5862
Ind. Eng. Chem. Res. 2004, 43, 5862-5873
Fused Chemical Reactions. 3. Controlled Release of a Catalyst To Control the Temperature Profile in Tubular Reactors Duc A. Nguyen,†,| Flavio Faria de Moraes,‡ and H. Scott Fogler*,† Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, and Departamento de Engenharia Quimica, Universidade Estadual de Maringa, Avenida Colombo 5790, Bloco D-90, Sala 102, 87020-900 Maringa, PR, Brazil
The feasibility of an exothermic fused chemical reaction (FCR) system to control the temperature profile inside a tubular reactor is demonstrated in a 300-m laboratory-scale flow loop. The exothermic FCR system consists of a reactive slug containing the two reactants, ammonium chloride and sodium nitrite, and the citric acid catalyst contained in polymer-coated capsules. The citric acid catalyst is designed to only release to the reactive slug when the slug reaches the destination region. Once the acid catalyst is released, it catalyzes an exothermic reaction, generating a substantial amount of heat in situ. The location or time of the heat release is controlled by the thickness of the polymeric coating, while the maximum temperature and the rate of heat release are determined by the in situ concentrations of the reactant and acid catalyst. The in situ concentrations of the reactants were found to be controlled by the fluid Peclet number and the ratio between the reactive slug length and the flow length. Meanwhile, the in situ concentration of the acid catalyst depends on the flow of the capsules and the controlled release of the acid catalyst from the capsules. The turbulent dispersion of the capsules depended on the capsule Stokes number, while the flow of the capsule relative to the reactive slug was correlated to the capsule Reynolds number and the capsule Froude number. Good agreement between experimental and simulation temperature profiles was achieved. Simulations can then be used to determine all required system parameters for a desired temperature profile inside the tubular reactor. I. Introduction Fused chemical reactions (FCRs) are reactions that can be delayed from taking place by either physical or chemical means.1 FCRs have a wide range of applications, such as controlled release of drugs in the pharmaceutical industry,2 fertilizer management in the agriculture industry,3 and remediation of paraffin, asphaltene, and hydrate deposits1,4,5 and reservoir stimulation6 in the oil industry. An exothermic FCR system can provide a controlled temperature profile in a tubular reactor with the characteristic delay time. Effective removal of wax deposits in subsea pipelines is a potential application of such a delayed temperature profile. Because oil wells are drilled further offshore in deeper water, the phenomenon of paraffin, asphaltene, and hydrate deposition becomes more severe and extensive as a result of the extremely low temperature on the ocean floor. The U.S. Mineral Management Service reported that 51 subsea pipelines were plugged in the Gulf of Mexico from 1992 to 2002 as a result of wax deposition. Figure 1 shows a pipe section plugged by wax deposits that was recovered from a pipeline on the ocean floor. Wax precipitation and deposition occurring during production and transportation of crude oils, especially in deeper water and further away from the shore, are responsible for losses of millions of dollars annually.7 * To whom correspondence should be addressed. Tel.: (734) 763-1361. Fax: (734) 763-0459. E-mail:
[email protected]. † University of Michigan. | Current address: Research and Development Center for Petroleum Processing, Petrovietnam. ‡ Universidade Estadual de Maringa.
Figure 1. Paraffin plugging a subsea pipeline.
One of the more feasible solutions to the paraffin deposition problem is to melt and redissolve the deposit. However, if a hot liquid is introduced right at the entrance of the pipeline, the hot liquid will cool prior to reaching the wax deposit because of the low subsea temperatures. Therefore, the primary challenge in clearing the pipeline blockages is to supply heat to regions further down the pipeline (e.g., 70 km; Figure 2) that are more susceptible to wax deposition.8 Because of the ability to provide an extremely large amount of heat in a controllable fashion, an exothermic FCR system emerges as a feasible solution for this billion dollar problem of wax deposition.1,5 A FCR system that can provide a desirable temperature profile in a tubular reactor essentially consists of a highly exothermic reaction and a controlled-release system to fuse the reaction at the proper location in the reactor. These locations will vary from pipeline to
10.1021/ie049933k CCC: $27.50 © 2004 American Chemical Society Published on Web 07/29/2004
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5863
Figure 2. Desirable temperature profile in a sub-sea pipeline.
Figure 3. Timed release of the acid catalyst by encapsulation.
pipeline. Consequently, the release system must be designed for each individual pipeline. The reaction chosen for the FCR system is the one between ammonium chloride and sodium nitrite:
NaNO2(aq) + NH4Cl(aq) f NaCl(aq) + 2H2O + N2(g)v (1) This reaction is highly exothermic (∆HRx ) -334.2 kJ/ mol at 25 °C), irreversible (Keq ) 3.9 × 1071 Pa at 25 °C), and, more importantly, controllable. Nguyen et al.4 found that the reaction is strongly catalyzed by hydrogen ions. Therefore, a delayed temperature profile in the tubular reactor can be obtained by using an appropriate controlled-release method for the acid catalyst to fuse the reaction at the proper location in the reactor. Nguyen et al.1 successfully controlled the release of citric acid by applying a polymeric coating on the gelatin capsules encapsulating the citric acid (Figure 3). The delay time was obtained because the polymeric coating must dissolve before the acid catalyst is released into the solution. Batch experiments showed that the timed release of the citric acid can be totally controlled for up to 20 h by simply varying the thickness of the polymeric coating.1 Therefore, the FCR system for the tubular reactor will consist of a reactive slug containing the two reactants and the citric acid encapsulated in polymer-coated capsules (Figure 4). The reactive slug is followed by the carrier liquid (e.g., water or oil). The acid catalyst is only released to the reactive slug right before it reaches the destination region, catalyzing the exothermic reaction and thereby providing the desirable delayed temperature profile. The dissolution of the polymeric coating in the presence of the reactants was found to be independent of
Figure 4. FCR system to control the temperature profile in a tubular reactor.
