Mass Transfer and Kinetics in Spray-Tower-Loop Absorbers and

Oct 12, 2000 - Therefore, in this work, the basis for modeling spray-tower-loop absorbers .... corresponding on a plane to a greater concentration at ...
0 downloads 0 Views 255KB Size
Download PDF
4082

Ind. Eng. Chem. Res. 2000, 39, 4082-4093

Mass Transfer and Kinetics in Spray-Tower-Loop Absorbers and Reactors A. Dimiccoli, M. Di Serio, and E. Santacesaria* Dipartimento di Chimica, Universita` di Napoli “Federico II”, via Cintia, 80126 Naples, Italy

In this work we have studied the kinetic and mass-transfer behavior of spray-tower-loop absorbers and reactors. For this purpose, we have investigated mass-transfer rates in the physical absorption of CO2 in water and the effect on mass transfer of respectively a moderately fast reaction, such as the ethoxylation of nonylphenol or fatty alcohol, and an extremely fast reaction, such as the one between CO2 and an aqueous NaOH solution. Spray nozzles used have been geometrically and fluid-dynamically characterized. Experimental results have been interpreted by considering the drops to be internally stagnant or well-mixed. We have shown that an internally well-mixed drops model is more reliable. Moderately fast reactions, such as ethoxylation, (Hatta number of about 1) do not affect mass transfer, while extremely fast reactions require the introduction of an enhancement factor. We have proposed a semiempirical correlation for calculating this factor. Our work could be useful for modeling and simulating spray-towerloop absorbers and reactors. 1. Introduction Spray nozzles are widely used industrially for many types of applications such as drying, surface coating, solvent evaporation, gas stream depuration, gas-liquid reactions, etc. In all of these applications, the masstransfer exchange among the gas and liquid phase is a crucial factor of the process. Despite the importance of spray nozzles in industry, few papers dealing with the mass-transfer mechanisms occurring when a liquid is sprayed in a gaseous atmosphere have been published,1 especially when a chemical reaction is also involved. In many papers2-4 empirical correlations have been proposed to describe and characterize the mass-transfer operation for a given spray nozzle. A more general approach was given by Crank,5 Johnson et al.,6 and Srinivasan and Aiken,7 who studied the mass transfer for the absorption of a gas into a liquid drop, with the first two considering the drop to be internally stagnant and the last considering the drop to be internally wellmixed. Only one paper published by Hall and Agrawal8 describes mass transfer and kinetics in a spray-loop well-stirred reactor in which an ethoxylation reaction occurs. However, mass-transfer and kinetic models, adopted by the authors, have given poor results in simulating the performed experimental runs. In this work we have studied, therefore, the masstransfer occurring in spray nozzle devices operating in both the absence and presence of a chemical reaction. For this purpose we have, first of all, considered the mass-transfer rates in the physical absorption of CO2 in water and then the effect, on the mass-transfer rate, respectively of a moderately fast reaction, such as fatty alcohol and nonylphenol ethoxylation, and an extremely fast reaction, such as the one between CO2 and an aqueous solution of NaOH. Drops emerging from the used spray nozzles have been characterized as (i) the drops average diameter and cone angle amplitude of the emerging drops system, (ii) * To whom correspondence should be addressed. E-mail: [email protected].

the mean flight path of the drops, and (iii) the average flight time of the drops. Then, experimental data, in all of the mentioned cases, have been interpreted by adopting two different approaches: one considering droplets to be internally stagnant, according to the Crank5 and Johnson et al.6 approaches, and another one considering droplets to be internally well-mixed, in agreement with the suggestion of Srinivasan and Aiken,7 on the basis of Levich’s9 theory. By comparison of the results obtained in simulating experimental runs with the two mentioned approaches, it is found that drops must always be considered to be internally well-mixed, for an efficient spray nozzle. When gas absorption into the drops occurs in the presence of a very fast reaction, such as the one between CO2 and an aqueous solution of NaOH, mass-transfer rates are enhanced. A semiempirical correlation will be proposed, in this work, that is able to reproduce quite well the collected experimental data. Therefore, in this work, the basis for modeling spraytower-loop absorbers and reactors is given. The mathematical model developed can also be used for simulating the behavior of the mentioned absorbers and reactors. 2. Experimental Section 2.1. Reagents, Methods, and Apparatuses. 2.1.1. Physical Absorption in the Absence of a Chemical Reaction: The Physical Absorption of CO2 in Water. Runs of physical absorption of CO2 in water have been made in the laboratory spray system, schematized in Figure 1, by using the following reagents: a bottle of CO2-grade 4.5, at 50 atm of pressure, supplied by the Sol Gas SpA, bidistilled water. The employed apparatus of Figure 1 is a semibatch plant having a 4.0 L spray-tower reactor with a single spray nozzle (for details, see Table 1), a heat exchanger, a gear pump, a feed line of the gaseous reagent with a flowmeter interfaced to a hardware/software automation system, and an acquisition board interfaced to a personal computer with “LabTech Control” automation software. The spray tower and heat exchanger have been made

10.1021/ie000137y CCC: $19.00 © 2000 American Chemical Society Published on Web 10/12/2000

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4083 Table 2. Operative Conditions of Experimental Runs of CO2 Physical Absorption in H2O run

h (cm)

h - hL (cm)

liquid vol. (dm3)

temp (°C)

PCO2 (atm)

CO20 CO21 CO22 CO23 CO24 CO25

26.6 26.6 26.6 20.8 20.8 20.8

13.6 12.4 12.1 6.2 7.8 6.5

1.640 1.776 1.820 1.833 1.638 1.796

28 17 17 26 28 28

2.06 2.33 4.00 5.87 2.24 3.92

Table 3. Operative Conditions of Experimental Runs of CO2 Chemical Absorption in NaOH(aq), All Made at 46 °Ca

Figure 1. Laboratory spray-tower-loop plant. Table 1. Spray Nozzle Main Features (Nozzle Type: DAU.1118.B3.PNR) and Characteristic Mean Valuesa of Drop Diameter with Liquid Flow Rate Q ) 1.18 L/min thread drops cone type cone width angle (deg) D10 (µm) D20 (µm) D30 (µm) D32 or DS (µm)

1/8 in. BSPT full cone 90 108.3 116.7 122.9 136.3

a The mean values are estimated with the following equation: Dmn ) ∑iniDim/∑iniDin.

