Nonequilibrium Behavior in Heterogeneous Hydrocarbon Systems

California Institute of Technology, Pasadena, Calif. A thermodynamic criterion for the ... The foregoing proposition has been adequately established f...
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Nonequilibrium Behavior in Heterogeneous Hydrocarbon Systems J. B. HATCHER AND B. H. SAGE California Institute of Technology, Pasadena, Calif.

Quiescent Conditions

A

thermodynamic criterion for the existence of bubbles in supersaturated liquids is discussed. Diffusion constants for n-butane in a hydrocarbon liquid were determined experimentally as a function of composition a t 100” F. The influence of natural and forced convection upon the transfer of n-butane to and from hydrocarbon liquids was investigated. The usual concept of an effective film thickness was found to be a useful means of comparing experimental results of this nature. Experimental work indicated that turbulence was of nearly controlling importance in determining the degree of supersaturation obtainable in hydrocarbon liquids. The results are presented in graphical form.

N MANY of the operations encountered in the production 1under and refining of petroleum, fluids undergo changes in state such conditions that the probability of attainment of equilibrium is small. For the most part the prediction of the behavior of such fluids has been based upon equilibrium measurements. If significant departures from this condition are encountered, the equilibrium data are not directly applicable. For these reasons a study of the factors influencing the nonequilibrium behavior of homogeneous and heterogeneous systems was initiated with the specific objective of ascertaining the factors pertinent to the formation and growth of bubbles in supersaturated hydrocarbon liquids. This investigation included a study of the behavior of mixtures of propane and n-butane with a water-white hydrocarbon oil at pressures from below atmospheric nearly to the vapor pressure of the component in question. The greater part of the measurements was carried out a t 100’ F. The influence of agitation upon the attainment of equilibrium in both homogeneous and heterogeneous systems was ascertained as a function of the physical properties of the system in question. I n a quiescent phase the migration of a component as the result of a concentration gradient takes place in accordance with the following general relationship, proposed by Fick:

Under quiescent conditions Equation 1 has been integrated (13, 23) to yield the following expression for the weight of material transferred from a one-component gas phase into a liquid as a function of time:

This equation is based upon the assumption of an infinite depth of fluid, quiescent conditions, constancy of the diffusion constant, and a negligible partial volume of the diffusing component. It indicates that the quantity of material diffusing into a quiescent liquid is directly proportional to the square root of the time if the state of the gas phase is maintained invarient. Hill and Lacey (8) proposed a correction to the diffusion constant to account for the change in volume of the liquid phase upon solution of a gaseous component under these conditions of restraint. It is possible, therefore, to employ Equation 2 with this correction in evaluating the diffusion constant experimentally. The following expression is obtained for this case: (3)

Equation 3 may be applied to situations in which the change in the concentration of the diffusing component is small; this reduces the uncertainty resulting from the assumption of constancy of the diffusion constant.

Convective Influences Quiescent conditions are not often encountered in practice and therefore are usually only a limiting case which is sometimes approached. A knowledge of the influence of turbulence (3, 8, 14) in the transfer of material between phases is of industrial importance. The work of Sherwood and Woerte (21) is of special significance in this regard. Other experimental investigations relating to the influence of convection also directly relate to this problem (7, 19,10). For some purposes it is advantageous in fluid mechanics to consider the velocity distribution in a turbulent fluid in an idealized fashion. For example, the absolute velocity may be considered to vary in a linear manner with the distance from the boundary of the system throughout an “effective boundary layer” of thickness 6, such as is indicated in Figure 1. Beyond this point the velocity is considered to be independent of the space coordinates normal to the direction of flow. Although there are several means whereby the apparent boundary thickness may be evaluated, a useful concept is obtained from the following expression:

The foregoing proposition has been adequately established from both a theoretical (12, 22) and an experimental (10, 13) point of view. The solution of Equation 1for various boundary conditions permits the evaluation of the rate of mass transfer of a component in a quiescent phase under different conditions of restraint. Furthermore, this concept may be readily applied to the mass transfer between phases upon $he assumption of equilibrium a t the interface.

6~

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INDUSTRIAL A N D ENGINEERING CHEMISTRY

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As a matter of interest, an experimental velocity distribution (2)is included in Figure 1which conforms closely to the classical Blausius solution ( 1 ) for the velocity distribution in the boundary layer. I n the case of turbulent boundary layers, von K&rm&n(9) proposed relations which apparently describe the behavior with satisfactory accuracy.

RELATIVE

VeLocirY

?/A

FIGURE1. VELOCITY DISTRIBUTION ADJACENT TO THE BOUNDARY OF A FLOWING FLUID

The foregoing discussion related to the velocity distribution adjacent to the boundary of a system and this bears only indirectly upon the distribution of a component within a phase. However, in an analogous fashion i t is possible to relate the variation in the concentration of material with distance from an interface across which a concentration gradient exists. It has been indicated (21) that concentration gradients in the main body of a turbulent gas are small, although not negligible as compared to the gradients immediately adjacent to the interface. It is to be expected that the boundary region would exhibit a somewhat larger portion of the total concentration difference in the case of liquids which have a much smaller rate of diffusion than gases. On this basis an effective boundary layer thickness (24) may be ascribed for a particular situation in the following fashion: 6,

=so”-

c -c

Vol. 33, No. 4

nection with the concentration variations indicated in Figure 2. If this situation is at all descriptive of actuality, the effective film thickness should be relatively independent of the concentration gradients imposed upon the solution and would be primarily a function of the convective conditions obtaining. If it is possible to correlate the effective concentration film thickness established from measurements upon the rate of material transfer across the interface with that predicted for velocities from considerations of fluid mechanics, it will then become possible to correlate directly the rate of material transfer across the phase boundary with the flow conditions obtaining. Many such correlations have been proposed ( I S ) and apparently are of value in connection uTith the behavior of gaseous films. Therefore, it seems likely that such methods may be applied to liquids with comparable success when a sufficient experimental background is available. For present purposes the effective film thickness affords a useful means of recording the rate of material transfer over a wide variety of conditions. This parameter is primarily a function of the convective conditions obtaining near the interface and the properties of the phase in this region.

