Reactor Modeling Tools – Multiple Regressions

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Reactor Modeling Tools – Multiple Regressions

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Process Engineering Guide: Reactor Modeling Tools – Multiple Regressions

CONTENTS 0 INTRODUCTION 1 SCOPE 2 THEORY 3 EXCEL 2007: MULTIPLE REGRESSIONS

3.1 Overview 3.2 Multiple Regression Using the Data Analysis ADD-IN 3.3 Interpret Regression Statistics Table 3.4 Interpret ANOVA Table 3.5 Interpret Regression Coefficients Table 3.6 Confidence Intervals for Slope Coefficients 3.7 Test Hypothesis of Zero Slope Coefficients ("Test of Statistical

Significance") 3.8 Test Hypothesis on a Regression Parameter

3.8.1 Using the p-value approach 3.8.2 Using the critical value approach 3.9 Overall Test of Significance of the Regression Parameters 3.10 Predicted Value of Y Given Regressors 3.11 Excel Limitations

4 SPECIAL FEATURES REQUIRING MORE SOPHISTICATED

TECHNIQUES 5 USER INFORMATION SUPPLIED A SUBROUTINE

B DATA C RESULTS

6 EXAMPLE



0 INTRODUCTION Excel Multiple Regression can be used for parameter fitting for algebraic and ordinary differential equation models. It can be tailored for finding the values of rate constants which give the best fit of measured data to rate expressions. It carries out a statistical analysis of the results. 1 SCOPE This guide summarizes the application of Excel Multiple Regressions; using the Data Analysis Add-in as a fitting program for rate expression data. 2 THEORY Reaction rate expressions are usually functions of concentrations, temperature and pressure. For a homogeneous reaction, one might have:

For a homogeneous gas phase reaction, a similar type of equation might apply but one might have:



In the case of homogeneous catalysis, the catalyst concentration would be incorporated in the rate expression. Heterogeneous catalysis also gives rise to equations which incorporate the catalyst concentration, e.g.:

It is common to express the rates of this type of reaction as e.g.:



Whatever the form of the rate expressions, they contain “rate constants” i.e. the k, K1, K3 in the above rate expressions. These are themselves usually correlated with temperature by the Arrhenius equation:



Values of k, or k0 and E for the various reactions taking place are required before reactor analysis or design is possible by means of a mathematical model. These are analogous to heat or mass transfer coefficients in the analysis or design of a heat exchanger or gas absorber. In the case of mass and heat transfer, reasonable a priori calculation of coefficients can be made from mature correlations. Unfortunately there is no similar method for calculation of reaction rate constant. These have to be determined from measurements of the reaction rates, and the "fitting" of the k, K1, K2, K3, k0, E to the measurements of r, C1, C2, f1, f2 etc.. Hopefully, the data will be available from a reactor, either in the laboratory or on the works, which approximates acceptably to one of the ideal types. Data from a batch reaction will be in the form of reactant and product concentrations, and temperature (and pressure if necessary) at a number of points in time through the batch. Fitting of rate constants using a computer will require numerical integration of rate expressions devised for the particular reactions under study. Data from an ideal plug flow reactor will be in the form of reactant and product concentrations temperature (and pressure) at various residence times. These data are treated similarly to the batch data remembering that (for constant density):



It would then be feasible to fit the rate constants to r, using the exit concentrations and temperature in the rate expressions. If the reactors do not conform to one of the ideal types, then the model is complicated by having to take residence time distribution into account. The principle of fitting rate constants to a model of the reactor remains the same. The fitting process is an optimization, with the rate constants thought of as the variables and the objective function being some function of the difference between measured data and the corresponding data calculated from the reactor model incorporating "rate constant variables". An example would be to find the values of the rate constants which minimized:

In principle, any numerical optimizer could be tried. However, Excel Multiple Regressions using the Data Analysis Add-in as a fitting program, provides a reasonably flexible package which can be tailored specifically for your application. 3 EXCEL 2007: MULTIPLE REGRESSIONS 3.1 Overview

Multiple regressions using the Data Analysis Add-in.

Interpreting the regression statistic.

