Reaction Kinetics Modeling of CaC2 Formation From Coal and Lime

Article pubs.acs.org/IECR

Reaction Kinetics Modeling of CaC2 Formation From Coal and Lime Steven L. Rowan,*,† Ismail B. Celik,† Jose A. Escobar Vargas,† Suryanarayana R. Pakalapati,† and Matt Targett‡ †

Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506, United States Head of Research and Development, LP Amina, Pacific Century Place, Building B, 11th Floor, 2A Gong Ti Bei Lu, Chaoyang District, Beijing 100027, China

‡

ABSTRACT: Two new models for predicting the reaction kinetics for the formation of calcium carbide (CaC2) from coalderived coke and calcium oxide (CaO), or lime, are proposed and compared to four traditional reaction rate models found in literature. The first model follows the progression of the ratio of the current mass fractions of the involved chemical species and their equilibrium mass fractions until chemical equilibrium is reached. The second model instead solves the differential equation representing the change in the moles of the reactants and products over time. Unlike traditional reaction models, these models allow a given reaction to progress in either the forward or reverse direction in order to reach equilibrium. Comparison between the model predictions and experimental data found in literature shows that the second model provides accurate predictions of reactant consumption, CaC2 production and weight loss. The effective reaction rates estimated from this model indicate a reaction regime change around 1750 °C.

1. INTRODUCTION For many years, acetylene has been the preferred precursor for production of industrial chemicals such as vinyl chloride monomer and other unsaturated C2 products.1 One of the more common methods for acetylene generation is reducing lime into calcium carbide (CaC2) and then treating the calcium carbide with water to form acetylene. Numerous authors have studied the formation of calcium carbide from coal and lime. Müller presented a detailed study of the properties and reactions of calcium oxide (CaO) in burnt lime, including the diffusion of carbon into solid lime and presented composite reactions of CaO and C in solid and liquid states.2−4 Tagawa and Sugawara5 studied the formation of calcium carbide in a solid state reaction of calcium oxide with carbon, determining that the reaction followed parabolic kinetics and that the reaction rate was governed by the diffusion of gas through the product layer, as well as the diffusion of solids into the solid and product layers. Brookes et al.6 developed a zero dimensional model for the formation of calcium carbide in solid pellets assuming that heat transfer across a growing product layer controls the rate of movement of a reaction front through the solid pellet. Mu and Hard1 examined the process of making acetylene from calcium carbide created using a rotary kiln process, finding that it required less than 30 min to achieve lime conversion rates of 50−80%. Finally, Li et al.7,8 explored the reaction mechanisms of CaC2 production from pulverized coke and CaO at low temperatures, finding that there appeared to be either 2 or 3 distinct reaction pathways taking place, depending upon the initial ratio of carbon to CaO. They also studied the influence of the major minerals present in coal-derived coke upon the reaction of CaO with coke. In this second study, they found that, in general, the presence of minerals in coal-derived coke reduced the formation of the desired CaC2. © 2014 American Chemical Society

In general, the previously mentioned authors agree that the formation of calcium carbide from carbon and lime is governed by the following global reaction: 3C + CaO ⇔ CaC2 + CO

(1)

It is also widely accepted that once formed, CaC2 can either dissociate into calcium and carbon via the reaction in eq 2, or react with CaO per the reaction in eq 3. CaC2 → Ca + 2C

(2)

CaC2 + 2CaO → 3Ca + 2CO

(3)

Most, if not all, of the previous authors have studied the reactions listed in eqs 1−3 in an effort to determine the reaction rates of the chemical kinetics involved in the formation of CaC2. In these cases, these previous authors have attempted to fit traditional reaction rate models to experimental data with varying levels of success. However, most of these rate models are limited to a single global reaction, and it is becoming widely accepted that the formation of CaC2 involves a multistep reaction mechanism. In this paper, two new methods are presented for modeling the chemical reaction mechanisms associated with the formation of calcium carbide from coke and lime. The first of these new models is referred to as the progress variable model. This model utilizes equilibrium mass fraction data to determine the direction that the reaction must proceed in order to reach a state of chemical equilibrium. The second model, known as the multistep reaction model, is based upon modeling the kinetic rates of the calcium carbide formation and disassociation reactions. Received: Revised: Accepted: Published: 2963

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commonly used. Hence, the k values shown in Table 9 for eq 7 should be interpreted cautiously (see the Appendix). Of these four models, the two reacting solids model is the one that most accurately predicts the time-based reduction of CaO into CaC2, which will be discussed in greater detail later in this paper. However, all of these models suffer from the same limitation in that they only allow for the reaction to proceed in a single direction, thus not allowing for reverse reactions to occur. Moreover, they cannot be extended to multistep reaction mechanisms.

In the current study, a general modeling paradigm is retained such that once validated it can be implemented into readily available multidimensional computer codes to simulate CaC2 formation in complex reactors.