Nonequilibrium Behavior in Single-phase Systems The foregoing treatment has related entirely to the nonequilibrium behavior of heterogeneous systems in which there was always a change of the system toward equilibrium although the rate of this change might be markedly a function of the conditions obtaining. If a homogeneous system is brought to such a state that a t equilibrium a second phase will exist, there is a thermodynamic tendency for this second phase to appear. However, i t may not appear even though the homogeneous phase is maintained at such a state for an extended period of time, Such a phase is said to be in a

dX

The two apparent boundary layer thicknesses indicated in Equations 4 and 5 are not equal although they may be simply related. The resulting concentration gradient is illustrated in Figure 2 . The linear variation in the concentration of the diffusing component indicated in the effective laminar boundary layer results directly from Equation 1 while i t is assumed that there is sufficient turbulence in the main body of the phase to yield substantial homogeneity in that region. If this hypothesis is considered to apply to the transfer of material from a one-component gas phase to a liquid phase under convective conditions, i t is possible to ascribe a thickness to this apparent laminar film which is obtained from the rate of material transfer and the diffusion constant of the components in the phase in question, as indicated in the following expression in which the diffusion constant applies to the average conditions existing in the boundary layer:

This film thickness estimated from Equation 6 does not have any true physical identity other than that depicted in con-

c. CO NCENTR AT10 N 0 F COMPONENT

F I Q U R2.~ COKCESTRATION GRADIENT ADJACENT TO AN INTERFACE

supersaturated state. The present discussion will relate only to the behavior of a liquid phase supersaturated with respect to a gaseous component. Furthermore, i t will consider only a binary system with one nonvolatile component. From mechanical considerations it follows that the initial configuration of a separated gas phase is a spherical bubble. The fugacity of the material within the bubble mustflbe

INDUSTRIAL AND ENGINEERING CHEMISTRY

April, 1941

greater than that in a liquid phase with which it would coexist if the radius of the interface was infinite. This difference is indicated in the following general thermodynamic expression which may be derived in the same fashion as was done for the special case presented by Lewis and Randall (11) for a liquid drop of constant specific volume: bkT In

6yV~’

=

3VBTB

fkg

(7)

bV + 2Y($)*

This equation indicates that the fugacity of the material in the bubble increases with a decrease in the size of the bubble; therefore, in the case of small bubbles the pressure within

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FIGURE3. APPROXIMATEMINIMUM RADIUSOF A STABLEBUBBLEIN A SUPERSATURATED LIQUIDPHASE

5 f!L

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Except a t very low pressures and in cases involving exceedingly small bubbles, the last term in Equation 11 is usually small in comparison t o the others. It may then be neglected and thus yield the following approximate solution for the limiting size of a stable bubble:

Equation 12 relates approximately the minimum size of stable bubble, the interfacial tension between the phases, and the “supersaturation pressure”. The approximate variation in the size of stable bubble with supersaturation pressure for several values of interfacial tension is depicted in Figure 3. The data in Figure 3 indicate that small bubbles are relatively stable a t high supersaturation pressures but become unstable near equilibrium. Therefore, regions of high supersaturation-i. e., relatively low pressures-are conducive t o the formation of small bubbles since probability considerations indicate t h a t the frequency of their formation is much larger than for large bubbles. Although localized regions of low pressure are encountered in laminar flow, they are not so frequent as the low-pressure regions associated with vortices (14) which are widespread in turbulent flow. From this it is t o be expected that conditions which are conducive to high vorticity will enhance the separation of a gas phase from supersaturated homogeneous liquid. Furthermore, such conditions are normally associated with rather high rates of microscopic shear which increase the probability of bubble formation from a kinetic viewpoint.

Experimental Results

the bubble is markedly in excess of that in the remaining parts of the system a t the same elevation. I n order t h a t a bubble may be thermodynamically stable-i. e.; may continue t o exist-it is necessary t h a t the following relation exist: fkB

2

445

(8)

The experimental program may be divided conveniently into three categories, one directed toward a study of the diffusion constants of n-butane in a hydrocarbon liquid of relatively high molecular weight, a second relating t o the influence of natural and forced convection upon the absorption and desorption of n-butane by a hydrocarbon liquid, and a third covering the factors pertinent to the formation and growth of bubbles in supersaturated hydrocarbon liquids.

