a cross-flow reactor: theoretical model for first order kinetics

A CROSS-FLOW REACTOR: THEORETICAL MODEL FOR FIRST ORDER KINETICS

ROLF DE VOS* and C. E. HAMRIN, JR.t

Department of Chemical Reaction Engineering, Chalmers University of Technology, 412 96 Giiteborg, Sweden

(Received 6 March 1981; in final form 22 December 1981)

Abstract-Three-ph&e reactions (gas, liquid, solid catalyst) have traditionally been carried out in slurry, fluid-bed, and trickle-bed reactors. In this Paper a cross-flow reactor has been modeled in which liquid flows through channels in one direction and fills the catalytic porous walls, while gas flows at right angles in alternate channels. The walls are considered as idealized pores with reactant gas at one end and liquid reactant at the other. Flow through the channels is laminar and concentration profiles are given by the Leveque or Graetz solutions. Coupling of the pore and channel equations occurs at the walls, where diierent boundary conditions are examined. Operation of the reactor at total liquid recycle is treated and the concentration of reactant as a function of time is presented at a value of the ‘Ihiele modulus, initial reactant ratio, and Peclet number representative of experiments for the hydrogenation of nitrobenzoic acid.

INTRODUCTION In recent years, reactors of different geometry have received increasing attention. Two examples are the trickle-bed reactor in petroleum processing[l] and the monolith reactor for exhaust control[2]. Three-phase reaction systems were traditionally operated with slurry reactors, but the trickle-bed is finding more application in this area.

A novel type of reactor for three-phase flow which gives good gas-liquid contact with the catalyst has been fabricated. This reactor contained a’ solid catalyst con- sisting of a battery of parallel thin porous plates saturated with liquid. Through the interstitial channels gas and liquid are in cross-flow through every second channel (see Fig. 1). The reactor was operated as a batch differential reactor with the liquid flow as a total reflux stream, while the gas flowed through the channels without recycle. Experimental results will be presented in a second paper and compared with the theory developed herein.

The governing equation for this system is

d C,i Vr=N,,.S (1)

where V is the total liquid volume and NAw. S is the number of moles of A transported per unit time from the liquid bulk into the porous catalyst.

Only first-order kinetics are considered. As a model reaction hydrogenation of A

vnA + v,H,+products (2)

is chosen. Hydrogenation takes place at the catalyst surface and follows a first order rate dependence on Hz.

*Author to whom correspondence should be addressed. tPresent address: Department of Chemical Engineering, Uni-

versity of Kentucky, Lexington, KY 40506, U.S.A.

In order to express NAw as a function of the inlet concentration of A(G). the differential equations for the concentration distribution in the porous plates and in the liquid-filled channels are solved. The purpose of this work is to solve eqn (1) and evaluate the effects of some process parameters on the reactor behavior.

Pore equation

THEORY

Consider the one-dimensional pore shown in Fig. 2 where the reaction A + 3Hz+products is taking place. Such stoichiometry is typical for the hydrogenation of nitrobenzoic acid[3] where the reaction is first order with respect to hydrogen and zero order with A: i.e.

r = kc,. (3)

Hydrogen is dissolved in the liquid at one end (z = 0) and reactant A is at the other end with aqueous concentration CA, (z = 1). The hydrogen concentration at z = 0, Go, is the equilibrium solubility in the liquid.

For this system two cases.must be considered: first, both reactants are present throughout the pore (Case I) and secondly the concentration of A goes to zero (at z = A), somewhere in the pore (Case II).

For Case I a mass balance for hydrogen is

d2G 7 = d&X

with boundary conditions:

CH=CHo at z=O (5)

$$=O at z= 1 (6)

The symbol $ is the Thiele modulus, Lv((- v~)k/D~).

R. DE VOS and C. E. HAMIUN, JR. 1712

I Liquid

d

C’

Qr

GPS

Fig. 1. Gas-Liquid flow in section of cross-flow reactor.

i= 0

Fig. 2. Idealized pore in cross-flow reactor.

The mass balance for A is

$ = lja’vc,

where Y = (vADN)/(vwDA) and the boundary ~~~ditim~:

Cn=CAw at r=l. (9)

The solution to eqn (4) is the familiar

cH = cm0 co& +(I- 2) cash Q

and that of eqn (7) is

c, = c.4, - (1 - z)grCHo tanh 9

+ a [cash &l-z) - 11. (111

The relation between the fluxes of A and hydrogen is

NAL-I = N,w = -zN&o

(12)

For case II A is defined as the coordinate where Cm vanishes. This occurs first at z = 0. Examining eqn (11) one finds that case II occurs when the wall concentration is less than

C.4W I .&PO

=Gr,, dtanhd+---&,-1 ( > . (13)

When CAw r CawJ,=0 the equations for case I are valid, while for CAw C CAwI*-O the following equations must be applied.