2. DISCUSSION OF EXISTING REACTION RATE MODELS Prior to discussing the new proposed models for calcium carbide formation, a number of existing reaction rate models will be discussed and applied to the experimental results previously reported by numerous authors.1,5−7 The first three of these existing models are predicated upon the idea of a contracting reaction front that travels through the reacting solid. The fourth, and final, existing model to be considered is that of two reacting solids, based upon three-dimensional diffusion of the solids. All of these models are described in greater detail by House et al.9 The first of these models is for the case of a spherical solid in which the reaction begins on the surface of the sphere and progresses uniformly inward toward the center of the sphere. Due to the model’s assumption that the reaction proceeds in the radial direction only, it is often referred to as the 1-D contraction model, given by 1 − (1 − α)2/3 = kt

3. NEW REACTION RATE MODELS As previously stated, the existing models most commonly used for the modeling of chemical reactions in solids share a similar limitation in that they only consider situations in which the reaction only proceeds in the forward direction. In addition, all of these previously discussed models only consider a single global reaction. In actuality, chemically reactive systems will proceed toward a state of chemical equilibrium in which the mass fractions of each the reactants and their products will be a function of the temperature at which the reaction occurs. Depending upon the temperature and amounts of reactants and products present at any given time, the reaction may proceed in either the forward or reverse direction to achieve equilibrium. Additionally, many reactions, including that of calcium carbide formation, will involve a multi-step reaction mechanism that needs to be considered when formulating a kinetics model. In an effort to take these factors into account and improve upon the existing chemical kinetic models, two new models are proposed for the formation of calcium carbide from carbon and lime. 3.1. Progress Variable Model. The first model to be proposed is the progress variable model. In this model, a progress variable is introduced for each chemical species included in the reaction. These progress variables are defined as the ratio of the current species mass fraction over the species mass fraction at a designated equilibrium condition. In the case of the commonly accepted global reaction for calcium carbide production presented in eq 1, the reactants are solid carbon and lime (C and CaO, respectively) and the resulting products are solid calcium carbide and gaseous carbon monoxide (CaC2 and CO, respectively). The presence of these four chemical species then leads to the formulation of the following four progress variables: mC(S) mCaO(S) XC(s) = ini , XCaO(s) = ini , XCaCs(S) mC(S) mCaO(S) mCO(g) mCaC2(S) = theor , XCO(g) = theor mCaC2(S) mCO(g) (9)

(4)

where α is the extent of the reaction (or in the case of calcium carbide formation, the fraction of CaO that is converted into CaC2), k is the reaction rate, and t is time. The second existing model is for the case of a long cylindrical solid in which the reaction occurs on the long curved surfaces (ignoring the small end surfaces) and proceeds uniformly in the radial direction toward the central axis of the cylinder. This model, known as the contracting area model, is given by 1 − (1 − α)1/2 = kt

(5)

As with the first two models, the third one is also based upon the physical geometry of the solid undergoing a chemical reaction. The contracting volume model considers the more generic case of a noncylindrical, nonspherical solid in which the reaction begins at each face of the solid surface and proceeds inward. This model is given by 1 − (1 − α)1/3 = kt

(6)

The fourth and final existing reaction rate model considered in this paper is based upon the three-dimensional diffusion rate law, and is known as the two reacting solids model. This model is typically used when considering cases where different solid particles are brought into contact with one another and heated in order to bring about a chemical reaction. This model is given by [1 − (1 − α)1/3 ]2 = kt

where mi and miini are the current and initial mass fractions of the ith chemical species; and mitheor is the theoretical production or consumption of the species i, as determined from the chemical equilibrium conditions for a given reaction temperature. The time evolution of these variables is computed by solving a system of coupled differential equations of the form:

(7)

This is the model that has been traditionally used in calcium carbide literature to describe the kinetic properties of the calcium carbide formation. Tagawa and Sugawara5 assumed k to be an Arhennius rate expression of the form:5

k = Ae−Ea / RT

dXCaO(S)

(8)

dt

where A is a pre-exponential factor, Ea is the activation energy of the reaction, R is the universal gas constant, and T is the temperature at which the reaction is taking place. However, in solid−solid reactions, this Arhennius rate expression is not

keff f,1

eff eff XC(S)XCaO(S) + k r,1 XCaC2(S)XCO(g) = −k f,1

(10)

keff r,1

where and are the effective chemical reaction rates in the forward and reverse directions. Similar rate expressions apply to the other species. Equation 10 is then integrated using 2964

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eq 1, whereas the multistep reaction model assumes that the weight loss is due to the release of both carbon monoxide and calcium gases during the reactions specified in eqs 1 and 3, and is the sum of these.

a second-order Runge−Kutta Method where the value of each of the progress variables is updated at each time step via Xinew = Xiold + 0.5Δt(RHSold + RHSnew ) RHS =

(11a)

