CHEMICAL REACTION IN THE ENTRANCE LENGTH OF A TUBULAR REACTOR Laminar Flow DEAN L . ULRICHSON' AND
ROGER
A . S C H M I T Z
Department of Chemistrj and Chemical Engineering, UniL'erszty of Illinois, C'rbana. Ill.
Numerical solutions for first-order chemical reaction and simultaneous development of the parabolic velocity distribution are presented for the cases of homogeneous reaction and reaction catalyzed on the tube wall. The approximate solution of Langhaar for the entrance velocity profiles is employed to provide the velocity data for numerical solution of the component material balance. It is shown that the effect of the entrance region on conversion depends on the value of dimensionless groups K = kra'//v for homogeneous reaction and K, = k,ro/v for wall-catalyzed reaction. Results of the numerical solutions show that the entrance effect is negligible for most practical purposes if K or K , has a value less than 4.
analysis of laminar-flow tubular chemical reactors has considerable attention for both homogeneous reactions (2, 5, 7, 72-75) and reactions catalyzed a t the wall of the tube (7, 4, 6, 8, 9, 74). The previous studies, however, have been limited to either plug flow or fully developed parabolic flow situations. I n most cases neither situation is realistic, since the velocity may be nearly uniform at the inlet of a tube and develop into the parabolic distribution while chemical reaction and diffusion occur through an entrance length. The residence time distribution in the entrance length is very corpplex, since the hydrodynamic problem is a two-dimensioqal one. No attempt has been made to estimate quantitatively the effect of the hydrodynamic entrance length on chemical conversion and thus to establish conditions under which it may be neglected in kinetic studies or in reactor design. This paper presents the results of a theoretical study of the entrance effect in tubular reactors operating isothermally with constant fluid properties and first-order chemical reaction (1) occurring homogeneously and (2) catalyzed a t the tube wall. HE
Tattracted
Entrance Velocity Field
Under the assumption that fluid properties are constant, the velocity field is independent of the processes of mass diffusion and chemical reaction, and the reactor entrance length problem may be solved by numerical solution of the component material balances once the complete velocity field in the entrance region is specified. A number of approximate solutions for the development of parabolic flow from uniform inlet velocity are available in the literature. The solution of Langhaar ( 7 7) for the axial component of velocity provides a convenient and suitably accurate source of velocity data. In Langhaar's treatment the hydrodynamic entry problem is solved by an integral method using a velocity profile obtained from a linearized version of the differential boundary layer momentum equation. The resulting single expression for the axial component of the velocity over the entire flow regime is u - Io@) - _ U
- I&r/ro)
Id@)
(1)
Present address, Engineering Department, E. I . du Pont de Nrmours Br Co.. Inc., iYilmington, Del. 2
I&EC
FUNDAMENTALS
where u and zi are local and average velocities, respectively, ro is the tube radius. and I , is the modified Bessel function of order n . The parameter j3 is related to the axial distance, x , through an integral momentum balance. The relationship is given by
where
(3)
The argument of the modified Bessel functions is p . The integral in Equation 2 was evaluated numerically by Langhaar and tabulated for various values of j3. The above description of the entrance velocity field was used by Kays (70) in the study of entrance length heat transfer and by Bosworth and Ft'ard (3) to study mass transfer in the entrance length of a pipe. In both cases the appropriate conservation equation was solved numerically, neglecting the radial component of velocity. .4ccording to boundary layer theory, the radial convection term is not negligible, and significant errors may result, a t least very near the inlet. if this term is neglected. T o meet the objectives of this work an expression for the radial velocity component was obtained from Equation 1 and the continuity equation. The continuity equation in cylindrical coordinates is
