DYNAMICS OF A CATALYTIC SYSTEM

DYNAMICS OF A CATALYTIC SYSTEM

Joyné Contreras Decanato de Agronomía- Dpto.de Gerencia y Estudios Generales, Universidad Centroccidental “Lisandro Alvarado", 400 Barquisimeto, Venezuela, (joynecontreras @ucla.edu.ve) Wilfredo Angulo Departamento de Física, Universidad Centroccidental “Lisandro Alvarado", 400 Barquisimeto, Venezuela, ([email protected]) Postgrado en Matemática, Facultad de Ciencias, Universidad Central de Venezuela,Caracas, Venezuela

Abstract: We presented some cualitative aspects concerning to the solution for a mathematical model governing the dynamical behavior of the reversible chemical 1 carried out in a catalytic reactor used in the process of SO O SO 2( g )

2

2( g )

3( g )

sulfuric acid production. Keywords: dynamical behavior, chemical reaction, catalytic reactor

1.

INTRODUCTION

Constantly, the engineering of processes faces the problematic to determine optimal ranks of operation, control systems on the processes that allow improving them continuously and make sure the good operation of the equipment that operates in the production plants, in order to guarantee, at any moment, the quality of their products. In the search of methods that contribute solutions for this problematic, the engineers have seen themselves in the necessity to deepen the knowledge on the dynamics of the processes in order to understand the significant aspects of their behavior, and to be able to raise this behavior in mathematical models that allow to describe and to study the processes in detail. In this sense, the contribution that can give the application of mathematical methods in the formulation and resolution of these models is fundamental and due to the great development of the computation the numerical methods allow to obtain numerical and graphical solutions of the problems that model real situations; but these solutions must be validated experimentally and complemented with a qualitative study of the process behavior to confide in the numerically obtained results and to be able to use them as reference to determine operation ranks, to predict the behavior of the systems, to develop new processes or improve existing ones using the dynamical simulations based to the mathematical model of the process. The present work was realized to validate the numerical solutions of a differential equations problem that models the dynamic behavior of a catalytic reactive system.

2. MATHEMATICAL MODEL The studied catalytic system was the oxidation of sulfur dioxide to sulfur trioxide. The stoichiometric 1 equation SO . This reaction is exothermic in the direct sense, , therefore energy is O SO 2( g )

2

2( g )

3( g )

freed and an increase of temperature in the bed of the catalytic reactor is originated. The reaction is a homogenous mixture, its reactants and products are in gaseous phase to the conditions of operation. The speed of this reaction has studied widely and the expression that adapts well is the Eklund's equation (Mars, P. and Maessen, J. 1968). For the sake of simplicity, we consider a fixed volume element of catalyst bed, with cylindrical geometry of finite length L and radio R, in which the reaction is carried out. We assume that gradients of concentration and temperature in the radial direction of the catalyst bed do not exist. We consider only the variation of the concentration and temperature with respect to the time, and we assume that the changes in the total pressure of the system with respect to the time are negligible in the exit of each bed into the catalytic reactor.

The mathematical model was obtained by a dynamic balance of matter and caloric energy; it is described by the following equations:

rA 1

dX A (t ) dt

X A (t ) 1 C A0

C

(1)

rA 1

dT (t ) dt

C A0

X A (t ) 1 i

i

Cpi

C

X A (t ) Cp

Hr CI CpI

with the initial data XA(0)=XA0 and T(0)=T0 for the state variables XA(t) and T(t), conversion and temperature respectively. The subscript A it was used to denote component SO2 and the subscript I to denote inert presents in the mixture as bimolecular nitrogen. Therefore, XA represents the molar f ( X A , T ) is the Eklund's expression (Carberry, J. 1976, conversion of SO2 in the mixture, rA Fogler, S. 2001) written in terms of the molar conversion and temperature of the system. On the other hand , , C , C A0 , CpI , , Cp, CI , CpI are (constant) given physical parameters. This model is complemented with the following relations: Coefficient of chemical equilibrium:

kp

11829.44 11.24 T (t )

exp

(2)

