Ind. Eng. Chem. Res. 2007, 46, 7367-7377
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RESEARCH NOTES Aging of Crude Terephthalic Acid Crystals at High Temperatures Qinbo Wang,*,† Youwei Cheng,† Haibo Xu,‡ Lijun Wang,† and Xi Li*,† Department of Chemical Engineering, Zhejiang UniVersity, Hangzhou, 310027 Zhejiang, People’s Republic of China, and Zhejiang Hualian Sunshine Petrochemical Co., Ltd., 312000 Zhejiang, People’s Republic of China
By using a specially contrived experimental technique, crude terephthalic acid (TPA) crystals have been aged in their own saturated 90 mass % acetic acid + 10 mass % solution at 467, 477, 487, 497, and 507 K. The time evolution of crystal size distribution, as well as contents of the contaminant 4-carboxybenzaldehyde (4-CBA) in both solid and liquid phases, has been experimentally measured. Experimental results reveal the following: (1) The effect of stirring rate on the aging of crude TPA crystals is negligible under the conditions of this study. (2) The crystal size distribution is observed to broaden with time, accompanied by a decrease in number and an increase in averaged crystal size during aging. The aging rate increases with the increasing of temperature. (3) Under the conditions of this study, substantial purification occurs during aging. The rate of purification increases as the temperature increases. The Ostwald ripening mechanism is proposed to model the effect of aging on crystal size distribution. A purification model based on Ostwald ripening and molecular diffusion of 4-CBA in TPA crystal is proposed to simulate the effect of aging on the purification of crude TPA by crystal aging. The model parameters have been determined in a nonlinear optimization. The fit of the simulated results is fairly good. Introduction Terephthalic acid (TPA) is a major commodity chemical manufactured by the air oxidation of para-xylene (PX). TPA is a crystal that contains two carboxylic groups in the paraposition on a benzene ring. It is reacted with ethylene glycol via a condensation reaction in the manufacture of polyethylene terephthalate (PET), which is widely used in the polyester industry.1 Worldwide production of TPA has increased at an annual rate of >10% for the last 5 years, and the demands of TPA exceeded 30 million tons in the last year.2 Commercially, the majority of TPA is produced by the air oxidation of PX in acetic acid (HAc), promoted by cobalt (Co), manganese (Mn), and bromine (Br) catalysts at 150-210 °C.3-6 The impurities present in TPA crystals are generally intermediates, oxidation byproducts, and catalyst. The intermediate 4-carboxybenzaldehyde (4-CBA) is one of the most difficult contaminants to remove and, unfortunately, probably one of the most deleterious.1,7 The difference between 4-CBA and TPA is that 4-CBA contains an aldehyde group and a carboxylic group in the para-position on a benzene ring. The aldehyde group is unable to undergo the condensation reaction with ethylene glycol and acts as a chain terminator in the PET polymerization. This results in fibers that break easily as well as a slower rate of polymerization. The removal of 4-CBA from TPA is, therefore, of great commercial importance. Beside being impurity enriched, the reactive crystallized TPA consisted of polydispersed crystals that are usually small. The size of the TPA crystals affects the subsequent load of filter * To whom correspondence should be addressed. Phone: +86-57187952210. Fax: +86-571-87951227. E-mail:
[email protected] (Qinbo Wang),
[email protected] (Xi Li). † Zhejiang University. ‡ Zhejiang Hualian Sunshine Petrochemical Co., Ltd..
and dry operation, which directly affects the production yield and energy consumption. The larger the crystal size is, the easier the filter and dry operation can be.1 The enlargement of TPA crystal size is, therefore, also of great commercial importance. Crystal aging refers to all changes that occur after nucleation and growth.8,9 The aging of TPA crystals has been studied for a number of years by Myerson and co-workers.9-14 Brown and Myerson performed aging experiments using deuterated TPA (d-TPA) as a tracer. 9,12,14 In their experiments, saturated solutions of TPA were prepared at aging temperatures of 413493 K. Crystals of d-TPA were injected into the well-mixed saturated solution. The temperature as a function of time was recorded. Samples of solution and crystals were withdrawn periodically, and the percentage of d-TPA in the solution and the crystals was determined by using 13C NMR. The change in the percent of d-TPTA was used as a measure of the extent of aging. These studies demonstrated that the purity was found to be improved substantially with respect to 4-CBA when TPA crystals were suspended in their own saturated solution at temperatures ranging from 413 to 493 K. A mechanism of crystal aging based on crystal growth and dissolution due to small temperature fluctuations was used to estimate the aging rate. This estimation employed the experimentally determined growth and dissolution kinetics as well as the temperature vs time data. A comparison of the experimental and calculated results showed that, while the deviation was large in some cases, the general trend indicated that the temperature fluctuations could account for the crystal aging rate. The aging of TPA crystals at a variety of conditions has also been studied for several years by the authors.1,15,16 As opposed to the results of Myerson and co-workers,9-14 from the scanning electron microscope (SEM) photographs of the fresh and aged crystals (refer to Figures 1 and 2), we found that the crystal shape seemed unchanged when the TPA crystals were suspended
10.1021/ie0616139 CCC: $37.00 © 2007 American Chemical Society Published on Web 09/27/2007
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Figure 1. Fresh TPA crystals (×500) for aging at 487 K.
