A general analysis of multicomponent mass transfer with simultaneous reversible chemical reactions in multiphase systems

Pergamon Press.

Chemical Engineering Science, 1969, Vol. 24, pp. 553-569.

Printed in Great Britain.

A general analysis of mutticomponent mass transfer with simultaueous reversible chemical reactions in multiphase systems LAWRENCE L. TAVLARlDESt and BENJAMIN GAL-OR+ Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania

152 13, U.S.A.

(First received 19 January 1968; in revisedform 6 September 1968)

Abstract- A general analysis is presented

for describing the dynamics of multicomponent mass transport with simultaneous reversible or irreversible first-order chemical reactions in particulate systems consisting of large numbers of drops, bubbles or solid particles with residence time and particle size distributions. The method of approach and the general fundamental formulations are demonstrated by using matrix notation and appropriate transformations. The resulting partial differential equations for multicomponent, multiphase systems are first decoupled and then transformed into ordinary differential equations by employing an integral operator whose kernel takes into account the residence time distributions. Interaction among neighbor particles are taken into account by appropriateboundary conditions corresponding to an ensemble of spherical cell models. The total average transfer rates from the entire particle population are evaluated in terms of reaction rate constants, diffiaivities, partition coefficients, average residence times, dispersed-phase holdup fraction and particle size distribution. The results of this general analysis are useful in predicting the yields and etBuent concentrations of reacting dispersed systems as a function of the physical and operating variables. The practical application of the general solution presented is illustrated for the two-phase continuous stirred tank reactor and for the case of a three-component dispersed system undergoing pseudosteady-state diffusion in both phases with simultaneous reversible reaction in the continuous phase. One interesting general conclusion from this analysis is that the error involved in assuming a uniform particle size in lieu of a distribution of sizes is small when calculating total average transfer rates. It is found also that a reversal of the transfer may take place under certain conditions which depend on the average residence time and the relative rates of the backward to the forward reactions. INTRODUCTION

of transport phenomena in multiphase systems consisting of large numbers of drops, bubbles or solid particles, encompass broad lines of disciplines in engineering and sciences. Clouds of particles (i.e. drops, bubbles, or solid particles) undergoing chemical changes simultaneously with a multicomponent mass transport are of common occurrence in many industrial and biological systems and processes. For example, one can name the combustion of fuel drops and metallic powders in rocket engines, polymerization in emulsions or suspensions, gas-absorption with chemical reaction, fermentation, chemical changes taking place in blood, aerosols, cloud droplets, biological cells or solid and liquid fluidized beds. Most of the fundamental studies available at _ STUDIES

present in these fields have been limited, however, to transfer in one phase of a binary system when the transfer is to or from a single particle. Recently some work has been done on transfer involving clouds of equal size spherical particles [l, 2,11,14]. More recently these works have been extended to include non-uniform particle size and residence time distributions in nonreacting systems[3-5,10,16]. So far, no work is available to describe the behavior of multiphase systems undergoing reversible or irreversible chemical reactions simultaneously with a multicomponent mass transport. Hence, it is the purpose of the present study to fill up this gap in the knowledge of multiphase systems by evaluating the general behavior of such reacting systems with non-uniform particle size and residence time distributions. In this study we try also to answer

tPresent address: Laboratories of Chemical Technology of University of Delft, Delft, The Netherlands. #Also (as a joint appointment) at the Department of Aeronautical Engineering of the Technion-Israel Technology, Haifa, Israel.

553

Institute

of

L. L. TAVLARIDES

a fundamental question; whether or not one needs to employ the particle size distribution in evaluating transport phenomena in particulate systems (i.e. what is the error involved if one replaces the particle size distribution by a mean size). The total average transfer rates to or from the entire particle population will be evaluated to allow the practical application of the analysis in the prediction of yields and effluent concentrations. The total average transfer rates in multiphase systems will be expressed first in terms of a general normalized particle size distribution. A practical normalized particle size distribution will be employed later in a numerical analysis of a ternary system undergoing pseudo steady-state diffusion in both phases with a simultaneous reversible reaction in the continuous phase. ,. THE METHOD FUNDAMENTAL

OF APPROACH AND THE GENERAL FORMULATION

Transfer is considered in a multiphase system consisting of a large number of spherical particles (i.e. drops, bubbles or solid particles). In the case of dispersions of two immiscible fluid phases a continuous breakup and coalescence of drops and bubbles may take place. However, under constant operating conditions, a dynamic equilibrium is ultimately established between breakage and, coalescence rates and a time invariant spectrum of particle sizes results. This time invariant size distribution may’be described by a normalized particle size ‘distribution f(a) as described in ‘previous’ works [3-5,161. The description of the extremely’ complicated mechanisms of breakage and coalescence themselves lies ,outside the scope of the present formulation. Thus, we assume that the direct effect of coalescence and breakage on the’mass transfer mechanism is negligible and indirect effects are reflected by changes in the ‘particle’ size distribution under different operating ‘conditions. This can be done by expressing the size distribution in terms of the average size of the ‘population and following its variations as a ‘function of the Weber number[5]. Some other indirect effects of coalescence and breakage will be encountered in the following formulation by expressing the

and B. GAL-OR

residence time distribution in terms of the average residence time whose functional dependence on the operating conditions is assumed to be known experimentally. The following formulation is given for a pseudo steady-state mass transfer in an n-component multiphase system undergoing first-order reversible chemical reaction in any phase. It is assumed that (i) the solutions are sufficiently dilute so that the densities and diffusion coefficients are constant, (ii) thermodynamic coupling among the various fluxes involved are negligible, (iii) heats of reaction and solution are small and (iv) isothermal and isobaric conditions prevail. For a multiphase system consisting of r phases one needs (n - l)m continuity equations since only (n - 1) diffusional fluxes are independent. ‘Using state vector notation the continuity equations involved can be written as DC”

K=

D”V2c”+R*

(CY= 1,2,. . .,T)

(1)

where

is the substantial time derivative ,and ,C =. (cla, c,q. . ., &) ’ is the transposed (n - 1) dimensional state vector whose (n - 1) components are the molar concentrations of (n - 1) species in phase (Y. Here D* is a diagonal diffusion coefficient matrix,, R a an (n - 1) dimensional source vector whose components are the molar rate of production of the (n - 1) species in phase (Y where u independent chemical reactions take place, and v is the local velocity vector. The superscript (Y refers to the phase considered (continuous phase is denoted by c and discontinuous phase by d in a two-phase system). For first-order or pseudo first-order. reversible reactions the source state vector may be decomposed to R”=Nae

where N* is the matrix of reaction efficients for a first-order reacting 554

(2) rate cosystem.

