ARTICLE pubs.acs.org/IECR
Stochastic Modeling for the Formation of Activated Carbons: Nonlinear Approach L. T. Fan* and Andres Argoti Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506, United States
Song-Tien Chou Department of Finance and Banking, Kun Shan University, Yung-Kang City, Tainan Hsien, 71003 Taiwan
bS Supporting Information ABSTRACT: Activated carbons (ACs) have been widely deployed in the purification of gases and liquids or the separation of their mixtures. The formation of ACs entails the modification of the original internal surfaces of carbonaceous substrates, for example, coal or biomass, which can be effected by a variety of chemical or physical methods, thereby augmenting the carbonaceous substrates’ adsorbing capacities. The formation of ACs tends to proceed randomly or stochastically in view of the discrete and mesoscopic nature of the carbonaceous substrates, as well as the random encounters between the activation agent and carbon on the carbonaceous substrates’ internal surfaces; in addition, the carbonaceous substrates’ internal surfaces exhibit an intricate morphology or structure. Naturally, these traits of the formation of ACs render the process to vary incessantly with time. Thus, it is highly desirable that the analysis, modeling, and simulation of the formation of ACs from carbonaceous substrates be performed in light of a stochastic paradigm. Herein, a stochastic model for the formation of ACs is formulated as a pure-death process based on a nonlinear intensity of transition. The model gives rise to the process’ nonlinear master equation whose solution is obtained by resorting to a rational approximation method, the system-size expansion. This solution renders it possible to compute the mean as well as higher moments about this mean, for example, variance or standard deviation, which reveal and quantify the process’ inherent fluctuations. The results of modeling are validated by comparing them with the available experimental data.
’ INTRODUCTION Activated carbons (ACs) have long been deployed effectively for the purification of gases and liquids, as well as for the separation of a wide variety of their respective mixtures.13 Examples include the separation of carbon dioxide from air.4 Moreover, ACs can also serve as catalysts or catalyst supports.5,6 The manufacture of ACs can be accomplished by resorting to various carbonaceous substrates ranging from agricultural residues7,8 to wheat.9 The formation of ACs entails the development of porosities on the original internal surfaces of the carbonaceous substrates by means of various chemical or physical methods, thereby augmenting the carbonaceous substrates’ adsorbing capacity.3,10 Physical activation of a carbonaceous substrate is mainly performed by carbonizing it in an inert atmosphere at a temperature below 700 °C and subsequently activating it in the presence of steam, carbon dioxide, or air at a temperature between 800 and 1000 °C.8,11 In contrast, chemical activation of a carbonaceous substrate is generally carried out by impregnating it with a strong dehydrating agent, for example, phosphoric acid, followed by heating of this mixture to a temperature between 400 and 800 °C.12,13 The resultant porosities on ACs can roughly be classified as micropores (50 nm) depending on the lengths of their diameters.10,14 The formation of ACs occurs randomly or stochastically, which is mainly attributable to the following factors. First, the discreteness and mesoscopic nature of the carbonaceous r 2011 American Chemical Society
substrates whose shapes are highly irregular and their internal surface configurations extremely intricate.3,10 Second, the random encounters between the activation agent, for example, carbon dioxide or phosphoric acid, and the carbon on the surfaces of the carbonaceous substrate: This is a chemical reaction, which proceeds randomly.1517 Third, the incessant variation of the process with time. The randomness or stochasticity of any discrete or particulate process, including the formation of ACs, manifests itself in the form of ceaseless fluctuations in the macroscopic variables describing the process.18,19 These inherent fluctuations impede the prediction with certainty of any temporal or spatial variation in the evolution of the process. Nevertheless, such uncertainty can be accommodated by taking into account the probability distribution of the variables describing the process, thereby giving rise to the definition of random variables.18,19 Hence, the analysis and modeling of a process described by random variables must be performed in light of a stochastic paradigm. Unlike a deterministic model, a stochastic model renders it possible to estimate and predict the inherent fluctuations of a system or process; moreover, the mean Special Issue: Churchill Issue Received: November 7, 2010 Accepted: February 25, 2011 Revised: February 23, 2011 Published: March 16, 2011 8836
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Figure 1. Schematic of the progression of the formation of ACs from carbonaceous substrates.