state, since a great deal of energy is associated with the large amount of interfacial area present and the mixture is not initially in thermodynamic equilibrium. The total energy of the twophase system can be decreased via an increase in the size of the TPA crystals and, thus, a decrease in total interfacial area. Such a process is termed Ostwald ripening.8 Its driving force is the difference in solubility between the polydispersed TPA crystals, as can be given by the well-known Gibbs-Thomson equation.8 The solubility difference quantitatively described by the Gibbs-Thomson equation establishes a concentration gradient between the smaller and the larger crystals, which leads to the growth of the larger crystals at the expense of the smaller ones, and the substance is transported through the liquid phase. Ostwald ripening can have a significant influence on the final properties of the product TPA crystals, especially on the crystal size distribution and the impurity content. A search of the literature shows that no public reports are available on the effect of aging on the size distribution of TPA crystals. It has been demonstrated that not only can impure TPA crystals be purified but also the crystal size can be enlarged during crystal aging. It is the purpose of this work to experimentally examine the rates of crystal growth and purification of TPA by crystal aging and attempt to explain the experimental phenomena by appropriately modeling the experimental results. Experimental Work
Figure 2. Aged TPA crystals (×500) after 150 min at 487 K.
in their own saturated aqueous acetic acid solution at 467 to 507 K. In agreement with the results of Myerson and coworkers,9-14 we found that the purity levels of TPA crystals were improved substantially, but with levels of the impurity 4-CBA declining as much as 90%. In the work of Myerson and co-workers,9-14 they found a transformation of crystal shape from globular to faceted needles, which might result from a series of dissolution and growth cycles. On the basis of this phenomenon, the mechanism of crystal aging based on crystal growth and dissolution due to small temperature fluctuations was proposed. In our work, as shown in Figures 1 and 2, the transformation of crystal shape was not found; however, the levels of the impurity 4-CBA declined. This means that the mechanism proposed by Myerson and co-workers might not be the major mechanism for crude TPA purification in aqueous acetic acid under our experimental conditions, and there must be some other major mechanism for the purification of TPA crystals. On the other hand, the system of reactive crystallized TPA crystals and the saturated solution is not in its lowest energy
Chemicals. Crude TPA crystals were obtained from a domestic manufacturer. The crystals were initially examined by a Malvern 2000S laser particle size analyzer and a highperformance liquid chromatograph (Shimadzu 6A HPLC). The average crystal size was measured to be (58.0 ( 2.3) µm, and the content of impurity 4-CBA in TPA crystals was estimated to be (4500 ( 135) ppm. Methanol and acetonitrile were obtained from USA Tedia Company, Inc., and had a purity of 0.999 (mass fraction). All other chemicals used in the experiments were obtained from Hangzhou Chemical Reagent Co. and had a purity higher than 0.990 (mass fraction). Apparatus and Procedure. The apparatus employed to determine the rate of crystal aging appears in Figure 3. The 2.0 L aging vessel was constructed from titanium to resist corrosion and was designed for temperatures up to 533 K and pressures up to 60 atm. A paddle-type agitator with a turbine impeller is used for agitation. The system was equipped with three condensers in order to ensure complete condensation and recycle of the evaporated compounds. The pressure was controlled by a back-pressure valve to be 22 atm. A Pt100 thermal resistance thermometer was inserted into the vessel for the measurement of temperature, and the thermometers had an uncertainty of (0.1 K. In each experiment, 450 g of crude TPA crystals, 1260 g of acetic acid, 140 g of water, and catalysts Co/Mn/Br were deposited in the titanium vessel. The slurry in the vessel was heated at 10 K/min to the experimental temperature. The temperature was maintained within (0.5 K of the desired temperature by a PID controlling system shown in Figure 3. When the temperature was higher than the desired point, the computer program would start the peristaltic pump, and cooling oil would flow through the spiral cooling coil in the vessel to cool the contents of the vessel. When the temperature was lower than the desired point, the computer program would start the heating controlling circuit, and the vessel wall would be electrically heated. When the slurry temperature was heated to the experimental point, the mixtures of initially added TPA crystals and solvents