From general thermodynamic relations as applied t o the metastable equilibrium, Equation 8 may be expressed in the following form in which is the partial volume of component k in the supersaturated liquid :

vLl

It follows then that, for a bubble t o be stable, its radius must satisfy the following expression:

If the difference in pressure between the interior of the bubble and the main body of the solution is small, the change in the specific volume of the gas phase may be neglected. Furthermore, if the specific volume of the gas phase is large, corresponding t o conditions of low pressure, the partial volume of the component in the liquid phase may be neglected and the gas phase may be assumed to follow the perfect gas law. Under these conditions Equation 10 becomes:

FIGURE 4. APPARATUSFOR ABSORPTION AND DESORPTION OF GASEOUS HYDRO~ARBONS

MATBRIALS.The oil employed throughout this investigation was that used in an early study (16). It was a waterwhite oil refined from Pennsylvania crude stock, with a specific gravity a t 100” F. relative to water a t its maximum density of 0.8244, a viscosity-gravity factor (6) of 0.7979, and an average molecular weight of 342, determined from the freezing point lowering of benzene extrapolated t o infinite dilution. The n-butane was obtained from the Phillips Petroleum Company whose special analysis indicated the material to contain 99.7 mole per cent n-butane and 0.3 mole per cent isobutane. This hydrocarbon was used without fur-

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INDUSTRIAL AND ENGINEERING CHEMISTRY

ther purification. The propane was obtained from the same source and contained less than 0.03 mole per cent impurities.

Diffusion Constants A simplified schematic diagram of the arrangement of the glass apparatus employed for the investigation of the diffusion of nbutane into the hydrocarbon liquid is presented in Figure 4:

Vol. 33, No. 4

phase. The pressure over the quiescent liquid was then rapidly raised by a predetermined amount and maintained at the higher value by throttle valve 3. The quantity of gas entering the absorption chamber was ascertained by the change in pressure in one of three vessels, C, D, or E (Figure 4). The proper sized vessel was employed to yield such a pressure change that it could be measured with suitable accuracy by means of manometer F. The change in volume of the liquid phase was determined from time to time during the course of the measurements from the elevation of the oil-gas interface. From these measurements the quantity of gas diffusing across the gas-liquid interface was determined as a function of time.

A t'ypical set of results is shown in Figure 6, and the data closely follow the linear relation indicated by Equation 2. I n arriving at these values, corrections were made for the deviation of n-butane from perfect gas behavior (17) and for the increasing volume of the liquid phase. Values of the diffusion constant as a function of the concentration of n-butane in the liquid phase are presented in Figure 7. These data indicate a nearly linear increase in the diffusion constant with an increase in concentration of the diffusing component. The simple relation indicated in Figure 7 cannot persist to high concentrations since t h e diffusion constant becomes effectively infinite under isothermal conditions when the liquid phase is pure n-butane. The results serve to illustrate the large influence of the nature of the phase upon the migrational characteristics of a component.

Essentially, the equipment consisted of the absorption cell, A , which was connected through a throttle valve, B , to one of three isochoric chambers of different capacity, C, D , and E. The latter were maintained a t uniform temperature by a liquid bath and permitted the measurement of the quantity of gas withdrawn from absorption chamber A in terms of the indications of the mercury in glass manometer F . The pressure existing within absorption chamber A was determined by means of the mercury-in-glass manometer, G. Provision was made a t H for the .ivithdrawal of gas from Convective Influences vessels, C, D , and E after measurement by means of a mercury difThe apparatus described in connection with the measurefusion pump backed by mechanical ments of the diffusion constants for n-butane was employed equipment. The entire apparatus was composed of glass since the for the convective measurements. It is similar in some pressures involved were not more respects to that used by Hutchinson and Sherwood (7). than 15 pounds per square inch I n the study of the influence of natural convection, the liquid above that of the atmosphere. Dhase was brought to equilibrium initially with gaseous The detailed construction of the FIGURE 5. DETAILS O F ABSORPTION absorption chamber is Presented n-butane at s u c h a pressure as to yield the desired concentraCHdYBER in Figure 5. It consisted Of a tion of the lighter hydrocarbon. The pressure was then glass chamber, A , approximately 3 inches in diameter, which was lowered rapidly to a predetermined value and maintained placed within the agitated oil there by manual control of throttle valve B (Figure 4). The bath, J, in order to maintain the fluidso under investigaquantity of gas withdrawn was determined as a function ~ ~ o ~ a ~ ~ ~ ~ ; ; m m F.F;f , ~ ~ ~ t ~ ~ of time from the measured changes in pressure of one of t h e isochoric reservoirs c, D ,or E. Throughout these measuretemperature, the actual temperature being determined by a ments, stirrer L of Figure 5 was located below the mercury mercury-in-glass thermometer. Arrangements were made for the addition and withdrawal of mercury from the chamber surface. The influence of natural convection was investithrough valve K , which permitted the effective volume of the gated only with regard to the desorption of n-butane from the chamber to be varied over wide limits. A stainless steel agitator, liquid phase, since quiescent conditions obtained during t h e L, approximate~y inch in diameter, was inserted within the ababsorption process. This difference in behavior resulted from sorption chamber and was driven by a small steel shaft through packing gland M. Provision was made for raising and lowering this stirrer to accommodate changes in elevation of the mercury surface. The cell was rigidly mounted so that incidental vibraTABLE I. ABSORPTION AXD DESORPTION OF ?+BUTANE . 4 ~100" F. UNDER tions would not induce convective effects which QCIESCEXT A N D NATURAL CONVECTIVE CONDITIONS might impair the results. The connections a t the top of the chamber were Time of Initial Depth employed for the addition or withdrawal of gas or Test Conon., Lb./Cu. Ft. Measurement, of Liquid Weight of Av. Conon.b, liquid and for the measurement of the pressure NO. Initial Equilibriums Seo. Phase, Ft. Oil, Lb. Lb./Cu. Ft. prevailing in the gas phase. A vertical-comAbsorption ponent cathetometer was employed to measure the 31 4.09 5.59 12,500 0.155 0.1378 4.84 elevation of the mercury-liquid and liquid-gas 32 2.08 5.59 11,670 0.144 0.1378 3.84 4.09 5.59 10,300 0.153 0.1378 4.84 37 interfaces, as well as the differences in elevation 0.157 38 4.09 5.59 10,670 0.1378 4.84 of the mercury surfaces in the manometers used 2.08 5.59 4,083 0.143 0,1376 3.84 41 2.08 10,270 0.089 0.0927 2.08 to establish both the pressure and the quantity 40 0.00 4.09 7,680 0.096 0,0927 3.09 2.08 of gaseous material added to the diffusion chamber. ... .... 3.09 4.09 4,687 42 2.08 70 It is believed that the weight of gas withdrawn 71 2.08 6,600 ... .... 1.04 0.00 was established with an uncertainty of not more 72 0.00 4.69 5,467 ... .... 2.35 than 4 X 10-8 pound (2 mg.), while the quantity Desorption of liquid employed was ascertained within 1 X 4.09 5,580 0.053 0.0459 .. 5.59 20 pound (0.05 gram). .. 21 5.59 2.0s 5,000 0.054 0.0459 For the measurement of the diffusion constant 2.08 3,420 0.054 0.0459 5.59 22 of n-butane, a known quantity of oil was intro26 4.09 10,000 0.166 .. 0.1388 5.59 .. 0,1388 2.08 8,670 0.161 duced into chamber A of Figure 4. This liquid 27 5.59 0.0984 .. 5.59 2.08 4,210 0.113 56 was saturated with n-butane at the desired initial 62 4.09 2.08 18,710 0,106 0.0984 concentration, C". After equilibrium had been a Correaponds to conditions on liquid side of gas-liquid interface. attained, agitator L of Figure 5 was lowered b Based upon initial and equilibrium conditions. beneath the oil-mercury interface in order to avoid complication in the configuration of the liquid

