Dimensionless

Ratio of transport Mass

transfer

Pore

diffusion

for predicting parameters

processes

the dynamics

of continuous-flow

for estimating the presence limitations

Criterion

Bi

from Q

or absence

Criterion

=k,R

Bi

chemical

systems--I

of mass

3527

transport

from the literature

=k,R D,

2%D,

Masamune and Smith (1964) Mayfield and Do (1991)

Flow process Mass

transfer

Flow

process

47 =

sRNu,

P

3(1 - &)k&,

ERM, ‘m = 3( 1 - E)kbLA

Masamune

Pore

P, =

diffusion

and Smith (1965) ER2U,

3(1 ~ e)D,LA

Pcy = &

Do (1985)

Cen and Yang (1986) P A

Flow

process

Surface

4” =

diff.

CR%,

ER’Nu, 6(1 -&)KD,LA

PRR=(I

-&)KD,LA

Rhaghavan

and Ruthven

(1983)

eR*u, PR.V =

3(1 - E)KD,L/,

Rasmuson Flow

process

Micropore

qp =

diff.

eR2Nu, 6(1 - e)KD,L.d

Pm =

and Neretnieks ER2U,

3(1 - z)KD,LA

Eic and Ruthven

eq. (2.5), Q, = 4” mechanism

which

equivalent

is for

the

to PRN (Table

surface

diffusion

2). The difference

is, however, that PRN neglects any axial dispersion contribution. With increasing adsorber length (from LA = 0.025 up to 2.5 cm) the literature parameter P,,V varies from 20 to 0.02 while the MIA parameter remains constant at 4” = 10. From Fig. 4 it becomes clear that 4” is the more reliable parameter because the difference between the FDSe and the FEQ models does not essentially change with rising LA. When looking only at Fig. 4(a) one could conclude that this difference is reduced. The fallacy becomes obvious if the X-axis is modified in a manner leading to the same spatial extension of the FDSe breakthrough curves for different N [Figs 4(bt(d)]. This modification clearly shows that the difference between the two models is hardly influenced by changing N, even if N is raised to values as large as N = 10,000 [Fig. 4(d)]. The parameter PRN, however, predicts a considerable reduction of the contribution of the diffusion process when elongating the adsorber. This is not confirmed by Fig. 4. In Fig. 5 the third conclusion as well as a corresponding comparison with PRN is visualized. BCs calculated for increasing N (due to decreasing 0.J are shown for the models FDSe and FEQ. If N is raised from 1 to 1000, @ increases from 0.1 to 100 while PRN remains constant at 0.2. Figure 5 demonstrates a strong enhancement of the difference between the

(1980)

(1988)

FDSe and the FEQ breakthrough curves with increasing N. The difference is negligible for N = 1 (i.e. for 0 = 0.1) and very large for N = 1000. This behaviour is well predicted by @, but not predicted by PRN. Because of including axial dispersion the parameter Q, is also appropriate to assess the contribution of axial dispersion to the adsorber dynamics. This becomes obvious from Fig. 3. According to this figure the conditions under which axial dispersion may be generally neglected are found to be 1 10 N>

100 1 1000

Combining condition

for @’= 10 for (U = 1 for @’= 0.1

(27)

for (P = 0.01.

the latter relations

NO’=@>

one obtains the overall

10.

(28)

The consequence of the criteria (23) and (28) is that under conditions close to equilibrium (i.e. 0 < 1) axial dispersion can never be neglected. A neglect is not possible even if the adsorber is extremely elongated because Q, does not depend on L,. If Q, > 10 (i.e. the process is far from equilibrium) axial dispersion becomes negligible, but the process remains economically feasible only if the criterion (24) is fulfilled. This criterion is the more satisfied the larger one chooses

3528

R.

GUmHE

and K. HARTMANN

0

0

Fig. 5. Effect of N (LA = const.) and CD,respectively, on the difference between the linear models FTDSe and FEQ.

the parameter N. Hence, the criterion for a feasible sorption process with negligible axial dispersion is obtained by combining the inequalities (24) and (28) to get 10 < @ Q 0.1 N.

