Fused Chemical Reactions. 2. Encapsulation: Application to

Fused chemical reactions are reactions that undergo a delay before significant amounts of product are produced. One class of fused chemical reactions ...
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Fused Chemical Reactions. 2. Encapsulation: Application to Remediation of Paraffin Plugged Pipelines† Duc A. Nguyen,‡ H. Scott Fogler,*,‡ and Sumaeth Chavadej§ Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, and The Petroleum and Petrochemical College, Chulalongkorn University, Bangkok, Thailand 10330

Fused chemical reactions are reactions that undergo a delay before significant amounts of product are produced. One class of fused chemical reactions is the class of reactions triggered by an abrupt release of a catalyst. Fused chemical reactions have the potential of solving the problem of organic deposition in sub-sea pipelines which is a problem with enormous economic consequence (Singh, P.; Fogler, H. S. Ind. Eng. Chem. Res. 1998, 37 (6), 2203). The reaction between ammonium chloride and sodium nitrite catalyzed by citric acid was chosen as an example of a fused chemical reaction system where the acid was encapsulated in polymer-coated gelatin capsules. The timed release of the acid catalyst was achieved by putting an additional polymeric coating on the gelatin capsules. The reaction kinetics and the polymer dissolution kinetics were investigated in an adiabatic batch reactor. An excellent agreement between simulation and experimental results in the batch reactor was achieved. Experimental results in a flow-loop reactor demonstrated that the fused chemical reaction could provide a substantial amount of heat in situ. This amount of heat is sufficient to overcome the high heat loss to the surroundings and to raise the temperature of the fluid above the effective temperature to soften and melt the wax deposit. The delay in heat release was found to depend on the thickness of the polymeric coating, while the amount and rate of heat release depended on the in situ reactant and acid concentrations. Introduction Fused chemical reactions are reactions which can be delayed from taking place by either physical or chemical means. Fused chemical reactions have a wide range of applications, such as in the pharmaceutical industry (controlled release of drugs2), agricultural industry (fertilizer management3), and oil industry (remediation of paraffin, asphaltene, and hydrate deposits1 or reservoir stimulation4). As oil wells are drilled further offshore in deeper water, the phenomenon of paraffin, asphaltene, and hydrate deposition becomes more severe and extensive because of the extremely low ocean floor temperature. Figure 1 shows a pipe section plugged by wax deposit that was recovered from the pipeline on the ocean floor. Removal of wax from wells and pipelines have accounted for significant additional operating costs. The direct cost of using deep-sea divers to cut and remove paraffin blockage from a pipeline 40 000 ft in length and 6 in. in diameter is about $6 000 000, while the production loss during downtime is approximately $40 000 000 (as reported by Elf Aquitaire). In some extreme cases such as of the Lasmo field, U.K., the entire field was abandoned with the cost of over $100 000 000 because of recurrences of paraffin blockage. † Submitted for the special issue of Ind. Eng. Chem. Res. to be published in conjunction with the United Engineering Foundation-CRE VIII conference to be held June 23-29, 2001 in Barga, Italy. * To whom correspondence should be addressed. Phone: (734) 763-1361. Fax: (734) 763-0459. E-mail: sfogler@ engin.umich.edu. ‡ University of Michigan. § Chulalongkorn University.

Figure 1. Paraffin plugging a pipeline.

One of the solutions to the paraffin deposition problem is to melt and redissolve the deposit. The primary challenge in clearing the pipeline blockages is in supplying heat to regions further down the pipeline that are more susceptible to wax deposition.5 Because of the characteristic delay time of the encapsulation technique, a highly exothermic fused chemical reaction system using encapsulation is very promising to provide a substantial amount of heat at the desired location in the pipeline.

10.1021/ie0009886 CCC: $20.00 © 2001 American Chemical Society Published on Web 04/26/2001

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Figure 3. Dissolution of paraffin deposits due to a separate feed of reactant pulses. Figure 2. Coiled tubing to control the location of the fused chemical reaction.

