ࡱ > ~ Z H a Z . 9
{ I
t bjbj 8 n, , , , , , , , 8 , 2 , 4
? " ? ? ? @ @ @ % t , A @ @ A A , , ? ? K W W W A Z , ? , ? W A W W n u @ x ? @\C , 8C v p a 0 v P @ x , x @ L @ 6 W A , 7A @ @ @ &T @ @ @ A A A A @ @ @ @ @ @ @ @ @ ( 4+ : Chapter 3. Numerical solution of the direct problem in chemical kinetics from the book V.Korobov, V.Ochkov Chemical Kinetics: Introduction with Mathcad/Maple/MCS - Moscow, 2009
Previously discussed analytical methods for solving the direct problem in chemical kinetics are not sufficient for analysis of different reaction kinetic schemes. First, even given the mathematical model represented by an ODE system, it is not always possible to integrate the equations analytically. The reason for that may be just an absence of such a solution. This refers primarily to a large number of kinetic models in which differential equations are non-linear relative to the sought functions.
Second, if the analytical solution is obtained, it is often to lengthy and awkward.
Finally, there is a large class of real mathematical models that are described by partial differential equation sets, which cannot be integrated numerically. Thus, the series of mathematical problems that can be solved with the previously discussed numerical integration methods is quite narrow. That is why, in order to solve the direct problem, we have to rely on more universal approaches. Such approaches are based on using numerical integration of differential equations and systems.
3.1. Given/Odesolve solver in Mathcad system
Among the built-in tools of the Mathcad suite an important role belongs to those designed for the numerical solving ordinary differential equations (ODEs) and their systems. Lets look through these tools starting with the solving unit GIVEN/ODESOLVE. The ODESOLVE function first appeared in Mathcad 2000 Pro. On default this function used Runge-Kutta method of the fourth-level of precision.
The organization of the GIVEN/ODESOLVE solver resembles greatly that of the GIVEN/FIND solving block: it starts with the GIVEN keyword. An ODE or a system as well as the initial conditions should be placed in the solver body. The solving is performed with a call up of built-in function ODESOLVE using the following format:
ODESOLVE(x,b,[steps]),
where x is the unknown, and b is the upper integration limit. The last parameter, steps, determines the number of integration steps and is optional. If this parameter is absent, the number of steps is set up automatically.
As an example, lets calculate a kinetic curve for an intermediate in a consecutive reaction where the second step is of the second order:
EMBED Equation.3 ,
EMBED Equation.3 .
Analytical solution of this problem was discussed in the chapter 1.3.2, so we can compare numerically- and analytically-obtained results.
The way of using GIVEN/ODESOLVE solver and the results of its work are shown in fig. 3.1. The following values were used: initial concentration A = 0.5M, rate constants EMBED Equation.3 =0.05s-1and EMBED Equation.3 =0.1M-1.s-1. Referring to fig. 3.1, the kinetic curve calculated numerically (dots) matches the curve calculated using analytical formula (line). The authors want to point out that during the assigning of the results of symbolic computation one should use only the name of the desired function (without argument). For the construction of a graph both function name and its argument are specified fig. 3.1).
Fig.3.1. Kinetic curve calculation for an intermediate in a consecutive second-order reaction using ODESOLVE function
The results of this example allow one to think that the ODESOLVE function is a sufficient tool for solving the direct kinetic problem. Now we proceed to a numerical solution of an ordinary differential equation (ODE) set using a solver. Solving process for an ODE set does not differ much from that for a single ODE: all equations in the set as well as the starting condition equalities are given in the solver. The only difference is in the number of arguments that ODESOLVE function should have. An array of desired function names is required for solving sets of equations. The variable, where the calculation result is stored, is an array of function names as well. Below is an example of a numerical calculation of all component concentrations in a hypothetical multi-step reaction.
EMBED Equation.3 . (3.1)
We can see that the process goes through 6 elementary steps with 5 compounds; and most of the reactions have order higher than first. Finding the analytical solution of the direct problem is barely possible in this case. However, having numerical integration tools we do not need an analytical approach.
The solution of this example is shown in fig. 3.2. One can see that the use of ODESOLVE function is rather simple.
Fig.3.2. Calculation of the kinetic curves for all components in a multi-step reaction using ODESOLVE function
Working with ODESOLVE the user can choose an algorithm for the numerical ODE solution. If the cursor is above ODESOLVE keyword, the right-click will activate the corresponding context menu (fig. 3.3). In Mathcad 20 0 1 "1 3 t h e u s e r c o u l d c h o o s e b e t w e e n i n t e g r a t i o n w i t h a f i x e d ( F i x e d ) o r a d a p t i v e ( A d a p t i v e ) s t e p a s w e l l a s a s p e c i a l m e t h o d f o r s t i f f s y s t e m s i n t e g r a t i o n ( S t i f f ) . A n e w o p t i o n , A d a m s / B D F , w a s a d d e d i n M a t h c a d 1 4 .
F i g . 3 . 3 . C h o o s i n g a n O D E S O L V E a l g o r i t h m in different Mathcad versions.
An obvious advantage of the GIVEN/ODESOLVE solver is that differential equations and ODE sets are written in the usual manner. That is why Mathcad documents designed for kinetic problems are rather clear. Another feature of the ODESOLVE solver is the possibility to use not only differential equations but also usual algebraic ones.
For example, lets consider solving the direct problem for a parallel reaction with the following mechanism:
EMBED Equation.3 ,
EMBED Equation.3 .
Here reagent A is consumed in two parallel steps through interaction with reagents B and C. Each step has the second order. To find kinetic curves for all the compounds in this reaction it is enough to use a mathematical model consisting of two differential and three algebraic equations (the latter represent the material balance of the reaction system):
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 .
All equations can be put in the solver without changes. The corresponding numerical solution is shown in fig. 3.4.
Fig.3.4. Solving a system of differentialalgebraic equations
So far, we have considered the problem of calculating kinetic curves for compounds. In mathematics this is called a Cauchy problem. It is known that for the Cauchy problem entry conditions are given, i.e. the value(s) of the desired function y(x) at point x=x0. In kinetics, as a rule, this point corresponds to the starting point of a reaction. Often one needs to find a partial solution of a differential equation using known function values in several points. Such kind of problems is called a boundary-value problem (boundary problem).
The two-point boundary problem may be exemplified by the calculation of the kinetic curve for an intermediate in a consecutive reaction EMBED Equation.3 using known values of concentration at two moments of time (fig. 3.5). Assume the initial A concentration EMBED Equation.3 =1 M, rate constants EMBED Equation.3 = 0.008s-1, k2 = 0.004s-1, and the concentration of an intermediate B after 7.2 and 445.1 s from the beginning of the reaction Cb=0.05M. To calculate the kinetic curve for the compound B we will use the equation:
EMBED Equation.3 ,
that is converted into:
EMBED Equation.3 .
