Ind. Eng. Chem. Res. 1988,27, 238-242
238
Scale-up of Two-Impinging-Streams(TIS) Reactors Abraham Tamir* and Beni Shalmon Department of Chemical Engineering, Ben Gurion University of the Negev, Beer Sheua 84 105, Israel Scale-up rules for the hydrodynamics, the mean residence time, and holdup of the particles and the residence time distribution curve were developed and tested successfully for two-impinging-streams reactors. The two-impinging-streams(TIS) reactor has proven to be a very efficient defice for performing heat- and masstransfer operations (Tamir et al., 1984; Luzzatto et al., 1984; Tamir and Luzzatto, 1985a,b; Kitron et al., 1987; Tamir and Hershkovitz, 1985; Tamir and Sobhi, 1985; Tamir, 1986; Tamir and Grinholtz, 1987). In the method of impinging streams, two coaxial, oppositely directed fluid-particle suspension streams are made to impinge. In the impingement zone, particles penetrate into the opposing stream due to their inertia, decelerate, reverse their direction, and possibly penetrate into the original stream. Having undergone several dumped oscillations, particles drop out of the system. The flow pattern thus developed enhances heat and mass transfer due to the (a) significant increase of the relative velocity between the phases (b) inertia forces and increased particle residence times caused by particle oscillations, and (c) turbulent mixing caused by jet impingement. The applications of the TIS reactors in industry require the knowledge of appropriate scale-up rules. Therefore, the objective of this paper is to develop and verify scale-up rules, based on a theoretical model and a nondimensional analysis. The scale-up analysis is performed with respect to the following aspects of the TIS reactors; hydrodynamics, mean residence time, holdup, and determination of the residence time distribution (RTD)curve of the particles.
Hydrodynamics Figure 1 shows a scheme of the TIS reactor used in the present experiments (Shalmon, 1985). The reactor consists of a cyclone chamber with an annular space 3. Two suspension streams are fed tangentially through two symmetrically positioned channels 2 and impinge in region 3 where heat and mass processes are performed very efficiently. Particles leave the reactor at position 5 , and air leaves the reactor at position 4 of the annular gas outlet pipe. The variables associated with the hydrodynamical behavior of the reactor and their values for the small reactor are shown in Figure 1. Due to the complexity of the hydrodynamics, recourse was taken to dimensional analysis so that the critical governing dimensional groups could be identified. It is assumed that the pressure drop on the reactor is given by
AP = (Pa,Pp,Ya,WptWa,Ua~,dpPl-D4,Hl-HS,Ll) (1) Applying the Buckingham pi theorem yields E u p = f(Re,~J+,dp/Dl,Di / D , J f j / D l ,Ll/D,) i = 1-3 and j = 1-5 (2)
where M
=
w,/w,
(3)
For p = 0, namely, in the absence of particles flow, Eu, is obtained. On the basis of E u , and Eu,, the following quantity is defined:
q = Eu,/Eua =
AP,/AP,=
f(Re,~,Fr,dp/D1,Di/Dlfij/D,,LI
/ D l ) (4)
The conditions for a complete hydrodynamical similitude between a large-scale reactor (subscript 1) and a laboratory-scale (subscript s) are e=c
E = 1
and p = l
(5)
where c is a constant. The bar designates scaled quantities, namely, the ratio between the magnitude in the large-scale reactor, divided by the respective magnitude of the small-scale reactor. I t should be emphasized that the relevant physical properties of the suspension are maintained constant due to the condition p = 1. Neglecting the effect of gravity because of the high inertia of the particles (i.e., Fr = 0) and considering eq 2 and 5 yields
Eu,=Eu,=l
(6)
Then, from eq 4, it follows that ii = 7 / % = 1
(7)
The significance of eq 7 is that a correlation of the kind q = f ( p ) , determined from small-scale reactor measure-
ments, could be used to predict the pressure drop with particles flow in a large-scale reactor. The above results were verified as follows. Pressure drop measurements between positions 2 and 4 of Figure 1,for yellow seeds with d , = 1.9 X m and pp = 1158 kg/m3, were performed on a small reactor and on a geometrically similar reactor with = 2. Results of these measurements were presented in terms of the dimensionless groups appearing in eq 2. Figure 2 demonstrates the correlation E u = f ( R e ) with the parameter W,, the particle mass flow rate which is proportional to p. As seen, each reactor is characterized by a different set of curves. When the same data of Figure 2 were presented in terms of the variables appearing in eq 4, it was found that, within experimental accuracy, all the curves were transformed into a single curve of 7 = q ( p ) , as shown in Figure 3, which verifies also eq 7 . It should be noted that in the experiments, all length quantities were multiplied by 2. The single exception was particle size d,, which remained unchanged because larger particles were not available. On the other hand, eq 2 and 4 indicate that d,/D, should also change. The fact that the experiments confirm eq 7 without changing d, may be explained by the fact that the volume fraction of the particles in the reactor is on the order of 0.01. In other words, for constant p , which maintains the physical properties of the mixture, the effect of d, on the curve in Figure 3 is not detectable within experimental accuracy. The application of Figure 3 for the determination of PP for a large-scale reactor is performed as follows. For a prescribed value of p, which fixes also the physical properties of the suspension, the value q = AP,j AP, is obtained for the large reactor. As can be seen from Figure 2 , the values of Eu, (and also of APa) are different for the two
0SS8-5S85/SS/262~-0238$01.50~0 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 239
a0-
p-LI'O.22
1 I
U I L +2
SMALL REACTOR LARGE REACTOR L=2
, I
/
Figure 3. Euler numbers ratio versus mass flow rates ratio for small and large two-impinging-streams reactors.
