TWO-PHAS€ REACTOR DESIGN TUBULAR REACTORS
P. T. CICHY T. W. F. RUSSELL
Reactor
Mode
Yarameters he model equations have been derived so that the T parameters have some fundamental significance with regard to previous experimental studies in gasliquid cocurrently flowing systems. In addition, it is felt that some of the parameters chosen to characterize the system can be reasonably estimated from experimental studies in two-phase systems different from pipeline contactors. It is the purpose of this section to discuss the parameters used in the model equations and to propose methods by which their magnitude and importance may be determined for each flow configuration. The areas in which additional information is needed will be designated and the best current methods of estimation discussed. The first step in the design or analysis of any twophase reactor is to determine the detailed flow configuration which is expected to be present in the reactor. This is accomplished, as explained in the preceding section, through the use of the Baker-Govier charts to estimate the flow pattern present and through a reasonable selection of one of the configurations found in that flow pattern. When a configuration has been chosen, the flow regime classification to which it belongs may be identified and thus the appropriate model equations determined. Examination of the model equations then determines which parameters are needed to complete the analysis. I t should be realized that the present state of the art is not well enough developed to allow for accurate determination of the flow pattern transitions and some care must be taken at this step. As a second step in a reactor-mass contactor analysis, it is helpful to designate three characteristic times; tp, the total length of time an average element of fluid remains in the reactor; to, the length of time that a volume element of the liquid is exposed to the gas-liquid interface; and tR, the time needed for the liquid phase reaction to proceed to an appreciable extent. Determination of these quantities aids in parameter evaluation and helps in our initial understanding of the overall reactor performance. These characteristic times have
been discussed by Aktarita (6) and their use illustrated for tank-type reactors by Schaftlein and Russell (33). The average length of time an element of fluid remains in a tubular reactor can be estimated from a knowledge of the holdup, RL, a parameter to be discussed in the next section. R LACL tpL = (33) 4 (1 - RL)AcLP tpa = (34) GR T The characteristic diffusion time, tD, is defined by Astarita as follows : tD
DL
= -
(35)
Since there has been little experimental work done to compute kLo for gas-liquid tubular systems, it is not as easy to assign limits to this quantity as it is for tank reactors. The experimental studies of Lamont and Scott (26) in horizontal bubble flow yield values of t D between 0.002 sec and 0.03 sec, in close agreement with the range of to reported for tank-type reactors (0.005 < tD< 0.04). I n the annular flow regime, it appears that to may be as small as 0.001 sec. This value, however, is suspect because of the problems of obtaining k L separately from kLa when experimental data are used. Wales (42) has also attempted to obtain values of to for the annular flow region, but some of his reported values are greater than the liquid residence time in the apparatus he used, and it appears that there is some basic error in his data reduction. The time for the reaction to proceed to a reasonable extent in the liquid phase, tR, must be determined from kinetic rate data and is defined in general as follows:
where Cgis the reactant concentration, C,’ is the equilibrium concentration, and r(C,) is the reaction rate expression. For first-order systems or systems which may be assumed pseudo-first-order, r(C,) = k,C, and (37) A knowledge of t R and tD enables one to calculate the effect of the liquid phase reaction on k L , a procedure discussed by Astarita (6) and Schaftlein and Russell (33)* The task of evaluating the parameters in the model equations is simplified if the symbols used are organized in a systematic manner. There is only one independent variable, z, and three dependent variables, y,, C, and Cei; the remaining symbols can be conveniently divided into three parameter groups as shown in Table IV. The material parameters are quantities whose value is 16
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
not dependent on the gas-liquid flow configurations. The value of a material parameter may be determined from a handbook, or by the standard estimation techniques, or experimental procedures. Its magnitude depends on the gas and liquid used and usually on one more of the design parameters, such as temperature or pressure. The design parameters include all those quantities which are known if an existing system is to be analyzed, or which may be set by the designer to achieve optimal conditions in a design problem. The flow configuration expected is determined once the design parameters have been set and the necessary material parameters calculated. The value of the parameters in the two-phase group is the most difficult to determine, since their magnitude depends on the flow configuration and the detailed fluid mechanics. In some cases, a reasonable estimate of the parameters can be obtained if the phase velocity and Reynolds number are known. I n other cases, direct experimental measurement is essential. The following four sections discuss evaluation of the two-phase parameters for each regime defined in Table 111. (A)
Holdup and Phase Velocities An essential primary piece of information needed for the analysis of any gas-liquid system is a knowledge of the in situ volume fraction of the liquid and gas. Much of the early research in gas-liquid flow was devoted to gathering data and developing correlations to predict this quantity which has come to be called the holdup, R,. Two correlations, one developed by Hughmark (22) and one by Lockhart and Martinelli (28),have been used with some success over the past few years. Except for laminar-laminar stratified flow in horizontal conduits, these two correlations, and some others not so widely used, are the only means of estimating holdup available at the present time. A comparison of the holdup correlations by Lockhart and Martinelli (28), Hughmark (22), and Hoogendoorn (79) has been made by Dukler et al. (75). Other methods of obtaining holdup are described in the review by Scott (34). The Lockhart and Martinelli correlation is the simplest to use and has a semitheoretical basis. The Hughmark correlation is entirely empirical, and involves a trial and error calculation procedure, as illustrated by Anderson and Russell ( 4 ) , but seems to give the best results over a variety of flow patterns. At low values of RL, the holdup is difficult to measure accurately. All the correlations work rather poorly when RL is less than 0.10.