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Figure 3. Experimental apparatus for crude terephthalic acid crystals aging: (1) nitrogen cylinder; (2) air cylinder; (3) mass flowmeter; (4) nitrogen/air switch valve; (5) air buffer vessel; (6) thermal resistance thermometer; (7) aging vessel; (8) vessel wall heating control circuit; (9) spiral cooling coil; (10) cooling oil tank; (11) peristaltic pump; (12) agitator; (13) condenser; (14) back-pressure valve; (15) gas-liquid separator; (16) flowrator; (17) vent washing tank; (18) dewatering cotton; (19) nitrogen sweeping valve; (20) sampling valve; (21) atmospheric valve; (22) solid-phase sample collector; (23) sintered metallic filter; (24) liquid-phase sample collector. Table 1. Experimental Conditions for Crude Terephthalic Acid Crystals Aging run
T (K)
mTPA (g)
mHAc (g)
mH2O (g)
stirring rate (rpm)
CCo (106 kg/kgHAc)
CMn (106 kg/kgHAc)
CBr (106 kg/kgHAc)
CS0 (106 kg/kgTA)
1 2 3 4 5 6 7 8
487 487 487 467 477 487 497 507
450 450 450 450 450 450 450 450
1260 1260 1260 1260 1260 1260 1260 1260
140 140 140 140 140 140 140 140
500 700 900 900 900 900 900 900
200 200 200 200 200 200 200 200
14 14 14 14 14 14 14 14
214 214 214 214 214 214 214 214
4500 4500 4500 4500 4500 4500 4500 4500
remained isothermal and aging began. Air was continuously fed into the vessel at 3 L/min. Aged slurry was sampled every 15 min. Each aging experiment lasted 150 min. To simultaneously determine the content of impurity 4-CBA in the liquid and solid phases, we separated the liquid phase from the slurry at the experimental temperature by using the specially designed sampling system shown in Figure 3. When the sampling valve (valve 20) was open, the slurry was transferred from the aging vessel into a titanium pipe between the atmospheric valve (valve 21) and the nitrogen sweeping valve (valve 19). The titanium pipe was electric-heated around the wall, and the wall temperature was controlled to be approximately equal to the slurry temperature. After this was done, the sampling valve was closed, and the atmospheric valve and the nitrogen sweeping valve were opened. The slurry was then pressed toward a porous sintered metallic filter with an internal aperture size of 1 µm. The filtered solution was then collected and cooled in a liquid-phase sample collector, and the concentration of 4-CBA in the liquid phase was determined by the chromatograph method. The filtered crystals in the solid-phase sample collector were taken out. The concentration of 4-CBA in the solid was determined by the chromatograph method, and the size distribution of TPA crystals was measured by the Malvern 2000S laser particle size analyzer. About 10 mL of saturated solution and 3.5 g of solid was sampled each time. The experimental runs’ reproducibility was verified by repeating each of them at least twice.1,15 The experimental conditions are listed in Table 1.
Analytical Techniques. 4-CBA concentration in the TPA crystals was determined by Agilent-1000 high-performance liquid-phase chromatography (HPLC). The Hypersil SAX ion chromatographic column was used. The mobile phase consisted of water and acetonitrile. During the analytical process, the water content was maintained at 90 mass % and the content of acetonitrile was maintained at 10 mass %. The external reference method was used, and each analysis took ∼9 min. 4-CBA concentration in the solution was determined by HPLC and gas chromatography (GC). The internal standard method was used in the analysis. The mass ratio of 4-CBA and the internal standard substances in the liquid were determined by HPLC. Agilent-1000 HPLC with column of Diamonsil C18 (250 mm × 4.6 mm, Φ5 µm) was used. Gradient elution was used for complete separation of the analytes at room temperature. The mobile phase consisted of three eluents (i.e., water + acetonitrile + methanol), and the following three-component gradient evolution program was adopted: at 0 min, 5 mass % acetonitrile and 95% water; from 0 to 5 min, the mixture composition changed linearly with time to become 85 mass % water, 5 mass % methanol, and 10 mass % acetonitrile; from 5 to 8 min, the mixture composition changed linearly with time to become 55 mass % water, 10 mass % methanol, and 35 mass % acetonitrile; from 8 to 12 min, the mixture composition changed linearly with time to become 15 mass % water, 10 mass % methanol, and 75 mass % acetonitrile; from 12 to 14 min, the mixture composition changed linearly with time to
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Figure 5. Volume-averaged crystal size L43 and dimensionless total crystal number N/N0 for experimental run 6.
