Chemical Kinetics Development with Response Surface Methodology (RSM)
The kinetic parameters of runaway reactions are typically assessed by trial and error, one at a time. If some knowledge of the system exists, such as the reaction order, then the simplest case would require two parameters: the pre-exponential factor and the activation energy. Even this unpretentious condition presents hurdles. How do we get the optimum of both parameters?
In this newsletter, Enio Kumpinsky explains the best approach to obtaining kinetic model parameters using the Response Surface Methodology (RSM) [1] with a Central Composite Design (CCD) [2]. Process Safety Office? SuperChems? software is ideal for evaluating upset scenarios with runaway chemical reactions.
This method needs laboratory testing to develop kinetic parameters. Let us assume that we have adiabatic calorimetry data and that our system is the simple case of a runaway reaction with these unknowns:
There are two reactants in this example with reaction orders m and n. This system can be expressed by Equation 1 as follows:
Let us assume that the reaction orders are identical, m = n. Figure 1? below shows the inverse Arrhenius plot of a lumped factor versus temperature. The lumped factor comes from chemical kinetics, expressed by the left-hand side of Equation 2:
The experimental outcome represented by the symbols was plotted in Figure 1 to choose the reaction order. The three curves come from the same experiment. They were plotted with different values of reaction orders, assuming that m and n are equal. A straight line was attached to each reaction order. Reaction order 2 is the closest to a straight line, so a second-order reaction was assumed.
The straighter curve in Figure 1, overall order 2 in this example, provides approximate values of the kinetic parameters, with the linear regression of Equation (2). These values are approximations and should not be used in simulations of industrial projects. Furthermore, for simulations, the kinetic rate needs a correction. Dynamic simulations should use k′ and k°′ as expressed by Equations 3 and 4, not k and k° established by Equation 2:
and
The groundwork to determine the pre-exponential factor k°′ and the activation energy term B is based on the best straight line in Figure 1. Then, Process Safety Office? SuperChems? software is used to establish the actual values of these two parameters.
The traditional way to obtain the kinetic parameters k°′ and B is by changing them by trial and error until an acceptable match to the experimental data is achieved. The reality is that neither parameter is optimized in this manner. When fixing the pre-exponential factor to determine the activation energy or vice-versa, one is adjusted for the fixed value of the other, which most likely is not the true optimum. It is virtually impossible to optimize multiple variables by trial and error.
One effective method to optimize multiple variables is using Experimental Design, a statistical technique that can simultaneously identify the optimum of all model factors under analysis. An experimental design organizes, conducts, and interprets the results for the best outcome based on the smallest number of trials.
The word trial usually refers to performing experiments. However, when developing a kinetic model for a runaway reaction, there may be a single experiment and the word trial has a different intent. It represents a simulation with assigned kinetic parameters in the design. Depending on the project, a simple experimental design works with squares, cubes, and hypercubes, possibly having a central point. Such designs are improvements to trial and error. However, a superior experimental design technique can be applied to establish kinetic parameters. It is known as Response Surface Methodology (RSM) [1], carried out with a Central Composite Design (CCD) [2].
RSM is a collection of mathematical and statistical techniques for modeling and analyzing complex relationships between factors (input variables) and responses (output variables).
We cannot effectively develop kinetic parameters for a runaway chemical reaction by trial and error.
Trial and error is still useful. A few trials assuming reasonably valued design factors can help narrow the search for their optimized values.
More Complex Designs for Runaway Reactions
A simple runaway reaction RSM contains two factors: the pre-exponential factor and the activation energy. More complex designs can be developed as exemplified next.
Example 1: Autocatalysis
Potential factors for this design include the reaction orders m and n of the reactants as shown in Equation (1), the order j of an auto-catalytic reaction product, the pre-exponential factor, and the activation energy, as shown in Equation 5.
Example 2: Binary Interaction Parameters (BIPs)
The design may include pressure factors besides temperature and concentrations. This case has one or more BIPs in the design.
Example 3: Variable Chemical Compositions
Examples 1 and 2 are based on simulations intended to match the outcome of a single adiabatic experiment. However, variations in composition cannot be analyzed based on a single experiment. Usually, the presumed worst-case scenario is evaluated, representing a line of products. It may be difficult sometimes to assess which composition is the worst case. This is where compositional experimental design becomes handy. The solution is to run a design with actual experiments where the compositions are established by statistical software, such as a CCD. The experimental results are analyzed with statistical software and the design factors are applied to simulations of this product line. The compositions in simulations can vary, as long as they are within the tested range for which the parameters were calculated.
