214
J. Phys. Chem. 1980, 84,214-219
Convective Diffusion with Homogeneous and Heterogeneous Reactions in a Tubet Vi-Duong Dang* and Meyer Steinberg Process Sciences Division, Depattment of Energy and Environment, Brookhaven National Laboratory, Upton, New York 1 1973 (Received December 4, 1978; Revised Manuscript Received September 17, 1979) Publication costs assisted by Brookhaven National Laboratory
A new complete solution for laminar flow with axial diffusion and first-order homogeneous and wall reactions in a tube is presented. The solution is in terms of an eigenfunction series expansion. Effects of axial diffusion, homogeneous reaction, and wall reaction on the concentration field are discussed. Comparison of the simplified results derived from the present analysis by neglecting axial diffusion and wall reaction with those of previous workers shows good agreement. Conditions are established for the necessity of applying the present results to describe the system accurately.
Introduction Convective diffusion with homogeneous reaction inside a tube and heterogeneous reaction a t the tube wall has been investigated extensively in the past under rather simplified assumptions. Lauwerierl investigated the homogeneous reaction without axial diffusion in a tube. K a t 9 analyzed the catalytic wall reaction in a tube, Kaufman3obtained a solution for the extreme ideal case that the rate of reaction was equal to the convective motion carried by the reactant, although he pointed out the importance of radial and axial diffusion along with the wall reaction in the system. Cleland and Wilhelm4 obtained experimental results and a numerical solution for convective diffusion with the first-order reaction but negligible axial diffusion. DranofPf obtained eigenfunction solutions for convective diffusion of the reactant with catalytic wall reactions in a tubular or annular reactor. Wissler and Schechter7 solved a problem similar to that of DranofP for turbulent pipe flow. Homsy and Strohman8 studied convective diffusion of non-Newtonian fluid. Subramanian, Gill, and Marragpresented an unsteady dispersion model for tubular reactor. Poirier and CarrlO repeated the solution of Cleland and Wilhelm4 but extended their results to second-order reactions. Ogrenl' obtained a solution for laminar tube flow with radial diffusion and first-order homogeneous and tube wall reactions. Walker12obtained an asymptotic solution downstream of the tube for chemical reaction with axial and radial diffusion in a catalytic tubular reactor. Solomon and Hudsonx3studied heterogeneous and homogeneous reactions in a tube without axial diffusion. A recent review of the convective diffusion problem was given by H 0 ~ e r r n a n n . lA~ summary of the previous work is given in Table I. Previous researchers generally analyzed this system by means of orthogonal eigenfunction series expansions. For any changes in the system such as those from laminar to turbulent flow, considering homogeneous and/or heterogeneous chemical reactions in the reactor, one has to develop a new set of eigenvalues, eigenfunctions, and their associated constants according to these changes. Therefore, there are extensive tabulated and/or graphical eigenfunctions, eigenvalues, and their associated constants reported by the researchers such a8 those listed in Table I for different physical and/or chemical problems treated. Yet, careful examination of Table I shows that previous researchers usually neglect axial diffusion in their system or if axial diffusion was considered by W a l k e P and +This work was performed under the auspices of the United States Department of Energy under Contract No. EY-76-C-02-0016. 0022-3654/80/2084-02 14$01.00/0
Kaufmann3 only simple asymtotic results were reported. Axial diffusion cannot be neglected when the axial velocity is low or the diffusivity is large. However, inclusion of axial diffusion in the conservation equation not only changes the conservation equation from parabolic to elliptic type but also causes the subsequent eigenfunctions to be nonorthogonal. This is believed to be the major difficulty in obtaining a complete solution of this problem previously. The present paper extends the work by the previous researchers in this field by considering axial and radial diffusion and homogeneous and heterogeneous chemical reaction in laminar pipe flow. The method of nonorthogonal series expansion is used to solve this problem. The specific contributions of the present paper are (1)to present a complete new solution to laminar tube flow with axial and radial diffusion and first-order homogeneous and wall reactions, (2) to show the effect of axial diffusion and homogeneous and wall reactions on the concentration field of the reactant, (3) to develop conditions for the negligible effect of heterogeneous reaction with respect to homogeneous reaction and vice versa when axial diffusion is significant, and (4) to establish ranges of parameters (Pe, a, and K ) where only the present results can describe the system correctly.