Figure 4. (a) Experimental and (b) model calculated volume-based crystal size distribution Φ(L,t) for experimental run 6: (1) 0 min; (2) 15 min; (3) 30 min; (4) 45 min; (5) 60 min; (6) 75 min; (7) 90 min; (8) 105 min; (9) 120 min; (10) 135 min; and (11) 150 min.
become pure acetonitrile; from 14 min on, 100 mass % acetonitrile. Each analysis took ∼20 min. The mass ratio of solvent HAc and the internal standard substances in the liquid was determined by GC using a Kexiao GC-1690 GC with a hydrogen flame ionization detector. The SE-54 (30 m) capillary chromatographic column was used. Toluene was used as the internal standard substance to correlate the data obtained from GC and HPLC analysis. To verify the uncertainty of the concentration measurement, two TA + toluene + HAc solutions of known concentration were analyzed. Compared with the known concentration, the uncertainty of the concentration was Lc(t), G(L,t) > 0, and the following backward difference scheme is used,
Φ(Lj,tm) - Φ(Lj-1,tm) Φ(Lj,tm+1) - Φ(Lj,tm) + G(Lj,tm) + tm+1 - tm Lj - Lj-1 b(Lj,tm)G(Lj,tm) ) 0 (11) For crystals with size L(t) ) Lc(t), G(L,t) ) 0, and eq 7 becomes
dΦ(Lc,t) + b(Lc,t)Φ(Lc,t) ) 0 dt
(12)
(5)
where λ represents the mechanism controlling the crystal growth process,18 c(t) represents the mean concentration of solute in solution, and csat(L) represents the solubility of solute with size L. From eq 5 and the well-known Gibbs-Thomson equation,8 we can obtain the critical radius Lc at time t as
Lc(t) ≡ R/ln(c(t)/csat ∞)
where
(7)
which can be by solved by the fourth-order Runge-Kutta method. In the above simulations, a time step of 0.005 min is used. The minimum size Lmin is set to 2 × 10-8 m, and the maximum size Lmax is set to 2.5 × 10-4 m. Four-hundred discrete points in the L direction are used. The model structure for Ostwald ripening and the initial CSD determine the approximate shape for the time-evolution curve of CSD during aging, and only one other characteristic variable is needed to determine the specific time-evolution curve of CSD.1 We chose to focus on the model’s ability to capture the volume-averaged crystal size rather than the correct prediction of the volume-baseed CSD, since there is not a dramatic change in the overall shape of the CSD and the averaged crystal size is more concerned than CSD in industrial aging vessel. The volume-averaged crystal size can be calculated from the discrete volume-based CSD by jmax
L43(tm) )
LjΦ(Lj,tm)∆Lj ∑ j)1 jmax
(13)
Φ(Lj,tm)∆Lj ∑ j)1 where Lj is the average crystal size of the jth given size interval. The kinetic parameters are determined in a nonlinear optimization, minimizing the relative difference between the simu-
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Figure 6. Content of impurity 4-CBA in TPA crystals and solution at different aging times for experimental run 6.