by D’Arco and Lazzarini Co. (Pontecagnano, Italy); the gas control system and flowmeter were supplied by Bronkhorst Co.; the gear pump was supplied by Pompe Cucchi; Swagelock pipes and fittings were supplied by Technofittings; automation hardware was supplied by Analog-Device Co.; and the spray nozzle was supplied by PNR Italia SpAsthe main nozzle features are reported in Table 1. The spray nozzle can be set at different distances from the bottom of the reactor. A preset temperature is reached by heating the system through the heat exchanger. When a valve of the gaseous reagent line is opened, the reactor head is filled with a fixed pressure of CO2; during such a step, the recirculation pump is not working to minimize the absorption of CO2 due to the start-up. The loaded liquid is continuously recirculated by the gear pump passing through an external loop until the head of the reactor, where the nozzle nebulizes it, forming a large number of droplets. These drops continuously absorb carbon dioxide from the gaseous atmosphere, and absorbed CO2 is promptly replaced by keeping the pressure constant. The carbon dioxide consumption is measured and recorded by the flowmeter apparatus. In Table 2, operative conditions of the CO2 in waterabsorbing runs, performed in the described spray system, are reported. In all of the runs, the inert pressure was PN2 ) 1 atm while the recirculating liquid flow rate was Q ) 1.18 dm3/min. 2.1.2. Absorption in the Presence of a Moderately Fast Reaction Not Affecting the MassTransfer Rate: Ethoxylation of Nonylphenol and Fatty Alcohols. Kinetic runs of ethoxylation have been performed starting respectively from nonylphenol and fatty alcohols (a dodecyl alcohol-tetradecyl alcohol mixture) in the presence of a traditional alkaline catalyst such as NaOH or KOH. The reactions were

run

[NaOH] initial (mol/dm3)

h - hL (cm)

liquid vol. (dm3)

PCO2 (atm)

CO11 CO12 CO13 CO14 CO15 CO16 CO17 CO18

0.588 0.556 0.591 0.748 0.764 0.713 0.719 0.260

10.9 10.9 11.0 11.0 11.0 14.7 18.9 10.9

1.954 1.961 1.941 1.943 1.941 1.503 1.011 1.956

0.28 0.43 0.82 0.27 0.54 0.24 0.27 0.70

a

In all cases, h ) 26.6 cm.

performed in a pilot plant similar to the one schematized in Figure 1 and differing only in plant size. The heat exchanger on the external recirculating loop, initially used for heating the liquid, is also used to cool the reacting mixture during the run, with the ethoxylation reaction being highly exothermic. Two experimental runs have been performed in a pilot plant of 150 L (for details, see Table 5) with the following conditions: (i) Nonylphenol (15.20 kg) ethoxylation in the presence of NaOH (15.0 g). The temperature was kept at about 170 °C, the total pressure at about 5 atm, and the inert pressure (N2) at about 0.2 atm. (ii) Dodecyl alcohol-tetradecyl alcohol mixture ethoxylation (EPAL1214, MW = 195, 15.00 kg) in the presence of KOH (25.0 g). The temperature was kept at about 160 °C, the total pressure at about 6 atm, and the inert pressure (N2) at about 3.0 atm. 2.1.3. Absorption in the Presence of an Extremely Fast Reaction Affecting the Mass-Transfer Rate: Chemical Absorption of CO2 in an Aqueous NaOH Solution. In this case the experimental procedure is similar to the one used for physical absorption of CO2 in water, with the only difference being the liquid phase loaded into the reactor, that is, a sodium hydroxide solution of known concentration. Also, in this case carbon dioxide consumption is measured and recorded by a flowmeter interfaced with a computer. The NaOH reagent was supplied by the Sigma-Aldrich in pellets. In Table 3 operative conditions of the experimental runs, performed in the laboratory spray system of Figure 1, are reported. In all of the runs, the inert pressure was PN2 ) 1 atm and the recirculating liquid flow rate was Q ) 1.18 L/min. 2.2. Geometric and Fluid Dynamic Characterization of the Drops System. 2.2.1. Drops Size Distribution in the Reactor. A geometrical parameter of a spray nozzle, normally furnished by the supplier, is the form of a drops cone (full cone or hollow cone), as well as the cone width angle (R ) 60°, 90°, and 120°). A photo (see Figure 2) of the spray cone can be useful to better define these geometric properties and the uniformity of the cone drops system. A laser-

4084

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

Considering case I, it is possible to assume, for example, that the number of droplets collected in a circular ring, with radii between r and r + dr, is proportional to r itself. In this case, corresponding to a uniform distribution of the drops on the surface of the liquid column, the average path of the droplets can be calculated with the following relation:

dm )

∫0r rx(h - hL)2 + r2 dr

2 rL2

L

(1)

Another possibility is to assume that the number of droplets collected in a circular ring, with radii between r and r + dr, is constant and independent of r. This corresponds to a uniform distribution on the radius r and to a drops concentration over the orthogonal section that is higher at the center and lower at the periphery of the surface. Always considering case I, the average flight path, in this case, will be

dm )

Figure 2. Drops distribution detail for the full cone spray nozzle installed on the laboratory plant of Figure 1.

Figure 3. Typical numerical distribution of drops size of a full cone spray nozzle with R ) 90° and for a feeding liquid flow rate of 1.2 dm3/min.

scattering technique has been used to estimate the numerical distribution of the drops diameter in relation to the angle R amplitude and the liquid flow rate feeding the nozzle (see Figure 3). From this distribution, it is possible to evaluate some characteristic mean values such as, for example, the Sauter mean diameter DS (see Table 1). 2.2.2. Mean Flight Path of the Drops. Different approaches have been tested to evaluate the mean flight path of the drops. In our case, drops could be approximately considered to be uniformly distributed on the spherical surface of the cone nozzle, as can be appreciated in Figure 2. A drop falling into the tower can impact both on the free liquid surface (case I in Figure 4) and over the internal walls of the tower (case II in Figure 4).