component of the stochastic model is equivalent to the deterministic model.18,19 The formation of ACs is an instance of fluidsolid reactions for which various mathematical models have been formulated;2024 their complexity ranges from linear22 to highly nonlinear.23 In such models, the experimentally measurable variable is usually the extent of conversion of the solid material as it reacts with the chemical species present in the fluid phase. Naturally, the kinetics of formation of ACs has been formulated in light of these models;2530 however, they are deterministic in nature. Apparently, no attempt has been made to model the formation of ACs as a stochastic process. The current contribution might represent the first attempt for stochastic analysis and modeling of the formation of ACs. As such, the model is formulated as a pure-death process with a single random variable and a nonlinear intensity of transition, and therefore, the scope and predictive capability of the resulting model are modest: The intricacy of the phenomena under consideration requires the formulation of a series of models of increasing complexity involving more than a single random variable. Such efforts will be undertaken in our future work. The resultant model gives rise to the process’ master equation18,31 whose solution is exceedingly complex to obtain in view of its nonlinearity. Such complexity is circumvented by resorting to the system-size expansion, a rational or logical approximation method, of the master equation of the process. The solution of the master equation renders it possible to compute the mean, or average, value of the random variable characterizing the process as well as higher moments about this mean, for example, the variance. These higher moments are collectively a manifestation of the process’ inherent fluctuations; their estimation could play a useful role in designing and implementing improved control strategies for the manufacture of ACs. For validation, the analytical solution of the master equation is compared with the experimental data for the formation of ACs obtained by Guo and Lua.28
’ MODEL FORMULATION Figure 1 depicts the system of concern, which comprises the obtainable pores in a carbonaceous substrate. The obtainable pores are identified as discretely and randomly distributed parts, or portions, of carbon in the substrate, which are not yet activated; upon activation, they generate open pores on the substrate’s internal surfaces. The formation of open pores is considered to be attributable entirely to the chemical reaction between the obtainable pores and an activation agent. The mass loss from the substrate concomitant with the generation of the open pores gives rise to the reduction in its weight. Naturally, the number of obtainable pores decreases temporally as they are
being activated; furthermore, the open pores generated from the obtainable pores give rise to a convoluted pore network.3,10 To facilitate the formulation of the model, it is assumed that the obtainable and open pores have uniform morphology; specifically, they are regarded as cylindrical. In addition, it is assumed that no open pore is blocked and that the reaction terminates prior to the collapse of the internal structure of the activated substrate because of the accelerated loss of substrate’s structural rigidity. The formation of ACs as described is to be modeled as a pure-death process,18,19 a class of Markovian process, in which the death event is identified as the activation of a single obtainable pore to form an open pore. The available experimental data for the formation of ACs are given in terms of the extent of conversion of a carbonaceous substrate into ACs at different temperatures.28 It is logical to equate the substrate’s weight loss to the reaction between the obtainable pores and the activation agent. Hence, the number of obtainable pores present on the carbonaceous substrate’s internal surfaces per unit weight of the activated substrate up to time t is taken as the random variable of the process, N(t), whose realization is n. All possible values of N(t) are the states of the process, and their collection, {n0, n0 1, ..., 2, 1, 0}, is its state space where n0 is the initial number of obtainable pores that could be activated per unit weight of the activated substrate. Identification of Intensity of Transition. The intensity of transition, or intensity function, is defined as the instantaneous rate of change of transition probability;18 it signifies the driving force, that is, momentum, for change. It is worth noting that the intensity function in stochastic modeling can be identified as or recovered from the expression