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Turbulent flow of gas through a circular tube with chemical reaction at the wall
Chemical Engineering Science, 1962, Vol. 17, pp. 937-948. Pergamon Press Ltd., London.
Printed in Great Britain.
Turbulent flow of gas through a circular tube with chemical reaction at the wall E. H. WISSLERand R. S. SCHECHTER Departmentof ChemicalEngineering, The University of Texas,Austin12, Texas ,
(Received 12 November 1961)
Abstra&--A scheme is developed for computing concentration profiles when an arbitrary reaction occurs at the wall of a tubular reactor. Solutions are constructed using complete sets of orthogonal functions. For reactions which are not first order, it is necessary to solve an integral equation for the concentration of reactant at the wall. A convenient numerical scheme is developed for doing this. Eigenvalues and Fourier coefficients are presented together with computed concentration profiles for several typical cases.
IN ORDERto study a chemical reaction occurring at a solid-gas interface one must be able to evaluate the resistance to mass transfer near the interface. Tubular reactors having a catalytic or reactant material at the wall appear to offer unique advantages over such systems as packed beds. The surface area is well defined; most of the heat of resiction can be supplied or removed through the external surface of the reactor; and, at least in principle, the resistance due to diffusion can be computed. The tubular reactor with laminar flow has been used by several investigators [l-4]. Diffusional resistance is often evaluated by assuming that plug flow exists and the reaction at the wall is first order and irreversible. Then the concentration at any point in the reactor can be expressed in terms of a Fourier Bessel series. BARON [l] has considered first-order reversible reactions at the wall, and HOELSCHER[5] presented an approximate evaluation of the effect of a parabolic velocity profile. CHAMBRB[6] and KATZ [7] have derived integral equations which can be used to compute the concentration when the rate of reaction is an arbitrary function of the concentration at the wall. WEGER and HOELXHER [8] have sounded a word of warning, pointing out that highly reactive surfaces are generally rough and that it is nearly impossible to remove all of the heat of reaction through the external surface. Both of these effects disturb the laminar velocity profile and facilitate the transfer of material to the wall.
Reactors with turbulent flow have also been used. SATTERFIELD er al. [4] based their analysis of the decomposition of hydrogen peroxide and water on a macroscopic model using the j-factors for heat and mass transfer. WESTKAEMPER and WHITE [9] studied the evaporation of carbon tetrachloride into air in a rectangular channel. Although they were forced to make many assumptions in their analysis, they were able to calculate successfully the rates of evaporation for Reynolds numbers between 7000 and 15,000. SCHSVARZand HOELSCHER[lo] found an inlet length of six diameters for a Reynolds number of 25,000 in a wetted-wall column. The computational problem presented by the tubular reactor with turbulent flow is somewhat more complex than the corresponding problem for laminar flow, especially if laminar plug flow is used instead of a parabolic velocity profile. LONGWELL [l l] has shown how the graphical Schmidt method can be used to solve the diffusion equations for turbulent flow. The purpose of this paper is to present some analytic solutions for both first order and arbitrary reactions occurring at the wall. ANALYTICALSTATEMENTOF THE PROBLEM
The co-ordinates and geometry are shown in Fig. 1. Fluid flows from left to right through the circular tube with a fully developed turbulent velocity profile. For z < 0 no reaction occurs and the fluid has a uniform composition. For 0 < z an arbitrary irreversible reaction occurs at the wall and
937
E. H. WISSLER and R. S. SCHECHTER
CO .? -0
FIG. 1. Geometry and co-ordinate system.
,=D+E
the concentration becomes a function of r and z. The concentration is to be determined subject to the following restrictions : (1) A steady state has been attained. (2) Diffusion in the z-direction is negligible in comparison with the convective transport of mass. (3) The radial transport of mass can be adequately described using an eddy diffusivity. (4) The physical properties p (density), ,u (viscosity), D (molecular diffusivity) and k (reactionrate constant) are independent of position. Conservation of mass for the reactant is expressed by the equation
UacSr a; aZ
[
r(D + E) g
1
0) = CO,
(2)
the inlet concetitration which is constant,
E (0, z) = 0
(3)
and -
,
v
in which z. is the shear stress at the wall and v is the kinematic viscosity. Making these substitutions in equations (l)-(4), we obtain ut
ac
R+ a
(10)
z=r+arf
C(r+, 0) = Co
(11)
(12)
-I@+)
aW+, 5) ar+
h
(13)
=Jo
(1) To complete the statement of the problem the
in which U(r) is the velocity, D is the molecular diffusivity, E(r) is the eddy diffusivity and C(r, z) is the concentration of the reactant. This equation must be solved subject to the following boundary conditions : W,
v
[(D+E)z],=,= h
where h is the rate of reaction. Since most of the equations for E and U are given in terms of u+ and y+, it is convenient to make the following changes of variables :
functions ut(rt) and E(r+) must be given, and i.t is in this regard that the problem is least well defined. For Reynolds numbers of 20,000 or less there seems to be rather general agreement that U+ expressed as a function of yf is only mildly dependent on the Reynolds number and the dependence is significant only near the centre of the pipe. However, we wanted to compute compositions for the lowest possible Reynolds numbers so that readily measurable conversions could be obtained in tubes of reasonable length. The data obtained by DEISSLER [12] indicate that the generalized velocity distribution for smooth tubes can be used for Reynolds numbers at least as low as 8000. We went a little further than that and made our computations at Reynolds numbers of 7000 and 14,000. The velocity profile is computed from du+ -= dy+
1 1 + (O*124)2u+y+[1 - exp(-0*1242u+y+)]
o
426
u+(y+ =O)=O
U+
938
(14) (15)
26
(16)
Turbulent flow of gas through a circular tube with chemical reaction at the wall
which are the equations presented by DEISSLER[121. C(r+, 5) = Co Cn&(r+)e-28n2C’Re (21) The choice of a function to represent E is more difficult; and, indeed, there is no sound basis for making a choice between the several functions which in which the /3.‘s and &(r+)‘s are the eigenvalues have been proposed [13-l 71. The principal point of and eigenfunctions of the Sturm-Liouville system contention concerns the ratio of the eddy diffusivity or thermal conductivity to the eddy viscosity. This (22) ratio has been found to be about 1.4 near the wall by several investigators [13, 14, 16, 171 who have dM-9 differentiated measured velocity, concentration and (23) dr+= temperature profiles, but large uncertainties are introduced by the inherent inaccuracy of differen_y(R+) ddb(R+) tiation. On the other hand, good agreement has -=&h(R+) (24) dr+ been obtained between computed and measured Nusselt numbers in both the thermal entry and fully developed regions by several authors [15, 181 who From equation (17), y = D/v at r+ = R+, and assumed that the eddy conductivity equals the eddy equation (24) can be written in the form viscosity. We have based most of our computations on the assumption that the ratio is unity, but we (25) have checked the sensitivity of the assumption by repeating some of the computations with a ratio of Since the &‘s form a complete set of orthogonal 1.4. Following SPARROWet al. [ 181 we have used functions, a reasonably well-behaved function f(r +) can be expanded into a series of 4,,‘s. In this proby = ; + O-124’u+y+[l - exp(-0*1242u+y+)] lem the C,‘s are chosen so that
“El
o
0 < y+ < 26
(17)
C(r+,0) =
5 CA(r+)
(26)
?I=1
.
