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International Journal of Chemical Engineering
Volume 2012 (2012), Article ID 654321, 13 pages
The Effects of Mixing, Reaction Rates, and Stoichiometry on Yield for Mixing Sensitive Reactions—Part II: Design Protocols
1Department of Chemical and Materials Engineering, University of Alberta, 9107-116 Street, 7th Floor ECERF, Edmonton, AB, Canada T6G 2V4
2Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, AB, Canada T6G 2G8
Received 29 April 2012; Revised 24 July 2012; Accepted 2 August 2012
Academic Editor: Shunsuke Hashimoto
Copyright © 2012 Syed Imran A. Shah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competitive-consecutive and competitive-parallel reactions are both mixing sensitive reactions where the yield of desired product depends on how fast the reactants are brought together. Recent experimental results have suggested that the magnitude of the mixing effect may depend strongly on the stoichiometry of the reactions. To investigate this, a 1D, dimensionless, reaction-diffusion model was developed at the micromixing scale, yielding a single general Damköhler number. Dimensionless reaction rate ratios were derived for both reaction schemes. A detailed investigation of the effects of initial mixing condition (striation thickness), dimensionless reaction rate ratio, and reaction stoichiometry on the yield of desired product showed that the stoichiometry has a considerable effect on yield. All three variables were found to interact strongly. Model results for 12 stoichiometries are used to determine the mixing scale and relative rate ratio needed to achieve a specified yield for each reaction scheme. The results show that all three variables need to be considered when specifying reactors for mixing sensitive reactions.
Mixing-sensitive reactions are reactions which are particularly sensitive to the rate at which the reactants are brought together, that is, how fast they are mixed. These reactions are of two main types: the competitive-consecutive (C-C) reaction scheme, which involves two competing reactions where the second unwanted reaction consumes the desired product from the first reaction, and the competitive-parallel (C-P) reaction scheme, where two reactions compete for a limiting reagent, forming a desired and undesired product. The effects of mixing and relative reaction rates of the competing reactions have been investigated previously and it is known that mixing can affect the product distribution significantly. Past work has concentrated on the investigation of a single classical stoichiometry for each of the reaction schemes [1–25]. This work investigates whether the stoichiometry of the reaction plays a role in determining the maximum final yield of desired product and how the three reactor design variables—mixing, reaction, rate and stoichiometry interact.
In a previous paper by Shah et al. , a model was developed to capture the effects of reaction stoichiometry, mixing, (characterized by the Damköhler number (Da)), and relative reaction rates, (characterized by a dimensionless reaction rate ratio (/)). General forms of the reactions as given in Table 1 were used to derive mass balance equations. From the equations a single Damköhler number that is common to both reaction types and all stoichiometries (Da) and a dimensionless reaction rate ratio (/) that is specific to each reaction type were obtained. Both Da and / depend on the stoichiometric coefficients of the reaction scheme, making the potential impact of stoichiometry immediately evident. The expression for the general Damköhler number is:
The effect of stoichiometry on the Damköhler number is given by , the stoichiometric coefficient of the limiting reagent (B) in the desired reaction. The diffusion time in the smallest eddies is determined by the mixing scale, , the thickness of the smallest striation of concentration homogeneity, and the molecular diffusivity, , which completes the variables needed to determine the diffusion time at the smallest scales of segregation.
The effect of the relative reaction rates of the competing reactions is also of interest. The model provides dimensionless reaction rate ratios for the C-C and C-P reactions as follows:
These dimensionless reaction rate ratios are specific to the type of mixing sensitive reaction, C-C or C-P, and incorporate the effect of the relative reaction rates of the competing reactions as well as the effect of stoichiometry. The model was validated for the classical reaction stoichiometry where all coefficients are equal to 1. This work extends the simulations to ten new stoichiometries where one or more of the coefficients are greater than one.
For both the C-C and C-P simulations, the Damköhler number was varied from 0.01 to 10 000 in 100x increments, where 0.01 is the best and 10 000 is the worst mixing. The dimensionless reaction rate ratio (/) was varied from 1 to 0.00001 in 10x increments, where 1 is the worst ratio and 0.00001 is the best ratio. For the C-C reactions, was always 1 and were given a value of either 1 or 2. For the C-P reactions, and were either 1 or 2. Table 2 shows the different C-C reaction scheme stoichiometries investigated, with the corresponding dimensionless reaction rate ratios. Table 3 shows the same terms for the C-P reaction schemes.