I n the laminar-laminar region, where the holdup varies from 0.10 to about 0.50, the Hughmark correlation is not in as good agreement with theory as the Lockhart and Martinelli approach. Since the correlations cannot be compared with theory outside this region, the best approach is to compute the holdup using both methods and thus to determine reasonable limits on
1 co occ
13.c30 -14
103c
RL. With the holdup determined, it is possible, in some flow configurations, to obtain the average phase velocities and phase Reynolds numbers defined in terms of a hydraulic diameter. The phase velocities are parameters needed in the model equations, and a knowledge of the phase Reynolds numbers can be helpful in determining mass transfer coefficients. REGIME I. There are seven flow configurations which fall into this classification (Table 111). The holdup, R,, can be calculated by solving the equations of motion for laminar-laminar stratified flow, Yu and Sparrciw (45). I n all the other configurations in this regime, either the Lockhart and Martinelli or the Hughmark correlation must be used. The laminar-laminar region in stratified horizontal flow can be outlined on a Baker chart for any pipe size. This region is shown in Figure 4 for air and water flowing in a 1-in. pipe. The phase Reynolds numbers, defined using average velocities and a hydraulic radius, are rather insensitive to the height of liquid in the pipe over the range of RL encountered for laminar-laminar flow. As a result, the position of the laminar-turbulent transition lines on the Baker chart, calculated for each phase from the holdup, pipe diameter, and transition Reynolds number (2100), may be predicted equally well from holdups found theoretically or from any of the holdup correlations. In most of the flow patterns defined on the Baker chart, either one phase or both is in turbulent motion, a characteristic adding to our difficulties in parameter evaluation. The detailed flow configurations 1, 2, 3, and 4 defined in Table I, are identified in Figure 4 for air and water flowing in a 1-in. pipe. These regions can be obtained for any pipe diameter. I t has been found that the laminar-laminar region becomes smaller as the pipe diameter is increased. The identification of the wavy flow regime (configuration 5) is not readily accomplished, but studies by Kerr (25), using the holdup correlation by Hughmark and a modified form of the classical Kelvin-Helmholtz stability
AUTHORS P. T. Cichy and T.W. F. Russell. Workfor this paper was supported in part by funds from the U.S.Department of Interior under the Water Resources Research Account of 7964, Public L a w 88-379.
I3C
Figure 4.
