TWO-PHASE REACTOR DESIGN TUBULAR REACTORS
Reactor
Model Development a previous article (33), design equations were Ithenproposed for two-phase tank reactors, evaluation of model parameters was discussed, and a comparison of the model predictions with experimental results was made. The same format is followed in the present discussion of gas-liquid cocurrent tubular reactors, and some limited data on the ethylene oxide vapor and liquid water system are presented and discussed. Little experimental work has been reported for twophase reactions in tubular systems, and it is hoped that the model equations developed here will be an aid in the planning and execution of needed experimentation. Until some extensive testing of two-phase m a s transfer and reaction models has been undertaken, and until more information is available on the model parameters, the rational design of industrial scale two-phase tubular reactors will be, at best, a most difficult task. The analysis of two-phase pipeline devices is complicated because of the variety of configurations which each phase may assume in the pipelme. The in sih configurations range from small droplets of liquid die persed in a turbulent gas to small bubbles of gas d i p p e r s c d in a turbulent liquid (Table I). The configuration within the pipe depends on the phase properties, the phase flow rates, and the pipe size and orientation. Both horizontal and vertical systems have been studied, and the various flow configurations have been visually classified into flow patterns. The following descriptive terms are commonly employed, and a n attempt is made to give a rough picture of the flows observed in each pattern. DISPERSED FLOW: This flow pattern occurs when most of the liquid is flowing as droplets supported by a high velocity turbulent gas stream. There is also a thin liquid film on the wall, difficult to measure or even detect. ANNULARFLOW:In this flow pattern, the liquid travels as a thin film completely circumscribing the wall of the conduit. This annulus of liquid is of uniform thickness in vertical systems but shows marked circumferential 6
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
P. T. ClCHY J. S. U L T M A N 1. W. F. RUSSELL
variation, due to gravitational effects in horizontal systems. The interface between the gas core and the annular liquid is highly disturbed, and large circumferential ring waves may be seen traveling on the liquid surface at irregular intervals. The spray of multisized droplets produced from these waves passes through the turbulent gas core and is redeposited on the film. It is possible, in the vertical case, to obtain annular flows without entrained droplets, and in the limit, a smooth interface may exist between the phases. STRATIFIED FLOWS : This flow pattern is observed only in horizontal cases and is a configuration in which the denser fluid flows entirely in the lower portion of the conduit. The interface is relatively undisturbed over a range of gas and liquid rates. However, when the gas velocity reaches a value of approximately 22 ft/sec, waves are observed. Depending on the intensity of the turbulent gas stream, the disturbances may range from ripples to large cresting waves which produce a multisized droplet spray and result in entrainment of a portion of the flowing liquid. When waves occur in stratified flows, the resulting pattern has sometimes been designated wavy flow. SLUGFLOW:This pattern is characterized by the alternating flow of liquid and gas slugs which occupy the entire cross section of the duct. The length of the liquid slugs varies with flow conditions from short frothy slugs in which a large number of bubbles is entrained, to long fairly uniform slugs with entrained bubbles present principally in the end regions. BUBBLE FLOW:I n this flow pattern, the gas travels in discrete volumes of various sizes and shapes. In horizontal systems, when these volumes are nearly spherical, they are called bubbles; otherwise they are referred to as plugs. I n vertical systems, very nonspherical but discrete volumes are referred to as bubbles, and the term plug is used to describe bubbles with dimensions approaching those of the conduit. A wide variety of bubble size distributions and spatial configurations have been observed. These range from single bubbles VOL. 6 1