1 rL

∫0r x(h - hL)2 + r2 dr L

(2)

When the two relations are applied to our case, relation (1) gives values that are greater by about 6% than the ones obtainable with relation (2). Both the relations could be, therefore, used with acceptable approximation, but we preferred relation (2), because, according to us, it better approximates a system of drops uniformly distributed on the spherical surface of the cone nozzle, corresponding on a plane to a greater concentration at the center. To easily extend the use of relation (2) both to case II of Figure 4 and to any kind of reactor geometry, we solved this relation numerically by using a Monte Carlo method. A random variable pi (0 e pi e 1) was introduced with a uniform probability density function. Any generated number corresponds to a point of drop impacting, in case I of Figure 4, between 0 and rL and hence to a definite trajectory. When a large number of trajectories is obtained, an average is made to obtain the average flight path. This computation can be applied to the different cases I and II of Figure 4 as follows: Case I. When the liquid level (hL) is such that hL g h - h0, the drops will always meet the free liquid surface and will never coalesce over the tower walls at the end of their flight. The rL value depends on the liquid level hL; considering the triangle with side h0 and r, the area of its surface is equal to the sum of the areas of triangles A and B and of rectangle C (see Figure 4); therefore,

rh0 rL(h - hL) (r - rL)(h0 + hL - h) ) + + 2 2 2 rL(h0 + hL - h) (3) Elaborating relation (3), we have then

rL ) r

(h - hL) h0

(4)

From Pythagora’s theorem, the following particle path results:

di ) x(h - hL)2 + (rLpi)2

(5)

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4085

Figure 4. Scheme of the spray-tower section. The drops spraying geometry is shown. On the right is shown the case for hL g h - h0 (case I); on the left is shown the case for hL < h - h0 (case II).

of drops (N f ∞), we obtain the mean value of the drops flight path dm:

that is

di )

x

(h - hL)2 +

[

r (h - hL)pi h0

]

2

(6)

Computing the numerical average on the individual drop paths for a large number of drops (N f ∞) and for a given value of hL, we obtain the mean flight path as N

dm )

di

∑ i)1 N

N

dm )

(7)

Case II. When the liquid level hL is such that hL < h - h0, the drops can impact both on the free liquid surface of the liquid (case IIa) and over the internal walls of the reactor (case IIb) (see Figure 4). By assuming, as in the previous approach, a uniform distribution of the drop on the line along which the drop can impact, we can again define a random variable pi (0 e pi e 1), according to which

h# ) h0 + h*pi

(8)

h* ) h1 + r

(9)

dj ) xh#2 + r2

(10)

Case IIb. If, instead, h# > h - hL, that is, h# > h0 + h1 and h*pi > h1, then the drop impacts on the free liquid surface in a point with a radius rj:

rj ) r(1 - pi)

(11)

(13)

F′TOT ) F′P - F′G - F′R

(14)

where F′TOT is the resultant force, F′P the weight force, F′G the floating force, and F′R the resistance of the medium, all normalized with respect to the particle volume. The second equation is the rate expression for the uniformly accelerated motion:

v ) a′t + v0

(15)

Variations of the drop falling speed and of the drop flight path during the flight are calculated, then, by solving the system of these two differential equations (the problem of Cauchy):

{

(FP - F) d2x dx 2 F )g - 3CD 2 F dt 4FPDP dt P

(16)

dx d2x ) t 2 + v0 dt dt

(17)

( )

with the boundary conditions

x

at time t ) 0 x ) 0 and v ) v0 ) φ

In this case the flight path of the drop is

dj ) x(h0 + h1)2 + rj2

∑ j)1 N

2.2.3. Average Flight Time of the Drops. The average flight time of the drops is calculated by solving a system of two differential equations; the first is the expression of the resultant force, acting on a drop, normalized with respect to the volume of the particle. To simplify force balance calculations, we have considered only the vertical falling path of the drop; the final result of the model is not affected by this simplification:

where

with h* representing the maximum extension of the line along which the drop can impact (wall + radius). Case IIa. If h# E h - hL, that is, h# e h0 + h1 and h*pi e h1, then the drop impacts over the wall of the reactor and the particle flight path can be determined as

dj

2(∆P)nozzle FP (18)

(12)

Again making a numerical average on a great number

The coefficient of form CD is calculated by successive approximations as a function of the Reynolds number

4086

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

(NRe ) DPvFP/µL), which, in turn, depends on the drop speed (v) that we are estimating. The relation between CD and NRe, for a spherical particle falling across a fluid under the influence of an outside force, can be evaluated from data reported by Brown et al.10 The resulting correlation is

log(CD) ) 1.355 - 0.806 log(NRe) +

However, Astarita11 and Danckwerts and Kennedy12 suggested that drops emerging from a spray nozzle are, normally, internally well-mixed. Srinivasan and Aiken7 have, recently, developed a mass-transfer model for internally well-mixed drops, based on Levich’s theory,9 and by examining many experimental data have given the following correlation between dimensionless numbers for calculating kL:

Sh ) 0.16Sc1/2We1/2Re5/16

0.0817[log(NRe)]2 (19) The v0 expression is Bernoulli’s equation applied between the inlet and outlet of the nozzle:

1 F v 2 + FPgzIN + PIN ) 2 P IN 1 2 F v + FPgzOUT + POUT (20) 2 P OUT with zIN ) zOUT and vIN2 , vOUT2. The problem solution of our interest is that where x(t) ) dm (the mean flight path of the drop); the corresponding time (t) is the average flight time (tflight). 2.2.4. Drops Interfacial Area and Mass-Transfer Parameters. The interfacial surface area estimation can be made with the relation

aG ) 6Qtflight/D32

(21)

For an internally stagnant drop, according to Crank4 the gas diffusion into the spherical drop can be determined by integrating the following equation:

(

)

∂2Cr 2 ∂Cr ∂Cr ) PA + 2 ∂t rP ∂t ∂rP

(22)

Crank4 has given an analytical solution to relation (22), furnishing, for a given time, the concentration profile of the gaseous species inside the drop: Cr - C b Ci - C b

)1+

DS

∞

∑

-1

trP n)1 n

sin

( ) ( 2πnrP DS

exp -

)

4PAn2π2t DS2

(23)

The mean absorbed gas concentration after the flight time can be calculated with the relationship

CFV - Cb Ci - Cb

)1-

6

∞

∑

1

π2 n)1 n2

(

exp -

)

4PAn2π2tflight DS2

(24)

The previous approach is based on the molecular transport theory. Another approach to study the masstransfer phenomenon is based on the classical liquid film theory considering the resistance, to the mass transfer, confined in the quiescent hydrodynamic film. On the basis of this theory, Hall and Agrawal,8 recently, have proposed a model for a spray reactor in which the drops were considered to be internally stagnant. Within that model kL was estimated on the basis of the Johnson’s correlation:6

kL ) -

[

]

πxPAtflight DP ln 1 tflight D /2 P

(26)

and hence

( )(

PA(0.16) µL kL ) DP F P PA

1/2

)( )

v2FPDP σ

1/2

DPvFP µL

5/16

(27)

