2021 National Higher Education Cup Mathematical Modeling
B. Ethanol Coupling to Prepare C4 Olefins
Reproduce the original title
C4 olefins are widely used in the production of chemical products and medicines, and ethanol is the raw material for the production of C4 olefins. During the preparation process, the catalyst combination (i.e. the combination of Co loading, Co/SiO2 and HAP loading ratio, and ethanol concentration) and temperature will have an impact on the selectivity of C4 olefins and the yield of C4 olefins (see the appendix for terminology) . Therefore, it is of great significance and value to explore the process conditions for the preparation of C4 olefins by catalytic coupling of ethanol through the combination design of catalysts.
A chemical laboratory conducted a series of experiments on different catalysts at different temperatures, and the results are shown in Appendix 1 and Appendix 2. Please complete the following questions through mathematical modeling:
(1) For each catalyst combination in Appendix 1, study the relationship between ethanol conversion rate, C4 olefin selectivity and temperature, and for the given catalyst combination at 350 degrees in Appendix 2 The test results at different times in an experiment were analyzed.
(2) To explore the effects of different catalyst combinations and temperatures on the conversion of ethanol and the selectivity of C4 olefins.
(3) How to choose catalyst combination and temperature to make the yield of C4 olefins as high as possible under the same experimental conditions. If the temperature is lower than 350 degrees, how to choose the catalyst combination and temperature to make the yield of C4 olefins as high as possible.
(4) If it is allowed to add 5 more experiments, how should it be designed and give detailed reasons.
Overview of the overall solution process (abstract)
This paper mainly studies the influence of different catalyst combinations and temperatures on ethanol conversion and C4 olefin selectivity during ethanol coupling to prepare C4 olefins, and the experimental design of the catalyst combination and temperature optimization when the C4 olefin yield is the highest.
Aiming at problem 1, analyze the data law of each catalyst combination in Annex 1, and establish the quadratic regression model of ethanol conversion, C4 olefin selectivity and temperature respectively. Based on the least squares method, use the polyfit() and regress() functions in Matlab to solve the model to obtain the regression coefficient, and then use the discriminant coefficient R^2 to judge whether the model is accurate. If the discriminant coefficient is less than 0.6, it is considered that the model is not good. Accurate, it is necessary to further optimize the model by eliminating data with large deviations from the overall and optimizing the equation model, and then draw a conclusion by analyzing the changing law of the obtained curve: within a certain range as the temperature rises, the conversion rate of ethanol and C4 The selectivity of olefins will increase, but if the temperature is too high, the selectivity of C4 olefins may decrease; comparative analysis of the data in Annex 2, combined with the chemical reaction mechanism to draw a conclusion: ethanol conversion rate with time As the reaction time increases, the selectivity of each product tends to be stable, that is, the chemical reaction reaches an equilibrium state.
For the second question, it is necessary to analyze the influence of different catalyst combinations and temperatures on ethanol conversion and C4 olefin selectivity. First, use the control variable method to classify and compare the data in Annex 1, and obtain corresponding conclusions through qualitative analysis. It was obtained that the Co loading had the greatest effect on the C4 olefin selectivity, and the Co/SiO2 and HAP loading ratio had the least effect on the C4 olefin selectivity. Finally, combined with the actual consideration of the interaction between factors, the multiple linear regression models of C4 olefin selectivity, ethanol conversion rate and each factor and their interaction were established, and the comparison and analysis of the second problem was obtained: Co loading and Co/ The combined action of SiO2 and HAP loading ratio has the greatest impact on ethanol conversion, and temperature has the greatest impact on the selectivity of C4 olefins.
For problem 3, firstly perform data preprocessing on the basis of problem 2, remove a set of data in group A11 with quartz sand and no HAP, and give the functional expression of C4 olefin yield on the basis of problem 2 multiple regression model . Secondly, temperature, the mass sum of Co/SiO2 and HAP, ethanol concentration, and Co loading are used as decision variables, the highest C4 olefin yield is used as the objective function, and the value range of each influencing factor is used as constraint conditions to establish the C4 olefin yield The highest optimized model. Then based on the iterative algorithm, the least squares method was used to fit, and then the Newton iterative method was used to solve the model, and then the selection sorting method was used to obtain: the optimal yield of C4 olefins was 53%. Then change the temperature constraints so that the maximum temperature does not exceed 350 degrees, and the optimal yield of C4 olefins at this time is 12%.