bu + 1- b (ru) bx
rbr
=
0
where u is the radial velocity component. Thus
From Equation 2 dj3-
-v
1
u t = w([,
Evaluation of the integral in Equation 5, followed by the indicated differentiation u i t h respect to /3 and substitution for dp du from Equation 6, gives the folloMing relationship for 2:
K, R)
which upon substitution into Equation 12 gives
bc b[
0 -
where the argument of the modified Bessel functions is /3 unless otherbvise indicated. T h e right-hand side of Equation 7 vanishes at Y = 0 and r = ro. as expected. Equations 1 and together with Equations 2 and 3 give a n approximate description of the complete entrance velocity field. T h e expressions for u and u may be written in functional form, convenient for later use as
where functions w and (C have their obvious counterparts in Equations 1 and 7: respectively. Homogeneous Reaction
'r'
bc
+ - - - = c y
KbR
T h e bulk average concentration. C,. is defined as rl
C, = 2 J
wCRdR 0
A noteworthy point concerning Equation 16 is the presence of the dimensionless group, K . This group has no counterpart in the material balance for homogeneous reaction in fully developed flow. Thus when the entrance effect is included, the average concentration, C,. a t any distance from the tube inlet depends on t\vo parameters, a and R or a and Sc. \vhereas the average concentration assuming fully developed flo~v throughout depends only on a T h e parameter K may be related to the dimensionless entrance length. According to Langhaar (77), the entrance length, uE. defined as the distance from the tube inlet required for the centerline velocity to reach a magnitude of 1 98 a.is given by .rEu - _ Uro2
Basic Equations. Under the usual assumptions a material balance on a differential element in the entrance region of a tubular reactor can be written as
-
0.23
or
0.23 K
(E
where c is the concentration of a component undergoing firstorder irreversible chemical conversion. Equation 13 may be Lyritten in dimensionless form by making the following substitutions:
R =rr0 [
= xk/B
c
= CdCo
(11)
u t = u,.u U I
c,,'B
=
\\.here co is the uniform inlet concentration. form of Equation 10 is
T h e dimensionless
Lvhere
D
a = kro2
(13 )
\\here Sc is the Schmidt number T h e boundary conditions for Equation 1 2 are
E =O.C=l R
=
OandR
(1 4) =
1 . X bR
=
0
In terms of quantitie5 defined above. Equations 8 and 9 for the velocit) component5 ma\ be Mritten as
(1 8 )
\There = .uEk U . T h e qualitative effect of K on conversion is apparent. If K is large, either the entrance length. y E : or the ratio of reaction velocity to stream velocity is large. I n either case a major portion of the chemical conversion is occurring in a region of nearly plug flow. If K is small. entrance effects are negligible. Thus the chemical conversion at any distance along the reactor? for a given value of a : will lie somewhere between the values predicted for parabolic flo~vand plug flow, depending on the magnitude of the dimensionless group: R. Numerical Solution and Discussion. T h e finite-difference form of Equation 16 was solved numerically on a digital computer using the difference forms of the derivatives employed by Cleland and LVilhelm ( 3 ) . who solved the finite-difference equations assuming parabolic flow over the entire reactor length. Point velocities were computed by means of Equations l . 2: 3 , and. 7 . This \vas facilitated by numerically integrating Equation 2 and storing values of B us. K . Solutions were carried out on 30 equally spaced radial increments with a step size in ( of 0.01. T h e integral in Equation 17 was evaluated by Simpson's rule. As a result of the smaller increment sizes used in this study, accuracy \vas improved over that of Cleland and Nilhelm in some cases for the fully developed flow situation. Figure 1 shobvs the values of C, us. for various values of a that result from the numerical treatment outlined above for w = 2 (1 R2)and (C = 0. T h e effect of the entrance region. characterized by the parameter K : is shown for values of a = 0 , 0.01, and 0.1 in Figures 2, 3 , and 4. respectively. These figures show that as K becomes large: the conversion in a tube approaches that given by the plug flow solution. T h e solutions for K = 0 correspond VOL.