Detailed flow cony7gurations
Boundaries predicted by Hughmark correlation - Lockhart-Martinelli - Numbers refer to detailedyow configuration tabulated in Table I Boundaries shown are for a ?-in. pipe
-
-
analysis derived by Sontowski (36),show that the transition to wavy flow occurs at a gas velocity of about 22 ft/sec. Until a more detailed analysis and more data are available, it is recommended that this value be used to define the transition from stratified to wave flow. The transition to a flow in which droplets are generated from the wave crests is not well defined [con figuration 6). Only three experimental studies have been reported, and different gas velocities have been observed at the point of droplet generation. Van Rossum (40),working with a small wind tunnel, reports a critical air velocity of 58 ft/sec at the point of droplet generation. Kerr (25) reports critical air velocities between 50 and 58 ft/sec for an air-water system in a 1-in. pipe, and Woodmansee (44) observed droplet formation at a gas velocity of 38 ft/sec in a 1-in. parallel plate duct using air and water. While the upper bound for configuration 5 is not clearly established, a rough estimate can be made, and, until further work is reported, it is felt that a gas velocity of 50 ft/sec should be used. I n configurations 1 through 5, the calculation of the phase velocities is a trivial task, once the holdup is known, and, as discussed above, Reynolds numbers are easily computed. For the two vertical configurations which are classified in Regime I the chart by Zhivaikin (46),presented and discussed by Scott (34),is recommended for determining holdup and the transition between configurations 8 and 9. Average phase velocities can be readily determined once the film thickness is known. REGIME11. I n this regime, there are two horizontal configurations, 6 and 7, and two vertical configurations, 10 and 11. The transition gas velocity for configuration 6 has been discussed, but, a t the present time, the Baker chart is the only means of defining configuration 7. VOL. 6 1
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Configurations 10 and 11 are designated on the Govier plot as annular and froth flow. Holdup measurements in this regime are difficult to make accurately since the holdup has a small value, ranging from 0.02 to 0.2. The isolated volume method of measurement includes the volume of the entrained droplets. Conductivity probe studies suffer from the difficulty in averaging liquid heights for the highly chaotic interface and, of course, do not measure the entrained liquid. I n the development of the model equations, the holdup, RL, is defined to include only the liquid in the continuous stream. Correlations such as Hughmark’s and Lockhart and Martinelli’s, based on isolated volume data, can be expected to give high results unless corrected for the amount of entrained liquid present. Swanson (37) presents an attractively simple method of correlating the continuous liquid holdup in horizontal annular flow. He finds that a plot of RL us.
@Lo)OJ ~
flG
is linear with a slope of 28.0. Data from his own work and that of Alves (7) and Omer and Govier (37) were used in the correlation. The vertical annular configuration has been studied by Zhivaikin (46),and a plot of the average film thickness as a function of flow conditions is presented by Scott (34). The average holdup can be obtained from the film thickness as discussed under Regime I. Since the liquid holdup is so low for the flow configurations in this regime, a good estimate of the average gas velocity can be made by assuming that the gas occupies the entire cross section of the conduit. Average liquid phase velocities are not as easy to calculate since the area occupied by the continuous liquid stream in the pipe is so small. Some actual measurements of the liquid film axial velocity have been made by Swanson for horizontal annular flow in 1- and 2-in. pipes. He found a dependence on circumferential position and average film velocities varying from 5 to 12 ft/sec. In the absence of further studies, it is recommended that the liquid velocities be estimated by assuming a uniform film around the inner periphery of the pipe in configurations 7, 10, and 11. The depth can be calculated from the RL obtained by the correlations after correcting for entrainment. Methods of estimating entrainment are discussed in a following section. I n configuration 6 the area occupied by the liquid is computed in the same manner as for Regime I. REGIME111. A knowledge of the holdup for this regime does not give the same insight into the fluid mechanics as it did for Regimes I and 11. The periodic nature of the flow makes accurate experimental data difficult to obtain and interpret. The best information on the fluid mechanics of this regime is presented by Hubbard and Dukler (27). REGIMEI V . The detailed flow configurations in this regime are characterized by discrete gas phase units flowing in a continuous liquid stream. I t is difficult to define the transition boundaries between these nine configurations, because the transition is strongly de18
INDUSTRIAL A N D ENGINEERING CHEMISTRY
pendent on the way in which the gas is introduced into the liquid. Use of the holdup correlation yields average values for the fraction of the pipe cross section occupied by each phase and thus does not give much useful insight into the behavior of the discrete phase. An average gas phase velocity is a meaningless number and probably not worth calculating. The liquid phase velocity can be closely estimated by assuming that the area available for flow is that which would be predicted if the gas phase were continuous. REGIMEV. Four configurations are included in this regime: two, 12 and 13, are configurations in which droplets of liquid are dispersed in the gas; and two, 18 and 26, are fine dispersions of gas in a continuous liquid. A knowledge of the holdup is almost of no value since it is a meaningless parameter if the flowing material can be considered as having the properties of a homogeneous mixture. Individual phase velocities also have no meaning if the homogeneous approach is assumed. (B)
Interfacial Area