General analysis of multicomponent mass transfer

Substituting now (2) in (I), premultiplying [D”] -l and rearranging one obtains V2c” = - [D”]-‘NY+

by

[D=]-lg.

(3)

This system of partial differential equations can be reduced to a set of ordinary differential equations by introducing initial conditions for c and an integral transformation with a suitable kernel K”(P, t) defined by . ‘*

t>c”(t)dt

(c”) = j-,mP(P, ‘. ’

,

(4)

where’ (c*) is the expected value of co! over the distribution density function P(P, t) . The time coordinate in Eqs. (3) and (4) is the time a particle in a feed group has spent in the dispersed system. The advantage of this method is that the kernel P(P, t) can be chosen to be the residence time distribution function for the particles as well as for the continuous phase, so that~ ( c~) becomes the expected value of the concentration fields in both phases. Therefore, P can be interpreted as a parameter of the ‘residence time distribution function of phase CX. For the general case of residence time distribution in real particulate systems ‘one can’ choose the recently proposed normalized kernel [5, .16]:

The proposed method is illustrated here for the use of perfectly-mixed two-phase system where nc = qd = 1 and there is no obstruction of the flow of either phase in the exit. Under this domain one obtains from Eq. (5) the normalized distribution function for either phase Kd(+d,

K”(i-,7)“, where

t)

F

,’

* “=Pexp[-lnna]. 7.

KC(+C,

t)

exp

(-t”)

(;a)

‘:

= Q”(P) +?[P]-‘v

I

* V(c”) -, [pq-1cab

(5)

,‘.

Qa,= [Da]-‘[I-PN*]

,

and the initial conditions I/

( 46)

,,.’

where

Here P is the holding time for either phase and r) is a measure of the efficiency of mixing[5]. Thus Pa can be considered as a characteristic time parameter of the system. For a well-mixed dispersion v” = 1 and ?*a becomes the average residence time of phase (Y. For a well-mixed dispersion with partial plug flow or dead spaces r) > 1 while for well-mixed vessels with partial short circuiting n < 1.

=

where P = ‘id = e. Note that Eq. (5a) has the same form as Eq. (5) which means that the proposed method can be easily extended to any real system for which r) is known. ’ The integral transformation (4) with kernel (5) or (5a) is thus accomplished by taking the Laplace transforms of Eqs. ‘(3) with exp (- st) , where s = qa/v, and dividing the transformed quantity by +. a. Since this method gives directly the expected values of Cu one does not need to perform the tedious (and usually impossible) inverse transformation. Consequently, the proposed method removes a considerable mathematical difficulty in obtaining analytical solutions for multicomponent mass transfer in multiphase .’ systems [ 51. Substituting (5) in (3), integrating and rearrang’ ing one obtains

. p 3 -v F

=

t

P72(c”) exp (- 59) 6,. O1

t)

c”=CO

”

,’ ‘at

,,‘l

‘I=()

(6a)

” * (6b)

have been introduced. In obtaining’(6) in terms of the expected values of cn we have assumed that v depends on the space coordinates but is independent of the time variable. To solve the set of Eqs. (6) one should evaluate first the dependency ofv on the space coordinates. Such evaluation by experimental or theoretical methods is usually quite diffichlt. Nevertheless, 555

L. L. TAVLARIDES

it is possible to estimate the relative contribution of the convective term by employing a simplified cell model. If the concentrations, temperatures and shear flow stresses around any two adjacent particles in an ensemble of many particles are superimposed, the resulting projiles must exhibit an extremum value at a point between any two adjacent particles. In dispersions, emulsions, suspensions, etc., where a large number of spherical particles are present, a particle chosen at random can be expected (statistically) to be surrounded by a spherically symmetric cluster of particles. If the radius of such a spherical “cloud” is P, then one can approximate the locus of points at which the resultant profiles exhibit an extremum, by a spherical surface of radius b = P/2 (i.e. all the profiles exhibit an extremum value at points midway between adjacent particles). This assumption is expected to be a good approximation especially in the case of evenly distributed particles (i.e. when the local dispersed-phase holdup fraction is the same everywhere in the system and is equal to the overall dispersed-phase holdup fraction - @) . Hence, each surface with extremum conditions effectively isolates each particle. The result is an ensemble of cell models which has been used successfully by Happel[8,9] in his “free surface model” for the prediction of the terminal velocity of an ensemble of spherical solid particles. The ensemble of spherical cell models has been used also[ 1-3, 11,121 in predicting mass and heat transfer in dispersions as well as in a recent generalization by Gal-Or and Waslo[ 131 of Levich’s work [ 141 (on the velocity of a single drop) to that of an ensemble of drops moving in. the presence of surfactants. These authors show that for an ensemble of “free surface” cell models, the terminal velocity of a particle in an ensemble of particles is

u ensemble

=

2bd--f9ga2 9cLc

x

and B. GAL-OR K*=

2roa

Here y is the retardation coefficient due to the presence of surfactants. It should be emphasized now that dispersions and emulsions with fluids having clean interfaces are relatively rare in actual practice. Many unsuspected impurities in a system can be surface-active agents and even the equipment itself can supply enough impurities to make a profound decrease in the transfer rates. It was shown [ 131 that the terminal velocity of drops or bubbles is decreased in the presence of traces of surfactants. For high Q, values Eq. (7) also predicts that the particle Stokes velocity (which is already very small for relatively fine particle populations) should be further reduced to a negligible value due to hindering effects of neighbor particles (for @ = 0.5, for example, u ensemble/ ~st&?S is reduced from 1.5 to O-027 [ 131). Hence, in the domain of high @ values and in actual practice where surfactants are present we assume that the particles are completely entrained by the continuous phase eddies resulting in a negligible effect of the convective term in Eq. (6). Evidently this assumption does not preclude the possibility of finite relative velocities among the eddies themselves.? In any case, the solution of Eq. (6) without the convective term provides the designer of a multiphase system with the lowest bound of the transfer rates to be expected under given operating conditions. This result is also in good agreement with some experimental studies [ 13, 161 which show that the mass transfer coefficient (based on measured interfacial surface area) in agitated dispersions

3@( 1 - @1’S)( 1 - (P5’3)+ ( y + jAd)(3 - %I+‘3+ &I+‘3- 3cp2) 2@(1 -@5’3) + (y+/~~)(3+2@~‘~)

where

(7) r=-+K*-g

(8)

tNumerical solutions of the nonlinear differential Eq. (6) with the convective term will be reported in due time.

