CRITICAL EVALUATION OF BOUNDARY CONDITIONS FOR T U B U L A R FLOW REACTORS LIANG-TSENG FAN AND YONG-KEE AHN Department of Chemical Engineering, Kansas State Univerrify, Manhattan, K a n . The three sets of boundary conditions frequently employed for the design and analysis of isothermal tubular flow chemical reackrs are critically reexamined. The differences and the relationships among the concentration profiles and conversions obtained b y use of the different sets of boundary conditions are numerically evaluated. HE ANALYSIS of various boundary conditions for tubular Tflow chemical reactors in which fluids are dispersed in the direction of flow has attracted considerable attention ( I , 5, 6, 8. 9 ) . The previous studies, however, have been primarily limited to the mathematical and physical justification of the boundary conditions and their applications to particular experiments (1. 4, 6-8). Because of mathematical complexities, no attempt has been made to clarify numerically the quantitative relationships among the solutions obtained by use of different sets of boundary conditions, and the relationships between the solutions and the physical limitations of the isothermal flow chemical reactor without segregation and bypassing This report presents the results of such an attempt with respect to the three sets of boundary conditions most frequently employed in the analysis of tubular flo\v reactors with accompanying axial dispersion. They have been employed often as approximations for each other without much quantitative justification (4-6).
Differential Equation and Boundary Conditions
the reactor, where x = 0, by neglecting the diffusion within the entrance of the reactor: x =
- d2r
dc
dx2
dx
D--f7-
661
= zicx-o+
dc -= dX
x=L,
(4a)
0
This set of boundary conditions is designated as (11). Another set of boundary conditions employed by several investigators (5, 6) represents a n approximation of a reactor with finite length by a reactor with infinite length:
ac,
x = 0, x
+ m,
lim x-
(ja)
=o ,,c,zi
C(X)
=
0
(jb)
m
This set of boundary conditions is designated as (111). Mathematical Treatment
T h e solutions of Equation 2, which satisfy the three sets of boundary conditions, are obtained through the following transformations of the variables into the dimensionless forms (1. 8, 9):
A material balance on a section of differential length of a homogeneous tubular flow reactor can be bvritten as ( 7 ) :
where c is the concentration of reactant at cross-section x . For a reactor with homogeneous steady flow and a first-order chemical reaction, Equation 1 simplifies to (71 :
0,
*, i
c
’ ,c
X
9 =
where
-kc=O
rvhere D,a, and k are the mean axial dispersion coefficient the flow velocity. and the reaction rate constant. respectively. Boundary conditions \\-hich have been proposed for use Lvith Equation 2 are as folio\\-s. DanckLrerts ( 7 ) introduced the follo\ving set: x =
0,
ac,
= 2ci,o+
-
b
(g) Y
x+o+
(3a)
where c1 is the concentration of reactanT in the entering stream This set of boundary conditions is designated as (I). Hulburt and others (?. 3. 8) proposed a set of boundary conditions Lvhich approximates Danchverts’ set at the entry to 190
l & E C PROCESS D E S I G N A N D DEVELOPMENT
The solutions of Equation 6 which satisfy the boundary conditions, Equations 7a through 9b. are obtained by means of Laplace transformation.
Laplace transform of Equation 6 with respect to q is: =
+ y'(O+) - 2 M Y ( 0 + )
sr(0+)
- 2Ms
s2
Solving for y' ( 0 ' ) Equation 18 yields: (10)
- 2MR
For the solution using (I), Equation 7a is substituted in Equation 10 to obtain:
By means of partial fractionation, Equarion 11 is separated into two parts as: =
(1 A P ) Y ( O + ) - 2 2PIS - M(l p)]
+
- (1
- P)r(O+) - 2 2P[s - M(1 - P)1
in Equation 21 and substituting it into
*,(v)
exp[.bl(l - 6)vI
=
(22)
where
8 = ,,l+,
2R
For the solutions of the same differential equation with nonlinear, nth order chemical reaction, Equation 1 is transformed in terms of dimensionless variables as ( 8 ):
(12)
where
where p=J1+:
T h e inverse transform of Equation 12 gives:
T h e numerical solutions of Equation 23 for the first set of the boundary conditions have been presented by Fan and Bailie (2). T h e solutions for the other two sets of boundary conditions, by use of a three point finite-difference method (2), are given here.
