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0098-1354/94 $6.00+0.00 Copyright © 1993 Pergamon Press Ltd

NUMERICAL SIMULATION OF CATALYTIC INERT MEMBRANE REACTOR A.B. Bindjouli t

Z. Dehouche t

,

,

B. Bernauert

t, J.

Lieto t

tABORATOIRE D'AUTOMATIQUE ET DE GENIE DES PROCEDES URA CNRS D1328 UNIVERSrrE CLAUDEBERNARD LYONJ

43 bd du 11 Novembre 1918 69622 VilJeurbanne Cedex - FRANCE

ttINSTITUT OF CHEMICAL TECHNOLOGY DEPARTMENT OF INORGANIC TECHNOLOGY TECHNICKA5

166 28 Praha 6 - CZECHOSLOVAKIA

ABSTRACT We are developping a new reactor model able to perform catalytic activation and separation simultaneously : the catalytic membrane reactor. This paper analyses the influence of gas diffusion through the membrane on mass transfer in the catalytic reactor with an inert membrane wall. A mathematical description of these reactors often leads to partial differential equations (PDE) with complex boundary conditions. Two efficient techniques are used to solve these equations. The method implies an orthogonal collocation on finite elements in the spacial dimension followed by a global orthogonal collocation. Theoretical comparisons are made between the concentration profiles in the radial direction obtained for both reactors. The effect of radial dispersion on the performance of the inert membrane reactor is examined. The applicability of approximate methods for membrane transport model is also discussed.

KEYWORDS Membrane, catalytic, model, simulation

INTRODUCTION A catalytic membrane reactor is made of two coaxial tubes separating the space in two parts devoted to reaction (internal) and separation (external). The outside tube is non porous whereas the internal porous tube is covered with a selective 'Yalumina membrane and packed with an heterogeneous catalyst. Because a continuous and selective removal of one product can be obtained, such a reactor can be used for reversible reactions such as alkane deshydrogenation to produce yields higher than that corresponding to the equilibrium. The reactor design must take into account chemical, and physical kinetics (transports through the membrane) and hydrodynamics (flow pattern in reactive and separation sides). Several studies exist which propose flow patterns such as plug flow (Itoh ~., 1988), perfect mixing (Sun et al., 1988, 1990). None of these hydrodynamics proposition are based on experimental data. We carried out RID experiments and proposed a complete mathematical modelling of the reactor based on mass balances on the different reactor regions. Solving the obtained set of equations implied the use of two numerical methods " orthogonal collocation" and " orthogonal collocation on finite elements" which are described here.

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c=o

_~f~o~--tI'1

If-I-

L.~--+-+-~

Co -+ I

I

JHl

ill

IBm

Catalyst pellet"

""""""'"

Mem.~ra:D.C

Figure 1. Catalytique membrane reactor view MEMBRANE REACTOR EQUATIONS

The steady-state mass transfer problem and the boundary conditions of this type can be described by the equations for diffusion presented in table 1. Kinetics and physical data are summerized in table 2. The difference of total pressure between two sides is assumed to be zero (no convective flow in radial direction ). Table 1 Mass balance in membrane reactor

~

r=O (axe of membrane tube)

ar =0 2 ac1 a Cl 1 aC1 ED [--+--1- U1- -D .f(C1)=0 1 al r ar at m

0< r < Rl (inner side)

ac1

r = Rl (interface membrane/inner side)

aC3

D 1Tr=D3(Tr) et

a raC3

a;<~) =0

Rl < f < R2 (membrane)

r C1 ln[R9 - +C2In [R] r 1 R2 In[R1 1

C3

;

ac2 aC3 D2Tr = -D3ta:'") et C3=C2 2 a C2 1~ ac2 D2[ ar2 +; ar ] - U2ae =0

f = R2 (interface membrane/outer side) R2 < r < R3 (outer side)

a C2

f = R3 (reacteur wall)

D2Tr =0

e=o

et

C1 = Co

. alda ta Ta bie 2. K'mettc an dPh~YSIC

fp(A)

E

2 -1 D1(m s )

13

0,57

4,05.10- 5

C3=Cl

2 -1 D2(m s ) 1,2.10

-5

R1(m)

R2(m)

R3(m)

e.o(m)

3.10- 3

5.10- 3

7.10- 3

0,1

2 -1 D3(m s )

Dm

3,01.10- 7

0,01

-1 u1(m.s ) -2 3,6.10

-1 u2(m.s ) -2 2,3.10

C2 = 0 f(C1)

..fCi

[Levenspiel 0., 19721 -3, C o( mol.m ) 0,0625

Numerical solutions of concentration fields for both sides of membrane are combined with the analytical solution corresponding to the separative wall so as to satisfy the continuity of the concentration and the mass flux at both sides of the membrane. Then, the differential equations describing the dimensionless concentration profiles in the membrane reactor as function of dimensionless radial position (inner side, inside the pores,and outer side), to be : for