556

General analysis of muiticomponent mass transfer

is unaffected by changes in agitation intensity. Under this domain Eq. (6) takes the form eaV2 (c”) = Q* (c”) - [Da]-+“.

be solved with appropriate boundary conditions to give the expected values of the state vector as a function of the space coordinates., DeGroot and Mazur[7] have shown for a similar system that the eigenvalues are real and non-negative when Onsager reciprocal relations, stability conditions and the second law of thermodynamics are employed. For the reacting systems studied here, i.e. first-order reversible reactions in each phase, the eigenvalues were found (see examples following Eq. (41)) to be real and non-negative within parametric ranges of practical interest.

(10)

To solve Eq. (10) it is recalled that any nonsymmetric matrix with distinct eigenvalues [6] can be diagonalized by a similarity transformation. The eigenvalues are assumed distinct since they are obtained by experimental techniques with a certain range of error (i.e. even if repeated roots are found one is always entitled to make a negligible change, of the order of, say, l/l0000 in the absolute values of the elements, to cause repeated roots to disappear). Therefore, one can make the similarity transformations [p”]-‘Q*p” = Aa

(111

where the roots &*(i= 1,2,..., n-l) obtained from the characteristic equation

are

]Q”-A”I] = 0 and the eigenvectors

TOTAL

In a two-phase the instantaneous typical spherical population with f(a) is given by (Ja(a))

(12)

of PO:from the relation

RATES SYSTEMS

IN MULTIPHASE

system the expected value of flux across the surface of a particle with radius a in a normalized size distribution

=--D”

or in terms of the transformed

vP1@=AAIaPta i= 1,2,...;n-1.

state vector (6”))

(13)

To execute the transformation, (c”) is transformed to a new variable (p) by employing the modal matrix p so that

Evidently (17) accounts for the variations in the residence times of all the particles. The total average interfacial transfer rate from a population of particles in a size range a + da is then expressed by

(c”) = P%*) and

TRANSFER

(14)

dW = ZV4ma2(J(a))f(a)da

c”O=P* f &.

(18)

where Premultiplying (10) by [PI-l and substituting (14) yields now a new system of uncoupled ordinary differential equations eaV2(&=) = Au{&*) +G”e&

3@V N=47r(&#

(CZ= 1,2,. . .,m)

(15)

(19)

is the total number of particles in the multiphase system whose total volume is V, and ri3 is the mean volume radius defined by

where G” = - [P=]-‘[D”]-‘p.

(16)

l/3 (20)

Since the variables are now separated,

(15) can 557

L. L. TAVLARIDES

When (18). is summed over the entire size range of the particle population, one finally obtains the ,total average interfacial transfer rate of each component expressed in terms of the state vector. ,, ,, .,

and B. GAL-OR

particles are uniformly distributed (a is constant throughout the vessel) one obtains141 for each spherical cell model of the ensemble .*: I, .1 &ra3 = &rb3Q,

i.e.

,.‘S_

,,‘.

,,‘*‘1.

‘,,’

./ ..,b,,

?‘.‘,,

;.

.I

,,

,lL

,(..”

_

.*’

.._

,‘, 'I'

,, .,i,,

:*,

a (24) ti

,,

For this ensemble’ of spherical cell models’ ‘we employ now the following boundary conditions:

.,I’

,

,(

‘,

* b='~@-l/3!

,,.

.’

,,,, at r=d,

‘,

,

’

,.

,

The total rate W represents the quantities
(22)

.

An average interfacial flux of the population (averaged over residence time as well as over particle size distributions) may now be defined as

s;a’(.&) I_f(a)da

JAVC =

(23)

JmaTf(n)da 0

ANALYTICAL

SOLUTION

ENSEMBLE

OF SPHERICAL

FOR

AN

CELL

MODELS

.To obtain analytical solutions to the characteristic equations describing the system one recalls that between ‘any two adjacent particles in the ensemble an extremum condition exists in the concentration profiles. Thus an extremum surface exists around each particle through which no transfer of species occurs. When the

at at

r=a,

r=

.a&,

,,.

i’$O .,,

.z.. . .

a@-1/3,’ f’$ ,,I

0 ,.I

.,,I(,.

at’

.”

‘.

r=a,

ta0

t 3 0

L

-0

r :

(25)

$0

(2(j) ,,f

,

cd = MC”

(27)

Dd($)==D=(g).

(28) ..I

Boundary condition (25) is a result of the radial symmetry of the models when the origin of the coordinate system is fixed at the center of a typical particle. Boundary condition (26) follows directly from the extremum condition at the outside surface of the spherical shell of each cell model. Boundary condition (27) relates the concentration at the interface by their equilibrium partition coefficient MI where M = diag{M*}. The last boundary condition demands the equality of the fluxes for each component at the interface (i.e. no accumulation exists at the interface). Solution of (10) subject to the subsidiary conditions (6b) and (25)-(28) with transformations (14) is given ‘in Appendix A. Expected values of the concentrations profiles in either phase are then given by (cc) = ~P%Sc(r)[L-U]-‘Y +P”[hc]-‘[~]-l[DC]-lcC~

(29)

and (Cd)

=

fPSd(r)L[L-U]-lY +pd[Ad]-l[pd],-l[Dd]-lcdo

558

(30)

General analysis of multicomponent

where,. y(r)

transformation matrices pa can be evaluated in order to predict the total average multicomponent mass transfer rates and the performance of the particulate system considered:,The application of the general solution obtained will be demonstrated now for several cases.

.

= diag{sinh J($)(r-b)

+b,/(k$)cosh,/(k$(r--b)} ’. #I, i = 1,2, . . . , n’~ 1 Sd(r).=diag{sinhd($)r}

(30a)

.,

;,diag{r - sinh

J(s) Aid J( ;“r >1

cash J($)r i=1,2,...,n-I (3Oc)

WC(r) =diag[(!?$-l)sinhJ@(r-b)+ + ,/($)

(r-b)

cash J(k$)

”

FLOW (CFSTR)

The total average transfer rates in a two-phase CFSTR can be obtained from a macroscopic mass balance. Let F represent the ,volumetric flow, rate of the dispersed phase with inlet and effluent concentrations cd0 and CF. Let Q denote the continuous phase volumetric flow rate with inlet and effluent concentrations cc0 and SE. The accumulation of species in the continuous phase of a typical cell model (a < r Q b) can be expressed in vector notation by

(r-b)]

i= 1,2,. . . , n-

/.