Hence, R+
26 < y+ < G
(18) c,=
R+
2
u+r+&(r+)dr+
co
s R+”
,
(27)
u+r+[&,(r+)]2dr+
(19)
s0
or y = Mean of the values computed from equations (17) and (18) at y+ = 26.
(20)
In the preceding equations SC = v/D is the Schmidt number.
(28)
in which N. is the norm of 4,. The values of /?., &(r+), C,, and N. have been determined numerically using an iterative procedure and the Runge-Kutta method on the I.B.M. 650 DETERMINA~ONOF THE CONCENTRATION PROFILES Computer. The constants are listed in Table 1, and In evaluating the effect of the eddy diffusivity on the 4”‘s are available on I.B.M. cards. Using these values one can compute the conthe rate of reaction it is convenient to study the special system in which the reaction is first order and centration of the reactant at any point in the irreversible. For this case h = kC(R, z) in which k reactor. However, in analysing data from a tubular is the reaction-rate constant. Then the solution of reactor, one is usually concerned only with the mean concentration, the set of equations (l&20) has the form 939
E. H. WISSLERand R. S. SCHECHTER
Table 1. Eigenvalues and Fourier coeficients for jirst-order reaction R+ = 240.6
(a) Re = 7003
(b)
No.
Bn
1 2 3 4 5 6
3.983 17.34 28.11 38.33 48.36 58.17
Re=
SC = 0.7
3.276 4.719 1.690 9,361 6.030 4.368
x x x x x x
lo5
0.4873
104
- 0.2675
lo4 103 lo3 lo3
0.2083 -0.1980 0.1888 -0.1765
kR/D = 100
c?I
N,
1 i
5,579 29.18 18.31
1.2490 -0.4535 0.3976
2601 x 105 4.172 1.529 x lo4
4 5 6
39.68 49.99 60.03
-0.3832 0.3818 - 0.3745
8.506 x lo3 5.499 x 103 4.132 x lo3
R+ = 240.6
Re = 7003
1 2 3 4 5 6
(e)
1.1288 - 0.2270 0.1879 -0.1735 0.1616 -0.1442
Bn
No.
(d)
kR/D = 10
-A@+)
R+ = 240.6
7003
No.
(c)
SC = 0.7
2.585 16.42 26.66 36.05 45.13 54.11
No.
Bn
1 2 3 4 5 6
4.076 17.07 27,43 37.14 46.50 55.57
3.740 4.695 1.535 7.980 4.806 3.154
SC = 2
c?z
G
4.369 22.56 36.65 50.19 63.55 76.67
1.0891 -0.1567 0.1272 -0.1171 0.1113 -0.1061
105 lo4 104 lo3 lo3 103
-h@+) 0.6294 - 0.3228 0.2545 - 0.2486 0.2328 -0.2018
x x x x x x
MR+) lo5 lo4 lo4 lo3 lo3 103
0.1430 -0.0793 0.0627 - 0.0626 0.0634 - 0.0623
kR/D = 10 __-N?Z 1.265 1.667 5.559 2.802 1.695 1.191
940
0.0840 - 0.0527 oa430 - 0.0426 ow35 - 0.0462
kR/D = 100
3.167 4.266 1.396 7.189 4.369 3.008
SC = 0.7
Bn
x x x x’ x x
NfZ
1.1452 -0.2686 0.2518 -0.2663 0.2837 -0.2834
R+ = 430
Re = 14,017
1 2 3 4 5 6
1.0613 -0.1075 0.0985 -0.1013 0.1011 - 0.0930
R+ = 240.6
kR/D = 10 N,
C*
Bn
Re = 7003
No.
SC = 2
VW+)
x x x x x x
dn(R+) 106 lo5 104 lo4 lo4 104
0.6110 -0.3089 0.2223 -0.1919 0.1762 -0.1720
Turbulent flow of gas through a circular tube with chemical reaction at the wall
Table 1 (continued) (f)
R+ = 430
Re = 14,017 No.
(9)
6.866 23.86 37.97 51.67 65.23 78.67
1 2 3 4 5 6
1 2 3 4 5 6
2.778 21.79 35.50 48.47 61.07 73.25
1a0378 - 0.0679 0.0548 - 0.0595 0.0624 -0.0635
1
w =-gq s
5.012 22.58 36.33 49.48 62.34 74.88
R
1.398 1.661 5.130 2.424 1.392 9.322
SC=2
(29)
It is easy to show that (30)
The variation of concentration with position is illustrated in Figs. 2-5, in which the solid lines refer to an eddy viscosity to eddy diffusivity ratio of 1-Owhile the crosses refer to a ratio of 0.7. Figs. 2 and 3 show, in order of decreasing composition, the centre-line concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration. The circles pertain to the simple macroscopic model described later. In Figs. 4 and 5 the values of 41 are plotted in the vicinity of the wall. In the region where the concentration proI
0.1318 - 0.0776 0.0596 -0.0534 0.0506 -0.0517
x x x x x x
ddR+) 106 105 104 104 lo4 108
0.7430 -0.3551 0.2544 - 0.2242 0.2122 -0.2091
kR/D = 100
1.185 1.526 4.790 2.262 1.289 8.636
0
Co n$1C~N,e-2un2FIRe
105 105 104 104 104 104
kR/D = 10
1.1228 - 0.2248 0.2026 -0.2051 0.2174 -0.2321
2xrUC(r, z)dr.
x x x x x x
MR+)
N,
c,
R+ = 430
9.879 1468 5.081 2.593 1.573 1.107
SC=2
Bn
Re = 14,017
kR/D = 100
N,
1.2186 -04001 0.3505 -0.3325 0.3258 -0.3255
Rf = 430
Re = 14,017 No.
(h)
cn
Bn
1 2 3 4 5 6
sc = 0.7
x x x x x x
106 105 104 104 lo4 103
0.2217 -0.1161 0.0850 -0.0753 0.0720 - 0.0744
files are fully developed, the function 41(~‘) may be interpreted as the concentration at y+ divided by the concentration at the centre of the reactor. As one would expect the discrepancy between the two sets of computations increases as the Schmidt number and kR/D increase. It does not seem to be strongly dependent on the Reynolds number. Although the difference between the mean concentrations is rather striking on semi-logarithmetic paper, it should be observed that the differences in the percentage of reactant converted is only 6-10 per cent. If one were to base one’s computations on an eddy-viscosity to eddy diffusivity ratio of 0.85, the computed conversions should be in error by no more than 3-5 per cent. Although additional experimental work is needed in this area, the information which is currently available can be used without introducing excessive errors.
941
E. I-L
WISSLER
and R. S. S~HECHTER
1.0 8.: 07 06 05 CO4 C*
03
Ai OS 07 06 05 c c
O4 03
0
FIG. 2. Variation of concentration with axial position. The centreline concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration in order of decreasing concentration.
