Conversion in Turbulent Tubular Reactors with Unmixed Feed D. Phaneswara Rao and louis 1. Edwards' Department of Chemical Engineering, liniuersity of Idaho, ~lfoscow, Idaho 83843
The Monte Carlo coalescence model (MCM) of Rao and Dunn i s used to obtain root-mean square concentration fluctuations as a function of distance for a tracer input to a tubular flow reactor. These results are used to calculate the conversions for fl = 1 .O, 1.4, and 3.0, for a second-order very rapid chemical reaction from Toor's relations for conversion in terms of root-mean square concentration fluctuations and fl = CBo/nCAo. The calculated conversions agree well with both the experimental data of Vassilatos and Toor and the conversionsfound b y the direct use of the MCM. Also it i s shown that MCM agrees with the recent generalized approach of Toor for fl = 1 and for various values of y = kncAoZ,/U. Use of MCM for obtaining CX as a function of y and fl i s suggested.
T o o r (1962) classifies chemical reactions taking place in turbulent tubular reactors as very rapid, rapid, and slow, depending on whether the conversion is diffusion controlled, both diffusion and reaction rate controlled, or reaction rate controlled, respectively. For the case of a diffusion-controlled second-order reaction, A nB + products, taking place between the two components A aiid B fed separately into the reactor through many inlet tubes, Toor developed the following expressions for fraction conversion, F , in terms of a stoichiometric factor, p, aiid a scalar mixing factor, ro/rOo, which is the ratio of root-mean square concentration fluctuations without chemical reaction a t a particular axial position, 2, to the root-mean square concentration fluctuations a t the feed, 2 = 0. For feed in stoichiometric proportions ( i e . , P = 1)
+
p=1--
-0 1
roo
and for p # 1
where
z CY
ierfc
1
I _
=
P
~-
and
Equations 1 and 2 were verified experimentally by Vassilatos and Toor (1965) using an indirect method. Their method assumed that eq 1 was valid for a single experimental run and F values were calculated from conversion data. With the resulting information 011 the variation of r with axial position, they verified other experimental runs for various p values. Keeler, et al. (1965), verified eq 1 and 2 by actual measurements of the root-mean square concentration fluctuations in a turbulent flow reactor. To whom correspondence should be sent. 398
Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
Kattaii and hdler (1967) successfully modelled the reactor of Vassilatos aiid Toor with the use of a Monte Carlo simulation technique similar to that of Spielman and Levenspiel (1965) for the dispersed phase of a two-phase liquid-liquid system. Kattaii and ddler essentially replaced the tubular flow reactor by a batch reactor consisting of a finite number of fluid elements which are regarded as similar to drops in a two-phase system in the sense t'hat they coalesce and redisperse a t a particular rate. The rate of coalescence and the subsequent redispersion was determined by matching experimental data of Vassilatos and Toor (1965) for one particular run and then used to verify other runs for the cases of very rapid, rapid, and slow chemical reactions. The same technique was also used independently by Harris and Srivastava (1968) to verify the experimental data of Vassilatos and Toor. Rao and Dunn (1970) extended the work of Spielman and Levenspiel (1965) to simulate a tubular flow reactor with axial dispersion. The reactor is divided into a finit,e number of wafers of fluid, each L I D thick, where L is the length of the reactor and D is the number of wafers. Each wafer is assumed to be represented by a disk containing 200-300 droplets situated on a finite number of radial lines. Axial dispersion is obtained by allowing coalescence of droplets between adjacent disks. The coalescence parameter, I , is defined as I = u ~ / l Vwhere , u is the coalescence rate, 7 is the residence time, and Ar is the number of drops in the reactor. For a homogeneous system, t'he droplet's are taken to represent fluid elements or eddies. The value of the parameter I for the reactor of Vassilatos and Toor was determined by matching the experimental run for p = 1 and the same value of I was used to check other runs made for both very rapid and rapid reactions. For I = 10.8, the agreement, was found to be good and the results were found to be insensitive to the number of disks. The details of this work have already been reported elsewhere by Rao and D u m (1970). l l a o and Toor (1970) successful11 verified t'he experimental data of Vassilatos and Toor by using a diffusion model based oii the simultaneous interdiffusion and reaction between alternate slabs of reactants. Toor (1969) also showed that for stoichiometric feed, i . e . , p = I , the decay law for concentration fluctuations is the same for very slow and very rapid reactions. This led him to postulate that the decay law is
0.7
o
Experimental data o f
0
MCM (140.8)
X
Equations (1) and (2)
0.6
tI %
0.4'
'
'
'
'
'
'
'
'
'
'
'
0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 09 1.0 1.1
'
I
I
1.2 1.3 1.4
Z Figure 3. Conversion vs. axial position for very rapid reactions ( p = 1 A )
.lot .05
,
,
,
z/zm\j
0 0.1 0.20.30.4 03 0.6 OX 0.8 0.9 1.0 Fraction unconverted vs. Z/Z, ( p = 1 )
Figure 1.