According to Levich’s9 theory, very near to the free surface of the turbulent liquid there is a region mathematically equivalent to a diffusive substratum. In this region, of thickness δ1, the mass transfer occurs only by molecular diffusion. Larger than this region, and more internal, there is a zone, of thickness δ2 (with δ2 . δ1), in which the mass transfer occurs for whirling motion. The Srinivasan and Aiken mass-transfer coefficient is valid only for a falling drop. Certainly, in a spray system other contributions to the global mass transfer exist and could have their effect in various spray-tower zones, such as, for example, the following: (i) Mass transfer through the free surface of the liquid column; we have estimated that this contribution is less than 2% of the overall mass transfer. (ii) Mass transfer through the liquid film falling along the tower walls, formed when the drops impact over the walls; also this contribution has been neglected because the corresponding kL is of about 2 orders of magnitude lower than the kL of the drops. (iii) Mass transfer through the conical liquid layer, formed immediately near the nozzle outlet, before its fragmentation to give drops; in this case Hall,13 suggests that this contribution is negligible. (iv) Mass transfer effects due to coalescence or breaking of the flying drops; again estimating this contribution, Hall found that it is small. However, laser scattering examination of the drops, in different points of the spray cone, shows that distribution does not change during the time and changes very slightly in different points of the cone. This suggests that coalescence would not have a remarkable role. Therefore, by simulating the experimental runs, apart from the model in examination (internally stagnant or well-mixed drops), we will assume that the only important contribution to mass transfer occurs during the drops flight. The practical difference existing between the two considered models, beyond the different theoretical assumptions, is the mass-transfer rates: those calculated with the internally well-mixed drop model are much greater than the ones calculated with the internally stagnant drop model: typical kL values calculated on the basis of the internally well-mixed drop model range from 1 × 10-2 to 3 × 10-2 m/s. 3. Results and Discussion

(25)

3.1. Physical Absorption in the Absence of a Chemical Reaction: Physical Absorption of CO2 in Water. The mass balance on a single drop, related

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4087

Figure 5. Solubility data of CO2 in water at different pressures and temperatures taken from the literature.14,15 Table 4. Simulation Parameters for the CO2 Physical Absorption in H2Oa run

DCO2 (m2/s)

FP (kg/m3)

µL [kg/(m s)]

σ (kg/s2)

µG (Pa s)

CO20 CO21 CO22 CO23 CO24 CO25

2.12 × 10-9 1.55 × 10-9 1.55 × 10-9 1.95 × 10-9 2.12 × 10-9 2.12 × 10-9

996.2 998.8 998.8 997.0 996.2 996.2

8.327 × 10-4 10.81 × 10-4 10.81 × 10-4 9.40 × 10-4 8.33 × 10-4 8.33 × 10-4

7.14 × 10-2 7.31 × 10-2 7.31 × 10-2 7.20 × 10-2 7.14 × 10-2 7.14 × 10-2

1.50 × 10-5 1.48 × 10-5 1.48 × 10-5 1.60 × 10-5 1.50 × 10-5 1.50 × 10-5

a For all of the experimental runs, V 3 LOOP ) 94 cm , DREACT ) 12.3 cm, and h0 ) 6.15 cm.

to CO2, in the volume element included between the flight coordinates z and z + dz, is

dnGCO2 dz ) NCO2 dz dt

(28)

In all of the examined experimental runs, the contribution of the gas-side transport resistance can be neglected; therefore, the mass flow expression for CO2 can be written as

NCO2 ) kLaG[CiCO2 - CGCO2]

(29)

By combining the two relations (28) and (29), multiplying, and dividing the second member by mean drop volume VG, we obtain

dnGCO2 dt

)

kLaG (VGCiCO2 - nGCO2) VG

(30)

By integration of this equation, the trend of CO2 consumption during the time can be obtained. Integration is, however, subordinate to the evaluation of (1) the interfacial area (aG), (2) the liquid-side mass-transfer coefficient (kL), and (3) the solubility of CO2 in water. The interfacial area and the mass-transfer coefficient can be calculated by using the previously defined procedures (see section 2.2.4). CO2 solubilities in water, at different temperatures and pressures, are reported by the literature.14,15 These data are plotted in Figure 5. Parameters employed in the simulations are represented in Table 4. To simulate the experimental runs

Figure 6. Examples of comparison of the performances of the two considered models in the simulation of CO2 consumption in the runs CO22 and CO25.

of Table 2, it is possible to distinguish two internal zones of the reactor: the flying drops zone of conical form and the liquid pool containing the liquid collected in the cylindrical reactor. a. Flying Drops Zone. For the simulation of the absorption, occurring in this zone, in the case of an internally well-mixed drop model, eq 30 is integrated with respect to the mean flight time, that is, from time t ) 0 to t ) mean flight time (internal integration). Internal integration must be repeated up to the overall process time, that is, when the experimental run is stopped (outside integration). However, mass-transfer parameters must be continuously updated during the internal integration because they are depending on the instantaneous drop speed, which, in turn, changes during the flight. In the case of the internally stagnant drop model, we used the original Crank’s5 procedure to evaluate the mass-transfer rate. Drops flying with a very high speed rate generate gas waves perturbing and mixing the gas phase that would have, for this reason, a relatively uniform composition. b. Liquid Column Zone. Drops fall into the liquid column with a CO2 concentration that is greater than the one present, at the same time, in the liquid column. Because drops impact on the free liquid surface with high speed and considering that the liquid level in the reactor is relatively low (about 15 cm) and the liquid recirculation rate is relatively high, we have assumed, for the calculations, the liquid to be well-mixed. Consequently, the reactor outlet concentration will be considered to be equal to that present, at the same time, in each point of the liquid bulk mass, and it will be updated after any flight time, during the integration. Examples of simulation, obtained by integrating eq 30 and by assuming drops to be internally well mixed, can be appreciated in Figure 6 related to runs CO22 and CO25, respectively. In the same figure simulation results obtained by assuming drops to be internally stagnant are also reported for comparison. As can be seen, the model considering the drops to be internally well mixed is more reliable, in agreement with the suggestions of Srinivasan and Aiken,7 and is able to simulate also those runs in which the drops saturation with CO2 is incomplete, because of the short mean flight path limited by lowering of the distance between the spray nozzle and the liquid surface. In Figures 7 and 8, the drop CO2 saturation levels, achieved during the flight, at different absorption times

4088

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

the drop system having an elevated interfacial area and a high mass-transfer coefficient. Chemical reaction, on the contrary, occurs mainly in the liquid column. Ethylene oxide dissolved is gradually consumed along the liquid column from the top to the bottom, with a conversion depending on both the chemical reaction rate and the liquid recirculating flow rate. According to our estimation, in the experimental adopted conditions, ethylene oxide (EO) mass transfer is liquid-side-limited, with the gas-side contribution being very small; so, making a material balance on a single drop, related to EO, results in

dCEO kLaG i ) (C EO - CEO) dt VG Figure 7. Drop CO2 saturation level achieved during the flight in the run CO22 at different run times calculated by a well-mixed drop model. An example of the saturation level calculated by a stagnant model has shown for run time of 0 min. In the small figure the evolution of the drop speed during the flight is also reported (average flight time ) 6.5 × 10-3 s).