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April, 1941

INDUSTRIAL AND ENGINEERING CHEMISTRY

the existence of a stable specific-weight gradient during an absorption process, the specific weight of the solution being less a t greater concentrations of dissolved ta-butane; but an unstable situation results when this hydrocarbon is withdrawn from the liquid phase across a horizontal gas-liquid interface. I n the investigation of the influence of agitation, stirrer L of Figure 5 was located in the liquid phase and was rotated a t a uniform rate of approximately 300 r. D. m. The rate of n-butane transfe; to and *from the liquid phase was determined under known concentration differences between the interface and the main body of liquid. I n the case of transfers to the solution the resulting specific-weight

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TABLE11. ABSORPTIONAND DESORPTION O F %-BUTANE AT 100' F. UNDER FORCED CONVECTIVE CONDITIONS Test

No. 33 36 43 51 53 55 60

Time of Initial Depth Measureof Liquid Weight of Equilibrium4 ment, Sec. Phase, Ft. Oil, Lb.

Conon., Lb./Cu. Ft. Initial 2.08 2.08 4.09 2.08 2.08 2.08 4.09

5.59 5.59 5.59 5.59 4.09 5.59 5.59

Absorution 960 1410 4750 1125 1020 2920 2633

0.142 0.143 0.100 0.098 0.098 0.098 0.105

0.1378 0.1378 0.0927 0.0984 0.0984 0.0984 0.0984

Depth of Stirrer below Surfaoe, Ft. 0.083 0.133 0.083 0.059 0.059 0.059 0.066

Desorption 34 35 54 61

a

5 59 2 08 7250 0 160 0 1377 5 59 2 08 5830 0.160 0 1378 4 09 2 08 2690 0.105 0 0984 5 59 4 09 1920 0 110 0 0984 Corresponds to conditions on liquid side of gas-liquld interface.

0 0 0 0

OS3 083 066 071

The measurements indicate a fairly regular increase in the effective film thickness as the concentration of the n-butane in the liquid phase approaches that a t the interface. This tendency is marked in the case of the behavior of tests 20 and 26 in which the concentration a t the interface was 4.10 pounds per cubic foot. The effective film thickness increased approximately fourfold during a decrease in the average concentration of the liquid phase from 6.0 to 4.55 pounds per cubic foot of liquid. I n this instance there was little if any bubble formation except immediately adjacent to the gasliquid interface. I n the case of tests 22, 27, and 56 there 10

20

30

40

50

60

70

80

90

(SECONDS)'

FIQURE6.

ABSORPTIONOF ?+BUTANE

CONDITIONS A T 100" F.

UNDER

QUIESCENT d U

'b'

gradients tended to reduce the rate of material transfer while they enhanced it during the withdrawal of the lighter hydrocarbon. Owing to the random appearance of bubbles and the inherent mechanical instability of the system, rather poor reproducibility was expected. However, the agreement between duplicate measurements, which is indicated in Figure 8, is typical of the data except for one or two instances when a large number of bubbles made their appearance simultaneously. I n these few cases the disagreement between essentially duplicate measurements was somewhat greater than that shown in Figure 8. Two typical sets of results a t different concentration differentials are presented in Figure 9. These show the quantity of n-butane evolved under natural convection conditions as a function of time. The rate of gas evolution is greater in proportion in the case of the larger concentration difference between solution at the interface and in the main body of liquid. This results from the larger corresponding difference in specific weight, coupled with the fact that a few more bubbles were formed during the early part of the process in the measurements involving the larger concentration difference. The results obtained under natural convection conditions are given in Table I. The general trends indicated in Figures 8 and 9 prevailed over nearly all of the conditions recorded in this table. The effective film thickness adjacent to the gas-liquid interface was computed by application of Equation 6 to each set of the experimental measurements in Table I. The results obtained for the desorption of n-butane under the influence of natural convection are presented in Figure 10 as a function of the concentration of this hydrocarbon in the liquid phase.