(29)

In the sense of eq. (29) axial dispersion is expected to be negligible if the adsorber is sufficiently long. This conclusion was referred to by several authors [e.g. Raghavan and Ruthven (1983) and Kast and Otten (1989)], but was not found to be restricted to conditions far from equilibrium. The parameter Q includes a number of special parameters which are compared in Table 2 with the corresponding literature parameters. The comparison reveals that the parameters 4’ and 4” include the quantity N additionally to the process coefficients already involved in the parameters from the literature. The consequence is that the MIA parameters yield a more reliable prediction of the adsorber dynamics than the literature parameters, as discussed above in regard of the comparison of 4” and PRN. Furthermore, according to eq. (22) the Biot criterion representing the ratio between the mass transfer intensity AT and the diffusion-sorption intensity $” is involved in the parameter @. According to eq. (25), however, the parameters 4’ and 4” are equivalent to Bi with respect to establishing the absence of mass transfer and diffusion limitations. On the other hand, the significance of @ is independent of the value of Bi. The reason is that the BC is much less sensitive to Bi than

to @ [Fig. 2(c)]. Additionally, this sensitivity is reduced with decreasing Q, i.e. with approaching the equilibrium dynamics. These two properties are visualized in Fig. 2(c). The shaded areas in this figure include all possible BCs of the region 0 d Bi < cc for a given value of Q. Because of the identity of PRS and PER (Table 2) one inclines to conclude that also PER does not involve any mixing criterion. In this case the criterion PER (as PR.V) should differ from the corresponding I$~ with respect to its physical significance. Eic and Ruthven (1988) derived the parameter PER for the zero length column (ZLC). According to their model equations the ZLC is characterized by a gradientless fluid phase. In the light of the series-of-stirred-tanks model this assumption is satisfied by setting N = 1 so that for the ZLC the parameter 4” is identical to the parameter PER (neglecting the difference with respect to the constants in the denominator of the two parameters, see Table 2). Hence, the criterion P,, is physically equivalent to 4” provided the condition N = 1 is fulfilled. The parameter $D is suitable to give another interpretation of a result obtained by Cen and Yang (1986). The authors demonstrated that in the case of a negligible external mass transfer a constant BC is attained for large values of the parameter P,, (Table 2). According to the definition of PC, this constant BC is approached with increasing adsorber length LA. Therefore, the authors concluded that a constant pattern behaviour as previously observed only for a favourable isotherm (Cooney and Lightfoot,l965; Rhee

Generalized criteria for predicting the dynamics of continuous-flow chemical systems-1 et al., 1971; Cooney and Strusi, 1972) is possible also in the linear case. This conclusion follows from a plugflow model used by the authors to simulate the flow process. The plug-flow model is characterized per definition by N = const = z If N is assumed to be constant, the conclusion that a “constant pattern” behaviour is approached with increasing LA is confirmed by the parameter 4”. However, this approach is due to the decreasing contribution of the internal diffusion (local property) rather than to balanced effects caused by elongating the adsorber (integral property). 5.2. Non-linear FTDSe model The validity of the parameter @ and the derived criteria is examined also for the non-linear case (nonlinear isotherm, concentration-dependent diffusivity). For this purpose computer simulations were performed which show the change in breakthrough behaviour as well as @ with increasing degree of non-linearity (DON). To intensify the effect of non-linearity the external mass transfer was neglected, i.e. the calculations were carried out by using the FDSe model. Two diffusion mechanisms were taken into account: the pore and the surface diffusion mechanisms, The non-linear equilibrium behaviour is represented by the Langmuir isotherm equation so that

K(cis) =

qsBL 1 + B,,

(30a)

Cis

and

qs&

K(CJ = ~ 1 + BLc, for the complete solution and the MIA, respectively. Therefore, with reference to the total BC the DON is quantitatively described by the parameter ‘J = B,,c.,. The concentration dependence of Fickian diffusivity for the surface diffusion mechanism is given by (Gutsche et al., 1985) dlnc, D.T(qi) = D,, __ dlnq * 4*=Yi