Types of Fuses Chemical Fuses. Chemical methods to fuse include the following: a. Series Reactions: A f B f C f D. The formation of species D is delayed because of the slow reaction rate of one of the series reactions. b. Autocatalytic Reactions: A + B f 2B. Initially, the concentration of B in the solution is very low; the initial reaction rate, therefore, is slow. As the reaction progresses, more B will be formed, leading to an everincreasing rate of the reaction. c. Retardants: A + 2B f X (Slow); X + C f D (Rapid); X + B f P (Moderate). Here a scavenger, C, reacts with the intermediate, and the reaction will not proceed to form the final product until C is consumed. As the initial concentration of C in the solution increases, the delay time before a significant amount of product P formed increases. Physical Fuses. Physical means to fuse chemical reactions are normally used for reactions in which the reaction rate strongly depends on the concentration of either one reactant or a catalyst: catalyst

A + B 98 products + heat The delay of the reaction can then be achieved by physically separating either one of the reactants or the catalyst from the reactive system. An example of a chemical reaction which can be fused is the aqueous exothermic reaction between ammonium chloride (NH4Cl) and sodium nitrite (NaNO2) catalyzed by an acid such as citric acid6 to produce a substantial amount of heat: H+

NH4Cl(aq) + NaNO2(aq) 98 NaCl(aq) + 2H2O + N2(g) ∆Hrx ) -79.95 kcal/mol at 25 °C In the absence of the acid catalyst, this reaction proceeds at an extremely slow rate. Physical methods to fuse chemical reactions include the following: a. Physical Separation. The two reactants are separated physically so that they only meet at the wax deposition region. An example is the use of coiled tubing which is shown in Figure 2. Essentially, a tube is inserted into the pipeline up to the wax deposition region. Reactant A is introduced to the tube, whereas reactant B is flowed into the annulus. The two reactants will only meet at the exit of the tube, where they react and release heat to melt the paraffin deposit. This technique works well for very short pipelines. However, if the distance of the plug is more than 8 km from the platform, one cannot effectively use coiled tubing.

Figure 4. Dissolution of paraffin deposits due to a reactive mixture of water/oil emulsions.

b. Dispersion. Singh and Fogler1 developed an axial dispersion model to simulate the flow of alternately injected pulses of reactants separated by an inert (Figure 3). There is a delay in reaction and heat release because the reactants have to disperse through an inert spacer to react. Simulation results show that the heat released can be delayed for moderately long periods of time. However, a large amount of inert is required for a long delay time, which in turn lowers the maximum temperature reached during reaction because the inert slug acts as a heat sink. As a result, the temperature will not be sufficiently high to melt and dissolve the wax deposit. c. Emulsification. Here an oil-in-water emulsion of an aqueous-based fluid which contains the reactants and an oil-based fluid which consists of aliphatic and/ or aromatic hydrocarbons is used to delay the reaction (Figure 4).7 Each reactant was emulsified in the oil separately in the oil base system, and the emulsion is stabilized by the addition of a suitable surfactant. The surfactant sterically stabilizes the drops, thereby retarding the coalescence of the individual A and B drops and delaying the reaction. Several dewaxing operations performed at the Campos Basin (Brazil) have reported success using this process for short pipelines. However, using this process for long pipelines is still a challenge, because the heat release has to be delayed for a considerably longer time. d. Encapsulation. With this technique, either one of the reactants or the catalyst is encapsulated. The delay time is obtained from the controlled release of the encapsulated substance to the bulk solution. Encapsulation is very widely used in the pharmaceutical,2,8 agricultural,3 and oil industries4 because it can provide various types of release profiles (zero-order, first-order, pulsed, and timed). Consequently, the encapsulation technique was chosen in this work as the primary method to delay the exothermic reaction between NaNO2 and NH4Cl by controlling the release of the encapsulated catalyst. The heat generation from the reaction will be delayed because the polymeric coating has to dissolve before the acid catalyst is released into solution. This release mechanism is shown schematically in Figure 5. The thickness of the polymeric coating will determine the extent of the delay in the heat release. The catalyst release parameters of dissolution time and release time were determined for a variety of experimental conditions and used in the mathematical modeling phase of this work. Experimental Section Materials. Sodium nitrite (NaNO2) and ammonium chloride (NH4Cl) were used as reactants. Hydrochloric

