1D and 2D modelling of a chemical process using the FM-II model: Rotary drum filter

Abstract

This work treats the physical one-dimensional (1D) and two-dimensional (2D) modelling of a chemical process of filtration of slurry, which is the second step of phosphoric acid manufacture. This work focuses on the different physical laws involved in the filtration stage in order to obtain a simulator of the filter. In this work, we are interested in the 1D modelling of a rotary filter using all parameters and physical phenomena established in the filtration phase. Then, a 2D model emerged from the previous model by choosing both a temporal and a spatial variable. These two parameters are quite necessary for the construction of a 2D model based on the method of Fornasini–Marchesini (FM-II) to describe the dynamics of the system.

Keywords

Filtration Darcy law state-space model 1D system 2D system FM-II model

Introduction

In recent years, membrane filtration technology, especially low pressure membrane technology, has been more widely applied in liquid aspiration fields (Pei et al., 2007).

Well known for some time, the membrane separation processes are considered advanced, efficient, and beneficial technologies to use. These methods are used to separate suspended particles. The filtration technique is recognized in porous membranes and follows the pore size. There are four levels of filtration: macrofiltration, microfiltration, ultrafiltration, and nanofiltration (Zart, 2008). Filtration processes can be classified into two categories:

discontinuous process: the press filter and the Nutshe filter are the most prevalent;

continuous process: the rotary drum filter and the pass band filter are the most prevalent.

In our case, a rotary drum filter will be described which ensures the separation between the phosphoric acid and the phosphogypsum. In the literature, few works have dealt with the filtering modelling due to its complexity and the number of parameters interfering in the filtration process. The problem is therefore to successfully model the system to be studied. So, we want to develop a model close to the real system.

The modelling filter is essentially based on Darcy’s law treated by Nicolas (2003) which applies to a homogeneous and isotropic porous middle. This middle is traversed by a flow at low speed in order to calculate the rate of phosphoric acid produced. Darcy’s law is also processed by Chassagne (2010) and a similar strategy has been discussed by Darcy (1938). Then a relationship frequently quoted was proposed by Robert and Obertin (2003) and modified by Carman (1938). The multiple physical equations that are based on the parameters such as specific resistance using the method of Wu and Zhang (2010) are exploited. As Robert and Obertin (2003) has treated the coefficient of permeability and permeability models whilst seeking to establish an expression depending on the geometry of the pore network. Finally the porosity (the main parameter) for describing a porous middle is outlined by Nicolas (2003).

In this research a pseudo-homogeneous two dimensional model was proposed to describe the dynamic behaviour of a rotary filter. The interest in two-dimensional (2D) systems goes back to the early seventies (Fornasini and Marchesini, 1978), and has gained momentum due to its extensive use. 2D systems have drawn considerable attention because of their wide range of practical applications and the significant theoretical importance. 2D system theory can be used efficiently in many fields such as multidimensional signal processing and digital filtering, multi-variable network modelling, digital picture processing, seismic data processing, thermal processes, and batch processes (Kaczorek, 2006; Liu and Wang, 2012; Renard, 1996; Tse, 1973). Therefore, considerable interest has emerged in the theoretical analysis and synthesis of 2D systems, resulting in a large number of compelling research results in the literature. For instance, investigations on 2D systems have been conducted related to the state-space model realization (Chen, 1999; Kaczorek, 2006), stability and stabilization (Chesi and Middleton, 2014; Souza and Osowsky, 2013), estimation and observers (Bisiacco, 1985a; Liang et al., 2014; Xu et al., 2012), and filtering (Gao, 2013; Wu and Wang, 2015; Zhao et al., 2014).

More recently, many other interesting contexts where 2D systems prove to be the appropriate setting have been enlightened (Bisiacco and Valcher, 2015; Xu et al., 2004). Indeed, in the one-dimensional (1D) context, there has been a long stream of research on this subject, which originated in the seventies and flourished in the eighties (Du and Xie, 2002; Kurek, 1983), but still represents a very lively topic of research. During the last thirty years, research interests in 2D system theory have been focused on most of the classical topics already investigated in the 1D setting (Bisiacco, 1985b; Fornasini and Zampieri, 1990; Kaczorek, 2002; Liang et al., 2014). More generally the most interesting point of the 2D systems is related to the fact that they depend on two invariant variables where generally one of them is time and the other is spatial. However there are many 2D systems such that neither of their variables are time or spatial (Kaczorek, 2008; Liu, 2015; Paran and Adloo, 2004; Shong Hong, 2010). Moreover, one of the fundamental issues in 2D systems theory is the realization of a given transfer function or transfer matrix by a certain kind of 2D local state space model, typically by the Roesser model or the Fornasini–Marchesini second (FM-II) model.

