NOTES
Improving Linear Relations to Obtain Kinetic Parameters Dominic M. Di Toro Environmental Engineering & Science Program, Manhattan College, Bronx, N.Y. 10471
or w An iterative method is presented which corrects the truncation error inherent in the linearized approximations commonly used in obtaining reaction kinetic parameters. This correction extends the applicability of these convenient methods.
U
se of graphical techniques t o analyze kinetic data is an established and useful part of the methods that are routinely applied to water quality data. Two examples are the Thomas slope method (Thomas, 1950) for obtaining the first-order reaction coefficient, KI, and ultimate BOD concentration, Lo,from data obtained at times t,, i = 1, . , N , and assumed to follow the reaction model L(t) = L,(1
-rKIt)
(1) and a method suggested by Pessen (1961) t o obtain both the order, n, and the reaction coefficient, K,, from data which is assumed to foliow the reaction model dc dt
=
Since the coefficient of ,f2 in this expansion is not small, this expansion does not lead to a good approximation directly. What is required is t / f expanded in powers of t. This can be accomplished using Equation 10 and an expansion for f i n terms o f f . To obtain the latter expansion, the technique of reversion of series (Abramowitz and Stegun, 1964) is used. This is, if
+ a2x2+ u 3 x 3+
y = a,x
,
..
(11)
then
x
1
= -
a2
y - a,3 y z f
(2
01
- ala3) a15
+.
Thus Equation 9 is inverted t o give: n 2
f ' = at - -(af)2
n - 1) + n(2 ____ 6
(af)3
+
,
.
.
And substituting this equation into Equation 10 gives the required expansion:
-Kncn
or, in integrated form: c(t)
=
+ (n - l ) t K , ~ , ~ - ~ ] ~ ' ~ - ~
c(o)[l
(3)
where c(o) is assumed known. Both of these techniques rely on approximately linear relations that exist between expressions involving time and the measured concentrations vs. time. The Thomas slope method depends on the approximate equation and the nth-order method is based on the relationship t l n -=-+-t f a 2 where
and cy
=
Kn~(~)n-l
(7)
Since a derivation of this relationship is not readily available, one is presented below. The linearization of the nthorder reaction solution is based on the expansion of the solution in an appropriate way. For both n = 0 and n = 2 the linear relation given by Equation 5 holds exactly. Thus it is reasonable to investigate whether t,y vs. t is approximately linear for other reaction orders. Toward this end the solution for a n nth-order reaction is: (1
- f)'-"
Expanding the binomial (1
- 1 = at@ - 1)
(8)
the first two terms of which are an approximation of the exact solution. These linear relations are the basis for the simple graphical methods since plots of ( f / L ) l / 'and ( [ i f ) vs. time are approximately linear with respect t o time and the slopes and intercepts of the fitted straight lines are used to calculate the unknown kinetic parameters. A difficulty with this procedure stems from the approximate nature of the equations used. These equations are the first two terms of exact infinite series expansions and the resulting truncation error makes them approximate. It is the purpose of this note t o suggest a method of iteratively reducing the truncation error to make these methods asymptotically exact. Estimating Truncation Error For the sake of being explicit, the nth-order method will be used in this exposition; a similar analysis applies to the Thomas method. The expansion that relates t/f to f can be written as t
-
f
1114 Environmental Science & Technology
af
l
-
a
+ n2 t + R(n, a , f) -
(15)
where R(n, a , 1) is the truncation error which is a function of n and a , the unknowns. However, a first estimate of these unknowns, nl and cy1, can be obtained by using the approximate linear relation and ignoring the truncation error R. Thus nl and a l , can be obtained by fitting a straight line to f , i f , calculated from the dataf,, vs. f * . Thus fitting
- f)'-" and simplifying yields : =
=
(16 )
(9)
gives nl and
cyl.