We caution the reader that this model is highly simplified. Elementary reactions involving 4 molecules do not exist, so the kinetic expressions for Cases 7 and 8 for the C-C reaction schemes and for Case 4 for the C-P reactions will assuredly have a different and more complex form for real reactions with similar stoichiometries. The mixing model is equally flawed when considering real systems. The simple striation model does not allow for variation of striation thickness with time as mixing progresses or between eddies; it includes no consideration of the effects of stretching and no mixing between layers. The dynamics of a more realistic model of turbulence would be much more complex. Keeping these limitations in mind, the goal of this work is to investigate whether a hypothesis that the reaction stoichiometry can have a significant impact on the course of mixing sensitive reactions is reasonable.
2. Numerical Details
Simulation of the systems of five or six PDEs for the C-C and C-P reaction schemes, respectivelyr (shown in Tables 4 and 5), was carried out using COMSOL 3.4, a finite element PDE Solver. It is worth noting that for the C-C reaction scheme only four of the equations are independent while for the C-P reaction scheme there are five independent equations. The 1D, transient, convection, and diffusion mass transport model was used, with the mass fractions for each species specified as the independent variables. The default Lagrangian-quadratic element was chosen. The specified 1D geometry line of unit length was split equally into two subdomains and a mesh of 2048 equally spaced elements with 2049 nodes was generated. The mesh was tested for grid independence, and it was found that 1024 elements was sufficient resolution to ensure repeatable results to within the required tolerance of the solver, which was set to 10−6. Since the geometry was only 1D and the computational cost was minimal, a finer mesh than the minimum required resolution was used. The total time taken per simulation was approximately 30 seconds. The boundary conditions were specified for no net mass transfer across the boundaries.
Figure 1 shows the initial conditions for the C-C and C-P schemes. For the C-C cases, the initial conditions were chosen such that all of the mass initially present could be converted to desired product . This was done by specifying a ratio of A : B as 1 : 1 in all the simulations, with and B being present in pure striations, . For the C-P scheme, owing to the parallel nature of the reactions, the initial conditions were a bit more complicated. They were chosen such that either or could consume the entire limiting reagent by themselves, that is, the initial ratios depended on the stoichiometric coefficients in the reaction scheme. In order to satisfy the constant mass concentration assumption of the model, it was necessary to include the inert in the C-P simulations. and were always set to 0.5, and were calculated accordingly.
For most cases the limiting reagent, B, was completely consumed by so the transient simulation results are shown from to Da·t*= 500. is equivalent to a dimensionless reaction time where , so running all simulations to is the same as running all simulations for 500 reaction times.
The modelled equations allow for specification of the Damköhler number, the dimensionless reaction rate ratio and the stoichiometry. All possible combinations of the values of reaction rate ratio, Damköhler number, and stoichiometry for the C-C reaction scheme (stoichiometries given in Table 2) and the C-P reaction scheme (stoichiometries given in Table 3) were investigated. This resulted in 192 converged cases for the C-C and 96 converged cases for the C-P reaction scheme.
3. Results and Discussion
The results obtained for the simulations are profiles of mass fraction for each of the species over the space for all dimensionless times . Samples of these results were given in Shah et al.  (Figures 2 and 3, resp.).
Since the main process objective is to maximize the production of desired product P, the profiles of mass fraction of P are of most interest. These profiles were integrated over the domain to obtain the total mass of desired product and the instantaneous yield using the formula:
Following over time gives the progression of yield over time. The total production of as time progresses can be observed in a plot of versus dimensionless time . Figures 4 and 5 in Shah et al.  show examples of such plots.