Figure 8. Drop CO2 saturation level achieved during the flight in the run CO25 at different run times calculated by a well-mixed drop model. An example of the saturation level calculated by a stagnant model has shown for run time of 0 min. In the small figure the evolution of the drop speed during the flight is also reported (average flight time ) 4.8 × 10-3 s).

for runs CO22 and CO25, calculated by adopting the well-mixed drop model, are reported. In the same figures the evolution of the drop speed during the flight is also reported. Then, for comparison, the behavior of the stagnant drops is reported, at run time 0. As can be seen, the mass transfer is much lower in this last case. On the basis of these findings, we can conclude that for any kind of spray nozzle a minimal mean flight path of the drops is necessary in order to achieve the complete drops saturation. It is very important to know this characteristic value for optimizing the absorber size for a given spray nozzle or, on the contrary, for selecting the more suitable spray nozzle for an absorber of a given size. 3.2. Absorption in the Presence of a Moderately Fast Reaction Not Affecting the Mass-Transfer Rate: Ethoxylation of Nonylphenol and Fatty Alcohols. Because ethoxylation is a moderately fast reaction with a Hatta number of about 1,16 the reaction occurs to a negligible extent in the sprayed drops, with the average flight time being very short. Therefore, we assumed that chemical reaction and mass transfer occur in two different zones of the reactor. Mass transfer occurs almost exclusively during the drops flight, with

(31)

By integrating this differential equation, in the case of an internally well-mixed drop model, that is, by calculating kL with relation (27) from time t ) 0 to t ) mean flight time, we can obtain the average value of the drops EO concentration after the mean flight time. In the case of an internally stagnant drop model, we used the original Crank’s5 procedure to evaluate the masstransfer rate. After EO absorption, drops impact on the free liquid surface more slowly than in the case previously examined because of the different size of the reactor, forming a layer that begins to react and move toward the bottom of the reactor. The EO concentration profile and temperature profile along the liquid column must be calculated together by integrating the mass and heat balance equations from the top to the bottom of the liquid column. We assume, for this system, a plug-flowlike fluid dynamic behavior (even if we are slightly far from the fluid dynamic ideality) and, for simplicity, we neglect axial mixing produced, through convective motions, by thermal gradient; on the other hand, for a liquid level of about 50-60 cm, the temperature rise is lower than 10 °C. Considering a liquid element with a cylindrical geometry and an infinitesimal thickness dz, we can write the following mass and heat balance equations:

{

dCEO 4πr2 ) rEO dz Q

(32)

d(FPCPT) dCEO ) ∆H dz dz

(33)

Balance equations (31)-(33) are interdependent, so discretization must be applied in the numerical integration. A detailed description of the mechanism and chemical kinetics of the considered ethoxylation reactions has been reported elsewhere.17 In the same paper17 are also reported relationships giving EO solubility as a function of temperature and composition. It must be pointed out that composition, in the reactor, changes during the time, as a consequence of ethoxylation first of the substrate and successively of the formed oligomers. Therefore, EO solubility changes too, and its value must be adjusted at any integration step, on the basis of the reached average ethoxylation degree, as has been shown in the mentioned paper.17 The same occurs for the liquid density. By integrating equations (31)-(33), we can follow the evolution with time of EO concentration and of the

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4089 Table 5. Simulation Parameters for the Ethoxylation of EPAL1214 and for the Ethoxylation of Nonylphenola ethoxylation DEO (cm2/s) µL [kg/(m s)] σ (kg/s2) D32 (µm) F (kg/m3) µG (Pa s) h (cm) ki0 [cm3/(mol s)] kp0 [cm3/(mol s)] Eatti (kcal/mol) Eattp (kcal/mol) ∆HREACT (J/g of EO)

EPAL1214

nonylphenol

2.3 × 10-5 1.0 × 10-3 5.4 × 10-2 211.2 1.83 2.07 × 10-5 101.5 7.4 × 108 7.4 × 108 12.8 12.8 -2091

2.34 × 10-5 1.0 × 10-3 5.38 × 10-2 211.2 1.83 2.07 × 10-5 101.5 1.5 × 1011 2.0 × 1012 18.3 19.3 -2091

a For all of the experimental runs, V LOOP ) 5 L, DREACT ) 42 cm; h0 ) 21 cm, and (∆P)nozzle ) 3.14 × 105 Pa.

the best performances in fitting the experimental data. The great difference between the calculated values of the EO overall consumption in the two considered models is essentially due to the difference in the masstransfer rate. In fact, at the end of the flight, internally well-mixed drops always reach saturation of EO, while stagnant drops were not, as in the case already seen, of CO2 diffusing in water. As a consequence, the total EO consumption, corresponding to the EO fed, cannot be reproduced by the internally stagnant drop model. It is worth mentioning that EO mass-transfer rates to the well-mixed drops of organic reagents have been interpreted with the approach identical with that used for the absorption of CO2 in water, without introducing any adjustable parameter and with kL being calculated in both cases directly with relation (27). Moreover, the simulation of the runs performed in the described pilot plant has been made by determining all of the model parameters in an independent way and without using any adaptive parameter. 3.3. Absorption in the Presence of an Extremely Fast Reaction Affecting the Mass-Transfer Rate: Chemical Absorption of CO2 in an Aqueous NaOH Solution. When CO2(g) is absorbed in a NaOH aqueous solution, the following reactions occur: K1

CO2 + OH- 798 HCO3K2

OH- + HCO3- 798 CO32- + H2O Figure 9. Ethoxylation run of an alcohol mixture of C12-C14 in a spray-loop-tower reactor.