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Figure 5. Encapsulation technique for the controlled release of the acid catalyst.

acid (HCl) and sodium hydroxide (NaOH) were used as pH-adjusting agents. Citric acid (C6H8O7) was used as the catalyst. All chemicals used were of reagent grade and were purchased from Aldrich. Hard gelatin capsules size 4 (13 mm in length and 5 mm in diameter, natural/ transparent) were provided from Capsugel. A copolymer of poly(methylmethacrylic acid) and poly(methyl methacrylate) with the trade name Eudragit S100 (Rho¨m Pharm, Darmstadt, Germany) provided by RohmAmerica was used as the coating polymer. This polymer is soluble in the reactive system (aqueous medium) in the working pH range (5-8). Ethanol and acetone used as solvents to prepare the polymer solution were purchased from Aldrich. Reaction Kinetics Studies. The kinetics of this reaction were investigated using an adiabatic batch reactor (Dewar flask, Fisher Scientific, Pittsburgh, PA). The temperature and pH of the solution were monitored using a digital thermometer with a type J thermocouple and a pH meter purchased from Cole Parmer Instruments, Chicago, IL. An initial rate method was applied to find the order of the reaction as well as the specific reaction rate constant, k. Preparation of Acid-Filled Capsules. The 18 wt % polymer solution used for coating capsules was prepared by stirring 17.27 g of Eudragit S100 pellets in a mixture of 60 mL of ethanol and 40 mL of acetone for at least 1 h until a clear solution was obtained. Hard gelatin capsules were filled with approximately 0.001 mol (0.192 g) of citric acid. After the capsules were filled and the halves were locked back together, the core capsules were coated with the polymer solution by means of alternately dipping half of the capsule into the polymer solution and then drying it for about half a day. The procedure was then repeated for the other half of the capsule. The thickness of the polymeric coating was controlled by varying the number of dips and the drying time. Measurement of the Coating Thickness. The thicknesses of each capsule before and after coating were measured using a dial gauge (SIS-6 from Peacock, Japan). The dial gauge can measure a thickness to an accuracy of (0.03 mm. The thickness of the Eudragit S100-coated layer was then deduced from the difference of the two thicknesses. Four different positions were measured for each capsule to calculate the least thickness difference. Batch Experiments. To determine the kinetics of polymer dissolution as well as to obtain parameters for evaluating the efficiency of the encapsulation technique

Figure 6. Definition of parameters deduced from the temperature-time and pH-time trajectories.

to delay the fused chemical reaction, experiments were conducted in a batch reactor. Temperature-time and pH-time trajectories were recorded for each experiment. The effects of stirring speed, thickness of the polymeric coating, initial temperature, initial pH of the solution, and number of capsules used per unit volume of the reactive solution were studied. From the pH-time trajectories monitored, values of the dissolution time and the release time were deduced and used in the polymer dissolution kinetics study. From the temperature-time trajectories, values of the delay time and the rate of temperature increase were computed and compared with simulation results. The temperature-time and pH-time trajectories are shown in Figure 6. Essentially, the dissolution time is the time when the pH of the solution begins to drop because of the release of the catalyst inside the capsules (the time when the polymeric coating and the gelatin capsule have been dissolved). The release time is computed from the dissolution time to the time when the pH of the solution does not change significantly. Regarding the parameters obtained from temperature-time trajectories, the delay time gives us an idea as to when the temperature starts increasing significantly while the rate of increasing temperature describes how rapidly the temperature rises after the delay time. In all batch experiments, the concentrations of the two reactants NaNO2 and NH4Cl were fixed at 2.5 M. The concentration of capsules is normally 2 capsules/ 100 mL of reactive solution unless specified otherwise. Results and Discussion Reaction Kinetics. a. Method of Initial Rates. The combined mole balance on species A, rate law, and energy balance for an adiabatic batch reactor is9

