RICHARD W. SCHAFTLEIN
T. W. FRASER RUSSELL
two-phase reactor design TANK-TYPE REACTORS
Gas-liquid tank reactor design via simple mass transfer models and judicious assessment of model parameters 12
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
reactor design and analysis have been Single-phase greatly aided by a categorization of reactors as either tubular plug-flow or well-mixed tank reactors (continuous flow, semibatch, and batch), Experience has shown that these classifications accurately represent physical reality in a number of situations of practical interest. In reactor design problems which do not fall into the tubular or tank categories, analysis using the simple reactor models is necessary since it provides performance limits which are useful in evaluating and checking design procedures for a more complex reactor configuration. Since the “ideal” reactor concept has proved so useful for single-phase reactors it seems reasonable to extend this approach to two-phase reactors and to develop a set of simple model equations which are tractable enough to be tested by experiment and industrial practice. I t is the purpose of this paper to present the design equations for “ideal” gas-liquid tank-type reactors, to outline the difficulties in parameter evaluation with a review of the present state of the art, and to compare experimental results from a semiflow-batch reactor with the “ideal” model predictions. In a second paper of this two-part series, gas-liquid tubular reactor analysis will be considered following the same format as for the tank-type reactor analysis.
REACTOR MODEL EQUATIONS The two-phase tank-type reactors of interest can be classified in three groups as follows : Continuous flow tank reactors (CFTR) Semiflow batch reactors (SFBR) Batch reactors
The liquid-phase behavior in any of the well-stirred tank-type reactors listed above can be represented by the following general component mass balance :
qcoj
- qc5 + KQa‘PVbNVL
(
yj
3
- cj -
-
d dt
75VL = - (VLC,)
(1)
where a’ = ratio of surface area to volume of a single bubble, length-’ v b = volume of a single bubble, length3 N = number of bubbles per unit volume of liquid r g = rate of reaction for componentj VL = volume of liquid phase In the above equation, which is written for the j t h component, convective flow into and out of the reactor, transfer between phases, changes by chemical reaction, and accumulation are accounted for by the respective terms from left to right. The liquid-phase behavior of any of the two-phase tank reactors can be readily obtained from Equation 1 by deleting the appropriate terms. For a steady-state continuous flow tank reactor, Equation 1 is applicable if the time derivative is set equal to zero. For a batch or semiflow batch reactor the terms containing the input and output flow rate, q, are eliminated. T o describe the gas-phase behavior, two different models are proposed. I n the absence of mechanical agitation the bubbles of gas will rise in a swarm through the reactor and their spatial variation over the reactor length must be considered; thus the total mass of j in the bubble, Pvbyj/ RT, is dependent upon both z and t. For a single
VOL. 6 0
NO. 5 M A Y 1 9 6 8
13
bubble the differential change in mass between any two pointa in the liquid (21, tJ and (ZZ, t ~is) equal to the mass of j t r a d d to the liquid, -K&PVb(yj "2, ..7 the bubble is a-funca,+;., . C f l / P ) , and since the mass of j . -.r. :-. kLo(c, CJ (47) a Assuming D to be 10-6 cm*/sec and using kLo from Equation 44 yield:
-
a
>>> 2.5
(10)-4 em
V O L 6 0 NO. 5 M A Y 1 9 6 0
(48) 19
which is a condition easily encountered in practice. Kc is simply H / k L a for this case, and the liquid-phase reaction does not affect the mass transfer coefficientLe., k , = kLO. The “kinetic subregime” occurs when the reaction is very slow and the over-all driving force is used entirely by the reaction. The following criterion applies: (49)
I n this case the term in the reactor model equations representing interphase transfer may be dropped since the driving force is negligibly small. The reader is reminded again that the conditions of Equation 4.5 must also be met. The third subregime is that which is intermediate between the diffusional and kinetic subregimes-Le.,
sumed a t the reaction plane per mole of absorbed component). Astarita (7) further discusses chemical reaction near the gas-liquid interface, and also Sprow (37) has recently discussed reversible reactions and gas-liquid interfaces. I n the instantaneous reaction regime the concentration distribution of the liquid-phase reactant influences the over-all absorption rate and it is often found that the rate-controlling step is the diffusion of the liquid-phase reactant toward the interface. I n such situations, the over-all driving forces (29) used to develop Equation 40 do not apply, since the liquid-phase driving force is not simply (Ce, - CJ. An analysis of the moving boundary problem for the instantaneous reaction regime is discussed by Astarita, who shows that the average absorption rate in terms of liquid-phase parameters is expressed as :
B5 = kL0(Ce5 Astarita (7) discusses this “intermediate subregime” for zero-, first- and nth-order reactions. For a first-order reaction, k , takes the particularly simple form, kL=
-+-
( L o
9-l
The application of Equation 51 will be shown in the section which discusses the experimental work. The second major regime categorized by Astarita is the “fast reaction regime” for which tD
>> t R
(53)