Interpreting the ANOVA table (often this is skipped)

Interpreting the regression coefficients table

Confidence intervals for the slope parameters

Testing for statistical significance of coefficients

Testing hypothesis on a slope parameter



Testing overall significance of the regressors

Predicting y given values of regressors

Excel limitations

There is little extra to know beyond regression with one explanatory variable. The main addition is the F-test for overall fit. 3.2 Multiple Regression Using the Data Analysis ADD-IN This requires the Data Analysis Add-in: see Excel 2007: Access and Activating the Data Analysis Add-in. Note1: The only change over one-variable regression is to include more than one column in the Input X Range. Note2: The regressors need to be in contiguous columns. If this is not the case in the original data, then columns need to be copied to get the regressors in contiguous columns.



The regression output has three components:

o Regression statistics table

o ANOVA table

o Regression coefficients table 3.3 Interpret Regression Statistics Table This is the following output. Of greatest interest is R Square.

Explanation Multiple R 0.895828 R = square root of R2 R Square 0.802508 R2 Adjusted R Square 0.605016 Adjusted R2 used if more than one x variable

Standard Error 0.444401 This is the sample estimate of the standard deviation of the error u

Observations 5 Number of observations used in the regression (n) The above gives the overall goodness-of-fit measures: R2 = 0.8025 Correlation between y and y-hat is 0.8958 (when squared gives 0.8025).

Adjusted R2 = R2 - (1-R2 )*(k-1)/(n-k) = .8025 - .1975*2/2 = 0.6050. The standard error here refers to the estimated standard deviation of the error term u. It is sometimes called the standard error of the regression. It equals sqrt(SSE/(n-k)). It is not to be confused with the standard error of y itself (from descriptive statistics) or with the standard errors of the regression coefficients given below.



R2 = 0.8025 means that 80.25% of the variation of yi around ybar (its mean) is explained by the regressors x2i and x3i. 3.4 Interpret ANOVA Table An ANOVA table is given. This is often skipped.

df SS MS F Significance F Regression 2 1.6050 0.8025 4.0635 0.1975 Residual 2 0.3950 0.1975 Total 4 2.0

The ANOVA (analysis of variance) table splits the sum of squares into its components. Total sums of squares

= Residual (or error) sum of squares + Regression (or explained)

sum of squares. Thus Σ i (yi – yo)2 = Σ i (yi – y1i )2 + Σ i (y1i – y0)2 where y1i is the value of yi predicted from the regression line and y0 is the

sample mean of y. For example: R2 = 1 - Residual SS / Total SS (general formula for R2)

= 1 - 0.3950 / 1.6050 (from data in the ANOVA table) = 0.8025 (which equals R2 given in the

regression Statistics table). The column labeled F gives the overall F-test of H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. Aside: Excel computes F this as: F = [Regression SS/(k-1)] / [Residual SS/(n-k)] = [1.6050/2] / [.39498/2] = 4.0635.



The column labeled significance F has the associated P-value. Since 0.1975 > 0.05, we do not reject H0 at significance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors including the intercept. Here FINV(4.0635,2,2) = 0.1975. 3.5 Interpret Regression Coefficients Table The regression output of most interest is the following table of coefficients and associated output

Coefficient St. error t Stat P-value Lower 95% Upper 95% Intercept 0.89655 0.76440 1.1729 0.3616 -2.3924 4.1855 R1 0.33647 0.42270 0.7960 0.5095 -1.4823 2.1552 R2 0.00209 0.01311 0.1594 0.8880 -0.0543 0.0585 Let βj denote the population coefficient of the jth regressor (intercept, R1 and R2). Then

Column "Coefficient" gives the least squares estimates of βj.

Column "Standard error" gives the standard errors (i.e. the estimated standard deviation) of the least squares estimates bj of βj.

Column "t Stat" gives the computed t-statistic for H0: βj = 0 against Ha: βj ≠ 0.

This is the coefficient divided by the standard error. It is compared to a t with (n-k) degrees of freedom where here n = 5 and k = 3.

Column "P-value" gives the p-value for test of H0: βj = 0 against Ha: βj ≠ 0..