( )

R β dT -∆Hrx(T0) kCACB ) dt Cps CA0

(1)

Here, changes in the overall heat capacity of the solution have been neglected. The initial rate of the temperature rise which can be calculated from experimental temperature-time trajectories can be related to the rate parameters as follows: β

kt)0[-∆Hrx(T0)]CB0 R-1 dT ) CA0 dt t)0 Cps

( )

(2)

Consequently, a log-log graph of the slope calculated

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Figure 7. Effect of [NaNO2] on the reaction rate.

versus the initial concentration revealed the coefficient R - 1 following eq 2. Here, Figure 7 gives the reaction order with respect to sodium nitrite, R, to be approximately 3.1. The order with respect to ammonium chloride was analyzed in a similar manner and found to be 2.5. b. Effects of pH and Temperature of the Solution. The rate constant, k, for oxidation-reduction reactions is typically expressed as

k ) k1 + k2[H+]m + k3/[H+]n

(3)

where [H+] is the concentration of proton in the solution and k1, k2, and k3 are the rate constants in the medium, high, and low pH regions, respectively. The rate constant ki can be expressed by the Arrhenius equation:

ki ) k0ie-Ei/RT

Figure 8. Reaction rate constant as a function of pH.

Figure 9. Typical temperature trajectories for batch reactors for a delay time of up to 12 h.

(4)

The rate constants and activation energies were then obtained by using nonlinear regression for values of kt)0 at different initial pH values and initial temperatures following the model specified in eqs 3 and 4. The complete equation for the specific reaction rate constant was found to be

k ) k1 + k2[H+]1.8 + k3/[H+]0.7

(5)

k1 ) 8 × 109e-12000/T

(6a)

where

13 -7000/T

k2 ) 5 × 10 e

5 -13000/T

k3 ) -8 × 10 e

(6b) (6c)

c. Summary of Reaction Kinetics. The reaction rate is basically third-order with respect to sodium nitrite and second-order with respect to ammonium chloride: 3.1 C2.5 -γNaNO2 ) kCNaNO 2 NH4Cl

Figure 10. Typical temperature trajectories in batch reactors for a delay time of up to 20 h.

(7)

Such a fifth-order reaction usually is comprised of several intermediate steps. This equation also reveals that the reaction rate strongly depends on the concentration of both reactants. Figure 8 plotting the reaction rate constant as a function of the solution pH, calculated from eq 5, clearly shows that the reaction rate is a strong function of the pH in the solution. The reaction rate is very slow at high pH and very rapid at low pH. Therefore, we can control

the rate of reaction by controlling the acid concentration in the solution. Batch Experiments. Typical temperature-time trajectories from batch experiments are shown in Figures 9 and 10. When no coating was applied, the temperature rose after a very short delay of only about 6 min. However, with only a 0.04 mm coating, the reaction was delayed for 2 h. The delay time increased as the thickness of the coating increased. For a coating thickness of 0.4 mm, a delay of approximately 20 h was achieved before the reaction occurred. This 20-h delay time corresponds to a flow distance in a pipeline of approximately 70 km under normal operating conditions! In other words, results from batch experiments show that the encapsulation technique can definitely provide a desired and significant delay time! d. Polymer Dissolution Process. Because the polymer used [a copolymer of poly(methyl methacrylate) and poly(methylmethacrylic acid)] can be considered as

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Figure 12. Effect of stirring speed on the dissolution rate. Figure 11. Effect of thickness of the coating on the dissolution time.

a polyacid, the dissolution of the polymeric coating on the capsules’ surface essentially consists of three steps: 1. Ionization of the polymer chain:

2. Detachment of the ionized polymer chain from the polymer matrix. 3. Diffusion of the ionized polymer chain from the polymer surface to the bulk solution. The mechanism of polymer dissolution is currently being studied in great depth using a rotating-disk apparatus, and the results will be presented at a later date. e. Effect of the Coating Thickness. To examine the effect of the thickness of the Eudragit S100 layer on the dissolution time, citric acid-filled capsules were coated with a 18 wt % solution of Eudragit S100 in various coating thicknesses from 0 to 0.37 mm. The temperatures were kept at 24 °C, while other parameters such as concentrations of the two reactants, size and number of the capsules, volume of the reactive solution, stirring speed, and the pH of the solution were maintained constant throughout the experiments. Figure 11 reveals an excellent linear relationship between the coating thickness and the dissolution time. Therefore, the rate of dissolution of the polymer can be described by the ratio between the thickness of the coating and the dissolution time:

dL ) dt thickness of the coating ) γ(T,pH,Re) (8) dissolution time

dissolution rate (γ) ) -

where L is the thickness of the coating. This dissolution rate is independent of the coating thickness. Hence, hereafter, the dissolution rate calculated from the above equation will be used to derive the rate expression of the polymer dissolution instead of the dissolution time. f. Effect of the Mixing Degree. The effect of the mixing degree on the dissolution rate was studied in order to investigate the mechanism of the polymer dissolution. Experimental results (Figure 12) showed that the dissolution is reaction-limited and that the polymer dissolution rate is independent of the flow regime of the surrounding solution. The dissolution of the polymeric coating is therefore limited by the ioniza-

tion reaction at the polymer surface which can be expressed as

γ ) -dL/dt ) kp[OH-]a

(9)

where kp is the specific reaction rate and a is the reaction order with respect to the hydroxide ion concentration. g. Effect of the Solution pH on Polymer Dissolution. To determine the reaction order with respect to the concentration of proton in the solution, the pH of the solution was varied from 6.19 to 8.06 by adding a prescribed amount of sodium hydroxide (from 0 to 0.01 mol/L of the reactive solution). h. Summary for Polymer Dissolution Kinetics. On the basis of the preliminary results, it can be concluded that the polymer dissolution is limited by the ionization of the polymer at the surface; its rate is a function of temperature and the pH of the reactive solution as follows:

γ ) -dL/dt ) 570.4e-1158/T[OH-]0.411 (mm/h)

(10)

Simulation Results Assumptions. 1. The dissolution of a particular capsule is not affected by the presence of other capsules; i.e., the capsules behave independently. However, a capsule, once completely dissolved, decreases the solution pH and thereby affects the dissolution of other capsules. 2. Once the coating dissolves to the point where there is a breakthrough in the capsule coating, the catalyst is released linearly from the capsule. The released catalyst is immediately dissolved in the solution. Methodology. For a well-mixed batch reactor, the following apply: (a) Mole balance:9

rA dX )dt CA0

(11)

where CA0 is the initial concentration of reactant A, rA the rate of appearance of A, X the conversion of reactant A, and t the time. (b) Energy balance:9

dT UA(Ta - T) + (rAV)[∆Hrx(T)] ) dt NA0Cps

(12)

where A is the heat-transfer area of the reactor and U is the overall heat-transfer coefficient of the reactor.

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Figure 13. Comparison between simulation and experimental delay times at 24 °C.

Figure 14. Comparison between simulation and experimental temperature profiles.

(c) Reaction kinetics: eqs 5, 6a-c, and 7. (d) Polymer dissolution kinetics: eq 10. (e) Catalyst concentration:

{

0 if t e tl ncN0 t - tl if tl < t < tl + tr Ccatalyst ) V tr ncN0 if t g tl + tr V

(13)

where L is the thickness of the polymeric coating, nc the number of capsules introduced into the reactive solution, N0 the number of moles of catalyst encapsulated in one capsule, tl the dissolution time, and tr the release time. (f) pH of the solution:

pH ) f(CNaNO2,CNH4Cl,CNaOH,Ccatalyst)

Figure 15. Experimental setup for flow conditions.