THE TW,-,~AS,“CONT’,NUO;I’S’ STIRRED TANK REACTOR

‘i=1,2,...,n-1 (3W

y:(r)

mass transfer

E(u, t) =

(32)

1 (30d) or in terms of expected values

L = [Wd(a)]-l[pd]-l[Dd]-lDCpCWC(a) u = [Sd(a)]-i[P]%P%C(a)

(30e)

Y = [Sd(a)]-l{[pd]-lMPCIAC]-l[PC]-l[DC]-lcCo - [Ad]-l[Pd]-‘[Dd]-‘cdo} (I = unit matrix)

For a perfectly mixed CFSTR the above becomes

(30g)

(5) applies and

(30h)

The total average interfacial transfer rate of each species can now be obtained by taking the derivative of (29) or (30) with respect to the space coordinate r (at r = a) and substituting the result in (21) to’yield W= -- 3@VD” m uf(u)PWc(u) I0 (&)3

Kc(P, t)

(3Of)

[L-U]-‘Y

The total accumulqtion AC in the continuous phase is then obtained by an integration over the entire particle population (i.e. by the zeroth moment)

da. (31)

Given the properties of the system (a, h3, +“, Di, Mi, /cl, etc.) the eigenvalues &a and the

AC= $!&WWu,P))du.

Thus one finally obtains

559

L. L. TAVLARIDES

and B. GAL-OR THE THREE

+-

3@V

m b

f(a)P(cC)

fc (id3 II0

d&z.

(33)

a

The macroscopic mass balance for all chemical species in the CFSTR is then given by Qc@ = QP + AC.

(34)

Since the nominal average residence time of the continuous phase is V(1 -@)/Q = +“, (33) and (34) may be combined to give an expression for the concentration of material quantities in the effluent stream

COMPONENT SYSTEM

The application of the general solution to the case of a first-order or pseudo first-order reversible reaction in both phases for a three component system is examined here with more details. When c3” is in excessive amounts its concentration varies slightly during the reaction and the diffisional process so that the reaction is represented by (39) For this case the reaction rate coefficient matrix becomes N”=

Q, --ccE - 1-a

3

mb f(a)r2(cC)

(c?3)3

II o

“f(a),.2(cd)

phase

&.&

kyfl”+ l+&?+ 12 ++I[ DIQ

(36)

D2=

-cdE(ii:)3 IfO” af(a)r”(Cd) o

d&z.

_4

(37)

(ky,?+

1)

pa=

af(a)rP%“(r)

x [L-‘U]-lYdr&+PIAc]-‘[F]-‘[Dc]-‘cco. (38) Given the particle size distribution function, average residence times, physical parameters, and initial concentrations (38) is useful for the prediction of yields in multicomponent, multiphase systems undergoing reversible or irreversible chemical reactions.

1)

kyb+“+ 1 -&“D2nkAiU+ ky,+ a 1

--

(k:bT”+

I

_

(ia)2k$kyb

D,aD2a

DlaDzu

o

Effluent concentrations can be evaluated for dispersed phase CFSTR when (29) and (30) are substituted in (35) and (37) ‘respectively. Thus for the continuous phase =

WW

The expressions for the eigenvalues of Q and the suitable transformation matrix obtained from (12) and (13) are then given by the simplified form:

and

ccE

[-2 -21.

(35)

drdu.

a

A similar balance for the discontinuous yields P. For this case

DISPERSED

1 --2aD2a ky,P 1

1’2

II (40) . 1 (41)

As expected all the values of hIa(i = 1,2) calculated for the wide range of applicability; (1 X 1O-3set-‘) s P,, 1-Oset < P c 10.0 set, (1 X 1O+3set-l), (04 X 10m5cm2-set-‘) ‘%b s < Die =S (3.3 X 10v5 cm2-set-‘), and (0.08 cm2set-l) < Did s (0.30 cm2-set-l) were found to be real, non-negative, non-zero and distinct. To illustrate further the application of the genera1 solutions, we consider now the restricted cases of three-component systems undergoing

560

General analysisof multicomponentmaSstransfer reversible reaction only in the continuous phase while the diffusion takes place in both phases. Hence for the non-reacting dispersed phase one obtains from ( 10) +dV2(~d) = [Dd]-‘(cd)

- [Dd]-l~do

W2=- DzdM# DgdSzdW 2d

Pi2

(L-u)-’

(42)

s H =&

where I = PCnPE2-Pf,Ph.

(46)

Likewise by using (30e) and (3Of), the expression for the last matrix on the right side of (44) becomes Wzd (47)

-Ph

D2CS2dWlc - D,dM2S,CW,d DzdSzdW 2 d

where (cd) is a two-component concentration vector (n-l equations are independent). Since no reaction occurs in the dispersed phase Pd = I, and from (42) it is obvious that Ad = diag { l/DICd} for i = 1,2. Using these matrices the expected values of the concentration of any component in either phase can now be expressed by (29) and (30). An expression for the total average interfacial transfer rate of component one will be derived for this case. Here (3 1) gives

where 7) = D,dD2dS,dS2dW,dW2d

(47a)

and E* = r ( D,cD,cS,dS2dW,CW,c + DldDzdM, MzS;cS2cWldW2d) - D2cD,dM,S2d Wld x

(47b)

x (P~1P:2SlCW2C-PBF2PElS2cWlC) - DlCD2dM2S,dW2d (p~,p~&WIc -PEPE1S1CW2c) *

~$(a) D:$ItiYi

da

(43)

where summation is implied by repeated latin subscripts and B,, are the elements of the matrix product B = PcWc (a) [L-U]-‘.

The elements of Eq. (47) will be denoted now as Hu. Substitution of (44). (45) and (47) in (43) and performing the scalar multiplication gives the final expression

(44)

It is necessary to obtain an expression for the scalar product in the integrand of (43). From (30g) and some manipulations the vector Y is expressed as (continued on next page)

561

CJS.S.Vd.24No.3--1

L. L. TAVLARIDES

and B. GAL-OR

transfer accompanied a single homogeneous

with a chemical reaction in phase.