For O<z<A

and for A c: z < 1 one finds

d=C, _ dr’- 9’vCr.r.

The boundary conditions are

CH=Go at z=O

C, and 2 continuous at z = h

C =S=OaIsoatz=* A dz

C,=CAw %=Oatz=l. ’ dz

(14)

(15)

(16)

(17)

(10

( 19920)

(21,22)

(23,24)

Boundary conditions (18) and (20) were put into eqn (1% and (19) and (24) into eqn (16) to give the resulting hydrogen concentration profiles for 0 < z < A

where

cw = C&l - Bz) (25)

B=l tanhdl(l-A)

(26) T+A tanhq5(1-A)

andforh<z<l

CH = C&l - AB) . [ Ez;; $I; 1 ;;I. (27)

The concentration profile of A in the interval A < z < 1 is given by (Zl), (23), and (17):

C,=,cHo cosh&(I-z)_l cash +(l - h) 1 ; z--A

l-h

x cAw+1_ [

1 VCHO cash d(1 - A) 1 . (28)

A cross-flow reactor 1713

Boundary conditions (22) and (24) say that

N,. VA

r-1 =---$r . I r-o

(29)

This condition gives a relation between C,, and A. Another relation for CAw and A is given by the channel equation developed in the following section. Since A varies slowly with C,, a mean value for h can be used. A CAw in the actual interval gives A.

Channel equation The flow distribution in the channel was modeled as

fully developed onedimensional laminar flow between parallel flat plates (see Fig. 1).

A momentum and mass balance for flow and com- ponent A is the well-known[4,5]

Neglecting diffusional transport in the direction of flow, eqn (30) becomes

The solving of (31) and its coupling with the pore equations will depend upon which boundary condition is chosen for the wall (y = (d/2)).

The boundary conditions used are:

C, = C,, at x = 0, all y (32)

F=Oatallx,y=O

G=const. at y=: dY

+$=k,C,at y=$ (35)

To solve eqn (31), boundary conditions (32), (33), (36) are used in conjunction with either (34) or (35). The choice between (34) and (35) is dictated by the situation in the pore.

For Case I both reactants are present throughout the pore. Examining eqn (12) one sees that the molar fluxes are independent of Cnw ; therefore boundary condition (34) is used instead of (35).

For Case II we rearrange SC (36) and get

The factor (dC.Jdz)l,=l is then found from eqn (28) as the linear approximation

(38)

Equation (37) can then be identified with (35)

~ I

=- &CAW dy y=wm D. L (l-*)-$I

(39)

which gives

k, = - D..%

D.L (l-1)-$]’ [

(40)

When Case II is occurring in the pore the situation is more complicated. The molar flux at the wall is a function of C,, Boundary condition (34) cannot be used, as C,, is varying along the channel. The flux N..,w will then be a function of the channel length and determined by the equations (28) and (31).

The solution for eqn (31) in dimensionless form is then obtained by separation of the variablesIf:

CA = C~r~AIYdy’) exp (- $:xr) (41)

yi =x a.(cx*Xy 3” n

where

x’=xD/& and +f_

The parameter x’ can also be expressed as x/dNP= where Npr = N&Jz=. The eigenvalues (I* were calculated from the equations (35) and (36).

Making these boundary conditions dimensionless yield

N +dY’l=o ml dy Y,

where

NBI = - k,d = L(l _y_ 1,+) . $. (4)

Solving (43) for a,, one obtains the eigenvalues wanted for an actual Ns, (a process parameter).

It should be pointed out that CAw catt be determined both by Cases I and II at the same time in the channel. The boundary between the two cases may be situated somewhere in the channel and moves upward with time. This intermediate state will also be discussed in the Results section.

R. DE Vos aad C. E. HAMFUN, JR.

Fig. 3. Cross-flow reactor system showing liquid recycle.

Design equation The design equation for the reactor system shown in

Fig. 3 is

where NAw is determined by the pore and the channel equations.

The change of Cai with time is very slow so that steady state can be considered; i.e. the concentration distribution of A and HI in the pore is established. A rough estimation of the fubiIlment of this assumption can be checked by a calculation of the penetration time for molecule A going from CA = C,W at I = 1 to C, = 0 at .z = 0.

