Solution of the convection–dispersion–reaction equation by a sequencing method

doi:10.1016/S0098-1354(02)00095-9

Computers & Chemical Engineering

Volume 27, Issue 5, 15 May 2003, Pages 615-629

https://doi.org/10.1016/S0098-1354(02)00095-9 Get rights and content

Abstract

A new approach for solving convection–diffusion–reaction equations is presented. The method is based on the separation of the different phenomena. At each time step, convection, diffusion and reaction are applied successively on the reactor mesh. This sequencing method allows us to solve both hyperbolic and parabolic equations and to extend the frequency response of the numerical solution. Simulation results are given for linear and nonlinear reaction kinetics. These are compared with the exact solution for the high diffusion case and with up-wind finite difference methods for all cases.

Introduction

Tubular reactors have a distributed nature and their mathematical formulation leads to a system of partial differential equations (PDEs). Modeling of an isothermal tubular reactor, using mass balances, leads to convection–dispersion–reaction equations. The resulting well-known dispersion model for a non-ideal reactor is composed of second order parabolic equations. Several common numerical methods have been used to approximate PDEs by ordinary differential equations (ODEs) for numerical simulation and control design: finite difference methods (FDM), finite element methods (FEM) and orthogonal collocation methods (OCM) (Gerald & Wheatley, 1990, Villadsen & Michelsen, 1978, Varma & Morbidelli, 1997, Reddy, 1993). In the specific case of hyperbolic PDEs, the method of characteristics give an exact transformation of the PDEs into ODEs (Farlow, 1993). Recently, other developments occurred to address specific problems such as shock wave propagation: moving finite element method (Sereno, Rodrigues & Villadsen, 1991, Sereno, Rodrigues & Villadsen, 1992), up-wind finite element method (Park, 1995), wavelet transform approach (Kosanovich, Moser & Piosovo, 1997) and many others.

Traditional numerical methods lead to satisfactory results in many cases, especially when the system has a clear behavior such as that of a plug-flow reactor (PFR) or a continuously stirred tank reactor (CSTR). Unfortunately, none of the above could be applied for the whole range of behaviors. The accuracy of the solution depends on the relative importance of the convection, dispersion and reaction terms. Moreover, for low dispersion systems, many problems may arise such as numerical diffusion or oscillation and even instability caused by inappropriate use of the Danckwerts’ boundary conditions.

The proposed method has been developed in the context of an on-line application, i.e. the numerical simulation of a PDE model within a control scheme for an industrial application. Therefore, accuracy is not the only key issue of the problem: stability, ease of use, CPU time demand, and adequate frequency response of the numerical scheme are all other important issues.

This paper presents a novel approach for solving partial differential equations. The solution is based on the time occurrence of phenomena instead of using a variational formulation. Here each phenomenon is solved successively at each time step. This simple approach exhibits stability and rapidness properties. Moreover, the frequency response of the scheme is enhanced by treating the convection phenomena separately. Therefore, our intent in this paper is not to show the superiority of the novel method over classical method regarding accuracy. The benefits of this method are more on the simplicity and real-time applicability front for a method that will capture a higher frequency content of input variations.

The next section of the paper describes the main concepts of this approach. It is followed in the third section by a detailed description of the algorithm. In the fourth section, the results from the novel approach are compared with the finite difference method and the analytical solution for the linear case. The fifth section shows an application to the nonlinear PDEs of a chlorine dioxide bleaching reactor application.

Section snippets

Description of the sequencing method

The dynamics of a tubular reactor can be modeled by transient mass balance on a thin transverse section. The resulting unsteady state convection–diffusion–reaction equation may be written as follow: $∂x(z, t) ∂t =−v ∂x(z, t) ∂z +D ∂^{2} x(z, t) ∂z^{2} −r(x(z, t))=0$ with the following closed–closed Danckwerts’ boundary conditions and initial concentration profile in the reactor: $D ∂x(0, t) ∂z =v(x(0, t)−x_{in} (t))$ $∂x(l, t) ∂z =0$ $x(z, 0)=x_{0} (z)$

In those equations, x represents the concentration in the reactor, v the superficial

Sub-problems resolution

For a time step Δt, N elements of dimension Δz are used to define the reactor mesh. The following values are used: $Δ z= l N$ $Δ t= Δ z v$

So the choice of the mesh dimension determines time and space discretization. This approach implies an equally spaced mesh but extension to a variable mesh is also possible. For varying flow, the time step is adjusted according to the preceding equations.