The final yield of at was plotted against Da and to assess the effects of each of the variables on the final yield of desired product. Representative figures are presented here. The results ( versus Da and ) make up a 3D surface for each stoichiometry. Slices of the results are presented as follows: a base case set of six slices of versus Da is presented to show the effects of mixing (Da) and dimensionless reaction rate ratio () on yield of desired product. Where a point needs to be clarified, the results are replotted by taking a slice along the axis or by expanding the time variable. The results are then revisited to focus on the effect of stoichiometry and how it interacts with mixing and reaction rate. The same set of plots is repeated for the C-P reaction schemes. The results are then summarized to show a possible design specification space of Da and for a specified yield.
3.1. Competitive-Consecutive (C-C) Reaction
Figures 2(a) to 2(f) are semilog plots of the final versus Damköhler number for decreasing dimensionless reaction rate ratio, . The curves on each of the plots represent the eight C-C stoichiometry cases. The effects that are of interest are the effects of the Damköhler number (mixing), reaction rate ratio (chemical kinetics), and the reaction stoichiometry (Cases 1 to 8). The Damköhler number is plotted on the -axis and the results are first presented from this perspective. Next, the effect of ratio is observed by comparing graphs (a) to (f). Finally the effect of stoichiometry (Cases 1 to 8) within each graph is reviewed and summarized. At each step, selected results are replotted either to illustrate a point more clearly or to provide a perspective on the data.
3.1.1. The Effect of Damköhler Number (Da): The Well-Mixed Limit
The plots in Figure 2 show a decrease in yield of desired product with increasing Damköhler number. This trend is true for all stoichiometries and at all values of . A larger Damköhler number represents worse mixing (larger striations) and a smaller Damköhler number represents improved mixing (smaller striations). The yield for and is the same for all stoichiometries at all . The yield at decreases for the cases with unfavourable dimensionless reaction rate ratios () but remains for same for the favourable ratios (). By the yield always decreases, regardless of the value of .
The data is replotted in Figures 3(a) and 3(b) for the best (Case 7) and worst (Case 2) stoichiometry cases to confirm the versus trends. This shows that the results for and lie exactly on top of one another. When Da is increased from 1 to 100 the curves diverge over part of the domain: for Case 2 when and for Case 7 when . It is concluded that is the well-mixed limit for large , while is the well-mixed limit for small and that the meaning of “large ” depends on the stoichiometry.
3.1.2. The Effect of Dimensionless Reaction Rate Ratio (): Can Chemistry Solve the Problem?
Looking at the changes in Figure 2 from (a) to (f), as decreases the yield increases. This result is as expected, since a smaller gives a slower undesired side reaction. As is decreased it is also clear that the curves for the different stoichiometric cases approach, collapsing onto one curve by . The results are replotted to directly compare the effect of on the 8 cases in Figure 4(a): (well mixed), (b) and (c) (badly mixed).
increases to 1 as decreases for all stoichiometries, with the 8 cases collapsing at about for the well-mixed cases and at for , the badly mixed case. The yield of the badly mixed case is about half of what was obtained in the well-mixed results. This brings up an interesting question: “Is it possible to get perfect yield of desired product, that is, , for , a badly mixed case?”. The simulations for the Da =10000 cases were rerun for a hundred times longer to instead of . Figure 5 shows the results of these simulations. To obtain a perfect yield with insufficient mixing, both a 100x longer reaction time and 100x better chemistry are required. This clearly illustrates the need for an improved understanding of mixing.
3.1.3. Effect of Stoichiometry
Returning for a final time to Figure 2 there is a clear effect of stoichiometry on the final yield of desired product, and the effect of stoichiometry shows interactions with both the mixing scale and the dimensionless reaction rate ratio.
In Figure 2(a) the curves are widely separated, coming closer together as Da increases and the effect of bad mixing becomes dominant. There is a clear distinction between stoichiometries, with Cases 7 and 8 having the highest yields and therefore the most favourable stoichiometry, and Cases 1 and 2 having the lowest yields, meaning they have the least favourable stoichiometries. Cases 1 and 2 require only two molecules for the side reaction, while Cases 7 and 8 require four molecules. This places a mass transfer and collision probability limitation on the side reaction for Cases 7 and 8, hence making them more favourable for the desired reaction. Cases 3–6 all consume 3 molecules in the undesired reaction, making the mass transfer limit similar; however, at the large values of , Cases 3 and 4 have a much lower yield of than Cases 5 and 6, forming two bands.