CstoMi eff eff (k f,1 (XCXCaO) − k r,1 (XCaC2XCO)) mieq

4. RESULTS AND DISCUSSION A number of comparisons between the previously discussed models and published experimental data for calcium carbide can be used in order to assess the predictive capability of the models. Among these are comparisons of the reported experimental data on consumption of CaO and weight loss as functions of temperature and time. 4.1. CaO Conversion. A number of authors, including Mu and Hard,1 Tagawa and Sugawara,5 and Brookes et al.6 have published experimental data showing the consumption of CaO over time for calcium carbide formation reactions occurring at different temperatures. In each of these publications, 3:1 stoichiometric mixtures of carbon and lime (CaO) were pelletized and heated together in order to initiate the production of calcium carbide (CaC2). Mu and Hard carried out their experimental studies in both an induction furnace and a rotary kiln. In the induction furnace, the samples were heated at 500 and 1000 °C for 10 min at each temperature, before being heated to the selected temperature at a heating rate between 100 and 150 °C/min, and then holding that final temperature for a specified period of time before allowing the sample to cool. The heating rate within the rotary kiln reactor was not specified, but it was stated that the alumina tube within the rotary reactor could not exceed 300 °C/min. No heating rates were specified by either Tagawa and Sugawara or Brookes et al. The following discussion presents a comparison between the experimental results of Mu and Hard1 for the conversion of CaO into CaC2 with the extent of CaO conversion predicted by the existing reaction rate models presented earlier in this paper (eqs 4−8). In the experiments conducted by Mu and Hard, carbon and CaO were pelletized with a 3:1 molar ratio and then heated at temperatures of 1580, 1630, 1670, 1740, 1800, and 1870 °C. In the experimental work, no evidence of CaO conversion was noted at a temperature of 1580 °C. As a consequence, a comparison of the experimental data with the reaction rate models for that temperature has been omitted. For each of these models, the corresponding equations (i.e., eqs 4−8) can be used to solve for the value of α as a function of time (t), using a constant reaction rate value, k, for each temperature. For each model and temperature, a “best fit” reaction rate was found that resulted in the best match between the model results and the experimental data. The “best fit” value of k was obtained by first taking a first order central difference derivative of the reaction data presented by Mu and Hard, and then iterating over a range of k values in order to minimize the error between the calculated α and the experimental data via a least-squares method. Figure 1 shows the comparison of the results of 1-D contraction, contracting area, and the contracting volume models and the experimental data. Upon review of this data, it is seen that all of these models fail to provide an accurate representation of the power law nature of the consumption of calcium oxide that is exhibited in the experimental data. All three of these models are intended to represent the decelerating nature of geometrically based reaction models in which the reaction front progresses through a material, leading to a contraction of the surface area over which the reaction takes place. It can easily be seen from the comparisons in Figure 1 that these models do

(11b)

where Csto is the stoichiometric coefficient from eq 1 for species i, Mi is the molecular weight of species i, and meq i is the mass of i present at equilibrium conditions. Chemical equilibrium is attained when the values of these progress variables approach a value of 1.0. This allows the reaction to proceed toward equilibrium in either the forward or reverse directions. 3.2. Multistep Reaction Model. Although the progress variable model allows a given reaction to occur in either the forward or reverse direction in order to achieve equilibrium, the model itself still has limitations. The first such limitation is the fact that the model necessitates that the masses of each species present at equilibrium be known in order to define the progress variables. Given the limited amount of thermodynamic data currently available for calcium carbide at high temperatures, these cannot currently be estimated, thus requiring the use of experimental data. The second limitation is the fact that the progress variable model is still a single reaction mechanism model. This limits the ability of the progress variable model from accurately modeling mechanisms that involve more than one reaction. This is of particular importance in the case of calcium carbide formation, as it is commonly believed that once formed, calcium carbide is somewhat unstable and may disassociate into its constituent elements. In an effort to address these limitations, the multistep reaction model was developed. In this model, the formation of calcium carbide and its subsequent disassociation into solid carbon and gaseous calcium (eqs 1 and 2) are combined into a single differential equation (eq 12). dni = v(i , 1)(k f1nCnCaO − k r1nCaC2nco) dt + v(i , 2)(k f2nCaC2 − k r2nCnCa)

(12) th

where n denotes the number of moles of the i species, kf1, kr1, kf2 and kr2 are the forward and reverse reaction rates for the two equations, and v(i)’s are the stoichiometric coefficients, respectively. As with the progress variable model, eq 12 is integrated numerically using a second order Runge−Kutta method in which the number of moles of each species present is continuously updated at each time step, thus allowing the prediction of the instantaneous number of moles of each species present as a function of time. Additionally, by assuming that weight loss is caused by the release of gaseous calcium and carbon monoxide, it is possible to predict weight loss as a function of time. With both models (i.e., progress variable and multistep reaction models), the initial number of moles of each chemical species, furnace temperature profile, effective reaction rate values, and the time step and number of iterations are specified. This information is then used in conjunction with eqs 9−12 to numerically solve for the amount of each chemical species present after each time step, as well as update the values of the progress variable (in the progress variable model). The progress variable model assumes that the weight loss is due to the release of carbon monoxide produced during the reaction specified in 2965



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Figure 1. Comparison of traditional models with experimental data of Mu and Hard, 1987.