external flow characteristics (i.e., Reynolds number) owing to the facilitated diffusion phenomenon.1,9 Therefore, the release profiles of the encapsulated acid catalyst in batch conditions can be applied directly to flow conditions. However, because of dispersion, the in situ concentrations of both the reactants and capsules, and hence of the acid catalyst, could be much lower in a flow system than in a batch system. The reactants disperse out of the reactive slug as the reactive slug moves along the tubular reactor. Moreover, the capsules can flow and disperse faster or slower than the reactive slug, thereby decreasing the effective in situ concentration of the capsules, and hence the in situ concentration of the acid catalyst. The decrease in the in situ concentrations of the reactants and the acid catalyst results in a significant decrease in the in situ reaction rate, and hence the rate of heat release. Therefore, a thorough understanding of the dispersion of the reactants and the flow and dispersion of the capsules is necessary. An important aspect in the design of the FCR system is to be able to predict the reactant and capsule distribution in the tubular reactor. Therefore, the goal of this paper was fourfold. First, the capsule flow and dispersion were investigated in a laboratory-scale flow loop. Second, the characteristic dimensionless groups that affect the in situ release of heat and the effectiveness of the FCR system were identified and analyzed. Third, experiments and theoretical simulations for the FCR system in a laboratory flow loop were compared. Finally, the effectiveness of the FCR method to remove wax deposits from the pipe wall was demonstrated. II. Experimental Section 1. Materials. Sodium nitrite (NaNO2) and ammonium chloride (NH4Cl) were used as reactants. Solid citric acid (C6H8O7) was used as the acid 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) were provided by Capsugel. The copolymer of poly(methylmethacrylic acid) and poly(methyl methacrylate) (trade name Eudragit S-100 provided by RohmAmerica) was used as the coating polymer. The capsules were filled with citric acid and then polymer-coated as previously described in Nguyen et al.1 Candle wax, which has a melting point of 60 °C, was deposited on the wax observation section. 2. Flow-Loop Setup. The flow loop is a 300-m transparent poly(vinyl chloride) (PVC) pipe (inside diameter of 0.0254 m) flow system shown in Figure 5. The flow loop consists of four pipe sections. The first pipe section (section 1) was a detachable section that was used to introduce the reactive solution and the polymer-coated capsules. The second pipe section (sec-
5864 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 6. Tubular reactor model to simulate FCR in flow conditions.
Figure 5. Flow-loop setup with each pipe section numbered.
tion 2-244 m in length) was circled in the two water tanks. The flow loop was set up so that a flow length longer than the total length could be achieved by connecting union 1 with union 3 and union 2 with union 4 when the chemical slug was in the pipe section of the second tank. Then, instead of going to the measurement section, the chemical slug was pushed back to the first tank by the mobile phase coming from the entrance of the pipe section in the second tank. Eight thermocouples were placed equally along the measurement section (section 3-45 m in length) to monitor the temperature profile of the chemical slug. The distance between any two adjacent thermocouples was 6 m. A 3-m dewax observation pipe (section 4: transparent, rigid PVC pipe) was placed in parallel with the measurement section from thermocouple number 7 to thermocouple number 8. When it is desired, the solution can be flowed through the observation pipe so that the dissolution of the wax predeposited on the pipe wall can be observed. 3. Flow of Capsules. The velocity of the capsules was determined by measuring the time taken by the capsules to travel through the pipe section in the two tanks (244 m) of the flow loop. Two different sets of capsule flow experiments were conducted: 1. A single capsule was injected into the flow loop. The velocity of the capsule was correlated with the flow rate of the carrier liquid and the density of the capsule. 2. A group of capsules was injected into the flow loop to study the effect of capsule-capsule interactions on the dispersion of the capsules. 4. FCR System in Flow Conditions. Water was used as the mobile phase to transport the reactive slug in the system. Because the capsules were introduced after the pump, adverse shearing effects of the pump on the capsules were avoided. The reactive solution and the capsules flowed through the pipe sections in the first and second tanks and then, depending on the designed flow distance, either recycled through the pipe sections or flowed directly to the measurement section. However, reactant dispersion limits the number of cycles passing by the entrance section, and hence the flow length. After the reactive slug reached the measurement section, the reactive slug was shut in and temperature profiles were recorded using a data acquisition system and the Labtech software. III. Theoretical Analysis The entire flow loop is modeled as a tubular reactor (Figure 6). Because the length of the flow loop is much
larger than its diameter, the one-dimensional dispersedplug-flow model10 is used to simulate the concentration and temperature profiles along the flow loop. For simplicity, while still complying with the intent and purposes of this study, the following two assumptions are made: 1. The acid catalyst dissolves immediately into the solution after being released from the capsule. 2. Radial variations in the concentration and temperature are small. Batch experiments show that the dissolution time of the free citric acid catalyst in aqueous solutions is virtually instantaneous compared with the release time of the citric acid catalyst from the capsule. Therefore, assumption 1 is valid. Assumption 2 is appropriate for turbulent flow and for tubular reactors with a high length/diameter aspect ratio. 1. Dispersion of Reactants (Either NaNO2 or NH4Cl). Because the reactants dissolve into the carrier liquid, their dispersion follows the Aris-Taylor dispersion model.10 The mass balance for the reactants can be written as 2 rRx ∂Ψreactant 1 ∂ Ψreactant ∂Ψreactant ) -Da 0 (2) + 2 ∂θ Pem ∂λ ∂λ r Rx
where λ ) z/L and θ ) tVf/L are respectively the dimensionless length and time, t (s) is the time from the start of the flow, z (m) is the distance from the injection point, L (m) is the distance from the injection point to the center of the destination region (cf. Figure 6), Vf (m/s) is the average liquid velocity, Ψreactant ) 0 C h reactant/Creactant denotes the dimensionless concentration of the reactant, C h reactant (mol/m3) is the average axial 0 (mol/m3) is the concentration of the reactant, Creactant initial concentration of the reactant, Pem ) VfL/Da is the Peclet number for mass transfer, Da (m2/s) is the dispersion coefficient of the reactant, Da ) L(-r0Rx)/ VfC0A is the Damkohler number, C0A (mol/m3) is the initial concentration of the limiting reactant, and -r0Rx and -rRx (mol/m3‚s) are respectively the initial and current reaction rates. For turbulent flow, the dispersion coefficient Da depends mostly on fluctuating velocities in the axial directions. These axial fluctuating velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for Da is used for each component in a multicomponent system.23 2. Flow of Single Capsules. In turbulent flow, capsules not only flow with the liquid stream but also disperse with the turbulent eddies. Because the capsules are not generated or consumed, the mass balance on capsules in tubular reactors can be expressed as