(34) (35)

in which equilibrium constants, at 20 °C and infinite dilution, are11 K1 ) 6.1 × 107 M-1 and K2 ) 5.9 × 107 M-1. For high free OH- concentrations, greater than 10-2 M, reaction (35) is entirely shifted to the right. Reaction (34), instead, is completely shifted to the right until that OH- concentration is lower than 10-7 M. Because in our experimental absorption runs free OH- concentrations change in the range 10-2÷1.0 M, the following global reaction can be considered:

CO2 + 2OH- f CO32- + H2O

(36)

Reaction (36) kinetically is a consecutive-parallel reaction: consecutive with respect to CO2 and parallel with respect to OH-: Figure 10. Ethoxylation run of nonylphenol in a spray-loop-tower reactor.

profiles of both the EO concentration and the temperature, along the liquid column. The EO consumption contribution in the external loop line has been neglected because the corresponding volume is much smaller than the one of the reactor. It is possible, by consideration of the kinetic model18 and introduction of the WeibullNycander distribution,19 to evaluate the evolution with time of the oligomers distribution too. Parameters employed in the simulations are reported in Table 5. Kinetic results obtained for ethoxylation of respectively fatty alcohols and nonylphenol are reported in Figures 9 and 10. Both runs have been simulated by considering drops to be internally either well-mixed or stagnant. As can be seen, also in this case the model considering the drops to be internally well-mixed gives

+OH-, k1

+OH-, k2

CO2 98 HCO3- 98 CO32- + H2O (37) The first reaction is the rate-determining step, and the rate law for the overall process is11,20

rG ) k1[OH-][CO2]

(38)

with an activation energy11 of Eatt ) 13.25 kcal/mol. The kinetic constant (k1) depends not only on the temperature but also on the solution ionic strength according to the following relation:20

log k1 ) log(k)inf.dil. + AIi

(39)

whereas

Ii )

1 2

∑i zi2Ci

(solution ionic strength)

(40)

4090

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

in which the kinetic constant to infinite dilution is (k)inf.dil. ) 4700 m3/(s kmol) (at 20 °C) and A is a constant corresponding, for NaOH solutions, to A ) 0.13 m3/kmol. The whole CO2 absorption process requests that reagents, being in two different phases, meet each other so that reaction can take place. Reaction rate, masstransfer rate, reagent solubilities, and gas-liquid contact conditions are factors influencing the kinetic equation form and its complexity. However, only some factors are determinant for the overall process rate. Reaction takes place, as is well-known, exclusively in the liquid phase because only CO2 diffuses into both phases. An estimation of the gas-side mass-transfer contribution has been made, and it was 2 orders of magnitude smaller than the liquid-side one.21 For this reason, the gas-side mass-transfer contribution has always been neglected. Kinetic conditions range between “very fast reaction” and “instantaneous reaction” according to the CO2 partial pressure and the free OH- concentration (see eq 38).11,20 Diffusion times will always be much greater than reaction times, and so diffusion of CO2 and OH- species toward a thin reaction plane will control the overall process rate: that is, the process is in the diffusive regime.11 In other words, because of the high reaction rate, a liquid element can contain CO2 or OHbut not both of the reagents, and reaction will occur on the plane (reaction plane) separating the liquid layer containing CO2 from the one containing OH-, on the reaction plane CO2 and OH- concentrations annul each other by instantaneous reaction. Reagent diffusion toward this reaction plane is the rate-determining step. Moreover, any variation of the relative amount of CO2 and OH- will shift the reaction plane toward the CO2 zone or the OH- zone (see Figure 11). Between the gasliquid interface and the reaction plane there could exist a small concentration of Na2CO3, which hydrolyzes according to the reaction

CO32- + H2O T HCO3- + OH-

(41)

and then a small amount of OH- could exist between the reaction plane and the gas-liquid interface, reacting here with CO2. This problem has been studied by Danckwerts and Kennedy12 but with a conclusion that the correction is negligible. The temperature increase of the drops is negligible, considering the very short flight time (5-6 ms), as suggested also by Danckwerts.20 The presence of an instantaneous reaction involves the following effects on the system: (i) The reacting system is in a diffusive regime, allowing one to point out, specifically, the mass transfer. (ii) The concentration gradient is increased close to the reaction plane. (iii) In comparison to the absorption with a not very fast chemical reactionsin which the absorbed species diffuse completely into the liquid film before the reaction deletes their concentrationshere the layer thickness, in which the reacting species concentration drops to zero, decreases and then the mass-transfer coefficient will be enhanced, being inversely proportional to the thickness (see Figure 11), that is

kLe x0 R )E kL x

(42)

The enhancement factor (E) can be defined as “a rate

Figure 11. Concentration profile near the gas-liquid interface for an instantaneous reaction.

increase in CO2 absorption due to the presence of NaOH in the liquid”; that is,

E ) enhancement factor ) x0 rate in the presence of reaction R (43) rate in the presence of mass transfer only x As in the experimental conditions, mass transfer is limited by the liquid-side gradient, the mass balance, for an internally well-mixed drop, on the whole spray system, related to CO2 and OH- species, reminding us reaction (36) stoichiometry, can be written as follows:

-V

d[OH-]b ) 2EkLaGCiCO2 dt

(44)

As has been seen, the previously described approach is similar to the classical one based on the film theory11,20 when the gas phase is the dispersed phase. However, in the case of spray systems, drops are very often smaller than the thickness of the hydrodynamic liquidside film, and none of the relations derived from this theory for calculating the enhancing factor can directly be applied to our case. In this system, absorption and chemical reaction occur together during the drop flight, with the reaction being very fast. After the flight every drop does not contain CO2 at all, and so in the liquid, collected on the bottom of the reactor, no chemical reaction occurs. a. Flight Drops Zone. For experimental runs simulation, then we must integrate eq 44 with respect to the mean drop flight time (internal integration), repeating the internal integration from time t ) 0 to t ) mean flight time until the integration total time is equal to the experimental final time (outside integration). On the other hand, mass-transfer parameters must be continuously updated during the internal integration because it depends on both the instantaneous OH- concentration and the instantaneous drop falling rate, which, in turn, change during the flight. b. Liquid Column Zone. Drops fall into the liquid column with an OH- concentration smaller than that present, at the same time, in the bulk. For the same reasons as those shown in section 3.1, we can consider that the reactor outlet concentration will be equal to that present, at the same time, in each point of liquid bulk, and it will be updated after every flight time. Numerical integration, by consideration of only the internally well-mixed drop model, requires one to know (1) the interfacial area (aG) and the liquid-side mass-

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4091

transfer coefficient (kL), (2) the solubility of CO2 in a NaOH aqueous solution, and (3) the enhancement factor (E). (1) The interfacial area and mass-transfer coefficient can be calculated by following the previously defined procedure; for kL it is possible to estimate its mean value directly by calculating the value corresponding to the Sauter diameter (D32). (2) A lot of literature data of CO2 solubility in water are available, while very few data for the CO2-NaOH system are provided, because the experimental determination of the solubility of a gas in a liquid that reacts quickly with the absorbed gas is very difficult. Nijsing et al.22 and Astarita11 have suggested the use of an empirical correlation that also takes into account the solution ionic strength:

log

(

CiCO2

CiCO2(in H2O)

)

) -kSIi

(45)