Using these estimates it is possible to estimate the truncation error by simply subtracting the exact value of t / f evaluated using the known solution f ( n , cy, t )
=
1 - [I
+ (n - l ) ~ ~ t ] ~ ’ l - ~
Table I. Kinetic Data for n
0.5
and the approximation, Equation 16. Thus
1. o
is the truncation error for nl, CYI at ti. This truncation error can be removed and new estimates, n2, CY^ are obtained by fitting a straight line to the improved equation:
1.5 2.0 2.5 3 .O 3.5 4.0
1.5, cy
=
1.0 R(1.5, 1.0,
fi
tilfi
ti)
0.640 0.444 0,327 0.250 0.198 0.160 0.132 0.111
1.388 1.800 2.227 2.667 3,115 3,571 4.033 4,500
0.013 0.050 0,102 0.167 0.240 0.321 0.408 0.500
ti
(17)
=
Table 11. Iterations of the Algorithm With the improved estimates n2, cy2 the process can be repeated by recalculating the truncation error R(nZ,c y 2 , ti) using Equation 18, fitting the data using Equation 19, and so on. As shown in the accompanying example the process appears to converge in a few iterations. The result is exact in the sense that the method converges to the correct order and reaction coefficient if the data used are exact. The method also appears to work well with actual data although the effects of measurement errors can be troublesome if the errors are large and the data scattered. For such cases it is questionable whether kinetic models that result from any fitting technique such as direct least-square error minimization (Ravimohan, 1969) have much utility. In any case the statistical analysis of the resulting kinetic parameters becomes an important adjunct to the fitting technique. Example Synthetic data, f i , for a reaction order of n = 1.5 and for which CY = 1.0, calculated for t i = 0.5, 1.0, 1.5, . . ., 4.0 is shown in Table I together with t J f t and the exact truncation error R(1.5, 1.0, ri). The iteration procedure described above is applied to this data. The straight line fit is accomplished using a least-mean-square error criteria. The resulting order and reaction rate at each iteration is shown in Table 11. Convergence is obtained after 19 iterations. Algorithm The algorithm used in the above example is as follows: For the data, the concentration fraction remaining at each observation time f , , t i , i = 1, . . , ,N (1) calculate the ordinate values dt
=
ti/fi
i
=
1, . . ., N
(20)
(2) least-mean-square error estimate of the slope to calculate order
(3) least-mean-square error estimate of the intercept to calculate reaction coefficient cy
=
N / ( E1 d i -
Eti)
2 i
(4) exact value off at f a for n and CY calculated above f ( n , CY, t J = 1
- [I
+ (n - l ) ~ ~ t ~ ] ~ ”1,.- ~ i, N
(5) truncation error at t , for n and
=
CY
calculated above
(23)
Iteration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
n
cy
1.78218 1,65589 1.58484 1.54575 1.52452 1.51310 1 ,50699 1.50372 1.50198 1.50106 1.50056 1.50030 1,50010 1.50008 1 ,50005 1.50002 1,50001 1 ,50001 1.50000
1.10141 1 ,05719 1.03158 1.01718 1 ,00926 1 ,00496 1 ,00265 1 ,00141 1 ,00076 1 .00040 1 ,00022 1.00012 1 .00006 1 ,00003 1.00002 1 .00001 1 .00001 1 . 00000 1 . 00000
(6) ordinate values adjusted for truncation errors. di
=
t. 2 - R(n,
cy,
ti) i = 1, . . , , N
(25).
.fi
(7) Go to (2) until procedure converges. Conclusion
The technique described above provides an iterative improvement for reducing the truncation error in approximately linear relations that are used to estimate kinetic parameters. The technique is general and can be applied to any such formulation. Literature Cited Abramowitz, M., Stegun, I. A., “Handbook of Mathematical Functions,” National Bureau of Standards Applied Mathematics Series 5 5 , p 16, Washington, D.C., 1964. Pessen, H., Science, 134, 8, September 1961. Ravimohan, A. L., Znd. Eng. Chem., 61 (5),76-77 (1969). Thomas, H. A., Water Sewage Works, 97,123 (1950). Receiced for reciew April 6, 1972. Accepted August 2, 1972.
Volume 6, Number 13, December 1972
1115