0<

Rm< 1

(1)

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2

Dr2~a Cm2 + 1 aC m 2] _ aC m2 = 0 aR2 R R2 aRm aL m m+s (2)

by using the C3 expression of tablel, we easily obtain the expressions of aC m3 . d . aRm near Rm=O Ie r=R2 an Rm =1 Ie r=RI. The problem is completely defined by making use of the following boundary conditions: At Rm=O ie ( r=O and r=R2) aCml d aCm2 aRm =0 an aRm = -A2(Cml-Cm2) (3)

The residual for each component is satisfied at the collocation points, but not necessary in between. Note that although the residual of the differential equation is satisfied at a collocation point, the solution C, is not necessarily correct at that point. C is the concentration of the solute which is not rejected selectively by the membrane. In our case, orthogonal collocation is being used, and the collocation points are the zeroes of a series of shifted Legendre polynomials on [0,1], mapped into each element. In practice, however, we do not solve for the polynomial coefficients, but for the concentrations themselves. We consider the concentration of each species at any point in the element in terms of a Lagrange polynomial interpolated solution:

At Rm=1 ie (r=RI and r=R3)

(8)

aCml OC m2 aRm = AI(Cml-Cm2) and aRm = 0 (4)

Dimensionless variables are Ci .h . L_t Cmi = Co Wit 1 = I. 2 or 3 to (5)

for 0 S r SRI,

R

for R2 S r S R3.

~ Rm = R3-R2

-..!..... m- RI'

i

2

_

= 1.2

and k

= 1. P

(())

+ R I· C x k

p VC .. = L CikA' k I) k=1 )

v

The above differential equations (1) to (4) can be solved by two mathematical methods, 9J:thog.Q.Q.aLc.ullo~atiQ.u and or.thogunaJ ~Qll!:~c.a.ti.Qo..Q.1J.[J.o.ite..eJ~mSlnts. The orthogonal collocation method was developed by Villadsen ~, 1978. We first divide the domain into elements, and within each element the concentration of each species is approximated by a polynomial. The criterion for choosing the coefficients of the aPl?roximating polyno~ials i~ that the ~eslduals. of the dIfferentIal equatIons, shows 10 equatlO.n (7) as an example, ~re zer? at certam po1Ots, known as collocatIon po1Ots, X . k·

= D1·V c ik + uvc 1'k

The first and second derivatives can be expressed as a summation of derivative weights and the species concentration within each element:

(6)

RESOLUTION METHODS

Res

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= 0

(7)

2

P

(9)

C .. = L CikB'k I) j(=1 )

where the A jk and B jk are the derivatives of the Lagrange interpolation polynomial at the coll~ation point, j. The derivative weights are obta.1Oe~ fro~ code JCOBI. based upon the routme 10 Villadsen and MIchelsen (1978). Seve~~l elements are usually used, With the cond.Itlon t~at at each element boundary, the flux IS contmuous: J

I - I J

i x+ -

i x-

(10)

where x+ and x- denote the approach to the elements boundary from the right and left sides, respectively. These numerical techniques transform the partial differential equations into algebraic and ordinary differential equations, and then using a general-purpose program to solve the . . known as the resu I·tmg set 0 f equatIons, IS method of lines.

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Pea ... 41

- L - 0.00 +H++L - 0.33 L - 0.50

U

***** ...... L -

membrane

2.2 10-2m.s-1

Outer .Ide

0.50

1.00

~~~~TrlTMnTnT~~~~

~IWIUIS(-)

"b.00

Fig. 2 Concentration profiles in membrane react (Orthogonal collocation method)

In ...aotlon aide

1.00

F1g.4.

DtmoDitoDI... .&wrago Coac:ODlrattOD

1.2,..--------=------------,

U

I ......

Pea ... 41

10-2m.s- 1

L ... 1.05 10-1m

- - ooIluiatod

0.1110, CoiL ..olllod 0.1110, CoiL o .....

~

exFtsrimental

1

______ 1L ______ 1L ____ _

I

1

1

1 1

1 1

1

1 -I-

1 1

I

0.4

;

(R.T.D.)

~:~~~~~~~~~~~~r

0.' I-

2.00

RESI~(nonl1Io1Et~IIUTION poroUII tube)

= 2.2

..............................................

0.6

L ... 1.05 10-1m

;

~~~~~~~~~~~~~~

0.87 ..... L - 0.83 - - Interface (membrane/lnner aide) - - Interface (membrane/ouler aide) Inner .id.

...