The rotary filter is a fairly giant industrial process that works in real time. In this context, it is impossible to stop the system to do tests, to take measures or to make variations on the control variables. So here we need a 1D model close to the real system that allows us to perform simulations and to better understand the operation of the actual process and to control it. Regarding the 2D model, it is more accurate than the 1D model because it is based on two spatial and temporal parameters.

Our contribution is mainly based on the modelling of a rotary drum filter using physical laws and with the realization problem for a given 2D Multi-Input Multi-Output (MIMO) system by FM-II model. This paper is organized as follows: Section ‘1D non-linear modelling of rotary drum filter’, describes the 1D non-linear modelling of the filter using several laws. Section ‘2D modelling based on FM-II method’ describes the 2D modelling of the rotary filter and the implementation of the FM-II method algorithms. Finally, concluding remarks are made.

1D non-linear modelling of rotary drum filter

The literature generally distinguishes between two types of approach to model filtration: macroscopic models based on the material conservation equations and a filtration rate which is determined empirically, and microscopic models describing the grain transportation process. Concerning macroscopic models, these types are not based on the physical modelling of the particle deposition mechanisms. The parameters of these models do not necessarily have a clear physical meaning but the results are usually close to the experimental results.

Developing a model first requires us to write mathematical equations that relate state variables descriptive of physical process. When compared to other procedures the filtration system of fixed cultures has the complementary particularity to include an additional model of physical filtration, which increases the difficulty of resolution and integration of system equations. For activated sludge processes like the filter, the problem of the complexity of the models make the control possibilities in practice very limited (Gomez–Quintero, 2002).

The model we’re looking for is a model for a control objective. So we will look for the physical equations that contain the state variables X and control variables U.

The rotary drum filter description

These are mainly devices for running empty including: rotary drum filters and belt filters. They have the same applications but the band filters deal thicker slurries (50% solids). Among the filters that are considered continuous we find the rotary drum filter. The rotary drum filter is constituted by two coaxial cylindrical drums, the outer drum carries a filter canvas (see Figure 1).

Figure 1.

Rotary drum filter.

In recent years, membrane filtration technology, especially low pressure membrane technology, has been more widely applied in liquid aspiration fields. Based on this technology, the phosphoric acid plant acquires a huge filter from the ”AOUSTIN-UCEG” company. This filter is composed of 36 flat cells covered by 36 membranes with 36 tubes where each tube is connected to a cell, a vacuum box, and a driving motor which is responsible for the transition of the slurry by five sectors which are: the pre-sector, the strong acid sector, the acid medium sector, the weak acid sector, and the sector of wash canvases. Figure 1 shows a photo of the filter.

Formulation of the mathematical model

Calculation of the height of the cake equation ‘h’

According to the mass balance, the quantity of the slurry in the filter is equal to the difference between the quantity of slurry injected into the filter and the quantity of acid get from the slurry (see Figure 2)

M_{b} = M_{filter} + M_{A}

(1)

where $M_{b}$ is the mass of the slurry injected into the filter, $M_{filter}$ is the mass of the slurry after filtration, and $M_{A}$ is the mass of the phosphoric acid filtrated.

Figure 2.

The flowchart and the block diagram of the filter.

The mass of the slurry injected from the reactor to the filter $M_{b}$ is modelled by the volume of the slurry $V_{b}$

M_{b} = ρ_{b} V_{b}

(2)

where $ρ_{b}$ is the volumetric density of the slurry and $V_{b}$ is the volume of the slurry filled the nacelle.

After filtration, the mass of the slurry in the filter $M_{filter}$ is modelled by the height of the cake h and the surface S of the nacelle

M_{filter} = ρ_{b} S h

(3)

The mass of the filtrated acid $M_{A}$ is modelled by the volume of this acid V

M_{A} = ρ_{A} V

(4)

where $ρ_{A}$ is the volumetric density of the phosphoric acid $P_{2} O_{5}$ .