This equals the Pr{|t| > t-Stat}where t is a t-distributed random variable with n-k degrees of freedom and t-Stat is the computed value of the t-statistic given in the previous column. Note that this p-value is for a two-sided test. For a one-sided test divide this p-value by 2 (also checking the sign of the t-Stat).

Columns "Lower 95%" and "Upper 95%" values define a 95% confidence interval for βj.

A simple summary of the above output is that the fitted line is y = 0.8966 + 0.3365*x + 0.0021*z 3.6 Confidence Intervals for Slope Coefficients 95% confidence interval for slope coefficient β2 is from Excel output (-1.4823, 2.1552). Excel computes this as b2 ± t_.025(3) × se(b2) = 0.33647 ± TINV(0.05, 2) × 0.42270 = 0.33647 ± 4.303 × 0.42270 = 0.33647 ± 1.8189 = (-1.4823, 2.1552) Other confidence intervals can be obtained. For example, to find 99% confidence intervals: in the Regression dialog box (in the Data Analysis Add-in), check the Confidence Level box and set the level to 99%.



3.7 Test Hypothesis of Zero Slope Coefficients ("Test of Statistical Significance")

The coefficient of R1 has estimated standard error of 0.4227, t-statistic of 0.7960 and p-value of 0.5095. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. The coefficient of R2 has estimated standard error of 0.0131, t-statistic of 0.1594 and p-value of 0.8880. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. There are 5 observations and 3 regressors (intercept and x) so we use t(5-3)=t(2). For example, for R1 p = =TDIST(0.796,2,2) = 0.5095. 3.8 Test Hypothesis on a Regression Parameter Here we test whether R1 has coefficient β2 = 1.0. Example: H0: β2 = 1.0 against Ha: β2 ≠ 1.0 at significance level α = .05. Then t = (b2 - H0 value of β2) / (standard error of b2 ) = (0.33647 - 1.0) / 0.42270 = -1.569. 3.8.1 Using the p-value approach

p-value = TDIST(1.569, 2, 2) = 0.257. [Here n=5 and k=3 so n-k=2].

Do not reject the null hypothesis at level .05 since the p-value is > 0.05.



3.8.2 Using the critical value approach

We computed t = -1.569

The critical value is t_.025(2) = TINV(0.05,2) = 4.303. [Here n=5 and k=3 so n-k=2].

So do not reject null hypothesis at level .05 since t = |-1.569| < 4.303.

3.9 Overall Test of Significance of the Regression Parameters

We test H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. From the ANOVA table the F-test statistic is 4.0635 with p-value of 0.1975. Since the p-value is not less than 0.05 we do not reject the null hypothesis that the regression parameters are zero at significance level 0.05. Conclude that the parameters are jointly statistically insignificant at significance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors including the intercept. Here FINV(4.0635,2,2) = 0.1975. 3.10 Predicted Value of Y Given Regressors

Consider case where x = 4 in which case R2 = x^3 = 4^3 = 64. yR1 = b1 + b2 x2 + b3 x3 = 0.88966 + 0.3365×4 + 0.0021×64 = 2.37006



3.11 Excel Limitations Excel restricts the number of regressors (only up to 16 regressors ??). Excel requires that all the regressor variables be in adjoining columns. You may need to move columns to ensure this. E.g. If the regressors are in columns B and D you need to copy at least one of columns B and D so that they are adjacent to each other. Excel standard errors and t-statistics and p-values are based on the assumption that the error is independent with constant variance (Homoskedastic). Excel does not provide alternatives, such as Heteroskedastic-robust or autocorrelation-robust standard errors and t-statistics and p-values. More specialized software such as STATA, EVIEWS, SAS, LIMDEP, PC-TSP, is needed. 4 SPECIAL FEATURES REQUIRING MORE SOPHISTICATED

TECHNIQUES The program has many features which offer facilities in special cases, which the ordinary user is unlikely to need. See worked example. (a) The basic application of Excel is to a single model, where all data values

are known at all measurement stages. Residuals at different measurement stages have unknown variances and are assumed to be independent. However, departures from this basic application can be accommodated (see next points).