The solution of this boundary problem is given in fig. 3.5 and it does not require additional explanation.
Fig.3.5. Solving a boundary-value problem with solver
3.2. Built-in Mathcad integrators
In addition to Given/Odesolve solver, one can use built-in Mathcad suite integrators for solving the direct kinetic problem. These integrators are designed specifically for the numerical solving of ODEs and ODEs sets. There are several built-in functions of this kind. It is important to mention that each of the integrators requires right-hand member array of the ODEs set. Recall that the mathematical model for a complex reaction is obtained through multiplying the stoichiometric array by the rate array. The result of this operation is the array of the ODE set right-hand values. That is why built-in integrators are very convenient for solving kinetic problems with stoichiometric arrays of high dimensionality.
We will begin the learning of the integrators with the rkfixed function. It implements the fourth-order Runge-Kutta method (rk) with fixed step of integration (fixed). According to Mathcad syntax this function has five required arguments:
rkfixed(v,x1,x2,npoints,f),
where v is the intial values vector; x1 and x2 independent variable values that determine the interval of integration; npoints the integration steps quantity; f the array function of the ODE set right-hand values.
Creation of the array function f requires following a specific procedure. Usually this function is declared right before rkfixed using the following syntax:
f(t,x):=[array of the ODE set right-hand values],
where t is the independent variable and x is the desired functions array. The right-hand part of this construction is an array of the ODE set right-hand values; and each of these functions is represented by an index variable x, i.e. x0, x1 etc.
The quantity of the array x elements as well as the quantity of the right-hand values is equal to the quantity of equations in the system. Lets clarify how rkfixed works by an example of a numerical solution of the direct problem for a three-step consecutive chemical reaction (fig. 3.6).
In the beginning, the right-hand value array Model(k,x) is defined. Current concentrations of the compounds A, B, C, D are given as x1, x2, x3, x4 (here 1, 2, 3, 4 are vector indices, not lettered ones!) After assigning the rate constant values, a function F(t,x) is declared. The first argument must be an independent variable time. This declaration allows the rkfixed integrator to understand the function F(t,x). The numerical calculation result is stored in a matrix S. Based on the rows and columns quantities in this matrix one may conclude: the solution matrix has (n+1) columns if the set has n equations. It is easy to see that the first column starts with a value tn=0 and ends with a value tk=100. The matrix has (N+1) rows for N integration steps. The second column contains the value of the variable x2 at each integration step. In this case these are the values of the reagent A concentration. Correspondingly, the 3rd, 4th and 5th columns represent concentrations of the reagents B, ! a n d D . T h u s , t h e m a t r i x S g i v e a p i c t o r i a l v i e w o f t h e r e a c t i o n m i x t u r e c h a n g e s o v e r t i m e . F i n a l l y , w e c a n m a k e a g r a p h i c a l r e p r e s e n t a t i o n o f t h e c a l c u l a t i o n r e s u l t s a s k i n e t i c c u r v e s f o r e a c h r e a g e n t .
T h e r e b y , M a t h c a d t o o l s f o r n u m e r i c a l s o l u t i o n o f d i f ferential equation sets allow one to calculate quickly kinetic curves of all reactants in a complex chemical process. As the independent variable in chemical problems is time t, discussed methods can be used in modeling any time function (having postulated a differential model for the process). One can find many analogies of the kinetic models of chemical reactions in other fields of knowledge (microbiology, sociology etc.) It is useful to discuss the corresponding examples in order to form some practical skills in creating differential models.
Lets consider one of such problems. Certain microorganisms propagate in proportion to the colony size (with an aspect ratio k) but at the same time produce some excrement, which is a poison for them.
Fig3.6. Numerical solution of the direct problem for a consecutive reaction with two intermediates
The rate of the colony disappearance is proportional to the amount of poison and current microorganism population with an aspect ratio k1. The poison formation rate is proportional to the number of microorganisms (with a ratio k2). Suppose the initial colony size equals N0, and the amount of poison Z is 0 at the beginning. One is required to make the corresponding set of differential equations and solve it numerically, and present graphically the microorganisms population trends along with the amount of poison in the system. Assume k=0,1; k1=0,0001; k2=0,01, N0=2000.
First, we should create a differential equation set in accordance with the problem specifications. Changes in the microorganisms population is determined by an increase kN as a result of reproduction and a decrease k1NZ due to poisoning. Therefore, the first differential equation of the system will be of this form:
EMBED Equation.3 .
The rate of poison amount change will be described as
EMBED Equation.3 .
A differential model for the process has been defined; now we can create a corresponding Mathcad document to solve the problem (fig. 3.7).
Fig3.7. Microorganism population and poison amount trends
As the figure 3.7. implies, the number of microorganisms first increases with time, achieves the highest value at some point, and after that the colony becomes extinct. The curve Z(t) is a typical saturation one. At the beginning, the poison accumulation rate is small, but it increases with the lapse of time until it reaches the maximum. Certainly, after full disappearance of the microorganisms the amount of poison stabilizes and becomes constant. A chemical analog for this model would be a complex chemical reaction where compounds N and Z participate in an intermediate step. In this case, the compound Z is an autocatalyst for the decomposition of N in accordance with discussed mathematical correlations.
Further in the book we will see that a minor (at first sight) modification of the starting differential equations set and initial conditions may cause a significant change in the dynamic outlook of the solution.
One may often have a case when in order to obtain enough accurate result it is necessary to use a variable integration step: decreasing in the area of large derivative changes and, vice versa, increasing when the derivative changes slowly. This approach is implemented in an algorithm that function Rkadapt uses. During the work of this function the step of integration is adapted in accordance with the derivative trend in the selected interval of integration. Lets consider the following ODE system as an example:
EMBED Equation.3 ,
EMBED Equation.3 .
Further, we will discuss minutely this system which is a mathematical model of the widely-known kinetic scheme bruesselator. Now we will compare the results of its numerical solution using the functions rkfixed and Rkadapt.
The corresponding plots are shown in fig. 3.8. They show that using a fixed step can lead to an instable solution which can be interpreted wrongly from a physical point of view (dashed line). The function Rkadapt, as we can see, allows us to eliminate the mistakes of rkfixed, and reveals the true behavior of the desired function in the given independent variable range (due to an adaptable step of integration). In practice, Rkadapt is preferable in the solving of many direct problems, especially in cases when the starting kinetic model is non-linear.