lated and experimentally measured volume-averaged crystal size. The objective function, F, is the sum of squares of all residuals. imax mmax
F)
∑ ∑ i)1 m)1
(
)
L43,cal(tm,i) - L43,exp(tm,i) L43,exp(tm,i)
2
(14)
where tm,i is the mth time point in the ith experiment, imax represents the number of batch experiments, and mmax represents the number of samples in each experiment. The Nelder-Mead Simplex method has been used in the optimizations.19 The method is implemented in the Matlab Optimization Toolbox (The Mathworks). The simulation program is written in Matlab to use its optimization routine. Purification Mechanism and Model Purification Mechanism. In the work of Myerson and coworkers, they found a transformation of crystal shape from globular to faceted needles, which might result from a series of dissolution and growth cycles. On the basis of this phenomenon, the mechanism of crystal purification based on crystal growth and dissolution due to small temperature fluctuation was proposed.9-14 In our work, from tens of experiments, we did not find the transformation of crystal shape; however, the levels of the impurity were found to substantially decline. According to the results of Myerson and co-workers, the main driving force for the purification process was the very small temperature fluctuations. If the operation temperature remained unchanged and did not fluctuate, the driving force would disappear and the purification process would stop. In the Ph.D. dissertation of Wang,1 it had been proved that, even if the operation temperature remained unchanged and did not fluctuate, the purification process would still continue and the impurity levels in the solid would still decline. Further, in an industrial aging tank, the temperature is controlled to be as steady as possible for safety. For these reasons, the mechanism proposed by Myerson and co-workers might not be the major mechanism for TPA purification at industrial conditions. The experimentally measured 4-CBA content in TPA crystals and solution at different aging times for experimental run 6 are shown in Figure 6. It can be seen that substantial purification occurs, which might be attributed to the following two reasons. (1) Myerson and co-workers suggested the mechanism of crystal purification based on crystal growth and dissolution due to small temperature fluctuation.9-14 We agree with the mechanism of crystal purification based on crystal growth and dissolution. However, we do not agree that the driving force is the very small temperature fluctuations; we think the driving
force is Ostwald ripening, which refers to the fact that crystals larger than the critical size grow while smaller crystals decay. The overall dissolution of the smaller crystals is caused by the fact that the equilibrium concentration of solute TPA at the interface on any such crystal is greater than the concentration of TPA in the bulk of the solution. Along with the dissolution of TPA crystals, the incorporated impurity 4-CBA will also dissolve into the solution and be quickly oxidized into TPA. By this means, the amount of impurity 4-CBA in both solution and TPA crystals declines. Conversely, the overall growth of the larger crystals is caused by the fact that the equilibrium concentration of solute TPA at the interface on any such crystal is less than the concentration of solute TPA in the bulk of the solution, leading to a net flow of solute TPA toward the large crystal. Along with the growth of TPA crystals, the impurity 4-CBA in solution will also incorporate into TPA crystals. Since the concentration of impurity 4-CBA in solution is very low because of the oxidation of 4-CBA to TPA, the amount of incorporated 4-CBA is much less than the dissolved 4-CBA from TPA crystals into solution accompanied with the dissolution of smaller crystals, leading to a net flow of impurity 4-CBA from TPA crystals to solution. (2) By using the Ostwald ripening kinetics determined by the experimentally measured CSD and the crystal growth model proposed in the section Crystal Growth Mechanism and Model, the content of 4-CBA in the solid was predicted by using the crystal growth and dissolution mechanism due to Ostwald ripening. The predicted results are shown in Figure 6. A clear inconsistency between the experimental and predicted contents of 4-CBA can be found. The predicted results show that the content of 4-CBA in TPA crystals does decline due to crystal growth and dissolution, but the predicted decline amplitude is much smaller than the experimental results. It indicates that, besides the crystal growth and dissolution mechanism, some other mechanism also attributes to the decline of impurity 4-CBA content in TPA crystals. The concentration of 4-CBA in solution is very low due to oxidation, which leads to a very small solid-liquid equilibrium concentration for 4-CBA at the interface on TPA crystals. However, the concentration of 4-CBA in TPA crystals is relatively higher. In the r direction of a TPA crystal, a concentration gradient exists, which makes the molecular diffusion of impurity 4-CBA from the inside of TPA crystals to the solution possible. In this work, we assume the molecular diffusion mechanism in TPA crystals also attributes to the decline of impurity 4-CBA content in TPA crystals. Model. The basic steps simultaneously occurring in the purification of TPA crystals due to molecular diffusion are (i) removal of 4-CBA from the inside to the surface of TPA crystals, (ii) dissolution of 4-CBA from the surface of TPA crystals to solution, and (iii) oxidation of 4-CBA to TPA in solution. Compared with the experimental results shown in Figure 6, we can conclude that step 3 is much faster, and it might not be the control step of the purification process. Step 1 was assumed to be achieved by dynamic molecular diffusion, and the diffusion rate had been proved to be much slower than that of step 2 in the Ph.D. dissertation of Wang.1 That is to say, the removal of impurity 4-CBA from the inside to the surface of TPA crystals may be assumed to be the control step of the purification process. By the assumptions that 4-CBA concentration is in partition equilibrium at the interface on TPA crystals and TPA crystals are spheric, the concentration distribution of 4-CBA in TPA
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{
crystal with size L can be described by the following diffusion equation
( )
∂cS ∂cS 1 ∂ ) DS 2 r2 ∂t ∂r ∂r r cS(r, 0) ) cS0 ∂cS | )0 ∂r r)0 L cS , t ) φSLcL(t) 2
(15)
( )
where cL(t) is the concentration of 4-CBA in solution. We find the following correlation can be used to correlate the experimentally measured cL(t)
cL(t) ) p exp(-κt) + q
(16)
Figure 7. Volume-averaged crystal size at different aging times and different stirring rates.