According to the boundary condition, Equation 7b, differentiation of Equation 17 and setting it to zero gives:
Results and Discussions
The concentration profiles for the various combinations of
M and R for the first-order reactions \\ere calculated by use of Equations 15. 20. and 22. TJpicdl results corresponding to are shown in Figure 1. The figure shows that the R = concentration profiles obtained b\ use of (11) and (111) are
Substitution of Equation 14 into Equation 13 upon simplification gives:
inconsistent with the limits of both a plug flow reactor and a uniformly stirred tank reactor for all values of .M and R ;
For the solution using (11). Equation 8a is substituted in Equation 10 and simplified to:
whereas the profiles computed by use of (I) are consistent with such limits. The term inconsistent means that a computed conversion which is either less than th3t of a uniformly stirred tank reactor or greater than that of a plug flow reactor is unreasonable for the dispersion model considered in this Lvork. The numerical values of the limits of inconsistency \\ere calculated by the follouing two equations:
By means of partial fractionation, Equation 16 is separated into tivo parts as: =
--.M(l
- 3 ) + -/YO+)
2 M P [ S - M(l
+ 811
+ +
- -M(1 8) 2MP[s - 'Zf(1
Y'(O+)
- 811
(17)
Y.,,=, = exp ( - R v i
T h e inverse transform of Equation 17 yields:
(25)
According to the boundary condition, Equation 8b, Equation 18 is differentiated to solve for y' (O+). I t gives:
Figure 2, howe\w. indicates that the over-all conversions computed by use of the different sets of boundary conditions are numerically in agreement for large values of M and increasinaly deviate from each other as M becomes smaller. (111) give values lo\ver than those obtained in the uniformly stirred tank reactor \\.hen ,M becomes very small especially at
Substitution of Equation 19 into Equation 18, upon simplification, gives:
Again, the over-all conversions computed by use of (11) and (111) give values lower than those obtained in the uniformly
1.(v) =
(1
+ P) e x p [ W 1 + 4 ) + M(1 - P)vl - (1 - P ) e x p [ W 1 + P ) 7 + M(7 - P ) ] (1
+ 6)exp[M(1 + P)l - (1 - P)exp[.+f(I - 1311
For the solution using (111), substitution of Equation 9a in Equation 10 \vi11 give the same result as Equation 18. As q approaches infinity in Equation 18, the second term in the righthand side of Equation 18 vanishes as: O =
M(6
-
+
1) -/YO+) 2MP
(20)
stirred tank reactor \+hen M becomes very small, especially a t low values of R This is also indicated further in Figure 3. The curves in the figure show the locus representing: (jll)?=,
- fr/=o
(,h,lh-$ VOL.
- / L O
1
=
0
=
0
NO. 3
JULY
1962
191
0.8
c
ZiI 1 0.6
0.8
o.21
I .o
0.8
hI)
0. I
0.6
0'5b
M
Parameter:
h
Of2
014
0:s
I
Ol8
'
1.0
77 Figure 1. Concentration profile of reactant a t various M values for first-order reaction with R = 2
O
L
i
h
S
4
5 R
6
;
8
9
3
Figure 2. Comparison of final conversions as a function of R for first-order reaction
-
M
-
Parameter, R
-
-
0 1
O
Figure 3. Locus indicating regions in which conversions are inconsistent with limit of a uniformly stirred tank reactor for first-order reaction 192
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
b
'
;
I
4I
I
6
I
I
8
I
I
IO
I
I
12
.