AIR AND PARTICLES
+
-It2I
Figure 1. Small TIS reactor and the main parameters. __
__7
D t
01 S c h e m a t i c modelof vessel5 ond f l o w
I-
X
S M A L L REACTOR
0
LARGE
4
Re=$
Figure 4. Model of a single-stage TIS reactor.
REACTOR
L-2
l0C
blvesseis locotion tn the r e a c t o r
Du
Figure 2. Euler number versus Reynolds number of small and large two-impinging-streams reactors.
reactors, with identical Reynolds numbers corresponding to p = W , = 0. Thus, in order to determine AP, for the large reactor, it is necessary to measure only the pressure drop (AP,)in the absence of particles flow and then to calculate AP, = qAP,. It is possible to bypass the measurement of AP, on the large-scale reactor for large Re, assuming that for large values of Reynolds numbers %, x l/t.This correlation is supported by the data presented in Figure 2 where Eu becomes practically constant for large Re. Another possibility is to use a value of AP,from a small-scale reactor. Such an approach has some advantages because the value of AP, which is obtained by this procedure will be the maximal and therefore a safe value of AP, for design purposes. Finally, it should be noted that the effect of the physical properties of the particles on the pressure drop can be evaluated with the aid of the procedure described
elsewhere (Tamir et al., 1985).
Basis for Scale-up Rules The procedure for scaling up the mean residence time 7 , the holdup V, and the RTD of the particles is based on a model suggested by Luzzatto et al. (1984) which was extended by Kitron (1984). They visualized the twoimpinging-streams reactor to consist of equal volume vessels, CSTR and plug flow types where particles enter by feed or recycle flows. This is shown schematically Figure 4. The reactor was assumed to be composed of three perfectly mixed vessels, 1,2, and 3, and a plug flow reactor, 4. Particles enter vessels 1and 2 at flow rates of W J 2 ; from there, at a flow rate r / 2 they pass through vessel 3 and are recycled into the entrance vessels 1and 2. Ultimately, the particles leave the reactor through a tubular reactor. Transition of a particle between two vessels occurs instantaneously, while the minimal time it resides in a vessel it enters is At. The latter quantity can be looked upon also as the minimal time for a particle to cross a mean distance Lt while crossing a mixed vessel, namely, At = L,/V, (8) V, is the particle mean velocity within a mixed vessel, which is only slightly affected by air or particle flow rates as the result of velocities cancelling effects caused by the impinging-streams configuration. L, is slightly increased with increased particle flow rate, due to lengthening of the particle path caused by inter-particle collisions. The dimensionless form of the parameter At is defined by A0 = A t / ? (9) where 7 is the mean residence time of the particles in the two-impinging-streams reactor. An additional parameter
240
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
incorporated in the flow vessel model is the recycle flow rate r (see Figure 4) with its dimensionlessform R defined by R = r/Wp (10)
' -
4
3
W , is the overall flow rate of the particles through the reador. Kitron et al. (1987) developed an expression which relates R to an operating parameter by describing it as the ratio between the drag force acting on a particle in a certain recycling zone in the reactor to the flux of the momentum entering particles in this zone. The final expression obtained for R reads R
a:
( p p / p a ) 1 1 6 d p 5 / 6 / p= 1 /K 3 /p113
100
f(R,W
m
(13)
Ciati i=l
Assuming that the Ati% are constant values, as in the experiments of Luzzatto et al. (1984), gives m
7
m
= C ( C i / C ) t i = Cse,,mAt i=l
i=l
(14)
When eq 9 is applied, the above equation reads m
1 = A0
C mse,,(R,AO)
(15)
m=l
Thus, for a given value of R and by use of Markov chain equations to calculate s ~for, a~specific model, A0 is obtained. For the model depicted in Figure 4, the R-A8 relation is given in Figure 5 , which may be used also for scale-up purposes.