As stated previously, one of the primary limitations of using the reaction kinetics models discussed so far is that these models only allow the reaction to proceed toward equilibrium conditions in the forward direction. This is a significant limiting factor in the applicability of these traditional reaction kinetic models. It is expected that if more products are present than would be present at equilibrium, then a reverse reaction is likely to occur in order to restore equilibrium, provided that the temperature is sufficiently high to start activation. The progress variable model, which was developed in an effort to address this

not adequately predict the consumption of lime (CaO); therefore, more robust models must be considered. Another model considered was that of two reacting solids, as described in eq 8. Once again following the previously described least-squares method for determining the best k value for matching the experimental data, the predicted CaO reduction is compared to the experimental data in Figure 1. As can be seen in the figure, the two reacting solids model provides a more accurate prediction of the fraction of CaO reacted, α, than the three previous models. 2966



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Figure 2. Comparison of experimental data (Mu and Hard, 1987) and progress variable model.

limitation, was also compared to the experimental data presented by Mu and Hard,1 as shown in Figure 2. As can be seen, the progress variable model results in CaO conversion values that closely approximate the experimental data for all of the temperatures presented, and it is at least as good as the predictions by using eq 7. So far, it has been shown that the progress variable model, as described in eqs 9−11, is able to accurately predict the reduction of lime as a function of time as presented in literature. In order to do so, the equilibrium mass fractions of each of the chemical species must be known so that the progress variables can be determined. For the comparisons presented so far, the mass fractions have been estimated using the stoichiometric

relationships given in eq 1, assuming that the experimentally reported data are representative of the fraction of lime (CaO) that has been consumed in order to reach chemical equilibrium. Unlike the models that have so far been compared to previously published experimental data, the multistep reaction model, detailed in eq 12, considers the case in which multiple reactions occur. In the first reaction mechanism, carbon and lime react to form solid calcium carbide (CaC2) and gaseous carbon monoxide (CO) via eq 1. In the second reaction mechanism, the calcium carbide that has formed via eq 1 disassociates into solid carbon (C) and calcium gas (Ca) via eq 2. As a consequence, in order to validate the model against experimental data, information on the decomposition of 2967



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Figure 3. Comparison of model results and experimental data (Tagawa and Sugawara) for CaO consumption.

calcium carbide is necessary. For this purpose, the experimental data published by Tagawa and Sugawara5 is better suited than the data presented by Mu and Hard.1 The data provided by Tagawa and Sugawara5 includes not only information on the amount of lime (CaO) consumed but also the amount of residual solid calcium carbide (CaC2) remaining in the sample, as well as the ratio of CaC2 decomposed/theoretical CaC2 formed. As in the case of the Mu and Hard1 experiments, Tagawa and Sugawara5 heated pellets composed of a 3:1 carbon/lime molar mixture at temperatures of 1640, 1680, 1720, 1760, and 1800 °C. As can be seen in Figure 3, the CaO consumption predicted by the reaction rate model closely matches the data provided by Tagawa and Sugawara.5 Tagawa and Sugawara5 also provide information on the rate of decomposition of calcium carbide (CaC2) in the form of the “decomposition ratio”, which is defined as the ratio of the amount of CaC2 that decomposes divided by the theoretical maximum amount of CaC2 that can be formed if eq 1 is allowed to go to completion so that no reactants remain. Tagawa and Sugawara report that the decomposition ratio follows a linear profile; however, this was not seen to be the case with the reaction rate model predictions, as seen in Figure 4. Additionally, Tagawa and Sugawara provide data on the amount of solid calcium oxide found within their test samples following heating. This “residual ratio” is an expression detailing the net calcium carbide remaining after decomposition and is expressed as a ratio of solid calcium carbide present divided by

the total theoretical carbide production. At the lower temperature regimes, this value appears to follow a parabolically increasing pattern. However, at higher temperatures, the data exhibits an initial increase followed by a gradual decrease in the total amount of calcium carbide present. This decrease can most likely be attributed to the fact that formation of calcium carbide stops once all of the lime has be converted, but decomposition of the calcium carbide will continue until the system reaches a state of equilibrium. Figure 5 depicts the comparison of the residual CaC2 predicted by the multistep reaction model, and the experimental data presented by Tagawa and Sugawara. As might be expected, it was found that the data presented in Figures 4 and 5 were highly dependent upon the rate constants listed in eq 12. For the purposes of this preliminary investigation, it was decided that greater importance would be placed upon finding rate constants that would result in the best matches between the model and experimental data for the rates of CaO consumption and total weight loss, while allowing some variation between the model and experimental data for the decomposition and residual ratios. 4.2. Analysis of Reaction Rates. Figure 6 shows the forward reaction rate values for eq 1 (normalized by dividing each rate by the maximum value used for each model) utilized by the three primary models of interest (two reacting solids, progress variable, and multistep reaction models) for the formation of calcium carbide via eq 1. With the two reacting solids 2968



