In the Classroom
Parameter Sensitivity Study of the Unreacted-Core Shrinking Model: A Computer Activity for Chemical Reaction Engineering Courses Ignacio Tudela* and Pedro Bonete Departamento de Química Física, Universidad de Alicante, Ap. Correos 99, 03080 Alicante, Spain *
[email protected] s Fullana and Juan Antonio Conesa Andre Departamento de Ingeniería Química, Universidad de Alicante, Ap. Correos 99, 03080 Alicante, Spain
Fluid-solid reactions are common chemical processes that take place in a heterogeneous media. These processes are of considerable importance, especially in chemical and metallurgical industries (1, 2). Fluid-solid processes in which the solid particle size does not effectively change are the roasting or oxidation of sulfide ores to yield the metal oxides and the nitrogenation of calcium carbide to obtain cyanamide, among others. Other fluid-solid processes involve the variation of particle dimensions such as combustion of solid fuels, gasification of coal or oil shale, and the production of sodium thiosulfate from sulfur and sodium sulfite. For heterogeneous, noncatalytic reactions of solids with surrounding fluid where the particle inner dimensions vary, Yagi and Kunii (3) developed the unreacted-core shrinking (UCS) model using spherical particles and an isothermal system. During the reaction, the reaction zone advances from the outer skin of the particle into the solid, leaving behind completely converted solid material or ash (Figure 1). A sharp interface between the ash layer and the unreacted core is assumed. Five steps occur during the overall reaction: (i) diffusion of gaseous reactant through the thin gas layer surrounding the solid particle, (ii) diffusion of the gaseous reactant through the ash layer, (iii) reaction of the gaseous reactant with solid at the unreacted core surface, (iv) diffusion of gaseous product through the ash layer back to the inner gas layer, and (v) diffusion of gaseous product through the gas layer back to the bulk fluid. Since it was introduced, the UCS model has become one of the most studied models for heterogeneous reactions courses, receiving continuous attention from several research groups. As an example, the cyclic reduction-oxidation of NiO/Ni mixed with yttriastabilized zirconia as a medium material for chemical-looping combustion (4) is well described by the UCS model, showing that the reduction of NiO is nearly controlled by the chemical reaction, whereas the oxidation of Ni is in the intermediate regime between chemical reaction control and ash-layer diffusion control. Other authors have used modified UCS models accounting for the deposition of terephtalic acid on unreacted poly(ethylene terephthalate) (PET) particles to estimate the kinetics of the hydrolysis of PET powder in nitric acid (5) and accounting for the formation and growth of pores and crack on unreacted PET particles to model the hydrolysis of PET powder in sulfuric acid (6). More recently, Lee and Koon reported the kinetic modeling of flue gas desulfurization using CaO/Ca(SO)4/coal fly ash sorbent at low temperatures (7) using the chemical reaction as the rate-limiting step coupled with surface coverage effect to take into account the diffusion-controlling step at the end of the reaction. Daggupati et al. have recently used the 56
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Figure 1. Visualization of the size reduction of the unreacted core and growth of the ash layer for a certain period of time.
UCS and uniform-conversion models for the analysis of CuCl2 hydrolysis in a reactive spray-drying process and in a thermochemical copper-chlorine cycle of hydrogen production (8), showing that the reaction is controlled by both chemical reaction control, as well as diffusion through product layer control, for particles around 200 μm. Students must also be aware that more rigorous mechanistic models shall be proposed, including new rate-limiting steps, to obtain a satisfying fit to experimental data for processes that cannot be described with the UCS model such as the production of hydrogen by sorption-enhanced steam methane reforming. Johnsen et al. found that the UCS model predicted lower conversions (up to 30%) well (9), whereas Rusten et al. decided to employ an empirical model as mechanistic models because the UCS model did not explain the experimental data when using Li2ZrO3 (10) and Li4SiO4 (11) as CO2-acceptors. Other processes where the reaction may occur along a diffuse front between the ash layer and the unreacted core or where the heat released may be high enough to cause significant temperature gradients within the particles or between particle and the bulk fluid may also not be properly described by the UCS model (1). Wen suggested that the UCS model for liquid-solid reactions could also give erroneous results unless the reactant concentration was very low (2). For the latter case, some criteria have been summarized in order to use the UCS model in hydrometallurgy (12). The incorporation of personal computers in the classroom allows teachers to easily show the influence of the variables in the UCS model that affect the system behavior. Several systems can be simulated by using the theoretical equations governing the processes, avoiding long and difficult experimental testing that could distract the students. The aim of this activity is to provide a
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Vol. 88 No. 1 January 2011 pubs.acs.org/jchemeduc r 2010 American Chemical Society and Division of Chemical Education, Inc. 10.1021/ed100302n Published on Web 10/18/2010
In the Classroom