∂2 C h capsule ∂C h capsule ∂C h capsule a - Dcapsule ) 0 (3) + Vc 2 ∂t ∂z ∂z
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5865
where C h capsule (number of capsules/m3) denotes the a average axial capsule concentration, Dcapsule (m2/s) the dispersion coefficient of the capsule, and Vc (m/s) the average velocity of the capsules. As shown later, the average velocity of the capsule is determined by drag and capsule neutral buoyancy while the dispersion coefficient is influenced by the turbulent eddies in the liquid. In dimensionless form, eq 3 becomes
where Fc (kg/m3) is the density of the capsule, dd (m) the drag-equivalent capsule diameter, and µf (kg/m‚s) the dynamic viscosity of the liquid. Meanwhile, the characteristic time of the flow field is the time characteristic of the most energetic and large eddies and can be approximated as the ratio of the timeaveraged eddy length scale, Le (m), and the timeaveraged eddy velocity scale, Ve (m/s), i.e.13
τf ) Le/Ve
(10)
2
∂Ψcapsule 1 ∂ Ψcapsule Vc ∂Ψcapsule )0 + ∂θ Pec Vf ∂λ ∂λ2
(4)
0 where Ψcapsule ) C h capsule/Ccapsule is the dimensionless 0 capsule concentration, Ccapsule (number of capsules/m3) the initial capsule concentration in the reactive slug, a and Pec ) VfL/Dcapsule the Peclet number for mass transfer. By using a coordinate moving with the center of the reactive slug, an analytical solution of eq 4 for the average axial capsule concentration with the initial condition
{
rs rs eλe 2 2 otherwise for -
Ψcapsule(λ,0) ) 1 0
(5)
and boundary condition
Ψcapsule(0,θ) )
{
1
for 0 e θ e
0
otherwise
was found as (Appendix C)
Ψcapsule(λ,θ) )
[( (
1 erf 2
erf
xPec
xPec
)
Vc rs λ- θ+ Vf 2
)]
2xθ Vc rs λ- θVf 2 2xθ
rs Vf 2 Vc
(6)
for θ > 0 (7)
(8)
(9)
Le D/10 D ) ≈ Ve Vf/10 Vf
(11)
Therefore, the primary significance of the capsule Stokes number is that it dictates how readily a particle can follow the fluctuations of an eddy.16 Capsules with very small Stokes numbers (e0.1) have ample time to respond to changes in the flow velocity. Thus, the capsule and fluid dispersivities and velocities will be a ≈ Da, Vc ≈ Vf).12 Therefore, eq 7 nearly equal (Dcapsule becomes
-
The capsule momentum response time τc relates to the time required for a capsule to respond to a change in the local flow velocity. Crowe et al.12 defined the capsule momentum response time τc as the time required for a capsule released from rest to achieve 63% of the liquid velocity:
τc ) Fcdd2/18µf
τf )
Ψcapsule(λ,θ) )
where rs ) ls/L is the length ratio and ls (m) is the length of the reactive slug. If the capsule velocity Vc is different from the fluid velocity Vf, the center of the capsule slug does not coincide with the center of the reactive slug. a. Dispersion of Capsules in Turbulent Flow. The fluctuation in the capsule velocity is caused by the underlying turbulence of the continuous phase. The effect of the dispersion of turbulent eddies on the fluctuation in the capsule velocity is characterized by the Stokes number of the capsule. The capsule Stokes number is defined as the ratio of the capsule momentum response time (τc in s) to the characteristic time of the flow field (τf in s):11
Stc ) τc/τf
Ve is approximated as the root mean square of the isotropic turbulent velocity fluctuations14 and therefore approximately equals one-tenth the average liquid velocity,15 whereas Le is approximately one-tenth the diameter of the pipe (D in m).13 Consequently, the characteristic time of the flow field approximately equals the ratio between the diameter of the pipe and the average liquid velocity:
[( (
1 erf 2
λ-θ+
xPem
)
rs 2
)]
2xθ rs λ-θ2 erf xPem x 2 θ
-
for θ > 0 (12)
On the other hand, capsules with very large Stokes numbers (.1) have essentially no time to respond to the changes in the liquid velocity. Therefore, the capsule velocity will be little affected by the dispersion of turbulent eddies during the capsule’s passage in the a ≈ 0). Equation 7 then becomes pipeline (Dcapsule
Ψcapsule(λ,θ) )
{
1
0
(
)
(
)
rs rs Vc Vc θeλe θ+ Vf 2 Vf 2 otherwise (13) for
The minimum capsule Stokes number in the FCR system in the flow loop can be estimated from the minimum capsule momentum response time and the maximum flow characteristic time in the flow loop. The minimum capsule response time for 5 × 13 mm capsules with a density of 900 kg/m3 used in this work is
(τc)min )
Fcdd2 900 kg/m3 (0.005 m)2 ) ) 1.25 s (14) 18µf 18(0.001 kg/m‚s)
Meanwhile, because the velocity of the liquid in the flow loop was always kept higher than 0.3 m/s to reduce the effect of reactant dispersion, the maximum flow characteristic time in the flow loop can be calculated as
5866 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
(τf)max )
D 0.0254 m ) 0.08 s ) Vf 0.3 m/s
(15)
Therefore, the minimum capsule Stokes number for the FCR experimental flow conditions is
(Stc)min )
(τc)min (τf)max
)
1.25 ) 15.6 0.08
(16)
Because Stc . 1, the capsule slug is basically transported without distortion and the capsule distribution can be described using eq 13. b. Capsule Flow. The flow of capsules with Stokes numbers much larger than 1 in liquids is primarily influenced by drag and gravity.14 When the flow is initiated, the capsule is retarded by the static friction between the capsule and the pipeline wall. The liquid then accelerates the capsule up to the average suspension velocity as a result of the drag force exerted by the liquid on the capsule. There are basically two approaches in modeling the flow of capsules with Stokes numbers much larger than 1 at steady state: (1) using a force balance to directly solve for the capsule velocity;17 (2) using Buckingham’s Pi Theorem (dimensional analysis) to develop a correlation between characteristic dimensionless groups in the system.18,19 Both models have been able to fit experimental data well. The common conclusion from both models and experimental observations is that, as the fluid velocity increases, the capsule migrates toward the pipe center and the mean capsule velocity is greater than that for the fluid.20 The force balance model17 is more fundamentally sound because it is derived from first principles. However, this approach is only appropriate for straight pipelines and perfect cylindrical capsules. Moreover, the model based on force balances also includes a number of coefficients (i.e., drag, lift, entrance and exit head loss, pressure, and static friction), many of which are very empirical in nature. For the coiled pipeline system with changing elevation slope used in this work (Figure 5), the model correlating dimensionless groups is more suitable. Molerus and Wellmann18 used Buckingham’s Pi Theorem to develop a correlation between the capsule velocity/liquid velocity ratio and two dimensionless groups as
(x|
Vc )φ Vf
|
)
4dd , ) φ(Frc,Rec) (17) 3cDD
Vf Fc - Ff Dg Ff
where Ff (kg/m3) is the density of the liquid, Frc ) Vf/ x|(Fc-Ff)/Ff|Dg is the capsule Froude number (square root of the ratio between the inertial force and gravitational force), and cD is the capsule drag coefficient, which depends on the capsule Reynolds number [Rec ) (Ff|Vf - Vc|dd)/µf ) ratio between the inertial force and viscous force]. Similar approaches have been used successfully for other systems.18,19,21,22 Equation 17 gives a reasonable basis for processing the experimental data. For a constant capsule size and pipe diameter, the average velocity of the capsule in the flow loop is solely a function of the average velocity of the liquid and the neutral buoyancy of the capsule. In fact, the experimental velocity ratio in the flow loop correlates well with
Figure 7. Hydraulic conveying of single capsules in the flow loop.
