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RESEARCH NOTES Stochastic Modeling and Simulation of the Formation of Carbon Molecular Sieves by Carbon Deposition L. T. Fan,* Andres Argoti, Walter P. Walawender, and Song-Tien Chou† Department of Chemical Engineering, Kansas State University, 105 Durland Hall, Manhattan, Kansas 66506

Molecular sieves, including carbon molecular sieves (CMSs) manufactured from activated carbons, can be variously applied in the purification and separation of gaseous and liquid mixtures and can also serve as catalysts. CMSs are formed by depositing fine particles of carbon on the mouths of pores of such activated carbons. These carbon particles are generated by decomposing a gaseous carbon source. The formation of CMSs proceeds stochastically, which is mainly attributable to the mesoscopic sizes and complex motion of the depositing carbon particles and random distribution of the pores on the activated carbons. In this work, CMS formation is modeled as a pure-birth process with a linear intensity of transition. The resultant model gives rise to the governing equations for the mean and variance. The model is validated by comparing the analytical solutions of these equations with available experimental data. The mean value of the model is in excellent accord with the data. In addition, the kinetic constants resulting from the model have been found to obey the Arrhenius law. Introduction One type of molecular sieves, i.e., carbon molecular sieves (CMSs), has been known to be profoundly effective for the purification and separation of fluid mixtures1-3 and also serves as a catalyst.4,5 CMSs can be manufactured by depositing very fine carbon particles on the pores’ mouths of activated carbons to narrow them.6 Activated carbons, in turn, can be manufactured from various sources including biomass.3,6-9 The fine carbon particles are generated by decomposing gaseous organics of low molecular weights, for instance, methane and methanol. Because of the very complicated motion of depositing carbon particles and the highly complex geometry of participating material species including carbon particles and activated carbons in terms of their size and configuration, the formation of CMSs appears to occur irregularly and randomly. It is natural, or even desirable, that various aspects of a process involving mesoscopic, discrete entities undergoing complex motion and behavior, e.g., the CMS formation of interest, be dealt with by resorting to the statistical framework or a stochastic paradigm. Such aspects can be the analysis and modeling of the phenomena involved in the process as well as the design and control of the process itself.10-16 This work aims at stochastic analysis and modeling of CMS formation on the basis of the pure-birth process, which is a subclass of the widely adopted birth-death processes.11,17 The stochastic analysis and modeling are capable of revealing inherent fluctuations, i.e., internal * To whom correspondence should be addressed. Tel.: (785) 532-5584. Fax: (785) 532-7372. E-mail: [email protected]. † Permanent address: Department of Finance and Banking, Kun Shan University of Technology, Yung-Kang City, Tainan Hsien, 71003 Taiwan.

noises, of the process.10,11,17 The analytical solutions of the model are validated by comparing them with available experimental data for CMS formation through carbon deposition onto activated carbons obtained by Freitas and Figueiredo.18 Model Formulation The system under consideration comprises aggregates of fine carbon particles, termed “carbon packets” or simply packets, generated by the decomposition of a carbon source and mesopores, or simply pores, on activated carbons, as illustrated in Figure 1. At the outset, these pores’ mouths are completely open. Subsequently, they are narrowed as time progresses by the deposition of packets onto them to a size suitable for effecting molecular sieving; naturally, the number of narrowed pores increases temporally. It is considered that a pore is narrowed by a single packet. Moreover, the fraction of narrowed pores that become blocked by more than one packet is assumed to be negligible. This process, which is to be modeled as a pure-birth process, will be termed “pore narrowing” hereafter for simplicity. The number of packets that have already been deposited onto the pores’ mouths, thereby causing them to narrow, at time t is taken as the random variable of the process, N(t), whose realization is n. Thus, N(t) ≡ {0, 1, 2, ..., nM - 1, nM}, where nM is the number of packets that would have been deposited onto the maximum number of pores susceptible to narrowing. Definition of Transition-Intensity Functions. The intensity function (intensity of transition), λn, is defined as the instantaneous rate of change of the transition probability.11,17 In this work, the driving force, or potential, of the pore-narrowing process by carbon deposition is considered to be linearly proportional to