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FEBRUARY 1965
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I
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Figure 1. Conversion in fully developed flow with homogeneous reaction
Figure 2. Conversion in developing flow with homogeneous reaction and cy = 0
to those for fully developed flow. For the sake of curiosity some solutions were carried out neglecting the radial component of velocity. The solution thus obtained for a = 0 and K = 10 is shown in Figure 2. For practical purposes the effect of the radial velocity in this case may be negligibly small. Hoivever, a serious error may result if this component is neglected for the case of a wall-catalyzed reaction. The types and magnitudes of error that one could commit by neglecting entrance effects can be readily estimated from Figures 2, 3, and 4. T o take a hypothetical example, suppose diffusion effects are negligible, so that a = 0:and a conversion of 807,: or C, = 0.2. is obtained in a laminar flow tubular reactor of a given length being used for kinetic studies. According to Figure 2 an error of about 7 7 , in t , or thus in the reaction rate constant, k? would result if entrance effects were neglected and K were to have a value of 10. Similarly if one designs a reactor for a given k to yield C, = 0.2, a 7y0shorter length is actually required as a result of entrance effects. If entrance effects are neglected in the design, C, will be about 77, below the design value. Calculations carried out for K values smaller than 10 show that a linear interpolation for C, between the curves for K = 0 and K = 10 is valid in Figures 2, 3, and 4. For K = 4 an error generally of less than 2% in C, would result in assuming fully developed flow throughout, even for very small values of a . Thus for most practical considerations, entrance effects are negligible if K is less than 4. From Equation 18 this criterion may be roughly written as tz < 1 or x E < 6 : k . In terms of the Schmidt number this criterion is Sc > 0.25 ’ a . In the experimental work of Cleland and Wilhelm (5) on the hydrolysis of acetic anhydride, the largest value of f E involved was less than 0.1. Entrance effects in these experiments were certainly negligible, as was conservatively estimated in the original publication. Figure 5 shows the effect of the developing velocity on the concentration profile at a distance of about one quarter of the entrance length from the tube inlet. Concentrations near the
wall are greater than those for flow assumed to be fully developed from the inlet, because residence times there on the average are shorter than they would be over an equivalent length of fully developed flow. Conversely, concentrations near the tube axis are lower than concentrations predicted under the assumption of fully developed flow, since average residence times in this region are longer.
4
l&EC FUNDAMENTALS
Wall-Catalyzed Reaction
If chemical reaction is activated by a catalytic wall, entrance effects may be eliminated entirely by a “calming” length of noncatalytic material, when practical. I n this work it is assumed that reaction begins at the inlet to the tube, where velocity and concentration are uniform. Basic Equations. The material balance on a component undergoing first-order chemical conversion only at the surface of the tube is
bc
u bx
ac +ubr = D
with boundary conditions r = ro. D bC -
(E+ E)
+ k,c
=
0
br
bC br
r = o , - = o x = 0,c =
cg
Equations 19 and 20 may be written in dimensionless form by introducing R, C: u ’ , and u ‘ from Equation 11 together with t S ,a,,and K,, defined for their convenient analogy to E. a , and K above as xk, Es =
0
K,
2.,
= -
=
D
k,ro k
2’ = v
1 ~
a,sc
Figure 3. Conversion in developing flow with homogeneous reaction and a = 0.01
The dimensionless equations are
bc
R=l,a,-+C=O
bR
E,
=
o,c=1
where u' and u' have been replaced by Equation 15, which may now be written as u' =
u(Es!K,, R )
Figure 4. Conversion in developing flow with homogeneous reaction and a = 0.1
infinite value of cy,. Damkohler (6) and Baron. Manning, and Johnstone ( 7 ) have considered first-order reactions a t the wall under the assumption of plug flow and finite diffusivity. The solutions obtained numerically in this work for plug flow are shown in Figure 6. The numerical results for the solution in the entrance length lie between the parabolic and plug flow solutions and approach the solution for plug flow as the entrance length, or K,, increases. From curves of this type an estimate of the error in neglecting entrance effects is readily available. Results of the numerical work indicate that for practical purposes the error in neglecting the entrance effect is negligible if tsE< 1 or Sc > 0 . 2 5 , ' ~ ~These ~. conclusions are analogous to those stated earlier for a homogeneous reaction. Although the effect of neglecting the radial component of the velocity was shown to be small when the reaction takes place homogeneously, large errors and inconsistent results are ob-
As was the case for homogeneous reaction, an additional parameter, in this case K,, is involved in the reactor problem when entrance effects are not neglected. K , may be related to the dimensionless entrance length as in Equation 18. ESE
where
=
0.23 K 8
(24)
tSE = xEks/Cro.