Substituting (2), (3) and (4) into (1), we get

ρ_{b} V_{b} = ρ_{b} S h + ρ_{A} V

(5)

If we divide (5) by $ρ_{b}$ and we suppose that $\frac{ρ_{A}}{ρ_{b}} = m_{ρ}$ , we find the following formula

V_{b} = S h + m_{ρ} V

(6)

The filter has the shape of a ring with external radius $R_{1}$ and internal radius $R_{2}$ as shown in Figure 3. So the surface S of the nacelle is modelled by the angle θ

S = \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) θ

(7)

we introduce (7) into (6), we obtain

V_{b} = \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) θ h + m_{ρ} V

(8)

Figure 3.

Decomposition of a filter cell.

If we differentiate with respect to time, we obtain

\frac{{dV}_{b}}{dt} = \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) θ \frac{dh}{dt} + \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) h \frac{d θ}{dt} + m_{ρ} \frac{dV}{dt}

(9)

In fact, the angular deviation with respect to time $\frac{d θ}{dt}$ is equivalent to the angular speed of the filter $w_{f}$ , and the slurry volume deviation with respect to time $\frac{{dV}_{b}}{dt}$ is equivalent to the volumetric flow of the slurry $q_{b}$ , so from (9) we obtain

q_{b} = \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) θ \frac{dh}{dt} + \frac{1}{2} (R_{1}^{2} - R_{2}^{2}) h w_{f} + m_{ρ} \frac{dV}{dt}

(10)

We take $\frac{dV}{dt} = q_{AF}$ . From (10), we can get the height deviation with respect to time $\frac{dh}{dt}$

\frac{dh}{dt} = \frac{2}{(R_{1}^{2} - R_{2}^{2}) θ} q_{b} - \frac{h}{θ} w_{f} - \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) θ} q_{AF}

(11)

Finally we get

\frac{dh}{dt} = f_{1} (X (t), U (t))

(12)

Calculation of the flow rate of the phosphoric acid equation $q_{AF}$

To describe the flow of a fluid through a porous medium, we use Navier–Stokes law and Darcy’s law. Darcy’s law is a relation between the flow of a fluid and the loss of charge (Chassagne, 2010). The mass conservation equation coupled to the fluid motion equation gives the incompressible Navier–Stokes equations. The main unknown parameters in this equation are the mass density, the velocity, the pressure, and the temperature, but this list could be longer depending on the case studied. The purpose of this part is to describe the corresponding fundamental solution for the equations of an incompressible fluid model (Thomann and Guenther, 2006). The Navier–Stokes equation, on the other hand, has received renewed interest provided by the clay mathematics institute (Peter, 2015). The Navier–Stokes equation is

ρ \frac{\partial ϑ}{\partial t} + ρ (ϑ \cdot Δ) ϑ = - \nabla P + μ \cdot Δ ϑ

(13)

\nabla \cdot ϑ = 0

(14)

with

$ϑ$ : velocity field;

P: pressure;

$ρ$ : fluid density;

$μ$ : kinematic viscosity;

t: time;

$Δ$ : Laplace operator.

where $Δ = \sum_{j = 1}^{1} \frac{\partial^{2}}{\partial z_{j}^{2}}$ stands for the Laplace operator.

If we multiply equation (13) by S, we obtain

ρ \frac{\partial ϑ \cdot S}{\partial t} + ρ \cdot ϑ \cdot \frac{\partial ϑ \cdot S}{\partial z} = - S \cdot \frac{\partial P}{\partial z} + μ \cdot \frac{\partial^{2} ϑ \cdot S}{\partial z^{2}}

(15)

Then if we take $ϑ \cdot S = q_{AF}$ so we find

\frac{\partial q_{AF}}{\partial t} = - ϑ \cdot \frac{\partial q_{AF}}{\partial z} - \frac{S}{ρ} \cdot \frac{\partial P}{\partial z} + \frac{μ}{ρ} \cdot \frac{\partial^{2} q_{AF}}{\partial z^{2}}

(16)

Now, we want to search for the following two terms

\frac{\partial q_{AF}}{\partial z}, \frac{\partial^{2} q_{AF}}{\partial z^{2}}

To find these two terms, we must use Darcy’s law. The flow of the strong acid is characterized by a flow rate $\vec{ϑ}$ , according to Darcy’s law, this flow rate is proportional to the pressure $Δ P$ and the permeability K, although $\vec{ϑ}$ is inversely proportional to the dynamic viscosity μ of the fluid. After expressing the pressure drop through the filter, the term is derived from Darcy’s law

q_{AF} (z) = \frac{K . (P_{0} - P (z)) . S}{μ \cdot z} = \frac{K . P_{0} \cdot S}{μ \cdot z} - \frac{K . S}{μ} \frac{P (z)}{z} = \frac{β_{1}}{z} - β_{2} \frac{P (z)}{z}

(17)

with $β_{1} = \frac{K . P_{0} \cdot S}{μ}, β_{2} = \frac{K . S}{μ}$ .