10.1021/ie049020q CCC: $30.25 © 2005 American Chemical Society Published on Web 02/23/2005

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Moments about the Mean. Equation 4 renders it possible to compute the mean, or the first moment, and higher moments about the mean, e.g., the variance.21-26 a. Mean. The mean, E[N(t)] or m(t), is defined as21-23

m(t) ) E[N(t)] )

∑n npn(t)

(5)

For the pore narrowing, the mean is, from eqs 4 and 5,

m(t) ) nM(1 - e-Rt)

(6)

Note that m(t) ) 0 at t ) 0 and m(t) f nM as t f ∞. b. Variance and Standard Deviation. The variance, Var[N(t)] or σ2(t), is defined as21-26

σ2(t) ) E[(N(t) - E[N(t)])2] )

∑n (n - E[N(t)])2pn(t)

(7)

For the pore narrowing, the variance is, from eqs 4, 6, and 7,

σ2(t) ) nM(1 - e-Rt)e-Rt

(8)

Thus, the standard deviation, σ(t), is Figure 1. Visualization of the progression of CMS formation: side view.

the number of carbon packets, or packets, yet to be deposited onto the pores’ mouths, which decreases with time. Consequently, the corresponding intensity function can be defined as

dn ) λn ) R(nM - n) dt

(1)

where R is a proportionality constant whose dimension is inverse time (t-1). Derivation and Solution of the Master Equation. The master, i.e., governing, equation is derived as the gain-loss equation resulting from probability balance,11,19 thereby leading to

d p (t) ) R[nM-(n - 1)] pn-1(t) - R(nM-n) pn(t), dt n n ) 1, 2, ..., nM - 1, nM (2) where pn(t) is the probability that n packets have already been deposited onto the pores’ mouths, thus narrowing them at time t. For n ) 0, the term R[nM(n-1)] pn-1(t) in the above expression is absent; thus

d p (t) ) -RnMp0(t) dt 0

(3)

Equation 2 can be solved recursively, thus yielding20,21

pn(t) )

nM! n!(nM - n)!

(1 - e-Rt)n(e-Rt)nM-n

(4)

This expression indicates that the distribution of random number N(t) is binomial. Further details are given as Supporting Information.

σ(t) ) xnM(1 - e-Rt)e-Rt

(9)

Note that σ2(t) ) 0 at t ) 0 and σ2(t) f 0 as t f ∞. Similarly, σ(t) ) 0 at t ) 0 and σ(t) f 0 as t f ∞. c. Coefficient of Variation. The coefficient of variation, CV(t), is defined as24-26

CV(t) ) σ(t)/m(t)

(10)

For the pore narrowing, therefore, eqs 6 and 9 yield

CV(t) )

x

e-Rt nM(1 - e-Rt)

(11)

Note that CV(t) f ∞ as t f 0 and CV(t) f 0 as t f ∞. Analysis of Experimental Data The available experimental data are presented in terms of the temporal increase in the amount of carbon deposited per unit weight of activated carbons at 11 temperature levels; specifically, they are given in units of milligrams of carbon per milligram of activated carbon, i.e., mg of C/mg of AC.18 The model derived in this work is validated with these data. To fit the model to the data, the number of carbon packets that have narrowed the pores, which is the random variable, N(t), needs to be related to the experimentally measurable variable, W(t), representing the weight of carbon already deposited on the activated carbons. At any time t, W(t) should be proportional to N(t); thus

W(t) ) ωN(t)

(12)

where ω is the weight of a single packet of carbon. The mean value of W(t), denoted by mW(t), is obtained from the above expression as follows:

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mW(t) ) E[W(t)] ) E[ωN(t)] ) ωE[N(t)] ) ωm(t) Substituting eq 6 for m(t) into this equation yields

mW(t) ) ωnM(1 - e-t) or

mW(t) ) WM(1 - e-Rt) where WM, or ωnM, is the weight of carbon that would have been deposited onto the maximum number of pores susceptible to narrowing. When Rt is lumped as dimensionless time τ, the above equation can be rewritten as

mW(τ) ) WM(1 - e-τ)