c'

'L

5:

"2 x c

P

B

2

0 3

k CONCENTRATION OF &BUTANE

LS. PER CU. FT

FIGURE 7. DIFFUSION CONSTANT FOR %-BUTANE AT 100' F.

was a sufficient difference in concentration to permit the formation of a few bubbles throughout the course of the desorption process which resulted in somewhat smaller initial film thicknesses and also a lower rate of increase in film thickness with decrease in n-butane concentration. It is probable that the appearance of bubbles caused the more nearly linear increase in film thickness with concentration indicated for the latter measurements. I n the case of test 62 there was little bubble formation, and the film thickness increased a t an increasing rate with a decrease in the concentration of *butane. The concentration a t the interface for all tests except 20 and 26 was 2.08 pounds per cubic foot. Under forced convection somewhat better agreement between duplicate measurements was obtained, as Figure ll indicates. All of the experimental conditions obtained

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

448

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Vol. 33, No. 4

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2000

1500

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SECONDS

DUPLICATE MEASUREMENTS

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2000

O F %-BUTANE

4000 TIME

FIGURE 9.

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D E S O R P T I O N O F n-BUT.4NE AT T W O COrVCEKTR4TION DIFFERFNCES

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FIGURE 10. EFFECTIVE FImi THICKNESSES IN THE LIQUID P H A S E UNDER A-.4TUR.IL CONVECTIVE COrVDITIONs AT 100" F.

with forced convection for both the absorption and desorption of the n-butane are recorded in Table 11. The relative rates of absorption and desorption of n-butane under otherwise identical situations with forced convection are presented in Figure 12. The variation in the rate of mass transfer across the interface is significant and indicates in a qualitative fashion the influence of specific-weight gradients. I n the absorption measurements the specific-weight differences tended to decrease the effectiveness of the agitation, while in the desorption study the specific-weight difference and the few random bubbles that were found enhanced the influence of the agitation. Effective film thicknesses for conditions of forced convection are given in Figure 13, which indicates that this quantity remains relatively constant throughout the entire absorption or desorption process. This type of behavior is to be expected from the simplified analysis of the mechanism of the process. Some variation in the film thickness is to be ex-

1000

2000

3000

TIME

SECONDS

4000

FIGCRE 11. DUPLICATE MEASUREMENTS OF THE ABSORP~IOW O F %-BUTANE UNDER FORCED CONVECTIVE COVDITIONS

pected since the viscosity and other properties of the liquid phase within the boundary layer undergo significant changes from the beginning to the end of a particular set of measurements. The effective film thickness appears to be roughly independent of the direction of the transfer, as would be expected from the nature of the process. There was a distinct tendency for the film thickness to decrease in cases where agitator L of Figure 5 was located near the gas-liquid interface. The experimental measurements were insufficient to permit a direct correlation of the effective film thickness based upon the rate of absorption and desorption with the hydrodynamic situation obtaining. However, the constancy of this thickness over a wide variation in the rate of mass transfer indicates the possibility of such a correlation. The information presented in Figures 10 and 13 permits the calculation of the rate of mass transfer across the liquidgas interface under either transient or steady-state condition.; by suitablr application of Equation 6. It is believed

April, 1941

INDUSTRIAL AND ENGINEERING CHEMISTRY

that this method of presenting the data is more useful than a detailed record of the experimental results. Suitable values for the concentration of n-butane a t the interface may be obtained from Table I11 as a function of pressure for a temperature of 100.0" F.

TABLE 111. SOLUBILITY OF 12-BUTANE IN CRYSTAL OIL AT 100" F. Abs.

Pressure. Lb./Sq. I n .

2

4 6 8 10

Abs. Pressuro,

Conccnrrat ion, Lh./Cu. Ft. 0.48 0.97 1.44 1.93 2.44

Lb., Sq. In.

12 14 10 18

Concentration. Lb.1 Cu. Ft. 2.95 3.51 4.15 4.84

Bubble Formation The apparatus for the investigation of the formation and growth of bubbles is shown in Figure 14: Essentially it consisted of a heavy glass-walled vessel, A , equipped with a throttle valve, B,so arranged that the fluid leaving the valve constriction was visible through the plane glass walls of chamber A . The fluid was circulated through the chamber by a ear pump, C, located within pressure vessel D and mechaniGayly driven by a rotating shaft through packing gland E . A suitable high-pressure mercury-in-glass manometer, F , was utilized to ascertain the change in pressure across throttle valve B . After leaving the observation chamber A , the fluid was returned t o the lower part of ressure vessel D, whence it passed upward through the l a b y r i d agitator, B, and into the inlet of circulating pump C . Provision was made for the transfer of mercury in either direction between pressure vessel D and chamber H . This permitted the effective volume of the working section to be varied in order to ascertain the influence of pressure independently. The pressure within observation chamber A was determined by a ressure balance, J, connected to the fluid within the chamber Ey a mercury seal at K through an oilfilled circuit. The entire apparatus was located within an air thermostat, L,whose temperature was automatically maintained within 0.15' F. of the desired value. A weighed quantity of the same oil employed in the convective and diffusion experiments was introduced into pressure vessel D, along with a weighed amount of propane. Circulation through the system was started, and the total effective volume was decreased by the addition of mercury from chamber H until the system was entirely liquid. The desired pressure differential was then set up across valve B, the corresponding flow rate being ascertained by a nozzle-type orifice meter, M , with mercury-in-glass manometer N . The by-pass valve, P, was employed to make these adjustments. The pressure upon the air-