(14)

where CNaNO2, CNH4Cl, CNaCl, CNaOH, and Ccatalyst are the concentrations. The above system of coupled ordinary differential equations was solved using the fourth-order RungeKutta method.10 Comparison between Simulation and Experimental Results for a Batch Reactor The delay time, which is the time when the temperature of the reactive solution starts to rise rapidly, was chosen as the main parameter to compare simulation results to experimental results. In addition, some temperature-time trajectories were also used to evaluate the similarities between the experimental and simulation results. As shown in Figures 13 and 14, excellent agreement in the delay time and temperature-time trajectories was achieved. The simulated temperature-time trajectories have virtually identical delay times and rates of increasing temperature with the experimental trajectories. Because the exothermic reaction is very rapid at low pH and slow at high pH, it was necessary to study the reaction at low temperature for low pH and at high temperature for high pH in order to obtain good experimental results using the method of initial rates. Therefore, for extrapolation for medium pH and room temperature using the kinetic model, there were some deviations which resulted in the early temperature rises in the simulated temperature-time trajectories. The

Figure 16. Typical temperature profiles in flow conditions.

reaction mechanism and kinetics are being studied further to reduce this deviation. Demonstration of the Fused Chemical Reactions in a Flowloop System The ability of encapsulated fused chemical reactions to deliver a substantial amount of heat at the desired location, a location far from the entrance of the pipe was also examined in a flow-loop system. The system is a 300 m, 1 in. PVC pipe flow system built at Conoco, Production Research Division, and is shown in Figure 15. The temperature profiles at different times in the pipeline of a typical experiment are shown in Figure 16. The temperature profiles show that the reaction was delayed until the chemical slug reached the point approximately 750 m from the injection point. The

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reaction generated a substantial amount of heat that raised the temperature to 82 °C, which is higher than the melting point of the wax deposit. The hot fluid was then redirected through the dewax observation pipe section where some candle wax had been deposited. The combined effect of high temperature and high shear stress was observed to sweep the wax deposit off the pipe wall as large particles. The addition of dispersant is desirable to disperse the melted wax in the solution, preventing it from redepositing on the pipe wall. More flow experiments are being conducted to achieve a complete understanding of the controlled release of the encapsulated acid catalyst as well as to address some other aspects such as neutral buoyancy of the capsule and the dispersion of the reactants in flow conditions. Conclusions The reaction kinetics was found to be a strong function of the pH of the solution, which is typical for oxidation-reduction reactions. This important characteristic provides great flexibility to control the rate of reaction and the release of heat by simply changing the pH of the solution. The release of the catalyst encapsulated in Eudragit S100-coated hard gelatin capsules is controlled by the dissolution of the polymeric coating. The polymer dissolution is limited by the ionization of the polymer at the surface. Encapsulation of the catalyst by a soluble polymer such as Eudragit S100 is a very promising technique to delay the release of heat from exothermic reactions. Experimental results clearly show that the release of heat can be delayed as long as 20 h which, in normal operating conditions of the pipeline, is equivalent to a distance from the platform of approximately 70 km. More importantly, both the delay time and the rate of heat release are controllable so that a desired temperature-length profile in the pipeline can be achieved easily. The feasibility of the fused chemical reaction to remove wax deposit in pipelines was demonstrated in a laboratory flow loop. The release of heat was delayed to the desired location in the pipeline. The in situ release of heat was sufficient to raise the temperature of the fluid sufficiently so that the removal of wax was observed. Acknowledgment The authors acknowledge Capsugel and RohmAmerica for providing the capsules and coating materials. We also thank Dr. Steve C. K. Tsai and the staff at the Conoco, Production Research Division, Ponca City, for their advice and laboratory support in building the flowloop. We also gratefully acknowledge the financial support from Petrovietnam and the continuous support of our affiliate companies: Baker Petrolite, Chevron, Conoco, PDVSA-Intevep, Halliburton, Phillips Petroleum, and Schlumberger. Nomenclature R, β )reaction orders γ ) dissolution rate (mm/h) ∆Cp ) overall change in the heat capacity with respect to the limiting reactant (J mol-1 K-1) ∆Hrx(T) ) heat of reaction at temperature T with respect to the limiting reactant (J mol-1) a ) polymer dissolution rate order with respect to the hydroxide ion concentration