1

1 -----1-----1 I

I

7 oollooatlon poInto 11 flnlto olo_to, " ooll_llon polnto In oaol! olo_nt ~~~HTMM+r~~~~~~T+

"b.00 o~--~--~--~--~--~

o

0.2

0.4

0.6

0.1

DImeDlloDI... "'DCJIJa 01 .... a-tor ...... Anrago ......trallo. plOftlOl . . . 1.0,1 _ . _ _ ....., (radIal ClI.ponlo. aocIol WIUI . .r-u)

1.00

2.00

F1g.5. RESlDEHCE l1Io1E t&::l'BlI11ON IIU.p.) IN INNER SlDEiO'(pOROUS ruBE)

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RESULTS AND DISCUSSION Figure 2. show~ t~e dimens~onless concentration profiles 10 three reglOns of membrane reactor calculated by solving the equations (6), (7), (8), and (9) using the global orthogonal collocation method. It is apparent from these graphs that the presence of the membrane has a small effect on the ~adial concentration profi~es. It .is also Important t.O nO.te that the radi~ gradIents for concentration !n membra~e Imply that the mass .transfer IS strongly 1Ofluenc~d by the POroSIty of the membrane and the diametre of the pores. In order to evaluate the accuracy and infuence of the choice of approximate solutions, two numerical solutions were obtained and compared A comparison of the . . . average concentratlOn profiles obtamed by using both global orthogonal collocation and orthogonal collocation on finite elements methods in reaction side is presented in figure 3. As can be seen, the agreement between the two numerical solutions is very good. Figures 4 and 5 demonstrate experimentally that the radial diffusional phenomena in the gas phase membrane reactor can be neglected. These experimental results are quantitatively valid~t~d throu~h ~he model calculati~ns. The theonttcal predICtiOnS are seen to be 10 good agreement with the experimental data in the both cases.

CONCLUSIONS Two numerical techniques based on the "orthogonal collocation" and "orthogonal collocation on finite elements" methods of weighted residuals, have proved to be very efficient and accurate for solving the complicated differential equations of membrane transport model. Satisfactory results have been obtained even for small number of collocation points. Results from · d' h h f h h od e1 10 tern Icate t ~t, t e presence 0 t. e ~em~rane has not 1Ofluenced. ~he radIal diffuslOnal phenomena. The condItionS under which the radial diffusional release of a solute from such systems have been arrived at theoritically and validated experimentally with model systems. LIST OF SYMBOLS D3 _ S D3 A_ 1- (D3+Dl)So ~- (D2-D3)R2So Ci: Gas concentration (mole/m 3), with i=1 : Gas concentration in reaction side, i=2 : Gas concentration in separation side, and i=3 : Gas concentration into

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membrane Ct=Cm1 Dimensionless concentration Di: dispersion coefficient (m 2.s- 1), with i =1,2, and 3 Ome Dan=~' with Dm: Kinetic constant UI C . D . D' o. I R d' I d' . ffi . 'th riO ImenSlon ess a 13 IsperSlOn coo IClent, WI D _ Dl.£.eo D _ D2.eo rl - Ul.R 12' r2 - u2.s2 Dm f(Cl): Kinetic expression e: Reactor length (m) , with de: Differential length , and Total reactor length (m) 0 . ~: ~:,brane e;c'.IUS . ea· .131 pee et , . er.. Radial peelet. ~i: radius (~) , With 1=1 Inner ra~lUs of mem~rane, 1=2 Outer radius of membrane, and 1=3 Inner radiUS of reactor shell

e:

r

time (s) . . . ._ _ : DImensionless ume , With t: Mean t residence time offluide in a reactor (s) t=

.-1

u: Mean velocl~ (m.s) . £:Extemal poroslty:f catalyuc bed S =R3-Rl ,So=ln[R2] 1

REFERENCES 1- FINLAYSON ' BAN . m . .., on I'mear Ana IYSls Chemical Engineering, McGraw-Hill, New York, 1980. 2- ITOH N. "A Membrane Reactor Using Palladium" AIChe. J. ,n, 9, 1576-1578, (1987) 3- ITOH N., Y. SRINDO, "A Membrane Reactor Using Microporous Glass for Shifting Equilibrium of Cyclohexane Dehydrogenation", J. of Chem. Engng. of Japan 21,4,399-404 (1988) 4- ITOH N., Y. SRINDO, and K. HARA YA, "Ideal Flow Models for Palladium Membranes Reactors" J. of Chern. Engng. of Japan,~: 4 ,42~-426, (1990) 5- LEVEN SPIEL O. , Chemical ReactIOn Engineering",. Second Edition JOHN WILEY $ SON New-York Chichester, 1972. 6- SUN YI-MING and SOON-JAI KHANG "Catalytic Membrane for Simultaneous Chemical Reaction and Separation Applied to a Dehydrogenation Reaction", Ind., Eng.,Chem., Res., 21, 1136-1142, (1988) 7- SUN YI-MING and SOON-JAI KHANG "Catalytic Membrane Reactor: Its Performance in Comparison with other Types of Reactors", Ind., Eng., Res., 22,232-238, (1990) 8- VILLADSEN J.V., M. L. MICHELSEN "Solution of Differential Equation Models by Polynomial Approximation", Printice-Hall, Englewood Cliffs, 1978.

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