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traveling separately to macroscopically uniform bubble clouds. These distributions and configurations are strongly dependent on the entrance and mixing sections of the two-phase contactor. The bubbles in horizontal systems tend to travel at the top wall of the pipe after entrance effects have disappeared. In vertical systems the bubbles tend to travel at the centerline of the pipe. FROTHFLOW:This flow pattern describes vertical flows which are similar to annular flows except that a large number of s m a l l bubbles are entrained in the liquid film. A large body of data for oil-air and water-air systems in 1- to &in. pipes has been examined by Baker (7) for horizontal systems, and Govier et al. (77) for vertical systems. Baker observed that when superficial gas flow rates were plotted against liquid-gas ratios the coordinates of a particular flow pattern grouped in certain areas on the plot (Figure 1). Govier treated vertical systems in a similar manner and a modification of his flow pattern chart is presented as Figure 2. They then designated rather arbitrary but useful dividing lines on these plots to delineate approximate boundaries for the observed flow patterns. Corrections for physical pmpexty variations were proposed with the understanding that the number of different gas-liquid systems actually investigated was limited. These charts, though not general or widely teated, serve as the best basis for flow-pattern prediction now available and their use is discussed in a review article by Anderson and Russell (4). The flow patterns described above have served to classify a large proportion of the previous work done in the field of two-phase flow. C a r A l examination of each flow pattern shows that it may consist of several distinct flow configurations (Table 11). For example,
I
TABLE II. FLOW PATTERNS OBSERVED IN TWOPHASE FLOWS VISUAL GROUPING OF DETAILED FLOW CONFIGURATIONS
(Numbers Refer to Tab- -,
w-4
..
. . .... ~ i
0 ’
9 10 3 4
5 6
Iv. UVDFlnr 14
V.
b b U e Clew 16 17 18
15 19 20 21
22 23 24
25 26
the stratilied flow pattern includes flows with smooth interfaces, wavy interfaces, and wavy interfaces with droplet generation, each a separate flow configuration. The detailed flow configurations more closely describe the specific fluid dynamical characteristics of the system. A knowledge of this specific configuration plays an important role in the analysis of hamport and reaction processes occurring in two-phase tubular reactors. It now remains for careful fluid dynamic studiea to estab lish the boundariea of the detailed flow configurations on the Baker-Govier charts. Some initial work in this area will be discussed in the section on Reactor Mdel Parameters. At present, the best that can be done to determine the specific configuration present in the system is to estimate the flow pattern from the existing
,’” ,
INDUSTRIAL AND ENGINEERING CHEMISTRY
7
SknIlhdHrr 1 2
I
8
13
U. Annulanow
,,.,IO*@
21’;
n.w
12
charta, using the appropriate operating conditions
This narrows the configurations to those found in tha pattern. Some of the detailed flow configurations have similar fluid dynamical characteristics even though they belong to di5erent flow patterns. These common fluid dynamic Characteristics form the basis for the new flow regime classifications presented in Table I11 and briefly discuwd below. kc well as grouping the similar flow configurations, these regimes serve to categorize the modeling process, inasmuch as all the configurations within a regime may be treated by the same general modeling equations. REGIMEI. “Continuous Fluid Phases with a WellDdined Interface” is defined to include all configurations which may be considered to have continuous gas and continuous liquid phases flowing simultaneously in the reactor. Furthermore, the interface between the phases must be amenable to a simple description. REGIME11. “Continuous Fluid Phases with Complex Interfaces and Fluid Interchange” is defined to include those configurations in which the gas and liquid phase flows are eatentially continuous, but where a portion of one phase may leave its continuous flow stream and enter the other phase in the form of entrained droplets or bubbles. The effect of this phase exchange process, called interchange when applied to liquid drops, is to create a chaotically shaped interface that is not easily described except on an average bash. REQIME111. “Alternating Discrete Fluid Phases” is defined to include those configurations where the fluid
, umbers Refer
to
Table
,
Continuous Fluid Phases with a Wall-De13n.d InMac. 1 3 5 2 4 8 11.
Contlnuovi fluid Phases wHh Complex 1nhrtmr.s and fluid Phol. Int.rrhclnpe
6 7
10 11
111.
AWnnmllng D I s r r h fluid Phases
IV.
On. Continuous fluid ?hare and On. Dktr.1. Fluid Ph.8.