Numerical Solution and Discussion. Equation 22 was solved for various values of a , and K , by the finite-difference method outlined earlier. The resulting values of C, are shown graphically LIS.5, in Figure 6. In Figure 6 the curves for K , = 0 correspond to an entrance length of zero, or fully developed flow from the inlet. Solutions under this assumption have been obtained theoretically by Chambr6 ( 4 ) and Katz (9). Cnlike the case of homogeneous reaction, the assumption of plug flow does not give the same result as an infinite value of diffusivity. or in this case, an
---DEVELOPING FLOW
\\ \
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1965
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0.6 C
04
02
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0.2
0.4
R
0.6
0.8
I.o
Figure 6. Conversion in developing flow with wall-catalyzed reaction
Figure 7. Concentration profiles for developed and developing flow and effect of radial velocity for cys = 0.10, K, = 10
tained if this component is neglected when reaction occurs only at the tube wall. T h e solution resulting for cy, = 0.1, K , = 10, and u = 0 is shown for comparison as the dashed curve in Figure 6. From this curve one would conclude that the effect of the entrance region is to reduce conversion while the true effect is to increase conversion over that for parabolic flow. \Vhen the radial velocity component is neglected, a n over-all material balance is not satisfied, since dilute reacted material near the wall is not convected to the core. T h e error involved in neglecting this convective component depends on the parameters cy, and K,. In Figure 7 are shown concentration profiles for cy, = 0.10 resulting from parabolic flow-i.e., K , = 0-and developing flow with K , = 10 a t two axial locations. For comparison the profiles for u = 0 in developing flow are also shown. In this case the profiles for parabolic flow lie between those for u = 0 and u # 0.
version lies between that obtained assuming parabolic flow and that assuming plug flow-the exact value depending on parameter K or K,. As a result of the calculations, rather conservative criteria have been established under which entrance effects are negligible for practical purposes: that K and K , have values roughly less than 4. These criteria are expressed also in this work in terms of a dimensionless entrance length and in terms of the Schmidt number. For the sake of comparison some solutions were carried out neglecting the radial velocity component. T h e results thus obtained are inconsistent with an over-all material balance and may lead to incorrect conclusions regarding the effect of the entrance region, particularly in the case of a wall-catalyzed reaction. Acknowledgment
Fellowship support for one of the authors from the Sun Oil
Co. is acknowledged.
Su m mary
T h e partial differential equation governing the transport in laminar flow of a component undergoing first-order chemical conversion in the entrance region of a circular tube has been solved numerically for homogeneous reaction and for reaction catalyzed on the tube wall. Langhaar's ( 7 7 ) approximate solution was used for the axial velocity component, and an expression for the radial component was obtained by means of the continuity equation. T h e assumptions of uniform velocity and concentration a t the tube inlet, constant fluid properties, and isothermal operation were made throughout this study. T h e study centers principally on the effect on conversion of two dimensionless groups. K for homogeneous reaction and K , for a wall-catalyzed reaction, both of which may be interpreted in terms of a dimensionless entrance length and neither of xvhich arises Lvhen the flow is assumed to be fully developed from the inlet. T h e numerical solutions show that the con6
l&EC
FUNDAMENTALS
Nomenclature c c0
C C, D
f g
Z,, k k,
K K, r TO
R Sc u
=
concentration of reactant, moles/volume
= inlet concentration of reactant, moles/volume = 6 / 6 0 , dimensionless reactant concentration
dimensionless bulk mean reactant concentration molecular diffusivity, (length)2/time = function of p defined in Equation 3 = function of defined in Equation 3 = modified Bessel function of order n = homogeneous reaction rate constant, (time)-' = heterogeneous reaction rate constant, lengthltime = kro2/v, dimensionless = k,ro/w, dimensionless = radial coordinate, length = tube radius. length = r/rO, dimensionless radial coordinate = W I D ,Schmidt number, dimensionless = axial velocity component, length/time = =
u
=
u u ur x xE
= u , ’ ~ dimensionless . axial velocity component
average flow velocity, length/time
LY,~
= radial velocity component. length,/time = z! ‘u, dimensionless radial velocity component = axial coordinate. length = hydrodynamic entrance length, length = b o 2 . dimensionless = D;k,ro. dimensionless
fl
= dimensionless
v {
= =
[E
=
t7
=
literature Cited
D!
LY
=
v
=
w
=
axial distance parameter defined in Equation 2 kinematic viscosity, (length)* ’time xk,‘zi, dimensionless axial distance for homogeneous reaction xEk,’u, dimensionless entrance length for homogeneous reaction xk,,’Uro, dimensionless axial distance for heterogeneous reaction xEk,5,, dimrnsionless entrance length for heterogeneous reaction dimensionless radial velocity function defined in Equations 1 5 and 23 dimensionless axial velocity function defined in Equations 15 and 23
(1) Baron, T.. Manning. \V. I