Comparing the stoichiometries of Cases 1–4 with Cases 5–8, for the first 4 cases, 1 molecule of B is consumed in the second reaction, and in the last 4, 2 molecules of B are consumed. Comparing Case 3 to Case 6 and Case 4 to Case 5, the desired reactions are identical, whereas the stoichiometric coefficients on the undesired reaction are different. Of particular interest is the larger stoichiometric coefficient on reagent in Cases 5 and 6. An examination of the local concentration profiles showed that is the limiting reagent locally in the second reaction. Since Cases 5 and 6 require two molecules of the limiting reagent in the undesired reaction, making it harder for the reaction to proceed, it makes sense that the yield of will be higher for Cases 5 and 6 as compared to Cases 3 and 4.
Now comparing the stoichiometries within pairs, in three of the four pairs producing more in the first reactions leads to an increase in consumption in the second reaction and a drop in yield. In other words, at large ratios, the two reactions compete effectively for reagents. Under these conditions, when less is produced in the desired reaction, the side reaction is slowed down. The trend is reversed for Cases 3 and 4. These are the only cases where more is consumed than in the undesired reaction, so in this case it appears that the mass transfer limit on relative to overwhelms the effect of producing more .
These trends continue in Figure 2(b) and Figure 2(c) (. For the curves collapse onto each other and the effect of stoichiometry vanishes. For the very bad mixing condition () that was allowed to proceed for a very long time, Figure 5, the effect of stoichiometry creeps back into the picture, affecting yield at all the way to .
3.1.4. Summary of Results for the C-C Reaction Scheme
For competitive-consecutive reactions, we have the following.(a) is the well-mixed limit when and is the limit when .(b) with provides yields >95% at .(c) The effect of stoichiometry can be large if the is unfavourable. This effect vanishes at for short reaction times and good mixing. At longer reaction times with bad mixing conditions, the effect of stoichiometry reappears and requires a much smaller value of to eradicate.(d) The most favorable stoichiometry is one where the desired reaction consumes fewer molecules than the undesired reaction, thus placing a mass transfer limitation on the undesired reaction.
3.2. Competitive-Parallel (C-P) Reaction
The analysis proceeds through the same questions posed for the C-C reaction.(i)What is the well mixed limit (Da)?(ii)How much can chemistry contribute ()?(iii)Is the stoichiometry important?
3.2.1. Effect of Damköhler Number (Da)
The plots in Figure 6 show a large drop in when Damköhler number increases from 100 to 10000 with the exception of Case 3 at where the yield increases steadily with increasing Da. Case 3 is the only stoichiometry where the desired reaction consumes more molecules than the undesired reaction, so when the reaction rates are equal and mixing is good, the mass transfer limitation on the desired reaction will push the yield toward the secondary reaction. Since the reactions are parallel, there is no barrier to the production of over P.
For the C-P reaction pair, yield is constant for , giving a well-mixed limit of . Figures 7(a) and 7(b) confirm this result for Cases 1 and 3, respectively. The curves show the four values of Da investigated. The curves for and lie exactly on top of each other and the curve for is very close, but increasing Da to 10 000 drops the yield significantly. These results are very similar to the C-C results. It is concluded that is the well-mixed limit for C-P reactions.
3.2.2. Effect of Dimensionless Reaction Rate Ratio ()
Looking at Figure 6 from the perspective of reaction rate ratio, as is decreased, the curves for all the stoichiometries move upwards, increasing the yield for all cases at all values of Da. As expected, a smaller increases the yield of desired product. For Cases 1 and 2 the maximum yield is approximately 1 and for Cases 3 and 4 it is 0.88. The curves for Cases 1 and 2 collapse at (Figure 6(d)) and the curves for Cases 3 and 4 collapse at (Figure 6(e)). The largest change with improving is seen for Case 3, while Case 2 improves the least. These trends are confirmed in Figure 8, showing a persistent effect of stoichiometry on the maximum attainable yield. The poorly mixed condition ( 000) may not have reached completion at since the yield is only about 50%.