3529

Starting with the linear case (y = 0.001) the DON and ;: respectively, is enhanced by increasing the inlet concentration c, at a constant Langmuir parameter B,, as well as by increasing BL at constant c, (Table 3). In the latter case the change in BL is associated with a corresponding variation of the Langmuir parameter q.,.In Table 3 the dependence of the parameter 4” on the DON is presented for the two diffusion mechanisms considered. The equations for calculating 4” for the pore and the surface diffusion mechanism are given in Table 2. In the non-linear case the constant quantities K and D, are replaced by K(c,) and D,(q,), respectively. As shown in Table 3 the parameter 4” predicts that for the pore diffusion mechanism the contribution of diffusion is independent of the DON. This prediction is confirmed by Figs 6 and 7 which demonstrate that the difference between the BCs of the FDSe and the FEQ models does not significantly alter with increasing DON. As in Fig. 4 this conclusion becomes more obvious when looking at the plots (b)gd) of Figs 6 and 7. The curve pairs of the plots (b)-(d) are the same as in plot (a). However, in the plots (b))(d) the X-coordinates are modified in a manner entailing the same spatial extensions of the breakthrough curves so that a correct comparison becomes possible. As far as the surface diffusion mechanism is concerned the diffusion contribution is predicted to be independent of the DON only for the case that the DON is raised by increasing cL,at constant BI,. However, if the DON is enhanced by increasing BL at constant or the parameter 4” predicts a decrease of the diffusion contribution. These predictions are confirmed by Figs 6 and 8. It becomes obvious from Fig. 6 that the difference between the BCs of the FDSe and the FEQ models is not influenced by an increase of cr. In Fig. 8 this difference is strongly reduced as the result of rising DON (increasing BL at constant ce). The preceding results demonstrate that the parameter @ is able to predict the affect of the DON on the overall adsorber dynamics.

(31a) Table 3. Variations in 4” with :’(degree of non-linearity) the pore and the surface diffusion

for

mechanisms

and Pore diffusion

dlnc, D,(q,) = D.W~ dlnq * 4*=4< ’

@lb)

respectively. The validity of eq. (31) is confirmed by a number of authors (Higashi et al., 1963; Pope, 1967; Ruthven and Derrah, 1972; Kapoor and Yang, 1991; Gutsche and Yoshida, 1994). According to eq. (31) the concentration dependence of Fickian diffusivity is determined by the course of the sorption isotherm. Thus, the parameter y may be regarded as an overall quantitative measure for the DON including both the effect of the isotherm non-linearity and the concentration dependence of D,.

qT = 2000 mm01 cm ~’ BL = 0.01 cm3mmol-’

C, = 0.1 mmolcm~3

y(DON)

9”

q;mmol

0.001 10 100

10 10 10

Surface diffusion 0.001 10 100

10 10 10

cm- A

;‘(DON)

4”

2000 2.93 2.222

0.001 2.15 9.0

10 10 10

2000 2.222 2.0202

0.001 9.0 99.0

10 1 0.1

R. GLTSCHE and K. HARTMANN

3530

thick lines: thin lines:

0.6

1.0

0.8

0.4

1.2

0.6

1.0

0.8

1.2

1.4

1.6

FEQ FDSe

1.8

0.8 0.6

C

I

0.4 0.2 0.0

0

1.1

1.0

0.9

1.2

0

0

Fig. 6. Effect of ;’(changed by varying c, at constant B, and 4.) on the difference between the non-linear models FDSe (all curves are valid for the pore as well as the surface diffusion mechanisms) and FEQ.

0.6 1.0

1.0

0.8

1.2

:

,/’ : 1’ / : // : 1 I/ /IV :

y=2.15

0.8 0.6 -

C

0.4 0.2 -

/ (C)

o.o+--

’

y=9.0

; :

:

I

/

-

:

(d)

j”

1.0

1.1

0.90

;

/r[

,’ 0.95

0 Fig. 7. Effect of 7 (changed

i :

.:

:’

,),

0.9

..T

iI : ;.’

1’ :

_/ _’ /’

1.6

1.2

0.8

0.4

,/-- _---

by varying qb and B, at const. c,) on the difference models FDSe (pore diffusion mechanism) and FEQ.

1.00

1.05

0 between

the non-linear

1.10

Generalized

criteria

for predicting

the dynamics

of continuous-flow

chemical

systems-I

3531

If the laminar velocity profile controls axial dispersion (D,, = R~u~/48D,), eq. (36) becomes (37) 0.6

C

For the case that axial dispersion is determined by molecular diffusion (D,, = D,), eq. (36) is modified to

0.4

Q,=-

0.2

I

if!