Fig3.8. Comparison of the results for calculations with fixed step of integration (solid line) to an adaptive one (dashed line)
We also want to mention that the function Rkadapt requires the same five arguments specified in an rkfixed body. Even though integration utilizes a changeable step, the result will still be represented for evenly distributed points as specified by user.
There is one more circumstance related to a variety of built-in integrators. It is the existence of so-called stiff ODE sets. The concept of stiffness may be illustrated by the example of the kinetic equation for a multi-step reaction
EMBED Equation.3 .
It is considered that the mathematical model is stiff if among the eigenvalues i o f t h e r a t e c o n s t a n t m a t r i x E M B E D E q u a t i o n . 3 t h e r e e x i s t s u c h e i g e n v a l u e s f o r w h i c h E M B E D E q u a t i o n . 3 . U s u a l l y t h i s c o n d i t i o n h o l d s i f t h e r a t e c o n s t a n t s m a t r i x h a s e l e m e n t s v e r y d i f f e r e n t i n m o d u l o ( t h r e e a n d m o r e o r d e r s ) .
L e t s c o n s i d e r t he following kinetic scheme:
EMBED Equation.3 ,
that can described with only two differential equations in accordance with the stoichiometric matrix rank. The corresponding ODE set can be written as
EMBED Equation.3 .
The eigenvalues vector for this matrix is equal to
EMBED Equation.3 ,
which allows one to consider the mathematical model as a stiff one. In this case the integration should be performed using special built-in functions for stiff systems. Mathcad suite has integrators Stiffb, Stiffr, Radau, and starting from Mathcad 14 an integrator AdamsBDF. Fig. 3.9 shows the Mathcad document illustrating the solution of the direct kinetic problem for a stiff model using several integrators.
Fig3.9. An example of a kinetic scheme described with a stiff set of differential equations (Mathcad 11.2)
As fig. 3.9 shows, the integrator rkfixed can not solve the problem at all. It shows a diagnostic message Found a number with a magnitude greater than 10^307 while trying to evaluate this expression. In this case the integrators Rkadapt, Radau, Stiffb, and Stiffr do work but give different computation results for the chosen step of integration. One can find that a five-fold increase in the number of steps virtually levels the results. However, the integration of more complex stiff sets still should be done by using the specially designed built-in functions
The functions Stiffb and Stiffr require an additional argument J(t,x), which is a matrix of partial time (t, zero column) and the x vector derivatives for a kinetic function. A way to form the vector J(t,x) is shown in fig. 3.9. It is worth mentioning that, in the early versions, this stage was done manually (paying attention to the fact that the symbol editor cannot differentiate expressions with index variables). The user had to transform the index variables into lettered ones and vice versa (see the fig. 3.9 where the index and lettered variables are formatted with different styles). Because of that the Mathcad 2001i version included the Radau function that did not require the argument J(t,x). Although it was very convenient, the user had to accept some loss of precision. In the Mathcad 14 version the functionality of the Radau was expanded. In addition, this version had tools for automatization of the matrix J(t,x) symbolic calculation.
3.3. The Maple system commands dsolve, odeplot in numerical calculations
The command dsolve of the Maple system was previously discussed as a method of analytical solution of the direct problem in chemical kinetics. This command also can be used for a numerical solving of ODEs or their sets. In this case one should use the following syntax:
dsolve({ode,ic},numeric,vars,options)
Here ode is the differential equation (or ODE set) with the initial conditions ic. The option numeric is a directive for numerical computations (one may use the construction 'type=numeric' instead of the keyword numeric); vars is the desired function (or the desired function set in case of ODE set); options are additional options given in the keyword=value form.
The option numeric (or type=numeric) indicates that dsolve will return a numerical calculation result. The most important additional option is the method. It determines which numerical method will be used in the desired function search. The Maple system gives a choice among a variety of methods (see the list in table. 3.1).
On default, during a solution of the Cauchy problem the dsolve function automatically utilizes the Runge-Kutta method of the 45th order of accuracy. The major options are given in the table 3.2.
Table 3.1 Numerical methods for solving ordinary differential equations in Maple suite.
No.The method value
Numeric method used by dsolve solver
1.rk45RungeKuttaFehlberg method of the 45th order of accuracy
2.classical or
classical[foreuler]Euler method3.classical[heunform]Modified Euler method using Heuns formula
4.classical[impoly]Euler method subkind
5.classical[rk2]Runge-Kutta method of the 2nd order of accuracy
6.classical[rk3]Runge-Kutta method of the 3rd order of accuracy
7.classical[rk4]Runge-Kutta method of the 4th order of accuracy
8.classical[adambash]AdamsBashforth method
9.classical[abmoulton]AdamsBashforth-Moulton method
10.rosenbrock Rosenbrock method
11.bvpNumerical method to solve the boundary value problem
12.dverk78RungeKutta method of the 78th order of accuracy
13.lsode or lsode[choice],
where choice can be adamsfunc, adamsfull, adamsdiag, adamsband, backfunc, backfull, backdiag, backbandModification of the Adams method for the solving of stiff ODE and their sets
14.gear, gear[bstoer], gear[polyextr]Gear method and its modifications15.taylorseriesMethod utilizing the Taylor series expansion of integrand
Table 3.2 Some major options for the dsolve command used in numerical calculations
OptionPurpose'output' = keyword or arrayControls the output order of the calculation results. Can have the symbol values procedurelist, listprocedure, as well as array or Array. By default 'output'= procedurelist.'stop_cond' = listControls the process of computation finishing when the list conditions are met.'range' = numeric..numericDetermines the independent variable range (the length of the integration interval).'stiff'=booleanThis option is given as a Boolean expression. For example, if 'stiff'=true, the equation is considered stiff and the default method value is set to rosenbrock instead of rk45.
An important component of using the dsolve function in numerical calculations is the output order of the results. In particular, when the following construction is being used:
>dsol:=dsolve({sys1,invalues},numeric);
only a message about the successful procedure implementation is displayed:
dsol:=proc(rkf45_x)...end proc
After that, the user has to decide how the results should be visualized. It is possible to print the answer in the form of individual values of the desired function, an array, etc. However, the most visual output form is the graphical one. The plotting of the results is provided by the command odeplot from the graphical library plots. Figs. 3.10 and 3.11 show a solution of the differential equation set, which describes the kinetics of the first-order reversible reaction EMBED Equation.3 with arbitrary rate constant values.
Fig3.10. Numerical solution of the direct kinetic problem using Mathcad tools
Fig3.11. Kinetic curves for reversible reaction participants calculated using numerical calculation results
3.4. Oscillation processes modeling
In some reactions, one can see the periodic changes of the reagent concentrations over time. Correspondingly, the rate of the reaction has an oscillating character. Such reactions are called oscillating or periodic. Nowadays several dozens of homogeneous and heterogeneous oscillating reactions have been explored. Investigations of the kinetic models for these complex processes have allowed formulating a series of general conditions, which are required for the stable oscillations of the reaction rates and intermediate concentrations:
Stable oscillations appear usually in open systems, where reagent concentrations can be maintained constant.