1
14
M Figure 4. Differences between conversions a t reactor inlet (where fli = fill = 0) computed b y use o f boundaryconditions (11) and (Ill) and those computed b y use of boundary conditions (I) as functions o f M for first-order reaction
Parameter
:
R
Figure 5. Differences between conversions a t center of reactor computed by use of boundary conditions (11) and those computed b y use of boundary conditions (I)as functions of M for first-order reaction
Figure 6. Differences between conversions a t center of reactor computed b y use of boundary conditions (Ill) and those computed b y use of boundary conditions (I) as functions of M for first-order reaction
T h e domains below the curves, then, correspond to the conditions: (jX1)W
(JIII)V=I
-fAL0 < 0 - f.w=o < 0
T h a t is, the over-all conversions calculated for the conditions which are represented by these regions are inconsistent with the limit of the uniformly stirred tank reactors. T h e inconsistency of over-all conversions by use of (11) has been indicated hy Wehner and Wilhelm ( 8 ) . However, it has never been established quantitatively as a function of the system characteristics-i.e., M and R. The case of (111) has been hitherto unknown. The over-all conversions computed are consistent with the upper limit of a plug flow reactor for all cases. T h e design of an isothermal reactor should not be based only on the over-all conversion when the reactions to be carried out in the reactor are exothermic or endothermic. The estimation of heat transfer area to be provided requires a knowledge of the concentration profile, or the conversion, along the length of the reactor. Figures 4 to 8 show quantitativelv the relationships among the profiles and the conversions computed by use of (I), (11), and (111). T h e differences between the conversions computed by use of (11) and (111) and those computed by use of (I) are given a t three positions of the reactor as functions of the parameters of the system. The reason that (I) is used as a basis of the comparison is that, as pointed out previously, the use of (I) yields conversions which are consistent with both limits of the reactor a t any point along the reactor. The following observations are made from the figures :
0 .I 6
Parameter
R
0 12
0.08
0.04
0 M
Figure 7. Differences between final conversions computed by use of boundary conditions (11) and those computed by use of boundary conditions (I) as functions of M for firstorder reaction
0.20
n
I
I
I
1
1
I
I
Parameter : R
While the differences increase with increase of R in the vicinity of the inlet, they decrease with increase of R a t the center and exit of the reactor. T h e differences decrease with increase of M a t any position along the reactor. Except a t the inlet of the reactor, (11) yields consistently greater deviations than (111). Figures 9 and 10 are typical concentration profiles for secondand third-order reactions. These figures show that the use of (11) and (111) again gives profiles and conversions inconsistent with the limits of the tubular reactor under certain conditions. It is known that the use of (I) gives profiles and conversions consistent with the physical limits of the tubular reactor
Figure 8. Differences between final conversions computed b y use of boundary conditions (Ill) and those computed by use of boundary conditions (I) as functions of M for firstorder reaction VOL.
1
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JULY 1962
193
1
0.6
;
6
.
0.5 o
0.2
0.4
0.6
0.8
1.0
77
1
i
0.6
1
i
0.6
0.5k1
0.2
I
0.4
0.6
I
I
0.8
I
1.0
77
0.6
1
1
tl--uLA
0.5 o.6
0.2
0.4
0.6
0.8
1.0
77
Figure 9. Concentration profile of reactant at various M values for second-order reaction with R =
Figure 10. Concentration profile of reactant a t various M values for third-order reaction with R = I/'*
for nonlinear nth order reactions (2). Since the relative numerical values of the solutions obtained by use of (I), (11). and (111) are similar to the case of the first-order reaction. only the final conversions computed are illustrated in Figures 11 and 12 for the case of R = 2. These two figures show that increasing the order of reaction decreases the differences of both (11) and (111) from (I), and that (11) deviates consistently more than (111).