The penetration time may be estimated as

t&,=&. (45)

With the nitrobenzoic acid system as a reference, tP is of the magnitude of 30s at adequate temperatures, which is small compared to the reaction times calculated.

ItESUL’rs ANII DECUSSEON

Concentration distributions in the reactor contain parameters and variables that are governed by either the pore equations (4), (7) (16) and (13, or the channel equation (31). The design equation is built up by the coupling of these. If both reactants are present in the pore, the channel equation doesn’t affect the rate of reaction and only the pore equation for hydrogen affects the design equation (Case I). This is due to the zero order kinetics with respect to A.

If 0 < A < 1 (see Fig. 2) the channel equation must be used to account for the mass transfer resistance in the channel which affects the rate of reaction and mass transfer in the pore. In the nitrobenzoic acid example under study the concentration of A in the system is expressed as a function of time. The hydrogen pressure, initial concentration of A, the dimensions of the catalyst, the difIusivities, the rate constant, the liquid load, and the liquid flow rate are regarded as process parameters in this analysis.

In the solving of eqns (4), (7), (16), (17) and (31) the parameters can be arranged in dimensionless groups which include the Thiele moduli, $ and PI& a dimension-

less length (in the direction of flow) variable, x’, containing a Peclet number Np., and a Biot number ND;.

Pore equation Dimensionless concentration profiles of hydrogen,

C,lC,,, and A, C.JCAw in the pore are presented in Fig. 4 for vCHJCnW = 0.0267, 2.67 and 26.7. The Thiele modulus is a parameter on each of the plots varying from 5 to 80. Values shown are those for which little hydrogen reaches the channel end of the pore. The concentration of A can reach three different end points depending upon the value of vC&C..,, and 4. In the top figure the value of C.JCAW decreases to 0.9 and 0.5 at z = 0 indicating the presence of A throughout the pore for 4 = 5 and 20, respectively (Case I). The limiting end point occurs when C..JGW = 0 at z = 0. This case is approximated by the $ = 80 curve in the top figure. The third end point occurs when CA/CAw reaches 0 at some finite value in the pore, A (Case Il). With a decrease in C,, and an increase in 4, the value i increases as shown by comparing the profiles.

Another observation is the occurrence of a reaction zone between h and the value of z at which C, ap- proaches zero. This zone becomes narrower as 4 increases as shown in the middle figure. Also the concentration profiles become more linear in the pore. In this

Cl&l0 C

Fig. 4. Dimensionless concentration profiles of hydrogen and A in the pore at various stoichiometric ratios: (5 5 I$ s 80).

A cross-flow reactor

reaction zone the rate of reaction is given by a tist order expression; however, the order is relatively unimportant since the process is diffusion controlled.

The limits discussed for vC,dC_+w and the Thiele modlllus are summarized in Fig. 5. The upper curve presents the limits of the process variables for Case II; i.e. below this curve the concentration of hydrogen goes to zero (less than 1% of C,O) within the pore. The lower curve gives the upper concentration limits of Case I (CA+0 at z = 0) as a function of #. This figure presents a convenient way to distinguish between the two cases and will be referred to later.

The channel equation The situation in the channel is analyzed using the

solutions of equation (31) for different boundary conditions. With tist order kinetics the constant flux condition is appropriate as long as CA remains finite in the pore. If C, goes to zero within the pore, boundary condition (35) can be used since the flux at the wall (z = 1) can be approximated as proportional to CAw with the constant k1 given by eqn (40). This proportionality can also be seen in Fig. 4. In Fig. 6, C,dC..,, is given as a function of channel length x’ at various values of 4. The profiles were calculated using boundary condition (34) with three eigenvalues and asymptotic expressions for another 97 eigenvalues for Case I. Boundary condition (3.5) and five eigenvalues were used for Case II. Prom Fig. 6 it can be seen that the boundary condition of constant wall concentration can be used instead of (34) and (35) as an approximation for some cases. Notice . . that m Fig. 6, CAw for the curve I#J = 30 will cross the

Case II

0.001- * 10 20 30 40 50 60 70 80

.