Mathematical Analysis The treatment described here is restricted to laminar Poiseuille flow. For an isothermal system, the convective diffusion equation with axial and radial diffusion and first-order homogeneous reaction in a dilute system can be written as
U[1
-(:)"]E
= D[
k ( r 5 )
+
$1
-hC
(1)
where C is the concentration of the reactant, U is the maximum velocity of the fluid, and x and r are axial and radial coordinates of the tube, respectively. The boundary conditions of the system are C(0,r) = Co (2) aC -(x,O) = 0 (3) ar aC -D --(x,a) ar h,C(x,~) (4) Equation 2 assumes the inlet condition of the reactant to be constant. This condition will lead to an infinite mass flux at x = 0. However, this inlet condition will not change the results significantly for the investigation of the effect 0 1980 American Chemical
Society
The Journal of Physical Chemistry, Vol. 84, No. 2, 1980 215
Analysis of Diffusion and Reactions in a Flow Tube
TABLE 1: Summary of t h e Previous Work o n Convective Diffusion with Chemical Reactions
______
_l_l_l__
ref
_ _ _ _ l _ I c l l _
-_-
state
_
diffusion considered
_
_
_
_
l
_
_
l
_
~
-
reaction I _
radial
axial
homogeneous
heterogeneous wall
flow field I__
l_l_-__--
ll_l_l_l_____-_---___l_ll
4 1 2 3 12 5 7 6 13 8 10
9 11
steady steady steady steady steady steady steady steady steady steady steady unsteady steady
yes yes yes no yes yes yes yes yes yes yes yes yes
no no no yes yes no no no no no no no no
first order first order
no no arbitrary reaction rate first order first order
first order first order
first order first order first and second order first order first order
laminar laminar laminar laminar laminar laminar turbulent laminar 1ami n a r non-Newtonian laminar laminar laminar laminar
first order first order first order no first order
of axial diffusion and chemical reactions on the concentration field downstream from the reactor. A rigorous treatment of this inlet condition can be obtained by tlhe method of D~lllg.'~Equation 3 is the symmetric boundary condition at the center of the tube. Equation 4 describes the surface condition of the tube wall where diffusional flux is equal to the wall reaction. Because of the chemical reactions both inside the tube and at the wall, it is possible to assurne that C 0 as x: a. By introducing a set of dimensionless parameters 6 = C/C, r = r/a t = x / ( a P e ) Pe = a U / 8 K = ka2/D 01 = h,/(ka)
by using the Runge-Kutta method in a CDC 7600 computer. Functions R,(r) are not orthogonal, as can be seen from eq 10. Therefore, the coefficients of expansion A,, cannot be determined through orthogonal series expansions. However, one can determine the coefficients by means of the following equation through eq 13
one can translorm eq 1-4 to
where
- -
m#n
(l4a) After substituting eq 14a into eq 14, one gets the following equation:
('7) a6 -([,I) a17
f KO16([,1)
=0
(8)
The solution of eq 5-8 can be taken in the following form:
Walker12 proposed a form similar to eq 9 without the coefficients of expansion A,. He only proposed a form for the functions h:,(11) in principle without really determining the functions :numerically. One can then substitute eq 9 back into eq 5-8. By collecting terms properly, it is possible to obtain a set of ordinary differential equations for R,(y) which are d2R, ---++-----+
dn2
1 dR,
o d?t
'-) Pe
dv (141.b)
The last summation term in the numerator of eq 14b ran be further simplified by the following procedures. Multiply eq 10 for R,, by R, and eq 10 for R, by R,, respectively, and subtract the two equations. The resulting difference equation is then integrated with respect to 7 from 0 to 1 and eq 11 and 12 are utilized for further simplification. One can get
K
=1
+ &)2]qR;
$(K
( + +;) + %( +
m
Z: A,$n(a)
+
K
$J2
]rRmR, dq =
(13)
R=l
Equations 10-12 can be considered as eigenvalue problems with /32 as eigenvalues, and they can be solved numerically
By substituting eq 14c into eq 14b, one can get a set of
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The Journal of Physical Chemistry, Vol. 84, No. 2, 1980
linear algebraic equations to determine the coefficients A,. m
And,
+ mC= l
= In
(15)
mfn
where J,, I'n,m, and I , are given by the following expressions:
In =
(16)
J, =
In addition to the radial concentration distribution, one is dso interested in the average concentration in the tube. This can be defined as x a U [ l - (r/a)2]Cr dr Cb(X,?+) =
I____
(19)
L a U [ l - (r/aI2]r dr By means of eq 9-12, one then obtains
Results and Discussion Numerical solutions of eq 10-12 have been performed, and Tables 11-XI1 (see paragraph at end of text regarding supplementary material) give the eigenvalues and related constants for K = 10; Pe = 1, 5 , 10; and cy = 1, 2, 4. Eigenvalues and their related constants for K = 0.1,1,5, 10, 100; I'e = 1, 5, 10, 100; and a = 0, 0,001, 0.1, 1, 2, 4, 10, 100 have also been obtained but are not reported here to save space. Up to 50 eigenvalues and their related constants are reported in Tables 11-X for values of K , Pe, and a. One method to check the validity of the numerical solution of eq 10-12 is to substitute the values calculated for A, and R , back into eq 13 and determine its satisfaction. Eleven points of q with intervals of q = 0.1 from 0 t o 1have been used to check the validity of eq 13. When K 5 5 and cy 5 4,the summation of the series for the 11 points of v at intervals of 0.1 from 0 to 1 have been used to check the validity of eq 13. When h 2 5 and a R 4, the summation of the series for the 11 points of q at intervals of 0.1 from 0 to 1 calculated in eq 13 is 100, the effect of axial diffusion on the concentration field becomes insignificant. However, it is noted that the eigenvalues will still be affected in the third or fourth decimal point w e n when Pe = lo6 if one wants to neglect axial diffusion by merely taking a high value of Pe for the system. Also given is the dimensionless average concentration along the axial distance of the reactor for Pe = lo5 to show the practical reactor length required for the dimensionless average concentration to decay. For the present set of parameters, K = 1, Pe = lo5, and a = 4, the length-to-radius3 ratio of the reactor has to be 8850 in order for the dimensionless average concentration to decrease to half its inlet value. Cleland and Wilhelm4 reported the dimensionless average concentration to be 0.592 experimentally and 0.594 theoretically for x / a = 1470 ( x = 463 cm, a = 0.31 crn), Pe = 1.42 X lo5, and K = 27.78. Using these parameters and present method, we found that the present calculation gives the same value of 0.594 as the finite difference result of Cleland and Wilhelmq4 Hence it can be seen that an increase in the axial diffusion will
Figure 4. Radial concentration distribution for different axial position.