(13)

From this expression, the dimensionless form of the mean, m j W(τ), is

m j W(τ) ) mW(τ)/WM ) (1 - e-τ)

(14)

Similarly, the variance of W(t), σW2(t), is obtained from eq 12 as

σW2(t) ) Var[W(t)] ) Var[ωN(t)] ) ω2Var[N(t)] or

σW2(t) ) ω2[σ2(t)] Substituting eq 8 for σ2(t) into this expression yields

σW2(t) ) ω2nM(1 - e-Rt)e-Rt

(15)

Naturally, the standard deviation of W(t), σW(t), is, from this equation

σW(t) ) ωxnM(1 - e-Rt)e-Rt or

σW(t) ) WM

x

(1 - e-Rt)e-Rt nM

In terms of the dimensionless time, τ, this expression can be transformed into

σW(τ) ) WM

x

(1 - e-τ)e-τ nM

(16)

From this equation, the dimensionless form of the standard deviation, σ j W(τ), is

σ j W(τ) )

σW(τ) ) WM

x

(1 - e-τ)e-τ nM

(17)

From eqs 13 and 16, the coefficient of variation, CVW(τ), is obtained as

CVW(τ) )

σW(τ) mW(τ)

)

x

e-τ nM(1 - e-τ)

(18)

Table 1. Values of r and WM for the Pore Narrowing at Different Temperatures temperature, K (°C)

R, s-1 × 104 (min-1)

WM (mg of C)

873 (599.85) 923 (649.85) 948 (674.85) 973 (699.85) 1023 (749.85) 1048 (774.85) 1073 (799.85) 1098 (824.85) 1123 (849.85) 1173 (899.85) 1223 (949.85)

2.18 (0.0131) 1.73 (0.0104) 2.55 (0.0153) 5.10 (0.0306) 7.82 (0.0469) 9.28 (0.0557) 12.20 (0.0732) 11.33 (0.0680) 10.80 (0.0648) 8.05 (0.0483) 3.73 (0.0224)

0.0672 0.2464 0.3073 0.2602 0.2399 0.2381 0.2396 0.2780 0.2941 0.2938 0.4622

Note that this equation is identical with eq 11 for CV(t): Rt ) τ. Results and Discussion The model formulated, in terms of the temporal mean given in eq 13, has been fitted to the available experimental data18 through regression without linearization by resorting to the Levenberg-Marquardt method.27 The regression has resulted in the values of WM and those of R, listed in Table 1. These values have rendered it possible to evaluate the dimensionless time, Rt, i.e., τ, and the corresponding dimensionless mean, m j W(τ), from eq 14, the values of which are plotted in Figure 2 as a function of τ. Each of the experimentally measured weights in milligrams of carbon deposited per milligram of activated carbon is rendered dimensionless by dividing them by WM; the values of the resultant quantity, denoted as w(τ), are superimposed in the same figure for comparison. Note that m j W(τ) increases and asymptotically approaches to 1 as τ progresses; in other words, the process is bounded. The dimensionless standard deviation, σ j W(τ), as computed by eq 17, signifies the deviations attributable to the internal or characteristic noises of the process as predicted by the stochastic model. The values of σ j W(τ) around m j W(τ), i.e., m j W(τ) ( σ j W(τ), are plotted in Figure 2. Clearly, many of the experimental data18 lie beyond the expected variation, or scattering; this is almost always the case. The overall deviations of the experimental data include not only those attributable to the internal noises of the process as predicted by the model but also those attributable to the external noises due to unavoidable measurement errors and instrumental deficiencies that can never be totally eliminated. The coefficient of variation, CV, as defined by eq 10, provides a more meaningful relative measurement of the variability, spread, or dispersion of the values of a random variable about their mean than the standard deviation, σ. In general, the smaller the values of random variable N(t), i.e., the population size, the greater the extent of the expected fluctuations about their mean. As is evident from eqs 17 and 18, the expressions for σ j W(τ) and CVW(τ) depend on the number of packets that would have been deposited onto the maximum number of pores susceptible to narrowing per milligram of activated carbon, i.e., nM; thus, this number must be estimated. This is carried out by dividing the total volume of mesopores, or pores as termed earlier, per milligram of activated carbon by the volume of a single pore, which is assumed to be cylindrical. The numerical values of the former and latter have been estimated to be approximately 5 × 10-4 and 3 × 10-17 cm3, respec-