NUMERICAL

ANALYSIS

The equations describing the behavior of the ternary system were analyzed numerically ‘by using a digital computer. Table 1 lists the values of the physical parameters, initial conditions and operating conditions employed in the calculations. The parameters employed were estimated from data on gas-liquid dispersions[ 151. The behavior of Ji(i= 1,2), (Ji)/(Ji)*, and WI/W: was studied as effected by variations in holdup, particle size distribution, average residence time, average particle size, and the reversible reaction coefficient ratio K = k,/k,. The effects of Q, and a3 on (Ji) for various values of K are shown in Fig. 1. In general, the expected interfacial flux is reduced for larger values of dispersed phase holdup and as the particle size decreases. For a given value of a’, a decrease in particle size increases the number of particles per unit volume. The distance among particles decreases and, as a result of the diffusional and chemical processes, the concentration gradients in the

X[af (a) - [p,1w1cH1>~~‘2w2cHa da 1 (48) where the eigenvalues Aicand the elements of the transformation matrix P” are given by (40) and (41) for (Y= c. The elements of Wa(a) and S”(a) are obtained from Eqs. (3Oa)-(3Od). This example illustrates the procedure to evaluate WI. A similar procedure should be employed for the evaluation of W,. Thus, knowing the values of the system parameters, the total average interfacial transfer rate is evaluated for each component. Alternatively, one may use (38) or its analog in the dispersed phase to obtain effluent concentrations in a CFSTR. It can be easily shown[ 151 that the characteristic Eqs. (45)-(48) are a generalization of the solutions of Huang and Kuo[l7] for mass

Table 1. Parameters employed in numerical analysis of ternary system Fig. 4 Curve B

Fig. 1.

Fig. 2

Fig. 3

Curve A

cldo, (g-mol/cm”) czdo,(g-mol/cm3) cIco, (g-mo1/cm3) czeo,(g-mol/cm? Physical parameters

l.OO(--6)t O*SO(-6) l+lO(-7) O-50(- 7)

l-50(- 7) 7.50(- 8) l*OO(- 12) 2.50(- 8)

l-50(-7) 7*50(- 8) l+O(- 12) 2.50(- 8)

l-46(- 7) lXiO(-- 8) l*OO(- 12) l.OO(- 12)

1*50(-7) 7.50(- 8) l.Oo(- 12) 2*50(- 8)

M, M, Did, (cm%ec) Dzd, (cm%ec) Die, (cm%ec) Dzc, (cm’/sec) k,, (set-‘) kb, (set-‘) Operating parameters +. (set) &, (cm) @

1.0 1.0 3+lO(- 2) l+O(- 2) 3*00(- 5) l+lO(- 5) Variable 050

O-8

60-o

20.0 1.80(- 1) 860 (- 2) 2.50(- 5) 1.25(-5) 1.00 Variable

1-o 1.0

0.8

0.5 1*80(- 1) 8.60(- 2) 2.50(- 5) 1.25(- 5) Variable 0.50

303(- 2) l@O(- 2) 2.30(- 5) l&q- 5) l.OO(+ 4) l.OO(- 5)

0.5 1.80(860(2*50(1.25(1.0 0.25

2.0 Variable Variable

Variable 0.05 0.20

Variable 0.01 0.20

2.0 Variable 0.064

2.0 Variable 0.20

Parameters Initial Conditions

tparenthesis notation is

used

to express powers of ten, i.e. 1-O(- 9) = 1.0 X 10e9.

562

1) 2) 5) 5)

General analysis of multicomponent mass transfer

_ - .-. - ---

FARTICLE INTERACTION EFFECT ON CJ,(d,D, KnO.5 K EFFECT ON
0.0

020

@, MSPERSED

CONSTANT

090 PHASE

= 05

060 HOLDUP

SEC-’

0.80

la0

0.0

FRACTION

DO

200

30.0

40.0

Y ,O

P, SEC

Fig. 1. Effect of holdup, average particle size and reversible reaction rates on expected values of the interfacial fluxes in a ternary system. (System data are given in Table 1.)

continuous phase between any two particles are “flattened” out. This, of course, reduces the expected interfacial flux. A similar effect is observed for increasing Q, values at a given particle size. Hence, a formulation that does not take into account these interaction effects as a function of @ and & may introduce a considerable error especially in multiphase systems with relatively small particle sizes. The effect of increasing forward rates of the chemical reaction in the continuous phase are also shown in Fig. 1. A twenty fold increase in kf values at constant kb and & gives about twice to four times increase in the value of ( J1) depending on the value of a. As expected, a similar, but opposite effect is observed for the behavior of (J,) . Denoting (J1) * as the expected value of the interfacial flux of physical diffusion (i.e. without chemical reaction) the dimensionless flux ( Jl)/( J1) * is shown in Fig. 2 to change rapidly at short residence times and attain an extremum. The direction of the change and the nature of the extremum depends on whether K is less or greater than unity. For long residence times, as physical and chemical equilibrium are approached, the dimensionless flux asymptotically

Fig. 2. Effect of average residence time and forwar-d reaction on the ratio of expected interfacial fluxes with and without chemical reaction respectively. (System data are given in Table 1.)

tends to one. For K > 1 larger fluxes and a maximum are observed as K increases since the product is consumed more rapidly as it transfers into the continuous phase and there is a negligible generation due to the backward reaction. When K < 1, the backward reaction controls and the reactant is generated rapidly in the continuous phase giving rise to a minimum in the dimensionless flux. Eventually this accumulation may cause a reversal in the direction of the flux as shown in Fig. 3. In Fig. 3 large values are

Fig. 3. Illustration of the reversal point for the direction of mass transfer. (System data are given in Table 1.)

563

L. L. TAVLARIDES

assigned to the equilibrium partition coefficients M, and MS which indicate that both species are sparingly soluble in the continuous phase. When K = 1-O the backward reaction becomes significant and the reactant is generating rapidly in the continuous phase. Eventually when MICIC > Cld a reversal of the flux of component one occurs. Thus for a given set of operating conditions the reversal point depends upon the average residence time and K in addition to the initial conditions and the equilibrium partition coefficients. Such allocation of the reversal point is important in the design of reacting dispersed systems. For estimation of the total average transfer rate WI, the normalized particle size distribution

A

32 =

1*148d,.