numerical resolution of the UCS model to study the effect of the different steps that control the reaction rate throughout the whole process. The solution of the model is achieved by a set of different MatLab script and function files that are included in the supporting information for teachers and students. The UCS Model Equation Solved In this activity, the UCS model has been employed to study the calcination of spherical particles of zinc blende (ZnS) using a constant oxygen concentration gas flow as a classroom activity based on problem proposed by Levenspiel (1) according to the equation: 2ZnSðsÞ þ 3O2 ðgÞ f 2ZnOðsÞ þ 2SO2 ðgÞ Several mathematic expressions for the UCS model have been derived by various authors (1, 13, 14). Among these, the following general eq 1 has been considered, which was successfully employed by Ishida et al. in previous works (4): CO2 b drc FZnS ¼ 2 ð1Þ dt rc ðR - rc Þrc 1 þ þ k 00 R2 k g RDO2 where rc is the unreacted core radius, b is the stoichiometric factor, CO2 is the concentration of O2 in the gaseous flow, FZnS is the density of the zinc blende, R is the (initial) particle radius, DO2 is the diffusivity of O2 through the ash layer, kg is the mass transport coefficient of O2 in the gaseous layer, and k00 is the kinetic constant of the chemical reaction step. This equation presents a unique feature that shows how the resistances vary with particle size: the first summand in the denominator is the resistance of the diffusion through the gas film surrounding the particle, the second is the resistance of the diffusion through the ash layer, and the third summand in the denominator is the resistance of the chemical reaction step. Therefore, a combination of three limiting steps is considered in the numerical simulation proposed in this activity, instead of the classical study of each separated resistance being defined as the controlling step. This way, students are aware of the evolution of the reaction through time, and the numerical solution becomes an effective tool for teachers to highlight that the controlling step of the process changes during the reaction course. The numerical solution of the eq 1 is obtained from the script and function files where MatLab embedded-functions fzero (15) and ode45 (16) are employed. The fzero function finds the root of a continuous function of one variable and is used in obtaining the value of tend, the time when the reaction is complete, whereas the ode45 function is a powerful tool to numerically solve differential equations. To solve eq 1, different parameter values were set, which are included in the MatLab script file (see the supporting information). Computer Activity for Students: Study of the Influence of the Particle Size The structure of the MatLab script file included as supporting information allows teachers and students to study the influence that the variables play in the overall process by introducing a numerical value for parameters R, DO2, kg, and k00 in MatLab's command window. As an example, different R values have been examined.
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Figure 2. Dimensionless unreacted core radius for different particle sizes: R = 0.01 cm (dashed line), R = 0.1 cm (solid line), and R = 1 cm (dotted line). Inset graph: zoom showing reaction time range from 0 to 800,000 s.
Variation of the unreacted core radius, rc, as a function of the reaction time, t, and the influence of the initial particle radius, R, on the ZnS calcination process for particle radii of 0.01, 0.1, and 1 cm are shown in Figure 2. The reaction time to completely calcinate an individual particle, tend, that is, when rc = 0, is also shown in Figure 2. For R = 0.1 cm (solid line), tend is 22,492 s. By reducing the radius 1 order of magnitude (dashed line), the complete calcination takes place more than 90% faster (tend = 1721 s for R = 0.01 cm). However, increasing the radius 1 order of magnitude (dotted line) leads to a significant increase in the total calcination time needed and more than a 3500% of the reaction time is observed (tend would be larger than 8 105 s for R = 1 cm). From a chemical engineering point of view, this observed feature is of vital importance because the addition of a grinding mill into a the overall industrial process to calcine zinc blende could significantly reduce the reactor size and manufacturing cost. The MatLab script also shows the error obtained when assuming that steps (i) and (ii) control the overall process. Therefore, when considering the smallest radius, the time estimated assuming diffusion through the gas film surrounding the particle and the ash layer as limiting steps reaches an error of 22%, whereas the greatest radius reaches an error lower than 5%. The variation of the three resistances involved in the process (the three summands in the denominator in eq 1) is displayed in Figure 3 for the initial particle radii, R, of 0.01, 0.1, and 1 cm. Initially, the gaseous surrounding layer is the main resistance to the process (dashed lines) for all cases because the unreacted core surface is relatively large and the ash layer is very thin. As time passes, the ash layer thickness increases while the unreacted core shrinks and the gaseous surrounding layer resistance remains as the main resistance depending on the particle radius. Later, the resistances related to the ash layer (dotted line) and the kinetics of the chemical reaction (solid line) increase compared to the resistance related to diffusion through the gas layer surrounding the solid particle. At this stage, the O2 diffusion through the ash layer could become the controlling rate step of the overall process, which is especially relevant for R = 0.1 cm (Figure 3B). For the biggest particle size studied (Figure 3C), the process is mainly controlled by the diffusion of O2 through the ash layer (dotted line), whereas for the smallest radius studied (Figure 3A), the diffusion of O2 through the gaseous film surrounding the particle (dashed line) remains as the predominant resistance. Finally, when the calcination process is near the end for the
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Figure 3. Change in dimensionless resistances related to diffusion through the gas film surrounding the particle (dashed lines), diffusion through the growing ash layer (dotted lines), and the kinetics of the chemical reaction (solid lines), related to time, for the process where: (A) R = 0.01 cm, (B) R = 0.1 cm, and (C) R = 1 cm.