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Figure 2. Dimensionless mean, m j W, and dimensionless standard deviation, σ j W, for the pore narrowing as functions of the dimensionless time, τ.

Figure 3. Arrhenius plot for R exhibiting the three kinetic regimes for the pore narrowing.

tively. As a result, the order of magnitude estimate of nM is obtained as 1.67 × 1013 pores per milligram of activated carbon.28,29 Naturally, R in eq 6 has the connotation of the kinetic constant, and thus it should obey the Arrhenius law, as illustrated in Figure 3. It exhibits the well-known three kinetic regimes, including external-diffusion, porediffusion, and chemical-kinetic regimes.30-32 In the external-diffusion regime below 9.35 × 10-4 -1 K (above 1070 K), carbon particles deposit predominately on the external surfaces of the activated carbons. In this regime, the carbon-particle movement toward the surface of the activated carbons is the rate-control-

ling, or slowest, step. Once the carbon particles reach the surface of the activated carbons, they tend to react, or induct, rapidly with it, thereby leaving only a relatively small fraction of carbon particles to penetrate into the pores’ interior. This eventually leads to the blocking of the pores’ mouths by the carbon particles cumulatively remaining on the activated carbons’ surfaces. Note that the value of R at the highest temperature, 1223 K (8.18 × 10-4 K-1), is excluded from Figure 3. This value of R is a statistical outlier from the standpoint of the Arrhenius plot, thus indicating that a mechanism profoundly different from that at lower temperatures is active at this highest temperature. The

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depositing carbon particles nearly instantaneously initiate the formation of a rough layer of carbon on the activated carbons’ surfaces. The layer’s roughness is increasingly magnified as it grows; this magnification of surface roughness accompanied by increased surface area, in turn, accelerates the deposition of carbon particles. In other words, the magnification of roughness and the deposition of carbon particles affect each other synergistically. In fact, the exceedingly high value of WM at the highest temperature, 0.4622 mg of C, seems to attest to the occurrence of such a mechanism. The reaction rate in the external-diffusion regime is temperature-invariant;30 hence, the regression of the values of R subject to the constraint that the activation energy be greater or equal to zero leads to a null value of the activation energy in this regime. In the pore-diffusion regime between 10.31 × 10-4 K-1 (970 K) and 9.35 × 10-4 K-1 (1070 K), carbon particles deposit mainly on the pores’ mouths. In this regime, the carbon-particle diffusion through the pores is the rate-controlling, or slowest, step. These carbon particles, therefore, tend to rapidly react with the pores’ internal surfaces in the vicinity of the mouths, thereby narrowing them. This pore narrowing, in turn, hinders further penetration of additional carbon particles into the pores’ interior. The regime’s activation energy calculated from the Arrhenius plot is 73.8 kJ‚mol-1. In the chemical-kinetic regime above 10.31 × 10-4 K-1 (below 970 K), carbon particles deposit uniformly over the entire surface of the pores’ interior. In this regime, the reaction between the carbon particles and the pores’ internal surfaces is the rate-controlling, or slowest, step. These carbon particles, therefore, have ample time to diffuse and accumulate into the entire surface of the pores’ interior. Note that the value of R at the lowest temperature, 873 K (11.45 × 10-4 K-1), is excluded from Figure 3. This value of R is also a statistical outlier as that of R at the highest temperature from the standpoint of the Arrhenius plot. This lowest temperature is unfavorable for the reaction, or interaction, to be effected; thus, the fraction of carbon particles deposited onto the pores’ internal surfaces is minute. The exceedingly low value of WM at the lowest temperature, 0.0672 mg of C, which is about 1 order of magnitude smaller than those of WM at higher temperatures, attests to this fact. The activation energy calculated for this regime is 160.7 kJ‚mol-1. As indicated earlier, the activation energy for the external-diffusion regime is essentially negligible. In this regime, the energy barrier that the carbon particles must overcome to reach the activated carbons’ external surfaces is minimal.32 Similar to molecules, carbon particles diffusing through the external gas phase encounter little resistance. In contrast, the carbon particles’ diffusion or motion through the profoundly confined space of pores is far more restricted than that through the external gas phase. This is attributable to the mutual collisions among the particles and porewall friction, especially in the vicinity of the pores’ mouths. Thus, the activation energy increases substantially to 73.8 kJ‚mol-1. The activation energy in the chemical-kinetic regime is typical of a chemical reaction, which is expected to be appreciably greater than that of a physical diffusional process through the pores where the reaction, or interaction, occurs. As is well-known, it is approximately twice as large as that of the porediffusion regime,30,31 as is the case in the current work.