Mean Residence Time and Holdup of Particles On the basis of eq 8, assuming hydrodynamic and geometrical similitude between the large- and the small-scale reactors, as well as L a: L,, where L is the geometrical length in the TIS reactor, it may be obtained that =1 (16) A t l / A t , = L1/L, or In terms of scaled up quantities, eq 9 reads At =le7
- 10-4 8 7
6
6-
R
5-
R
-5
4'
4
3 t
3
2 .
2
-
-,
IO 9-
103
-8 -
8 7 6 54 -
4
3-
-
2-
- 2
I
01
(12)
CCitiAti 1=1
-
7 -
(11)
sen-which depends on a specific flow vessel model-is the probability of a particle entering the two-impingingstreams reactor to reach the exit after m transitions. For the model in Figure 4, for example, m = 1-4. R and A0 are not independent, and the relating equation may be obtained as follows. The mean residence time is defined by
7 =
2
8-
It should be noted that the above proportionality, namely, the inverse dependence on N, was confirmed experimentally by Kitron et al. (1987). An alternative form of eq 11 is confirmed here, as discussed later. As shown in the following, the above parameters A8 and R (or At and r ) are also useful for predicting particle RTD's in scaled up reactors. The mathematical formulation of the aforementioned model is based on discrete Markov chains detailed elsewhere (Luzzatto et al., 1984; Kitron, 1984). The application of Markov chains allows us to determine the true RTD curve from pulse experiments and the key parameters R and A0 incorporated in a mathematical expression of the form Se3m =
-
(17)
Figure 5. R versus A8 for the model in Figure 4.
and, hence, eq 16 and 17 yield the following general scale-up criterion for 7: 7 =L/ae (18) It should be noted that the assumptions which helped in deriving the scale-up rule are justified by the good agreement with the experimental results. If A8 remains constant, namely, = 1, eq 18 is reduced to a simplified scale-up rule for the particle mean residence time, namely, (19) Equation 19 fit the operating conditions prevailing in the present research. I t is interesting to note that according to eq 19, the particle (noncontinuous phase) mean residence time is proportional to the first power of L, while it is well-known that the mean residence of the continuous phase (air, in the present experiments) is proportional to the reactor volume, namely, to L3 or 7 = L3. Since the holdup (V (kg)) for the particles in the reactor is proportional to 7 (V = 7 it follows that a simple scaled-up rule for the holdup would read
w,),
p=i;
(20)
A general procedure for scale-up of 7 (or V), when A8 1, is as follows. For a desired value of L , it is possible to calculate R = R 1 / R , from eq 24. From the curve of R versus A8, such as in Figure 4, it is possible to obtain the values of AO1 and A8, and hence 2,which results in 'i according to eq 18. It is noteworthy that the assumption of A0 = 1,which allows us to bypass the above-mentioned procedure and rcquires the R-A0 curve, would yield for a certain value of L, the minimal value of 7 to be calculated , from eq 18. In other words, for a certain value of T ~ the value of 71 would be also minimal because the maximal value that A8 can reach (for the model in Figure 4) according to Figure 5 is 0.8. #
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 241
16-
PARTICLES
AIR FLOW-RATE im311i
0015 0015 0010 0 OlO
\9
I
reoctor
wi 02
!
/
.
AX
/e
j PO
24
e :in
011 4
22
6
I
I
1
I
I
8
IO
12
14
16
I8
Figure 7. RTD curves for small and large reactors.
( A i r flow r o t e ) x 1 0 3 ( m 3 / s )
Figure 6. Mean residence time of the particles versus air flow rate for large and small TIS reactors.
Figure 6 shows the variation of T against the air flow rate for the two similar reactors with = 2. As can be seen, the value of t = r l / r , varied from 1.91 to 2.47 with a mean value of 2.2. A similar trend was also observed in RTD measurements. Therefore, it was concluded that eq 19 was verified within experimental accuracy, as well as eq 20.