Fig. 5. Regions of validity of the boundary conditions in terms of the stoichiometric ratio and Thiele modulus.

b’C,,/CAi =00267

1715

Lima

1.0

0.4

a I I

a05 a10 0.15 ’ x’

Fig. 6. Ratio of concentration of A at the wall to the inlet concentration versus dimensionless channel position. xW exists

only for 4 = 30 (at X’ = 0.036).

line CAw = CAWlhEO at x = xw. This indicates that above this crossing-point xw(0 < x C XW), Case I is determining the equations, and below this point (xw -Z x < H), Case II is the governing set of equations.

In Figs. 7(a) and (b) concentration profiles C&Z, in the channel are plotted at various channel lengths. At x’ = 0 the ideal flat protile given by eqn (32) is shown. As x’ increases the profile develops. During Case II the molar flux of A at the wall is directly proportional to C,, and in this case the apparent reaction order for A is approximately unity. As C,W decreases, however, h will move into the pore towards z = 1 and the apparent reaction order for A will also decrease. At the same time N,, = (DnJL(1 - A - l/+))(d/D) continuously increases. Combining eqns (25), (28) and (29), one finds for Q > 5 and h (0.8.

The rate of reaction is expressed as N_&=l and if C,&uC,, B 1 the reaction order for A is unity. Decreasing CAW will then reduce the role of COW for the rate. The concentration of A at the wall is varying with channel length (and time) as the reaction proceeds. This is illustrated in Figs. 5 and 6.

The design equation Assume the starting conditions are such that Case I

exists throughout the reactor. As the concentration of A

1716 R. oe Vos and C. E. H.tr.mm, JR.

t

vCHo/C,, =0.0267

+ ‘40 \’

0.7- V&,/C,, ‘0.0267

+.40

0 Y’ Y'

(a) iW Fig. 7. Concentration profiles in the channel as a function of x’.

S

goes down in the recycle reactor, it reaches the value C AW = CAW~~_O at x = H; i.e. at the end of the channel.

dary conditions. For example, if eqn (34) had been

As time goes on and C_, decreases further, the x(x = _+,) chosen as the only one, a straight line (dotted) below the

at which C__,W = G&-O moves upwards in the channel. correct curve would have resulted. In Fig. 8, xJH is

During this motion Case I is governing above xw and also plotted as a function of time. Only Case I occurs if

Case II below. If C,, is su5cientIy low Case II will x&I = 1 and only Case II if xdH = 0. Between t, and

govern the equations throughout the reactor. The region tz a mixture of Case I and II is present in the reactor and

discussed above are designated in Fig. 8. As can be seen the conversion drops a sign&cant amount. This inter-

in Fig. 8 it is important to choose the appropriate boun- mediate state in the reactor cannot be neglected and must be taken into account.

Fig. 8. Conceotration of A in the reactor as a function of time (----- calculated from incorrect condition), Np. = 9112, St V = 50 m-’ and H = 0.05 m.

A cross-tlaw reactor

This change from Case I to II (never the opposite way) can also be visualized in Fig. 5. For example, starting the reactor at vC,dGw = 0.01 and 4 = 20 one follows the course of reaction by moving upwards on the diagram. It starts in the area of Case I and goes into the area of Case II before reaching the limit of the analysis.

A comparison of the mass transfer resistance for A in the channel and the pore is done in Fig. 9. The concentration difference from the centre to the wall in the channel, AC= is calculated using either Leveque’s linear approach, or the complete solution of (31) with the boundary condition (35). The concentration difference in the pore (AC,) is taken as the concentration dierence over the pore length [L or L(l-A)]. Ln AC,,/AC, was plotted against ln (HDIW2) at various Thiele moduli. The comparison was made at the end of the reactor where AC= is at maximum. In both cases, the dependence is shown with both the extrapolated Leveque solution and the complete eigenvalue solution. The Leveque-solution is valid for 0 < (HD/#dd)) < lo+. With the aid of Figs. 9 and 10 it is possible to distinguish whether Case I or Case II is governing the equations, and to estimate the relative importance of the mass transfer resistances in the reactor.

It is obvious that the eigenvalue solution for AC,JA& looses precision for small x’. In this area the Leveque- solutions are more appropriate. At higher x’ the eigenvalue solution peels off the straight line of the Leveque solution, which is not valid in this area.

vCno /CAI = .0267

1 . -6 -7 -6. -6 -4 -3 -2 ’

wb

Fig. 9. Ratio of ~concentration ditbrence in pore to that in channel versus In Holid*. (a) Lmeque solution, + = 10, (b) Leveque solution 6 = 30; (c) eigenvalue solution, Case I + = 10;

(d) same 9 = 30; (e) cigenvalue solution, case II .$ = 50.