One of the most important parameters in the analysis of two-phase reactors is the specific interfacial area, a, defined as the area of the gas-liquid interface contained in a unit volume of reactor. The magnitude of this parameter is sensitive to the flow configuration present and varies significantly from configuration to configuration. The specific interfacial area and holdup are related through their common dependence on the in situ fluid distribution. In general, the area is not easily measured, but may be estimated in some cases if a knowledge of the approximate geometry of the interface can be obtained. An estimate of the holdup is often useful in determining the approximate geometric dimensions of the interface. Other methods of estimation are discussed under the regime in which they have been applied. REGIMEI. The holdup, conduit shape, and our knowledge of the flow configurations in this regime allow a reasonable estimate of the specific interfacial area to be made. For purposes of discussion the conduit will be taken as a circular pipe with diameter, D . For stratified flows with smooth interfaces (configurations 1 to 4)) the liquid flows in the bottom of the pipe, and its surface may be described by the chord of a circular arc. When the holdup is known, the in situ ratio of liquid to pipe cross sectional area is known, and the length of the chord separating the liquid and gas can be determined. The specific interfacial area can then be computed from
where the limits on the chord length are 0 < ch < D. As an example, a 1-in. diameter circular pipe with a holdup of 0.3 has a chord length of 0.95 in. and the specific interfacial area is 14.5 ft2/ft3. When waves are present on the surface (configuration 5 ) , an average chord length can be computed from the
holdup information as in the smooth surface case. The area of the wavy surface is greater than that of a flat surface having the same chord position. When the area, computed as though the surface were flat, is multiplied by an extension factor, E, an estimate of the specific interfacial area of configuration 5 may be obtained. Ch
a = EA,
(39)
Typical values of E in this regime probably range from 1 to about 3. The geometry expected in the vertical annular configuration 8 is a smooth symmetrical annular distribution of liquid. The holdup gives the ratio of liquid flow area to pipe area and, through the symmetry mentioned above, the thickness of the annular film may be determined. I n some cases this dimension has been determined directly (34). The diameter of the gas core is just the pipe diameter less twice the film thickness. When this is known, the circumference of the liquid interface, LI, can be computed and the specific interfacial area calculated from a = -
LI A,
For thin films, the diameter of the pipe can be used as an estimate of the core diameter. When the pipe diameter is used as the core diameter for flow in a vertical circular 1-in. pipe, the specific interfacial area is 48 ftz/fta. When ripples or waves are present on the liquid surface, such as in configuration 9, an average film thickness and core diameter can be computed, and an extension coefficient used to account for the increased area associated with this type of interface :
LI
a = EA,
The value of E must be estimated and in most cases probably lies between 1 and 3. The specific interfacial area of the configurations found in this regime are not expected to be a strong function of axial position except in those cases where large changes in holdup occur. I n general, an average value will be adequate. REGIME11. The presence of an entrained liquid phase in the configurations of this regime results in a significant increase in surface area over that available in the Regime I configurations. The transfer across this entrained area will take place by mechanisms independent of those controlling the transfer to the continuous liquid stream, thus the two types of interfacial area are best treated separately in conjunction with separate transfer coefficients. When a large fraction of the liquid is entrained, the surface area between the gas and the continuous liquid film may be very small compared to the area of the entrained drops, and in some cases it may be negligible.
An estimate of the interfacial area of the continuous liquid stream may be obtained by the methods of Regime I, for configurations 5 and 9. A larger amplification factor, E, is expected, due to the increased agitation of the surface caused by the formation and deposition of the liquid drops. This factor must be estimated and probably lies between 3 and 5. The value of a determined in this way represents the surface area of the continuous stream per unit volume reactor. The determination of the interfacial area present in the entrained phase is a difficult task. The entrained drops may be considered to flow in the area of the pipe not occupied by the continuous liquid phase, that is, R d , . The meaning of Ru in this regime is altered to mean the volume fraction of gas phase plus entrained liquid drops. The exact size and shape of the flowing droplets are seldom known, although some photographic information has been obtained for vertical annular systems [Arnold and Hewitt (5)]. The lack of detailed information leads to the definition of the average droplet volume, VD, also a difficult parameter to estimate. The shape of this volume, in the absence of more detailed information, is taken to be a sphere. The diameter of this average sphere is thought to lie between 0.001 and 0.05 in.; however, the actual value is speculative under current information and more work is needed in the determination of this parameter. The surface area of this average sized sphere is then given by 3W3
ad
= -
VDL
Now if the average number of such drops per unit volume gas phase, N D , can be obtained, the specific area due to entrained droplets may be determined. T o compute N D , the average velocity and volumetric flow rate of the droplets must be estimated, as well as a value for V D L . The volumetric flow rate of the entrained liquid, qe, will be discussed in Section D. If the droplets are assumed to travel at B times the superficial gas velocity, then (43)
B is most likely not much less than one.
