Reactor Modeling Tools – Multiple Regressions

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Process Engineering Guide:
GBHE-PEG-RXT-821

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
3.2
3.3
3.4
3.5
3.6
3.7
3.8

3.9
3.10
3.11

Overview
Multiple Regression Using the Data Analysis ADD-IN
Interpret Regression Statistics Table
Interpret ANOVA Table
Interpret Regression Coefficients Table
Confidence Intervals for Slope Coefficients
Test Hypothesis of Zero Slope Coefficients ("Test of Statistical
Significance")
Test Hypothesis on a Regression Parameter
3.8.1 Using the p-value approach
3.8.2 Using the critical value approach
Overall Test of Significance of the Regression Parameters
Predicted Value of Y Given Regressors
Excel Limitations

4

SPECIAL FEATURES REQUIRING MORE SOPHISTICATED
TECHNIQUES

5

USER INFORMATION SUPPLIED
A
B
C

6

SUBROUTINE
DATA
RESULTS

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/(nk)).
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
y1i is the value of yi predicted from the regression line and y0 is the
sample mean of y.

where

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 (nk) 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 pvalue 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(53)=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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