In Figure 9 the poorly mixed condition is allowed to progress 100x longer. At this point, the yield can increase to above 90% but only for a 100x smaller ratio. The curves seem to collapse at , but at lower ’s the effect of stoichiometry persists, even at long times. Once again, 100x more residence time and 100x better chemistry are required to eliminate all the effects of 100x worse mixing.
3.2.3. Effect of Stoichiometry
Figures 6(a) to 6(f) illustrate a significant effect of stoichiometry for the C-P case. At (Figure 6(a)) the difference in yield for the different stoichiometries is enormous, and there is a clear distinction between the most favourable and least favourable reaction stoichiometries (Case 2 and Case 3, respectively). This difference decreases as decreases (Figures 6(b) to 6(f)). For the C-P scheme, the effect of stoichiometry persists even at , where the difference in yield between Cases 1 and 2 and Cases 3 and 4 is more than 10%. This is different from the C-C stoichiometries where the effect of stoichiometry disappeared by .
Referring to Table 3, Cases 1 and 4 require the same number of molecules for both reactions, while Case 2 favors the desired reaction (fewer molecules) and Case 3 favors the side reaction. Case 3 is the only case of all 12 stoichiometries considered where the desired reaction consumes more molecules than the side reaction, imposing a mass transfer and collision probability limitation on the desired reaction. This explains the dramatically different results observed for Case 3 and suggests that this class of reaction stoichiometry must be handled more carefully than the others.
3.2.4. Summary of Results for the C-P Reaction Scheme
For competitive-parallel reactions, we have the following.(a) is the well-mixed limit for all .(b) For all cases with favourable stoichiometries, at , ≤ 0.001 provides the maximum yield, as for the C-C reaction scheme, but for the C-P reaction scheme the maximum yield depends on the stoichiometry. Case 3, the only case with an unfavourable stoichiometry, requires to achieve maximum yield.(c) Stoichiometry affects the maximum obtainable yield to some degree for all mixing conditions and all . (d) As for the C-C scheme, increasing the number of molecules in the side reaction imposes a mass transfer limitation on that reaction, making conditions more favorable for the desired reaction. Case 3 in the C-P scheme clearly illustrates that when the desired reaction consumes more molecules than the undesired reaction, it is more difficult to achieve a high yield of desired product.
3.3. Phase Plots of Variables for C-C And C-P Reaction Schemes for the Purpose of Design
The purpose of design graphs, in the context of mixing sensitive reactions, is to provide a direct method of predicting the yield of desired product without extensive calculations and investigation of the different variables. For our purposes, design graphs would incorporate the effects of stoichiometry, mixing, and and the combinations of the three that could possibly give a certain specified yield of desired product. The purpose of the graphs would be to assist in the design of the following problem.
“If one has a certain mixing sensitive reaction and a target yield of desired product in mind, what range of values of dimensionless reaction rate ratio and Damköhler would be required to ensure that the target yield of desired product will be obtained? If one of the variables is inadequate, is it still possible to obtain the target yield? If it is possible, how much do the other variables need to change to accommodate for that inadequacy, if at all?”
The figures should allow for a design space to be established within which decisions can be made based on different criteria and restrictions.
Figure 10 and Figure 11 show the required Damköhler number (-axis) versus the dimensionless reaction rate ratio ( -axis) for yields of desired product of (a) 85% or more, (b) 95% or more, and (c) 99% or more at a time of . They are set up such that the axis intersection (origin) represents the most favourable conditions for both and Damköhler number. The figures have marked regions of required and Da where a specified yield can be obtained for different stoichiometries. These figures are essentially a filtered summary of the results presented in the previous sections. The effect of stoichiometry quickly becomes evident as the specified yield is increased. For the C-C cases, the requirements on the and Da become more stringent as increases. For C-P Cases 3 and 4, yields greater than 90% were unachievable for .