decrease the reactor length for the reactant to decay. Figure 4 shows some results of radial concentration distribution at various locations along the length of the reactor. For the present set of parameters, K = 5, Pe =. 1, and a = 1,the reactant concentration becomes practically zero at x/a N 3. The concentration distribution is strongly affected by the flow field as can be seen in the figure. The inlet concentration is assumed to be uniform but is distorted from a flat profile away from the entrance of the tube. This is due to the effect of the velocity distribution and the reaction at the wall which acts as a sink. Radial diffusion also interacts in the concentration field further downstream from the reactor. This effect tends to flatt,en the concentration profile. Hence the combination effelcts of diffusion, convention, and chemical reactions cause the concentration profile to be more uniform downstream from the reactor. The effect of the homogeneous chemical reaction inside the tube on the radial concentration distribution is shown in Figure 5. When other parameters such as CY, Pe, and x / a are kept constant, an increase in the bulk reaction will decrease the value of the dimensionless concentration and distort the uniform concentration profile to close to a parabolic profile. When K = 0.1, the dimensionless concentration remains constant between 0.955 and 0.99 throughout the tube. However, the dimensionless concentration changes from 0.71 at the center of the tube to 0.2 a t the wall for K = 10. This seems to be the general characteristic of the concentration profile as the bulk reaction increases.
218
The Journal of Physical Chemistty, Vol. 84,
No. 2, 1980
Dang and Steinberg
1
t
03-
0 01 0
01
02
03
04
0-5
06 2
08
07
09
2
IO
I
0
I O
30
20
40
50
60
-
Figure 5. Effect of bulk reaction on radial concentration distribution reactant.
Figure 8. Dimensionless average concentration vs. axial distance for Pe = 10, a = 0, IO, 100, and K = I , IO, 100.
Pe: IO
0:011
I\
1
0
05
I5
IO
20
1
25
Figure 6. Dimensionless average concentration vs. axial distance for Pe = 1, a = 0, 10, 100, and K = 1, IO, 100.
,
\
I
0
100
200
300
400
500
-
Figure 9. Dimensionless average concentration vs. axial distance for Pe= 10,a=0.1,1,10,and~=0.1,0.01.
1 0
I 0
IO
30
20
40
50
-
Figure 7. Dimensionless average concentration vs. axial distance for Pe = 1, a = 0.1, 1, IO, and K = 0.1, 0.01.
Figures 6-11 show the dimensionless average concentration along the reactor length for Pe = 1, 10, 100; K = 0.01, 0.1, 1, 10, 100; and a = 0, 0.1, 1, 10, and 100. It is preferred to determine the controlling mechanism in the reactor by these graphs rather than from graphs with a dimensionless average concentration vs. a dimensionless radial coordinate at a fixed axial position. The latter kind of graphs may not lead to the correct conclusion. For example, r/a = 0.1 in Figures 6 and 7 ; the dimensionless average concentration is greater than -0.98 for any value
0
20
I -
30
40
50
Flgure 10. Dimensionless average concentration vs. axial distance for Pe = 100, a = 0, 10, 100, and K = 1, IO, 100.
of a and K 50.01, but it changes from 0.36 to 0.93 for 100 5 K I1 and 0 5 a 5 100. If one plots a dimensionless average concentration vs. a dimensionless radial coordinate, one finds that the dimensionless average concentration varies slightly for K 5 0.01 and any value of a and would conclude that the system is controlled by a homogeneous chemical reaction. However, this conclusion is obviously not true when one looks at Figure 7 because the rate of decay of the dimensionless average concentration along the axial distance of the reactor changes significantly for a different value of a. So the system cannot be concluded
Analysis of Diffusion and Reactions in a Flow Tube
The Journal of Physical Chemistry, Vol. 84, No. 2,
1980 219
axial diffusion, Pe should be greater than 100. To neglect heterogeneous chemical reaction, K should be greater tlhan 100.