The complex reaction must include autocatalytic steps as well as product-inhibited ones;
The reaction mechanism must include steps with the order higher than first.
These conditions are required but not sufficient for the oscillation to occur in the system. An important role is played also by the ratio between the rate constants of certain steps and starting reagent concentrations. An investigation of the oscillating reactions is still an important chemical kinetics problem because it is crucial in understanding catalysis, periodic process laws for living systems, and chemical technology.
Sometimes chemical problems can be answered using the knowledge from other sciences that are not related to chemistry at first sight. For example, some information about a complex reactions flow can be gained from the mathematical models of the interspecific competition. A classical example is the predatorprey model, which describes the population trends for predators and prey in living conditions (the LotkaVolterra model). The analogies between this model and many reaction schemes are evident.
The main point of the model is the following. Lets consider some closed ecological system that includes two populations: predators and herbivorous prey. The population sizes are K and B correspondingly. The prey population is reproduced by means of nutrition. The prey consume grass only. The amount of grass is characterized by the T values. Predators eat solely the prey. Their population trend is determined by the amount of herbivors. There are no natural enemies for the predators. Instead, the latter experience the natural loss (diseases, age-specific death). Thereby, the conditions can be expressed with the following scheme:
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 .
Here E c o r r e s p o n d s t o t h e n u m b e r o f d e a d p r e d a t o r s . U s i n g c h e m i c a l t e r m i n o l o g y , o n e c a n s a y t h a t t h e r e s u l t i n g p r o c e s s i s a c o n v e r s i o n o f t h e s t a r t i n g r e a g e n t T i n t o t h e p r o d u c t E. T h e p r o c e s s i s a c c o m p a n i e d b y t h e f o r m a t i o n o f i n t e r m e d i a t e s K a n d B , w h i c h s e rve for different kinetic functions.
The stages of population expansion are autocatalytic with the reproduction factors (rate constants) k1 and k2, but the presence of predator mortality (k3) prevents the unlimited growth of both populations. Undoubtedly, the overall kinetics of the process is affected by the amount of grass necessary for the prey population increase. Assume that we have an unlimited amount of grass, i.e. T(t)=const. Then, based on the given conditions, one can write the following differential equation set:
EMBED Equation.3 ,
EMBED Equation.3 .
If the constants for each step and initial population sizes are given, then the numerical solution will allow one to predict prey and predator population trends (fig. 3.12).
Fig.3.12. Population trends for predators (dashed line) and prey (solid line) in the LotkaVolterra model
As fig. 3.12 shows, the population trends for both populations exibit a continuous wave pattern. For the given initial conditions these oscillations have a constant period and amplitude. There is an interdependence between the two population sizes: increase of one of them impedes the growth of other. In the case of some chemical process descibed by the Lotka-Volterra model, the concentrations of the intermediates K and B would be oscillating.
Often it is more convenient to present the solutions of such systems not as the concentration over time trends but rather as the dependence of one of the concentration on the other. In the second part of fig. 3.12, we have shown the prey population as a function of the predator population the so called phase path. The population dynamics can be represented as a movement along the phase path. The position of the point corresponds to the population sizes at a given moment of time. One can see that the phase path for these initial conditions is a closed graph. In the case of continuous oscillations, the point follows the phase path repeatedly.
Now lets change the form of the Lotka-Volterra model by dividing both parts of th e e q u a t i o n s b y k 2 , a n d a s s u m i n g t h a t k 2 d t = d , k / k 2 = a , k 3 / k 2 = b :
E M B E D E q u a t i o n . 3 ,
E M B E D E q u a t i o n . 3 .
T h e s y s t e m h a s o n l y t w o p a r a m e t e r s n o w , 0 a n d b . B o t h o f t h e m a r e p o s i t i v e a c c o r d i n g t o t h e p h y s i c a l m e a n i n g . I f o n e s o l v e s t h e r e s u l t i n g s y stem for a number of initial conditions, he will end up with a phase path set called the phase portrait of the system. For the given case we will have the phase portrait as a set of closed concentric graphs (fig. 3.13). Due to the a>0, b>0 conditions, all the phase paths are situated in the first quadrant of the coordinate plane. The oscillations occur near some stationary point, which can be determined by putting EMBED Equation.3 and EMBED Equation.3 equal to zero. One can easily find that EMBED Equation.3 , EMBED Equation.3 . The corresponding point (Xst;Yst) is called the critical point of the system.
Fig.3.13. Phase portrait of the LotkaVolterra system with a critical point
In the case when all the phase paths are closed and incapsulated one into the other, the crytical point is called a center.
If the initial grass-eater and predator populations equal b and a correspondingly, the simulation will not reveal oscillations in the system. Any deviation from the stationary values will lead to oscillations.
Even though the predatorprey model is rather idealized, many kinetic models for real chemical systems are based on it. For example, D.A.FrankKamenetsky used the Lotka-Volterra model to explain the processes of higher hydrocarbon oxidation.
The hydrocarbon oxidation kinetics is extremely complex because it includes many consequent-parallel steps. Thus, the full mechanism description is problematic. In such cases researchers confine themselves to model descriptions. Each of the model steps may represent a series of elementary stages, and each of the model symbols may correspond to a whole set of compounds playing the same kinetic function.
For example, in the case of a continuous supply of a gasolineair mixture into the reactor (heated to certain temperature), one can see periodical flashes of the cold flame appearing with a constant frequency. In this case the full combustion does not occur. The oxidation products include aldehydes, organic peroxides and other compounds. Some regularities have been established for this process. In particular, the flash frequency increases with the increase of oxygen concentration and temperature. In order to explain this effect, D.A.FrankKamenetsky suggested the following kinetic scheme involving two types of intermediates, X and Y:
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3
Here A is the starting compound, B is the product, X is the superoxide type molecules or radicals, and Y means the aldehyde type molecules or r a d i c a l s . O n e c a n s e e t h a t t h e s c h e m e p u s t u l a t e s a n a u t o c a t a l y s i s w i t h t h e t w o i n t e r m e d i a t e s . L e t s a s s u m e t h e r e a g e n t c o n c e n t r a t i o n d o e s n o t d e p e n d o n t i m e ( = c o n s t ) , i . e . i t s c o n s u m p t i o n r a t e i s c o m p e n s a t e d b y i t s i n s e r t i o n i n t o t h e r e a c t o r . T h a t w i l l give us the following equation set:
EMBED Equation.3 ,
EMBED Equation.3
After the division of both equation parts by k2 we end up with the Lotka-Volterra type set:
EMBED Equation.3 ,
EMBED Equation.3 ,
where a=k1A/k2, b=k3A/k2, k2dt=d . I t w a s s h o w n b e f o r e t h a t t h e s o l u t i o n f o r t h i s s y s t e m h a s a b e h a v i o r o s c i l l a t i n g i n t i m e
L e t s s h o w t h e n u m e r i c a l s o l u t i o n o f t h e L o t k a - V o l t e r r a m o d e l u s i n g M a p l e s u i t e t o o l s ( f i g . 3 . 1 4 ) . H e r e t h e D E p l o t c o m m a n d f r o m t h e D E t o o l s l i b r a r y w a s u s e d i n a d d ition to the dsolve. In this case, in addition to the integral curves set for the phase paths, the directional field is visualized. The directional field is a series of arrows, each of which represents the motion direction along the phase path at the given point. In many cases the directional field increases the clearness of the phase portrait. The directional field can also be plotted using the Maple commands phaseportrait and dfieldplot.