The kS parameter depends on the solution in which gas is absorbed. For a NaOH solution, we have kS ) 1.49 × 10-4 m3 mol-1. In this way the evaluation of the CO2 solubility in NaOH(aq) is possible when corresponding data of CO2 solubility in water are known.14,15 (3) The enhancement factor (E) has been calculated by following the procedure described below. For internally well-mixed drops, assuming the enhancement factor E ) 1, the OH- consumption rate calculated is much lower than the one experimentally measured (see Figure 12). Because the model predicts very well both the physical absorption of CO2 and that of ethylene oxide, we attribute the experimentally observed discrepancy to an enhancement phenomenon induced by the instantaneous chemical reaction. Because in the literature there are no relations based on Levich’s theory for calculating the enhancement factor for systems such as the one examined in this paper, we have derived a semiempirical correlation for estimating this factor. As we have already said, the reaction occurs on a reaction plane separating two liquid zones: the one containing only NaOH and the other one containing only CO2; then, because reagents must diffuse up to this reaction plane, the global process rate is influenced from both the NaOH and CO2 gradients, in such a way that PCO2 and/or [OH-] variations can shift the reaction plane toward or away from the gas-liquid interface. We suggest, therefore, the following relation for calculating the enhancement factor for internally well-mixed drops:

( )

E)A+B

[OH-]b 2CiCO2

) 3.0 + 0.11

Figure 12. Run CO16 simulation with the internally well-mixed drop model. In the same figure is also reported the behavior obtained by neglecting the effect of the enhancement factor.

( ) [OH-]b 2CiCO2

(46)

The numerical values of A and B, appearing in the correlation, have been determined by mathematical regression analysis on the experimental data and are probably correlated with the particular device used. This correlation, although containing only two parameters, is able to simulate all of the performed experimental runs, and the functional dependence of E seems to be of general validity for instantaneous reactions. We are now investigating to find a more rigorous theoretical background to justify this correlation and to extend the correlation also to fast reactions.

Figure 13. Run CO11 simulation with the internally well-mixed drop model. Table 6. Simulation Parameters for the CO2 Chemical Absorption in NaOH(aq) run CO11 CO12 CO13 CO14 CO15 CO16 CO17 CO18

FP (kg/m3) 1024 1023 1024 1031 1032 1030 1030 1014

µL [kg/(m s)] 10-4

7.9 × 7.9 × 10-4 7.9 × 10-4 8.5 × 10-4 8.5 × 10-4 8.5 × 10-4 8.5 × 10-4 7.5 × 10-4

for all of the runs VLOOP ) 94 cm3 DREACT ) 12.3 cm h0 ) 6.15 cm h ) 26.6 cm DCO2 ) 2.2 × 10-9 (m2/s) σ ) 7.4 × 10-2 (kg/s2) µG ) 1.6 × 10-5 (Pa s)

The reaction plane shifting away from the drop core (i.e., toward the gas-liquid interface) reduces the x thickness (see Figure 11) and so, when the slope of the CO2 profile concentration is increased, enhances the global rate. This enhancement then is greater, the greater is the ratio COH-/CCO2. Parameters employed in the simulations are reported in Table 6. In Figures 12-14 are reported examples of simulation of some of the experimental absorption runs, reported in Table 3. In comparison to physical absorption, in which the concentration gradient of the absorbed species is extended in the whole diffusive layer (x0), in the case of chemical absorption, the layer thickness, in which the reacting species concentration is annulled (x), is smaller (see Figure 11) and so the mass-transfer coefficient is enhanced for a quantity inversely proportional to such a thickness. According to the model based on Levich’s theory, near the gas-liquid interface, the film thickness, in which the reaction plane is shifted leading to the enhancement phenomenon, has dimensions remarkably

4092

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000

List of Symbols

Figure 14. Run CO17 simulation with the internally well-mixed drop model.

lower than that foreseen by Lewis’ film theory, and then in the case of Levich’s theory, the enhancement factor value will not be so high, because the reaction plane is only slightly shifted. This behavior, however, is that observed in simulating the experimental runs of CO2 absorption in NaOH(aq) in which, during every single drop flight, the enhancement factor changes in the range from 8 to 6.5. At last, a less fast reaction would have an E value falling between 1 and 8, that is, between the limiting values, but the relation giving E could be much more complicated. 4. Conclusions In this work, the behaviors of the spray-tower-loop absorbers and reactors have been studied in three different situations: (i) in the absence of reactions when only the physical absorption of a gas occurs, (ii) in the presence of a relatively fast reaction not affecting the mass-transfer rate, and (iii) in the presence of an extremely fast reaction influencing the mass-transfer rate. By using an efficient spray nozzle, sprayed drops were internally well-mixed and consequently mass-transfer rates are very fast. This behavior can be well-reproduced by adopting the approach suggested by Srinivasan and Aiken7 derived from Levich’s theory9 and developed for determining the mass transfer at the interphase of a single internally well-mixed drop. This approach failed only in the description of the absorption of CO2 in a NaOH aqueous solution, because, in that case, the absorption rate is enhanced by the presence of an extremely fast reaction. We have derived a semiempirical correlation for calculating the enhancement factor, allowing, with the same approach, also the interpretation of this type of absorption. In conclusion, on the basis of our calculations, we observed that for any kind of efficient spray nozzle operating in a given system, a minimal mean flight path of the drops is necessary to achieve a complete saturation of the drops, and when this value is known, the zone of absorption of a spray absorber or of a spraytower reactor can be optimized. On the contrary, for absorber or spray-tower loop of a given size, the more suitable spray nozzle can more easily be selected. Acknowledgment PNR Italia is acknowledged for their aid in the characterization of the spray nozzle. Pressindustria SpA is acknowledged for their aid in performing the ethoxylation runs.