A ) heat-transfer area of the reactor (m2) CA ) concentration of reactant A in the solution (mol dm-3) CA0 ) initial concentration of reactant A in the solution (mol dm-3) CB ) concentration of reactant B in the solution (mol dm-3) CB0 ) initial concentration of reactant B in the solution (mol dm-3) CNaNO2 ) concentration of NaNO2 in the solution (mol dm-3) CNH4Cl ) concentration of NH4Cl in the solution (mol dm-3) CNaCl ) concentration of NaCl in the solution (mol dm-3) CNaOH )concentration of NaOH in the solution (mol dm-3) Ccatalyst ) concentration of acid catalyst in the solution (mol dm-3) Cps ) heat capacity of the solution with respect to the limiting reactant (J mol-1 K-1) E1 ) activation energy of the medium-pH region (J mol-1) E2 ) activation energy of the low-pH region (J mol-1) E3 ) activation energy of the high-pH region (J mol-1) Ep ) activation energy for the polymer dissolution (J mol-1) k ) rate constant of the exothermic reaction (mol-4.6 dm-13.8 s-1) k1 ) rate constant in the medium-pH region (mol-4.6 dm-13.8 s-1) k2 ) rate constant in the low-pH region (mol-6.4 dm-15.6 s-1) k3 ) rate constant in the high-pH region (mol-3.9 m-13.1 s-1) kp ) rate constant of the polymer dissolution (mm/h) L ) thickness of the polymeric coat (mm) nc ) number of capsules introduced into the reactive solution N0 ) number of moles of catalyst encapsulated in one capsule (mol) NA0 ) initial number of moles of the limiting reactant in the solution (mol) pH ) pH of the solution at the given condition rA ) rate of appearance of the reactant A (mol dm-3 s-1) T ) temperature of the solution (K) Ta ) temperature of the surrounding area (K) t ) time (h or s) tl ) lag time (h) tr ) release time (h) U ) overall heat-transfer coefficient of the reactor (J s-1 K-1 m-2) [OH-] ) concentration of OH- in the solution (mol dm-3) [H+] ) concentration of proton in the solution (mol dm-3) X ) conversion of the limiting reactant

Literature Cited (1) Singh, P.; Fogler, H. S. Fused Chemical Reactions: 1. The Use of Dispersion to Delay Reaction Time in Tubular Reactors. Ind. Eng. Chem. Res. 1998, 37 (6), 2203. (2) Hirayama, F.; Uekama, K. Cyclodextrin-based controlled drug release system. Adv. Drug Delivery Rev. 1999, 36 (1), 125. (3) Akelah A. Novel utilizations of conventional agrochemicals by controlled release formulations. Mater. Sci. Eng. C 1996, 4 (2), 83. (4) Economides, J. M.; Nolte, G. K. Reservoir stimulation, 3rd ed.; Wiley: New York, 2000. (5) Brown, T. S.; Niesen, V. G.; Erickson, D. D. Measurement and Prediction of the Kinetics of Paraffin Deposition. SPE 1993, 26548.

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5065 (6) Ashton, J. P.; McSpadden, H. W.; Velasco, T. T.; Nguyen, H. T. Method of and Composition for Removing Paraffin Deposits from Hydrocarbon Transmission Conduits. U.S. Patent 4,755,230, 1988. (7) Khalil, C. N. Process for the thermo-chemical dewaxing of hydrocarbon transmission conduits. U.S. Patent 5,639,313, 1997. (8) Langer, R. S.; Wise, D. L. Medical Applications of Controlled Release; CRC Press: Boca Raton, 1984; Vols. I and II. (9) Fogler, H. S. Elements of Chemical Reaction Engineering, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1999.

(10) Carnahan, B.; Luther, H. A.; Wilkes, J. O. Applied Numerical Methods, Reprint; Wiley: New York, 1990.

Received for review November 27, 2000 Revised manuscript received February 22, 2001 Accepted March 12, 2001 IE0009886