If we differentiate the previous equation, we obtain

\frac{\partial q_{AF}}{\partial z} = \frac{β_{1}}{z^{2}} - β_{2} \frac{\frac{\partial P (z)}{\partial z}}{z} - \frac{P (z)}{z^{2}}

(18)

If we differentiate equation (18), we get equation (19)

\frac{\partial^{2} q_{AF}}{\partial z^{2}} = \frac{2 β_{1}}{z^{3}} + β_{2} \frac{\frac{\partial P (z)}{\partial z}}{z} - 2 \frac{P (z)}{z^{2}} - β_{2} (\frac{\partial^{2} P (z) . z}{\partial z^{2}} - \frac{\partial P (z)}{\partial z})

(19)

If we substitute (18) and (19) into (16), we find

\frac{\partial q_{AF}}{\partial t} = \frac{\partial P (z)}{\partial z} (\frac{ϑ \cdot β_{2}}{z} - \frac{S}{ρ} + \frac{μ \cdot β_{2}}{ρ \cdot z} + \frac{μ \cdot β_{2}}{ρ}) - υ (\frac{β_{1}}{z^{2}} + β_{2} \frac{P (z)}{z^{2}}) + \frac{μ}{ρ} (2 \frac{β_{1}}{z^{3}} - 2 \frac{P (z)}{z^{2}})

(20)

The overall pressure P in the slurry mixture is the sum of the pressure of the solid pressure $P_{s}$ and the fluid pressure $P_{l}$ such as

P = P_{s} + P_{l}

The differential of the pressure with respect to the z-axis is as follows

\frac{\partial P}{\partial z} = \frac{\partial P_{s}}{\partial z} + \frac{\partial P_{l}}{\partial z}

(21)

The expressions of $P_{s}$ and $P_{l}$ are given by the following relations

{\begin{matrix} \frac{\partial P_{s}}{\partial z} = \frac{Q_{e}}{2 π {Kz}^{2}} (1 - E) \frac{P_{s}}{z} + (ρ_{s} - ρ_{l}) w_{f}^{2} z \\ \frac{\partial P_{l}}{\partial z} = - \frac{μ_{l} Q_{e}}{K} + μ_{l} w_{f}^{2} z \end{matrix}

with

$Q_{e}$ : speed of the strong acid ( ${ms}^{- 1}$ )

$ρ_{s}$ : solid density (dimensionless)

$ρ_{l}$ : liquid density (dimensionless)

$μ_{l}$ : liquid viscosity ( $Pa \cdot s$ )

E: ratio of lateral and normal stress (dimensionless)

So, the overall pressure is the following

\frac{\partial P}{\partial z} = \frac{Q_{e}}{2 π {Kz}^{2}} (1 - E) \frac{P_{s}}{z} + (ρ_{s} - ρ_{l}) w_{f}^{2} z - \frac{μ_{l} Q_{e}}{K} + μ_{l} w_{f}^{2} z

(22)

If we replace $\frac{\partial P (z)}{\partial z}$ by its expression, we obtain

\begin{array}{l} \lim_{z \to h} \frac{\partial q_{A F}}{\partial t} = \frac{1}{h^{4}} (\frac{Q_{e} \cdot P_{s} \cdot S (1 - E)}{2 π \cdot K} + \frac{Q_{e} \cdot P_{s} \cdot μ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ}) \\ + \frac{1}{h^{3}} (\frac{- Q_{e} \cdot P_{s} \cdot ϑ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ} + \frac{Q_{e} \cdot P_{s} \cdot β_{2} \cdot μ (1 - E)}{2 π \cdot K . ρ} + \frac{2 . μ \cdot β_{2}}{ρ}) \\ + \frac{1}{h^{2}} (ϑ \cdot β_{2} - ϑ \cdot β_{2} \cdot P - 2 \frac{μ \cdot P}{ρ}) \\ + \frac{1}{h} (\frac{- Q_{e} \cdot μ_{l} \cdot ϑ \cdot β_{2}}{K} - \frac{- Q_{e} \cdot μ_{l} \cdot μ \cdot β_{2}}{K . ρ}) \\ + w_{f}^{2} (ρ_{s} - ρ_{l} + μ_{l}) β_{2} (ϑ + \frac{μ}{ρ}) \\ + w_{f}^{2} \cdot h (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) + \frac{μ \cdot Q_{e} \cdot S}{K . ρ} \end{array}