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Figure 4. Comparison of msr model and experimental data (Tagawa and Sugawara) CaC2 decomposition ratios.

model, the value of the forward reaction rate initially decreases with increasing temperature until reaching a temperature of approximately 1700 °C. Beyond this point, the reaction rate appears to slowly increase linearly with increasing temperature. It should be noted that this reaction rate is not equivalent to those used in the current models (see the Appendix). In the case of the progress variable model, the opposite trend can be seen. In this case, the reaction rate initially increases with increasing temperature until a temperature between 1700 and 1740 °C, where it appears to decrease linearly with increasing temperature. Finally, a third trend can be seen in the plot of the calcium carbide generation forward reaction rates used with the multistep reaction model. The forward reaction rate for this model appears to exhibit an initially increasing trend with increasing reaction temperature. However, the reaction rate levels off and is nearly constant within the temperature range of 1720 through 1760 °C, before increasing again at higher reaction temperatures. The actual, non-normalized rate values can be found in the Appendix. In contrast, while not shown in Figure 6, the forward reaction rates for the disassociation of calcium carbide by eq 2 exhibited a trend that inversely mirrored that of eq 1 for the multistep reaction model, suggesting that the calcium carbide has a greater tendency to disassociate at lower temperatures. These reaction rates are presented in the Appendix.

It is believed that the reason for this behavior is that the reaction rate utilized by these three models is not the actual chemical reaction rate, but is instead an effective reaction rate, such that keff = k Σ

(13)

Where keff is the effective reaction rate, Σ is a correction factor, and k is the actual chemical reaction rate constant. The correction factor must account for varying experimental conditions as well as morphological and microstructural differences of the various mixtures used. A combination of the solid−solid surface contact area, pellet size, pellet compression, and other factors that have not yet been identified or quantified, could be included in the correction factor Σ. In the above equation, the reaction rate keff can also be interpreted as the combination of the chemical and diffusion reaction rates, or 1 1 1 = + eff k k k (14) chem diff As such, the effective reaction rate can be dominated by either the chemical reaction rate, or the effects of diffusion. Given that the solid−solid calcium carbide reaction occurs when calcium diffuses into the carbon,2−4 it is believed that the diffusion term dominates eq 14 when the reactants are in the 2969



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Figure 5. Comparison between msr model and experimental data (Tagawa and Sugawara) for residual CaC2.

dominated by solid−solid reactions to one that is dominated by liquid−liquid or liquid−solid reactions. From the data presented in Figure 6, it is concluded that this eutectic mixture forms somewhere in the range of 1720 to 1740 °C. A recent study published by Li et al.,10 likewise suggests the formation of a eutectic mixture of calcium carbide and lime around a temperature of 1700 °C. 4.3. Weight Loss. Another metric by which the proposed models may be evaluated is in their ability to predict weight loss as a function of time and reaction temperature. Many of the previous authors who studied the formation of calcium carbide from lime and coke have presented experimental weight loss data for the time period in which the reactor temperature was at a constant value,5,6 and others have done so including the initial temperature ramping (heating) of the furnace in addition to the constant temperature time period of operation.7 For the purposes of this discussion, although not completely accurate, the first case will be referred to as isothermal weight loss and the second case will be referred to as nonisothermal weight loss to differentiate the manners in which the experimental data is presented. In general, there are two sources of possible weight loss that can occur during the formation of calcium carbide. The sources for weight loss are commonly believed to be the production and release of carbon monoxide (CO) and calcium (Ca) gases.

Figure 6. Model reaction rate comparisons for calcium carbide production.

solid phase. As the lime and carbon begin to react in the solid− solid state, the formation of calcium carbide will first act as a barrier between the two materials that serves to limit the ability of the calcium to diffuse into the carbon structure. As more calcium carbide is formed, the carbide will react with the remaining lime and form a eutectic mixture once the molar ratios of calcium carbide and lime matches that required for eutectic formation for a given reaction temperature. In reality, keff will be changing with time as the ratio of CaC2/CaO changes and this will be reflected in the average value of keff. The resulting lower melting point of this calcium carbide/lime eutectic leads the reaction to transition from one that is 2970



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Figure 7. Weight loss comparisons between multistep reaction model and experimental data (Tagawa and Sugawara5).