for the kinetic rate law in deterministic modeling. It is, therefore, rational to presume that the intensity of transition be proportional to the number of obtainable pores remaining to be activated, n, which is akin to the concentration of reactant remaining to be consumed in a reaction in any homogeneous reaction mixture. If the substrate being activated is totally homogeneous and the obtainable pores are activated independently as well as randomly, the intensity of transition is expected to be linearly proportional to n. This is, however, hardly the case: The overwhelming majority of substrates for AC formation are solid materials, nonuniform or heterogeneous in their structures as well as in the distribution of their components. Thus, it is highly plausible that the obtainable pores activated at the early stage might induce appreciable structural change, for example, the formation of cracks, in the remaining substrate, which is rendered more readily activated. Under this circumstance, the intensity function would increase with a power of n greater than one. In light of this, the intensity 8837
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function for the pure-death process of interest at state n and time t, μn(t), is given by μn ðtÞ ¼
dn ¼ Rnβ dt
ð1Þ
where R is a proportionality constant and β > 1. For the process under consideration, it is most logical to regard the value of β in the above equation to be 2: The activation of the carbonaceous substrate involves the chemical reaction between two reactants, one of which is the activation agent and the other, the carbon contained in each of the obtainable pores randomly distributed on the internal surfaces of the carbonaceous substrate. Thus, eq 1 can be rewritten as μn ðtÞ ¼
dn ¼ Rn2 dt
ð2Þ
Note that the intensity of death, μn(t), in this equation is dependent on realization n but independent of time t. Master Equation. For the pure-death process, the probability balance around state n leads to d p ðtÞ ¼ μn þ 1 pn þ 1 ðtÞ μn pn ðtÞ, dt n n ¼ n0 , n0 1, :::, 2, 1, 0
ð3Þ
which is the master, that is, governing, equation of the process;18,31 see Appendix A in the Supporting Information. The term, pn(t), in the above expression denotes the probability that n obtainable pores have generated an identical number of open pores by time t. Naturally, the master equation, eq 3, represents a set of n0 ordinary differential equations. Note that for n = n0, the term, μnþ1pnþ1(t), in eq 3 lacks any significance; thus, this equation reduces to d p ðtÞ ¼ μn 0 pn 0 ðtÞ dt n 0
ð4Þ
Substituting eq 2 for μn(t) into eq 3 yields d p ðtÞ ¼ ½Rðnþ 1Þ2 pnþ1 ðtÞ ½Rn2 pn ðtÞ, dt n n ¼ n0 , n0 1, :::, 2, 1, 0
ð6Þ
Mean and Variance. The mean and higher moments about the mean can be computed from the master equation of the puredeath process, eq 3, via a rational approximation method, the system-size expansion;18 see Appendix B. Among these higher moments, the second moment about the mean, or the variance, is of special importance: It signifies the fluctuations or scatterings of the random variable about its mean,19,32 which should be the major focus of any stochastic analysis and modeling. Mean. The mean, E[N(t)] or m(t), of random variable N(t) is obtained as
mðtÞ ¼
n0 ðRn0 Þt þ 1
From this expression, the normalized form of the mean, denoted by m(τ), is given by mðτÞ ¼
ð7Þ
where (Rn0) is a proportionality constant whose dimension is t1; see Appendix C in the Supporting Information. In terms of
mðτÞ 1 ¼ n0 τþ1
ð9Þ
Clearly, this expression is solely a function of τ. Variance. The variance, Var[N(t)] or σ2(t), of random variable N(t) is given by ( ) n0 1 2 1 σ ðtÞ ¼ ð10Þ 3½ðRn0 Þt þ 1 ½ðRn0 Þt þ 13 In terms of dimensionless time τ, this expression becomes " # n0 1 2 1 ð11Þ σ ðτÞ ¼ 3ðτ þ 1Þ ðτ þ 1Þ3 The standard deviation, σ(τ), is the square root of the variance, σ2(τ); thus, #1=2 1=2 " n0 1 1=2 2 σðτÞ ¼ ½σ ðτÞ ¼ 1 ð12Þ 3ðτ þ 1Þ ðτ þ 1Þ3 From this equation, the normalized form of the standard deviation, σ h(τ), is obtained as " #1=2 σðτÞ 1 1 σðτÞ ¼ ¼ 1 ð13Þ n0 ðτ þ 1Þ3 ½3n0 ðτ þ 1Þ1=2 Note that this expression is a function of τ and n0. The standard deviation relative to the mean, termed the coefficient of variation,33 is defined as CVðτÞ ¼
ð5Þ
Correspondingly, for n = n0, the above expression reduces to d p ðtÞ ¼ ½Rn0 2 pn 0 ðtÞ dt n 0
the dimensionless time, τ = (Rn0t), this equation becomes n0 mðτÞ ¼ ð8Þ τþ1
σðτÞ mðτÞ
ð14Þ
Substituting eqs 8 and 12 for m(τ) and σ(τ), respectively, into the above equation yields the coefficient of variation, CV(t), of the pure-death process as #1=2 " ðτ þ 1Þ 1=2 1 CVðτÞ ¼ 1 ð15Þ 3n0 ðτ þ 1Þ3 Naturally, this expression is also a function of τ and n0.