(23)

Finally we get

\frac{{dq}_{AF}}{dt} = f_{2} (X (t), U (t))

(24)

The 1D nonlinear model of the rotary drum filter

Finally, the nonlinear model obtained is as follows

{\begin{matrix} \frac{dh}{dt} & = f_{1} (X (t), U (t)) \\ \frac{{dq}_{AF}}{dt} & = f_{2} (X (t), U (t)) \end{matrix}

(25)

where $X = [h q_{AF}]^{T} \in R^{2}$ is the state vector and $U = [q_{b} w_{f}]^{T} \in R^{2}$ is the input vector of the system.

The nonlinear equation systems implementation brings changes onto the evolution of events of industrial system that can not be addressed like linear systems. For controlling such systems, it is quite necessary to take into account the nonlinear phenomena.

The simulation of the non linear system gives the two following outputs:

the height of the cake h (m);

the volume flow rate of the strong acid $q_{AF}$ .

We note that the thickness of the cake increases over time until it reaches a value of 0.06 m, similarly the volume flow rate of the filtrate which increases to a value close to 74 $m^{3} . h^{- 1}$ . These results are close to the real system.

Linearization of the 1D model of the rotary drum filter

In this section, we use linearization around an operating point $(\bar{X}, \bar{U})$ .

We will linearize the following terms: $\frac{1}{h}, \frac{1}{h^{2}}, \frac{1}{h^{3}} and \frac{1}{h^{4}}$ .

We make a linear approximation with the first order Taylor’s formula for h

\frac{1}{h} \to \frac{h}{h_{0}^{2}}; \frac{1}{h^{2}} \to \frac{h}{h_{0}^{3}}; \frac{1}{h^{3}} \to \frac{h}{h_{0}^{4}}; \frac{1}{h^{4}} \to \frac{h}{h_{0}^{5}}

where $h_{0}$ is a small variation of the height of the cake.

So, we obtain

\begin{matrix} \frac{\partial q_{AF}}{\partial t} = ((\frac{Q_{e} \cdot P_{s} \cdot S (1 - E)}{2 π \cdot K . h_{0}^{5}} + \frac{Q_{e} \cdot P_{s} \cdot μ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{5}}) \\ + (\frac{- Q_{e} \cdot P_{s} \cdot ϑ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{4}} + \frac{Q_{e} \cdot P_{s} \cdot β_{2} \cdot μ (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{4}} \\ + \frac{2 . μ \cdot β_{2}}{ρ \cdot h_{0}^{4}}) + (\frac{ϑ \cdot β_{2}}{h_{0}^{3}} - \frac{ϑ \cdot β_{2} \cdot P}{h_{0}^{3}} - 2 \frac{μ \cdot P}{ρ \cdot h_{0}^{3}}) \\ + (\frac{- Q_{e} \cdot μ_{l} \cdot ϑ \cdot β_{2}}{K . h_{0}^{2}} - \frac{- Q_{e} \cdot μ_{l} \cdot μ \cdot β_{2}}{K . ρ h_{0}^{2}})) h \\ + (ρ_{s} - ρ_{l} + μ_{l}) β_{2} (ϑ + \frac{μ}{ρ}) w_{f}^{2} \\ + (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) w_{f}^{2} \cdot h + \frac{μ \cdot Q_{e} \cdot S}{K . ρ} \end{matrix}

(26)

We suppose that

\begin{matrix} α = ((\frac{Q_{e} \cdot P_{s} \cdot S (1 - E)}{2 π \cdot K . h_{0}^{5}} + \frac{Q_{e} \cdot P_{s} \cdot μ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{5}}) \\ + (\frac{- Q_{e} \cdot P_{s} \cdot ϑ \cdot β_{2} (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{4}} + \frac{Q_{e} \cdot P_{s} \cdot β_{2} \cdot μ (1 - E)}{2 π \cdot K . ρ \cdot h_{0}^{4}} \\ + \frac{2 . μ \cdot β_{2}}{ρ \cdot h_{0}^{4}}) + (\frac{ϑ \cdot β_{2}}{h_{0}^{3}} - \frac{ϑ \cdot β_{2} \cdot P}{h_{0}^{3}} - 2 \frac{μ \cdot P}{ρ \cdot h_{0}^{3}}) \\ + (\frac{- Q_{e} \cdot μ_{l} \cdot ϑ \cdot β_{2}}{K . h_{0}^{2}} - \frac{- Q_{e} \cdot μ_{l} \cdot μ \cdot β_{2}}{K . ρ h_{0}^{2}})) \end{matrix}