A third major regime is defined by Astarita (7) for systems which are intermediate between the slow and fast reaction regimes. For these cases both the k , and the driving force for mass transfer are affected by the liquid-phase reaction. For certain kinetic expressions a modification which incorporates this driving force effect can be made for the models discussed in the first section of this paper. The reader is referred to Astarita’s text (7) for the detailed development. A fourth regime must also be recognized if the reaction in the liquid phase is very fast. Astarita defines this regime as the “instantaneous reaction regime.” It may be encountered in practice when an acid is absorbed in a strong alkali (H2S in NaOH) or when a base is absorbed in a strong acid (“3 in HzSO4). The condition which must be satisfied is tD
tR
>> __
ace,
(54)
where Cotis the initial bulk concentration of liquid-phase reactant, C,, is the interface concentration of the absorbed component, and a is the stoichiometric coefficient (Le., the number of moles of liquid-phase reactant con20
(55)
provided that the liquid-phase reactant and adsorbed component diffusivities are equal. It should be noted that the driving force represents the total concentration of chemically combined (Coi/a) and physically dissolved (Ce5)absorbing component which would be present in the liquid at saturation. When the rate-controlling step is as described above, the gas-phase resistance to mass transfer may become important, in which case the following equation applies:
(52)
If the system properties are such that this regime is appropriate, and if the liquid-phase reaction is firstorder, then k L may be represented by:
kL = ( D k R ) l l z
+ %)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
If the interface equilibrium concentration, C, nated, Equation 56 becomes
is elimi-
(57) where K , is defined as in Equation 40. T h e problem of evaluating kG for gas bubbles has not been dealt with in the literature and it appears that hardly any work has been done in the area. The penetration theory approach offers one possibility-i.e.,
t , for the gas bubble must depend on the internal gas circulation rate and is presumably less than the corresponding liquid-phase value, which is approximately equal to db/vb for bubbling systems. Astarita also discusses the smooth transition which exists between the fast and instantaneous reaction regimes. E. Liquid-phase mass transfer coefficients. Applying Astarita’s approach to the problem of parameter evaluation for tank-type models yields a means for determining the effect of the liquid-phase reaction on k,, the chemical absorption coefficient. If kLO, the physical absorption coefficient, can be determined, k , can be readily found by the methods summarized above. There has been considerable experimental work done
for bubbling systems and a variety of correlations have been proposed to obtain kLO. Because of the two different gas-phase models proposed (Equations 2 and 3) the correlations for kLo will be discussed for each separate model. For those two-phase tank reactors which operate with no agitation other than bubble action (Cases 1-A-a, b, c and 2-A-a, b, c) one is interested in the data and correlations for bubbling systems in which the gas rises through the liquid in plug flow under the action of gravity. Several methods for estimating kLo in bubbling systems have been presented in the literature: Calderbank and Moo-Young (4), Danckwerts (7), Griffith (17), Higbie (72), Johnson and Akehata ( I S ) , Leonard and Houghton (ZO), Miller (24), So0 (30), and Teller (33). The correlations generally take the following form: kLO
- = al(LuL/vL)a~(vL/D)aa
(59)
UL
where uL and L are characteristic of the liquid velocity and a linear dimension, respectively; al, a2, and a3 are constants. I t is often difficult to assess experimental work in the area because investigators are careless about driving forces and the effect of any liquid-phase reaction. I t may be necessary to use a different model for the experimental situation employed by an investigator and to recalculate k,” if one is interested in a particular study. I n addition to the above references, the basic theory, experimental work, and correlations related to bubbling systems have recently been discussed by Hughmark (73) and Schaftlein (28). There will, no doubt, be more data and correlations proposed for kLo as work is done in the area. However, the recent correlation of Hughmark (73) seems to be one of the most useful. Hughmark’s correlation was developed from semitheoretical equations which have been previously used. The modification was proposed to correlate mass transfer data for single bubbles in liquids, for liquid drops in liquids, and for single spherical surfaces exposed to flowing air or liquid streams. The predictions are claimed to deviate from experimental data by about 15%. The general form of the correlation for liquidphase mass transfer coefficients is presented as:
for single gas bubbles for bubble swarms
a 0.061 0.0187
b -
1.61 1.61
For bubble swarms the velocity in the Reynolds number is represented by the slip velocity between the bubbles and the liquid. For cases where
NRe,Ns,, and
rg)
are
>> 2
the liquid-phase mass transfer coefficients for single
bubbles (SB) and bubble swarms (BS) are related as follows :
Hughmark’s correlation indicates that for a range of systems the mass transfer coefficient for bubble swarms is directly related to the mass transfer coefficient for single bubbles, and that mass transfer coefficients for single bubbles are greater than those for bubble swarms. For those two-phase tank reactors in which the gas phase is dispersed and mixed by mechanical action, the problem of obtaining kLo or kLoa is made difficult because all the single-phase mixing problems are still present and are further complicated by having a twophase system. Various correlations have been developed relating kLo to the power input, the vessel geometry, and the mixer configuration. An entry into the literature and some of the correlations are presented in Perry’s Chemical Engineers’ Handbook (5, sec. 14, p. 37 and sec. 18, pp. 77-81). 2.