N D may now
be estimated by
The specific area associated with the entrained droplets can be computed from =
ND~~VDL
(45)
The analysis of the entrained phase will be difficult until more information on the size distribution and typical droplet geometry can be determined. The size distribution, which determines the average volume, and the average velocity of the entrained phase are not expected to change significantly with axial position unless large changes in the liquid holdup occur. VOL. 6 1
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Wales (42)has determined interfacial area in a Regime I-type model for the horizontal annular configuration through the combined analysis of physical and chemical absorption data. He has lumped both annular area and entrained droplet area together for use with a single transfer coefficient. I t was found that in a 1-in. pipe the specific surface area could be expected to range from 100 to about 500 ft2/ft3 and could be smoothly plotted as a function flow rate, increasing with both liquid and gas rates. As was pointed out in the discussion on to, Wales's reported results appear to contain some significant errors. REGIME111. The interface between phases in slug flow exists principally at the beginning and end of each slug. Some additional transfer area is created when bubbles become entrained in the liquid slugs. The best current method to estimate the interfacial area for horizontal slug flows is presented by Gregory and Scott (78). They combined the analysis of physical and chemical mass transfer experiments in a 1-in. circular pipe to determine the specific interfacial area, a, as a function of an energy dissipation factor. The specific area increased from 18 to 65 ft2/ft3with increasing values of the dissipation factor. REGIMEIV. The flow configurations which belong to this regime all contain gas phase plugs or bubbles traveling in a continuous liquid stream. The specific interfacial area, which is based on a unit volume of reactor, is thus an average property taken over all the plugs or bubbles present in a unit volume of reactor. The specific interfacial area is given by the relation
where N B is the average number of gas phase units per unit volume of the reactor, u' is the interfacial area associated with a phase unit having volume V D G , and VDGis the volume of the average sized phase unit. The phase units, in fact, possess a distribution of sizes, but little information is available on these distributions, and averages are thus required. When the phase units are spheres, the interfacial area associated with any volume, VDG,may be easily determined. For other shapes, photographic studies combined with basic geometry will lead to an empirical relation between volume and surface area for any typical shape. The value of NB can be determined from careful photographic techniques or may be estimated by the relation (47)
where the holdup and phase unit volumes the entrance conditions as a first estimate. N B is not expected to change drastically direction unless coalescence takes place. The initial volume of the phase unit can from the relation
v,, 20
are taken at The value of in the axial
if Y, the frequency of the entering phase units, is known. The frequencies may be measured by interrupted light beam techniques using photocells, but for horizontal systems, little information is currently available on the frequency as a function of the design parameters. Lamont and Scott (26) have used this technique but only report that the range of frequencies investigated went from 160 to 1250 per minute. For vertical flow configurations, significantly more information is available ; however, most work has been done in stagnant liquid systems. The techniques for estimating initial bubble size in stagnant liquid systems have been reviewed by Schaftlein and Russell (33) and Valentin (39). The velocity of the phase unit is also an important parameter in the analysis of this regime and the average velocity may be estimated for horizontal configurations by the relation (49)
This velocity in fact will depend on the pipe orientation, the size, shape, and spatial distribution of gas phase units in the reactor, and it is difficult to predict. In horizontal systems, limited measurements indicate that flB is greater than the average liquid velocity. For vertical cocurrent flows, the Davies-Taylor ( 73) equation for v * ~ the , rise velocity, can be used as a first estimate. flB is then: flB
= fl&
+
u*b
and
When significant changes in the volume of a phase unit occur as it moves in the axial direction, the interfacial area and phase unit velocity will also be affected. As shown by Schaftlein and Russell (33) for tank-type systems, certain special cases may be treated and these parameters written as a function of the phase unit volume, V D G . A large variety of gas phase unit sizes and distributions can be obtained for the same flow conditions by altering the geometry of the phase mixing section of the reactor. The horizontal reactors are considered to have three important stages : the phase unit formation, the transient flow region, and the steady flow region. An understanding of each of these regions will be necessary to understand fully the performance of a reactor operating in this regime. Scott and Hayduk (35)have studied experimentally the interfacial area in 1/2- to 1-in. circular pipes for a wide variety of bubble flows and have determined the following correlation for specific interfacial area,
be estimated
GoRT
= -
Pv
INDUSTRIAL A N D ENGINEERING CHEMISTRY
(50)
where N varies from 2/3 to 1.0.