(27)

So the linear model of the rotary drum filter is given by the two following equations

f_{1} (X (t), U (t)) = \frac{dh}{dt} = \frac{2 . ρ_{g}}{(R_{1}^{2} - R_{2}^{2}) θ} q_{b} - \frac{h}{θ} w_{f} + \frac{2}{(R_{1}^{2} - R_{2}^{2}) θ} . q_{AF}

(28)

\begin{matrix} f_{2} (X (t), U (t)) = \frac{{dq}_{AF}}{dt} = α . h + (ρ_{s} - ρ_{l} + μ_{l}) β_{2} (ϑ + \frac{μ}{ρ}) w_{f}^{2} \\ + (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) w_{f}^{2} \cdot h + \frac{μ \cdot Q_{e} \cdot S}{K . ρ} \end{matrix}

(29)

The equations of the model include the product of a state variable and a control variable. In order to linearize the model, we must use Jacobian linearization.

We will select an operating point at steady state $P (\bar{X}, \bar{U})$ such as $f (\bar{X} (t), \bar{U} (t)) = 0$ .

Moreover, if we suppose that $\bar{U} = 0$ , so the operating point is $([\begin{matrix} \frac{μ_{l} Q_{e} S}{K ρ α} \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \end{matrix}])$ . The model is as follows

{\begin{cases} \dot{X} = F_{x} X + B_{u} U \\ Y = G_{x} X \end{cases}

(30)

where

X = (\begin{matrix} X_{1} \\ X_{2} \end{matrix}), Y = (\begin{matrix} Y_{1} \\ Y_{2} \end{matrix}), U = (\begin{matrix} U_{1} \\ U_{2} \end{matrix})

$F_{x}$ , $B_{u}$ are the Jacobian matrices of the partial derivatives of $f (X, U)$ with respect to X and U, the state variable and the input variable of the system, which is measured at point $P (\bar{X}, \bar{U})$ . The two matrices $F_{x}$ and $B_{u}$ change according to the operating point.

At the steady state we have

\begin{matrix} \bar{X} = (\begin{matrix} \bar{h} \\ {\bar{q}}_{AF} \end{matrix}), \bar{U} = (\begin{matrix} {\bar{q}}_{b} \\ {\bar{w}}_{f} \end{matrix}) \\ F_{x} = (\begin{matrix} \frac{\partial f_{1}}{\partial h} & \frac{\partial f_{1}}{\partial q_{AF}} \\ \frac{\partial f_{2}}{\partial h} & \frac{\partial f_{2}}{\partial q_{AF}} \end{matrix}) = (\begin{matrix} F_{x 11} & F_{x 12} \\ F_{x 21} & F_{x 22} \end{matrix}), \\ B_{u} = (\begin{matrix} \frac{\partial f_{1}}{\partial q_{b}} & \frac{\partial f_{1}}{\partial w_{f}} \\ \frac{\partial f_{2}}{\partial q_{b}} & \frac{\partial f_{2}}{\partial w_{f}} \end{matrix}) = (\begin{matrix} B_{u 11} & B_{u 12} \\ B_{u 21} & B_{u 22} \end{matrix}) \end{matrix}

with

\begin{matrix} F_{x 11} = \frac{\partial f_{1}}{\partial h} = - \frac{{\bar{w}}_{f}}{θ} \\ F_{x 12} = \frac{\partial f_{1}}{\partial q_{AF}} = - \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) \cdot θ} \\ F_{x 21} = \frac{\partial f_{2}}{\partial h} = α + (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) {\bar{w}}_{f}^{2} \\ F_{x 22} = \frac{\partial f_{2}}{\partial q_{AF}} = 0 B_{u 11} = \frac{\partial f_{1}}{\partial q_{b}} = \frac{2 . ρ_{g}}{(R_{1}^{2} - R_{2}^{2}) θ} \\ B_{u 12} = \frac{\partial f_{1}}{\partial w_{f}} = - \frac{\bar{h}}{θ}, B_{u 21} = \frac{\partial f_{2}}{\partial q_{b}} = 0 \\ B_{u 22} 2 (ρ_{s} - ρ_{l} + μ_{l}) β_{2} (ϑ + \frac{μ}{ρ}) {\bar{w}}_{f} + 2 (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) \\ (μ \cdot β_{2} - S) {\bar{w}}_{f} \cdot h \end{matrix}

