Construction of an optimal model for the preparation of C4 olefins based on multiple regression

. In this paper, based on the data obtained from a series of experiments conducted by a chemical laboratory for different catalysts at different temperatures, the optimization model and regression relationship model are established by using Gibbs free energy law, chemical equation energy analysis, chemical reaction equilibrium, the preparation of C4 alkenes by ethanol coupling was studied. By analyzing the temperature variation law of ethanol conversion and C4 olefin selectivity under different catalyst combinations, a regression relationship model was established, and the functional relationship between the variables was obtained. Then, the effects of different catalyst combinations and temperatures on ethanol conversion and C4 olefin selectivity were discussed. At the same time, in order to establish two optimization models, the first model is the C4 olefin yield model, which takes the minimum C4 olefin yield as the goal, and takes the range of factors such as temperature, and total mass as constraints. The second optimization model is based on the previous one with a constraint that the temperature is below 350°C, and then solved in MATLAB.


Question restatement
Ethanol is a raw material for the production of C4 olefins and is widely used in pharmaceutical production and chemical products. The combination of temperature and catalyst i.e. Co loading, Co/SiO2 and HAP charging ratio, ethanol concentration will affect the C4 olefin selectivity and C4 olefin yield during the preparation. Therefore, it is of great significance to design different combinations of catalysts to explore the process conditions for the catalytic coupling of ethanol to prepare C4 olefins.
For a series of experiments performed by a chemical laboratory on different catalysts at different temperatures, a mathematical model was established to solve the following problems: Question 1: For each catalyst combination in Annex 1, study the relationship between ethanol conversion, C4 olefin selectivity and temperature, and analyze the test results of the given catalyst combination in Annex 2 at 350 degrees at different times in one experiment .
Question 2: Discuss the effects of different catalyst combinations and temperatures on ethanol conversion and C4 olefin selectivity.
Question 3: How should the catalyst combination and temperature be selected to maximize the C4 olefin yield under the same experimental conditions. At temperatures below 350 degrees, how should the catalyst combination and temperature be selected to maximize the C4 olefin yield. Question 4: If 5 more experiments are allowed, how should it be designed and give detailed reasons. 1). The effect of pressure on the reaction process is not considered.

Model assumptions and notation
2). The concentration change after gas dissolution is not considered.
3). Does not consider human measurement error.  (1) where s1 is the ethanol conversion rate and T is the reaction temperature.
(2) Establishment of selectivity and temperature regression models for C4 olefins First, we determine the research object as the product C4 olefin, and determine the variable representing the research object, that is, the explained variable of the model is the selectivity of C4 olefin, and establish the function of the selectivity of C4 olefin as a function of temperature: (2) where s2 is the C4 olefin selectivity and f2 is the reaction temperature.

Relationship Analysis
(1) Establishment of time and ethanol conversion rate regression model For the data given in the title, the conversion rate of the reactant ethanol and the selectivity data of the product C4 olefin and other impurities are measured as a function of time under the same temperature and the same catalyst combination. To analyze the relationship of these test results, we build a time-to-ethanol conversion regression model: (3) where y1 is the ethanol conversion rate and t is the reaction time.
(2) Establishment of time and C4 olefin yield regression model According to the data, it can be obtained that different reaction conditions have an important impact on the yield of C4 olefins, and it is known that the yield of C4 olefins is the product of ethanol conversion and C4 olefin selectivity. In this experiment, the combination of temperature and catalyst was controlled unchanged, only the reaction time was changed, so the relationship model between time and the yield of C4 olefins was established: (4) where h is the ethanol conversion rate, and t is the reaction time. According to the calculation formula of C4 olefin yield, it can be obtained: (5) where y1 is the ethanol conversion and y2 is the C4 olefin selectivity. In summary, the time-C4 olefin yield regression model is established as follows:

Solution of the model
(1) Considerations for ethanol conversion Since the overall conversion rate can better reflect the relationship between the conversion rate and industrial conditions (catalysts, reactors, and reaction conditions), it is beneficial to analyze chemical data to optimize industrial conditions, thereby reducing economic costs. Therefore, this model adopts the one-way conversion rate and the conversion rate per unit time. The calculation formula is as follows: (7) where is the change in concentration per unit time, is unit time, is concentration, and is reaction rate. (2) The relationship between ethanol conversion and temperature Considering that the conversion rate of ethanol is the conversion rate per unit time, that is, the reaction rate per unit time determines the conversion rate per unit time (the amount of ethanol intake is constant and does not determine the functional relationship), the external factors affecting the reaction rate of this model are mainly external factors There are catalyst performance, reactant concentration, temperature (ignoring the consideration of pressure, because ethanol has less influence on pressure due to the amount of gas reduced by the reaction, so the degree of influence of pressure on the reaction rate per unit time is compared to temperature, reactant concentration. , the catalyst performance is significantly smaller.) Highlights in Science, Engineering and Technology