Gas-Phase Properties
Two gas-phase models are proposed for each of the two-phase reactor configurations (Equations 2 and 3). Because of the nature of the experimental work for bubbling systems, any discussion of the evaluation of parameters is most conveniently done separately for each gas-phase model. For a two-phase reactor in which the gas rises through the liquid under the action of gravity, knowledge of a’, v b , v b , and N is essential and information can be obtained from the vast amount of literature available on bubble fluid mechanics. For two-phase reactors in which the gas phase is well mixed an understanding of the bubble fluid mechanics is not as helpful; one must then resort to experimental studies of particular reactor systems in which parameters such as the total area for transfer and total gas volume are investigated. For the plug-flow gas reactors (Cases 1-A-a, b, c and 2-A-a, b, c) a knowledge of the bubble geometry as it enters the liquid phase will generally allow one to obtain expressions for the bubble volume v b , the rise velocity O b , the bubble surface area to volume ratio a’, and the number of bubbles per unit volume of liquid N . Many bubbling systems can be considered to have spherical bubbles, thus a knowledge of db immediately allows one to calculate a f and v b . With this information u b and N may be estimated from semiempirical formulations in the literature. If the bubble cannot be assumed to be spherical, a characteristic dimension cafi, as a first approximation, be used in the semiempirical expressions to estimate u b and N . A. Discrete bubble gas phase. Much more work has been done with systems in which there is no interaction between the bubbles than with systems where there is a strong interaction. The “single bubble’’ approach has generated a wealth of information and correlations which can be used to estimate the gas-phase VOL. 6 0
NO.
5
MAY 1968
21
parameters in the model equations, if it can be assumed that the bubbles move through the system with minimum interaction. I t is essential that the initial bubble size dob, be obtained for the system under study. An entry into the literature and some expressions developed for estimating do, can be found in “Perry’s Chemical Engineers’ Handbook” (5, sec. 18, pp. 68-82), in a recent book by So0 (30),and in an excellent review paper by Jackson (74). Three flow regimes of bubble formation from a single orifice have been visually identified-i.e., single-bubble, intermediate, and jet. At low flow rates where a single bubble is emitted from an orifice, a force balance, considering only buoyancy and interfacial tension, yields the following expression for dob:
(30), “Perry’s Chemical Engineers’ Handbook” (5, sec. 18, pp. 71 and 72), and a review paper by Jackson
(74). A particularly useful expression for u b is the TaylorDavies equation discussed by Davidson and Harrison
(8): Ub =
0.711 (gdb)’/2
(66)
where db is the equivalent spherical bubble diameter if the bubbles are not spherical. The correlation was developed for spherical cap bubbles rising in a large volume of liquid. Davies and Taylor ( 9 ) modified the original theory of Rippin (27) with the use of extensive experimental observations. With an estimate of the bubble rise velocity it is possible to obtain the number of bubbles per unit volume of liquid A’, which is a function of the bubbling frequency and bubble residence time as follows:
where D o = orifice diameter u = interfacial tension of gas-liquid film = densities of liquid and gas pL, The bubble frequency can be readily calculated from Equation 64 if the total gas rate is known and if the bubbles are assumed to be spherical.
where Q = volumetric gas flow rate. As the gas rate becomes high enough to cause the system to move into the intermediate flow regime, the bubble size becomes dependent on the gas flow rate. The effect of the gas and liquid properties on initial bubble size has not been fully investigated; in fact, the regime is not really well defined. Only one correlation has been developed for dob (5, sec. 18, p. 70) which is claimed to hold for gases at orifice Reynolds numbers below 2000.