’ ANALYSIS OF EXPERIMENTAL DATA The aforementioned available experimental data were obtained via physical activation of oil-palm shell char, a carbonaceous substrate, with carbon dioxide.28 These data are expressed in terms of the temporal evolution of the substrate’s extent of conversion into ACs at various temperatures. For validation, the current model has been fitted to the data. The model’s random variable, N(t), is related to the experimentally measurable variable, Y(t), i.e., the predicted, or 8838
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theoretical, extent of conversion of the carbonaceous substrate into ACs, defined as W 0 WðtÞ WðtÞ ¼ 1 Y ðtÞ ¼ ð16Þ W0 W0
Table 1. Values of the Parameter, (rn0), Recovered from the Available Experimental Data for the Formation of ACs at Different Temperatures28 temperature
where W(t) is the amount of carbon in the obtainable pores that have not yet been activated up to time t and W0 is the total amount of carbon available for activation. As mentioned earlier, the formation of open pores on the carbonaceous substrate’s internal surfaces is considered to be attributable solely to the reaction between the carbon present in the obtainable pores and the activation agent. Hence, WðtÞ ¼ δ 3 NðtÞ
ð17Þ
where δ is the amount of carbon present in a single obtainable pore. As a result, W0 can be related to n0 as W 0 ¼ δ 3 n0
mY ðtÞ
or mY ðtÞ ¼ 1
K
°C
¼ E½Y ðtÞ WðtÞ ¼ E 1 W 0 δNðtÞ ¼ E 1 δn0
10
min1
973
700
7.01
0.042
750
9.90
0.059
1073
800
13.84
0.083
1123
850
23.36
0.140
1173
900
44.55
0.267
given by σY ðtÞ ¼ ½σY 2 ðtÞ1=2 ( ¼
1 f3n0 ½ðRn0 Þt þ 1g1=2
In terms of τ, we have σ Y ðτÞ ¼
1 1 E½NðtÞ ¼ 1 mðtÞ ð19Þ n0 n0
s
4
1023
ð18Þ
The mean value of Y(t), denoted by mY(t), is determined from this expression in conjunction with eqs 16 and 17 as follows:
(Rn0) 1
1 1 ½ðRn0 Þt þ 13
( 1
½3n0 ðτ þ 1Þ1=2
1 1 ðτ þ 1Þ3
)1=2 ð24Þ
)1=2 ð25Þ
From eqs 21 and 25, the coefficient of variation, CVY(τ), is obtained as ( #)1=2 " 1 τþ1 1 σY ðτÞ ¼ ð26Þ 1 τ 3n0 ðτ þ 1Þ3
Substituting eq 7 for m(t) into this equation gives rise to mY ðtÞ ¼
ðRn0 Þt ðRn0 Þt þ 1
ð20Þ
In terms of dimensionless time τ, the above equation can be rewritten as τ ð21Þ mY ðτÞ ¼ τþ1 Similarly, the variance of Y(t), denoted by σY2(t), is obtained from eqs 16 and 17 as σ Y 2 ðtÞ ¼ ¼ ¼ or
σ Y 2 ðtÞ ¼
1 n0 2
Var½Y ðtÞ WðtÞ Var 1 W 0 δNðtÞ Var 1 δn0
Var½NðtÞ ¼
1 σ2 ðtÞ n0 2
ð22Þ
Substituting eq 10 for σ2(t) into the above expression yields ( ) 1 1 2 1 σ Y ðtÞ ¼ ð23Þ 3n0 ½ðRn0 Þt þ 1 ½ðRn0 Þt þ 13 Naturally, the standard deviation of Y(t), denoted by σY(t), is
’ RESULTS AND DISCUSSION To validate the model formulated, mY(t) as given by eq 20 has been regressed on the available experimental data for the formation of ACs at various temperatures obtained by Guo and Lua.28 The regression has been performed via the adaptive random search procedure;34 it has yielded the values of (Rn0) listed in Table 1 for the available experimental data.28 These (Rn0) values have rendered it possible to evaluate the dimensionless time, τ = (Rn0t), as well as the mean, mY(τ), from eq 21. The values of mY(τ) as a function of τ are plotted in Figure 2 for the available experimental data.28 The corresponding experimentally measured values of the carbonaceous substrates’ extent of conversion into ACs, denoted as Xexp(τ), are superimposed in the same figure for comparison. Note that, mY(τ) clearly increases asymptotically to 1 as τ progresses; this asymptotic value for mY(τ) can also be discerned from eq 21. Note that mY(τ) is in sufficiently good accord with the temporal trend of the available experimental data in spite of the fact that the nonlinear stochastic model incorporates no structural parameters of the carbonaceous substrates in its formulation. As indicated by Aranda et al.,30 fluidsolid reactions are strongly influenced by the solid’s structural properties, especially when the solid phase is a porous material. Thus, the nonlinear stochastic model formulated herein has its own utility for the preliminary exploration of the formation of ACs from carbonaceous substrates. The incorporation of structural parameters into the formulation of the stochastic model would 8839