Inserting eq 16 into eq 15, the solution of eq 15 in the form of infinite series is ∞
cS(r,t) )
∑
n)1
[
(
Zn exp(-κt) + (Yn - Zn) exp L 2r
2nπ
sin
L
4n2π2DS L2
)]
t
×
r + φSL(p exp(-κt) + q) (17)
where
Zn )
2φSLpκL2(-1)n+1
,
nπ(4n2π2DS - κL2)
(-1)n+1 Yn ) 2(cS0 - φSLp - φSLq) (18) nπ The average concentration of 4-CBA in the solid phase is more concerned than the concentration distribution of 4-CBA in single TPA crystals, which can be calculated from eq 17. The average concentration of 4-CBA in the solid phase can be provided by
∫L
Lmax min
cS(t) )
F
Φ(L,t) m(L,t) dL kvL3
∫L
Lmax min
(19)
F)
∑ ∑ i)1 m)1
(
jmax Φ(L ,t ) j m,i
∑ j)1
∫0L/2 cS(r,t)r2 dr
cS,cal(tm,i) - cS,exp(tm,i) cS,exp(tm,i)
m(Lj,tm,i)∆Lj kvLj3
cS,cal(tm,i) )
jmax
F
(20)
Equation 19 reveals two coupled attributors to the content decline of impurity 4-CBA in TPA crystals. One is the molecular diffusion in TPA crystals, which is represented by the item m(L,t). The other is the crystal growth and dissolution due to Ostwald ripening, which is represented by item Φ(L,t). Optimization Algorithm. The purification model parameters are determined in a nonlinear optimization, minimizing the difference between simulated and experimental contents of impurity 4-CBA in TPA crystals. The objective function, F, is the sum of squares of all residuals. imax mmax
where cS,cal(tm,i) is the calculated average content of 4-CBA in the solid phase at time tm and in experimental run i. From eq 19, it can be provided by
Φ(L,t) dL
where m(L,t) is the mass of 4-CBA in the TPA crystal with size L and can be provided by
m(L,t) ) 4kaF
Figure 8. Content of 4-CBA in solid at different aging times and different stirring rates.
)
2
(21)
(22)
Φ(Lj,tm,i)∆Lj ∑ j)1
The analytical solution for m(L,t) can be obtained from eqs 17 and 20. The time evolution of volume-based CSD function, Φ(L,t), can be directly obtained from the measured volumebased CSD at different sampling times. Results and Discussion Effect of Stirring on TPA Aging. Experimental runs 1-3 were carried out to determine the effect of stirring speed on the aging process. The experimentally determined evolution of volume-averaged crystal size with time is shown in Figure 7. It shows that the effect of stirring rate on the volume-averaged crystal size is insignificant under the conditions of this study. The experimentally determined evolution of 4-CBA content in solid with time is shown in Figure 8. It also shows that the effect of stirring rate on 4-CBA content in solid is insignificant under the conditions of this study. As discussed in the preceding section, the content of 4-CBA in solid is determined based on three simultaneous basic steps, i.e., (i) removal of 4-CBA from
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R)
() ( )
2kaM 2ka 2/3 M σ ) 0.414k N 3kvRTF 3kvR A F
1/3
ln
cS
c(t)
≡
k0 ln
Figure 9. Volume-averaged crystal size at different aging times and different temperatures.