Fig.3.14. Predatorprey model analysis using Maple
To construct the directional field in the Mathcad environment we recommend the user function field, which was created by T. Gutman (fig.3.15).
There are other types of critical points. For example, consider the kinetic scheme consisting of the elementary steps:
EMBED Equation.3 ,
in which the first step has the zero order. The concentration changes for the intermediates are described by the equation set:
EMBED Equation.3 ,
EMBED Equation.3 .
Assume EMBED Equation.3 , EMBED Equation.3 and EMBED Equation.3 . We can plot a series of phase portraits for the different initial concentrations of X and Y based on the numerical solution of the direct kinetic problem (fig. 3.16).
Fig.3.15. Phase portrait of the LotkaVolterra system using a directional field
The computation results, presented in fig. 3.16, show that all phase paths converge at one point. Its coordinates are determined by the values of the intermediates stationary concentrations, EMBED Equation.3 and EMBED Equation.3 :
EMBED Equation.3 ,
EMBED Equation.3 ,
which gives EMBED Equation.3 , EMBED Equation.3 , or, taking into account the given rate constant values, EMBED Equation.3 , EMBED Equation.3 . Such type of critical point is called node, and oscillations are impossible in this system.
Fig.3.16. Phase portrait of the system with node "t y p e c r i t i c a l p o i n t
N o w c o n s i d e r t h e f o l l o w i n g e q u a t i o n s e t :
E M B E D E q u a t i o n . 3 ,
E M B E D E q u a t i o n . 3 .
T h i s s y s t e m i s a l s o o f t e n u s e d i n t h e d i f f e r e n t i a l b i o l o g i c a l m o d e l s . I f o n e a n a l y z e s t h e c o r r e s p o n d i n g p h a s e p o r t r a i t w i t h t h e d i r e c t i o n a l f i eld (fig. 3.17), one can see that all the phase paths approach a critical point and then move away. In this case, we have the saddle critical point.
Fig.3.17. System with a saddle critical point
Finally, there is one more remarkable critical point type focus. In order to illustrate it, we will examine one of the kinetic models of photosynthesis.
In the past there was suggested a mechanism for the dark steps cycle of photosynthesis. Sugars with different numbers of carbon atoms, 3 to 7 (trioses, te t r o s e s , p e n t o s e s e t c . ) , p a r t i c i p a r e i n t h i s c y c l e . H a v i n g l a b e l l e d t h e n u m b e r w i t h a s u b s c r i p t , o n e c a n c r e a t e t h e s c h e m e o f t h e p r o c e s s :
!5 + !1 + % ! 2 C 3 ,
2 C 3 ( C 6 ,
C 6 + C 3 ( C 6 ,
C 4 + C 3 ( 2 C 7 ,
C 7 + C 3 ( 2 C 5 .
H e r e X m e a n s t r i p h o s p h o r p y r y d i n e n u c l e o t i de and C1 is carbon dioxide. This kinetic scheme was analysed by D.S.Chernavsky, who assumed some concentrations remaining constant and ended up with the following differential equation set:
EMBED Equation.3 ,
EMBED Equation.3 .
Lets solve the set with Mathcad tools (fig. 3.18), using the numerical integration method with an adaptive step. The constants a0, a1, a2, b1, b2, b3 values have been chosen arbitrarily. The results show that there are periodical concentration oscillations, which decay over time.
Fig.3.18. Modelling the photosynthesis kinetics
The phase path is of the spiral form in this case. The spiral wraps around a critical point called focus.
The investigation of the critical point character is closely related to the question of the system stability. Here the chemical kinetics borrows some terms from the dynamic system theory, such as Lyapunovs stability criteria.
Without a deep discussion of the mathematical apparatus, we will show how the mathematical suites allow one to determine the critical point type. Assume the mathematical model of a process described by a set of two differential eqations. In order to find the critical point type one has to:
calculate the critical point coordinates on a phase plane. For this one has to solve the corresponding algebraic equation set, which is obtained through the equating of desired functions derivatives to zero;
compute the Jacobian matrix for the system using the critical point coordinates;
find the eigenvalues EMBED Equation.3 , E M B E D E q u a t i o n . 3 o f t h e l a t t e r m a t r i x .
O b t a i n e d e i g e n v a l u e s a l l o w o n e t o e s t a b l i s h t h e c r i t i c a l p o i n t t y p e a n d t h e s t a b i l i t y o f t h e s t a t i o n a r y s t a t e . S i x c a s e s a r e p o s s i b l e h e r e . T h e y a r e o u t l i n e d i n f i g . 3 . 1 9 .
1 , 2 :><?;5:A=K5 8 R e ( i ) = 0
E M B E D P B r u s h
&5=B@ 1 , 2 459AB28B5;L=K5 8 @07=>3> 7=0:0
E M B E D P B r u s h
!54;> 1 , 2 459AB28B5;L=K5 8 >B@8F0B5;L=K5
E M B E D P B r u s h
#AB>9G82K9 C75; 1 , 2 459AB28B5;L=K5 8 ?>;>68B5;L=K5
E M B E D P B r u s h
5CAB>9G82K9 C75; 1 , 2 :><?;5:A=K5 8 R e ( i ) < 0
E M B E D P B r u s h
#AB>9G82K9 D>:CA 1 , 2 :><?;5:A=K5 8 R e ( i ) > 0
E M B E D P B r u s h
5CAB>9G82K9 D>:CA F i g . 3 . 1 9 . P o s s i b l e c r i t i c a l p o i n t t y p e s a n d p h a s e p o r t r a i t s v e r s u s d i f f e r e n t J a c o b i a n m a t r i x e i g e n v a l u e s
A s a n e x a m p l e w e w i l l c onsider the previously discussed microorganism propagation model with slight modifications. Lets assume that the poison produced during the microorganism life can decompose (for example, by means of the Suns radiation). The poison decomposition will represent an elementary zero-order reaction with the rate constant k3. The new mathematical model will look this way:
EMBED Equation.3 .
