AbstractThis work deals with the application of invariant imbedding to solve a particularly important design problem, namely, the duration of the process. A numerical example is used to illustrate the approach. The advantage of this approach is its straightforward nature and uses only the usual design data. It avoids any iterations and thus no convergence problems need to be considered

Lee, E.S

Naadimuthu, G

Wang, K.M

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Application of invariant imbedding to the estimation of process duration 

JOURNAL OF MATHEMATTCAL ANALYSIS AND APPLICAT,ONS 98, 236-242 (1984) 
Application of Invariant lmbedding to the 
Estimation of Process Duration * 
K.M. WANG 
Tsinghua University, Taiwan, China 
0. NAADIMUTHU 
Fairleigh Dickinson University. Teaneck, New Jersqv 07666 
AND 
E. S. LEE 
Kansas State University, Manhattan, Kansas 66506 
This work deals with the application of invariant imbedding to solve a 
particularly important design problem, namely, the duration of the process. A 
numerical example is used to illustrate the approach. The advantage of this 
approach is its straightforward nature and uses only the usual design data. It avoids 
any iterations and thus no convergence problems need to be considered. 
1. INTRODUCTION 
An important problem in process industry is the design of equipment. An 
example is to estimate the size of equipment to produce a desired product 
given the raw material. This forms a typical boundary value problem with 
the given raw material and the desired product as boundary conditions. 
Some sample design problems of the above type are to determine the 
following: length of a distillation column, length of a tubular reactor, time 
for the movement of a particle from one position to another, time for the 
concentration of a drug to reach a certain level in some part of our body, 
and the time required for a chemical reaction process. 
Chandrasekhar [3], Bellman and co-workers [4,5, 111, Lee [6], Casti and 
Kalaba [7], Mingle [8], Scott [9], and Shimizu and Aoki [lo] have made 
* This work was supported by Grant ENG-77-06757 from the National Science Foun 
dativn. 
236 
0022-247X/84 $3.00 
Copyrlghr 0 ,984 by Academic P,w..s, 1°C. 
All rights OF reproduction in any form reserved. 
APPLICATION OF INVARIANT IMBEDDING 237 
major contributions to the solution of boundary value problems by the 
invariance principles introduced by Amberzumian [ 1,2]. 
Meyer [ 121 applied the method of lines to linear one-dimensional 
parabolic free boundary problems. The author [ 131 extended the method of 
lines to multi-dimensional parabolic free surface problems. Meyer [ 141 also 
used the method of lines to solve Poisson’s equation with nonlinear or free 
boundary conditions and combined the method of lines and invariant 
imbedding for elliptic and parabolic free boundary problems [ 151. 
This paper involves the application of invariant imbedding and the method 
of characteristics to solve free boundary value problems. 
2. INVARIANT IMBEDDING AND THE 
METHOD OF CHARACTERISTICS 
Consider a system whose behavior can be represented by the following 
nonlinear second-order differential equation: 
g- = P, g + P,x2 
with input data 
x(0) = c, (2) 
x’(0) = c, 
x(T)= c,. (3) 
The problem is to find the length of the above process, T, as a function of 
the given boundary conditions. 
Equations (l)--(3) are transformed into the following first-order differential 
equations: 
dx -= 
dt Y =f, 
$P,. y+P,.x2 
(4) 
with input data 
x(0) = c, 
Y(O) = c2 
x(T)= c,. 
(6) 
238 WANG, NAADIMUTHU, AND LEE 
By using the standard procedure (61, we obtain the following imbedding 
equation for T as a function of C, and C, : 
T(C,, C,) = A + T(C, f A . j-1, C, + A . f,) (7) 
where T(C, , C,) = length of the process represented by Eqs. (4~(6) starting 
at time a and with x(a) = C, and y(a) = C,. 
Expanding Eq. (7) by Taylor series, we have 
T(C,,C,)=A+T(C,,C,)+~~A~f,+~~A~f;+O(A). (8) 
1 2 
In the limit as A tends to zero, Eq. (8) becomes 
&f,+$ * f2 = -1. (9) 
1 2 
There are various computational techniques available for solving Eq. (9). 
The two frequently used techniques are the various versions of finite- 