the capsule Froude number and the capsule Reynolds number through the following empirical equation (Figure 7):
0.42Rec0.03 Vc ) 1.27 ) Vf Frc
1.27 - 0.42
x|
|
Fc - Ff Dg Ff Ff|Vf - Vc|dd Vf µf
(
)
0.03
(18)
Figure 7 shows that capsules with very high or very low density relative to the fluid density (low Froude numbers) will flow slower than the liquid. As capsules become more neutrally buoyant (|(Fc - Ff)/Ff| decreases) or as the liquid velocity (Vf) increases, the capsules start moving along the axis of the pipe with a faster velocity than the average velocity of the carrier liquid (Figure 7). The capsule/liquid velocity ratio reaches a maximum value of about 1.27 when the capsule Froude number approaches infinity. The effect of the capsule Reynolds number on the velocity ratio is much less significant than the effect of the capsule Froude number. Equation 18 is consistent with the dimensional analysis (cf. eq 17) and other experimental studies.19,21,22 When the capsule Reynolds number is larger than 1000, the drag coefficient cD becomes constant12 and eq 18 can be simplified to
Vc 0.5 ) 1.27 ) 1.27 - 0.5 Vf Frc
x|
|
Fc - Ff Dg Ff Vf
(19)
3. Flow of Multiple Capsules. Experiments in the flow loop (Figure 8) with groups of 5, 10, 20, 50, and 100 capsules with initial volume fractions higher than 0.04 show that the flow of multiple capsules also follows the empirical equation for single capsules. The spreading of the capsules represented by the error bars in Figure 8 is within the experimental error range for single capsules. Therefore, capsule-capsule interactions are negligible, consistent with other numerical and experimental studies.16 Other numerical and experimental studies show that when the average intercapsule spacing (sc in m) exceeds 5 capsule diameters for spherical capsules, capsule-capsule interactions are negligible.16 For cylindrical capsules with a diameter dc
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5867
the acid catalyst after all of the capsules have completely dissolved, and rcatalyst (mol/m3‚s) the rate of catalyst release. In the laboratory flow loop, because capsules do not disperse but only flow with the reactive slug (cf. eq 13), the maximal acid catalyst concentration can be calculated as max 0 capsule Ccatalyst ) Ccapsule Ncatalyst
(24)
capsule where Ncatalyst (mol of acid catalyst/capsule) is the number of moles of acid catalyst contained in a capsule. The rate of catalyst release, rcatalyst, is equal to the concentration of capsules (source of catalyst release) times the release rate from each capsule:
Figure 8. Hydraulic conveying of multiple capsules in the flow loop.
(m) and length lc (m), the intercapsule spacing can be calculated as
sc )
()
lc dc D
2
(20)
where is the volume fraction of capsules in the mixture. For the 5 × 13 mm capsules used in this study, the 5-length constraint corresponds to
< 0.005
(21)
Because the reaction is strongly catalyzed by acid catalysts, the capsule volume fraction used in all FCR experiments is always smaller than 0.004. Therefore, the correlation for single capsules (eq 18) is expected to hold for the flow of a group of capsules in the FCR experiments in the flow loop. 4. Energy Balance. The energy balance for the liquid in the flow loop is
[
]
∂Θ 1 ∂2Θ ∂Θ L (-rRx)(-∆HRx) 4Uk + ) Θ ∂θ Pet ∂λ2 ∂λ VfFfCp D (Te - Ta) (22) where Θ ) (T h - Ta)/(Te - Ta) is the dimensionless temperature of the liquid, T h (°C) the average axial temperature of the liquid, Ta (°C) the ambient temperature, Te (°C) the effective temperature, that is, the minimum temperature required to melt the deposits in the remediation of wax, hydrate, or frost deposits in pipelines, Pet ) VfL/Rt the Peclet number for thermal dispersion, Rt (m2/s) the thermal dispersion coefficient of the liquid, ∆HRx (kJ/mol) the heat of reaction, Cp (kJ/ kg‚K) the heat capacity of the liquid, and Uk (kJ/m2‚s‚ K) the overall heat-transfer coefficient. 5. Controlled Release of the Acid Catalyst. The mass balance for the acid catalyst dissolved in the solution is 2 ∂Ψcatalyst 1 ∂ Ψcatalyst ∂Ψcatalyst L rcatalyst + ) ∂θ PeL ∂λ Vf Cmax ∂λ2
catalyst
(23) max where Ψcatalyst ) C h catalyst/Ccatalyst denotes the dimensionless concentration of the acid catalyst, C h catalyst (mol/ m3) the average axial concentration of the dissolved acid max (mol/m3) the maximal concentration of catalyst, Ccatalyst
0 rrelease(θ) rcatalyst ) ΨcapsuleCcapsule
(25)
where rrelease(θ) (mol of acid catalyst/capsule‚s) denotes the rate of acid catalyst release from a single capsule. Nguyen et al.1 have shown that a timed-release scheme for the acid catalyst can provide the desirable delayed temperature profile. Moreover, the delay time can be controlled by simply varying the thickness of the polymeric coating.1 In the timed-release scheme (Figure 3), the release rate of the acid catalyst can be approximated as
{
capsule Vf Vf Ncatalyst for td e θ e (td + tr) rrelease(θ) ) tr L L 0 otherwise (26)
where td (s) is the dissolution time of the polymeric coating and tr (s) the release time. 6. Exothermic Reaction Rate. The kinetics of the exothermic reaction (eq 1) has been investigated and delineated by Nguyen et al.4 The results show that the reaction rate can be expressed by the following power law rate equation:
h NH4Cl0.95 C h NaNO21.81 -rRx ) 5.3 × 1010e-7600/Th aH+0.91 C (27) where aH+ is the activity of the hydrogen ion in the solution. The activity of the hydrogen ion in the solution is a complex function of the solution temperature and the concentrations of the reactants and acid catalyst. The hydrogen ion activity can be calculated using the electroneutrality requirement with the activity coefficients of all species calculated using the Pitzer model.24 7. Initial and Inlet Boundary Conditions. The two reactants and capsules are introduced into the flow loop as a reactive slug, the center of which is selected as the injection point (λ ) 0; cf. Figure 6). Initially (θ ) 0), the chemical slug is contained within the interval -rs/2 e λ e rs/2. Therefore, we have the following initial and inlet boundary conditions:
a. Initial condition (θ ) 0)
{
Ψi(λ,0) ) 1 0
rs rs eλe 2 2 otherwise if -
(28)
5868 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
{
0 Ccatalyst
if -
Ψcatalyst(λ,0) ) Cmax catalyst 0 otherwise Θ(λ,0) )
rs rs eλe 2 2 (29)
T0 - Ta Te - Ta
(30)
b. Inlet boundary conditions (λ ) 0)
{
Ψi(0,θ) ) 1 0
{
if 0 e θ e
rs 2
(31)
otherwise
0 Ccatalyst
rs if 0 e θ e Ψcatalyst(0,θ) ) Cmax 2 catalyst 0 otherwise T0 - Ta Θ(0,θ) ) Te - Ta
(32)
(33)