In order to link these results to physical process variables, the diffusion time is replaced with an analogous mixing time scale for turbulent flow. The Batchelor time scale gives the diffusion time for a Kolmogorov sized eddy and is numerically equal to the Kolmogorov time scale, which is the lifetime of the same smallest eddy before the momentum is diffused by viscosity. Replacing the molecular diffusion time in a slab in our definition of Da with the expression for the Kolmogorov time scale in a turbulent eddy gives the following: where is the kinematic viscosity and is local rate of dissipation of turbulent kinetic energy (TKE) per mass. Rearranging to solve for gives the following:
This expression proposes a relationship between the rate of dissipation of turbulent kinetic energy (), the reaction kinetics of the desired reaction (), and the reaction stoichiometry () which might be used to compare bench scale performance with the Da and reaction rate ratio from the phase plot. For example, for a C-C reaction of the type Case 1 (, using typical values for water ( m2/s, kg/m3, kg/kmole), for Da = 1 becomes
This gives a first-order estimate for the required level of dissipation at the bench scale. Since the model is based on a static striation thickness rather than the dynamic stretching and cutting field present in both chaotic and turbulent flows, this should be considered a starting point for further investigations in the context of more accurate models of mixing and diffusion.
The three effects of initial mixing condition, dimensionless reaction rate ratio, and reaction stoichiometry were investigated in detail using a 1D, transient, reaction-diffusion model for competitive-consecutive (C-C) and competitive-parallel (C-P) reactions. It was found that smaller values of Damköhler number and dimensionless reaction rate ratio are desirable to maximize the yield of desired product for both the C-C and C-P reaction schemes. It was also found that the stoichiometry of the reaction can affect the final yield of desired product considerably and needs to be taken into consideration in the design of reactors for such reactions. The following limits were found for each of the variables.(i) is the well-mixed limit for the C-C reaction when . With , the well-mixed limit can be relaxed to .(ii) is the well-mixed limit for all for the C-P reaction.(iii) with provides the maximum yield at for both the C-C and the C-P reactions with favourable stoichiometries. For the C-P case with the most unfavourable stoichiometry is required. (iv) The effect of stoichiometry is significant. Where the secondary reaction consumes more molecules than the primary reaction, it is subject to a mass transfer constraint which favors the primary reaction. For the C-P scheme the effect of stoichiometry is persistent, affecting the final yield by more than 10% even for a well-mixed case with favorable .
The model used here is intended to illustrate the importance of the previously uninvestigated effect of stoichiometry on the yield of desired product and to provide general forms of the Damköhler number and reaction rate ratio for C-C and C-P reactions. The results are not intended to give accurate predictions of yield but to illustrate the trends in a quantitative way. It is hoped that the results will guide development of experimental reaction schemes which serve as mixing probes and development of more realistic reactor models which incorporate mixing, reaction kinetics, and stoichiometry.
|:||Rate constant 1|
|Rate constant 2[varies]|
|:||Dimensionless reaction rate ratio|
|:||Molecular weight [kg/kmol]|
|:||Yield of desired product .|
|:||Local dissipation of turbulent kinetic energy per mass|
|:||Mixing time [s]|
|:||Reaction time [s].|
|:||Species A (reactant)|
|:||Species B (reactant)|
|:||Species C (reactant)|
|:||Species , , , , , or|
|:||Species S (byproduct)|
The authors would like to thank NSERC Canada and MITACS for funding provided to carry out this research.
- J. Baldyga and J. R. Bourne, “Interactions between mixing on various scales in stirred tank reactors,” Chemical Engineering Science, vol. 47, no. 8, pp. 1839–1848, 1992.
- J. Baldyga and J. R. Bourne, Turbulent Mixing and Chemical Reactions, Wiley, Chichester, UK, 1999.
- J. Bałdyga and R. Pohorecki, “Turbulent micromixing in chemical reactors—a review,” The Chemical Engineering Journal and the Biochemical Engineering Journal, vol. 58, no. 2, pp. 183–195, 1995.
- J. Bałdyga, J. R. Bourne, and S. J. Hearn, “Interaction between chemical reactions and mixing on various scales,” Chemical Engineering Science, vol. 52, no. 4, pp. 457–466, 1997.
- S. Bhattacharya, Performance improvement of stirred tank reactors with surface feed [Ph.D. thesis], University of Alberta, Alberta, Canada, 2005.
- M. J. Clifford, “A Gaussian model for reaction and diffusion in a lamellar structure,” Chemical Engineering Science, vol. 54, no. 3, pp. 303–310, 1999.
