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 + !