dob
=
0.18 D00.5A‘~e0*33
(65)
I n the jet flow regime the gas stream exists as a jet for about 3 to 4 in. from the orifice, at which point it then disintegrates into smaller bubbles. Less is known about this regime than the other two regimes, and experimental studies seem to be in conflict. At the present time it can be concluded that the initial bubble diameter is not easily estimated unless one has a fairly low gas rate through the orifice. Experimental studies may have to be conducted to evaluate dob,unless the specific system of interest has been investigated experimentally and reported in the literature. A comprehensive listing of literature sources related to bubble motion is presented by Baker and Chao ( Z ) , Peebles and Garber (26),and Yates (35). If the bubble geometry can be determined, or if the bubbles may be assumed spherical, it is possible to estimate the bubble rise velocity ub. There is a great deal of literature available, and discussions and reviews of this work may be found in Davidson and Harrison (8), So0 22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Le., the product of the bubbling frequency and the bubble residence time divided by the liquid volume, Of course the estimation of iV is only as reliable as the values of v o b and u b . Equation 66 or a similar equation may be used to replace u b by a function of Vb. The presence of ub in Equation 67 indicates that N may vary for a unit volume of liquid at different heights in the liquid. The use of some of the correlations discussed above has already been demonstrated in some of the two-phase reactor models developed above. A particular problem may necessitate the use of different empirical or semiempirical expressions for a ’, uo, vb,and N . E. Swarm bubble gas phase. If a situation is encountered in which bubble interaction is suspected or confirmed from preliminary experiments, the problem is more complex, . and less information is available in the literature. The approach described above may be used as a first approximation to the real situation. The following treatment for bubble swarms which rise in plug flow provides some correlations from which the model parameter can be estimated. A recent study by Koide, Hirahara, and Kubata (18) provides correlations for estimating the initial average bubble diameter, the slip velocity, and gas holdup in bubble-swarm systems. Koide et al. conclude that the average bubble diameter, dob, is a function of the linear gas velocity, uG, and the liquid properties. The type of gas distributor was found to be important, and separate correlations are presented for bubbles emerging from perforated plates or porous plates. Koide et al. recommend the following equation and limits for computing the average bubble diameter of bubble swarms generated from perforated plates : 1
dob = 2.94 ( . ! ~ ~ ~ , / No’071(gp/U6) F,~/~) 0.7
< A‘jv,/iV~,’‘~< 64
The average bubble diameter of bubble swarms from porous plates generally ranges from 0.05 to 0.5 cm, and may be estimated by Equation 69 : do, = 1.35 (N~1./N~,"2)00.278(gp/a~) -1'3
(69)
T o obtain an estimate for N and an equation for v b , it will be necessary to determine the gas holdup. Koide et al. proceed in the following manner to estimate the gas holdup. The slip velocity, a,, for bubble swarms is defined by :
The ratio of bubble slip velocity to terminal velocity, u t , was correlated with the liquid properties for two ranges of average bubble diameter. Small bubbles: db < 0.4 cm us - = 0.27
+ 0.73 (1 - h)2*80
vt
db2gp/cT < 2.7, 0.240
< (1 - h) < 1.0
(71)
Large bubbles: d > 0.4 cm
5
= [l
+ 0.0167 (d:gp/~r.)~J~][O.27+ 0.73 (1 - h)2*80]
ut
d:gp/a
< 8.0, 0.250 < (1 - h) < 1.0
(72)
The terminal velocity may be determined experimentally, obtained from the literature (25) or estimated as 26 cm/sec (5, sec. 18, p. 79). The accuracy of the correlations for estimating average bubble diameter and slip velocity is claimed to be =t20Y0. The review paper by Jackson (74) should also be consulted for a general discussion and formulas related to bubble properties. T h e average gas holdup, h, of bubble swarms can now be estimated if negligible bubble coalescence is assumed. The procedure is: (1) determine do, from Equation 68 or 69, (2) determine a value of v t based on dob,and (3) estimate h by trial and error from Equations 70 and 71 or 70 and 72. Koide claims an accuracy for h of =!~307~. Another recent correlation for estimating gas holdup in bubble columns is that of Hughmark (73). Equations for N and v b may now be obtained. From the two-phase flow work of Nicklin (25) for swarm bubbles rising through a liquid, experimental evidence indicates that the bubble rise velocity can be expressed as ub = Q/hA,. For significant gas absorption, Q is a function of z and therefore a more desirable formulation is Q = (Q1/Vob) V,, where V, may vary. The bubble rise velocity may now be expressed as
.
(73) I n the manner of development as above, N may be determined from Equation 67 using Equation 73 for +,. A similar but alternate expression for N is (74)
The parameters for swarm-bubble behavior have been developed in a manner parallel to the discrete bubble treatment above, and may be used in the plug-flow gasphase models; for instance, Equation 73 would replace Equation 15. C. Continuum gas phase. An alternate but less satisfactory approach for the plug-flow gas-phase results from treating the gas as a pseudo-continuum. The discrete bubble parameters are then combined and treated as over-all system properties which may be determined by experiment. The area available for mass transfer may be written as a = afUbN. The coefficient and area for mass transfer may be combined as K,a and evaluated from mass transfer studies. One may also use the total gas volume NVLI'b, instead of attempting to determine individual bubble properties. A gas-phase model incorporating the above changes can be derived as a pseudo-continuum representation of the plug-flow bubbles. The model will be of the same form as Equation 2 with the bubble rise velocity replaced by an average gas velocity. D. Completely mixed gas phase. If the gas phase is mixed and thoroughly agitated, it is not so useful to consider discrete bubble properties as it is to group the parameters to reflect an over-all gas-phase description. The most useful work completed to date, which characterizes the well-mixed gas two-phase models, is that of Calderbank (3, 5, sec. 18, p. 79). Using an extensive collection of experimental data from tanks in which gas was dispersed by an agitator, Calderbank obtained the following correlation for predicting the interfacial area for mass transfer :
where P = agitator shaft power during dispersion V = volume of agitated mass u,, u t = superficial gas and bubble terminal velocities
If v t is not known, a mean value of 26 cm/sec is often assumed. For the well-mixed gas-phase models the product, &a (or k,a, etc.), must be determined. Miller (24) in a recent review presents several methods for predicting the mass transfer coefficient for liquid-phase controlled transfer in agitated vessels. Included in this review is a summary of the extensive work by Calderbank and Moo-Young, who have applied the Gilliland-Sherwood correlation to homogeneous turbulent systems and sieve plate columns. Calderbank (3, 5, sec. 18, p. 80) has also developed a correlation for gas holdup in mechanically agitated vessels. Depending on the system, the gas holdup may be useful for computing the liquid volume VL, and hence 7 , the average liquid residence time, which must be known for the liquid-phase models. The recent work of Gal-Or and Resnick (70) should also be useful for predicting gas residence time in gas-liquid agitated systems. Some further work of Calderbank has recently been discussed in a text edited by Uhl and Gray (34). VOL. 6 0
NO. 5
MAY 1968
23
The correlations of Calderbank have not been tested for high gas rates and as for all gas-liquid contactors, the agitator shaft power, P, during gas dispersion is unknown and may be as low as 30y0 of the no-gaspower.