(50)

If one replaces now the particles size distribution by the mean hs2, Eq. (2 1) reduces to W

(ci32)

=

(PV 3’544 -&(J@32))*

(5

..

CURVE A CURVE B

I .75

200

\a

6,=Ol mm 075 4 0.0

I 50

1 B 0.75

I l.0

to mm 6w 8 00075 mm a,* =

I IO0 I I.25 w, lOlo) IW,

Fig. 4. Illustration of total average transfer particle size in lieu of are

I 15.0 I If.0 C&l

I 200

250

I I.75

I 20

0.0

*

the error introduced in calculating rates when one assumes a uniform a distribution of sizes. (System data given in Table 1.)

plot for two different sets of conditions. Each calculated point was obtained by selecting d3 and calculating the ratios of total transfer rates by using Eqs. (2 l), (50), and (5 1). Details of the procedure are described elsewhere [ 151. Although the initial conditions, physical parameters and operating conditions differ widely between the curves the results are basically the same. In curve A, for a32 3 2-O mm the differences in the calculated values do not exceed 3.45 per cent. Similar results are observed in curve B. These results were also obtained for plots of WI vs. Wl(833) [ 151. Thus one can replace the particle size distribution by Li32without introducing a significant error in the values of the predicted total transfer rates in the system. This general, and somewhat surprising result, confirms a previous observation obtained by Gal-Or and Hoelscher[3] for convective mass transfer rates.

proposed by Bayens and modified by Gal-Or and Hoelscher[3] is employed. This distribution is quite practical since it depends only on one parameter, namely, &. It has been found to approximate size distribution data in gas-liquid as well as liquid-liquid dispersions [3, 51. The upper limit of the integration in Eq. (21) for which larger values of a do not alter Wi was found to equal &+5 standard deviations (15). Using (49) one easily finds the relationships between ci, and the surface mean radius ci,, a

and B. GAL-OR

1)

Consequently, a plot Of WI/W: VS. Wl(d33)/ WTd33) can illustrate the error introduced in assuming a uniform particle size in lieu of a distribution of sizes. Here W,* and WT(u3& represent total transfer rates without chemical reaction calculated with f(a) and with (632) respectively. Figure 4 is an example of such a

CONCLUDING

REMARKS

The general analysis presented describes the mechanisms of multicomponent mass transport accompanied by reversible or irreversible first-order chemical reactions in multiphase systems consisting of large .numbers of drops, bubbles or solid particles with non-uniform

564

.* General analysis of

&icomponent masstransfer

particle size and residence time distributions. Using approgiiate transformations and integral operators it is possible to decouple the characteristic equations of the system and to express them in terms of expected values. The resulting ordinary differential equations can only then be solved to yield-expected interfacial fluxes for each component. - When these results are combined with second moment equations of the particle ‘size distribution, the total average transfer rate for each component can be obtained. The resulting total transfer rate in the system can be .expressed in. terms of the properties of the ensemble, i.e. as a function of the volumetric holdup fraction of the dispersed phase, the particle size distribution and the residence time distributions. The mathematical formulation of the total transfer rate includes also such physical parameters of the system as the matrix of the reversible reaction rate constants, diffusivities of the chemical components in each phase and the interfacial partition coefficients. The results of this analysis are useful for the rational design of particulate systems and in predicting the yields and effluent concentrations as a function of the physical and operating variables. The application of the ,general solution presented is illustrated for the two-phase continuous stirred tank reactor and for the case of a three component particulate system undergoing steady-state diffusion in both phases with a simultaneous reversible reaction in the continuous phase. One interesting conclusion from this general analysis is that the error involved in assuming a uniform particle size in lieu of a distribution of sizes is small when calcuiating total average transfer rates. It is found also that a reversal in the direction of the mass transport may take place under certain conditions which depend on the average residence time and the relative rates of the backward to the forward reactions. Furthermore, it is concluded that, in general, the expected value of the interfacial flux for each component decreases with increasing dispersed-phase holdup fractions. This is mainly due to interaction effects among neighbor

particles corresponding to changes of Q, and a in the boundary conditions of the ensemble of spherical cell models employed. At short contact times the effect of the chemical reaction compared to pure physical diffusion is found to be very sensitive to variations in average residence times. An extremum in the ratio of the expected interfacial fluxes exists whose nature and position is strongly dependent upon the reaction rate coefficient ratio. For a given set of physical and operating conditions this phenomenon demonstrates the existence of (an optimum reactor volume with respect to effects of the chemical reaction on the transfer rates. Finally, it is noted that this analysis demonstrates how the cell model can be employed to model dispersed phase systems. It is to the discretion of the worker to determine when the model may not be applicable to some specific systems (uiz., the model may not be applicable in high velocity systems or whenever the deformation rate of the fluid is large). Acknowledgments-The authors wish to express their sincere gratitude to the National Aeronautics and Space Administration for a Predoctoral Traineeship granted to one of us (LLT), to the School of Engineering of the University of Pittsburgh for the general support given for conducting this research, and to The National Science Foundation for Grant G 11309 which partially supported the expense of the computations.

565

NOTATION

a

radius of typical spherical particle in a dispersion mean volume radius defined by Eq. (20) a32 surface mean radius A total interfacial area defined by Eq. (22) A” total accumulation in phase (Ydefined by Eqs. (32,35) b outer radius of a typical spherical cell Bid element of matrix B B matrix defined by Eq. (44) ith component of ‘the concentration Ci state vector C concentration state vector whose A

L. L. TAVLARIDES

and B. GAL-OR

N

components are the molar concentrations of (n - 1) species in an n-component homogeneous phase diffusion coefficient of species i in phase (Y diffusivity of surfactants in the bulk

Pu Pi

Db’ $OhltiOtl : :. surface dif%sivity of surfactants p*

I3 diagonal matrix ,of diffusion coefficients E accumulation of species in continuous phase of spherical cell defined by ._ Eq. (32) E* defined by Eq. (47b) F dispersed phase flow rate g local acceleration G; coefficients of matrix G G matrix defined by Eq. (16) Hi.i elements of matrix H I-I matrix defined by Eq. (47) unit matrix state vector of instantaneous inter: facial fluxes (the expected value of J is defined by Eq. ( 17)) J AVG,l average instantaneous flux of component one J AVG average instantaneous flux defined by Eq. (23) reaction coefficient for the ith chemical k reaction K ratio of forward to backward reaction rate coefficients for ith chemical reaction, k,/k, constant due to the K” retardation presence of surfactants function for the integral K(t, t) kernel transformation defined by Eq. (4) L matrix defined by Eq. (30e) Mt interfacial partition coefficient for the ith component (can be evaluated from Henry’s law or other equilibrium relationships) M diagonal matrix of interfacial partition coefficients M” matrix defined by Eq. (A 18) N total number of particles in a dispersion defined by Eq. (19)