Residence Time Distribution (RTD) of the Particles The resicence time distribution function of the particles, E@(eq 21), is very important for reactor design. The Markov chains models which have been employed wccessfully in evaluation of the results of RTD experiments (Luzzatto et al., 1984; Kitron, 1984) could not be applied for the present reactors because of the extremely short residence times of the particles which varied from 0.2 to 0.7 s as compared to r = 1to 5 s in previous investigations. Therefore, the parameter At9 = A t / ? (Kitron, 1984) was out of the range of applicability of the Markov chains model. The short residence times were cuased by the use of very small reactors in order to perform experiments with small and large reactors, which required overlappingin the operating parameters. The RTD tests detailed elsewhere (Shalmon, 1985) comprised pulse experiments using the device detailed by Luzzatto et al. (1984). The results obtained in these experiments for the two reactors are presented in Figure 7 (the data points correspond to the large reactor). The major conclusion drawn from the spread of the data points, which correspond to a rather wide range of flow rates of particles and air for a certain reactor, is that within experimental accuracy, they belong to the same population. This fact is important because the Eo curve can be safely determined from a single pulse experiment. The measured values of Eo can also be correlated by the following formula corresponding to tanksin-series model (Levenspiel, 1972, p 291):
The value N , which appears in eq 21, is the number of equal-sized ideal stirred tanks. This parameter has to be fit from the experimental data. The fitting process is quite simple and was performed as follows. It can be shown that for the accepted model (Levenspiel, 1972, p 291),
Table I. D a t a To Verify Equation 21 % flow rate of air from max value
small reactor 7,s (0/7).
large reactor 7,s (u/T)~
0.32 0.29 0.33 0.29
0.67 0.67 0.59 0.56
1o3w,, kds 8
40 % 40 % 60% 60 %
23 8 23
0.313 0.379 0.394 0.379
0.269 0.239 0.305 0.285
017
=
(0/7)1/
(u/T).
0.859 0.631 0.774 0.755
where u2 and T are the variance and the mean residence time, respectively. u2 can be calculated from m
m
C(ti - 7)'CiAti
u2
N
is1
2CiAti
C(ti- T ) ~ C ~
- i=l
(23)
ci ill
i=l
where r is given by eq 13. The validity of the terms appearing in the righth-and side of eq 23 was verified in the experiments reported by Luzzatto et al. (1984) where the values of A ti were constant. Thus, pulse experiments yield u2 and r , and hence, N can be computed from eq 22. Therefore, the RTD curve Eo is completely defined. In the present investigation, several experiments were carried out for each reactor for various operating conditions given in Figure 7. Each experiment yielded a value of uz0and hence N. The dotted and the full curves in Figure 7 were obtained, accepting the mean values for N , where N = 14 for the large reactor and N = 8 for the small reactor. The shortcoming of the tanks-in-series model, successfully used for correlating the RTD data, is its inability to provide a basis for scale-up. However, as will be shown in the following, the RTD data obtained in the present experiments support the validity of eq 11. From the detailed expression for R, developed by Kitron et al. (1987), the scale-up equation for R reads
R
(24) The data in Table I, obtained for the two reactors with t = 2, indicate clearly that 417 is decreasing when the reactor dimensions increases. Assuming that this can be expressed by U / T = l/.P (25) it has been stated elsewhere (Kitron, 1984) that t-516
(r/7 l/(AO,/AO,)@ = 1/he@ R
1 / 3 7
(26) (27)
242 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 t = scaled quantity defined by
Thus, from expressions 25-27, it follows that
R
0:
l/P
(28)
where a , @, y,and n are greater than zero. The value of the exponent n which appears in expression 28 can be estimated from the data presented in Table I, and it was found that n = 0.22-0.66. Thus, it may be concluded that the validity of expression 24 is supported by the experimental results, in spite of the difference in the value of the exponents. The importance of expression 24 is that it predicts the trend of the variation R; vis., it decreases when the geometrical dimensions of the reactor are increased.