the inside to the surface of TPA crystals, (ii) dissolution of 4-CBA from the surface of TPA crystals to solution, and (iii) oxidation of 4-CBA to TPA in solution. When the stirring rate increases, step 2 and step 3 increase. However, the control step for the purification process is step 1. The increase of stirring rate does not increase the removal rate of 4-CBA from the inside to the surface of TPA crystals. In a word, the effect of stirring rate on the aging of crude TPA crystals is negligible under the conditions of this study. The experimental runs to study the effect of temperature on the aging process can be chosen to be 900 rpm. Effect of Temperature on Crystal Growth by Aging. The experimentally determined evolutions of the volume-averaged crystal size with time at different temperatures are shown in Figure 9. As discussed in the preceding section, the mixed slurry of TPA crystals and solvents was first heated to the experimental temperature and then kept isothermal for aging in batch experiments. At the first period of the aging process, the TPA crystals will dissolve, which will result in a decrease in the volume-averaged crystal size. As TPA crystals continue dissolving, the solution will be saturated and then the aging will begin. After the aging has begun, the volume-averaged crystal size will increase due to Ostwald ripening. The higher the experimental temperature, the more the TPA crystals will dissolve into solution. It results in the fact that the higher the experimental temperature, the smaller the averaged crystal size will be at t ) 0. The scattered points in Figure 9 reveal it. The results shown in Figure 9 also reveal that the higher the temperature, the bigger the averaged crystal size will be at the same aging time. This is because the driving force for Ostwald ripening is the difference in solubilities between the polydispersed TPA crystals. The ripening rate is determined by CSD and the growth and dissolving rates. The higher the temperature, the higher the dissolving and growth rates of TPA crystals will be, and the faster the ripening rate will be. The effect of temperature on the ripening rate is notable. On the basis of the known fundamental relationships of thermodynamics and their integration across the interfacial layer, Mersmann derived a simple equation for the calculation of solid-liquid interfacial tensions as20
σ ) 0.414kT
( ) () FNA M
2/3
ln
cS cL
(23)
According to eq 23, the capillary constant (R) in GibbsThomson equation can be correlated as
( ) cS
(24)
c(t)
where cS ) 6.0241 represents the concentration of TPA in solid, mol/kgTA and c(t) is the mean concentration of TPA in solution calculated by eq 9, mol/kgHAc. In order to simulate the aging process of TPA crystals due to Ostwald ripening, three model parameters need to be optimized. One is the λ in eq 5, the other is the k0 in eq 24. The two model parameters are temperatureindependent. The third parameter is the temperature-dependent crystal growth rate constant (kg) in eq 5. In this work, the effect of temperature on kg is correlated by the Arrehenius relation as
( )
kg ) kg0 exp -
Eg RT
(25)
The experimental runs 4-8 are optimized together. The obtained model parameters are
(
kg ) (1.008 ( 0.050) exp -
)
(4.435 ( 0.222) × 104 RT
k0 ) (4.720 ( 0.236) × 10-8
(26)
λ ) (4.210 ( 0.211) × 10-12 with the object function F ) 0.1082, which means the averaged relative deviation between the experimental and calculated volume-averaged crystal sizes is 4.7%. Simulated and experimental crystal size distributions for experimental run 6 are compared in Figure 4. The fit of the simulated distribution is tolerably good even though there are systematic deviations, especially for larger sizes. The simulated distributions are generally less broad than the corresponding experimental ones, and the peak of the distribution is shifted to the right. The same disagreements have also been found in many other works.8,17,21 The major problem causing the disagreement is the mean field nature of the widely adopted kinetic equation, eq 5. Such a mean field approximation assumes that a crystal’s aging rate is independent of its surroundings, i.e., a crystal with nearest neighbors that are larger than itself will age at exactly the same rate as if it were surrounded by crystals that were of small radius.21 Equation 5 can only be valid at an unspecified low volume fraction of solid phase. As pointed out by Baldan in his review,21 this flaw (i.e., the diffusional interactions between particles) is recognized and advances as the cause for the disagreement between the theoretically predicted and experimentally measured crystal size distributions. The strength of the diffusional interactions between crystals stems from the long-range Coulombic nature of the diffusion field surrounding a crystal. As a result, crystal interactions occur at distances of many particle diameters and restrict the validity of eq 5 to the unrealistic limit of zero volume fraction of the solid phase. Efforts to modify works on extending eq 5 to nonzero volume fraction of the solid phase have been attempted for decades since the 1960s. Many of the attempts to determine the statistically averaged growth rate of a crystal either do not account for the long-range nature of the diffusion field surrounding the particles and/or employed ad hoc assumptions in an attempt to account for the diffusional interactions between crystals.21 In the manufacture of TPA, average crystal size, especially the volumeaveraged crystal size, is more important than crystal size distribution in practice.1 In this work, we try to establish a model
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7375 Table 2. Partition Coefficient of 4-CBA and Regressed Parameters in Eq 16 for the Correlation of Measured 4-CBA Concentration in Solution run
φSL
p (103 mol/kgHAc)
κ (min-1)
q (104 mol/kgHAc)
RMDa
4 5 6 7 8
6.0504 5.2166 4.6140 4.1327 3.6627
3.5567 3.6047 3.8593 4.1540 4.4620
0.1367 0.1601 0.1758 0.2025 0.2215
4.6000 2.8400 1.6133 0.9213 0.7547
4.12 3.18 2.91 2.66 2.82
mmax 2 a RMD ) 1/m max x∑m)1{[(cL,cal(tm)) - (cL, exp(tm))/cL, exp(tm)] × 100} , representing the correlating accuracy.