P Q Q r rj R Ri S*(r) t T u u

ensemble

U v V, V V WI W

Wi(&)

W(r) X Y, Y Zi Z

matrix of reaction rate coefficients see for example Eq. (39a) element of matrix P ith vector in the transformation matrix defined by Eq. (13) transformation matrix continuous phase flow rate matrix defined by Rq. (6a) radius in spherical coordinates rate of reaction i source strength vector due to chemical reactions ith component of the source strength vector R defined by Eqs. (30a, 30b) time variable diagonal matrix of average residence times defined by Eq. (8) number of independent chemical reactions terminal velocity of a particle in an ensemble of drops, bubbles or solid particles matrix defined by Eq. (30f) local velocity vector element of matrix V total volume of the multiphase system matrix defined by Eq. (A.8) total interfacial transfer rate of component one state vector of total average interfacial transfer rates defined by Eq. (21) total average interfacial transfer rate of the ith component evaluated with uniform particle size & matrix defined by Eq. (3Oc, 30d) vector defined by Eqs. (A.25, A.26) element of vector Y vector defined by Eq. (30g) element of vector Z defined by Eqs. (A.5), A. 12) vector with elements Zt

Greek symbols a0 constant of desorption kinetics constant of adsorption kinetics PO

General analysis of multicomponent mass transfer

retardation coefficient due to surfactants order of backward reaction with respect to componentj defined by Eq. (47a) constant defined by Eq. (46) surface concentration of surfactants equilibrium surface concentration of surfactants maximum saturation concentration of surfactants at the interface thickness of Nemest boundary layer of diffusion stoichiometric coefficient order of forward reaction with respect to componentj ith eigenvalue defined by Eq. (12) viscosity diagonal matrix of eigenvalues defined by Eq. (11) ith component of transformed concentration vector transformed concentration vector, expected value of defined by Eq. (14)

T

number of phases in a multiphase system p density fzr surface tension + average residence time @ volumetric dispersed phase holdup fraction

Superscripts a! refers to the homogeneous phase considered (a = 1,2,. . ., 7~) c refers to continuous phase d refers to dispersed phase CQ refers to initial value in phase (Y aE refers to etlluent in phase a! * refers to physical diffusion without chemical reaction ’ denotes transpose of state vector Subscripts b f i, j

refers to backward reaction refers to forward reaction dummy subscript

Special V’mbozs ( ) refers to expected value

REFERENCES [l] GAL-OR B., RESNICK W., Chem. Engng Sci. 196419 653. [2] GAL-OR B.,WALATKAV. V.,A.I.Ch.E. .I1 1967 13650. [3] GAL-ORB., HOELSCHER H. E.,A.I.Ch.E. Jl1966 12499. [4] GAL-OR B., PADMANABHAN L.,A.I.Ch.E..JlSeptember 1968. I51 GAL-OR B., Znt.J. Heat Mass Transfer 1968 11551. i6] STOLL R. R., LinearAlgebra and Matrix Theory. McGraw-Hill 1952. [7] DEGROOT S. R., MAZUR P., Non-Equilibrium Thermodynamics. Interscience 1962. [8] HAPPELJ.,A.I.Ch.E.JI19584 197. [9] HAPPELJ.,A.I.Ch.E.JI 1958 5 174. [lo] PADMANABHAN L., GAL-OR B., Chem. Engng Sci. 1968 23 63 1. 1111 GAL-OR B.. RESNICK W.. Znd. Erwnn Chem. Proc. Des. Deuel. 1966 5 15. [12] GAL-OR B.; HAUCK J. P.,‘HOELSCHER H. E., 1nt.J. Heat Mass Transfer 1967 10 1559. [ 131 GAL-OR B., WASLO S., Chem. Engng Sci. In press (CES 697). [14] LEVICH V. G., Physicochemical Hydrodynamics. Prentice Hall 1962. [IS] TAVLARIDES L. L., PhD Thesis, Department of Chemical and Petroleum Engineering, University of Pittsburgh 1968. [I61 RESNICK W., GAL-OR B., Gas-Liquid Dispersions: Advances in Chemical Engineering, Vol. 7, pp. 295-395. Academic Press 1968. 1171 HUANG C.J.,KUOC.J.,A.l.Ch.E.J11965 11901.

APPENDIX A

which has the general solutions

This appendix outlines the general analytical solution of Equations (15) subject to the subsidiary conditions (9) and (25)-(28). For spherically symmetric coordinates (15) becomes i.-&r@)

= Aa

+G”(rp)

(&,‘) =~[XBcsinh$$~(r-b) +Z~coosh~~)(r-b)]+VB.~~

(A.1)

567

(A.2)

L. L. TAVLARIDES

and B. GAL-OR Substitution of (A.3) in (A. 11) and evaluating in the limit as r + 0 yields

and

(A.12)

Z#d = 0. p= j-

1,2,...,n-1 1,2,...,n-1

(A3)

Equation (A.3) can be written as

. 1 (lad ) = ;Xad sinh

where or

(A.13)

Summation is understood to be indicated in (A.2) and (A.3) by the repeated Latin subscript. The form of the boundary conditions (25)-(28) is invariant under the transformations (4) as well as (14) so that (26) becomes

where Sd(r) = diag[sinh\Ig)r]

d(Z,$) -= dr

0 at

r = a@-113.

(A.4)

Vd = - [Ad] -rGd.

(A.14)

Hence,

Substituting (A.2) in (A.4) yields

(A,13

(A.3 where Bringing (A.5) to (A.2) gives

Wd(r) = diag[r J(z)

cash J(z)r

(tBc) =fXOc[sinhJ(($)(r-b)

(A. 16)

++)coshj/($$)(r-b)]+Kk&”

For the interface, the boundary condition is transformed to

or .

($) (I”) = +)Xc

= v%$*

=M*(gd)

at

r=a

(A.6)

M* = [PI-IMP.

where

Xd= UXc+aY

(A.19)

where

(A.7)

and Vc = - [AC]-‘G”.