Carbon monoxide is generated via eq 1 during the formation of calcium carbide. Calcium gas is generated via eq 2 as a result of the decomposition of the calcium carbide once it forms. In addition, Li et al7 postulated that an additional source of both carbon monoxide and calcium gas may arise during the conversion of lime rich mixtures via eq 3. As the primary focus of this work has been on modeling the conversion of stoichiometric mixtures of carbon and lime into calcium carbide, the reaction in eq 3 has been given less emphasis. Isothermal Weight Loss. The experimental works of Tagawa and Sugawara5 and Brookes et al.6 provided weight loss data for cases in which their carbide reactors were operated at constant temperatures. During comparison of these cases, it was determined that the two reacting solids model and the progress variable model were unable to accurately predict the reported weight losses. In both cases, the models only consider eq 1, and thus only allow for the production and release of carbon monoxide gas. On the basis of the stoichiometry of eq 1, the maximum amount of carbon monoxide that may be produced if the reaction is allowed to go to completion (i.e., only products remain) is 30.4% of the initial mass. The data provided by both sets of authors reported weight losses in excess of 50% of their starting mass in cases of less than 100% conversion of lime to calcium carbide. From this, it is evident that losses due to the production of calcium gas must be considered. As a consequence of this, the multistep reaction model provides a much more accurate prediction of weight loss when

compared to experimental data. However, given the interdependence between the forward and reverse reaction rates of eq 1 and eq 2 and the predicted results for CaO conversion, CaC2 decomposition, residual CaC2 and weight loss, there is some trade-off between the accuracy of those results. Figure 7 provides a comparison of the reaction rate model weight loss predictions and the experimental data reported by Tagawa and Sugawara. For each of the temperatures considered in the figure, the reaction rates were adjusted to allow for the closest fit possible. As can be seen, there is fairly good agreement between the two for a range of temperatures. Nonisothermal Weight Loss. The weight loss as a function of time has also been presented in calcium carbide literature. As a consequence, an effort has been made to show whether or not the progress variable and reaction rate models can also predict weight loss as a function of time. For this, it was decided that the progress variable model results would be compared to the experimental weight loss reported by Li et al.7 In the Li experiments, the lime and carbon mixture was heated from approximately room temperature to 1750 °C at a rate of 20 °C/min, and then held at that temperature for an additional 60 min. Li utilized a thermo-gravimetric analyzer (TGA) to measure the sample weight and the rate of weight loss for a wide range of mixtures with differing C/CaO ratios. A C/CaO ratio of 2.8:1 was selected for comparison due to that ratio being the one examined by Li that was closest to the stoichiometric ratio of 3:1 derived from eq 1. The progress variable and multistep reaction 2971



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models were then modified to allow the rate constants, and the equilibrium mass fractions (used to calculate the progress variables) to change as a function of temperature. The predicted weight loss from the progress variable model (modified for additional weight loss attributed to devolatilization of impurities reported by Li) is compared to experimental weight loss data from Li et al. for the selected carbon/CaO ratio is shown in Figure 8. As can be seen in the figure, the initial progress variable model does a poor job of predicting the experimental weight loss measurements. As stated previously, this is due to the fact that the global reaction used in the model, shown in eq 1, only allows for weight loss in the form of carbon monoxide gas. As a result, the progress variable model was modified to make use of a new global reaction derived from combining eqs 1−3. This is the resulting global reaction: 5C(s) + 3CaO(s) → CaC2(s) + 3CO(g) + 2Ca(g) Figure 8. Comparison of PVM and experimental weight loss.

(15)

In this global equation, much larger weight loss values can be expected. In addition to the release of 3 times as much CO gas as the previous model, there is also a release of calcium gas, which is not accounted for in the initial progress variable model. Use of the global reaction presented in eq 15 requires that the progress variable model equations be modified to account for the addition of calcium gas as a product of the reaction. In addition, the equilibrium mass fractions must be recalculated to reflect not only the presence of gaseous calcium but also to reflect the stoichiometric coefficients given in eq 15. After incorporating these changes that are necessitated by the use of the modified global reaction, the progress variable model is once again compared to the weight loss data reported by Li for the 2.8:1 C/CaO ratio case. The results of this comparison are shown in Figure 9. In this instance, the length of the model simulation is extended an additional 100 min to demonstrate that the weight loss will attain a maximum value. As shown in Figure 9, the modified progress variable model can indeed predict the trend observed in the experimental data reported by Li et al. closely. In summary, the progress variable model, in conjunction with the modified global reaction given in eq 15, is able to closely predict the weight loss and CaO conversion percentages reported in literature. However, due to the stoichiometry of the

Figure 9. Weight loss comparison between experimental data and modified progress variable model.

Figure 10. Comparison of multistep reaction model and experimental data (Li) for nonisothermal heating using effective reaction rates obtained from fitting of Tagawa et al. data for (a) 1720 °C and (b) 1760 °C. 2972



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reaction, the ratio of moles of calcium carbide produced to moles of calcium oxide is 1:3. This contradicts the 1:1 ratio that is generally accepted in both the previous literature and in industrial operations, and thus leads to the conclusion that use of eq 15 is not valid. Finally, the Li et al. data for the 2.8 stoichiometric case is compared to the multistep reaction model using the rate values derived from the Tagawa and Sugawara data for reaction temperatures of 1720 and 1760 °C, as shown in Figure 10. As can be seen from the figure, the weight loss prediction based upon the effective reaction rates at the higher temperature of 1760 °C did not produce as good agreement as the lower temperature of 1720 °C. It is believed that this is a result of differences in the conditions under which the experiments were conducted. These differences would lead to different Σ values in eq 14, thus resulting in effective rates that are higher in the Tagawa and Sugawara experiments than in those presented by Li et al.