449

culating system was decreased slowly until the pressure within the observation chamber had fallen t o the desired operating value. The velocity of flow within the observation chamber was varied from zero to approximately 4 feet per second while the pressure was changed from bubble point t o as low a value as the volumetric capacity of the circulating pump would permit with heterogeneous mixtures. The experimental work with this equipment was semiquantitative in nature. I n the initial part of the program the influence of solid surfaces upon supersaturation was investigated, and it was found that changes in solid surface area of a hundred fold exerted a significant influence only in the case of essentially quiescent liquids. Furthermore, marked angularities of surface such as are obtained with needle points, silicon carbide crystals, and fractured quartz became important only a t very low fluid velocities. Under quiescent conditions there was a distinct tendency for bubble formation a t points on the boundaries of the system characterized by exceedingly small radii. If the hydraulic radius of the homogeneous system was small, it was possible to obtain supersaturation of as much as 20 pounds per square inch under quiescent conditions a t all concentrations of propane less than 30 pounds per cubic foot of liquid at 100" E'. Above this concentration the behavior under quiescent conditions was no longer reproducible. It was found that temperature gradients induced sufficient convection in the liquid phase to reduce the degree of supersaturation attainable to a marked extent. Extraneous vibrations also were conducive to the formation of bubbles in an otherwise quiescent phase. I n general, slight movements of the liquid relative to the surroundings or to other parts of the system were of nearly controlling importance i n determining the degree of supersaturation obtained. I n the case of supersaturated liquids flowing through t h e observation chamber, the maximum turbulence of the stream determined the supersaturation attainable in this hydrocarbon system. Pressure differences of 3 to 5 pounds per square inch across the throttle valve reduced the minimum pressure obtainable without the formation of bubbles to less than 0.3 pound below bubble point. Furthermore, this behavior was roughly independent of the relative opening of the valve except when it was nearly closed. If the pressure difference

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1000 TIME

1500

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FIGURE 12. COMPARISON OF THE ABSORPTIONAND DESORPTION OF

BUTANE

FIGURE 13. EFFECTIVE FILMTHICKNESSES IN THE LIQUID PHASE UNDER FORCED CONVECTIVE CONDITIONS AT 100' F,

INDUSTRIAL AND ENGINEERING CHEMISTRY

450

across the valve was raised to approximately 10 pounds per square inch, exceedingly small bubbles were visible as a cloud when the static pressure within the observation chamber at the outlet of the throttle valve was from 0.2 t o 0.4 pound

FIGURE 14. APPARATUSFOR T H E FORM.4TION GROWTHOF BUBBLESIN SUPERSATCRATED

Vol. 33, No. 4

in B could be obtained at the same velocity by reducing the rate of flow from a higher value. At a flow rate of about 0.0168 cubic foot per minute of saturated liquid, the bubbles became exceedingly fine and the material was virtually a foam, as indicated in B. Furthermore, there was little if any change in bubble size throughout the length of the observation chamber while a significant growth of an average bubble resulted from its passage through the observation chamber a t the lower rates of flow, When the rate of liquid flow was increased to 0.0362 cubic foot of saturated liquid per minute, the fluid was made u p of a fine foam with irregular-shaped bodies of mist passing upward through the foam at somewhat higher velocities. I n this instance microscopic examination of the photograph failed to indicate any significant bubble growth during the passage of the fluid through the observation chamber. The variation in the appearance of a heterogeneous fluid under nearly constant degree of initial supersaturation shows the marked influence of velocity or turbulence upon the formation and growth of bubbles. At lorn velocities only a few bubbles were formed and these apparently increased rapidly

AND

LIQUIDS

per square inch above the equilibrium bubble point value with a concentration of propane in the incoming liquid of approximately 15 pounds per cubic foot. The foregoing qualitative description of the controlling influence of flow conditions upon supersaturated liquids is in accord with the earlier discussion of the probability of bubble formation. Apparently the vortices and other regions of low pressure associated with the disintegration of the jet leaving the nozzle-type throttle valve caused nuclei for the formation of bubbles. These localized regions of low pressure apparently were of sufficient magnitude to permit the existence of very small bubbles for a short period a t pressures in the main body of the solution slightly above the equilibrium bubble-point value. Motion pictures of the transient situation obtaining during the formation and disappearance of these small bubbles indicated that the foregoing conclusions were probably correct. However, the definition of the small bubbles was so poor as to render quantitative interpretation of the results impossible. The physical appearance of heterogeneous mixtures under various conditions is of interest in connection with the qualitative evaluation of the applicability of the earlier discussions. The influence of the velocity of flow in the section as a whole is indicated in the sequence of four photographs shown in Figure 15. The conditions obtaining in the section for mixtures of the oil and propane at 85" F. are given under each photograph. The equilibrium bubble-point pressure was approximately constant for all of the conditions shown. Likewise, there was only a small variation in the prevailing pressure, which was approximately 18 pounds per square inch below bubble point. A t a flow rate of approximately 0.0107 cubic foot of saturated liquid per minute through the rectangular section, 1 X 0.5 inch, the bubbles were large and somewhat regularly spaced, and rose through the liquid sufficiently fast to collect as a foam in the upper part of the observation chamber. When the velocity was increased only slightly to 0.0109 cubic foot per minute, the situation indicated in Figure 15B obtained. The bubbles nearly touched one another and were somewhat smaller than a t the lower rate of flow. However, this configuration was not entirely stable and tended to break down into that shown in A . I n the neighborhood of this transition it was possible to obtain the conditions indicated in A if the velocity of flow was approached from a lower value, while the situation indicated