independent of reaction rate constant' and he obtained conversion profiles for various values of y = kn(?~oZ,/C where C is the mean velocity, is the mean feed concentration of A, n is the sboichiometric coefficient, k is the reaction rate constant, and 2, is the mixing distance required for the relative root-mean square concentration fluctuations to fall to a small value. Toor used a value of 2 in. for Z,, corresponding to ro/roo = 0.02. The purpose of this communication is twofold: (1) to compare the hlonte Carlo coalescence model (MChl) of Rao and Dunn (1970) with Toor's recent theoretical development (1969) ; ( 2 ) to show how root-mean square concentration fluctuations generated using the hfCM can be used in eq 1 and 2 to predict conversion for very rapid reactions. Rao aiid Dum1 (1970) h a r e already shown t h a t the X C M agrees with the experimental data of Vassilatos and Toor (1965) for an interaction parameter value, I , equal to 10.8. Wit'h this same value of I , the hfChI was used to calculate coilcentration profiles for comparison with Toor's more recent' work (1969). Figure 1 shows excellent agreement for
all y values. However, it should be noted that X C h I shows a finite amount of axial dispersion, however small, for any given value of I , whereas the theory of Toor assumes plug flow. The RICbI results were found to be insensitive to the number of disks used in the model for 1 = 10.8. Root-mean square concentration fluctuations obtained from the M C X simulation of a tracer experiment ( L e . , no chemical reaction) were used in eq 1 and 2 to calculate conversions for very rapid reactions. Figure 2 shows the results for p = 1. The value of Too was calculated from the folloiviiig formula derived by Toor (1969) I-
where W B and w.4 are mass flow rates of B and A, respectively. As p B 'V P A for dilute systems and the number of tubes feeding A and B are equal
roo= CAo Also shown in this figure are the conversion data calculated directly from the MCM and the experimental data of Vassilatos and Toor. Figures 3 and 4 show the same results for p = 1.4 and p = 3.0, respectively. The curves obtained from the IICM tracer simulation results and eq 1 and 2 agree with the experimental data rather well. It appears the Monte Carlo coalescence model may be useful in extending the work of Toor (1969) by simulating the effect of y aiid p on the concentration fluctuation term, ~
CACB.
0.8
0.7
LL
0.8 "
F
!