0.0

0.8

I .o

0.9

1.1

1.2

0 Fig. 8. Effect of 1’(changed by varying qsand B, at const. c,) on the difference between the non-linear models FDSe (sur-

face diffusion mechanism) and FEQ.

6. APPLICABILITY PREDICTING

OF THE PARAMETER

THE DYNAMICS

Q FOR

OF A WALL ADSORPTION

IFTWSe) MODEL

The FTWSe model (Flow process, mass Transfer to the Wall and equilibrium Sorption at the wall) includes the following single-mass-transport processes: (1) Laminar flow through an adsorber tube superimposed by axial molecular diffusion. (2) Radial diffusion determining mass transfer to the interior wall of the tube. (3) Equilibrium adsorption at the tube wall. According to Taylor (1953) and Aris (1956) these processes may be described by an axial dispersed plug-flow model (Appendix D) with D,, = D, + 48 D, provided

(32)

that 14.44 L, ->> Pe RA

1.

(33)

The flow process is again the reference process while mass transfer represents the only remaining single process. Hence, approximating the dispersion model by the SST model one obtains for the parameter @ the expression (34) where A{” is the intensity of mass transfer from the fluid phase to the wall. According to eq. (5) this intensity is given by

Pe2 8Sh;

(38)

For these two limiting cases the Sherwood number is nearly independent of Pe (Sorensen and Stewart, 1974) so that in the former case the parameter CDis independent of Pe while in the latter one it is proportional to Pe2. Model simulations were carried out to demonstrate the dynamic behaviour of the FTWSe model (Appendix D) as the result of varying Pe for the two limiting cases. The corresponding equilibrium behaviour (FEQ model) is governed by the balance equation of Appendix E. The simulations were performed by using the software package PDEXPACK (Nowak et al., 1995) for linear as well as strongly non-linear sorption isotherms. Figures 9 and 10 represent the simulation results obtained with different values of Pe for the laminar (D,, = R~u~/48D,) and the diffusion-controlled (D,, = D,) FTWSe models, respectively. The linear runs are presented in the upper rows of Figs 9 and 10 while the non-linear runs are given in the lower rows. Figure 9 shows that (as predicted by @) the difference between the laminar FTWSe and FEQ model does not change with increasing Pe for the linear and the non-linear case as well. Furthermore, the difference is in a magnitude expected with a value of @ = 6.5. From Fig. 10 it becomes obvious that the difference between the diffusion-controlled FTWSe model and the FEQ model is strongly decreased with decreasing Pe. The difference is negligible for @ = 0.1 as already found for the FTDSe model. Hence, the predictive capabilities of the parameter @ are confirmed also for the wall adsorption model investigated. 7. CONCLUSIONS

The Markovian intensities approach is shown to be suitable to derive a general dimensionless parameter @ which is the basis for criteria predicting the dynamics of continuous-flow chemical systems. The general procedure for determining the parameter resembles a modular construction principle. Because of this property the proposed procedure may be applied to considerably complex systems. The application of the procedure to two non-linear adsorber models reveals that the parameter is characterized by the following properties.

so that with eq. (18) dN @z----.-=-.

~2RA

kwA,

%JLx

(36)

(1) It includes all single transport and accumulation processes occurring in the system under consideration.

R. GUTXHEand K. HARTMANN

3532 l.Oo.8_

Pe=S 0=6.5

0.6.

Y; thick lines:

thin lines:

"'a.8

0.9

1.0

1.1

1.2

0.5

1.5

I.0

Pe = 500

Pe=50

Q,= 6.5

@=6.5

[

.@=6.2

f

0.6 -

. i (e)

Kl _Yd/’

: : 0.5

1.0

0.9

1.0

Fig. 9. Effect of Pe on the deviation of the laminar FTWSe from the FEQ model (linear sorption row of plots, non-linear sorption lower row).

1

- upper

thin lines: FTWSe

,,,.j /

(c)

pi

..’ 0.8

1.0

0.8

1.2

1.2

1.0

,. 0.8.