where i denotes either NH4Cl, NaNO2, or capsule, T0 0 (°C) the initial temperature of the fluid, and Ccatalyst 3 (mol/m ) the initial concentration of the acid catalyst. The governing equations are four simultaneous coupled partial differential equations (eq 2 for NH4Cl, eq 2 for NaNO2, and eqs 22 and 23) for the dependent variables ΨNH4Cl, ΨNH4Cl, Θ, and Ψcatalyst as a function of independent variables λ and θ. The problem is highly nonlinear because eqs 2 and 22 are nonlinear owing to the Arrhenius dependence of the specific reaction rate and the high reaction rate order. These partial differential equations together with the equation for the capsule concentration (eq 13), the reaction kinetics (eq 27), and the initial and boundary conditions were solved using FEMLab, a PDE solver provided by COMSOL with the parameters listed in Appendix A. IV. Results and Discussion The heat release is only effective when the temperature of the fluid in contact with the destination region is higher than the effective temperature. Therefore, the two characteristics of the temperature profile used to remediate the wax, hydrate, or frost deposits in pipelines are the total amount of effective heat (He in kJ) and the average effective contact time (the in s). The total amount of effective heat is the total amount of heat released into the destination region when the liquid temperature is above the effective temperature. Appendix B shows that the total amount of effective heat can be calculated as
He )
πD2L2 FC 4lds f p
λ)1+0.5l /L ∫λ)1-0.5l /L ds
ds
θ)∞ f[T h (λ,θ)-Te] [T h (λ,θ) - Te] dθ dλ ∫θ)0
(34)
where lds (m) is the length of the destination region and f is the Heaviside step function:
f(x) )
{
1 0
if x g 0 otherwise
(35)
Meanwhile, the average effective contact time is the average contact time between the liquid and destination region, provided that the liquid temperature is greater
than the effective temperature (Appendix B):
ht e )
L2 ldsVf
λ)1+0.5l /L θ)∞ f[T h (λ,θ)-Te] dθ dλ ∫λ)1-0.5l /L ∫θ)0 ds
ds
(36)
As the total amount of effective heat or the average effective contact time increases, the total amount of wax, hydrate, or frost deposits that would be melted will increase. To maximize the total amount of effective heat and contact time, an efficient strategy is to flow the reactive slug to the destination region and then shut in so that the majority of the heat can be released right in the destination region. The flow of the reactive slug can then be divided into two periods: (1) the flowing period where the reactive slug flows to the measurement section; (2) the shut-in period where the reactive slug generates heat in the measurement section. In the flowing period, the dominating processes are the reactant dispersion and the capsule flow relative to the flow of the reactive slug. Both of these processes reduce the in situ concentrations of the reactants and the acid catalyst, thereby lowering the in situ rate of heat release. Meanwhile, the acid catalyst release and the exothermic reaction are the main processes in the shut-in period. If the in situ reaction generates sufficient heat to overcome the heat loss to the surroundings, the temperature of the reactive slug will increase, resulting in a higher reaction rate and leading to a greater increase in the liquid temperature up to the point where most of the reactants are consumed. 1. Flowing Period. Because the acid catalyst is not released during the flowing period, the reaction rate is negligible (-rRx ≈ 0). Equation 2 can then be approximated as 2 ∂Ψreactant 1 ∂ Ψreactant ∂Ψreactant + ) 0 (37) ∂θ Pem ∂λ ∂λ2
By using a coordinate moving with the center of the reactive slug, this equation was solved analytically with the initial and boundary conditions (eqs 28 and 31) to give the average concentration of the reactant as (Appendix C)
Ψreactant(λ,θ) )
[( (
1 erf 2
λ-θ+
xPem
)]
2xθ rs λ-θ2 erf xPem 2xθ
)
rs 2
-
for θ > 0 (38)
(θ ) 0 corresponds to the initial condition, eq 28). Therefore, the two dimensionless groups that govern the reactant dispersion process in the flowing period are the Peclet number for mass transfer Pem and the length ratio rs ) ls/L. Figure 9 shows that the maximum dimensionless in situ reactant concentration doubles as the Peclet number for mass transfer increases fourfold or the length ratio increases twofold. For a stoichiometric feed, a twofold increase in the in situ reactant concentration leads to nearly an eightfold increase in the in situ reaction rate (cf. eq 27) and thus in the rate of heat release. Therefore, for a certain desired target length L, we can adjust the initial reactant concentration
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5869 / dΨreactant D2 )dθ* RC0
(-rRx)
(43)
FfCp dΘ* -rRx ) Ngl max - Θ* 4DUk dθ* -r
(44)
reactant
Rx
Figure 9. Effect of the Peclet number for mass transfer and the length ratio on the in situ reactant concentration profile. 0 Creactant , the length of the reactive slug ls, or the liquid velocity Vf to obtain a sufficient in situ rate of heat release. 2. Shut-in Period. Because there is no flow in the shut-in period, the mass and energy balances, eqs 2 and 22, become
( )( )
/ ∂Ψreactant D ∂θ* L
2
/ De ∂2Ψreactant D2 ) R ∂λ2 RC0
(-rRx)
reactant
(39)
2 max D 2 ∂2Θ* D (-rRx )(-∆HRx) -rRx ∂Θ* ) ∂θ* L ∂λ2 RFfCp(Te - Ta) -rmax
()
Rx
4DUk Θ* (40) RFfCp where θ* ) R(t - ts)/D2 is the dimensionless time after / and Θ* are respectively the dithe shut-in, Ψreactant mensionless reactant concentration and temperature after the time at which shut-in begins, R (m2/s) is the thermal diffusivity, ts (s) is the time at which shut-in 3 begins, -rmax Rx (mol/m ‚s) is the maximum in situ reaction rate, and De (m2/s) is the diffusivity of the reactant. The maximum reaction rate (-rmax Rx ) is the reaction rate at the highest in situ reactant concentration [i.e., at the center of the reactive slug, C h reactant(L,L/Vf) ) 0 Ψreactant(1,1)] and the highest acid catalyst conCreactant max ). The initial conditions (θ* ) 0) for centration (Ccatalyst eqs 39 and 40 are / (λ,0) ) Ψreactant(λ,1) Ψreactant
(41)
Θ*(λ,0) ) Θ(λ,1)
(42)
An order of magnitude analysis shows that the rate of molecular diffusion of the reactants in an aqueous solution is much smaller than the rate of reaction, especially when the acid catalyst has been released to the solution. Similarly, the rate of thermal diffusion is negligible compared to the rate of heat generation from the reaction and to the heat flux to the surrounding environment. Therefore, the mass and energy balances at any location λ in the destination region during shutin can be approximated as