three particle radii studied, the unreacted core surface is so small that the overall process is controlled by the chemical reaction step and diffusion through both gaseous and ash layers does not practically influence the process. For additional activities, another interesting variable to study could be the mass transport coefficient in the gas film surrounding the particle, kg. Varying its value would show the effect that could take place if the diffusion of O2 through the gas film surrounding the particle was either enhanced or diminished inside the reactor due to changes in the gas flow regime (i.e., addition of turbulence promoters or baffles inside the reactor, increase or decrease in flow rate, etc.). Other parameters could be the kinetics of the chemical reaction, k00 , and diffusion of gaseous reactant through the ash layer, DO2. Conclusion Nowadays, personal computers in the classroom are common and MatLab software is helpful, easy-to-use, and extensively employed in many universities. The computer-aided resolution of the UCS model can help students to understand complicated heterogeneous processes such as fluid-solid reactions, where the influence of many variables in the reaction rate takes place. By employing the MatLab script and function files included in the supporting information, teachers have the opportunity to show students how chemists and chemical engineers can modify a reaction process with a great understanding of the influence of the reaction parameters quickly and easily. Therefore, the influence of each step on the overall process and their control on the reaction rate can be observed by the students. Acknowledgment I.T. thanks the Generalitat Valenciana for its financial support under grant FPA/2009/024. Authors also thank J. Gonzalez-García, V. Saez, and M. D. Ezclapez for their help and support.
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Literature Cited 1. Levenspiel, O. Chemical Reaction Engineering, 3rd ed.; John Wiley & Sons: Hoboken, NJ, 1999. 2. Wen, C. Y. Ind. Eng. Chem. 1968, 60, 34–54. 3. Yagi, S.; Kunii, D. Studies on Combustion of Carbon Particles in Flames and Fluidized Beds. In Fifth Symposium ( International) on Combustion; Lewis, B., Hottel, H. C., Nerad, A. J., Eds.; Reinhold: New York, 1955; pp 231-244. 4. Ishida, M.; Jin, H.; Okamoto, T. Energy Fuels 1996, 10, 958–963. 5. Yoshioka, T.; Motoki, T.; Okuwaki, A. Ind. Eng. Chem. Res. 1998, 37, 336–340. 6. Yoshioka, T.; Motoki, T.; Okuwaki, A. Ind. Eng. Chem. Res. 2001, 40, 75–79. 7. Lee, K. T.; Koon, O. W. Chem. Eng. J. 2009, 146, 57–62. 8. Daggupati, V. N.; Naterer, G. F.; Gabriel, K. S. Int. J. Heat Mass Transfer 2010, 53, 2449–2458. 9. Johnsen, K.; Grace, J. R.; Elnashaie, S. S. E. H.; Kolbeinsen, L.; Eriksen, D. Ind. Eng. Chem. Res. 2006, 45, 4133–4144. 10. Rusten, H. K.; Ochoa-Fernandez, E.; Chen, D.; Jakobsen, H.A.. Ind. Eng. Chem. Res. 2007, 46, 4435–4443. 11. Rusten, H. K.; Ochoa-Fernandez, E.; Lindborg, H.; Chen, D.; Jakobsen, H.A.. Ind. Eng. Chem. Res. 2007, 46, 8729–8737. 12. Liddel, K. C. Hydrometallurgy 2005, 79, 62–68. 13. Shen, J.; Smith, J. M. Ind. Eng. Chem. Fundam. 1965, 4, 293–301. 14. White, D. E.; Carberry, J. J. Can. J. Chem. Eng. 1965, 43, 334–337. 15. Forsythe, G. E.; Malcolm, M. A.; Moler, C. B. Computer Methods for Mathematical Computations; Prentice-Hall: Englewood Cliffs, 1976. 16. Dormand, J. R.; Prince, P. J. J. Comput. Appl. Math. 1980, 6, 19–26.
Supporting Information Available The MatLab script (script.m) and function files (fproyect.m, fauxproyect.m, fproyectR12.m and fauxproyectR12.m). For an adequate use, please copy the content of each .doc file in a new .m file and name it as indicated (script.m, fproyect.m, fauxproyect.m, fproyectR12.m, and fauxproyectR12.m). This material is available via the Internet at http:// pubs.acs.org.
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