Chemical Reaction Engineering-Houston

lysis of the governing steady state differential equations, i.e. the Navier-Stokes equation of motion and the diffusion equation for incompressible fl...
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7 Experimental and Theoretical Study of the Simultaneous Development of the Velocity and Concentration Profiles in the Entrance Region of a

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Monolithic Convertor M. A. M. BOERSMA,* W. H. M. TIELEN, and H. S. VAN DER BAAN University of Technology, Department of Chemical Technology, Eindhoven, The Netherlands

In c a t a l y t i c afterburning of waste gases the monolithic or ac­ tive wall reactor is becoming increasingly important. Most experi­ mental and theoretical studies on this reactor that have been pu­ blished so far consider the velocity field i n the separate channels of the monolith either undeveloped, i . e . plug flow (1-3), or fully developed, i.e. laminar flow (4-7). In r e a l i t y , however, the velo­ city p r o f i l e , which can be assumed to be more or less uniform at the entrance of the channel, will develop into a laminar P o i s e u i l l e p r o f i l e downstream the tube. This implies that the experimental measured conversions will not correspond to the theoretically calcu­ lated concentration profiles for either plug flow or laminar flow, i.e. concentration profiles calculated for a uniform v e l o c i t y pro­ file will result i n a conversion which is higher than the actual conversion, while a fully developed p r o f i l e gives r i s e to a con­ version lower than the actual one. Therefore, to get a more accu­ rate description of the phenomena taking place i n the entrance re­ gion of the active wall reactor we studied the simultaneous devel­ opment of the velocity and concentration profiles by numerical ana­ l y s i s of the governing steady state d i f f e r e n t i a l equations, i . e . the Navier-Stokes equation of motion and the diffusion equation for incompressible flow of a Newtonian f l u i d . Some work in this f i e l d has been performed e a r l i e r by Ulrichson and Schmitz (8). These i n ­ vestigators used the approximate solution of Langhaar (9) for the entrance velocity profiles to provide the velocity data for nume­ rical solution of the component material balance. The calculations apply to an isothermal reactor, a condition which generally is satisfied i n case of low concentration of the waste gas. Further it is assumed that the kinetics of the chemical reaction taking place at the tube wall can be described by Γ

=

, η * HC,wall C

* Present address: Koninklijke/Shell-Laboratorium (Shell Research B.V.) Amsterdam, The Netherlands © 0-8412-0401-2/78/47-065-072$05.00/0 In Chemical Reaction Engineering—Houston; Weekman, V., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

7.

BOERSMA

Velocity and Concentration Profiles

ET AL.

73

η being the order o f the r e a c t i o n . This r a t e expression a p p l i e s t o the o x i d a t i o n of s m a l l concentrations hydrocarbon i n an excess of air. To check the c a l c u l a t e d c o n c e n t r a t i o n p r o f i l e s i n p r a c t i c e we s t u d i e d the combustion of low concentrations of ethylene and i s o butene i n a commercial m o n o l i t h i c convertor s p e c i a l made f o r us by Kali-Chemie Engelhard K a t a l y s a t o r e n GmbH.

Downloaded by EMORY UNIV on August 26, 2015 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch007

Theoretical Part At f i r s t we t r i e d to s o l v e the problem i n study by a s i m u l t a ­ neous numerical a n a l y s i s of the Navier-Stokes equations and the mass t r a n s p o r t equation. Since by t h i s method s e r i o u s numerical i n ­ s t a b i l i t i e s were encountered we choose an a l t e r n a t i v e procedure. This c o n s i s t s of c a l c u l a t i n g f i r s t , at each g r i d p o i n t i n the tube, the a x i a l and r a d i a l v e l o c i t i e s , a f t e r which these values are used to s o l v e the mass t r a n s p o r t equation. I n s o l v i n g t h i s equation ca­ re i s taken that the g r i d used i s i d e n t i c a l to that f o r s o l v i n g the velocity field. The Entrance V e l o c i t y F i e l d For an i s o t h e r m a l , steady s t a t e , incompressible flow of a Newtonian f l u i d being symmetrical i n the azimuthal d i r e c t i o n , the governing equations are the Navier-Stokes equations and the steady s t a t e c o n t i n u i t y equation. I n dimensionl e s s form the equations are:

The pressure dependent terms i n equations (1) and (2) can be remo­ ved by d i f f e r e n t i a t i n g equations (1) and (2) w i t h respect t o ζ and r , r e s p e c t i v e l y , and s u b t r a c t i n g the r e s u l t a n t equations from each other ( L Q , U ) . According t o the method o u t l i n e d by Vrentas et. a l . ( 1 1) now a stream f u n c t i o n Ψ i s introduced which i s d e f i n e d as: u



V r3r

-

^

t

r3z

(4)

We f u r t h e r r e c a l l t h a t the only non zero component of the v o r t i c i t y v e c t o r i s the azimuthal p h y s i c a l component, which i s given by: ω

=

3 l • 3Ϊ

( 5 )

With the a i d of equations (4) and (5) we f i n a l l y get the f o l l o w i n g

In Chemical Reaction Engineering—Houston; Weekman, V., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

CHEMICAL REACTION ENGINEERING—HOUSTON

74 equations: _ω 3Ψ 2 3z r

J_ 3Ψ 3ω _ j_ 3Ψ 3ω r 3z 3r r 3r 3z

2

2_ Re L 2

1 3ω _ ω_ r 3r 2 r

3jo\ I

^2

w

2

Downloaded by EMORY UNIV on August 26, 2015 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch007

8Ψ 3 Ψ _ J_ 3Ψ . 2 * 2 r 3? 9z dr

(7)

When i n these equations the second p a r t i a l d e r i v a t i v e s of Ψ and ω w i t h respect to ζ are neglected,which i s j u s t i f i e d because i n the m o n o l i t h i c convertor the convective t r a n s p o r t of v o r t i c i t y i n the a x i a l d i r e c t i o n i s greater than the a x i a l d i f f u s i v e t r a n s p o r t , the e l l i p t i c d i f f e r e n t i a l equations change i n t o p a r a b o l i c equations. S o l u t i o n of the equations i s c a r r i e d out w i t h the f o l l o w i n g i n i t i a l and boundary c o n d i t i o n s : r = 0, ζ > 0 : | ^ = ν = ω = 0 , Ψ = | dr 0 < r < 1, z = 0

(8)

: ω = 0 , Ψ = \ (1-r ) 2

(9)

2 r = 1, ζ > 0:ϋ = ν = Ψ = | ^ = 0 , ω = ^-4 dr

(10)

In the numerical a n a l y s i s the r a d i a l and a x i a l d e r i v a t i v e s of the p a r a b o l i c equations are replaced by the c e n t r a l d i f f e r e n c e ap­ proximations and the backward d i f f e r e n c e approximations, r e s p e c t i v e ­ l y . Thus N-l s e t s of f i n i t e p a r a b o l i c d i f f e r e n c e equations are ob­ t a i n e d , Ν being the number of r a d i a l steps. The number* of e q u i d i s ­ tant g r i d p o i n t s i n r a d i a l d i r e c t i o n amounted to 40, w h i l e f o r the f i n i t e d i f f e r e n c e increment of ζ(= 2z/Re) a value of 1.25 10"^ was used. D e t a i l s of the numerical s o l u t i o n procedure are given elswhere (J_2). F i g u r e 1 i s a g r a p h i c a l r e p r e s e n t a t i o n of the development of the a x i a l v e l o c i t y obtained by the numerical s o l u t i o n procedure. This r e s u l t agrees q u i t e w e l l w i t h that obtained by Vrentas e t . a l . (11). The Tube Wall Catalyzed Reaction Assuming incompressible flow of a Newtonian f l u i d and no c o n t r i b u t i o n s i n the azimuthal d i r e c t i o n a mass balance f o r a d i f f e r e n t i a l element i n the en­ trance r e g i o n of the tube y i e l d s the f o l l o w i n g steady s t a t e d i ­ mensionless d i f f e r e n t i a l equation: 1^2 + !