APPLICATION OF REACTOR MODELS This section is presented to illustrate the use of the model equations for the analysis of a laboratory reactor, to test one of the model formulations, to compare the experimental value of kL with values computed from correlations, and to gain some insight into the utility of the discrete bubble approach. 1. Semiflow laboratory Reactor Model Experimental data were obtained from a laboratory scale s e d l o w batch reactor which was operated isothermally at 90% with an effective reactor volume of 445 cm'; th&reactor was cylindrical with a I-in. inside diameter and length of 32.4 in. The reactor materials were transparent so that bubble behavior, extent of saturation, etc., could be oherved. The two-phase system consisted of a d i s p d pure gas phase (ethylene oxide, B ) and a stationary continuous liquid phase (water, A), where the following pseudo-first-order irreversible reaction occurred in the liquid phase.
1+B+R Monoethylene glycol (R) was the only significant reaction product because of the large mol ratio of A :B. The liquid-phase concentration-time profiles of both ethylene oxide (C,, j = B ) and ethylene glycol (Ck,
k = R) were obtained by using standard analytical techniques (modiied Lubatti's method and periodic acid oxidation, respectively). The pm6h resulted from a suies of separate and successively longer runs all at the same fixed operating conditions. At the end ob each run the liquid phase was withdrawn from the reactor and an average concentration obtained. Thee terminal conditions were used to construct the concentration-time profiles presented in Figures 6 and 7. A summary of the experimental conditions is pnsented in Table I. In the initial period of operation, when the ethylene oxide bubbles are completely d m l v e d a few inches above the inlet orifice, the average concentration determined by emptying the reactor and mixing its contenrs will k lower than the true concentration in the effective part of the reactor. As the ethylene oxide concentration in the liquid increases, the bubbles of ethylene oxidc gar rise the total length of the reactor and the assumption of a well-mixed liquid phase is more realistic, especially over the time interval of the experiment. In a tanktype reactor in which there are bubbles generated at more than one site the assumption of a completely mixed liquid is well justified. The physical system of the SemiAow batch reactor is mmt similar to the two-phase reactor model of Case 2-A-b-i.e., a plug-flow gasphase model and a wellmixed batch liquid-phase model. For the single column of bubbles there is very little bubble interaction and each bubble may be considered to be in plug flow. The liquid phase is stationary, and so can be modeled as a batch phase. Thus, for this case, Equations 35 and 18 to 20 are the liquid- and gas-phase models, m p
1.1 1 .o
0.9
0.8 0.7 0.6
0.5 0.4 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 t, TIME HKS
Figurr 6. Expcrimcntol and moa2 ethylem oxidc conccnIration p o j l t s
24
INDUSTRIAL AND ENGINEERING CHEMISTRY
1, TM I E HKS
Figurc 7. Expm'mcnlal and modal cthylcm glycol conccnlrntian p1ofJar
TABLE I. EXPERIMENTAL CONDITIONS AND PUBLISHED DATA CeB = 0.01838 lb mol/ft3, reference 6 Do = 0.0825 in. dob = 0.4 cm, experimentally observed H = 0.0498 lb mol/ft3-atm, reference 6 kR = 0.789 hr-1, reference 22 L = 2.7 ft P = 1 atm Ql = 40 cm3/min T = 9OoC V , = 445 cma
tively. These general equations must be modified for the particular conditions of this problem-i.e., for the pure gas phase, y j = 1 ; therefore in Equation 40 the gas-phase resistance l/ko, is negligible, and KG = k J H . y j is then the gas concentration in equilibrium with the liquid, or from Henry's law, Cer = y j P / H = P / H . If these modifications are substituted in the general Equations 35 and 18 to 20, the following liquidand gas-phase models result:
F
=
'/2
kL
RT S - - (CeB - CB) P W
The assumptions pertaining to this model are discussed under Cases 1-A-b and 2-A-b. If we assume all the model parameters can be evaluated, the above three equations can be solved to give the gas-phase reactant concentration profile in the liquid phase, CB(t). The product concentration profile, C,(t), can be obtained from the following liquid-phase material balance : CR(t)
=
kR
s,'
cB(t)dt
(79)
The boundary conditions for Equation 76 are t = 0, CB = 0 and t = t, CB = CB; those for Equation 79 are t = 0, C, = 0 and t = t , C, = C., After selecting the two-phase reactor model, the most crucial task is evaluation of the model parameters. I n general, because of the lack of experimental data, it will be necessary to refer to the methods outlined in the previous section. The choice of a particular method should depend on the physical behavior of the system. The following procedure for the semiflow batch reactor investigated should be useful in developing any of the other two-phase models. Of the required parameters, several are fixed either in developing the models or in specifying the process conditions. For instance, S results from the spherical bubble assumption, W from the bubble rise velocity assumption