20
40
I,D
60
SO
100
FIG. 3. Variation of concentration with axial position. The centre-line concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration in order of decreasing concentration.
07 06
06 05 4, 04
03 02
UPPER
CURVES
kR/D
01
= IO
0
Y+
FIG. 4.
Fully developed concentration profiles plotted as ratio to centre-line concentration.
FIG. 5.
942
Fully-developed concentration profiles plotted as ratio to centre-line concentration.
Turbulent flow of gas through a circular tube with chemical reaction at the wall
$. (095)= 0
ARBITRARY REACTION RATE
If h depends in an arbitrary manner on the concentration at the wall, say in accordance with a power law or an absorption model, a solution can be obtained in the form of a non-linear integral equation of the Volterra type. This was discussed some time ago by CHAMEW[6] and more recently by KATZ [7] who presented the general theory for a tubular reactor in which the flow is laminar. The same procedure has also been used in the analysis of heat transfer under conditions of variable thermal flux at the wall [18]. Before presenting the integral equation, it is worthwhile to consider briefly a system in which the reaction varies as shown in Fig. 6. To the left of q1 the reaction proceeds at a constant rate h,, and the solution of equations 10-20 has the form
qr+,5) = c, + y
c’(r+, s’)
-$(R+,c)=
(35)
C’(r+, 5) can be expressed as the sum of two functions. The fist is a particular solution which satisfies the boundary condition at the wall, but does not vanish at the inlet. The second function is determined by a variables separable set of equations having zero gradient at the wall and being equal to the negative of the Grst function at the inlet. Hence, the sum has the proper gradient at the wall and vanishes at the inlet when
C’(r+,5) = 2
t +$f(r+)
+
+ 2 D,e-(2en2r/Re)R~(r+) II=1
(31)
in which C’(T+, r) satisfies the following set of equations :
-$
(34)
(36)
in which 4r(r+) and RA(r+) satisfy d
C’(r+, 0) = 0
(37)
dr+
(32) (33)
n
$(O)=O
(38)
$(R+) = -;$
(39)
(40)
(41)
3
(R+) = 0
(42)
The Dn’s are the coefficients resulting from the expansion of -I#I/ into an infinite series of Ris. R+ u ’ r ’ cjfRAdr+
D,=FIG. 6.
Stepwise
approximation reaction rate.
to
s ;+ s
943
(43) ufr+(R32dr+
arbitrary wall 0
E. H. W~%JLER and R. S. SCHECHTER in which
Since
c R+
(r+, CV+, 5) = XI’ z
ReRf
r+u+dr+
= 7,
the boundary condition at the wall is satisfied for any choice of 4/(O). In the interval q1 4 r < q2, the concentration can be computed by simply adding another term to the previously obtained solution. Hence,
5).
It can be shown that Cn(r+, t) is the change in concentration due to a reaction rate which varies as S(r), where 6 is the Dirac delta function. Hence, C”(r+, 5) satisfies (49)
w-+90 = co+ y +
c’(r+,
5) +
(h, - ho)R
w+, r - rll)
Y
Similarly a solution can be constructed interval, say vi < g < vi+19 C(r+, <) = Co + y
WI
for any
C’(r+, t) +
+ ~ (hj - hj-,)R C’(r+, 5 - ?j)
V
j=l
(45)
Cn(r+, 0) = 0
(50)
$
(51)
‘$(R+,t)=
(0, g) = 0
-$3(C)
(52)
Equation (48) is the one derived by KATZ but there is a good reason for using equation (47) in making computations. Both C’(r+, t) and C”(r+, 5) are most easily computed in series form, cl@+, <) being given by equation (36) and C”(r+, 5) by
Passing to the limit as each of the intervals (vi - vi-r) approaches zero, one obtains C(r+, 5) = Co -I- y
+
(53)
C'(r+, <) +
s
_R ‘d&d C’(r+, s’ - $dq v
o
4
(46)
Setting r+ = R+ and once again allowing h to be determined by the concentration at the wall, the following integral equation is obtained C(R+, 0 = Co +
+
s
ht-C(R+,
0)lR
V
R ’ dhCW+,~91 v 0
dv
C’@+, 5) +
C’(R+, < - rl)dy
(47)
Integrating once by parts, one obtains
W+, t> = Co + ’ o’ Rv
h(q)C”(r+, 5 - q)dq
(48)
s
944
Since the values of s,’ and RI, must be determined numerically, it is not easy to obtain useful accuracy for more than the first six or seven eigenvalues and eigenfunctions. Hence, the series must be truncated after the sixth or seventh term and this seriously limits the smallest value of 5 for which an accurate value of C’ or C” can be computed. Comparing equations (36) and (53) one sees that the series for C’ converges more rapidly than the series for C”. Hence, a smaller interval can’be used in the numerical integration of equation (47) than of equation (48) and better accuracy should be obtained. To check the accuracy of the method we computed the concentration at the wall for several of the linear systems described previously. The integral in equation (47) was evaluated by subdividing the interval from 0 to 5 into a number of equal sub-
Turbulent flow of gas through a circular tube with chemical reaction at the wall
Finally the solution can be rewritten in a form that is convenient for machine computation.
intervals and assuming that dh/dq was constant over each subinterval. One obtains
C(R’,
gj) = Co + (Sy’ + UiAg)~“’ + +
+ gD,exp II=1
2E25 -< ( 1
2At2 me [S'," + (2j-
l)aj]
+
X
+ f
1
[E,(T,S(ii,’ +
aj)
+_Py’TJ (56)
n=2
+
in which to start the computation S!l) = 0 SC,!/= 0
p(l) n
=
h,R D
V
n
R’(R+) n
4(l) = 4/ + D1 -
z
(54)
in which
E,=jl
(Ji= -R h[C(R+, 531- h[C(R+vri- 111 V (
Remembering that ci = iA and simplifying the previous equation one obtains C(R+,lj)=C,+y
&(R+)+D,-2
+ f!$
(
K
1
(2j
Scii,,=
S'i',-"T
+
Ss"
=
Ss'-"
pW
=
p(I-l)T n
n
"
a._ J
+Acfa._
J
l)aj_
1
1 1
n
RJR+) +
* Re D,R:(R+) n=2
x exp -
x
+
Sy) = Sy- ‘) +
2&25
5 D, exp - s n=2
+c
I
Re 2&,2D,R;(R+)
Thereafter,
I
A<
- T,)
&2
x
n
2&j - i)A<
Re
[I -exp(-%)]I
X
(55)
Usually an iterative procedure is required to determine aj. The accuracy of the second computational scheme was checked by repeating the computations for the two first-order reactions with a Reynolds number of 7003 and a Schmidt number of 2.0. Table 2b contains the eigenvalues and Fourier coefficients which were used in these computations. For the slow reaction, kR/d = 10, the concentration at the wall computed using the integral equation was consistently OGO2 larger than the concentration
945
E. H. WISSLERand R. S. SCHECHTER
Table 2. Eigenvalues and Fourier coe$icients for arbitrary reaction
(a)
Re = 7,003 NO.’