14 16 17 19
15 20
21 22
23 24
25
V. ho&!*nlous TwD.Pkos. M W r u 12 18 13 26
flow alternates periodically between principally liquid flow and principally gas flow. During its travel down the conduit, each individual phase fills the entire cmss section of the conduit and may be considered as a discrete single-phase unit of finite length. REGIME IV. “One Continuous Phase and One Discontinuous Phase” is defined to indude those configurations where discrete units of one phase are present in the other phase, and where the second phase can be considered to be continuous. The bubble and plug flows fall naturally into this regime. REOIMEV. “Homogeneous Two-Phase Mixtures” is defined to indude those configurations where the fluid phases are sufficiently intermixed that macroscopic spatial uniformity may be assumed. In summary, the specific fluid dynamical characteristics of a given system determine the detailed flow configuration present. These configurations have been grouped by visual similarities into classes called “Flow Patterns” and by fluid dynamic similarities into the “Flow Regime Classifications.”
REACTOR MODEL EQUATIONS The performance of a two-phase contactor in which a desired product is formed by the reaction of gas and liquid reactants can be best understood through the analysis of a reasonable mathematical model of the physical system. Our present understanding of in situ flow configurations is limited, and the mathematical descriptions which can be derived for the two-phase reactor consider only axial variations in the pertinent concentrations. Although there is some evidence of secondary flows in the circumferential or radial direction in some regimes, a lack of quantitative knowledge precludes model development taking these variations into account. The proposed models are tractable enough to be readily compared with experiment and should prove useful for design and analysis for those situations in which the parameters can be evaluated. Two general unsteady-state model equations are developed below for those situations in which the gas and liquid phases are continuous, and two general equations are developed for those cases in which the phases are discrete. These four general equations are then simplified and combined in the appropriate manner to obtain the physical situation model for each regime listed in Table 111. When the gas and liquid may be considered as continuous phases, the mass transfer and reaction processes can be depicted schematically, as shown in Figure 3. The volume element chosen has a length Az and a c r w sectional area equal to that of the conduit, A,. Masa transfer occurs acrosa the gas-liquid interface and in characterized by an overall coefficient, KO, defined V O L 61
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,
with a driving force equal to (yi- CiH/P), the difference between the gas phase concentration of component i and the gas concentration of component i which would be in equilibrium with a liquid stream of bulk concentration et. Provision is also made to include mass transfer from the gas phase to entrained liquid drops in a similar manner. The terms IoCt/po and I d C e i / p d allow for the convective transfer of component i to and from the continuous liquid stream by drops which are formed from, and deposited on, the major gas-liquid interface (interchange). After combining these terms, considering the possibility of reaction in both the gas and liquid phases, and allowing for axial dispersion to take account of deviations from the assumed plug flow behavior, the following component mass balances are derived. CONTINUOUS GAS PHASE :
CONTINUOUS LIQUID PHASE :
Figure 3. Two-phase mass transfer model description of the physical situation will consist of a component mass balance equation for each phase. For some configurations a droplet component mass balance and an overall mass balance are necessary to complete the description. The following sections present the models for each flow regime classification in Table 111. 1. Continuous Fluid Phases with a Well-Defined Interface
DISCRETE LIQUID PHASE :
The flow configurations which fall into this regime classification are : all horizontal stratified flows without droplet generation, and all vertical annular flows without droplet generation. The interface in each of these configurations may be simply represented, and the transfer area easily determined, if the in situ gas-liquid volume ratio is known. For the case of interphase mass transfer in laminar-laminar flows, the concentration profiles may be obtained analytically by simultaneous solution of the diffusion equation in each phase. The results of such a solution have been obtained and verified by Byers and King ( 9 ) for interphase absorption in a rectangular channel. The general model equations have been developed by assuming that all the transfer mechanisms possible in two-phase systems were taking place simultaneously within the same volume element. When the basic fluid mechanics of a specific regime are considered, we find that many of the terms included in the general model
A model to describe the isothermal behavior of a specific flow regime can be obtained by modifying the appropriate general model equations to fit the particular flow configurations found in that regime. The simplest
AUTHORS P. T . Cichy, J . S. Ultman, and T. W. F. Russell are at the University of Delaware, Newark, Del. 19711. Dr. Russell is Associate Professor, Department of Chemical Engineeering, P. T . Cichy and J. S. Ultman are graduate students. Workfor thispafler was supported in part by National Science Foundation Grant GK-830.