of Equation 15, and R, T , P, and L from process criteria; CeB and k , must generally be obtained from experimental data. Estimates for V,, and N depend on the initial bubble diameter, do,. The only additional parameter is kL, the chemical absorption coefficient, which may or may not be a function of the physical absorption coefficient, kLo. I t is here that the work of Astarita is so appropriate, for Astarita has clearly defined the effect of chemical reaction on the absorption process-i.e., the chemical absorption coefficient k L is defined for each of the major reaction regimes and subregimes discussed in the previous section. The major reaction regime can be determined by use of Equations 41 and 42. Equation 43 gives the usual magnitude oft,. From Equation 42 with C,, = 0 for an irreversible reaction, r3 = k,Ce, for pseudo-first-order reaction and k R = 2.20 (10-4)"sec-1 from the data of Lichtenstein and Twigg (22) at 90°C, t , = kR-I = 4.55 (lo4) sec
(80)
By comparing the results from Equations 80 and 43, t , >> t,, and the ethylene oxide-water system is in the slow reaction regime (Equation 45), or more specifically one of the subregimes defined by Equations 47, 49, or 50. For the conditions specified above, the subregime criteria reduce to a comparison between the following rate coefficients :
T o determine the inequality relation of Equation 81, it is necessary to estimate a and kLa. These estimates will be based on do, which can be predicted (Equations 63, 65, 68, 69) or determined experimentally. From Equation 63, do, = 0.426 cm, a may - be estimated from Equation 77 using ii' G 6/d,,, V , .rrdOb3/6and N from Equation 67, where u,(dob) is from Equation 66. kLo can be obtained from Hughmark's correlation (Equation 60) for the single-bubble case. The estimated values needed to define the subregime are : kL0 = 8.21 ft/hr,
N = 5940ft-3,
a
3.64ft2/ft3 (82)
If the values from Equation 82 and kR are used in Equation 81, the conclusion is that k,/a is the same order of magnitude as kLO-i.e., 0.218 G 8.21. Thus, the criterion of Equation 50 is satisfied, and the reaction regime is the slow reaction intermediate subregime, where the kinetic and diffusion rates are of the same magnitude, and the chemical absorption coefficient is defined by Equation 51; this is the correct coefficient for the model Equations 76-78. The model for the ethylene oxide-water SFBR is now completely defined by Equations 76-79 and 51, and is in terms of the usual parameters, including the physical absorption coefficient rather than the chemical absorpVOL. 6 0
NO. 5
MAY 1968
25
tion coefficient. For this particular case the implicit form, resulting from the dependence of k , on a in Equation 51, renders the model equations very unwieldy even when all the parameters are known. This situation does not arise when k , is not a function of a. Then, in most of the other regimes proposed by Astarita, the model equations are more tractable. A solution technique for the present model is presented in the following section. 2.
=
0.560 ft/hr,
N
=
5900 ft-3,
a
=
1.395 ft2/ft3 (83)
where a was computed (after the best-fit determination) by averaging values of a from Equations 77 and 78, using k , and from Equation 83 and the experimental range of values for CB. All the computations were executed using a Scientific Data Systems 9300 digital computer with an IBM System/360 Scientific Subroutine Package. Equations 76-78 and 79 were solved using Runge-Kutta integration and numerical quadrature schemes, respectively. The model profiles and experimental data are presented in Figures 6 and 7 with the tabulated data in Table 11. I n Figure 6 the model profile for the ethylene oxide concentration shows a maximum positive deviation of 12.8y0 (at t = 3 hr) and a maximum negative deviation of 6.7% (at t = 11 hr). In Figure 7 the model profile for the ethylene glycol concentration shows a maximum positive deviation of 2.6% (at t = 11 hr) and a 14.2y0 negative deviation at t = 5 hr. In Figure 7 the best agreement is for t > 4 hr; the percentage deviation increases at lower values of t , however the over-all compatibility of the experimental data and model are in good agreement for all t. The reliability of the parameter predictions can be examined by comparing the estimated and best-fit values, as listed in Equations 82 and 83, respectively. For the bubble number density, A’, the agreement is excellent-5940 predicted versus 5900 ft-3 for the bestfit value. The initial bubble diameter do3, estimated from Equation 63, agreed well with the experimentally observed value-0.426 us. 0.4 cm. I n Equation 82 the estimate of a was based on the initial bubble diameter, d03. Any value of a computed from the model should be smaller because of the continuous transfer of mass 26
A.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Experimental Data Ethylene O x ide
Conc. Run No
.