(b)
En
1
0-O 16.45 27.23 37.34 4733 57.22
Re = 7,003
1 2 3 4 5 6
En 0.0 16GO 26.17 35.43 44.44 53.48
Re = 14,017
D?&
1
0.0
2 3 4 5 6
21.79 35.94 49.44 62.74 75.80
0.9907 0.0159 -0.0123 0.0108 - 0.0096 0.0089
sn
105 104 lo4 10s lo3 103
1GOOO -04610 0.3372 -0.2998 0.2643 -0.2274
R’n(R+)
4.212 4.990 1.629 8.426 4984 3.195
x x x x x x
105 104 lo4 lo3 lo3 103
1GOOO -0*4812 0.3733 -0.3503 03068 - 0.2504
SC = 0.7
Dn
R+ = 430
x x x x x x
N’n
0.9823 0.0317 - 0.0280 0.0275 - 0.0254 0.0224
en
No.
4.212 5.247 1.826 9.936 6290 4.424
R’n(R+)
SC = 2.0
R+ = 430
Re = 14,017
N’n
0.9823 0*0305 -0.0256 0.0245 - 0.0230 0.0211
R+ = 240.6
No.
(d)
SC = 0.7 DTI
2 3 4 5 6
No.
(c)
R+ = 2406
N’n 1.507 1.791 5.826 2.907 1.751 1.222
SC = 2.0
x x x x x x
R’n(R+) 10‘3 105 lo4 lo4 104 104
-R’n(R+)
N’n
D!Z
1WOO -04472 03078 --O-2579 0.2302 -0.2165
1 2 :
0.0 21.45 48.07 35.17
0.9902 0.0174 -0.0148 0.0146
1.507 x 106 1.721 x lo5 2,490 5.275 x lo4
1GOO -0.4567 - 02824 0.323 1
5 6
60.58 72.68
- 0.0143 0.0145
1.432 x lo4 9.532 x 10”
0.2637 -@2518
computed using the series solution. Hence, the difference between the two solutions as far down stream as 100 diameters was less than 0.5 per cent. For the fast reaction, kR/D = 100, the integral equation yielded a concentration at the wall which was consistently 1 per cent larger than the concentration computed using the series solution. Computations were also made for a second order reaction, but there was no way to check the accuracy of these computations. Since a relatively short length is required for the concentration profile to develop near the inlet of
the reactor, it is reasonable to expect that a less detailed computation might yield acceptable values for the mean concentration and the concentration at the wall. In the particularly simple case of a firstorder reaction, a material balance for the reactant can be written in the form, nR2u g
= -2xRkC,,
in which C and C, are the mean concentration and the concentration at the wall, respectively. A second equation is obtained by observing that the
946
Turbulent flow of gas through a circular tube with chemical reaction at the wall
rate at which the reaction proceeds is equal to the rate at which the reactant is transferred to the wall; h’(C’ - C,) = kc,,,,
(57)
in which h’ is the mass-transfer coefficient. Solving these equations one obtains
rate of reaction is more sensitive to the rate constant in the turbulent system. Although the eddy diffusivity needs to be better defined, the chemical engineer who is interested in solid-gas reactions should find the system analysed in this paper worthy of his consideration. NOTATION
and
c c C” C
Using Fig. 3 in DEISSLER’S Note [15] to evaluate h’, excellent agreement is obtained between the concentrations computed using the two models. . This is illustrated in Figs. 2 and 3 where values computed using the simple model are presented as circles. The mean concentrations generally agree over the entire length of the reactor, but the wall concentrations differ markedly in the inlet region. It should be pointed out that the mass-transfer coefficients presented by DEI~~LER were computed on the basis of equal eddy viscosities and diffusivities. Hence, the simple model agrees best with the solid lines. CONCLUDING REMARKS
The tubular reactor having catalytic or reactant material at the wall and turbulent flow offers several advantages over the corresponding system with laminar flow. The transfer of a moderate amount of heat from the wall to the fluid does not affect a turbulent stream nearly as much as a laminar stream. Since the diffiisional resistance is smaller in a turbulent stream than in a laminar stream, the
GO CO G D D* E l k N, R’ R’n R+ r+ Re SC u lY I;: 8,” En Y ” P 70 +?I
Concentration of the reactant Function defined by equation (32-35) Function detined by equation (49) Mean concentration Concentration at the wall Inlet concentration Fourier coefficients Molecular diffusivity Fourier coefficients Eddy diffusivity Rate of reaction at the wall Mass-transfer coefficient Reaction rate constant Norm of 4. Radial co-ordinate Inside radius of the tube Eigenfunctions of equation (40) Dimensionless inside radius Dimensionless radial c.o-ordinate Reynolds number Schmidt number Velocity of fluid Mean velocity of fluid Dimensionless velocity of fluid Dimensionless distance from the wall Longitudial co-ordinate Eigenvalues of equation (22) Eigenvalues of equation (40) Dimensionless total diffusivity Kinematic viscosity Density of the fluid Shear stress at the tube wall Eigenfunctions of equation (22) Particular solution of equation (32) Dimensionless longitudinal co-ordinate
REFERENCE BARON T., MANNINGW. R. and JOHNSTONE H. F., Chem. Engng. Progr. 1952 48 125. BUTLERR. M. and PLEWESA. C., Chem. Engng. Progr. Symp. Ser. 1954 50 121. JOHNSTONE H. F., HOUVOURA~ E. T. and SCHOWALTER W. R., Industr. Engng. Chem. 1954 46 702. SAITERFIELD C. N., RESNICK H. and WENTW~RTH R. L., Chem. Engng. Progr. 1954 SO460. HOELXHERH. E., Chem. Engng. Progr. Symp. Ser. 1954 50 45. CHAMBRE: P. L., Appl. Sci. Res. 1956 A6 98. KATZ S., Chem. Engng. Sci. 1959 10 202. WEGERE. and HOELSCWER H. E., Amer. Inst. Chem. Engrs. J. 1957 3 153. WESTKAEMPER L. E. and Wr-rrr~R. R.. Amer. Inst. Chem. Enars. J. 1957 3 69. SCHWARZW. H. and HOEUCHERH. E., Amer. Inst. Chem. &grs. J. 1956 2 101. LONGWELL P. A., Amer. Inst. Chem. Engrs. J. 1957 3 353. DEI~~LER R. G., NACA Tech. Note 2138 1950.
947
E. H. WISSLER and R. S. SCHECH~ [13] [14] [15] [16]
[17] [18]
ALBRECHT P. H. and CHURCHILLS. W., Amer. Inst. Chem. Engrs. J. 1960 6 268. CQRCORAN W. H. and SAGEB. H., Amer. Inst. Chem. Engrs. J. 1956 2 251. DE~SLER R. G., NACA Tech. Note 3145 1954. SLEICHER C. A., Trans. Amer. Sot. Mech. Engrs. 1958 80 693. WOERTZ B. B. and SHERWOODT. K., Zndustr. Engng. Chem. 1939 31 1034. SPARROWE. M., HALLMANT. M. and SIEGELR., Appl. Sci. Res. 1957 A7 37.