Figure 10. Content of 4-CBA in solid at different aging times and temperatures.
that can accurately simulate the evolution of volume-averaged crystal size, instead of accurately simulating the crystal size distribution. Despite this, the fit of the simulated distribution shown in Figure 4 is tolerably good. The evolutions of the total crystal number in experiments and simulations (with optimized parameters) for experimental run 6 are compared in Figure 5. The total crystal number decreases steadily as expected, and the evolution of it is, in principle, well-predicted, even though the experimental data look a little oscillating. These oscillations in the experimental data are a combined consequence of the discretization in the model, the instrument accuracy of the Malvern 2000S, and the sampling and measurement deviations. Fortunately, these oscillations are not large enough to influence the results significantly. The fit of the simulated crystal number is tolerably good. The evolution of the volume-averaged crystal size in experiments and simulations (with optimized parameters) for experimental runs 4-8 are compared in Figure 9. Generally, the simulated results agree well with the experimental values. The simplex method is used to optimize the model parameters. A disadvantage of the simplex method is that no information about the confidence intervals of the parameters is obtained. Several optimizations are carried out starting from different initial parameter values. The results given by eq 26 show the obtained averaged parameter values and the deviations of parameters from different initial parameter values. We try to establish a model that can accurately simulate the evolution of volume-averaged crystal size, instead of accurately simulating the crystal size distribution. We still recognize that the model does not provide an optimal fit of the experimental data. There are still deviations that seem to be systematic. The experimental crystal size distribution is broader than the simulated one, and the mode of the model distribution is shifted toward larger sizes. This indicates that the model cannot fully describe the experiments and, thus, introduces deviations that propagate as the simulation proceeds. Despite this, the fit of both the simulated distribution and the simulated volumeaveraged crystal size is tolerably good. The simulated results show that the mechanism based on Ostwald ripening can account for the aging of TPA crystals in aqueous acetic acid at high temperatures. Effect of Temperature on the Purification of Crude TPA. The evolution of impurity 4-CBA content in the solid phase as a function of aging time is shown in Figure 10. It can be seen that, between 467 and 507 K, substantial purification occurs during aging. The rate of purification increases as the temperature increases. As discussed in the preceding section, the rate of purification rate is affected by the rate of Ostwald ripening and the rate of molecular diffusion for 4-CBA in TPA crystals.
The rate of Ostwald ripening increases as the temperature increases, and so does the rate of molecular diffusion for 4-CBA in TPA crystals. These all directly result in the rate of purification increasing as the temperature increases. The purification of crude TPA crystals can be simulated by the purification model established in the section Purification Mechanism and Model. The only purification model parameter is DS, which refers to the diffusion coefficient of impurity 4-CBA in TPA crystals. In this work, the effect of temperature on DS is correlated by the Arrehenius relation as
( )
DS ) D0 exp -
ED RT
(27)
The experimental runs 4-8 are optimized together. During the optimization, the solid-liquid partition coefficient of 4-CBA, φSL defined in eq 15, is directly cited from the dissertation of Wang.1 The concentration of 4-CBA in solution is directly correlated from the experimentally measured value by using eq 16. The correlated results and the solid-liquid partition coefficient are shown in Table 2. By a nonlinear optimization, the obtained model parameter in the purification model is
D0 ) (1.562 ( 0.078) × 10-2 ED ) (9.017 ( 0.451) × 104
(28)
Also, the model parameters are obtained at different initial parameter values. The time evolutions of the content of impurity 4-CBA in solid in experiments and simulations (with optimized parameters for experimental runs 4-8) are compared in Figure 10. The fit of the simulated content is fairly good. The maximum deviation between simulated and experimental results is