0-C

0-i

0.1

0-c

1717

r 6 5 4 3 2 -HD

‘“Fz

Fii. 10. Value of vC,,&CA~ which gives Case II at channel end

An increase in r#~ results in several effects. First, the curves for higher 4’s cannot be distinguished from each other. Secondary, the situation in the reactor follows Case II and finally the precision is worse in the low x’ range.

The level of vC,dC, does not affect the curves for Case I, but it lixes the limits for Case I. This is shown in Fig. 10 where &&CA, giving Case II at the end of the reactor is plotted as a function of 4 and In (HD/CdT. The curves for Case II in Fig. 9 have a weak dependence on vC&CAr, but this is not apparent in the figure.

The time scale of the analysis (reaction going to 90% conversion of A) is dependent on the capacity of the reactor. Besides the parameters mentioned earlier which affect the concentration distribution, the capacity is determined by the geometric catalyst surface available to mass transfer (S): i.e. length, width, and the number of plates in relation to volumetric load of reactant A.

CONCLUSIONS Solutions of the one-dimensional pore equation where

gas reactant is at one end and liquid reactant (A) at the other, have been solved for the stoichiometry typical of hydrogenations of nitrobenzoic acid. Two cases were found: one where the concentration of A at the gas end is zero or finite and the other where A is depleted within the pore. From the curves presented in Fig. 5 one can determine which case occurs for a given 4 and vC&C,w for first order kinetics in the gaseous com- ponent.

1718 R. DE Vos and C. E. HAMI~N, JR.

The depletion of A in the pore is also important in L determining the appropriate boundary condition for the n channel equation. In Fig. 6 an example of the coupling of NA, N,w the pore and channel concentrations are shown in terms of the wall concentration as a function of dimensionless NH reactor length (containing the flow velocity) for different NBi values of 4. Analysis of the curves allows one to determine the fraction of the reactor in which Case I is valid and that in which Case II is valid. NP.

With the design equation a cross-flow reactor for NR. three-phase reactions can now be chosen to carry out a given first order process knowing kinetics and mass NSC transfer characteristics of the system. In Figs. 8 and 9 the relative importance of mass transfer in the channel s’ and the pore can be found for various process t, t. parameters. Figure 8 again emphasizes the importance of using correct boundary conditions as noted above. ;

x, xw Acknowledgements-One of the authors (CEH) wishes to ac- knowledne the kind hosnitalitv extended to him bv Prof. N. H. Schtin \;hile on sabbatical leave in his department: The Swedish Board for Technical Development provided financial support.

NOTATlON

X’

Y Y’

coefficient n in the eigenfunction Yi

coefficient i in the solution of channel equation

constant defined by eqn (26) concentration of H+ in liquid, solu-

blllty of Hz in liquid concentration of A, concentration

of A at the inlet of the channel, concentration of A at tbe wall, CA0 at t=O

AC,, A& concentration difference in the

d D

DA DH

X-I k

k,

channel and in the pore channel height bulk diKusivity for A effective di#usivity for A effective ditfusivity for H2 channel length rate constant

[I] SatterBeld C. N., A.1.Clr.E.J. 1975 21 209. [2] Young L.C. and Finlayson B. A., A.I.Ch.E.J. 1976 22 331. [31 Andersson B., Ph.D. Thesis, Chalmers University of Tech-

nology, Gijteborg, Sweden, 1977. [4] Skelland A. H. P., Dflusional Mass Transfer. Wiley, New

York, 1974. [5] Colton C. K., Smith K. A., Stroeve P. and MeriB E. W.,

A1.Ch.E.I. 1971 17 713. proportionality constant in eqn (35)

pore length index molar flux of A, molar flux of A at

the wall molar flux of Hf Biot number characterizing the

mass transfer in the plate ( = D.&X)

Peclet number for the liquid Reynolds number containing 2d as

characteristic length Schmidt number for the liquid rate of reaction mass transfer surface in the reactor time, penetration time mean velocity in the channel liquid volume in the recycle system length coordinate in the channel, X-

coordinate where Case I switches to Case II

dimensionless length coordinate in the channel (xZWd3

height coordinate in the channel dimensionless height coordinate in

the channel (y/d) eigenfunction dimensionless length coordinate iu

the pore eigenvalue i dimensionless length value in the

pore characteristic number for the actual

process (vJM vHDA) stoichiometric coefficients for A

and HZ Thiele modulus (Lq( - v&/D&)

REFERENCES

a cross-flow reactor: theoretical model for first order kinetics

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