Model Predictions and Experimental Data
T o determine the applicability of the semiflow batch reactor model and to assess the reliability of the parameter estimation methods, the model equations for both liquid-phase and gas-phase concentration profiles (Equations 76-79) were fitted to the experimental data by determining “best-fit values” for k , and N. Such values of k , and N were obtained by trial and error exploration over a range of values estimated from Equations 82 and 51 and for an experimentally observed value of d03-0.4 cm. The following parameter values gave the best agreement between the model and experimental data for the range of CB and C , investigated : k,
TABLE I I
Run Time, hr
(16 m o l / f t 3 ) X 702
1. o 2.0 3.0 4.0 5 .O 6.28 7.5 9.0 11 .o
0.6880 0.8397 0.8597 0,9777 0.9783
... 0.9946 1.043 1.075
Ethylene Glycol Conc. (16 mol/ft3) X 702 0.7798 1.652 2.628 3.401 4,845 5.690 6.797
... 9.922
Model Results, Semiflow Batch Reactor Ethylene Ethylene Oxide Glycol Conc. Conc. Run Run (lb m o l / f t 3 ) T i m e , hr T i m e , hr x 102 B.
0.50 1 .oo 1.50 2.00 3.00 4.00 5 .OO 6.00 7.00 11 .oo
0,4677 0,6949 0.8245 0,9003 0.9697 0,9926 1 .oooo 1 .0023 1 ,0031 1.0035
1.50 3.25 5 .OO 6.75 8.50 10.25 11 .oo
0.8129 2.4305 4.1672 5.9206 7,6763 9.4323 10,1849
TABLE 111. LIQUID-PHASE MASS TRANSFER COEFFICIENTS
Method Penetration theory (72) Boundary layer theory (24) Gilliland-Sherwood (24) Calderbank-Moo-Young (4, 24) Turbulent system Calderbank-Moo-Young (4, 24) Large bubbles Small bubbles Leonard-Houghton ( 7 9 , 20) Griffith ( 7 7 ) Hughmark (73) Single bubbles Rubble swarms
kLo,
ftlhr 5.30 0,0304 1,829 0.204 7.02 2.38 0.184 5.31 8.21 2.54
Richard W . Schaftlein is a Graduate Research Fellow at the University of Delaware. T . W . Fraser Russell is Associate Professor of Chemical Engineering at the University of Delaware. The authcrs acknowledge Jnancial support of the UniversiQ of Delaware Research Foundation. The machine computations were performed by the University of Delaware Computation Center. AUTHORS
Dimensionless Parameters Npr = u2/g& Froude Number N R ~= u d p / p , Reynolds Number NsO = v / D , Schmidt Number N8h = kLodb/D, Sherwood Number Nwe = Su2p/c, Weber Number
from the bubble. These requirements are met by the model results-1.395 us. 3.64 ft2/fta. Solution of the model equations also allows one to evaluate the various methods for predicting the physical absorption coefficient. If the value of kLo from Hughmark’s correlation (Equations GO, 82) and the best-fit model value of a (Equation 83) are used in Astarita’s relation (Equation 51)’ the result is kL = 0.529 ft/hr, which compares favorably with the model prediction of 0.560 ft/hr. Other methods were used to estimate k,O and the results are presented in Table 111. Hughmark’s correlation is the most recent and gives the best agreement with the model. The predictions of Calderbank-MooYoung, penetration theory, and Griffith are reasonable, since ethylene oxide is very soluble in water and is not a typical gas for which the correlations were developed.
Subscripts A = liquid-phase reactant, water B = gas-phase reactant, ethylene oxide b = refers to a single bubble i = liquid-phase reactant j = gas-phase reactant ej = interface value for component j ; chosen as the equilibrium value from Henry’s law ej = equilibrium value for component j associated with reversible reaction; equals zero for irreversible reaction k = product of the chemical reaction o = refers to an initial state R = product of chemical reaction, monoethylene glycol 1 = reactor inlet condition 2 = reactor outlet condition
NOMENCLATURE
Superscripts - = quantity averaged over z o = quantity evaluated in the absence of reaction
= cross-sectional area of reactor = mass transfer area per unit volume of continuous phase, = = = = = = = =
= = = = =
= = = = = = = = = = = =
=
u.