- M. J. Clifford and S. M. Cox, “Simple model for a two-stage chemical reaction with diffusion,” IMA Journal of Applied Mathematics, vol. 63, no. 3, pp. 307–318, 1999.
- M. J. Clifford, E. P. L. Roberts, and S. M. Cox, “The influence of segregation on the yield for a series-parallel reaction,” Chemical Engineering Science, vol. 53, no. 10, pp. 1791–1801, 1998.
- M. J. Clifford, S. M. Cox, and E. P. L. Roberts, “Lamellar modelling of reaction, diffusion and mixing in a two-dimensional flow,” Chemical Engineering Journal, vol. 71, no. 1, pp. 49–56, 1998.
- M. J. Clifford, S. M. Cox, and E. P. L. Roberts, “Reaction and diffusion in a lamellar structure: the effect of the lamellar arrangement upon yield,” Physica A, vol. 262, no. 3-4, pp. 294–306, 1999.
- M. J. Clifford, S. M. Cox, and E. P. L. Roberts, “The influence of a lamellar structure upon the yield of a chemical reaction,” Chemical Engineering Research and Design, vol. 78, no. 3, pp. 371–377, 2000.
- S. Cornell and M. Droz, “Exotic reaction fronts in the steady state,” Physica D, vol. 103, no. 1–4, pp. 348–356, 1997.
- S. M. Cox, “Chaotic mixing of a competitive-consecutive reaction,” Physica D, vol. 199, no. 3-4, pp. 369–386, 2004.
- S. M. Cox and M. D. Finn, “Behavior of the reaction front between initially segregated species in a two-stage reaction,” Physical Review E, vol. 63, no. 5, Article ID 051102, 7 pages, 2001.
- S. M. Cox, M. J. Clifford, and E. P. L. Roberts, “A two-stage reaction with initially separated reactants,” Physica A, vol. 256, no. 1-2, pp. 65–86, 1998.
- R. O. Fox, “On the relationship between Lagrangian micromixing models and computational fluid dynamics,” Chemical Engineering and Processing, vol. 37, no. 6, pp. 521–535, 1998.
- R. O. Fox, Computational Models for Turbulent Reacting Flows, Cambridge University Press, Cambridge, UK, 2003.
- I. Hecht and H. Taitelbaum, “Perturbation analysis for competing reactions with initially separated components,” Physical Review E, vol. 74, no. 1, Article ID 012101, 2006.
- O. Levenspiel, Chemical Reaction Engineering, Wiley, New York, NY, USA, 2 edition, 1972.
- F. J. Muzzio and M. Liu, “Chemical reactions in chaotic flows,” The Chemical Engineering Journal and the Biochemical Engineering Journal, vol. 64, no. 1, pp. 117–127, 1996.
- G. K. Patterson, E. L. Paul, S. M. Kresta, and A. W. Etchells, “Mixing and Chemical Reactions,” in Handbook of Industrial Mixing—Science and Practice, E. L. Paul, V. A. Atiemo-Obeng, and S. M. Kresta, Eds., Wiley, Hoboken, NJ, USA, 2004.
- M. Sinder, “Theory for competing reactions with initially separated components,” Physical Review E, vol. 65, no. 3, Article ID 037104, 4 pages, 2002.
- M. Sinder, J. Pelleg, V. Sokolovsky, and V. Meerovich, “Competing reactions with initially separated components in the asymptotic time region,” Physical Review E, vol. 68, no. 2, Article ID 022101, 4 pages, 2003.
- H. Taitelbaum, B. Vilensky, A. Lin, A. Yen, Y. E. L. Koo, and R. Kopelman, “Competing reactions with initially separated components,” Physical Review Letters, vol. 77, no. 8, pp. 1640–1643, 1996.
- J. Villermaux and L. Falk, “A generalized mixing model for initial contacting of reactive fluids,” Chemical Engineering Science, vol. 49, no. 24, pp. 5127–5140, 1994.
- S. I. A. Shah, L. W. Kostiuk, and S. M. Kresta, “The effects of mixing, reaction rate and stoichiometry on yield for mixing sensitive reactions part I: model development,” International Journal of Chemical Engineering, vol. 2012, Article ID 750162, 16 pages, 2012.