It is interesting to note that the mechanistic interpretations as well as the values of the activation energies for various kinetic regimes are essentially in accord with those deduced indirectly by Freitas and Figueiredo18 on the basis of the first-order kinetic constants for the gas-phase decomposition of benzene, which has served as the source for the carbon particles depositing on activated carbons in their work. Conclusions A stochastic model has been derived for the formation of CMSs on activated carbons by the deposition of carbon packets as a pure-birth process based on a firstorder intensity function. The solution of the governing differential equation of the model, i.e., master equation, yields the probability distribution of the number of carbon packets depositing on the mouths of the pores of activated carbons. This number is regarded as the random variable of the process; the resultant distribution is naturally binomial. In addition to the mean of carbon packets that have narrowed the pores, the second moment about the mean, i.e., the variance, has been derived. The model presented has been validated with available experimental data. The mean value of the model is in excellent accord with these experimental data. Moreover, the kinetic constants resulting from the model have been found to obey the Arrhenius law. The Arrhenius plot reveals the three classical kinetic regimes, namely, chemical-kinetic, pore-diffusion, and external-diffusion regimes. Acknowledgment This is Contribution No. 04-326 from the Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS. The authors indeed appreciate the generous help of Dr. J. L. Figueiredo in providing them with the experimental data. Supporting Information Available: Appendix A includes the detailed derivation of the master equation. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Ruthven, D. M. Principles of adsorption and adsorption processes; Wiley: New York, 1984; pp 8 and 9. (2) Yates, S. F. Process for removing vinylidene chloride from 1,1-dichloro-1-fluoroethane. U.S. Patent 4,940,824, 1990. (3) Vyas, S. N.; Patwardhan, S. R.; Gangadhar, B. Carbon molecular sieves from bituminous coal by controlled coke deposition. Carbon 1992, 30, 605-612. (4) Foley, H. C.; Lafyatis, D. S.; Mariwala, R. K. Design and synthesis of carbon molecular sieves for separation and catalysis. Symposium on Advances in Zeolites and Pillared Clay Synthesis, New York, Aug 25-30, 1991; p 358. (5) Schmitt, J. L. Carbon molecular sieves as selective catalyst supports10 years later. Carbon 1991, 29, 743-745. (6) Hu, Z.; van Sant, E. F. Carbon molecular sieves produced from walnut shell. Carbon 1995, 33, 561-567. (7) Lizzio, A. A.; Rostam-Abadi, M. Production of carbon molecular sieves from Illinois coal. Fuel Process. Technol. 1993, 34, 97-122. (8) Wu, C. C.; Walawender, W. P.; Fan, L. T. Chemical agents for production of activated carbons from extrusion cooked grain products. Proceedings of the 23rd Biennial Conference on Carbon, University Park, PA, July 18-23, 1997; pp 116 and 117.

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Received for review October 10, 2004 Revised manuscript received January 25, 2005 Accepted February 2, 2005 IE049020Q