EMBED Equation.3 .
The solution of the direct kinetic problem is shown in fig. 3.20. We can see that in the case of the assumed rate constants the microorganism population is oscillating. The critical point type is the node, because all the Jacobian eigenvalues are imaginary.
Fig.3.20. Oscillation mode of the population trend in microorganism colony
The first chemically grounded model of a oscillating reaction is a model, which was proposed by I. Prigozhin and is called bruesselator. The model is based on a hypothetical reaction with the following mechanism:
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 .
It is assumed that the concentrations of the reagents A and B do not change over time. The concentrations of D and E are not included in the mass action law. That is why one need only two equations for the formal kinetic description of the reaction:
EMBED Equation.3
EMBED Equation.3 .
It is possible to reduce the number of the controlling parameters in this system by substituting some variables: EMBED Equation.3 EMBED Equation.3 , EMBED Equation.3 . After these changes the system takes the form:
EMBED Equation.3 ,
EMBED Equation.3 ,
where EMBED Equation.3 , EMBED Equation.3 .
A remarkable feature of the bruesselator is the variety of the critical point types and, consequently, of the phase portraits depending on the a and b parameters ratio (table 3.3).
Table 3.3 Critical point types for the bruesselator
Parameters a and b ratioCritical point typeb < (a-1)2Stable node
b < a2+1Stable focus
b = a2+1Centerb > a2+1Unstable focus
(limit cycle)b > (a+1)2Unstable node
The case b>a2+1 requires additional examination. The critical point type is the unstable focus. One can see the appearance of the so-called limit cycle in the phase portrait. In this case, any point in the phase plane will end up following the same closed phase path regardless of the initial conditions. This means that stable asymptotic concentration oscillations (auto-oscillations) of the same amplitude and frequency will appear with the course of time. Correspondingly, this case is essentially different from the Lotka-Volterra model. In the latter, one can find closed phase paths as well, but there is no the only path that does not depend on the initial conditions. The point set, which attracts all phase paths, was called by I. Prigozhin an attractor. Thus, the bruesselator has the attractor, while the Lotka-Volterra system does not have. The appearance of the bruesselator limit cycle can be seen in fig. 3.21. The Mathcad tools were used to plot the phase portrait assuming a=1, b=3,25.
Nowadays many real chemical systems are known, in which processes, accompanied by the concentration oscillations, take place. These can be both heterogeneous and homogeneous reactions. In particular, the hydrogen peroxide reduction on the mercury drop surface can progress periodically in specific conditions. The conjugate process of mercury surface oxidation is accompanied by a change of the surface tension. It leads to the drop shape changes. The oscillating mode of the reaction can be observed through the periodical changes of the mercury drop shape, which resembles a heartbeat (mercury heart).
Fig.3.21. Brusselator phase portrait with a limit cycle
The oscillating reactions in homogeneous aqueous media are of a special interest. Probably, the oxidation of organic acids and their esters by the bromate ion is investigated the most. B.P.Belousov (1951) observed the periodic color changes during the oxidation of citric acid by bromate ion in sulfuric acid solution in the presence of cerium ions. The detailed investigation of this process was done by A.M.Zhabotinsky. The discovery of this reaction stimulated the investigation of periodical processes in chemical systems. It became evident that homogeneous oscillating reactions underlie the most important biochemical processes: generation of biorhythms and nerve impulses, muscles contraction, etc. As of today, the reaction of catalytic oxidation of different reducing agents by bromic acid (HBrO3I), following the auto-oscillating mode, is called the Belousov-Zhabotinsky reaction. This reaction goes in the acidic water solution and is accompanied by the concentration oscillations for the oxidized and reduced catalyst forms and intermediates. As a catalyst one can use transition metal ions, such as manganese or cerium. The reducing agents can be different organic compounds (malonic acid, acetylacetone, etc.)
We want to mention that, in spite of many publications dealing with Belousov-Zhabotinsky reaction, the true mechanism of this process in still unknown. Many kinetic schemes were proposed to explain the existence of the concentration oscillations. One of the possible mechanisms is shown in the table 3.4.
Here one can find several important conjugated processes.
1. During the step (1) HBrO2 is formed. It acts as an autocatalyst in the following reactions.
2. The extensive chain reaction of the oxidant BrO3 " w i t h t h e a u t o c a t a l y s t p r o v i d e s t h e c o n d i t i o n s f o r t h e M e + i o n s o x i d a t i o n ( s t e p s 4 7 ) .
3 . T h e o x i d a t i o n i s i n h i b i t e d d u e t o t h e c h a i n t e r m i n a t i o n ( s t e p 3 ) .
4 . T h e o x i d i z e d f o r m o f t h e c a t a l y s t i s r e d u c e d d u r i n g s t e p 1 4 .
T h e w a y o t h e r r e a g e n t s r e a c t c a n b e deduced from the given scheme. We have to admit that in spite of the large number of steps, this kinetic model should be considered as simplified. However, the solution of the direct kinetic problem for this scheme at the given conditions (see table 3.4) shows the presence of stable concentration oscillations. A fragment of the corresponding Mathcad document is shown in fig. 3.22. Here the mathematical model was developed in compliance with the kinetic scheme given in the table 3.4. It was assumed that the h y d r o g e n i o n c o n c e n t r a t i o n i s c o n s t a n t d u r i n g t h e r e a c t i o n . O n e c a n f i n d t h e c o r r e s p o n d i n g d o c u m e n t o n t h e b o o k s s i t e .
T a b l e 3 . 4 P o s s i b l e m e c h a n i s m o f t h e B e l o u s o v - Z h a b o t i n s k y r e a c t i o n
! S t e p n u m b e r R e a c t i o n K i n e t i c p a r a m e t e r s v a l u e s 1 1 - 2 E M B E D E q uation.3 k1=2,1; k2=1,0.10423 EMBED Equation.3 k3=3,0.10634-5 EMBED Equation.3 k4=4,2 ; k5=4,2.10746-7 EMBED Equation.3 k6=8,0.104 k7=8,9.10358 EMBED Equation.3 k8= 3,0.10369-10 EMBED Equation.3 k9=8,0.109 k10=1,1.102711 EMBED Equation.3 k11= 4,6.103812 EMBED Equation.3 k12= 1.106...1.107913 EMBED Equation.3 k13=1,0.1061014 EMBED Equation.3 k14=2,0.10-11115 EMBED Equation.3 k15= 3,20.109
Fig.3.22. Concentration oscillations in the BelousovZhabotinsky reaction
Somewhat different scheme for the Belousov-Zhabotinsky reaction was suggested by Field, Kors and Noyes. The model is called oregonator. It includes following stages:
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 .