The development of the unsteady-state discrete phase equations follows a similar pattern, using the velocity of the discrete volume instead of an average phase velocity. DISCRETE GAS PHASE
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
are not likely to be important and may be deleted, resulting in considerable simplification of the model equations. The phases. flowing in the configurations defined by this regime are continuous, and thus, the continuous gasand liquid-modeling equations are employed. Since entrainment is not present, the interchange and entrainment terms are dropped. The concentration profiles are assumed to be established rapidly, and the transients are neglected by dropping the time-varying terms. The conduit is assumed to be of constant cross section. The liquid holdup, which is the in situ ratio of liquid volume to conduit volume, may change in the axial direction if large changes in the gas flow rate or density are encountered. I n the cases considered in this regime, unless otherwise noted, it will be assumed that the change is not significant and RL will be treated as having an average value, calculated using the average gas phase properties. (Large axial changes in the gas stream flow rate or density could bring about a change in the flow configuration at some point along the length of the conduit. A combination of models may be needed to describe such a system.) The reactions of major interest occur in the liquid phase ; therefore, gas phase reactions are neglected. T o obtain analytical solutions, the liquid phase reactions are assumed to be first-order with reaction constant, k,. The pressure and temperature along the length of the conduit are taken at their average values. Henry’s law constant, H, is assumed to be independent of concentration. The continuous gas- and liquid-phase equations pertinent to this regime are presented below with
This model for the tubular two-phase reactor is very similar to the model of a fluidized bed proposed by May (30). When the dispersion terms are deleted, the model is similar to the plug-plug model of a fluidized bed proposed by Davidson and Harrison (12). This plug-plug model is often used to describe twophase mass transfer problems when there is insufficient information on the detailed fluid configuration, the interfacial area, or the transport mechanisms to justify a more complex model. The solution of the Equations 5, 6, and 7 is difficult, and several special cases are now considered where further simplification leads to analytical solution and determination of the model behavior. Case 1
This case considers the situation where the gas flow rate, G, and solute concentration, C, are both functions of axial position. The liquid phase holdup is also expected to vary with axial position, but an average value is often used in the absence of additional information. The absorption of a very soluble solute from a concentrated gas stream is an example of the situation described by this case. When the dispersion terms are dropped, the model simplifies to the plug-plug case, but the equations are still first-order nonlinear and difficult to solve. Through the combination of the overall and component gas phase equations, the following equations result. Gas Phase:
Liquid Phase :
( -‘ f)- k,RLAcCf
dCt q - = KGaAcP y c dz
and
q Continuous Gas Phase :
RLAC~L
(9)
where
(5) Continuous Liquid Phase :
k7R~AcCi (6) The application of an overall mass balance to the continuous gas phase yields a third equation necessary for some situations. Overall Gas Phase:
Evaluation of the integral will allow the concentration profile to be obtained. This method was illustrated in Case 1-A-C in Part I of this article (33). Case 2
When dispersion is neglected and the rate of solute transfer is assumed not to affect the gas flow rate, G, the nonlinearity disappears and one has the following set of coupled ordinary first-order differential equations.