Table 1. 1-D Contracting Model Effective Reaction Rates 1630 1670 1740 1800 1870

0.0040 0.0080 0.0141 0.0178 0.0267

reaction temperature, °C

effective reaction rate, keff

1630 1670 1740 1800 1870

0.0030 0.0062 0.0111 0.0144 0.0223

Table 3. Contracting Volume Model Effective Reaction Rates reaction temperature, °C


1630 1670 1740 1800 1870

0.0020 0.0042 0.0078 0.0103 0.0165

whereas for the multistep reaction model [CaC2][CO] dX = kr − k f [C]X dt [CaO]initial

In response to reviewer comments, we included Figure 11, which shows the effects of varying initial molar amounts of CO on the fractional amount of CaO consumption at equilibrium using the effective reaction rates obtain from the experimental data provided by Tagawa and Suagawara5 for reaction temperatures between 1720 and 1800 °C using the multistep reaction model. As can clearly be seen, increasing the amount of CO present in the system leads to a noticeable decrease in CaO consumption, and thus CaC2 formation. The results presented in this figure are similar to and in agreement with those shown in Figure 5 of Mu and Hard.1 The authors are encouraged that this demonstrates that the model has the capability to predict the effects of varying partial pressures of CO, although a more complete experimental study under known conditions will be required before the model can be truly validated for the effects of CO partial pressure effects. As suggested by one of the reviewers, the forward effective reaction rates for eq 1 for the three models (as shown in Figure 6 and listed in Tables 4−6) were fit to an Arrhenius rate expression of the form:

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APPENDIX The following tables provide the effective reaction rate values that resulted in the best fits to the experimental data presented in the related model comparisons. From the definition of α in eq 7 X=


Table 2. Contracting Area Model Effective Reaction Rates

5. CONCLUSIONS Two new models are proposed for modeling the reaction kinetics of calcium carbide formation from carbon and lime. The first of these two models was the progress variable model that utilized progress variables calculated from equilibrium mass fraction data for each chemical species. The second model is the multistep reaction model that utilizes the forward and reverse reaction rates for the reversible reactions describing the formation and decomposition of calcium carbide. Both models possess an advantage over previous models in that they allow reactions to proceed in either direction in order to achieve chemical equilibrium. However, the multistep reaction model has the additional advantage in that it can incorporate more than a single reaction mechanism. By comparing the predictions from this model to experimental data previously published by multiple authors, it was shown that this model is able to accurately predict not only the rate of conversion of CaO into CaC2, but also the further decomposition of CaC2 and the reported weight loss data. However, examination of the effective reaction rates obtained from attempting to match the experimental data from various sources shows significant variation from experiment to experiment, suggesting that factors like surface contact area, pellet compression, impurities present in the mixtures, etc., may have an effect on the reaction rates. Further examination of the effective reaction rates used in these evaluations suggest that the rates are highly dependent upon whether or not the reactions are taking place in a solid− solid or liquid−liquid medium, and that a eutectic mixture of reactants and products forms in the temperature range of 1720−1760 °C, these observations are supported by data presented in other literature.10−12

α = 1 − X,


keff = Ae−ΔE / T Table 4. Two Reacting Solids Model Effective Reaction Rates

[CaO] [CaO]initial

dX 2 X2/3 =− k dt 3 (1 − X1/3) 2973



1630 1670 1740 1800 1870

0.0033392 0.0014 0.0044 0.0070 0.0146



Article

Table 5. Progress Variable Model Effective Reaction Rates reaction temperature, °C

Table 9. Arrhenius Rate Variables for Combined Temperature Ranges

eff

effective reaction rate, k 9.4706 × 10−5 1.6263 × 10−4 2.3231 × 10−4 2.005 × 10−4 1.301 × 10−4

1630 1670 1740 1800 1870

model

4.8 2.5 3.3 7.5

× × × ×

10−5 10−4 10−4 10−4

Table 7. Multistep Reaction Model Effective Forward Reaction Rates for Disassociation of CaC2 reaction temperature, °C


1680 1720 1760 1800

5.0 × 10−2 1.25 × 10−3 1.1 × 10−3 5.25 × 10−4

ΔE

Σ(error2)

86595 5191.8 31582

1.66315 × 10−8 1.15619 × 10−8 1.00197 × 10−5

However, the activation energies obtained for the progress variable model are significantly lower than those expressed in previous literature, whereas the multistep reaction model activation energies are much closer to the previously published activation energies. In addition to applying the Arrhenius rate expression to each region of the data presented in Figure 6, it was also applied over the entire temperature range, as shown in Table 9. Once again, it can be seen that the two models presented here lead to significantly smaller errors. However, as was the case with the previous treatment, the activation energies obtained for the progress variable model are questionable. The appropriateness of applying an Arrhenius rate expression to the formation of calcium carbide is seriously questioned by the authors. As stated earlier, the rate values obtained are effective reaction rates that consist of a combination of the effects of the chemical kinetics and mass diffusion. The Arrhenius rate expression applies to chemical kinetics and the authors feel that it is not appropriate in cases where mass diffusion serves as a rate limiting factor.