Here A corresponds to the BrO3 " i o n ; B c o r r e s p o n d s t o a l l o r g a n i c r e a g e n t s t h a t c a n b e o x i d i z e d ; P i s H O B r ; X i s H B r O 2 ; Y i s t h e B r " i o n ; Z i s t h e r e d u c e d f o r m o f t h e c a t a l y s t . T h e m a t h e m a t i c a l m o d e l c a n b e w r i t t e n a s a s e t o f t h r e e d i f f e r e n t i a l e q u a t i o n s :
E M B E D E q u a t i o n . 3 ,
E M BED Equation.3 ,
EMBED Equation.3 .
It is assumed that the concentrations of the compounds A and B remain constant during the reaction. By using dimensionless variables, one can transform the set:
EMBED Equation.3 ,
EMBED Equation.3 ,
EMBED Equation.3 ,
where EMBED Equation.3 ; EMBED Equation.3 ; EMBED Equation.3 ; EMBED Equation.3 . Now the model has three controlling parameters:
EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ,
values of which influence greatly the system dynamic behavior.
The document shown in fig. 3.23 can be used as a template in the computer modeling of the oregonator model. By changing the controlling parameters, one can see a variety of the reaction modes with different amplitudes and oscillation frequencies, an appearance of limit cycles, changes in phase paths trajectories, etc. The value of the stoichiometric factor f is also of great importance. Compared to bruesselator, this model is more complex in analysis of possible stationary states and plotting of the phase portraits. The reader can find more details in the specialized literature.
Fig.3.23. One of the direct problem solutions for the oregonator problem
3.5. Some points on non-isothermal kinetics
By this point, we were considering only the chemical kinetics problems in case of constant temperature of the reaction mixture. But the temperature can change due to ambient conditions (forced heating of cooling) as well as due to internal factors (heat liberation or adsorption during reaction). Previously discussed methods are not sufficient in this case. The mathematical model of the reaction becomes complicated because the temperature is now a function of time. While equations describing a material balance of the system are sufficient for isothermic kinetics, in case of altering temperature one have to consider energy balance as well.
If the reaction has a thermal activation character, changes in temperature lead to changes the rate constant. This relationship is often described by the Arrhenius equation:
EMBED Equation.3 ,
where EMBED Equation.3 is the activation energy (J/mol), EMBED Equation.3 is the pre-exponential factor. The Arrhenius equation is based on the collision theory. The theory exploits ideas of an energy barrier and effective collisions of the reacting particles, which happen in a unit of space over a unit of time. The k0 value is proportional to the total collision number. The activation energy determines the energetic conditions for an active collision the collision when a transformation of the reagents into the products is possible. It is a certain excess of energy in comparison with the average reactants energy, which have to be applied for the reacting species to react. The Arrhenius equation implies that the reaction rate increases when the temperature rises. The smaller the activation energy is, the greater such increase will be.
So, the rate constant depends on temperature. In the case of the altering temperature the rate constant also becomes a function of time. Consequently, when solving the direct kinetics problem, we have to add the corresponding equations (the temperature over time relationships) to the reaction model.
Lets consider one of the non-isothermic kinetics cases. Some amount of germanium (IV) chloride is being heated. The heating is accompanied by a consequent decomposition:
EMBED Equation.3 ,
EMBED Equation.3 .
Assume the heat exchange is organized in a way that the heating appears with a constant rate EMBED Equation.3 10 K/min. One is asked to establish how the gross mass of solids will change if 0.002 mol of GeCl4 is being heated. The initial temperature is 298 K. It is known that the Arrhenius relationships for the rate constants have the following form:
EMBED Equation.3 , EMBED Equation.3 .
In these equations the pre-exponential factors have dimensions of min "1 , a c t i v a t i o n e n e r g i e s a r e g i v e n i n c a l / m o l .
A m a t h e m a t i c a l m o d e l o f t h e p r o c e s s c o n s i s t s o f t h r e e d i f f e r e n t i a l e q u a t i o n s t h a t d e s c r i b e t h e c h a n g e s i n r e a c t a n t a m o u n t s :
E M B E D E q u a t i o n . 3 ,
E M B E D E q u a t i o n . 3
E M B E D E q u a t i o n . 3 ,
a s w e l l a s of a heating rate equation:
EMBED Equation.3 .
Initial conditions for the given ODE set are: EMBED Equation.3 =0,002; EMBED Equation.3 ; EMBED Equation.3 =0; EMBED Equation.3 .
To integrate this ODE set one can use any of the previously discussed mathematical suite built-in tools. In fig. 3.24 we have shown how to solve the problem using the Mathcad built-in function AdamsBDF. The plots allow one to track the trends of compound concentrations over time. As one can see, during the first 15 mins of heating the amount of the starting material virtually remains constant. After that, decomposition occurs with a notable rate. When GeCl4 has decomposed, the solid phase of the reaction mixture consists solely of GeCl2. This composition remains unchanged until approximately the 30th minute of heating, when the intermediate begins to decompose into the final product. Finally, some time later the mixture will consist of pure germanium. One can track the change in mass of the initial sample in the same way.
When describing the processes of non-isothermic kinetics, it is convenient to use a unitless variable conversion of the starting compound X. For example, if we have a single first-order reaction under the programmed temperature changes conditions, we can describe its kinetics with a set of two equations:
EMBED Equation.3 .
with initial conditions EMBED Equation.3 , EMBED Equation.3 . The numerical solution of this set for given kinetic parameters values and linear heating mode (fig. 3.24) shows that a conversion vs. time plot has a distinct S-shape. A slope of such a curve changes depending on the given temperature change rate. It important to note that such curve can be obtained experimentally with a special device called derivatograph. Information about the system behavior within a given temperature range is completely enough for solving the inverse problem, i.e. for the determination of the reaction kinetic parameters using experimental data.
Fig.3.24. Solution of the GeCl4 decomposition problem
Fig.3.25. Conversion vs. time and temperature for different heating rates
We want to point out that so far we considered only the cases when the temperature changes were controlled by external factors (i.e. was determined by the experiment conditions).