(A.18)

Substituting (A.6) and (A. 13) into (A. 17) and simplifying

Se(r) = diag (sinh d(g)(r-b) +b@)cosh@)(r-bj}

(A.17)

where

(A.8)

U = [Sd(a)]-‘M*Sc(a)

(A.20)

Y = [Sd(a)]-l[M*Vc~@-Vd~do].

(A.21)

Equating fluxes at the interface one finally gets

The derivative takes the form

(A.22)

(A.9) where

Substitution for the derivatives

WC(r) = diag[(F)

sinhJ(F)(r-b)

Xd = LXC

(A.23)

where (A.lO)

L = [Wd(a)]-l[P]-l[Dd]-lD(a).

Simultaneously solving (A.19) and (A.23) using matrix manipulations, the coefficients are finally represented as

Boundary condition (25) transforms to

dUd) -=O

at

r=O.

(A.24)

(A.1 1)

568

Xc= a[L-U]-‘Y

(A.25)

General analysis of multicomponent x = aL[L-LJ]-‘Y. Transformed

concentrations

($)

mass transfer

(A.26)

can now be written in the form

=+)L[L--U]-‘Y+

VdPO

Finally, in order to return to expected values of the concentration profiles one uses (14) to obtain

(cd) =+d(a)L[L-Ul-‘Y +pd[Ad]-l[pd]-‘[Dd]-ledO.

R&sum&- Une analyse gtn6rale est p&se&e pour d&ire 1’Ctude dynamique du transfert de masse d’un composant multiple avec des reactions chimiques simultanees, reversibles ou itreversibles, du premier ordre, dans des systemes particu1CS consistant en un grand nombre de gouttes, bulles ou particules solides, avec distribution du temps de residence et de la taille des particules. Le mode d’approche et les formulations g&t&ales fondamentales sont dCmontrees par I’emploi d’un systbme de notation-matrice et des transformations appropri6es. Les equations diff6rentielles partielles qui en r6sultent pour des systemes a phases et a composants multiples, sont tout d’abord d6coup16es et ensuite transformees en Cquations diff6rentielles ordinaires par I’usage dun 0pQateur integral dont le noyau tient compte des distributions du temps de residence. Les intereactions entre les particules voisines sont prises en consideration par les conditions appropriCes de limite correspondant a un ensemble de cellules spheriques. Les taux de transfert moyens de toutes les particules sont calcules en terme de constantes de taux de reaction, de diffisivites, de coefficients de separation, de temps moyens de r6sidence. de fraction de retenue dans la phase dispersCe et de distribution de la taille des particules. Les resultats de cette analyse g&&ale sont utiles pour prevoir les rendements et les concentrations qui se d6gagent des systemes disperses en reaction en tant que fonction de variables physiques et de variables de fonctionnement. L’application pratique de la solution g&&rale presentbe est illustree pour un r6acteur a deux phases a turbulence continue et pour le cas d’un systeme disperse a trois composants subissant une diffusion en pseudo &at stable dans les deux phases, avec r&t&on rtversible simultanee dans la phase continue. Une conclusion g&ttrale interessante de cette analyse, est que l’erreur commise en supposant une taille uniforme des particules au lieu dune distribution des tailles est faible quand on calcule les taux de transfert moyens. On trouve aussi qu’un renversement du transfert peut se produire dans certaines conditions qui d6pendent du temps moyen de residence et du taux de la reaction en arriere par rapport a la reaction en avant. Zusammenfassung-Es wird eine ahgemeine Analyse zur Beschreibung der Dynamik des Massentransportes in Mehrstoffsystemen bei gleichzeitigem Ablauf reversibler oder irreversibler chemischer Reaktionen erster Ordnung in Teilchen systemen, die aus einer grossen Anzahl von Triipfchen, Bibsthen oder Feststoifteilchen bestehen, mit Verweilzeit- und Teilchengriissenverteilungen, dargelegt. Die Untersuchungsmethode und die ahgemeinen, grundssltzlichen Formulierungen werden in Matrizenform und mit entsprechenden Transformationen veranschauhcht. Die sich ergebenden partiellen Differentialgleichungen fiir Mehrphasensysteme mit mehreren Komponenten werden zunachst getrennt und dann in gewiihnliche Differentialgleichungen umgewandelt und zwar unter Verwendung eines Integraloperators, dessen Kern die Verweilzeitverteilungen beriicksichtigt. Die gegenseitige Beeinflussung benachbarter Teilchen wird durch entsprechende Grenzbedingungen beriicksichtigt,. die einem Ensemble spharischer Zellenmodelle entsprechen. Die gesamten, durchschnittlichen Ubergangsgeschwindigkeiten aus dem gesamten tilchenbestand werden hinsichtlich der Reaktionsgeschwindigkeitskonstanten, des Diffisionsvermidgens, der Verteilungskoeffizienten, der durchschnittlichen Verweilzeiten, des hold-up-Bruchteils der dispergierten phase und der Teilchengrtisseverteilung ausgewertet. Die Ergebnisse dieser allgemeinen Analyse sind fir die Voraussage der Ausbeuten und Abflusskonzentrationen dispergierter Reaktionssysteme in Abhiingigkeit von physikalischen und betriebsbedingten Variablen von Nutzen. Die praktische Anwendung der angefuhrten Allgemeinlosung wird am Beispiel des kontinuierlich geriihrten Zweiphasen-Tankreaktors, sowie fdr den Fall eines Dreikomponenten-Dispersionssystemes mit pseudostationarer Diffusion in beiden Phasen und gleichzeitiger, reversibler Reaktion in der kontinuierlichen Phase erl;iutert. Eine interessante Erkenntnis allgemeiner Art, die aufgrund dieser Analyse gewonnen wurde, besteht darin, dass der durch Annahme einer gleichfiirmigen Teilchengriisse an Stelle einer Griissepverteilung bedingte Fehler verh%ltnism%ssig gering ist, wenn es gilt, die gesamten durchschnittlichen Ubergangsgeschwindigkeiten zu berechnen. Es wird femer festgestellt, dass unter gewissen Bedingungen eine Umkehr der Ubertragung stattfinden kann, in Abhangigkeit von der durchschnittlichen Verweilzeit und vom Verhaltnis der Geschwindigkeit der rtickwarts und der vorwarts gerichteten Reaktionen.

(A.28)