given numerically as a function of M and R. Differences increase as M and R decrease a t the exit of the reactor. and (11) always yields larger deviations than (111). Since the solution to (111) is in the form of a simple exponential function, the use of (111) to approximate (I) always yields a more convenient method and also a more accurate result than the use of (11). While the use of (I) yields concentration profiles consistent with the limits of a plug flow reactor and a uniformly stirred tank reactor, the use of (11) and (111) gives profiles inconsistent with such limits for all possible values of M and R. The use of (11) and (111) for predicting and estimating the concentration profiles therefore should be avoided. The diagrams (Figures 4 to 8) representing numerical comparisons of the various solutions offer a convenient means of computing numerical values for the exact solution (Equation 15) from the approximate, but simple solutions (Equation 20 or 22). Finally, the use of (11) for a homogeneous tubular flow reactor with axial dispersion is not desirable from the viewpoints of both accuracy and convenience. The use of (111), if its limit is recognized, offers a convenient means for computing the final conversion in such a reactor. especially for reactions of higher order. Figures 4, 6, and 8 offer a convenient means of estimating the degree of approximation as functions of system characteristics-i.e.. M and R.
Conclusions
The domains in which boundary conditions of (11) and (111) yield final conversions inconsistent with the limits of a uniformly stirred tank reactor have been established in the M-R plane. Such a domain does not exist for the boundary conditions of (I). The use of (11) and (111) should be avoided in and around such domains for computing final conversions. I n other words, the approximation of a reactor by a uniformly stirred reactor yields' not only a convenient means? but also a more accurate means, of finding the conversion than the use of (11) and (111) in and around such regions. Whenever possible, therefore, the use of (I) should be attempted. It is also shown that the inconsistent domain for (11) is larger than that for (111). Differences between conversions computed by the use of (11) and (111) and those computed by the use of (I) have been 194
I&EC PROCESS DESIGN A N D DEVELOPMENT
0 IO.
I
I
I
I
-
Porometer: order of
reactlon
008 -
-
-
Porometer' order of reaction
-
-
-- 004--
-
-
-
8-
006
Lc
I
H
LC
002
-
-
0
Nomenclature c
=
c1
=
1 1 ~-
=
n f
= =
JI
= fIl = frrI = f.vf=~= k = L = .bf = n =
y
concentration of reactant, massivolume inlet concentration of reactant, mass/volume axial dispersion coefficient, (length)2,'time mean value of D:(length)2,'time 1 - y,conversion 1 - y I >conversion computed by use of (I) 1 - y I r , conversion computed by use of (11) 1 - y r I I ,conversion computed by use of (111) conversion for a uniformly stirred tank reactor reaction rate constant length of reactor? length UL'2D: l,'? of axial Peclet number, dimensionless order of reaction rate of reaction kLiii, dimensionless reaction number time average flow velocity of reacting mixture, lengthitime coordinate in the direction of floiv, length
r
=
K t ii s
= = = =
P
=++g
I
I
LA
n
17
= c cl. dimensionless concentration = t L. dimensionless distance from the inlet
of reactor
Acknowledgment
T h e authors thank B. J. Jeng whose assistance in computer programming made it possible to draw Figures 11 and 12. literature Cited
Danckwerts, P. V., Chem. Eng. Sci. 2, 1-13 (1953). Fan, L. T., Bailie, R., Ibid., 13, 63-8 (1960). Hulburt, H. M., IND.ENG.CHEM.36, 1012 (1944). Kramers, H., Alberda, G., Chem. Eng. Sci. 2, 173 (1953). Levenspiel, O., Smith, W. K., Ibid., 6, 227-33 (1957). Otake, T., Kunugita, E., Chem. Eng. (Japan) 22, 144 (1958). Pearson, J. A . R., Chem. Enz. Sci. 10, 281-4 (1959). Wehner, J. F., Wilhelm, R. H., Ibid., 6,89-93 (1956). Yagi, S., Miyauchi, T., Chem. Eng. (Japan) 19, 27-34 (1955). RECEIVED for review June 26, 1961 ACCEPTED March 16, 1962
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