For question 4, combined with the analysis conclusions of the first three questions, it is not difficult to conclude that when the temperature reaches a certain level through the research of the previous questions, and the analysis of the first three questions can clearly conclude that when the temperature is within a certain range, the higher the temperature, the higher the temperature of C4 olefins The lower the conversion rate is, the yield of C4 olefins will drop significantly to a minimum point, and after this range, the conversion rate of C4 olefins increases with the increase of temperature, so the equation model is established here to get the minimum point and its change Therefore, by keeping other conditions constant and changing the temperature, the quadratic function equation model of whether the yield of C4 olefins and the temperature change in a quadratic curve model is obtained.
Model assumptions:
1. Ignore the influence of external factors on the experiment.
2. Ignore the influence of quartz sand and no HAP on the experiment.
3. The ambient temperature and pressure have no effect on the
experiment
. The coefficient R^2>0.6 is considered to be a valid fit
problem analysis:
Analysis of Problem 1.
Problem 1 analyzes the relationship of temperature to ethanol conversion and C4 olefin selectivity respectively. In the first part of question 1, first draw the scatter diagram of ethanol conversion rate, C4 olefin selectivity and temperature for each catalyst combination in Annex 1. Based on this, we preliminarily conclude that ethanol conversion rate generally increases The conclusion is high and increasing, but some values will decrease when they increase to a certain temperature. We need to deal with the abnormal points and then classify them. If the ethanol conversion rate, C4 olefin selectivity and temperature have a certain linear relationship, then establish a mathematical model and use MATLAB to obtain the correlation coefficient.
In the second part of question 1, we analyze the process and products of the chemical reaction based on image fitting, and obtain the change law of ethanol conversion rate, C4 olefin selectivity, and C4 olefin yield over time at 350°, and analyze The amount ratio of products in the reaction process and the equilibrium time of the reaction are shown.
Analysis of the second problem
The second problem is to analyze the influence of different catalyst combinations and temperatures on the conversion of ethanol and the selectivity of C4 olefins. First of all, by controlling variables, we combined and compared 20 groups of test data for intuitive quantitative analysis. Control the independent variable Co loading (weight ratio of Co/SiO2), Co/SiO2 and HAP charge ratio, the sum of Co/SiO2 and HAP mass, the influence of ethanol concentration and temperature on the dependent variables ethanol conversion and C4 olefin selectivity . In the meantime, we need to discover the important factors and secondary factors between different catalyst combinations and temperatures of independent variables, and conduct quantitative analysis on the basis of qualitative analysis to determine the relationship between variables and find out the appropriate mathematical expression.
The analysis of problem 3
is based on the model of problem 2. Firstly, the data is preprocessed to remove the experimental data of group A11 with quartz sand and no HAP. The functional expression of C4 olefin yield is calculated by temperature, Co/SiO2 and HAP The mass sum, ethanol concentration, Co loading, Co/SiO2 and HAP charging ratio are used as decision variables, the highest C4 olefin yield is used as the objective function, and the value range of each influencing factor is used as a constraint condition to establish the C4 olefin yield The highest optimized model.
Analysis of Question 4
Considering questions 1, 2, and 3 comprehensively, for experiments that do not change a certain experimental condition in the appendix, it may make the experimental results better. Through the analysis of the first three questions, it can be clearly concluded that when the temperature is within a range, the higher the temperature C4 On the contrary, the lower the conversion rate of olefins, the yield of C4 olefins will drop significantly to a minimum point, and after this range, the conversion rate of C4 olefins will increase with the increase of temperature. However, for the lack of data, the experiment is not convincing for a certain conclusion.