= = = = = = = =
uL, uG = ug
=
v b
= = = = = =
VL vb vt
W y z
=
area/volume ratio of surface area to volume for a gas bubble, length-’ average volumetric mass transfer area component molar concentration, mole/volume molecular diffusivity, area/time orifice diameter diameter of bubble, length bubble formation frequency, number/time molar gas flow rate, mols/time gravitational constant, length/time2 Henry’s law coefficient, press-volume/mols fractional holdup of gas over-all gas-phase mass transfer coefficient, mols/areatime-pressure individual gas-phase mass transfer coefficient, mol/areatime-pressure over-all liquid-phase mass transfer coefficient, length/ time individual liquid-phase chemical absorption coefficient, length/time individual liquid-phase physical absorption coefficient, length/time pseudo-first-order reaction rate constant, time-‘ length of reactor number of bubbles per unit volume of liquid average rate of mass transfer, mols/time-area total pressure, force/area partial pressure of a component in a n ideal gas phase, force/area volumetric gas flow rate, volume/time volumetric liquid flow rate, volume/time universal gas constant, pressure-volume/mols-temperature volumetric rate of reaction, mols/time-volume bubble shape factor; equals 4.84 for spherical bubbles absolute temperature time during any portion of a run diffusion time surface exposure time for penetration theory reaction time gas velocity through orifice, length/time superficial gas velocity, length/time linear liquG or gas velocity, length/time slip velocity, defined by Equation 70,length/time volume of a bubble volume of liquid phase rise velocity of a bubble, length/time terminal rise velocity of a bubble, length/time constant defined by Equation 16 mol fraction of component in gas phase vertical distance through the liquid phase
Greek Letters a = stoichiometric coefficient, the number of mols of liquidphase reactant consumed per mol of absorbed component 6 = pore diameter of porous plate gas distributor, assumed same size as plate particle diameters u = kinematic viscosity, length2/time u = surface tension, force/length = volume of liquid holdup per unit interface area; for tank-type reactors use the inverse of the interface area per unit volume liquid phase ~ L , Q = density of liquid, gas, mass/volume = average liquid residence time, VL/q 7
+
REFERENCES (1 ) Astarita, G., “Mass Transfer with Chemical Reaction,” Elsevier, Amsterdam, 1967. (2) Baker, J. L. L., Chao,B.T., A.I.Ch.E. J . 11 (Z),269 (1965). (3) Calderbank, P. H., Tranr. Inst. Chem. Engrs. 37, 173 (1759). (4) Calderbank, P. H., Moo-Young, M. B., Chem. Eng.Scr. 16,39 (1761). (5) “Perry’s Chemical Engineers’ Handbook,” 4th ed., McGraw-Hill, New York, 1963. (6) Coles, K. F., Popper, F., IND.ENO.CHEM.42, 1434 (1950). (7) Danckwerts,P. V., Ibtd., 43, 1460 (1951). (8) Davidson, J. F., Harrison, D., “Fluidized Particles,” Cambridge University Press, Cambridge, 1763. (9) Davies, R. M., Taylor, G., Proc. Roy. Soc. A, 200, 375 (1950). (10) Gal-Or, Benjamin, Resnick, W., IND.ENO.CHEM.PROOESS DESION DEVELOP 5 (l), 15 (1766). (11) Griffith, R. M., Chem. Eng. Sct. 12, 198 (1960). (12) Higbie, R., Trans. Am. Inst. Chem. Engrs. 31, 365 (1935). (13) Hughmark, G . A., IND.ENO. CHEM.PROCESS DESIGNDEVELOP. 6 (Z), 218 (1967). (14) Jackson, R., Trans. Inst. Chem. Engrr. 42 (4), 107 (1964). (15) Johnson, A. I., Akehata,T., Can. J . Ch.E. 43, 10 (1965). (16) King, C. J., IND.ENO.CHEM.FUNDAMENTALS 4, (2), 125 (1965). (17) Ibtd., 5, (l), 1 (1766). (18) Koide, K.T.,Hirahara, T., Kubata, H., Chem. Eng. (Japan) 30,712 (1766). (19) Leonard, J. H.,Houghton, G., Chem. Eng.Sct. 18,133 (1763). ( 2 0 ) Leonard, J. H., Houghton, G., Nature 190,687 (1961). (21) Levenspiel, O.,“Chemical Reaction Engineering,” Wiley, New York, 1962. (22) Lichtenstein, H.J., Twigg, G. H.,Trans.Far.Soc. 44, 905 (1948). (23) May, W. G., Chem. Eng. Prog. 55 (12), 49 (1759). (24) Miller, D. N., IND.ENO. CHEM.56 (lo), 18, (1764). (25) Nicklin, D. J., Chem. Eng. Scz. 17, 693 (1962). (26) Peebles, F. N., Garber, J. J., Chem. Eng. Prog. 49 (2), 88 (1953). (27) Rippin, D. W.T., Ph.D. Dissertation, Cambridge University, 1959. (28) Schaftlein, R. W., Master’s Thesis, University of Delaware, Newark, Del,, 1967. (29) Sherwood, T.K., Pigford, R. L., “Absorption and Extraction,” McGraw-Hill, New York, 1952. (30) Soo, S . L., “Fluid Dynamics of Multiphase Systems,” Blaisdell, Waltham, Mass., 1767. (31) Sprow, F. B., Prausnitz, J. M., A.I.Ch.E. J . 12, 193 (1766). (32) Szekely, J., Chem. Eng. Scr. 20, 141 (1965). (33) Teller, A. J., Chem. Eng., p. 111, July 11, 1960. (34) Uhl V. W., Gray, J. B., Eds., “Mixing,” Vol. 11, Academic Press, New York (1967).’ (35) Yates, R. A., Ph.D. Dissertation, Univ. of Delaware, Newark, Del., 1966.
VOL. 6 0
NO. 5
MAY 1968
27