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Volume 2 (2022) 292 At the same time, the temperature will affect the activity of the catalyst and the properties of the substance itself, the reaction rate, and whether it can still meet the temperature conditions for the reaction to occur. From the reaction rate formula: (8) Where c is the concentration,t is the reaction time,k(T) is the reaction rate constant and a function related to temperature, A and B refers to the reactants.
(3) Regression of ethanol conversion rate and temperature by chemical kinetics empirical equation According to the analysis in 2, in order to combine practice and theory, the empirical equation of chemical kinetics is established from empirical equation (8), the kinetic equation is regressed from the data of ethanol conversion rate and temperature ( ), and the 21 combinations are made by R language. Point diagram, according to the trend of the scatter diagram to judge whether it is a linear or nonlinear regression analysis, and get the regression equation. And the scatter points are fitted into a function image, and the functional relationship between the ethanol conversion rate and temperature of each combination is obtained, see the appendix.
(4) Test the regression equation using significance tests and goodness-of-fit criteria In the research of practical problems, we cannot conclude in advance that there is a linear (nonlinear) relationship between ethanol conversion rate and temperature. Through the significance test definition and experimental data, the functional relationship of each catalyst combination is obtained ( ), and the fitting degree is good [4].  At the same time, it is found that the temperature has a positive effect on the ethanol conversion rate. The higher the temperature, the higher the ethanol conversion rate, although the growth rate of different catalyst combinations is different. According to the image extrapolation, it is speculated that different catalyst combinations have different initial reactions. The lower the starting reaction temperature, the earlier the temperature at which the reaction is started is reached, and the earlier the reaction starts.
(6) Ethanol conversion versus time Because this model is designed for the catalyst, the ethanol conversion rate should first increase and then decrease as the reaction starts and ends. Because the ethanol conversion rate is the highest when the catalyst activity is the highest, the unit reaction rate is the largest, so the ethanol conversion rate also reaches the maximum, and then the unit reaction rate As the rate decreases, the ethanol conversion rate (unit time) decreases. It can be seen from the figure that the ethanol conversion rate Highlights in Science, Engineering and Technology

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Volume 2 (2022) has been in a declining state, which is the part of the curve after reaching the highest catalyst activity. From the downward trend of the curve, it can be found that the decreasing speed is getting slower and slower and tends to a stable value.After equilibrium, the equilibrium conversion rate was 29.9%.  (1) Determination of the objective function It can be seen from the title that in the process of preparing C4 olefins, the influence of temperature, the combination of catalysts and the separation of Co loading, Co/SiO2 and HAP charging ratio, and ethanol concentration on the selectivity and yield of C4 olefins [8]. The population is subdivided into five variables, namely "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", Highlights in Science, Engineering and Technology

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Volume 2 (2022) 294 "Co/SiO2 total mass". Taking the previous five variables as independent variables, the conversion rate of ethanol is the dependent variable, and the abstract functional relationship that the conversion rate of ethanol is changed due to the five independent variables is expressed. The formula is as follows: (9) Among them, Z1 is the ethanol conversion rate, x1 is the temperature, x2 is HAP:co/SiO2, x3 is Co/SiO2, x4 is the ethanol concentration, and x5 is Co/SiO2 total mass.
In the same way, taking the five variables of "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", "Co/SiO2 total mass" as independent variables, and taking the selectivity of C4 olefins as the dependent variable, it shows that the selectivity of C4 olefins is caused by these five variables. The abstract function relation expression that changes with the variable is as follows: (10) Therefore, a general functional relation expression can be established to express the above relation, as follows: (11) Considering the error between the estimated value and the true value, by the least square's method, we define a criterion that the sum of squares of the difference between the estimated value and the true value is the smallest, thus transforming the problem into an optimization problem, the formula is as follows: (12) (2) Limitation of reaction temperature First, the temperature determines whether the reaction can occur and continue in actual production. Second, the reaction temperature affects the reaction rate and catalyst performance, thereby affecting the ethanol conversion and C4 olefin selectivity. Considering the activity of the catalyst, there is a temperature range. At the same time, the reaction temperature cannot be increased indefinitely. At the same time, if the temperature is too high, the catalyst will be deactivated or the reaction will stop, and the reaction has a minimum reaction temperature limit. According to the data, the iron-chromium catalyst is a brown cylinder or Plate-like solid particles, the active temperature is 350 ~ 550 ℃, so the range can be set as follows according to the experimental data: (13) (3) Limitation of Co/SiO2 and HAP charging ratio Because the charging ratio should be selected within a suitable range. On the one hand, when there is too much Co/SiO2, the catalyst will be wasted; on the other hand, HAP as a carrier can increase the activity of the catalyst, but the activity of the catalyst will also increase to a certain upper limit and remain unchanged, so the range can be set as follows according to the experimental data: (14) (4) Proportional restrictions of Co/SiO2 According to the experimental data, the range of the ratio is as follows: (15) (5) Limitation of ethanol concentration The concentration of the reactant will affect the rate of the chemical reaction. If the concentration of the reactant is too low, it will not have a significant positive effect on the reaction, and if the concentration of the reactant is too high, it will inhibit the progress of the reaction. Therefore, ethanol can be set according to the experimental data. The concentration ranges are as follows: (16) (6) Total mass limit of Co/SiO2 Highlights in Science, Engineering and Technology