difference methods and the method of characteristics [ 16-211. In this work, 
the method of characteristics is used. The characteristic equation of Eq. (9) 
is [21] 
dC, dC, dT 
f,=fi=-T. 
From the first two terms of Eq. (lo), we have 
dC2 f2 
dC,=f,* 
Solving Eq. (11) either analytically or numerically, we can obtain 
c2 =f(C,)- 
From the second and third terms of Eq. (lo), we have 
dC, dT 
f,=--’ 1 
Integrating Eq. (13), we obtain 
T(C, , C,) = - jr’ 9 + I/I. 
(10) 
(11) 
(12) 
(13) 
(14) 
From the definition of T(C,, C,), we see that 
TW’h Y(T)) = 0. 
APPLICATION OF INVARIANT IMBEDDING 
Thus, Eq. (14) becomes 
-1 
x(T) dC, -+ly=o. 
0 fi 
From Eq. (3) and Eq. (15), we obtain 
Substituting Eq. (16) in Eq. (14), we have 
239 
(15) 
(16) 
(17) 
The computational procedure is summarized below: 
(i) With the given values of x(O), x’(O), and Eqs. (11) and (12), 
obtain the C, values at various C, values. 
(ii) Using C,, C,, and C, in Eq. (17), calculate the value of T. 
3. CHEMICAL REACTOR MODEL 
Consider the chemical reaction [6] 
A +A+B 
which is taking place in a homogeneous tubular flow chemical reactor with 
axial mixing. Assuming that the change of volume in the reactor is 
negligible, the following equation can be established easily by the use of 
material balance on reactant A [6]: 
-.e-dx_R .x2=() 1 
Npp dt= dt 
or 
d2x ---EN 
dt2 
pp 1% + NPr ’ R . x2 
with input data 
x(0) = c, 
x’(0) = c, 
x(T) = C, 
(18) 
(19) 
409/98/I 16 
240 WANG, NAADIMUTHU, AND LEE 
where 
NPe = the dimensionless Peclet group Lv/D 
R = the reaction rate group kL/u 
x = concentration of reactant A 
t = dimensionless reactor length, 0 < t < 1 
L = total reactor length 
v = flow velocity of the reaction mixture (constant) 
D = mean mass axial dispersion coefficient (constant) 
k = specific chemical reaction rate (constant). 
The notations and assumptions of this model are the same as those in 
Ref. [6]. The problem is to determine the length of the chemical reaction 
process, T, by using invariant imbedding and the method of characteristics. 
Equations (18) and (19) can be rewritten as 
dx 
-= 
dt ’ 
with input data 
x(0) = c, 
Y(O) = c2 (22) 
x(T) = C,. 
Using Eq. (9) for the above system, we obtain the invariant imbedding 
equation for T(C,, C,) as 
T.c,+2&. 
ac, (N,/C,+N,/RC;)=-1. 2 
As per Eq. (lo), the characteristic equation for Eq. (23) becomes 
dc, dG dT 
-= Npe.C2+Npe.R.C; =-I’ c2 
Applying Eqs. (1 l), (12), and (13) to Eq. (24), we obtain 
G 
-zz 
N&2 + &,RC: 
dc, c2 
(24) 
(25) 
(26) C,(C, =x(O)) = Y(O) 
APPLICATION OF INVARIANT IMBEDDING 241 
and 
dC, dT -=-- 
c* 1 
respectively. 
Thus, according to Eq. (17), we have 
(27) 
The numerical values used in the calculation are 
x(0) = C, = 0.83 129 
y(0) =x’(O) = c, = -1.0122 
x(T) = C, = 0.38727 
Np,= 6 
R = 2. 
This problem is solved using the above computational procedure. The 
results are listed in Table I. At cj = 0.38727, the value of T should be 1.0. 
Considering the various computational errors, the value of 1.026 obtained is 
very close to the correct one. 
It should be emphasized that the main advantage of this approach is its 
straightforward nature. No iterations are involved and thus no convergence 
problems need to be considered. This forms a big advantage when modern 
computers are used. 
TABLE I 
Length of Chemical Reaction Process, T 
c, c* 1 
x(0) = 0.83 129 x ‘(O)= -1.0122 0 
0.786888 -0.9 16744 0.046 114 
0.742486 -0.825 192 0.097187 
0.698084 -0.737563 0.154127 
0.653682 -0.653815 0.218097 
0.609280 -0.573752 0.290623 
0.564878 -0.49678 1 0.373812 
0.520476 -0.42 1124 0.470852 
0.476074 -0.345 160 0.580353 
0.431672 -0.226559 0.743211 
x(T) = 0.38727 = C, -0.006369 1.026337 = T 
242 WANG, NAADIMUTHU, AND LEE 
In actual applications, the number of conditions given is frequently greater 
than the number of unknowns to be determined. In this overspecified 
situation, the method of least-squares can be used. The other computational 
procedures remain essentially the same. 
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Application of invariant imbedding to the estimation of process duration

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