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Figure 2. Mean extent of conversion mY(τ) and standard deviation envelope [mY(τ) ( σY(τ)] as functions of dimensionless time τ for the formation of ACs. The experimental data have been obtained by Guo and Lua28 via the physical activation of oil-palm shell char with carbon dioxide.
entail the definition of new random variables or nonlinear intensity functions; such effort should await our forthcoming work. The standard deviation, σY(τ), as computed by eq 25, signifies the fluctuations attributable to the internal or characteristic noises of the process about their mean as predicted by the stochastic model. In general, the smaller the values of random variable N(t), that is, the population size, the greater the extent of the expected fluctuations about their mean. The envelope of standard deviation, mY(τ) ( σY(τ), is plotted in Figure 2; note that the majority of the experimental data28 in this figure lie appreciably beyond the expected variation, or scattering: 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 to the external noises because of measurement errors and instrumental deficiencies that can never be totally eliminated. Obviously, eq 25 for σY(τ) depends on n0, that is, the total number concentration of obtainable pores that could form open pores on the carbonaceous substrate’s internal surfaces per unit weight of the activated substrate. The number of pores on ACs is profoundly large;3537 thus, it is reasonable to expect that the value of n0 be enormous. In fact, the order of magnitude estimate of n0 falls within the range between 3.12 1015 and 1.56 1017 pores per milligram of ACs; see Appendix D in the Supporting Information. The coefficient of variation, CVY(τ), as defined by eq 26, provides a relative measure of the variability or dispersion of the values of a random variable about their mean. Hence, CV(t) could be a more meaningful measure of variability than the standard deviation, σ(t), especially when comparing populations differing significantly from one another in their size or the nature of their constituent entities.
’ CONCLUDING REMARKS A stochastic model has been derived for the formation of ACs from carbonaceous substrates. Specifically, the model is based on a puredeath process with a nonlinear intensity of transition; this model can serve as initial exploration of the kinetics of formation of ACs. The analytical expressions for the mean, variance, and standard deviation of the extent of conversion of a carbonaceous substrate into ACs have been obtained by solving the master equation of the pure-death process via a rational approximation method, the system-size expansion. The analytical solutions have been compared with the available experimental data. The temporal variation of the means as predicted by the model follows the trend of these data. As expected, the data’s fluctuations around the means are more noticeable than those predicted by the model: In addition to the process’ internal noises, the deviations of the experimental data also account for the external noises due to inherent measurement errors and instrumental deficiencies. ’ ASSOCIATED CONTENT
bS Supporting Information. Appendixes AD. This material is available free of charge via the Internet at http://pubs.acs.org. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This is contribution No. 08-75-J, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. 8840
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