1680 1720 1760 1800

1.14823 × 10 2.07 × 10−3 3.05 × 104

15

multistep reaction progress variable two reacting solids

Table 6. Multistep Reaction Model Effective Forward Reaction Rates for CaC2 Formation reaction temperature, °C

A

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AUTHOR INFORMATION

Corresponding Author

*S. L. Rowan. Tel: 304-293-3197. E-mail: steve.rowan@mail. wvu.edu. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This technical effort was performed in support of the US− China Clean Energy Research Center−Advanced Coal Technology Consortium’s (CERC-ACTC) ongoing research in clean coal conversion processes under the DOE contract DEPI0000017.

Figure 11. Effects of the presence of initial CO on the consumption of CaO for a stoichiometric mixture of carbon and lime.

where A is a pre-exponential factor and ΔE is the activation energy with units of J/mol. Table 8 presents the values of these parameters for the case where the lower temperature values (prior to the abrupt change in the slopes curves in Figure 6) and higher temperature values (after the change in slope) are treated separately. Also shown are the sum of the squares of the errors between the effective rate values listed in tables 4−6 and those calculated using the Arrhenius expression and the A and ΔE values provided. As can be seen from the data presented in Table 8, the errors obtained from the two reacting solids model are several orders of magnitude larger than those of the two models being proposed; which suggests that both the progress variable model and the multistep reaction model are better fits.

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REFERENCES

(1) Mu, J. J.; Hard, R. A. A rotary kiln process for making calcium carbide. Ind. Eng. Chem. Res. 1987, 26, 2063−2069. (2) Müller, M. B. Structure, properties and reactions of CaO in burnt lime: Part 1. Design of a new model of burnt lime. Scand. J. Metall. 1990, 19, 64−71. (3) Müller, M. B. Structure, properties and reactions of CaO in burnt lime: Part 2. Diffusion of carbon into solid lime. Scand. J. Metall. 1990, 19, 191−200. (4) Müller, M. B. Structure, properties and reactions of CaO in burnt lime: Part 3. Composite reactions of CaO and C in solid and liquid state. Scand. J. Metall. 1990, 19, 210−217.

Table 8. Arrhenius Rate Variables for Separate Temperature Ranges low temperature region ΔE

model

A

multistep reaction progress variable two reacting solids

2.46 × 10 7.66 × 102 1.53 × 10−21 31

160584 30111 −80353

high temperature region Σ(error ) 2

A −16

1.06609 × 10 6.14757 × 10−10 1.92 × 10−6 2974

4.78 × 10 1.61 × 10−8 6.73 × 107 8

ΔE

Σ(error2)

56545 −19370 47584

9.89641 × 10−9 1.13774 × 10−10 1.1441 × 10−6



Article

(5) Tagawa, H.; Sugawara, H. The kinetics of the formation of calcium carbide in a solid-solid reaction. J. Chem. Soc. Jpn. 1962, 35, 1276−1279. (6) Brookes, C.; Gall, C. E.; Hudgins, R. R. A model for the formation of calcium carbide in solid pellets. Can. J. Chem. Eng. 1975, 53, 527−535. (7) Li, G.; Liu, Q.; Liu, Z. CaC2 production from pulverized coke and CaO at low temperatures: reaction mechanisms. Ind. Eng. Chem. Res. 2012, 51, 10742−10747. (8) Li, G.; Liu, Q.; Liu, Z. CaC2 production from pulverized coke and CaO at low temperatures: influence of minerals in coal-derived coke. Ind. Eng. Chem. Res. 2012, 51, 10748−10754. (9) House, J. E. Principles of Chemical Kinetics, 2nd ed.; Academic Press: Saint Louis, MO, 2007; pp 229−261. (10) Li, G.; Liu, Q.; Liu, Z. Kinetic Behaviors of CaC2 Production from Coke and CaO. Ind. Eng. Chem. Res. 2013, 52, 5587−5592. (11) Healy, G. Why a Carbide Furnace Erupts. J. Met. 1966, 18, 643− 647. (12) Healy, G. The calculation of an internal energy balance in the smelting of calcium carbide. J. South. Afr. Inst. Min. Metall. 1995, 95, 225−232.

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Reaction Kinetics Modeling of CaC2 Formation From Coal and Lime

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