Pe = 7.9

Pe = 2.5

0.4 -

0.98

0.99

Fig. 10. E&ct of Pe on the deviation

f

/

:

:’

1.00 8

1.0

0.8

1.4

1.2

/’

j

Pe = 25

: : : : Q=l : : : : : : :: :: L’ I /‘: I’: (e) /’ : I‘ : _/* ,’ 8’ _____-----,z

0.6 -

C

0.6

cp=lO

fE: 1 ,.i : i .’ ;

(f-J 0.96

0.98

1.00

8

of the diffusion-controlled FTWSe from the FEQ model (upper and lower rows as in Fig. 9).

,.

Generalized

criteria

for predicting

the dynamics

(2) The parameter relates the Markovian intensity of the non-negligible single process of the system to the overall intensity of the remaining processes. In this sense if is found to be a quantitative measure for the dynamic behaviour between the two limits - fully developed equilibrium behaviour (no transport and kinetic limitations) and - an asymptote for which adsorption is still practically feasible. (3) It is applicable to the linear as well as the strongly non-linear case (non-linear sorption isotherm, concentration-dependent transport parameters). (4) The parameter is shown to be the common basis for special dimensionless quantities given in the literature for predicting the presence or absence of masstransport limitations.

of continuous-flow

&I

DP DS D D:O I

kl kb kw K

1,; LA N n nh

NOTATION

BI,

ratio of the tube wall area to the tube volume, cm-i external surface area of the bed particle, cm’ cylinder cross section, cm’ internal wall area of the tube, cm* external surface area of the diffusion particle, cm* Langmuir isotherm parameter, cm3 mmol-’

nd

ri &j Pij pcu PD

Bi

PER

Bo c co

Ci CS

CP Cd d C, CW

UXL.4 Bodenstein number = &X > ( fluid-phase concentration, mmol cmM3 fluid-phase concentration within the cell 0 (inlet) of the SST model, mmol cm-3 fluid-phase concentration in the equilibrium of sorption dynamics, mmol cmm3 fluid-phase concentration in the cell i, mm01 cme3 fluid-phase concentration at the external surface area of the bed particle coupled with cell i, mm01 cmm3 pore volume concentration, mm01 cm 3 concentration within the diffusion particle, mm01 cme3 diffusion particle concentration in equilibrium with c,, mmol cmm3 fluid-phase concentration at the wall, mmol

P MS P R.V PRR

Ci D D,, DC

D&l)

c/c,), dimensionless vector of dimensionless cell concentrations ( = Ci/C,),dimensionless effective diffusion coefficient, cm’ s-i axial dispersion coefficient, cm2 s 1 e,D,, effective pore diffusivity, cm* s 1 effective pore-surface diffusivity, f”=

= D, +

I 1

$D”(cf) 2 * ,*=c;

cm2sm1

systems-1

3533

molecular diffusivity, cm* s 1 pore diffusivity, cm2 s ’ surface diffusivity, cm* s- ’ D, for q Z- 0, cm2 s- ’ micropore diffusivity, cm* s- ’ unit matrix first-order reaction rate constant, s - 1 external mass transfer coefficient, cm s- 1 wall mass transfer coefficient, cm s ’ sorption equilibrium parameter at the outer surface of the bed particle distance between the centers of two neighbouring cylinder volume elements, cm adsorber length, cm cell number of a general cell model. cell number of the series-of-stirred-tanks model number of moles, mmol number of bed particles coupled with a cell of the series-of-stirred-tanks model number of bed particles coupled with the total SST model number of diffusion particles (e.g. zeolite crystals) in a bed particle (& = 1 for Rb = R) molar flow rate, mm01 s-i gross mole flux from cell i to cellj of a general cell model, mmol s- ’ transition probability from cell i to cell ,j of a general cell model dimensionless parameter defined by Cen and Yang (1986) dimensionless parameter defined by Do (1985) dimensionless parameter defined by Eic and Ruthven (1988) dimensionless parameter defined by Masamune and Smith (1965) dimensionless parameter defined by Rasmuson and Neretnieks (1980) dimensionless parameter defined by Rhaghavan and Ruthven (1983)