where Ngl ) D(-rmax Rx )(-∆HRx)/4Uk(Te - Ta) is the dimensionless ratio between the heat generation and the heat loss. Therefore, the temperature profile in the destination region is characterized by the dimensionless group Ngl. The heat generation/loss ratio Ngl depends not only on the initial in situ reaction rate, which depends on the in situ concentrations of the reactants and the acid catalyst, but also on the heat-transfer characteristics of the destination region. The in situ maximum temperature increases with an increase in the heat generation/loss ratio Ngl. For a reasonable in situ concentrations of the reactants of 2000 mol/m3 and the activity of the hydrogen ion of 2 × 10-5 (pH ) 4.7), the initial in situ reaction rate is 0.1 mol/m3‚s. The flow loop in this research would have a heat generation/loss ratio of 0.44. Meanwhile, a typical subsea pipeline with a diameter of 0.3 m and an overall heat-transfer coefficient of 0.04 kJ/m2‚s‚K and being exposed to an ocean temperature of 4 °C would have a heat generation/loss ratio of 11.2, more than 25 times higher. Therefore, the heat loss condition in the laboratory flow loop is more severe than that in typical subsea pipelines and as such will give a safety factor in the actual case. Equation 44 reveals that when the heat generation is larger than the heat loss (Ngl > 1), the in situ fluid can generally be raised to a higher temperature than the effective temperature Te. This criterion provides a constraint for the maximum in situ reaction rate as
-rmax Rx >
4Uk(Te - Ta) D(-∆HRx)
(45)
Figure 10 shows the maximum temperature that can be obtained in the destination region of the laboratory flow loop after 1 h of shut-in time as a function of the in situ concentrations of the reactants and the acid catalyst. A similar calculation can be conducted for subsea pipelines with various overall heat-transfer coefficients. If the reactant and catalyst concentrations are too low, the reaction rate will be small and, as a result, the temperature will increase very little. Figure 10 can be used to determine the required in situ concentrations of the reactant and the acid catalyst for an effective wax deposit treatment in the laboratory flow loop. The results can then be combined with eqs 38 and 18 to determine the required initial concentrations of the reactants and capsules in the reactive slug, the length of the reactive slug, and the flow velocity. 3. Experimental and Theoretical Simulations. To demonstrate the FCR system under flow conditions, let us consider a particular case of the FCR system having 0 0 0 ) CNH ) 5000 mol/m3, Ccapsule ) 16 875 CNaNO 2 4Cl 3 capsules/m (volume fraction ) 0.004), and a reactive slug length of 16 m. Experiments in an adiabatic batch reactor showed that the capsules used have a 1 h dissolution time and a 20 min release time. The center of the destination region is located at 263 m.
5870 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 10. Effect of the in situ concentrations of the reactants and the acid catalyst on the maximum fluid temperature in the laboratory flowloop after 1 h of shut-in time.
Figure 12. Comparison between experimental and simulation results for the 263 m run.
Figure 13. Wax deposit removed from the pipe wall after FCR treatment.
Figure 11. Dimensionless reactant and capsule concentration profiles for the 263 m run.
The dimensionless concentration profiles of the capsules and the two reactants, NaNO2 and NH4Cl, when the center of the reactive slug is at the injection point and at the center of the destination region were calculated using eqs 13, 18, and 38 and shown in Figure 11. Again, the capsule dispersion is negligible because the capsule Stokes number is much greater than 1 (cf. eq 16). Therefore, the capsule slug is basically transported without distortion from the injection point to the destination region. Because of the high dispersion coefficient in the flow loop, the reactant concentration decreases more than twofold when the reactive slug reaches the destination region. Moreover, because the capsules flow slightly more slowly than the reactive slug, the reaction zone where the released acid catalyst and the reactants mixed was primarily between 252 and 268 m and the resulting temperature profiles are expected to be asymmetric. Figure 12 shows the comparison between experimental and predictive temperature profiles in the destination region during the shut-in period. Good agreement between experimental and simulation results was obtained, showing that the predictions of the in situ concentrations of reactants and acid catalyst are accurate. Moreover, the delay time before the temperature increases significantly is similar to the delay time obtained in batch experiments, confirming that the dissolution rate of the polymeric coating does not depend on the external flow characteristics (cf. Nguyen et al.1,9). The rate and extent of the temperature increase in the
measurement section also shows that the FCR system can easily overcome the heat loss to the surroundings to raise the temperature of the liquid above the effective temperature required to soften and melt the wax deposit. 4. Effectiveness of the FCR System in Flow Conditions. When the temperature of the liquid was raised above the effective melting temperature of the wax deposits, the liquid was directed through the observation section where wax had been predeposited. The result (Figure 13) shows that the combination effect of a high temperature coupled with the high shear rate caused by the generated nitrogen gas can effectively remove wax deposits from the pipe wall. Wax particles, not soluble in the aqueous reactive slug, were then observed flowing with the reactive slug. The redeposit of wax particles further downstream is unlikely because particulate wax does not deposit on the pipe wall.25 Moreover, the addition of dispersant would further help disperse the wax particles in the solution. V. Conclusions An exothermic FCR system consisting of a reactive slug containing the two reactants and citric acid contained in polymer-coated capsules was shown to be able to manipulate the temperature profile in a tubular reactor. The location or time of the heat release is controlled by the thickness of the polymeric coating, while the maximum temperature and the rate of heat release are determined by the in situ concentrations of the reactants and the acid catalyst. The in situ concentrations of the reactants were found to be controlled by the fluid Peclet number and the ratio between the chemical slug length and the flow length. Meanwhile, the in situ concentration of the acid catalyst depends on the flow of the capsules and the controlled release of the acid catalyst from the capsules. The turbulent
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5871
dispersion of the capsules depended on the capsule Stokes number, while the flow of the capsule relative to the reactive slug was correlated to the capsule Reynolds number and the capsule Froude number. The FCR system under flow conditions was simulated using a dispersed-plug-flow model. Good agreement between the experimental results and the simulation was achieved. Finally, the FCR system was shown to effectively remove the wax deposit from the pipe wall. The ability to provide a large amount of heat in a controllable fashion at a desired location made the FCR technique a promising method to solve the billion dollar problem of wax deposition in subsea pipelines. The simulation model can be used to determine the reactive slug length, the inlet concentrations of the reactants and capsules, and other system parameters for a desired temperature profile inside a tubular reactor.