Model establishment and solution Overall paper thumbnail
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Program code: (code and documentation not free)
%对数据进行导入
%导入附件 1 中的数据
%wendu - 温度 yczhl - 乙醇转化率 wyl - 对温度与乙醇转化率进行拟合后得到的函数
wenduA1=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C2:C6");
yczhlA1=xlsread("C:\Users\ASUS\Desktop\数学建模\题目\B\附件 1.xlsx","性能数据表","D2:D6");
wenduA2=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C7:C11");
yczhlA2=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性能数据表
","D7:D11");
wenduA3=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C12:C18");
yczhlA3=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性能数据表
","D12:D18");
wenduA4=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C19:C24");
yczhlA4=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性能数据表
","D19:D24");
wenduA5=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C25:C30");
yczhlA5=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性能数据表
","D25:D30");
wenduA6=xlsread("C:\Users\ASUS\Desktop\ 数 学 建 模 \ 题 目 \B\ 附 件 1.xlsx"," 性 能 数 据 表
","C31:C35");
%k 为拟合次数
k=3;
wyA1=polyfit(wenduA1,yczhlA1,k); %求出拟合参数
A1=vpa(poly2sym(wyA1),10);%求出曲线 %画出拟合曲线为
y1=polyval(wyA1,wenduA1);
plot(wenduA1,y1,wenduA1,yczhlA1,"*r");
hold on ;
wyA2=polyfit(wenduA2,yczhlA2,k);
A2=vpa(poly2sym(wyA2),10);%曲线为
%画出拟合曲线为
y2=polyval(wyA2,wenduA2);
plot(wenduA2,y2,wenduA2,yczhlA2,"*r");
hold on
wyA3=polyfit(wenduA3,yczhlA3,k);
A3=vpa(poly2sym(wyA3),10);%曲线为
%画出拟合曲线为
y3=polyval(wyA3,wenduA3);
plot(wenduA3,y3,wenduA3,yczhlA3,"*r");
hold on ;
wyA4=polyfit(wenduA4,yczhlA4,k);
A4=vpa(poly2sym(wyA4),10);%曲线为
%画出拟合曲线为
y4=polyval(wyA4,wenduA4);
plot(wenduA4,y4,wenduA4,yczhlA4,"*r");
hold on
wyA5=polyfit(wenduA5,yczhlA5,k);
A5=vpa(poly2sym(wyA5),10);%曲线为
%画出拟合曲线为
y5=polyval(wyA5,wenduA5);
plot(wenduA5,y5,wenduA4,yczhlA5,"*r");
hold on
wyA6=polyfit(wenduA6,yczhlA6,k);
A6=vpa(poly2sym(wyA6),10);%曲线为
%画出拟合曲线为
y6=polyval(wyA6,wenduA6);
plot(wenduA6,y6,wenduA6,yczhlA6,"*r");
hold on
wyA7=polyfit(wenduA7,yczhlA7,k);
A7=vpa(poly2sym(wyA7),10);%曲线为
%画出拟合曲线为
y7=polyval(wyA7,wenduA7);
plot(wenduA7,y7,wenduA7,yczhlA7,"*r");
hold on
wyB7=polyfit(wenduB7,yczhlB7,k);
B7=vpa(poly2sym(wyB7),10);%曲线为
%画出拟合曲线为
y21=polyval(wyB7,wenduB7);
plot(wenduB7,y21,wenduB7,yczhlB7,"*r");
%{
legend("A1=0.3331893326*x - 84.08276818","A2=0.6629580734*x -
161.8909145","A3=0.4195648852*x - 95.88308918",...
"A4=0.5817118467*x - 144.5709001","A5=0.4078621995*x -
97.62301858","A6=0.5015334253*x - 119.8325796",...
"A7=0.3775367296*x - 74.26046445","A8=0.3396341323*x - 83.77607016",
"A9=0.2491040369*x - 65.67280331",...
"A10=0.1834135666*x - 49.64860852","A11=0.2067687555*x -
56.50163086","A12=0.2859331347*x - 74.81353909",...
"A13=0.3359113498*x - 86.68673211","A14=0.3359113498*x -
86.68673211","B1=0.2794712209*x - 73.14252889",...
"B2=0.2726886201*x - 70.9546115","B3=0.1313117907*x -
36.1563065","B4=0.2077582601*x - 56.81340574","B5=0.2715966155*x - 72.27286731",...
"B6=0.3837143453*x - 100.0040718","B7=0.4197142857*x - 109.3428571");
%}
legend('A1','A2','A3','A4','A5','A6','A7','A8','A9','A10','A11','A12','A14','A14','B1','B2','B3','B4','B5','B
6','B7')
xlabel("温度");
ylabel("乙醇转化率");
title("温度-乙醇转化率拟合图")
avgA1=(sum(yczhlA1))/length(yczhlA1);%乙醇转化率平均值
totA1=sum((yczhlA1-avgA1).^2);
resA1= double( sum(( subs(A1,wenduA1)-yczhlA1).^2));
R2A1=1-resA1/totA1;
avgA2=(sum(yczhlA2))/length(yczhlA2);%乙醇转化率平均值
totA2=sum((yczhlA2-avgA2).^2);
resA2= double(sum((subs(A2,wenduA2)-yczhlA2).^2));
R2A2=1-resA2/totA2;