0.5
0
0.5
1.0
1.5
2.0
z Figure 2. Conversion v5. axial position for very rapid reactions (0= 1 )
0
nExperimental d data Toorof(1965)
0
MCM (I=lO,8)
x Equations (I)and (2)
06
-3.0
0
0.1
0.3
0.5
0.7
Z Figure 4. Conversion vs. axial position for very rapid reactions ( p = 3.0) Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
399
Acknowledgment
T
The authors are grateful for a National Science Foundation grant (GK-10043) which supported this work. Nomenclature
C = concentration c = concentration fluctuation
D = number of disks
F - fraction of feed converted I = U T / N = coalescence parameter, average number of
k = L = n = N = u = U=
w =
z=
coalescence redispersions per fluid element taking place in one residence time reaction rate constant length of the reactor stoichiometric coefficient number of drops or fluid elements coalescence rate mean velocity in 2 direction mass flow rate axial distance
GREEKLETTERS = defined by eq 2a r = root-mean square concentration fluctuation j3 = stoichiometric factor
=
residence time
y = reaction parameter (or first Damkohler number)
SUBSCRIPTS A = species A B = species B 0 = inlet to reactor m = mixing distance SUPERSCRIPTS
- = bulk or time average O
=
absence of reaction
literature Cited
Harris, I. J., Srivastava, R. D., Can. J . Chem. Eng. 46, 66 (1968). Kattan, A., Adler, R. J., A.I.Ch.E. J . 13, 580 (1967). Keeler, R. N., Petersen, E. E., Prausnitz, J. M., A.I.Ch.E. J . 11,221 (1965).
Mao, K. W., Toor, H. L., A.Z.Ch.E. J. 16,49 (1970). Rao, D. P., Dunn, I. J., Chem. Eng. Sci. 25,1275 (1970). Spielman, L. A,, Levenspiel, O., Chem. Eng. Sci. 20,247 (1965). Toor, H. L., A.I.Ch.E. J . 8 , 7 0 (1962). Toor, H. L., IND. ENG.CHEM.,FUNDAM. 8,655 (1969). Vassilatos, G., Toor, H. L., A.I.Ch.E. J . 11,666 (1965).
CY
RECEIVED for review July 13, 1970 ACCEPTEDMarch 15, 1971
Reflection and Transmission of Thermal Waves at the Boundaries of a Fixed-Bed Regenerator G.Alan Turner Department of Chemical Engineering, University of Waterloo, Waterloo, Ont., Canada
The complication of end effects at the inlet and exit boundaries of a regenerator suffering a sinusoidal perturbation of temperature may be removed b y using two different bed lengths. This work examines the conditions under which the procedure i s valid and shows how to find the magnitude of errors resulting from the assumption of an infinite bed length. In so doing, it points out analogies with waves on electrical transmission lines and derives expressions for reflection coefficients at boundaries and for the propagation constants and impedances for both positive and negative waves. Unlike most traveling-wave systems the latter are not symmetrical in any flow system.
T h e system considered has three sections, with boundaries dividing the central packed regenerator section from the fore and after sections in which are located temperature measuring probes. This is a n arrangement often used in the experimental measurement of parameters. I n this measurement it is necessary that a descriptive differential equation be set up, the solution to which will depend upon the boundary conditions. With a certain amount of care it is possible to set u p an apparatus that permits simple conditions normal to the flow direction, but in the longitudinal direction the situation is different. Whatever kind of perturbation is used the measurements in any realistic experimental procedure must include the effect that this perturbation suffers in a bed of finite length. This means two things: there are both inlet and exit boundary conditions to be satisfied, and the detecting devices, with their own response characteristics, 400 Ind. Eng, Chem. Fundam., Vol. 10, No. 3, 1971
must be located before and after the active region. Measurements in the past have been made with the simplifying assumptions that the detector is infinitesimal in size and of perfect response; that the signal as measured b y the upstream detector is truly the signal a t the entrance to the bed and the signal a t the end of the bed is the same as that measured by the downstream detector; and that the bed behaves as though the exit were a t infinity (but with the downstream detector situated a t a finite distance from the entrance). The attraction of the latter assumption is that the mathematics is much simplified. One of the advantages of using a sinusoidal perturbation is that by making measurements on beds of two different lengths it is possible to overcome the difficulties mentioned in the last paragraph, and this has been done by McHenry and Wilhelm (1957), Turner (1959), Liles and Geankoplis (1960),