Kinetic coefficient of reaction rate:

exp

97782.22 110 ln(1.8T (t )) 912.8 T (t )

(3)

Hr

34923.286 65.395T (t ) 0.0725T (t ) 2

(4)

k Heat of reaction:

For the mathematical abstraction of the model we redefined the two state variables (conversion and temperature) of the following way:

u1 such that, for all time t

0,

variables to determine in the subset

u1 (t )

X A (t ), u2

, u (t )

u1 (t ), u2 (t ) is the vectorial function of the two state

 2 given by:

0, u1* with

u1* 1

, 0

u2

u2 (t ) T (t )

u2 , u2

1, and where are taken as:

min 0 T (t )

t 0,

, u2

max 0 T (t )

t 0,

The right side of each EDO in (1) is a real valued function defined on

f (u )

f1 (u ), f 2 (u )

3. SOLUTIONS OF THE MATEMATICAL MODEL We began the dynamic analysis of chemical reaction determining the states stationary. For the reaction, the steady-states are given by the following subset:

ue

u1e , u2e

; f ue

0, u1e

0, u1* , u1*

1

, u2e

h u1e

(5)

The subset

represents a curve of steady-states which the values of right side of each EDO are zero.

confined within the domain

and divides

Figure 1 illustrates the continuous

in two subset:

1,

of steady-states and the subsets

Fig. 1. Curve of steady-states

is

2.

1,

and subsets

2.

1,

2

Using numerical simulations we can see that curve divides to domain in two subset which the solutions o mathematical model present two behaviors. In order to validate the behavior of the solutions into the both subset 1, 2, we realized a qualitative analysis of the solutions of the problem for great times. The global existence and uniqueness of the solutions of dynamic state for the problem (1), are a direct consequence of the associated global Lipschitz property to the vectorial field (Amann H. 1990, Coddington E. and Levinson N. 1995). In this qualitative analysis the following propositions are demonstrated:  , i 1, 2, satisfies a global Lipschitz condition on Proposition 1: The scalar field fi : .

C1 0,

Proposition 2: The problem (1) has a unique solution u initial conditions

u 0 (t0 )

for all t0

0,

,

that verifies the given

.

The Proposition 2 says that curves trajectories of the system for any initial condition in are unique and they are not crossed; meaning that for any initial condition of the process a unique trajectory for the chemical reaction is had, thus any final condition or any point (XA ,T)can be determined by this trajectory. Great part of the dynamic analysis of the reactive systems centers its attention in predicting what will happen to the state variables when these evolve from an initial time t0 or, equivalent, to establish the dynamic behavior of the system for all time t≥t0. Our numerical simulations generated by a computer code based on the method of Runge-Kutta to fourth order suggest that exists sufficient conditions to affirm that all solution of the system whose initial value is into 1 or 2 remains confined on 1 or 2 and tends to steady-state in when the time tends to infinite. In order to verify these numerical results the following propositions were demonstrated. Proposition 3: Let be

u '(t )

,

2

a point in

f (u (t )) is strictly positive on

Proposition 4: Let be

u '(t )

1

1

,

2

1 for

a point in

f (u (t )) is strictly negative on

1

2 2 for

where a solution of problem (1) begins. The vectorial field all time t in the interval I

1

,

2

0,

.

where a solution of problem (1) begins. The vectorial field all time t in the interval I

1

,

2

0,

.

Based on the two previous propositions, the following corollary establishes the important property on the behavior of the solutions in 1 and 2. Corollary 1: For each i=1,2, solutions ui(t) of the problem (1) are increasing functions on 1 and decreasing functions on 2 for all time t≥t0. Finally, the following lemma was established: Lemma1: All solution u(t) of the problem (1) that begins in the region defined by subdomain i, for each i=1,2, at time t=t0 remains in that region for all future time t≥t0, and finally tends to the stationary solution in . Figure 2 illustrates Lemma 1 which guarantees solutions of the system that begin at subdomains, 1 or 2 they tend to a steady-state located on curve when the time tends to infinite whatever the initial condition of starting within each of region. In addition, the figure shown situations on each subdomain that cannot happen, these situations are illustrated for the curves drawn up by segments that start in 1

,

2

on

1

and

1

,

2

on

2.