948
Printed in Great Britain.
Turbulent flow of gas through a circular tube with chemical reaction at the wall E. H. WISSLERand R. S. SCHECHTER Departmentof ChemicalEngineering, The University of Texas,Austin12, Texas ,
(Received 12 November 1961)
Abstra&--A scheme is developed for computing concentration profiles when an arbitrary reaction occurs at the wall of a tubular reactor. Solutions are constructed using complete sets of orthogonal functions. For reactions which are not first order, it is necessary to solve an integral equation for the concentration of reactant at the wall. A convenient numerical scheme is developed for doing this. Eigenvalues and Fourier coefficients are presented together with computed concentration profiles for several typical cases.
IN ORDERto study a chemical reaction occurring at a solid-gas interface one must be able to evaluate the resistance to mass transfer near the interface. Tubular reactors having a catalytic or reactant material at the wall appear to offer unique advantages over such systems as packed beds. The surface area is well defined; most of the heat of resiction can be supplied or removed through the external surface of the reactor; and, at least in principle, the resistance due to diffusion can be computed. The tubular reactor with laminar flow has been used by several investigators [l-4]. Diffusional resistance is often evaluated by assuming that plug flow exists and the reaction at the wall is first order and irreversible. Then the concentration at any point in the reactor can be expressed in terms of a Fourier Bessel series. BARON [l] has considered first-order reversible reactions at the wall, and HOELSCHER[5] presented an approximate evaluation of the effect of a parabolic velocity profile. CHAMBRB[6] and KATZ [7] have derived integral equations which can be used to compute the concentration when the rate of reaction is an arbitrary function of the concentration at the wall. WEGER and HOELXHER [8] have sounded a word of warning, pointing out that highly reactive surfaces are generally rough and that it is nearly impossible to remove all of the heat of reaction through the external surface. Both of these effects disturb the laminar velocity profile and facilitate the transfer of material to the wall.
Reactors with turbulent flow have also been used. SATTERFIELD er al. [4] based their analysis of the decomposition of hydrogen peroxide and water on a macroscopic model using the j-factors for heat and mass transfer. WESTKAEMPER and WHITE [9] studied the evaporation of carbon tetrachloride into air in a rectangular channel. Although they were forced to make many assumptions in their analysis, they were able to calculate successfully the rates of evaporation for Reynolds numbers between 7000 and 15,000. SCHSVARZand HOELSCHER[lo] found an inlet length of six diameters for a Reynolds number of 25,000 in a wetted-wall column. The computational problem presented by the tubular reactor with turbulent flow is somewhat more complex than the corresponding problem for laminar flow, especially if laminar plug flow is used instead of a parabolic velocity profile. LONGWELL [l l] has shown how the graphical Schmidt method can be used to solve the diffusion equations for turbulent flow. The purpose of this paper is to present some analytic solutions for both first order and arbitrary reactions occurring at the wall. ANALYTICALSTATEMENTOF THE PROBLEM
The co-ordinates and geometry are shown in Fig. 1. Fluid flows from left to right through the circular tube with a fully developed turbulent velocity profile. For z < 0 no reaction occurs and the fluid has a uniform composition. For 0 < z an arbitrary irreversible reaction occurs at the wall and
937
E. H. WISSLER and R. S. SCHECHTER
CO .? -0
FIG. 1. Geometry and co-ordinate system.
,=D+E
the concentration becomes a function of r and z. The concentration is to be determined subject to the following restrictions : (1) A steady state has been attained. (2) Diffusion in the z-direction is negligible in comparison with the convective transport of mass. (3) The radial transport of mass can be adequately described using an eddy diffusivity. (4) The physical properties p (density), ,u (viscosity), D (molecular diffusivity) and k (reactionrate constant) are independent of position. Conservation of mass for the reactant is expressed by the equation
UacSr a; aZ
[
r(D + E) g
1
0) = CO,
(2)
the inlet concetitration which is constant,
E (0, z) = 0
(3)
and -
,
v
in which z. is the shear stress at the wall and v is the kinematic viscosity. Making these substitutions in equations (l)-(4), we obtain ut
ac
R+ a
(10)
z=r+arf
C(r+, 0) = Co
(11)
(12)
-I@+)
aW+, 5) ar+
h
(13)
=Jo
(1) To complete the statement of the problem the
in which U(r) is the velocity, D is the molecular diffusivity, E(r) is the eddy diffusivity and C(r, z) is the concentration of the reactant. This equation must be solved subject to the following boundary conditions : W,
v
[(D+E)z],=,= h
where h is the rate of reaction. Since most of the equations for E and U are given in terms of u+ and y+, it is convenient to make the following changes of variables :
functions ut(rt) and E(r+) must be given, and i.t is in this regard that the problem is least well defined. For Reynolds numbers of 20,000 or less there seems to be rather general agreement that U+ expressed as a function of yf is only mildly dependent on the Reynolds number and the dependence is significant only near the centre of the pipe. However, we wanted to compute compositions for the lowest possible Reynolds numbers so that readily measurable conversions could be obtained in tubes of reasonable length. The data obtained by DEISSLER [12] indicate that the generalized velocity distribution for smooth tubes can be used for Reynolds numbers at least as low as 8000. We went a little further than that and made our computations at Reynolds numbers of 7000 and 14,000. The velocity profile is computed from du+ -= dy+
1 1 + (O*124)2u+y+[1 - exp(-0*1242u+y+)]
o
426
u+(y+ =O)=O
U+
938
(14) (15)
26
(16)
Turbulent flow of gas through a circular tube with chemical reaction at the wall
which are the equations presented by DEISSLER[121. C(r+, 5) = Co Cn&(r+)e-28n2C’Re (21) The choice of a function to represent E is more difficult; and, indeed, there is no sound basis for making a choice between the several functions which in which the /3.‘s and &(r+)‘s are the eigenvalues have been proposed [13-l 71. The principal point of and eigenfunctions of the Sturm-Liouville system contention concerns the ratio of the eddy diffusivity or thermal conductivity to the eddy viscosity. This (22) ratio has been found to be about 1.4 near the wall by several investigators [13, 14, 16, 171 who have dM-9 differentiated measured velocity, concentration and (23) dr+= temperature profiles, but large uncertainties are introduced by the inherent inaccuracy of differen_y(R+) ddb(R+) tiation. On the other hand, good agreement has -=&h(R+) (24) dr+ been obtained between computed and measured Nusselt numbers in both the thermal entry and fully developed regions by several authors [15, 181 who From equation (17), y = D/v at r+ = R+, and assumed that the eddy conductivity equals the eddy equation (24) can be written in the form viscosity. We have based most of our computations on the assumption that the ratio is unity, but we (25) have checked the sensitivity of the assumption by repeating some of the computations with a ratio of Since the &‘s form a complete set of orthogonal 1.4. Following SPARROWet al. [ 181 we have used functions, a reasonably well-behaved function f(r +) can be expanded into a series of 4,,‘s. In this proby = ; + O-124’u+y+[l - exp(-0*1242u+y+)] lem the C,‘s are chosen so that
“El
o
0 < y+ < 26
(17)
C(r+,0) =
5 CA(r+)
(26)
?I=1
.