Figure 14. FCR system to control the temperature profile in a tubular reactor.
t is ∆He(t) ) πD2 h (z,t) - Te]∆z F C [T 4 f p 0
{(
)
(B.1)
otherwise
Using the Heaviside step function f (eq 35), eq B.1 can be simplified to
∆He(t) )
Acknowledgment
if [T h (z,t) - Te] > 0
πD2 F C [T h (z,t) - Te]f[T h (z,t)-Te]∆z (B.2) 4 f p
By integration over the destination region (from z ) L - lds/2 to z ) L + lds/2), the total amount of effective heat released to the destination region at time t is
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 former Conoco Production Research Division, Ponca City, for their advice and laboratory support in building the flow loop. We also thank Mr. Scott Thompson and Mr. Michael A. Iwaniw for their help in conducting some experiments. We also gratefully acknowledge the financial support from our affiliate companies: Baker Petrolite, ChevronTexaco, ConocoPhillips, PDVSA-Intevep, Schlumberger, Shell Oil, and Total Fina Elf & NPUSA Inc.
By integrating over time t ) 0 to time t ) ∞, we obtain the total amount of effective heat being released by the reactive slug in the destination region throughout the treatment as
Appendix A: Parameter Values in the Laboratory Flow Loop
He )
Heat capacity of the liquid: Cp ) 4.18 kJ/kg‚K. Density of the liquid: Ff ) 1000 kg/m3. Temperature of the injecting liquid: T0 ) 20 °C. Viscosity of the liquid: µf ) 0.001 kg/m‚s. Diameter of the pipe: D ) 0.025 m. Average overall heat-transfer coefficient: Uk ) 0.012 kJ/m2‚s‚K (experimental result). Diffusivity of NH4Cl and NaNO2 in aqueous solutions: De ) 10-9 m2/s. Thermal diffusivity of the reactive solution: R ) 2 × 10-6 m2/s. Dispersion coefficient of NH4Cl, NaNO2, and the acid catalyst in aqueous solutions: Da ) 0.2 m2/s (experimental result). In the case of turbulent flow, the thermal dispersion coefficient can be assumed to be equal to that for material dispersion:26
Rt ) De
(A.1)
Surrounding temperature: Ta ) 20 °C. Effective temperature: Te ) 60 °C. Appendix B: Derivation of the Criteria To Evaluate a FCR Treatment The heat release is only effective in melting the wax deposit when the temperature of the fluid in contact with the destination region [T h (z,t)] is higher than the effective temperature (Te). The amount of effective heat released by the reactive solution to a pipe section from z to z + ∆z in the destination region (Figure 14) at time
He(t) )
πD2 FC 4 f p
πD2 FC 4tds f p
z)L+l /2 f[T h (z,t)-Te] [T h (z,t) - Te] dz ∫z)L-l /2 ds
ds
(B.3)
z)L+l /2 ∫z)L-l /2 ds
ds
t)∞ f[T h (z,t)-Te] [T h (z,t) - Te] dt dz ∫t)0
(B.4)
where tds (s) is the characteristic time of the flow. A reasonable value for the characteristic time is the time it takes for the reactive slug to flow through the destination region:
tds ) lds/Vf
(B.5)
Therefore
He )
πD2Vf FC 4lds f p
z)L+l /2 ∫z)L-l /2 ds
ds
t)∞ f[T h (z,t)-Te] [T h (z,t) - Te] dt dz ∫t)0
(B.6)
Using the dimensionless length λ ) z/L and time θ ) tVf/L, eq B.6 can be converted to
He )
πD2L2 FC 4lds f p
λ)1+0.5l /L ∫λ)1-0.5l /L ds
ds
θ)∞ f[T h (λ,θ)-Te] [T h (λ,θ) - Te] dθ dλ ∫θ)0
(B.7)
which is eq 34. Meanwhile, the average effective contact time hte is the average contact time between the liquid and the destination region, provided that the liquid temperature is greater than the effective temperature Te. At any point z in the destination region, the effective contact time from time t to time t + ∆t is
5872 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
∆te(z) )
{
if T h (z,t) - Te > 0 otherwise
∆t 0
(B.8)
Ψ(η,θ) )
Again, using the Heaviside step function f (eq 35), eq B.8 can be simplified to
h (z,t)-T] ∆t ∆te(z) ) f[T
(B.9)
Integrating over time t ) 0 to time t ) ∞, we obtain the total effective contact time at point z in the destination region as
z)L+l /2 t)∞ f[T h (λ,θ)-Te] dt dz ∫z)L-l /2 ∫t)0
1 lds
ds
ds
(B.11)
∫λ)1-0.5l
λ)1+0.5lds/L ds/L
∫θ)0 f[Th (λ,θ)-Te] dθ dλ θ)∞
(B.12)
Appendix C: Derivation of the Analytical Solution for the Flow and Dispersion of a Reagent in a Tubular Reactor
2 ∂Ψ(λ,θ) 1 ∂ Ψ(λ,θ) ∂Ψ(λ,θ) + )0 ∂θ Pem ∂λ2 ∂λ
(37)
with the initial condition (eq 28)
rs rs eλe 2 2 otherwise
(28)
or
(
)
rs rs - Heaviside λ Ψ(λ,0) ) Heaviside λ + 2 2 (C.1) and the inlet boundary condition (eq 31)
{
Ψ(0,θ) ) 1 0
rs if 0 e θ e 2 otherwise
(C.4)
[ (x
x )
( x
x )]
Substituting η ) λ - θ into eq C.5 yields
[(
1 erf 2
λ-θ+
xPem
(
)
rs 2
2xθ λ-θ-
xPem
-
)]
rs 2
2xθ
for θ > 0 (C.6)
2 a ∂Ψcapsule 1 Dcapsule ∂ Ψreactant Vc ∂Ψcapsule )0 + ∂θ Pem Da Vf ∂λ ∂λ2 (4)
{
Ψcapsule(λ,0) ) 1 0
rs rs eλe 2 2 otherwise for -
(5)
and boundary condition (eq 6)
Ψcapsule(0,θ) )
{
1
for 0 e θ e
0
otherwise
rs Vf 2 Vc
(6)
By a change of variable, θ′ ) (Vc/Vf)θ, eq 4 becomes
(31)
Using a η coordinate moving with the center of the traveling chemical slug (η ) λ - θ), eq 37 can be transformed to 2 ∂Ψ(η,θ) 1 ∂ Ψ(η,θ) )0 ∂θ Pem ∂η2
)
with the initial condition (eq 5)
if -
)
(
which is eq 38. Similarly, we can obtain the solution for the equation of capsule flow (eq 4):
We will derive the analytical solution for eq 37:
(
)
rs rs - Heaviside ξ 2 2
1 Pem 1 Pem 1 erf rs + η 2 4 θ 2 θ 1 Pem 1 Pem for θ > 0 (C.5) erf r + η 4 θ s 2 θ
Ψ(η,θ) )
erf
{
for θ > 0 (C.3)
(
Ψ(λ,θ) )
which is eq 36.