Pe

Peclet number

P

R RA Rb

matrix of transition probabilities adsorbed phase concentration, mmolcm-3 adsorbed phase concentration in equilibrium with c,, mmol cm - 3 adsorbed phase concentration in the diffusion particle coupled with cell i of the series-ofstirred-tanks model, mmol cm- 3 Langmuir isotherm parameter, mmol cm-3 adsorbed-phase concentration at the wall, mmolcmm2 radial coordinate within the diffusion particle, cm radius of the diffusion particle, cm adsorber radius, cm radius of the bed particle, cm

Skv

Sherwood

number

t

time, s interstitial volumetric

velocity, cm s-l flow rate. cm3 s- ’

4 qe 4i

4. 4w

-3

C C

chemical

r

UX d

( =‘%I

( = y>

3534 Vb

vc vi

x

R. GUTSCHE and K. HARTMANN

volume of the bed particle, cm3 volume of a cylinder volume element, cm3 cell volume of the series-of-stirred-tanks model, cm3 axial coordinate, cm

Antia, F. D. and Lee, S., 1985, The effect of stoichiometry

proportionality factor BLc,, dimensionless nonlinearity parameter dimensionless parameter quantifying local sorption equilibrium at the pore surface time interval, s adsorber void fraction macroporosity dimensionless time ( = t/p’, ) Markovian intensity of molecules to enter cell j from cell i, s ’ Markovian intensity of molecules to leave cell i to cellj, s-’ Markovian intensity of the physico-chemialprocess X occurring within cell i, s- ’ matrix of Markovian intensities first moment about the mean, s special version of the general parameter Q local dimensionless parameter relating the reference intensity to the characteristic intensity integral version of @

k N

related to the cell, i, j moment k of the time scale related to the cell N

s

related

*

particle sorption

to the external

surface

area

of the bed

equilibrium

equantity

x, y

related to the single physic0 chemical processes X and Y F, T related to the flow process, to external mass transfer D, DS relatd to diffusion, to diffusion with local sorption equilibrium TDS related to mass transfer and diffusion with local sorption equilibrium

FDSe FTWSe

on Markov chain models for chemical reaction kinetics. Chem. Enynyl Sci. 40, 1969%1971. Aris, R.. 1956, On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Sac. 235, 67-77. Cen, P. L. and Yang, R. T., 1986, Analytic solution for adsorber breakthrough curves with bidisperse sorbents (zeolites). A.1.Ch.E. J. 32, 1635. Cho, B. K.. Hegedus. L. L. and Aris, R., 1983, Discrete cell model of pore-mouth poisoning of fixed-bed reactors. A.1.Ch.E. J. 29, 289-297. Chou, S. T., Fan, L. T. and Nassar, R., 1988, Modeling of complex chemical reactions in a continuous flow reactor: a Markov chain approach. Chem. Engng Sci. 43, 2807-28 15. Cooney, D. 0. and Lightfoot. E. N., 1965, Existence of assymptotic solutions to fixed-bed separations and exchange equations. Ind. Engng Chem. Fundam. 4, 233. Cooney, D. 0. and Strusi, F. P., 1972, Analytical description of fixed-bed sorption of two Langmuir solutes under nonequilibrium conditions. Ind. Engny Chem. Fundam. 11, 123. Crank, J.. 1964, The Mathematics of Diffusion. Clarendon Press, Oxford. Deans, H. A. and Lapidus, L., 1960, A computational model for predicting and correlating the behaviour of fixed-bed reactors: I. Derivation of model for nonreactive systems. A.1.Ch.E. J. 6, 656-663. Do, D. D., 1985, Discrete cell model of fixed bed adsorbers wtih rectangular adsorption isotherms. A.1.Ch.E. J. 31, 1329. Do, D. D. and Rice, R. G., 1987, On the relative importance

of pore and surface diffusion in non-equilibrium adsorption rate processes. C’hem. Engng Sci. 42, 2269. Eic, M. and

Ruthven,

D. M., 1988. A new experimental hiffusivity.

technique for measurement of intracrystalline

Superscrivts

FTDSe

SST

equilibrium adsorber model including flow process and equilibrium sorption (Appendix B, Appendix E) series-of-stirred-tanks model (Appendix C)

REFERENCES

Greek letters

i, .i

FEQ

adsorber model including flow process (SST model), external mass transfer, internal diffusion, local sorption equilibrium (Appendix A) as FTDSe but negligible mass transfer adsorber model including flow process (dispersion model), mass transfer to the wall, equilibrium sorption at the wall (Appendix D)