Fig. 2. Situations that cannot happen on each subdomain The numerical solutions of mathematical model shape existence of two clearly defined regions, 1 and 2, and these regions are understood as the regions where it is carried out the direct chemical reaction and the inverse chemical reaction, respectively. This is illustrated on numerical simulations for phase portrait (figure 3), and figures 4 and 5 show the numerical simulations of Conversion vs. time and Temperature vs. time for the four initial conditions taken within 1 and the four initial conditions taken within 2 on the phase portrait.

Figure 3. Numerical Solutions for phase portrair

5. CONCLUSIONS: A simple numerical model for the simulation of the transport water-petroleum in horizontal pipes for one

Fig. 4. Numerical solutions for initial values at

Fig. 5. Numerical solutions for initial values at

1.

2.

4. BRIEF DISCUSION: With the previous results we demonstrated that on Ω1 the solution of the model (1) is increasing, which implies that indeed, when conversion of SO2, XA(t), is increasing the system has an increase of temperature, that is translated as energy liberation; with Corollary one demonstrates that region Ω1 is the region where exothermic character of the reaction predominates. Similarly, was proved that the region 2 is the region where endothermic character of the reaction predominates (diminution of conversion of SO 2 and diminution of energy of the system). We demonstrated, with this Lemma, that the solutions of problem (1) tend to a steady-state whatever the starting state and they remain in the origin region i for each i=1,2. This is coincident with the physical phenomenon that studies, since, experimentally, we know at very great times the reaction tends to the equilibrium because infinitesimal changes always exist in the conversion and the temperature; which defines states calls quasi-steady states. On phase portrait we can study the behavior of the system in order to determine optimal operations ranks for a wished final state. Also we can study possibility of carrying out the reaction in series processes in which the initial conditions of each step are the final conversion of previous step and suitable temperature to take the reaction to another state. This is possible providing or removing energy of the system, which will locate the reaction in a new initial state on the same subdomain of origin or the other subdomain; in this last case, the change of subdomain don’t is a natural behavior of the system but it is an changes the initial conditions due external factor of the problem, therefore, the solutions of the mathematical model must remain on the origin region while they are not disturbed.

5. CONCLUSIONS: We demonstrated that the mathematical model, for the dynamical analysis of the of the reversible 1 chemical SO O2( g ) SO3( g ) carried out in a catalytic reactor, is a well-possed Cauchy problem; 2( g ) 2

i.e., there exits an unique solution for each initial condition related to the state variables. The trivial solutions of the mathematical model correspond with the steady-states of the reactive system and it conform a continuous of points on the phase portrait for conversion versus temperature. In fact, the phase portrait was divided in two separated regions, 1 and 2, by the continuous of steady-states. We demonstrated that in a region 1 the reaction advances exothermically in the direct sense and in the 2 it advances endothermically in the inverse sense as we must hope from the physicochemical point of view. Also, we are demonstrated when the time becomes sufficiently great, conversion of the SO2 and temperature of system reactive will remain near to some steady-state whatever initial condition in the phase portrait.

6. REFERENCES Amann H. (1990). Ordinary Differential Equations: An Introduction to Nonlinear Analysis. De Gruyter. Walter. Carberry, J. (1976). Chemical and Catalytic Reaction Engineering. Mc Graw Hill, USA. Coddington E. and Levinson N. (1995). Theory of Ordinary Differential Equations. McGraw-Hill, New York. Fogler, S. (2001). Elementos de Ingeniería de las Reacciones Químicas, Editorial Prentice Hall, 3ra Edition. Mars, P. and Maessen, J. (1968). The Mechanism and the Kinetics of Sulfur Dioxide Oxidation on Catalyst Containing Vanadium an Alcali Oxide, Jour. of Catal., 10 , pp. 51--55.