78
Chapter 3
whereas the calculated value of TM
is close to the measured value of 0.137 the calculated value of
Yn
is woefully different from the experimental value of 0.077. One can certainly determine the rate
of glycerol formation with an experimental error, which is smaller than 10% relative.
Experimentally measured rates beyond the TV
+ A/ -3 rates necessary to determine the
stoichiometric coefficients of (3.30) can be used either to validate the stoichiometry by a qualitative
assessment of the difference between measured and calculated values of the four remaining
stoichiometric coefficients - as was done in Examples 3.5 and 3.6 - or they may be used, partly or
as a whole to obtain better values for all the stoichiometric coefficients in a so called reconciliation
procedure. This is done by least squares fitting of the coefficients. Thus, if
L
measurements where
N+M-3 <L
-
N+M+
1 are used the remaining rates can be calculated as follows:
Ec
r (Emqm
+ Eeqe)= E ^E cqc + E X q m
= 0
(3.36)
Here Equation (3.33) is multiplied from the left by E^, i.e. by the matrix Ec where rows and
columns are transposed. When
L > N + M - 3 the
matrix Ec is not quadratic but contains 4 rows
and less than 4 columns. But the product of E
l
and Ec is a quadratic matrix of order less than 4 but
larger than or equal to 0:
[(/V+A/+ 1
-Z,) x 4] x [4 x
(N + M -\-L)] = [N + M+
1
-L]
x
[N + M+
1
-L]
Consequently
q e = - f c X n E X ] q
(3.37)
The requirement for application of Eq. (3.37) is that E^ E c has full rank, i.e., rank(E
l
E c) =
N
+
M
+ 1
-L,
ordet(EcEc)*0. By inserting Eq. (3.37) in Eq. (3.33), we get
R q m
=0
(3.38)
where
R = E m
- EC
(E IE c y1
E
l
E „
(3.39)
The matrix R is called the
redundancy matrix
(Heijden
et al,
1994a,b), and its rank specifies the
number of independent equations in Eq. (3.38). Thus, if there are / elemental balances the
redundancy matrix contains
I
- rank(R) dependent rows, and if these rows are deleted we obtain
rank(R) independent equations relating the
L
measured rates, i.e.,
R , q m= 0
(3.40)