(b) If the residuals have known variances and/or covariance’s, this information

can be taken into account. (c) It may be that in a multi-response experiment, all the responses are not

measured at the same time. This can be accommodated. (d) Single response experiments, where different series of data show different

experimental errors (e.g. due to an improved measurement technique being used part way through the experimental program) can be treated.

These features are controlled by "measurement pattern" parameters.



(e) It is not necessary that experimental data in the form of concentrations

should be supplied. A "measurable response" can be used. For instance, because of limitations in the analytical technique, it may be possible to measure only a sum of concentrations, or some function of them; or it may be possible only to measure a temperature profile. Provided it is possible to calculate this "response" from the calculated dependent variables for comparison with the 'measured response", these sort of data can be treated.

(f) It may be that certain information is known about the parameters which

can be usefully incorporated into the fitting routine. For instance, one may know a parameter's mean estimate or variance (e.g. from the literature). Again, certain parameters may be known to be normally distributed with known mean values or known covariance matrix (e.g. due to previous application of Excel to the same problem but with different data values).



5 USER INFORMATION SUPPLIED (a) Subroutines If the reactor model is described by algebraic equations, two subroutines are required: MODEL Given independent variables, constants (X), parameters (PAR),

calculate the model dependent variables (Y) (e.g. the concentrations). This is straight forward if the equations are explicit. If the equations are implicit, a solving routine has to be supplied by the user.

RESP Given independent and dependent model variables (Y) (e.g. the

concentrations calculated by MODEL), constants (X) and parameters (PAR) calculate the responses (W) for comparison with experimental responses.

If the reactor model is described by differential equations, three subroutines are required.

DMODEL Given independent variables, constants (X), parameters (PAR),

calculate derivatives (DY). RESP Given independent and dependent model variables (Y) (e.g. the

concentrations calculated by DMODEL), constants (X) and parameters (PAR), calculate the responses (W) for comparison with experimental responses.

DIFIN Given independent variables, constants, parameters, specify initial

values of dependent variables (YO).

If the differential equations are stiff, and a stiff equation solver is selected, a fourth subroutine is required.

DIFJAC Given variables, constants, parameters, calculate analytical

expressions for the Jacobian.



(b) Data 1 PROBLEM IDENTIFICATION User selected characters. 2 NO IND VAR Length of array X. 3 PAR INFO Number of parameters to be

estimated, and some control information.

4 RESPONSES Number of dependent variables

(Y) (E.g. concentrations) and measurable responses (W).

5 MODEL INFO Algebraic or differential equation

model flag, calculation control settings. 6 ERROR DIST Option to take advantage of any

prior knowledge of error distribution. 7 PATTERNS Number of different error patterns

in response. 8 MAX ITER Maximum iterations. 9 & 10 PAR Parameter initial estimates,

termination criteria and bounds. 11 PRIOR IND Flag indicating user will

supply information on means (and possibly covariance’s) or parameters.

12 PRIOR MEAN Values of mean estimates

of parameters. 13 PRIOR COV Values of covariance’s of parameters. 14 COV Known covariance’s (only supplied

for certain choices of ERROR DIST and PATTERNS).



15 PATTERN Definition of measurement patterns (only supplied for certain choices of ERROR LIST and PATTERNS).

16 RUN 1 Number of experiments. 17 IND VAR The X array for experiment. 18 EXP 1.01 Measurement pattern, time

variable, and then the measured responses (W) at this time. Repeat 18 as necessary.

Repeat from 16 until number of experiments is satisfied.

(c) Results The program output consists of three large blocks. (a) Input data; (b) Summary of the iteration sequence of the parameter search; (c) Optional parameters estimates and their statistical analysis. Comparison

of measured and calculated response.

Analysis of residuals.



6 EXAMPLE Experimental "Plug flow" reactor with fixed feed composition, varying feed rate and isothermal temperature. Sampling at reactor exit plus four intermediate points. Each reactor element contained 0.78gm catalyst. Experimental result was the % conversion of reactant.

Reactor Modeling Tools – Multiple Regressions

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