If an exothermic reaction takes place in an isolated system, in other words, when the heat exchange with environment is absent (adiabatic reactor), a temperature will apparently increase over time. The rate of this increase de p e n d s b o t h o n t h e k i n e t i c p a r a m e t e r s ( r a t e c o n s t a n t ) a n d o n t h e t h e r m o d y n a m i c p r o p e r t i e s o f t h e s y s t e m ( t h e r m a l c o n d i t i o n s o f t h e r e a c t i o n , h e a t c a p a c i t y ) . F o r a w e l l - m i x e d p e r i o d i c r e a c t o r , w h e r e a s i n g l e f i r s t - o r d e r r e a c t i o n A !B o c c u r s , t h e m a t h e m a t i c a l m o d e l i s d e s c r i b e d b y t h i s s e t o f e q u a t i o n s :
E M B E D E q u a t i o n . 3 ,
E M B E D E q u a t i o n . 3 ,
E M B E D E q u a t i o n . 3 .
H e r e i s d e n s i t y , k g / m 3 , a n d C p i s s p e c i f i c h e a t c a p a c i t y o f t h e r e a c t i o n m i x t u r e , J / ( k g . K ) . H i s t h e r e a c t i o n h e a t e f f e c t ( t a k i n g t h e sing into account), J/mol. To be specific, these parameters depend on temperature. In addition, heat capacity and density can change as the reaction goes. One should account for that when performing important calculations. In order not to overcomplicate, we assume that these values are constant. We will define an additional parameter: EMBED Equation.3 . One can see that
EMBED Equation.3 ,
An integration of this equation with the initial conditions T(0)=T0, EMBED Equation.3 gives:
EMBED Equation.3 .
In the case of adiabatic conditions, the system will warm up and reach the final temperature that corresponds to an exhaustion of the reagent:
EMBED Equation.3 .
The temperature Tad is called adiabatic temperature.
The modeling results for the behavior of this system are shown in fig. 3.26. The initial parameters were: EMBED Equation.3 =1.105 s-1, ET=Ea/R=5000K, EMBED Equation.3 kmol/m3, J=100K.m3/kmol. I K l n 4 > G O W d i ݷݷݷzezP;P (h h^ B*CJ \aJ mH ph sH (h hZ_p B*CJ \aJ mH ph sH (h hnK B*CJ \aJ mH ph sH (h hb B*CJ \aJ mH ph sH (h h -D M
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Fig.3.26. Temperature and reagent concentration changes in a periodic adiabatic reactor
In the real conditions, some heat from the reaction is liberated into environment through the reactor walls. A differential equation for the temperature changes is given in the following form in this case:
EMBED Equation.3 .
In order to follow operational trends of such non-adiabatic reactor, one have to introduce additional parameters: heat transfer coefficient h, W/(m2.K), reactor volume V, m3, wall surface S, m2. In fig. 3.27 one possible way to compute the reagent concentration and reaction mixture temperature as a function time is shown.
Fig.3.27. Operation dynamics of a periodic nonadiabatic reactor
For a well-mixed flow reactor working in an adiabatic mode:
EMBED Equation.3 ,
EMBED Equation.3 ,
Here EMBED Equation.3 is, and T0 is the reagent temperature when entering the reactor. The solution of Cauchy problem for this reactor type allows one to conclude: the dynamic portrait can change strikingly depending on temperature of the initial mixture. Such situation is illustrated on fig. 3.28. One can see that for the time (=60 s many different kinetic curves as well as temperaturetime relationships are possible, even though the initial temperatures differ only for 1 K. In both cases a stationary state is reached. However, for T0 = 274K the stationary conversion is low and does not exceed 18%. If the initial temperature e q u a l s 2 7 5 , o t h e r s t a t i o n a r y s t a t e i s r e a c h e d . T h e l a t t e r c o r r e s p o n d s t o r a t h e r h i g h c o n v e r s i o n ( > 8 4 % ) .
F i g . 3 . 2 8 . T e m p e r a t u r e a n d c o n c e n t r a t i o n t r e n d s i n a f l o w a d i a b a t i c r e a c t o r
T h u s , t h e p r o c e s s e s t a k i n g p l a c e i n t e c h n o l o g i c a l r e a c t o r s c a n h a v e a multistationarity even for relatively simple kinetic schemes (in this case we consider a simple non-reversible first-order reaction). In practice, reactors work usually in the conditions close to stationary. Therefore, a problem of optimal organization of the reaction conditions becomes of great importance. In the discussed example the first stationary state is undesirable from the efficiency point of view.
For example, assuming A=1.105 s-1, Ea/R=5000K, J=100 K.m3/kmol, (=60s, initial concentration CA0 = 1 kmol/m3, and initial temperature T0 = 270 K, we can find three stationary states. Their quantitative properties are determined by the solutions of an algebraic equation set, to which the differential equation set is transformed when both EMBED Equation.3 and EMBED Equation.3 equal zero:
EMBED Equation.3 ,
EMBED Equation.3 .
Fig.3.29. Computations of possible stationary states and analysis of their stability
The presence of multistationarity can be illustrated in other way. The last two equations allow one to conclude that
EMBED Equation.3 .
A left part of the obtained correlation depends linearly on the temperature T. Its value is proportional to the cooling rate caused by a hot (T>T0) airflow out of the reactor. A right part corresponds to the heat generation rate in the reactor due to the reaction exothermicity. It is a non-linear function of temperature. If one plots the temperature dependences of the equation right and left parts, one will see their interception in points corresponding to the calculated stationary temperatures and concentrations (fig. 3.30).
Fig.3.30. Graphical representation of possible stationary states
When considering systems with many stationary states, it is important to investigate the stability of the latter. Stability of a stationary state is directly connected to the thermal stability of the reactor. It may happen that a small perturbation of the system takes it out of the unstable state. The process will convert into the other one, now stable. In this case calculations (fig. 3.29) show that two out of three possible stationary states are stable: for them the Jacobian matrix eigenvalues are real and of the same sign (stable node). The third stationary state has real, but negative, Jacobian eigenvalues (saddle point). A comparison of these results with the plots shown in fig. 3.30 allows one to conclude: a stationary state is stable if a slope of the heat elimination curve is smaller than a slope of the heat liberation.
Finally, we can prove the conclusions by plotting a phase portrait of the system (fig. 3.31). Here dots correspond to the possible stationary states. The phase paths for different initial temperatures are marked with bold lines.
Fig.3.31. Phase portrait for exothermic reaction in an adiabatic flow reactor
Fig. 3.31 allows us to see that one or another stationary state is realized according to the initial conditions.
The discussed examples by no means cover all possible problems of chemical kinetics as well as other differential models for chemical-engineering processes, on which the reactor theory is based.
We do not discuss such mathematical models in this book.
In this book we will not discuss the essence of such and such numerical integration methods. This information is available in virtually all handbooks on numerical methods.
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