So, the final Jacobian matrices of the state space are

F_{x} = (\begin{matrix} F_{x 11} & F_{x 12} \\ F_{x 21} & F_{x 22} \end{matrix}), B_{u} = (\begin{matrix} B_{u 11} & B_{u 12} \\ B_{u 21} & B_{u 22} \end{matrix}), G_{x} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

Simulation results and validation of the model

The simulation is carried out by “MATLAB” which is among the most developed software for the simulation of production systems. We suppose that the rotary filter on the Tunisian Chemical Group (TCG) is a multi-input/multi-output nonlinear system, which has like inputs and outputs:

$U_{1}$ : flow rate of the slurry;

$U_{2}$ : speed of the filter;

$Y_{1}$ : height of the cake;

$Y_{2}$ : flow rate of the filtrate.

In order to investigate the filter and its changing states, our system was modelled with the intervention of the disturbances. By this method, we can get closer to the real system.

Validation of the model is the process of determining the degree to which a simulation model and its associated data are an accurate representation of the real world. The factory is supervised by the SCADA system in which all the experiments and measurements are taken directly from the SCADA database. These control and output measurements are mentioned respectively as U and $Y_{r}$ whereas the modelled outputs are referred to as $Y_{m}$ , in the following figures.

Figure 4 shows the control inputs of the system. These inputs refer to the flow rate of the slurry $U_{1}$ and the rotation speed of the filter $U_{2}$ .

Figure 4.

Evolution of the inputs of the filter.

Figure 5 presents the output $Y_{1} = Y_{r} (cm)$ which corresponds to the thickness of the cake and the output $Y_{m} (cm)$ of the model which are almost consistent. The error between the real output $Y_{r}$ and the model $Y_{m}$ belongs to $\pm 20 %$ as shown in Figure 6. The Mean Square Error (MSE) = 0.4%.

Figure 5.

Representative figure of the real and the estimate output (height of the cake).

Figure 6.

Representative figure of the error of the estimate output (height of the cake).

Figure 7 describes the output $Y_{2} = Y_{r} {(m}^{3} . h^{- 1})$ which corresponds to the volume flow rate of the filtrate and the output $Y_{m} {(m}^{3} . h^{- 1})$ of the model which are almost consistent. The error between the output of the real system $Y_{r}$ and the model $Y_{m}$ belongs to $\pm 0.47 %$ as shown in Figure 8. The mean square error: MSE = 0.313%.

Figure 7.

Representative figure of the real and the estimate output (flow rate of filtrate).

Figure 8.

Representative figure of the error of the estimate output (flow rate of filtrate).

We note that there is a gap between the real output and that of the model, this difference is due to the errors on the estimated model resulting from insufficient excitement. So it’s a problem of excitation.

2D modelling based on FM-II method

Implementation of the FM-II method algorithms

Consider the linear 2D system described by the following FM-II model (Fornasini and Marchesini, 1976)

\begin{matrix} X (h + 1, k + 1) = A_{1} X (h, k + 1) + A_{2} X (h + 1, k) \\ + B_{1} U (h, k + 1) + B_{2} U (h + 1, k) \end{matrix}

(31)

Y (h, k) = CX (h, k)

(32)

where $X (h, k) \in R^{n}$ , $U (h, k) \in R^{1}$ , and $Y (h, k) \in R^{m}$ are respectively the local state, input and output vectors, h is the time t and k is the angle θ, and $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ and C are real system matrices of suitable sizes. The system is also conventionally denoted by ( $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , C).

The two dimensional linear system is said to be asymptotically stable if the definition 1 in Kaczorek (2006) is checked.