Hence, R+
26 < y+ < G
(18) c,=
R+
2
u+r+&(r+)dr+
co
s R+”
,
(27)
u+r+[&,(r+)]2dr+
(19)
s0
or y = Mean of the values computed from equations (17) and (18) at y+ = 26.
(20)
In the preceding equations SC = v/D is the Schmidt number.
(28)
in which N. is the norm of 4,. The values of /?., &(r+), C,, and N. have been determined numerically using an iterative procedure and the Runge-Kutta method on the I.B.M. 650 DETERMINA~ONOF THE CONCENTRATION PROFILES Computer. The constants are listed in Table 1, and In evaluating the effect of the eddy diffusivity on the 4”‘s are available on I.B.M. cards. Using these values one can compute the conthe rate of reaction it is convenient to study the special system in which the reaction is first order and centration of the reactant at any point in the irreversible. For this case h = kC(R, z) in which k reactor. However, in analysing data from a tubular is the reaction-rate constant. Then the solution of reactor, one is usually concerned only with the mean concentration, the set of equations (l&20) has the form 939
E. H. WISSLERand R. S. SCHECHTER
Table 1. Eigenvalues and Fourier coeficients for jirst-order reaction R+ = 240.6
(a) Re = 7003
(b)
No.
Bn
1 2 3 4 5 6
3.983 17.34 28.11 38.33 48.36 58.17
Re=
SC = 0.7
3.276 4.719 1.690 9,361 6.030 4.368
x x x x x x
lo5
0.4873
104
- 0.2675
lo4 103 lo3 lo3
0.2083 -0.1980 0.1888 -0.1765
kR/D = 100
c?I
N,
1 i
5,579 29.18 18.31
1.2490 -0.4535 0.3976
2601 x 105 4.172 1.529 x lo4
4 5 6
39.68 49.99 60.03
-0.3832 0.3818 - 0.3745
8.506 x lo3 5.499 x 103 4.132 x lo3
R+ = 240.6
Re = 7003
1 2 3 4 5 6
(e)
1.1288 - 0.2270 0.1879 -0.1735 0.1616 -0.1442
Bn
No.
(d)
kR/D = 10
-A@+)
R+ = 240.6
7003
No.
(c)
SC = 0.7
2.585 16.42 26.66 36.05 45.13 54.11
No.
Bn
1 2 3 4 5 6
4.076 17.07 27,43 37.14 46.50 55.57
3.740 4.695 1.535 7.980 4.806 3.154
SC = 2
c?z
G
4.369 22.56 36.65 50.19 63.55 76.67
1.0891 -0.1567 0.1272 -0.1171 0.1113 -0.1061
105 lo4 104 lo3 lo3 103
-h@+) 0.6294 - 0.3228 0.2545 - 0.2486 0.2328 -0.2018
x x x x x x
MR+) lo5 lo4 lo4 lo3 lo3 103
0.1430 -0.0793 0.0627 - 0.0626 0.0634 - 0.0623
kR/D = 10 __-N?Z 1.265 1.667 5.559 2.802 1.695 1.191
940
0.0840 - 0.0527 oa430 - 0.0426 ow35 - 0.0462
kR/D = 100
3.167 4.266 1.396 7.189 4.369 3.008
SC = 0.7
Bn
x x x x’ x x
NfZ
1.1452 -0.2686 0.2518 -0.2663 0.2837 -0.2834
R+ = 430
Re = 14,017
1 2 3 4 5 6
1.0613 -0.1075 0.0985 -0.1013 0.1011 - 0.0930
R+ = 240.6
kR/D = 10 N,
C*
Bn
Re = 7003
No.
SC = 2
VW+)
x x x x x x
dn(R+) 106 lo5 104 lo4 lo4 104
0.6110 -0.3089 0.2223 -0.1919 0.1762 -0.1720
Turbulent flow of gas through a circular tube with chemical reaction at the wall
Table 1 (continued) (f)
R+ = 430
Re = 14,017 No.
(9)
6.866 23.86 37.97 51.67 65.23 78.67
1 2 3 4 5 6
1 2 3 4 5 6
2.778 21.79 35.50 48.47 61.07 73.25
1a0378 - 0.0679 0.0548 - 0.0595 0.0624 -0.0635
1
w =-gq s
5.012 22.58 36.33 49.48 62.34 74.88
R
1.398 1.661 5.130 2.424 1.392 9.322
SC=2
(29)
It is easy to show that (30)
The variation of concentration with position is illustrated in Figs. 2-5, in which the solid lines refer to an eddy viscosity to eddy diffusivity ratio of 1-Owhile the crosses refer to a ratio of 0.7. Figs. 2 and 3 show, in order of decreasing composition, the centre-line concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration. The circles pertain to the simple macroscopic model described later. In Figs. 4 and 5 the values of 41 are plotted in the vicinity of the wall. In the region where the concentration proI
0.1318 - 0.0776 0.0596 -0.0534 0.0506 -0.0517
x x x x x x
ddR+) 106 105 104 104 lo4 108
0.7430 -0.3551 0.2544 - 0.2242 0.2122 -0.2091
kR/D = 100
1.185 1.526 4.790 2.262 1.289 8.636
0
Co n$1C~N,e-2un2FIRe
105 105 104 104 104 104
kR/D = 10
1.1228 - 0.2248 0.2026 -0.2051 0.2174 -0.2321
2xrUC(r, z)dr.
x x x x x x
MR+)
N,
c,
R+ = 430
9.879 1468 5.081 2.593 1.573 1.107
SC=2
Bn
Re = 14,017
kR/D = 100
N,
1.2186 -04001 0.3505 -0.3325 0.3258 -0.3255
Rf = 430
Re = 14,017 No.
(h)
cn
Bn
1 2 3 4 5 6
sc = 0.7
x x x x x x
106 105 104 104 lo4 103
0.2217 -0.1161 0.0850 -0.0753 0.0720 - 0.0744
files are fully developed, the function 41(~‘) may be interpreted as the concentration at y+ divided by the concentration at the centre of the reactor. As one would expect the discrepancy between the two sets of computations increases as the Schmidt number and kR/D increase. It does not seem to be strongly dependent on the Reynolds number. Although the difference between the mean concentrations is rather striking on semi-logarithmetic paper, it should be observed that the differences in the percentage of reactant converted is only 6-10 per cent. If one were to base one’s computations on an eddy-viscosity to eddy diffusivity ratio of 0.85, the computed conversions should be in error by no more than 3-5 per cent. Although additional experimental work is needed in this area, the information which is currently available can be used without introducing excessive errors.
941
E. I-L
WISSLER
and R. S. S~HECHTER
1.0 8.: 07 06 05 CO4 C*
03
Ai OS 07 06 05 c c
O4 03
0
FIG. 2. Variation of concentration with axial position. The centreline concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration in order of decreasing concentration.