I L - , 0
I
500
000
-
1500
2000
Figure 11. Dimensionless average concentration vs. axial distance for Pe = 100, a = 0.1, 1, 10, and K = 0.1, 0.01.
to be controllled by a homogeneous chemical reaction. Hence, one can see the value of drawing conclusions from the present graph based on dimensionless concentration as a function of x l a rather than r l a in Figures 6-11. From the present results as well as from those established earlier, axial diffusion becomes insignificant when Pe > 100. For Pe < 100, a decrease in the value of the Peclet number will increase the effect of axial diffusion and hence reduce the reactor length to a certain concentration level. ‘The shape and pattern of the dimensionless average concentration curves for Pe = 1,10, and 100 are the same. When K I 100,O Ia I100, and 1 IPe I100, the system is controlled by the homogeneous chemical reaction as seen in Figures 6,8, and 10. For 10 IK < 100, the system is istill controlled by a homogeneous chemical reaction for a 1. 10, but heterogeneous and homogeneous reactions should be considered for a I0.001 as shown in Figures 6, 8, and 10. From these same figures and for K I1, a different mechanism is observed. for K 5 1, the dimensionless average concentration changes slightly for 0 Ia I0.1 and 10 Ia I100, but it is farther apart for these two regions when a = 1. So the system is controlled by a homogeneous chemical reaction for a I0.1, but is controlled by a heterogeneous chemical reaction for 10 5 a. At a = 1, both homogeneous and heterogeneous chemical reactions are important. Hence from this analysis, one can see that, when Pe C 100,O.l Ia I1,and K I1,the present result must be used to investigate this system where axial diffusion and homogeneous and heterogeneous chemical reactions are all important. Relaxing these conditions to K I1, a I0.1, or a I 10 by neglecting a heterogeneous or homogeneous chemical reaction may introduce some errors. To neglect
Nomenclature a radius of tube coefficient of series expansion in eq 9 A, C concentration of reactant Cb average concentration of reactant, Co inlet concentration of reactant D molecular diffusivity g function defined by eq 14a I, integral defined by eq 16 J, integral defined by eq 17 k forward reaction rate constant Pe aU/D r radial coordinate of tube R, eigenfunctions U maximum velocity of fluid in tube x axial coordinate of reactor tube Greek Symbols p, eigenvalues rn,m integral defined by eq 18 0 CICO Ob Cb/CO
! K
rla
x/(aPe) ka2/D
Supplementary Material Available: Tables 11-XI1 containing eigenvalues and related constants for various values of K , Pe, and a (20 pages). Ordering information is available on any current masthead page. References and Notes (1) Lauwerier, H. A. Appi. Sci. Res. 1959, A8, 366. (2) Katz, S. Chem. Eng. Sci. 1959, 70, 202. (3) Kaufmann, F. “Reactions of Oxygen Atoms in Progress in Reaction Kinetics”, Porter, G., Ed.; Pergamon Press: New York, 1961, Vol. 1, pp 11-13. (4) Cleland, F. A.; Wilhelm, R. H. AIChE J. 1956, 2,489. (5) Dranoff, J. S. Mathematics of Computation 1962, 75,403. (6) Lupa, A. J.; Dranoff, J. S. Chem. Eng. Sci. 1966, 27, 861. (7) Wissler, E. H.; Schechter, R. S . Chem. Eng. Sci. 1962, 77, !337. (8) Homsy, R. V.; Strohman, R. D. AICbE J . 1971, 17, 215. (9) Subramanian, R. S.; Gill, W. N.; Marra, R. A. Can. J . Chem. Eng. 1974, 52,563. (10) Pokier, R. V.; Carr, R. W. J. fhys. Chem. 1971, 75, 1953. (11) Ogren, P. J. J. fhys. Chem. 1975, 79, 1749. (12) Walker, R. E. fhys. Fluids 1961, 4 , 1211. (13) Solomon, R. L.; Hudson, J. L. AIChEJ. 1967, 13, 545, (14) Hoyermann, K. H. I n “Physical Chemistry an Advanced Treatise”, Jost, W., Ed.; Academic Press: New York, 1975; Vol. VLB, pp 931-1006. (15) Dang, V. D. Brookhaven National Laboratoty Report 22383, Jan, 1877. Also in Chem. Eng. Sci. 1978, 33, 1179.