AN OIL WELLDERRICK OF THE UNION OIL COMPANY OF CALIFORSIA

in size, while a t higher velocities a large number of bubbles were obtained. This large area of interfacial contact rapidly brought the phase to equilibrium and thereby inhibited further bubble growth except that due to the change in the specific volume of the gas phase.

April, 1941

INDUSTRIAL AND ENGINEERING CHEMISTRY

451

suddenly to 6 pounds per square inch, it is possible to estimate the time required for the average concentration of n-butane in the liquid phase to fall to any predetermined value, such as 2.44 pounds per cubic foot of solution. I n this instance it wil€ be assumed further that the body of liquid is initially 0.11 foot thick, that the partial volume of n-butane in the liquid phase may be taken as constant a t a value of 0.0282 cubic foot per pound, and that isothermal conditions obtain. For the purposes of this example Equation 6 may be rewritten in the following form for 1 square foot area of liquid-gas interface:

A material balance for a portion of the solution involving the square foot of area exposed t o the gas phase yields : ck

=

mlc

n2rc

-i10

B

A

Pressure lb./eq. in. Satn. pr&ure, 1b./sp. in. Flow rate, cu. ft./min.

D

C

A 32.4 50.3

0.0107

B 33.0 51.3 0.0109

C

33.4 51.4 0.0168

(14) mb)

Equation 14 involves the assumption of a constant partial specific volume of n-butane. As an approximation the data of Figure 7 may be expressed in the following form in which the average concentration obtaining- in the film has been employed :

D 33.6 50 5 0.0362

OF HETEROGENEOUS HYDROCARBON MIXTURES AT SEVERAL FIGURE 15. APPEARANCE RATESOF FLOW

Applica Lions

-+ vkz(mi -

Dk = 1

The diffusion constants for n-butane reported in Figure 7 find their primary application in the interpretation of experimental data relating to the absorption and desorption of this component in a liquid phase. However, they may be employed directly in connection with the evaluation of the rate of solution of n-butane in certain quiescent hydrocarbon liquids. Essentially quiescent conditions are sometimes encountered in underground petroleum reservoirs, and these diffusion constants may be of some value in connection with the estimation of the influence of repressuring operations upon the properties of the liquid phase. When used for such purposes, these data must be incorporated with similar information for the other important gaseous constituents of naturally occurring hydrocarbon mixtures. Information is required for both gas and liquid phases. Furthermore, a detailed knowledge of the configuration of the liquid and gas phases within the reservoir is necessary before such calculations may be applied to an actual situation. A comparison of the experimentally determined rates of absorption and desorption of n-butane under quiescent conditions with those under natural convective conditions indicated the relatively large magnitude of gravitational influences in inducing the circulation of liquid under desorption conditions. Data of this nature permit the estimation of the time required for a change in concentration of a body of liquid exposed to a gas phase exhibiting a smaller fugacity of n-butane than that obtaining in the liquid phase. For example, if an oil having an average molecular weight of 347 is saturated with n-butane a t a pressure of 16 pounds per square inch and the pressure in the gas phase is decreased

The apparent film thickness may be expressed as a linear function of the concentration without introducing significant uncertainties as long as the average concentration of the solution does not approach the equilibrium value a t the interface. On the basis of this assumption the data of Figure 10 for an initial depth of liquid of 0.11 foot may be approximated by

If the foregoing expressions for the concentration, diffusion constant, and apparent film thickness are substituted in Equation 13, a relation of the following form results:

_ de dmk

(a (D

- pmh) ( Y - e m ) - Xmd ( P - vmd

(17)

The coefficients of the terms of Equation 17 are recorded below along with the numerical values corresponding to the example in question; in addition, the information recorded in Table I11 was employed in evaluating the numerical values of the coefficients: C Y =

(lo

p

7~= 0.0282

=

VkLntt) =

0.1229

y = 16a-= 1.966 E = (16Vk 2.6) = 3.051

+

INDUSTRIAL AND ENGINEERING CHEMISTRY

452 7 = X = I.L = Y

=

+

(2.42 0.85C&)a = 0.4478 [0.85(1 - C K ~ V E-I )2.427k1 Ck,a = 0.1769 (1 Ck,Vai) = 1.0406

=

0.7472

+

If these numerical values are substituted in Equation 17 and the right-hand member of the equation is separated into partial fractions, we obtain: 2.858

8.868

dB

- = -1‘1065

0.5993

dmk

+

mb

-k 0.1700

- mk

(18)

The integration of Equation 18 yields: $1

- Bo

= 1.6065(%

-

mk)

0.5993 + mi + 8.868 In 0.5993 f mi 2.858 0.1700 - m6 (19) 0.1700

-

mt

mission to publish the results. The assistance of W. E.Lacey in connection with the direction of the experimental program and the preparation of the manuscript is acknowledged. H. H. Reamer carried out the measurements relating to the formation of bubbles in supersaturated liquids. Randlow Smith and I. F. German, Jr., contributed materially to the evaluation of the experimental findings relating to the formation and growth of bubbles and to the detailed checking of the calculations pertaining to the mass transfer between phases. L. M. Reaney and D. A. Emberson assisted with the calculations and the preparation of the figures.