20
40
I,D
60
SO
100
FIG. 3. Variation of concentration with axial position. The centre-line concentration, the mean concentration, the concentration at y+ = 26 and the wall concentration in order of decreasing concentration.
07 06
06 05 4, 04
03 02
UPPER
CURVES
kR/D
01
= IO
0
Y+
FIG. 4.
Fully developed concentration profiles plotted as ratio to centre-line concentration.
FIG. 5.
942
Fully-developed concentration profiles plotted as ratio to centre-line concentration.
Turbulent flow of gas through a circular tube with chemical reaction at the wall
$. (095)= 0
ARBITRARY REACTION RATE
If h depends in an arbitrary manner on the concentration at the wall, say in accordance with a power law or an absorption model, a solution can be obtained in the form of a non-linear integral equation of the Volterra type. This was discussed some time ago by CHAMEW[6] and more recently by KATZ [7] who presented the general theory for a tubular reactor in which the flow is laminar. The same procedure has also been used in the analysis of heat transfer under conditions of variable thermal flux at the wall [18]. Before presenting the integral equation, it is worthwhile to consider briefly a system in which the reaction varies as shown in Fig. 6. To the left of q1 the reaction proceeds at a constant rate h,, and the solution of equations 10-20 has the form
qr+,5) = c, + y
c’(r+, s’)
-$(R+,c)=
(35)
C’(r+, 5) can be expressed as the sum of two functions. The fist is a particular solution which satisfies the boundary condition at the wall, but does not vanish at the inlet. The second function is determined by a variables separable set of equations having zero gradient at the wall and being equal to the negative of the Grst function at the inlet. Hence, the sum has the proper gradient at the wall and vanishes at the inlet when
C’(r+,5) = 2
t +$f(r+)
+
+ 2 D,e-(2en2r/Re)R~(r+) II=1
(31)
in which C’(T+, r) satisfies the following set of equations :
-$
(34)
(36)
in which 4r(r+) and RA(r+) satisfy d
C’(r+, 0) = 0
(37)
dr+
(32) (33)
n
$(O)=O
(38)
$(R+) = -;$
(39)
(40)
(41)
3
(R+) = 0
(42)
The Dn’s are the coefficients resulting from the expansion of -I#I/ into an infinite series of Ris. R+ u ’ r ’ cjfRAdr+
D,=FIG. 6.
Stepwise
approximation reaction rate.
to
s ;+ s
943
(43) ufr+(R32dr+
arbitrary wall 0
E. H. W~%JLER and R. S. SCHECHTER in which
Since
c R+
(r+, CV+, 5) = XI’ z
ReRf
r+u+dr+
= 7,
the boundary condition at the wall is satisfied for any choice of 4/(O). In the interval q1 4 r < q2, the concentration can be computed by simply adding another term to the previously obtained solution. Hence,
5).
It can be shown that Cn(r+, t) is the change in concentration due to a reaction rate which varies as S(r), where 6 is the Dirac delta function. Hence, C”(r+, 5) satisfies (49)
w-+90 = co+ y +
c’(r+,
5) +
(h, - ho)R
w+, r - rll)
Y
Similarly a solution can be constructed interval, say vi < g < vi+19 C(r+, <) = Co + y
WI
for any
C’(r+, t) +
+ ~ (hj - hj-,)R C’(r+, 5 - ?j)
V
j=l
(45)
Cn(r+, 0) = 0
(50)
$
(51)
‘$(R+,t)=
(0, g) = 0
-$3(C)
(52)
Equation (48) is the one derived by KATZ but there is a good reason for using equation (47) in making computations. Both C’(r+, t) and C”(r+, 5) are most easily computed in series form, cl@+, <) being given by equation (36) and C”(r+, 5) by
Passing to the limit as each of the intervals (vi - vi-r) approaches zero, one obtains C(r+, 5) = Co -I- y
+
(53)
C'(r+, <) +
s
_R ‘d&d C’(r+, s’ - $dq v
o
4
(46)
Setting r+ = R+ and once again allowing h to be determined by the concentration at the wall, the following integral equation is obtained C(R+, 0 = Co +
+
s
ht-C(R+,
0)lR
V
R ’ dhCW+,~91 v 0
dv
C’@+, 5) +
C’(R+, < - rl)dy
(47)
Integrating once by parts, one obtains
W+, t> = Co + ’ o’ Rv
h(q)C”(r+, 5 - q)dq
(48)
s
944
Since the values of s,’ and RI, must be determined numerically, it is not easy to obtain useful accuracy for more than the first six or seven eigenvalues and eigenfunctions. Hence, the series must be truncated after the sixth or seventh term and this seriously limits the smallest value of 5 for which an accurate value of C’ or C” can be computed. Comparing equations (36) and (53) one sees that the series for C’ converges more rapidly than the series for C”. Hence, a smaller interval can’be used in the numerical integration of equation (47) than of equation (48) and better accuracy should be obtained. To check the accuracy of the method we computed the concentration at the wall for several of the linear systems described previously. The integral in equation (47) was evaluated by subdividing the interval from 0 to 5 into a number of equal sub-
Turbulent flow of gas through a circular tube with chemical reaction at the wall
Finally the solution can be rewritten in a form that is convenient for machine computation.
intervals and assuming that dh/dq was constant over each subinterval. One obtains
C(R’,
gj) = Co + (Sy’ + UiAg)~“’ + +
+ gD,exp II=1
2E25 -< ( 1
2At2 me [S'," + (2j-
l)aj]
+
X
+ f
1
[E,(T,S(ii,’ +
aj)
+_Py’TJ (56)
n=2
+
in which to start the computation S!l) = 0 SC,!/= 0
p(l) n
=
h,R D
V
n
R’(R+) n
4(l) = 4/ + D1 -
z
(54)
in which
E,=jl
(Ji= -R h[C(R+, 531- h[C(R+vri- 111 V (
Remembering that ci = iA and simplifying the previous equation one obtains C(R+,lj)=C,+y
&(R+)+D,-2
+ f!$
(
K
1
(2j
Scii,,=
S'i',-"T
+
Ss"
=
Ss'-"
pW
=
p(I-l)T n
n
"
a._ J
+Acfa._
J
l)aj_
1
1 1
n
RJR+) +
* Re D,R:(R+) n=2
x exp -
x
+
Sy) = Sy- ‘) +
2&25
5 D, exp - s n=2
+c
I
Re 2&,2D,R;(R+)
Thereafter,
I
A<
- T,)
&2
x
n
2&j - i)A<
Re
[I -exp(-%)]I
X
(55)
Usually an iterative procedure is required to determine aj. The accuracy of the second computational scheme was checked by repeating the computations for the two first-order reactions with a Reynolds number of 7003 and a Schmidt number of 2.0. Table 2b contains the eigenvalues and Fourier coefficients which were used in these computations. For the slow reaction, kR/d = 10, the concentration at the wall computed using the integral equation was consistently OGO2 larger than the concentration
945
E. H. WISSLERand R. S. SCHECHTER
Table 2. Eigenvalues and Fourier coe$icients for arbitrary reaction
(a)
Re = 7,003 NO.’