Resolution of the convection subsystem (, , ) is the simplest part of this algorithm. It is a delay system from an

Sequencing method analysis

A simple linear PDE equation is used as a first test for the sequencing method. Solutions are compared with the solution by FDM and to an analytical solution in the case of dispersive systems. The following system is considered: $∂x(z, t) ∂t =−v ∂x(z, t) ∂z +D ∂^{2} x(z, t) ∂z^{2} −kx(z, t)$ $D ∂x(0, t) ∂z =v(x(0, t)−x_{in} (t))$ $∂x(1, t) ∂z =0$ $x(z, 0)=x_{0} (z)$ where v=0.05 m min⁻¹, l=1 m, k=0.057 min⁻¹ and x_in=1 g l⁻¹. Three different values are considered for the Peclet number: 1, 10⁴, 10⁸. A Peclet number of 10⁸ represents a near

Nonlinear PDE: application to a bleaching reactor

Consider the following system of PDE, describing a chlorine dioxide bleaching reactor (Gendron, 1997): $∂C ∂t =−v ∂C ∂z +D ∂^{2} C ∂z^{2} −k_{C} C^{3} L^{3}$ $∂L ∂t =−v ∂L ∂z +D ∂^{2} L ∂z^{2} −k_{L} C^{3} L^{3}$ $∂L(0, t) ∂z = v D (L(0, t)−L_{in} (t))$ $∂C(0, t) ∂z = v D (C(0, t)−C_{in} (t))$ $∂L(1, t) ∂z =0$ $∂C(1, t) ∂z =0$ $L(z, 0)=L_{0} (z)$ $C(z, 0)=C_{0} (z)$ where the equations are space normalized to obtain a reactor length of 1 m. The parameters values are equal to: $k_{c} =5.5×10^{−5} g^{−3} l^{3} %^{−2} min^{−1}$ $k_{L} =5.7×10^{−7} g^{−2} l^{3} %^{−3} min^{−1}$ $v=0.05 m min^{−1}$

Typical industrial influent concentrations of this process are shown in

Conclusion

A new algorithm for simulation of convection–dispersion–reaction PDE systems has been presented. This algorithm is based on a time solution approach of the PDE model and is validated by phenomenological arguments. It implies the successive solution of a specific sub-problem describing each phenomenon. Its main characteristics include: solution of parabolic and hyperbolic problems, to enhance numerical model frequency response, to give adequate transient responses to sharp and noisy inputs and

References (14)

K.A. Kosanovich et al.
A new family of wavelets: the Poisson wavelet transform
Computers Chemical Engineering
(1997)
C. Sereno et al.
The moving finite element method with polynomial approximation of any degree
Computers Chemical Engineering
(1991)
C. Sereno et al.
Solution of partial differential equations systems by the moving finite element method
Computers Chemical Engineering
(1992)
J. Winkin et al.
Dynamical analysis of distributed parameter tubular reactors
Automatica
(2000)
R. Curtain et al.
An introduction to infinite-dimensional linear systems theory
(1995)
S.J. Farlow
Partial differential equations for scientists and engineers
(1993)
H. Fattorini
Boundary control systems
SIAM Journal on Control
(1968)

There are more references available in the full text version of this article.

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Solution of the convection–dispersion–reaction equation by a sequencing method

Abstract

Introduction

Section snippets

Description of the sequencing method

Sub-problems resolution

Sequencing method analysis

Nonlinear PDE: application to a bleaching reactor

Conclusion

Computers Chemical Engineering

Computers Chemical Engineering

Computers Chemical Engineering

Automatica

An introduction to infinite-dimensional linear systems theory

Partial differential equations for scientists and engineers

Boundary control systems

SIAM Journal on Control