dG - = -KGaAcP dz VOL. 6 1
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A solution to this coupled set of first-order equations is readily obtained by standard techniques. The liquid concentration profile is :
(J--) -
TI
r2
]
KoaAcP r2z:
e-T
(13)
let resides in the gas phase, solute may be transferred to it at a rate independent of transfer to the continuous stream liquid ; thus a separate transfer mechanism must be considered for the entrained phase. When the droplet returns to the continuous liquid stream, its concentration will be different from that of the liquid at the point of entry and will thus affect the concentration profile. The development of a model for this regime follows the same pattern as the last regime and the same assumptions are made with the exception that the entrainment and interchange terms are retained. A component balance on the entrained liquid results in a fourth equation necessary to obtain solutions. The pertinent modeling equations follow with q
where -rl
=
2, [-(1
1
+ P + a> + (1 + 2P + P2 + 2a -
1 2a
+ a 2 ) 1 / 2 ](14) + P + a) - (1 + 2P + P2 + 2a 2Pa + a2)1/2](15)
- qe E RLA,aL
Gas Phase:
2Pa
-r2
=
- [-(I
Liquid Phase :
a = -p4
HG
Case 3
When dispersion is neglected and a pure gas phase is assumed, a concentration profile may be obtained from the solution of the liquid phase equation withye = 1. dCt - = - - KaaAcP dz 4
y,
- krRLA;) +
4
4
c,
Overall Gas Phase: dG
- = dz
(16)
The solution is
Entrained Phase :
- A_. (HKoa + ~ T R L ) Z Ci = Coe
+
II. Continuous Fluid Phases with Complex Interfaces and Fluid Phase Interchange
The flow configurations which fall into this regime classification are : horizontal annular flow, stratified flow with droplet generation, vertical froth flow, and vertical annular flow with droplets entrained in the gas phase. The interface is highly disturbed in these configurations since droplets are continuously being torn off and redeposited on the liquid surface. While the drop12
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
As in the last regime, Equations 18, 19, 20, and 21 are difficult to solve and certain special cases are considered where simplification leads to analytical solutions and the determination of model behavior. Case
1
This case represents the situation where the solute leaves the gas phase at such a high rate that the gas flow rate, as well as the solute concentration, is a function of axial position. Due to the change in flow rates, the liquid phase holdup, RL, is also expected to change with
position, but since no simple relationship between G and RL is presently available, one can use an average value of RL as a first approximation. When the dispersion terms are dropped, a plug-plug model results which is described by the set of four nonlinear Equations 18 to 21. These equations are difficult to solve analytically and describe a situation similar to Case 1 in Regime I.
Entrained Phase :
Case 2
When the gas phase flow rate, G, does not change with axial position, the nonlinearity is removed, and if dispersion is neglected, the following coupled linear ordinary differential equations result. Gas Phase: dYi G - = -KKoaAcP dz
Liquid Phase :
Entrained Phase :
The variables in this set of equations are the liquid and gas phase concentrations, the concentration of the liquid in the entrained phase, and axial position. The other parameters are estimated by methods to be discussed in the section on Reactor Model Parameters. Rosenhart and Jagota (32) discuss a similar model for vertical annular flow with entrained droplets. Although a solution of this third-order system of equations could be obtained, there is a need, first of all, for more fundamental information on the interchange process. At the present time, this formulation only serves to point to areas where further work must be done. Although some initial experimental work has been done in this area, until more data are available the use of the Regime I equations is recommended for the approximate analysis of this case. Case 3
When we assume no dispersion and consider the case of a pure gas phase, only the concentration profiles of the continuous liquid phase and the entrained droplets need to be considered. The equations describing this situation are given below.
These two coupled linear ordinary differential equations may be solved for the desired concentration profiles. The limitations discussed in Case 2 also apply to this situation. 111.