(b)
En
1
0-O 16.45 27.23 37.34 4733 57.22
Re = 7,003
1 2 3 4 5 6
En 0.0 16GO 26.17 35.43 44.44 53.48
Re = 14,017
D?&
1
0.0
2 3 4 5 6
21.79 35.94 49.44 62.74 75.80
0.9907 0.0159 -0.0123 0.0108 - 0.0096 0.0089
sn
105 104 lo4 10s lo3 103
1GOOO -04610 0.3372 -0.2998 0.2643 -0.2274
R’n(R+)
4.212 4.990 1.629 8.426 4984 3.195
x x x x x x
105 104 lo4 lo3 lo3 103
1GOOO -0*4812 0.3733 -0.3503 03068 - 0.2504
SC = 0.7
Dn
R+ = 430
x x x x x x
N’n
0.9823 0.0317 - 0.0280 0.0275 - 0.0254 0.0224
en
No.
4.212 5.247 1.826 9.936 6290 4.424
R’n(R+)
SC = 2.0
R+ = 430
Re = 14,017
N’n
0.9823 0*0305 -0.0256 0.0245 - 0.0230 0.0211
R+ = 240.6
No.
(d)
SC = 0.7 DTI
2 3 4 5 6
No.
(c)
R+ = 2406
N’n 1.507 1.791 5.826 2.907 1.751 1.222
SC = 2.0
x x x x x x
R’n(R+) 10‘3 105 lo4 lo4 104 104
-R’n(R+)
N’n
D!Z
1WOO -04472 03078 --O-2579 0.2302 -0.2165
1 2 :
0.0 21.45 48.07 35.17
0.9902 0.0174 -0.0148 0.0146
1.507 x 106 1.721 x lo5 2,490 5.275 x lo4
1GOO -0.4567 - 02824 0.323 1
5 6
60.58 72.68
- 0.0143 0.0145
1.432 x lo4 9.532 x 10”
0.2637 -@2518
computed using the series solution. Hence, the difference between the two solutions as far down stream as 100 diameters was less than 0.5 per cent. For the fast reaction, kR/D = 100, the integral equation yielded a concentration at the wall which was consistently 1 per cent larger than the concentration computed using the series solution. Computations were also made for a second order reaction, but there was no way to check the accuracy of these computations. Since a relatively short length is required for the concentration profile to develop near the inlet of
the reactor, it is reasonable to expect that a less detailed computation might yield acceptable values for the mean concentration and the concentration at the wall. In the particularly simple case of a firstorder reaction, a material balance for the reactant can be written in the form, nR2u g
= -2xRkC,,
in which C and C, are the mean concentration and the concentration at the wall, respectively. A second equation is obtained by observing that the
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Turbulent flow of gas through a circular tube with chemical reaction at the wall
rate at which the reaction proceeds is equal to the rate at which the reactant is transferred to the wall; h’(C’ - C,) = kc,,,,
(57)
in which h’ is the mass-transfer coefficient. Solving these equations one obtains
rate of reaction is more sensitive to the rate constant in the turbulent system. Although the eddy diffusivity needs to be better defined, the chemical engineer who is interested in solid-gas reactions should find the system analysed in this paper worthy of his consideration. NOTATION
and
c c C” C
Using Fig. 3 in DEISSLER’S Note [15] to evaluate h’, excellent agreement is obtained between the concentrations computed using the two models. . This is illustrated in Figs. 2 and 3 where values computed using the simple model are presented as circles. The mean concentrations generally agree over the entire length of the reactor, but the wall concentrations differ markedly in the inlet region. It should be pointed out that the mass-transfer coefficients presented by DEI~~LER were computed on the basis of equal eddy viscosities and diffusivities. Hence, the simple model agrees best with the solid lines. CONCLUDING REMARKS
The tubular reactor having catalytic or reactant material at the wall and turbulent flow offers several advantages over the corresponding system with laminar flow. The transfer of a moderate amount of heat from the wall to the fluid does not affect a turbulent stream nearly as much as a laminar stream. Since the diffiisional resistance is smaller in a turbulent stream than in a laminar stream, the
GO CO G D D* E l k N, R’ R’n R+ r+ Re SC u lY I;: 8,” En Y ” P 70 +?I
Concentration of the reactant Function defined by equation (32-35) Function detined by equation (49) Mean concentration Concentration at the wall Inlet concentration Fourier coefficients Molecular diffusivity Fourier coefficients Eddy diffusivity Rate of reaction at the wall Mass-transfer coefficient Reaction rate constant Norm of 4. Radial co-ordinate Inside radius of the tube Eigenfunctions of equation (40) Dimensionless inside radius Dimensionless radial c.o-ordinate Reynolds number Schmidt number Velocity of fluid Mean velocity of fluid Dimensionless velocity of fluid Dimensionless distance from the wall Longitudial co-ordinate Eigenvalues of equation (22) Eigenvalues of equation (40) Dimensionless total diffusivity Kinematic viscosity Density of the fluid Shear stress at the tube wall Eigenfunctions of equation (22) Particular solution of equation (32) Dimensionless longitudinal co-ordinate
REFERENCE BARON T., MANNINGW. R. and JOHNSTONE H. F., Chem. Engng. Progr. 1952 48 125. BUTLERR. M. and PLEWESA. C., Chem. Engng. Progr. Symp. Ser. 1954 50 121. JOHNSTONE H. F., HOUVOURA~ E. T. and SCHOWALTER W. R., Industr. Engng. Chem. 1954 46 702. SAITERFIELD C. N., RESNICK H. and WENTW~RTH R. L., Chem. Engng. Progr. 1954 SO460. HOELXHERH. E., Chem. Engng. Progr. Symp. Ser. 1954 50 45. CHAMBRE: P. L., Appl. Sci. Res. 1956 A6 98. KATZ S., Chem. Engng. Sci. 1959 10 202. WEGERE. and HOELSCWER H. E., Amer. Inst. Chem. Engrs. J. 1957 3 153. WESTKAEMPER L. E. and Wr-rrr~R. R.. Amer. Inst. Chem. Enars. J. 1957 3 69. SCHWARZW. H. and HOEUCHERH. E., Amer. Inst. Chem. &grs. J. 1956 2 101. LONGWELL P. A., Amer. Inst. Chem. Engrs. J. 1957 3 353. DEI~~LER R. G., NACA Tech. Note 2138 1950.
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E. H. WISSLER and R. S. SCHECH~ [13] [14] [15] [16]
[17] [18]
ALBRECHT P. H. and CHURCHILLS. W., Amer. Inst. Chem. Engrs. J. 1960 6 268. CQRCORAN W. H. and SAGEB. H., Amer. Inst. Chem. Engrs. J. 1956 2 251. DE~SLER R. G., NACA Tech. Note 3145 1954. SLEICHER C. A., Trans. Amer. Sot. Mech. Engrs. 1958 80 693. WOERTZ B. B. and SHERWOODT. K., Zndustr. Engng. Chem. 1939 31 1034. SPARROWE. M., HALLMANT. M. and SIEGELR., Appl. Sci. Res. 1957 A7 37.
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