Zeolites 8. 40. Fan, L. T., Too, J. R. and Nassar, R., 1985, Stochastic simulation of residence time distribution curves. Chem. Enunu Sci. 40. 1743-1749. Gargy d. R. and Ruthven, D. M., 1973, Theoretical predictions of breakthrough curves for molecular sieve adsorption columns I. Asymptotic solutions. Chem. Engng Sri. 28, 791. Garg. D. R. and Ruthven, D. M., 1974a, The performance of molecular sieve adsorption columns: systems with micropore diffusion control. Chem. Enyny Sci. 29, 571. Garg, D. R. and Ruthven, D. M.. 1974b, The performance of molecular sieve adsorption columns: systems with macropore diffusion control. Chem. Engng Sci. 29, 1961. Gibilaro, L. G., Kropholler, H. W. and Spikins, D. J., 1967, Solution of a mixing model due to van de Vusse by a simple probability method. Chem. Enyny Sci. 22, 517. Gutsche. R.. 1993. Concentration-deoendent microoore diffusion analysed by measuring laboratory adsoiber dynamics ~ I. Study of the adsorber flow behaviour. Chem. Engng Sci. 48, 3723-3733. Gutsche, R.. Fiedler, K. and Klrger, J., 1985, Concentration dependence of Fickian diffusivity in solutions and sorption systems. J. Chem. Sot., Faraday Truns. I 81, 3103. Gutsche, R. and Yoshida, H., 1994, Solid diffusion in the pores of cellulose membrane. Chem. Engng Sci. 49, 179-188. Higashi, K., Ito, H. and Oishi, J., 1963, Surace diffusion phenomena in gaseous diffusion - I. Surface diffusion of pure gas. J. Atomic Energy Sot. Japan 5, 846.

Generalized

criteria

for predicting

the dynamics

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A

The FTDSe model is governed by the following differential equations (i = 1,2, _._ , N). Adsorber void volume:

(AlI Diffusion

particle:

642) Boundary

and initial conditions: kb(ci - c(s) = aD(cf) 3 ?r r=R r^c4

0

Sr r=O =

Sorption

(A3)

(A4)

co(r) = c,

(A5)

ci(to) = cf(r, to) = 0.

(A6)

equilibrium: c&(t) = Kc&t) c$(t) = K (cd cis(t) APPENDIX

(linear)

(A7)

(non-linear)

(A8)

B

The FEQ model for the FTDSe case is given by the differential equation (i = 1, 2, , N) (1 +r,!+)~+Ci_L

-+$!d$

(Bl)

with qi = Kci

WI

(linear)

q.BL qt = =+B,ciC’ (non-linear) Boundary

and initial conditions: co(r) = c,

(B4)

c&o) = 4&o) = 0

(B5)

APPENDIX

The 2,

SST .N)

(B3)

model

is

based

C

on

the

relation

(i=

1,

(Cl)

R.

3536 Boundary

and K. HARTMANN

GUTSCHE

and initial conditions:

Sorption

co(t) = c,

qw = Kc,

(C2)

c,(to) = 0.

APPENDIX

The FTWSe equation:

model

is based

equilibrium:

4w

a BL. =-+w

D

upon

balance

1

?t

Boundary

=

~ k,, a,,. (c - c,).

z=-

“,D x 8.x

case is given by

PC

ax &2

da,

- a,--.

dt

(El)

(Dl)

Boundary

and initial conditions: ;

and initial conditions: u,(c,=~

- c,) = D,,‘c iix x=o

k,>,(C- cw) = f$

(W

(7c 7

CY x=L,

ux(cx=o -c,)

(D7)

E

The FEQ model for the FTWSe u

c’c - xx z + Da, $

06)

(non-linear)

APPENDIX

the following

?C l%-

(linear)

= kx;

x=o

(D3)

= 0

c(x, 0) = 0 Sorption

(W

(E3) (E4)

equilibrium:

iiC =

z

x=r,,

0

c(x, 0) = 0

qw= Kc

(D4) (D5)

4w =-

4s BL 1 + BLcc

(linear) (non-linear)

(E5) (E6)