A C

D

f

m

If it is assumed, as was stated earlier, that the desired final

P

concentration of the liquid phase is 2.44 pounds per cubic foot, which corresponds to 0.2864 pound of n-butane per square foot of exposed area, Equation 19 indicates that 1.52 hours would be required for the average concentration to decrease to this value under the conditions stipulated in the example. The importance of turbulence in the formation of bubbles from supersaturated hydrocarbon liquids indicates a t least qualitatively the type of equipment which may be employed to approach equilibrium in certain types of processes. For example, the separation of liquid and gas phases in oil production might be expedited by inducing turbulence in the fluid prior to or in connection with its entrance into the separating chamber. Such a procedure may induce the formation of bubbles from a supersaturated solution and thereby permit their separation a t an advantageous point in the process.

T

Summary The diffusion constant of n-butane in a nonvolatile hydrocarbon liquid of relatively high molecular weight increased rapidly with an increase in the concentration of n-butane. The usual concept of an effective boundary layer adjacent to a gas-liquid interface afford3 a useful means of presenting the results of experimental measurements relating to the absorption and desorption of a one-component gas in a nonvolatile liquid. Under isothermal conditions natural convection yields a thickness of the effective boundary layer in the liquid phase which increases rapidly as the concentration of the diffusing component in the main body of the solution approaches the concentration of that component a t the gasliquid interface. The present experimental results indicate that the effective boundary layer thickness is relatively constant over a wide range of concentration differences between that in the main body of the phase and that a t the gas-liquid interface. Turbulence appears to be of nearly controlling importance in establishing the degree of supersaturation obtained in single-phase liquids. Under the conditions of flow normally encountered in circulation conduits, it is unlikely that marked divergences from equilibrium persist for extended periods. However, in laminar flow such as is usually encountered throughout a large part of petroleum reservoirs containing only liquids, rather large degrees of supersaturation are possible. Aclinowledgment This work was carried out as a part of a program of research sponsored by the Cnion Oil Company of California, t o whom the authors are indebted for financial assistance and for per-

Vol. 33, No. 4

T u

Ti

X

Y

6 y

6 8

Nomenclature area normal t o Concentration gradient, sq. ft. = concentration, lb./cu. ft. = diffusion constant, sq. ft./hr. = fugacity, lb./sq. in. = weight of material transferred acrow an interface, lb. = pressure, lb./sq. in. abs. = radius, f t . = thermodynamic temperature, ’ R. = velocity, ft./sec. = total volume of system, cu. ft. = partial volume, cu. ft./lb. = distance in direction of concentration gradient, ft. = distance normal t o direction of flow, ft. = correction to diffusion constant for expansion of liquid = interfacial tension, lb./ft. = effective film thickness, ft. = time, sec. =

Superscripts 0 = initial conditions ’ = supersaturated conditions Subscripts B = conditions within a bubble = bubble point = relating to concentration = equilibrium ronditions with an infinite radius of interface = film or boundary layer = gas phase = component k = liquid phase Literature References Blausius, Z . Math. und Physik, 56 (1908). Burgers, Proc. 1st Intern. Congr. A p p l . Mech., Delft, 1924. Dryden, IND.ENG.CHEM., 31, 416 (1939). Fick, P o g g . Ann., 94, 59 (1855). Hill and Coats, IND. EKG.CHEM., 20, 641 (1928). Hill and Lacey, Ihid., 26, 1327 (1934). Hutchinson and Sherwood, ISD.ENG.CHEM.,29,836 (1937). Kdrmdn, T.yon, J . Aeronaut. Sci., 1, 1 (1934). Ibid.. 4, 131 (1937). Lewis and Chang, Trans. Am. Inst. Chem. Engrs.. 21, 127 (1928). Lewis and Randall, “Thermodynamics”, p. 262, New York, McGraw-Hill Book Co., 1923. Maxwell, “Scientific Papers”, Vol. 2, p. 343, Cambridge Univ. Press, 1890. Pomeroy, Lacey, Scudder, and Stapp, IND.ENQ.CHEM.,25, 1015 (1933). Rouse, “Fluid Mechanics for Hydraulic Engineers”, p. 76, New York, McGraw-Hill Book Co., 1923. Rouse, Proc. Am. SOC.Civil Engrs., 62,21 (1936). Sage, Inman, and Lacey, IND. ENG.CHEM.,29, 288 (1937). Sage, Webster, and Lacey, Ihid., 29, 1118 (1937). 3herwood, “Absorption and Extraction”, p. 31, New Yolk, McGraw-Hill Book Co., 1937. Sherwood, Draemel, and Ruckman, IND. ENG,CHEX., 29, 282 (1937). Sherwood and Holloway, Trans. Am. Inst. Chem. Engrs., 36,21 (1940). Sherwood and Woertz, IND.ENG.CHEM.,31, 1034 (1939). Stefan. Sitzher. A k a d . W i s s . Wien, 63,( 2 ) 63 (1871). Ibid., 79, (2) 177 (1879). Whitman and Keats. J. IUD.END.CHEW,14. 186 (19‘221.