Nomenclature c = constant C = total number of tracer particles C, = number of tracer particles collected between times t, and t , 4- At, d = diameter of the particle L&D4 = diameters defined in Figure 1 Ed = residence time distribution function defined in eq 21 Eu = Euler number defined by AP/paU2 = scaled quantity defined by Eul/Eu, Fr = Froude number g = gravitational acceleratiog H1-H5 = heights defined in Figure 1 4, = mean distance for a particle while crossing a mixed vessel
Eu
L = scaled quantity defined by LI/L, m = number of transitions N = number of mixed vessels r = recycle stream mass flow rate R = dimensionless recycle ratio defined by eq 10 R = scaled quantity defined by RIIR, Re = Reynolds number defined by DIUapa/pa Re = scaled quantity defined by Rel/Re, s ~ = ,probability ~ of a particle to be in the exit of the reactor after m transitions from the entrance U, = air velocity at inlet pipe (see Figure 1) = reactor holdup, kg V = scaled quantity defined by Vl/V, V , = particle mean velocity within a mixed vessel Wa = mass flow rate of the air W , = mass flow rate of the particles Greek Symbols hp = pressure drop on the reactor between points 2 and 4
in Figure 1 At = minimal residence time that a particle resides in a mixed
vessel of the Markov model
Ati = time interval for collecting tracer particles in pulse -experiments At = scaled quantity defined by A t l / A t , A6 = defined by eq 9 A%= scaled quantity defined by AtI1/A%, e = t/T T
q/T,
defined by eq 4 ij = scaled quantity equal to Vl/7, pa, pp = density of air and solid particles, respectively p = defined by eq 3 p = scaled quantity equal to p l / p s pa = viscosity of air o = defined by eq 23 a/. = scaled quantity defined by ( u / T ) l / ( u / T ) S 7 =
= mean residence time of the particles in the reactor, eq 13
Subscripts a = in the presence of the air flow only; of the air
1 = of a large-scale reactor p = of the particles; in the presence of particle flow s = of a small-scale reactor Abbreviations
RTD = residence time distribution TIS = two impinging streams
Literature Cited Kitron, A. “Hydrodynamics, Particles Residence Time Distribution, and Solids Drying in Impinging Streams Multi-Stage Reactors for Gas-Solids Operations”. MSc. Thesis, Department of Chemical Engineering, Ben Gurion University, Beer Sheva, Israel, 1984. Kitron, A.; Buchmann, R.; Luzzatto, K.; Tamir, A. “Drying and Mixing of Solids and Particles Residence Time Distribution in Four Impinging Streams and Multistage Two Impinging Streams Reactors”. Ind. Eng. Chem. Res. 1987, 26, 2454. Levenspiel, 0. Chemical Reaction Engineering, 2nd ed.; Wiley: New York, 1972. Luzzatto, K.; Tamir, A.; Elperin, I. ”A New Two-Impinging-Streams Heterogeneous Reactor”. AIChE J . 1984, 30, 600. Shalmon, B., “Scale-up of Two-ImpingingStreams Reactors”. B.Sc. Thesis, Department of Chemical Engineering, Ben Gurion University, Beer Sheva, Israel, 1985. Tamir, A. “Absorption of Acetone in a Two-Impinging-Streams Absorber”. Chem. Eng. Sci. 1986, 41, 3023. Tamir, A.; Elperin I.; Luzzatto, K. “Drying in a New Two-Impinging-Streams Reactor”. Chem. Eng. Sci. 1984, 39, 139. Tamir, A.; Grinholtz, M. “Performance of a Continuous Solid-Liquid Two-Impinging-Streams (TIS) Reactor: Dissolution of Solids, Hydrodynamics, Mean Residence Time, and Holdup of the Particles”. Ind. Eng. Chem. Res. 1987, 26, 726. Tamir, A.; Hershkovitz, D. “Adsorption of C 0 2 in a New Two-Impinging-Streams Absorber”. Chem. Eng. Sci. 1985, 40, 2149. Tamir, A.; Luzzatto, K.; Sartana, D.; Surin, S. ‘A Correlation Based on the Physical Properties of Solid Particles for the Evaluation of Pressure Drop in the Two-Impinging-Streams Gas-Solid Reactor”. AIChE J. 1985, 31, 1744. Tamir, A.; Luzzatto, K. “Solid-Solid and Gas-Gas Mixing Properties of a New Two-Impinging-Streams Mixer”. AIChE J. 1985a, 31, 781. Tamir, A.; Luzzatto, K. “Mixing of Solids in Impinging-Streams Reactors”. J. Powder Bulk Solids Technol. 1985b, 9, 15. Tamir, A.; Sobhi, S. “A Two-Impinging-Streams Emulsifier”. AIChE J . 1985, 31, 2089.
Received for review July 7, 1986 Revised manuscript received August 18, 1987 Accepted October 8, 1987