Ψ(λ,0) ) 1 0
2
m
f(ξ) ) Heaviside ξ +
Using the dimensionless length λ ) z/L and time θ ) tVf/L, eq B.11 can be converted to
L2 ht e ) ldsVf
∫-∞∞[f(ξ) e-Pe (η-ξ) /4θ dξ]
where f(ξ) is the initial boundary condition written with η ) ξ as (eq C.1)
(B.10)
The average effective contact time in the destination region (from z ) L - lds/2 to z ) L + lds/2) then becomes
ht e )
Pem πθ
Substituting eq C.4 into eq C.3 and integrating give
t)∞ f[T h (z,t)-T] dt ∫t)0
te(z) )
x
1 2
(C.2)
A general solution for eq C.2 can be adapted from eq 1 in section 2.2 in Carslaw and Jaeger27 as
2 ∂Ψcapsule 1 Vf ∂ Ψcapsule ∂Ψcapsule + )0 ∂θ′ Pec Vc ∂λ ∂λ2
(C.7)
and the boundary equation becomes
{
Ψcapsule(0,θ′) ) 1 0
for 0 e θ′ e
rs 2
(C.8)
otherwise
Using solution (C.6), we obtain the solution for the concentration of the capsule as
Ψcapsule(λ,θ′) )
[( (
1 erf 2
λ - θ′ +
xPec
)]
2xθ′ rs λ - θ′ 2 erf xPec 2xθ′
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5873
)
rs 2
-
for θ′ > 0 (C.9)
Substituting θ′ ) (Vc/Vf)θ:
[(
Ψcapsule(λ,θ) )
1 erf 2
erf
xPec
(
λ-
xPec
)]
2xθ Vc rs λ- θVf 2 2xθ
)
Vc rs θ+ Vf 2
-
for θ > 0 (C.10)
which is eq 7. Literature Cited (1) Nguyen, A. D.; Fogler, H. S.; Sumaeth, C. Fused chemical reactions. 2. Encapsulation: Application to remediation of paraffin plugged pipelines. Ind. Eng. Chem. Res. 2001, 40 (23), 5058. (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 (Jun 2, 1996), 83. (4) Nguyen, A. D.; Iwaniw, A. M.; Fogler, H. S. Kinetics and Mechanism of the Reaction between Ammonium Ion and Nitrite Ion: Experimental and Theoretical Studies. Chem. Eng. Sci. 2003, 58, 4351. (5) Singh, P.; Fogler, H. S. Fused chemical reactions: The use of dispersion to delay reaction time in tubular reactors. Ind. Eng. Chem. Res. 1998, 37 (Jun 6, 1998), 2203. (6) Economides, M. J. N. K. G. Reservoir Stimulation, 3rd ed.; John Wiley: Chichester, England, 2000. (7) Moritis, G. Flow Assurance Challenges Production from Deeper Water. Oil Gas J. 2001, Jan (1), 66. (8) Brown, T. S.; Niesen, V. G.; Erickson, D. D. Measurement and Prediction of the Kinetics of Paraffin Deposition. J. Pet. Technol. 1995, 47 (4), 328.
(9) Nguyen, A. D.; Fogler, H. S. Facilitated Diffusion in the Dissolution of Carboxylic Polymers. AIChE J. 2004, in press. (10) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice Hall PTR: Upper Saddle River, NJ, 1999. (11) Crowe, C. T.; Gore, R. A.; Troutt, T. R. Particle Dispersion by Coherent Structures in Free Shear Flows. Part. Sci. Technol. 1985, 3, 149. (12) Crowe, C. T.; Sommerfeld, M.; Tsuji, Y. Multiphase flows with droplets and particles; CRC Press: Boca Raton, FL, 1998. (13) Mols, B.; Oliemans, R. V. A. A turbulent diffusion model for particle dispersion and deposition in horizontal tube flow. Int. J. Multiphase Flow 1998, 24 (1), 55. (14) Loth, E. Numerical approaches for motion of dispersed particles, droplets and bubbles. Prog. Energy Combust. Sci. 2000, 26 (3), 161. (15) Deen, W. M. Analysis of transport phenomena; Oxford University Press: New York, 1998. (16) Loth, E. Numerical Approaches for Motion of Dispersed Particles, Droplets, and Bubbles. Prog. Energy Combust. Sci. 2000, 26 (3), 161. (17) Liu, H.; Gao, X.; Xu, W. Flow Regimes and Equations for Predicting Capsule Flow through Pipe. HTD (Am. Soc. Mech. Eng.) 1998, 361-5, 343. (18) Molerus, O.; Wellmann, P. A New Concept for the Calculation of Pressure-Drop with Hydraulic Transport of Solids in Horizontal Pipes. Chem. Eng. Sci. 1981, 36 (10), 1623. (19) Ohashi, H.; Sugawara, T.; Kikuchi, K.; Ise, M. Average Particle-Velocity in Solid-Liquid 2-Phase Flow through Vertical and Horizontal Tubes. J. Chem. Eng. Jpn. 1980, 13 (5), 343. (20) Sastry, S. K.; Zuritz, C. A. A Review of Particle Behavior in Tube Flow: Applications to Aseptic Processing. J. Food Proc. Eng. 1987, 10, 27. (21) Kruyer, J.; Redberge, P. J.; Ellis, H. S. Pipeline Flow of Capsules. 9. J. Fluid Mech. 1967, 30, 513. (22) Vlasak, P. An Experimental Investigation of Capsules of Anomalous Shape Conveyed by Liquid in a Pipe. Powder Technol. 1999, 104 (3), 207. (23) Nauman, E. B. Chemical reactor design, optimization, and scaleup; McGraw-Hill: New York, 2002. (24) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, Fl, 1991. (25) Singh, P. Gel Deposition on Cold Surfaces. Ph.D. Thesis, University of Michigan, Ann Arbor, MI, 2000. (26) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport phenomena, 2nd ed.; Wiley: New York, 2002. (27) Carslaw, H. S.; Jaeger, J. C. Conduction of heat in solids, 2nd ed.; Clarendon Press: Oxford, 1959.
Received for review January 20, 2004 Revised manuscript received April 12, 2004 Accepted May 13, 2004 IE049933K