The parameters discussed with regard to Regime I V are all in need of further experimental determination so that meaningful correlations with design parameters can be developed. I t would be particularly helpful if initial bubble sizes, shapes, and frequency could be obtained as functions of the gas and liquid flow rates and entrance region geometry. REGIMEV. The specific interfacial area present in this regime is that associated with the highly dispersed discrete phase units. For the bubble-cloud configurations, it is the surface area of all the bubbles in a unit volume of reactor, and its magnitude is expected to be a function of holdup, bubble number density, and bubble size distribution. Very few data have been reported on the total surface area or bubble size distribution of bubble clouds, and further information on the characteristics of this configuration is needed. The interfacial area of the dispersed liquid phase is the total surface area of all the entrained droplets in a unit volume of reactor. This area is expected to depend on the holdup, the number density of droplets, and their size distribution. The measurements of these basic properties of entrained droplet flow are difficult, and little information has been reported on their determination. Wales (42) has used the analysis of physical and chemical absorption in dispersed liquid flows in 1-in. horizontal circular pipes to determine specific interfacial area as a function of gas and liquid flow rates. He found that the area ranged from 500 to 800 ft2/fta over the range of his investigation.
(C) M a s s Transfer Coefficient The mass transfer coefficient is an empirical parameter used in engineering studies when a fundamental analysis of species transfer between phases is precluded by a lack of information on the detailed fluid mechanics. The coefficient is defined by the system model equations in which it appears, and its value in relation to the basic transport mechanisms acting within the system is established by this mathematical description. I t is most desirable to relate the coefficient exclusively to the interphase transfer process, accounting for other transport processes through additional parameters or terms in the model equations. However, it is often necessary to model systems for which little information is available, and a simplified description must be employed. The effects of any transport processes not explicitly accounted for in the model become lumped into the coefficient, complicating its meaning, and limiting its usefulness in the analysis of other systems. The magnitude of such an empirical coefficient is determined by comparing the model predictions with experiment and choosing a value of the coefficient which yields the best correspondence. If the model equations are so derived that the mass transfer coefficient does not include our lack of knowledge of the gross fluid mechanics, the coefficient determined by experimental analysis becomes useful in a more general sense, and can be employed in physical situations where the gross fluid mechanics are entirely
different. I n developing the model equations by regime, an attempt has been made to achieve this goal. K G is related to the individual phase transfer coefficients by the usual combined resistance approach.
(53) If a pure gas is employed in a reactor, l / k G may be neglected. I n cases where more than one gas is present, l / k G may be small compared to H/kL and may again be ignored. I t is, of course, possible to have a situation where H/kLmay be neglected, or a system where both resistances are important. The liquid phase mass transfer coefficient, kL, may be affected by chemical reaction, and it is important to be able to determine this effect in a reacting system. This problem is discussed for tank-type reactors by Schaftlein and Russell (33),who based their discussion on the work of Astarita ( 6 ) . The same considerations apply for tubular reactors, except that it is more difficult to obtain estimates of t D . The problem of calculating or estimating K u resolves itself into one of estimating ku and kL. If kLo, the coefficient in the absence of reaction, can be calculated, k L can be estimated using Astarita's techniques. T h e gas phase reaction, if any, will generally not have any effect on kG, and for all cases which we consider kG = kGo.
A method of estimating k G and k L is available to us if we consider to as defined previously, and make use of the penetration theory boundary conditions to solve the diffusion equation. (54)
(55) I n the absence of experimental measurements, Equations 54 and 55 can be used for order of magnitude estimates by attempting to find reasonable limiting values for to, the length of time that an average element of fluid is exposed to the interface. Unfortunately, it is difficult to obtain measured values of the mass transfer coefficient. The experimental work which has been done in this area and that which is reported generally suffers from analyses which of necessity have been oversimplified, thus including in the coefficient all the unknown information on the fluid mechanics. Figure 5 shows the range of flow conditions where studies have been conducted in gas-liquid horizontal flow. The difficulties which can be encountered in attempting to obtain mass transfer coefficients from experimental data are illustrated by examining the data of Ultman (38), presented in Figures 6 , 7, and 8. These data were obtained in a 1-in. stainless steel pipe using ethylene oxide and water which react to form glycol. VOL. 6 1
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-6 Figurr 5. Two-~hanmass ban?fnslumbr
The studies were carried out at flow rates which pmduced an annular flow pattern as defined by the Baker chart. For this flow configuration, the Regime I1 model equations apply, but information on interchange, droplet area, and droplet movement in the core in not available, so the simpler Regime I model was used to analyze the data. For the system studied, the axial concentration profile is represented by the following expression
'
=
HG
A (HG+ Pq)z]} + qP (I - exp [- K 'q (56)
Equation 56 is a special case derivable from Equation 13 when a = 1.0 and j3