Alternating Discrete liquid Phases
The flow configurations of interest in this regime classification are the horizontal and vertical slug flows. The description of these configurations indicates that each phase flows essentially as though it were traveling alone in the conduit, but the flow alternates between phases periodically. This description, adequate for initial modeling, suggests that each phase be treated as a discrete volume element to which the discrete phase equations may be applied. However, Hubbard and Dukler (27) have recently shown that slug flows are, in fact, more complex than this and have offered a model of the fluid mechanics involved. Unfortunately, at this time there is not even enough detailed information available to solve the set of coupled differential equations describing the simple model. Jepsen (23) has proposed that a Regime I-type model be used to describe transfer in slug flows. Gregory and Scott (78) have also carried out some experimental studies following this approach. The parameters used in the model equations are correlated through the use of an energy dissipation factor. This approach seems to offer the best method currently available for the analysis of the complex configurations of this regime. IV. One Continuous Fluid Phase and One Discrete Fluid Phase
The flow configurations which are considered under this regime classification are all the horizontal and vertical bubble and plug flows except those of uniform bubble clouds. The discrete gas and continuous liquid general modeling equations apply to this regime. Characteristics of the gas phase, such as the size, spatial distribution, and frequency of the discrete volume elements, are dependent on entrance conditions as well as flow parameters. These characteristics will be further discussed in the section on Reactor Model Parameters. The simplifying assumptions made for Regime I also apply to this regime. When these assumptions are applied to the pertinent general modeling equations, VOL. 6 1
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and the discrete gas phase equation is multiplied by the number of discrete units per foot reactor, NBAC, the following equations result. Gas Phase:
(27) Liquid Phase :
An analytical solution to this coupled set of first-order equations can be readily obtained. Case 3
When dispersion is neglected and a pure gas phase is assumed, the solution to the liquid phase equation yields the liquid phase concentration profile. The pertinent liquid phase equation is:
with boundary conditions: Ct
where U
= NgU'VDG
An overall mass balance on the gas phase within a reactor volume element yields a third useful equation. Overall Gas Phase:
When large changes in the size of the gas phase volume element occur, the holdup, RL, would be expected to change in the axial direction. As previously mentioned, an average value can be used as a first approximation. V.
The equations, as written, are difficult to solve and, as before, three cases of interest are now considered. Case 1
When solute is transferred between the gas and liquid phases at such a high rate that the volume of a discrete phase unit varies significantly with axial position, the holdup, discrete phase velocity, and interfacial area also become functions of axial position. The relationship of these parameters to the phase unit volume or flow conditions is not generally known and reasonable estimates can be made for only a few simple cases. This is discussed further under Reactor Model Parameters. Even when the relationship between these parameters and the phase unit volume is obtainable and the secondorder dispersion terms are dropped, the remaining equations are highly nonlinear, and unless further simplifications are possible, they must be solved numerically. Case 2
If, in addition to neglecting dispersion, the volume and velocity of a gas phase element are assumed to remain constant as the element moves in the axial direction, the nonlinearity is removed and the following equations result. Gas Phase:
Liquid Phase :
14
INDUSTRIAL A N D ENGINEERING CHEMISTRY
= CO,z = 0
Homogeneous Two-Phase Mixtures
The flow configurations which fall into this regime are all the horizontal and vertical dispersed and bubble cloud flows. Examination of the phase distribution on a microscale reveals the presence of small discrete phase units suspended in the continuous second phase. When the system is examined on a macroscale, the spatial distribution of discrete phase units is found to be uniform, suggesting that the mixture may be thought of as a single phase fluid with unique physical properties. It is this characteristic that distinguishes this regime classification. The configurations of this regime may be modeled in two ways. When the dispersed phase is treated as discrete phase units, the modeling equations derived for Regime I V are applicable to this regime. The discrete phase unit in all Regime I V configurations was a gas. In this regime, the discrete unit will be a gas for the bubble clouds and a liquid for the dispersed flows. An alternative approach, suggested by the spatial uniformity of the mixture, is to consider both phases continuous, each exhibiting the physical properties of the homogeneous mixture. The model equations applicable for this viewpoint are identical to those developed for Regime I configurations with the single phase physical properties replaced by those of the mixture. This section has dealt with the classification of flow configurations and has presented the development of the pertinent model equations for the analysis of each flow configuration found in normal two-phase tubular systems. The model equations will be useful for design if the parameters can be estimated with sufficient accuracy to allow decision making. For those cases where little or no information exists, it is hoped that the mathematical models will aid experimental planning.