aG ) interfacial surface area of a drop (m2) a′ ) acceleration (m/s) CD ) coefficient of form Ci ) ith species concentration Cb ) gas concentration in the liquid bulk for r ) DS/2 (mol/ m3) FV C ) gas concentration inside the drop after the flight (mol/m3) i C ) gas interfacial concentration (mol/m3) Cr ) gas concentration inside the drop at the radius ) r (mol/m3) CiEO ) saturation concentration of ethylene oxide (mol/m3) CEO ) ethylene oxide concentration (mol/m3) CGCO2 ) CO2 drop bulk concentration (mol/m3) CiCO2 ) CO2 interfacial concentration (mol/m3) cP ) specific heat at constant pressure [(kcal/(kg °C)] Di ) ith drop diameter (µm) DCO2 ) diffusion coefficient of CO2 in the liquid (m2/s) DP ) drop diameter (m) DREACT ) reactor internal diameter (cm) D32 ) DS ) Sauter mean drop diameter (m) dm ) mean flight path of the drop (cm) di, dj ) see Figure 4 PA ) diffusion coefficient of the gaseous species A in the liquid phase (m2/s) E ) enhancement factor Eatti ) activation energy of the ethoxylation initiation step (kcal/mol) Eattp ) activation energy of the ethoxylation propagation step (kcal/mol) Eatt ) activation energy (kcal/mol) g ) acceleration of gravity (m/s2) h0, h, hL, h1 ) see Figure 4 Ii ) solution ionic strength kL ) liquid mass transport coefficient without chemical reaction (m/s) kLe ) liquid mass transport coefficient with chemical reaction (m/s) k1, k2 ) kinetic constants for CO2 chemical absorption in NaOH(aq) [m3/(mol s)] ki0 ) Arrhenius constant of the ethoxylation initiation step [cm3/(mol s)] kp0 ) Arrhenius constant of the ethoxylation propagation step [cm3/(mol s)] ni ) number of drops with diameter ) Di NCO2 ) mass-transfer rate of CO2 (mol/s) NRe ) Reynolds dimensionless number (DPvFP/µL) nGCO2 ) CO2 moles in the drop bulk (mol) [OH-]b ) OH- concentration in the liquid bulk (mol/m3) P ) pressure (Pa) Q ) recirculating liquid flow rate (m3/s) r ) reactor radius (m) rL, rj ) see Figure 4 rEO ) reaction rate of ethylene oxide formation [mol/(m3 s)] rG ) reaction rate [mol/(s m3)] rP ) drop radius ranging from 0 to DS/2 (m) Sc ) Schmidt dimensionless number (µL/FPPA) Sh ) Sherwood dimensionless number (kLDP/PA) T ) temperature (°C) t ) time (s) tflight ) average drop flight time (s) V ) liquid volume (m3) VG ) drop volume (m3) VLOOP ) reactor loop volume (cm3) v ) drop speed (m/s) z ) quote (m) zi ) ionic charge of the ith species We ) Weber dimensionless number (v2FPDP/σ)

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4093 Greek Symbols R ) cone width angle ∆H ) reaction enthalpy (kcal/mol) (∆P)nozzle ) pressure drop of the liquid through the nozzle (Pa) φ ) spray efficiency factor (0-1) µG ) gas viscosity (Pa s) µL ) liquid viscosity [kg/(m s)] x ) flying pathway (m) F ) gas density (kg/m3) FP ) liquid density (kg/m3) σ ) surface tension (kg/s2)

Literature Cited (1) Charpentier, J. C. Mass Transfer rates in gas-liquid absorbers and reactors. Advances in Chemical Engineering; Academic Press: New York, 1981; Vol. 11, pp 1-33. (2) Mehta, K. C.; Sharma, M. M. Mass transfer in spray columns. Br. Chem. Eng. 1970, 15 (11), 1440-1558 (3) Bendall, E.; Aiken, R. C.; Mandas, F. Selective absorption of H2S from larger quantities of CO2 by absorption and reaction in fine sprays. AIChE J. 1983, 29 (1), 66-72 (4) Edwards, W. M.; Huang, P. The Kellog-Wet air quality control system. A horizontal spray absorber, in combination with Magnesium promoted lime slurries, removes SO2 from boiler flue gases. Chem. Eng. Prog. 1977, (8), 64-65. (5) Crank, J. The mathematics of diffusion; Clarendon Press: Oxford, U.K., 1958. (6) Johnson, A. I.; Hamielic, A. E.; Ward, D.; Golding, A. End effect corrections in heat and mass transfer studies. Can. J. Chem. Eng. Sci. 1958, 8, 201-215. (7) Srinivasan, V.; Aiken, R. C. Mass transfer to droplets formed by the controlled breakup of a cylindrical Jet-Physical Absorption. Chem. Eng. Sci. 1988, 43 (12), 3141-3150. (8) Hall, C. A.; Agrawal, P. K. Separation of kinetics and masstransfer in a batch alkoxylation reaction. Can. J. Chem. Eng. 1990, 68, 104-112. (9) Levich, V. G. Physicochemical Hydrodynamics; PrenticeHall: Englewood Cliffs, NJ, 1962. (10) Brown, G. G. Unit Operations, John Wiley and Sons, New York, 1950.

(11) Astarita, G. Mass Transfer with Chemical Reaction; Elsevier Publishing Co.: New York, 1967. (12) Danckwerts, P. V.; Kennedy, A. M. The kinetics of absorption of carbon dioxide into neutral and alkaline solutions. Chem. Eng. Sci. 1958, 8, 201-214. (13) Hall, C. A. Kinetics and mass transfer effects in batch alkoxylations. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 1987. (14) CRC Handbook of Chemistry and Physics, 74th ed.; Lide, D. R., Frederikse, H. P. R., Eds.; CRC Press: Boca Raton, FL, 1993. (15) Stephen, H.; Stephen, T. Solubilities of inorganic and organic compounds; Pergamon Press: Oxford, U.K., 1963; Vol. 1, Part 1. (16) Santacesaria, E.; Di Serio, M.; Lisi, L.; Gelosa, D. Kinetic of nonylphenol polyethoxylation catalyzed by potassium hydroxide. Ind. Eng. Chem. Res. 1990, 29 (5), 719-725. (17) Di Serio, M.; Tesser, R.; Felippone, F.; Santacesaria, E. Ethylene oxide solubility and ethoxylation kinetics in synthesis of nonionic surfactants. Ind. Eng. Chem. Res. 1995, 34 (11), 40924098. (18) Santacesaria, E.; Di Serio, M.; Iengo, P. Kinetics and reactor simulation for polyethoxylation and polypropoxylation reactions. In Reaction Kinetics and the development of catalytic processes; Froment, G. F., Waugh, K. C., Eds.; Elsevier Science B.V.: Amsterdam, The Netherlands, 1999; pp 267-274. (19) Weibull, H.; Nicander, K. The distribution of compounds formed in the reaction between Ethylene Oxide and water, ethanol, ethylene glycol, or ethylene glycol monoethyl ether. Acta Chem. Scand. 1954, 8 (5), 847-858. (20) Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill Co.: New York, 1970. (21) Ranz, W. E.; Marshall, W. R., Jr. Chem. Eng. Prog. 1952, 48, 141-173. (22) Nijsing, R. A. T. O.; Hendriksz, R. H.; Kramers, H. Absorption of CO2 in jets and falling films of electrolyte solutions, with and without chemical reaction. Chem. Eng. Sci. 1959, 10, 88-104.

Received for review January 31, 2000 Revised manuscript received July 14, 2000 Accepted August 10, 2000 IE000137Y