It is important to note that the number of simulations grows geometrically as the number of factors increases. In a typical Central Composite Design (CCD) with no replicates, the number of trials is determined as follows:
For two factors, N = 22+2x2+1 = 9 simulations. For four factors, N = 24+2x4+1 = 25 simulations. Hence, choose the factors wisely to prevent too many simulations. The number 1 in Equation 6 comes from the central point. In a design based on experiments, that number would be higher than 1 to represent replicability. The results are automatically replicated in simulations, so a single central point suffices.
Experimental Design Example
Consider the simple case of a Di-Tert-Butyl Peroxide decomposition reaction for which only the pre-exponential factor and the activation energy must be determined. Earlier, Figure 1 showed the experimental results of this runaway reaction. Usually, the kinetic parameters are determined based on the self-heating rate vs. temperature output, graph “a” in Figure 2. However, temperature vs. time will be used in this demonstration, which is graph “b” in Figure 2.
Kinetic development based on temperature vs. time is too complex to cover in this newsletter. What is important at this time is to understand how to determine kinetic parameters using statistics.
The most common procedure is to match the experimental data of Figure 2a, together with other assessments, such as temperature vs. time (Figure 2b), self-pressurization rate vs. temperature, pressure vs. time, and pressure vs. temperature. Fundamentally, the curves generated by simulations must match the experimental data as well as possible.
A few simulations by trial and error can narrow down the boundaries of the experimental design, as shown in Figure 3 for two factors, the logarithm of the pre-exponential factor, and the activation energy divided by the universal gas constant. If the boundaries of the design were properly determined, the optimum would lie within the borders of the Central Composite Design.
Figure 4 is a repeat of Figure 2 with the addition of two points representing the optimum set of parameters. If Point 1 is the optimum, then the design limits were properly assessed before the statistical analysis because it lies well within the limits of the CCD. Now consider Point 2. Response Surface Methodology (RSM) does not extrapolate the optimum outside the boundaries of the design. In this example, the design limits of the B factor were properly assessed, as Point 2 lies between its minimum and maximum. However, ln(k°) was off-range because Point 2 is pegged at the low end of ln(k°).
A benefit of RSM is that the design does not need to be started from the beginning with lower values of ln(k°). It suffices to augment the current design with at least one additional point with the following characteristics: B within the current design limits and ln(k°) below its low limit, represented by the axial (star) point to the left. The optimum has been successfully identified if the calculations show that it is within the boundaries of the design. Otherwise, it may be necessary to add more points to the design. The Figure 4 situation is not uncommon and more points may have to be added involving the two factors to increase the chance of hitting the optimum within the design limits.
Conclusions
Experimental design has superior performance when compared to trial and error. It is difficult to achieve optimized conditions for all factors simultaneously without a systematic approach like the Response Surface Methodology (RSM) can provide. Changing variables one at a time is unlikely to lead to the optimum because it is difficult to guess the other variables correctly.
Three sources of information are needed for this type of study:
Figures 1-4 were developed with SigmaPlot Software Scientific Graphing and Statistics Software.
References
[1] Response Surface Designs. NIST Engineering Statistics Handbook, 5.3.3.6, National Institute of Standards and Technology at the U.S. Department of Commerce.
[2] Central Composite Design. NIST Engineering Statistics Handbook, 5.3.3.6.1, National Institute of Standards and Technology at the U.S. Department of Commerce.
Kinetics Development with RSM Part 1: Acetic Anhydride-Methanol Reaction White Paper
It is virtually impossible to optimize kinetic parameters by trial and error when two or more factors are present, so it makes sense to consider an alternative technique. One effective method is Experimental Design, a statistical technique that simultaneously identifies the optimum of all model factors under consideration. An experimental design organizes, conducts, and interprets the results for the best outcome based on the smallest number of trials. Read this white paper by Enio Kumpinsky which employs the exothermic reaction of acetic anhydride with methanol yielding methyl acetate and acetic acid, a known chemical reaction, to provide the background for kinetic development with Response Surface Methodology (RSM).
To read this white paper, visit our website at > https://bit.ly/3ZCGbUE
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