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Volume 2 (2022) 295 The reduction of the activation energy of the catalyst in the reaction has a certain limit, so the range of the total mass can be set as follows according to the experimental data: (17) In summary, the optimization model for problem 2 is established as follows: (18)

Solution of the model
(1) Algorithm steps Step1: Integrate data into tables and position them for easy calculations Step2: Enter the required data in rows and columns:xi,yi,i=1,2,3,4,5,j=1,2 Step3: Set the parameters by means of multivariate nonlinear polynomial fitting Step4: Bring in data to calculate the value of each parameter Step5: Judging the residual r based on the fitted value and the true value Step6: Select the appropriate parameters to calculate the fitting equation, if the residual is not within the reference range, go to Step3 Step7: Based on the fitted equations, analyze (different catalyst combinations and temperatures) and (ethanol conversion and olefin selectivity) Step8: Output result (2) Calculation results and analysis   Using the model established above, the set value in the question is brought into R and MATLAB to solve [7], and the functional relationship expression of the ethanol conversion rate with the five factors of "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", "Co/SiO2 total mass" and the C4 olefins are obtained. The selectivity of C4 olefins varies with "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", "Co/SiO2 total mass" and the functional relationship expression of the five factors of ethanol concentration and total mass.
The regression results of different catalyst combinations, temperature and ethanol conversion were obtained by using multiple linear stepwise regression (18), R2 is 0.791, which means 79.1% of the changes in the five variables of "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", "Co/SiO2 total mass". And through the F test (F=103.422, p=0.000<0.05), this model is valid. (19) Similarly, the regression results of different catalyst combinations and temperature and C4 olefin selectivity are as follows: (20)

Table 3. AIC and BIC guidelines
The AIC and BIC criteria are two important criteria for selecting models in multiple regression. In multiple regression analysis, in order to prevent problems such as overfitting (not only to make the model more explanatory, but also to have a little tension), Akaike (1978) and Schwarz (1978) proposed AIC and BIC respectively as regression models. standard. In regression models, both values are as small as possible.

Establishment of model without temperature constraint
(1) Determination of the objective function Because we need to determine the different catalyst combinations and temperatures that maximize the yield of C4 olefins under the same experimental conditions, that is, the goal is to maximize the yield of C4 olefins, and accordingly, the objective function can be obtained as follows: (22) Where Y is the yield of C4 olefins, y1 is the ethanol conversion rate, and y2 is the selectivity of C4 olefins.
(2) Determination of Constraints After analysis, the objective function will be constrained by the range of five factors: "temperature", "HAP:co/SiO2", "Co/SiO2", "Alcohol concentration", "Co/SiO2 total mass". This constraint is the same as the constraint in model two. (23) In summary, the optimization model for problem 3 is established as follows: (24)

Establishment of model with temperature constraint
(1) Determination of the objective function When the temperature is lower than 350 degrees, in order to determine the different catalyst combinations and temperatures when the yield of C4 olefins is the largest, the target is set as the maximum yield of C4 olefins [9], and the objective function is determined as: (25) (2) Determination of Constraints According to the description of the objective function, the difference between this question and the first model is that if the temperature is lower than 350 degrees, it is only necessary to add a constraint on the basis of the previous model: (26) In summary, the optimization model for problem 3 is established as follows:

) Calculation results and analysis
The data in Appendix is processed to obtain the yield of C4 alkene, and the chemical empirical equation is used to establish the multiple regression between the yield of C4 alkene and the combination of catalyst and temperature, and the regression equation function of the yield of C4 alkene is obtained, using matlab Solve the solution of the first model: the temperature is 400 ℃, the ratio of 1 wt%Co/SiO2-HAP to HAP is 1:1, and the yield is the largest; the solution of the second model: 350 ℃, -The ratio of 1 wt%Co/SiO2-HAP to HAP is 1:1.

Conclusion
Based on the results of a series of experiments on different catalysts at different temperatures in a chemical laboratory, the process conditions for the preparation of C4 alkene by ethanol catalytic coupling were explored. After the above analysis, the main conclusions are as follows: ethanol conversion rate and C4 alkene conversion rate Both are positively correlated with temperature, and different catalyst combinations will affect the growth slope and initial value. The charging method has little effect on the experimental results and will not cause too much deviation in the experimental results.The charging ratio has little effect on the ethanol conversion rate, and the ethanol concentration has little effect on the C4 olefin selectivity. The increase of temperature has a great influence on the ethanol conversion rate and C4 olefin selectivity, and the ethanol concentration will reduce the ethanol conversion rate.