Lemma 1 (Hmamed et al., 2008)

The 2D system is asymptotically stable if there exist matrices $P > 0$ , $Q > 0$ , $R = R^{T} > = 0$ , $Π > 0$ , $F \in R^{2 n . n}$ , and $G \in R^{n . n}$ such that

(\begin{matrix} - Π + FA + A^{T} F^{T} & - F + A^{T} G^{T} \\ * & P - G - G^{T} \end{matrix}) < 0

(33)

where $Π = (\begin{matrix} P - Q - 2 R & R \\ R^{T} & Q \end{matrix})$ , $F = (\begin{matrix} F_{1} \\ F_{2} \end{matrix})$ and $A = {(\begin{matrix} A_{1} \\ A_{2} \end{matrix})}^{T}$ :

the transfer matrix of the previous model is: $ψ = diag {ψ_{1}, \dots, ψ_{l}}$ where $ψ_{i}$ is the column vector defined as $ψ_{i} = [z_{2}^{k_{i}}, z_{2}^{k_{i} - 1}, z_{1}, \dots, z_{2} z_{1}^{k_{i} - 1}, z_{1}^{k_{i}}, \dots, z_{2} z_{1}, z_{1}^{2}, z_{2}, z_{1}]$

write $D (z_{1}, z_{2})$ and $N (z_{1}, z_{2})$ in the form of

D (z_{1}, z_{2}) = D_{HT} \cdot ψ

(34)

N (z_{1}, z_{2}) = N_{HT} \cdot ψ

(35)

with $N (z_{1}, z_{2}) = N_{R} (z_{1}, z_{2})$ , $D (z_{1}, z_{2}) = 1 - D_{R} (z_{1}, z_{2})$ . So, $F (z_{1}, z_{2}) = (\begin{matrix} N (z_{1}, z_{2}) \\ D (z_{1}, z_{2}) \end{matrix})$ where $D_{HT}$ and $N_{HT}$ are real matrices with sizes conformable to $ψ$ ;

construct the matrices $A_{i 0}, B_{i}; i = 1, 2$ such that

ψ = (I - A_{10} z_{1} - A_{20} z_{2})^{- 1} (B_{1} z_{1} + B_{2} z_{2})

(36)

the realization is finally obtained as $A_{i} = A_{i 0} + B_{i} D_{HT}$ , Bi, $i = 1, 2$ and $C = N_{HT}$ .

By thoroughly investigating the structural properties of FM-II model, we found that to meet the relations specified in (42), (43), and (44) so that a realization can be obtained.

ψ has to satisfy the following conditions.

$ψ_{i}$ contains all the power products occurring in the polynomial entries of the $i_{t} h$ column of $F (z_{1}, z_{2})$ .

$ψ_{i}$ contains $z_{1}$ or $z_{2}$ , or both $z_{1}$ and $z_{2}$ .

Let $n_{i} (i = 1, \dots, l)$ be the dimension of $ψ_{i}$ . For every entry $ψ_{i} (j) (j \in 1, \dots, n_{i})$ except for the entry that is either $z_{1}$ or $z_{2}$ , there exists another entry $ψ_{i} (h) (h \in 1, \dots, n_{i})$ such that $ψ_{i} (j) = z_{1} ψ_{i} (h)$ or $ψ_{i} (j) = z_{2} ψ_{i} (h)$ .

The above realization procedure produces a state space description ( $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , C) which has the following properties:

$[I - A_{1} z_{1} - A_{2} z_{2} B_{1} z_{1} + B_{2} z_{2}]$ is full rank in $C^{2}$ ;

$det (I - A_{1} z_{1} - A_{2} z_{2}) = det D (z_{1}, z_{2})$

Starting from the initial $ψ_{i} (i = 1, \dots, l)$ , the following algorithm is given below for constructing the final $ψ_{i} (i = 1, \dots, l)$ which all satisfy the conditions of (2) and (3).

Algorithm 1 (Xu et al., 2005)

Step 1: $i = 0$ .

Step 2: $i = i + 1$ ; $j = 0$ . If $I > l$ , exit. Otherwise, proceed to Step 3.

Step 3: $j = j + 1$ , $r = 1$ . Check whether there exists an entry $ψ (h) (j < h < = n_{i})$ such that condition (3) is satisfied. If yes, repeat Step 3. Otherwise, proceed to Step 4.

Step 4: If $j = n_{i} 1$ , check whether $ψ (j) = z_{2}$ (which implies $ψ_{i} (j + 1) = ψ_{i} (n_{i}) = z_{1}$ ). If yes, return to Step 2. Otherwise proceed to Step 5.

Step 5: If $j = n_{i}$ , check whether $ψ (j) = z_{1}$ or $ψ (j) = z_{2}$ . If the answer is yes, return to Step 2, and if the answer is no, go to Step 7. If $j \neq n_{i}$ , proceed to Step 6.

Step 6: $r = r + 1$ . Check whether there exists an entry $ψ (k)$ for $j < k \leq n_{i}$ that satisfies either $ψ_{i} (j) = z_{1}^{r} ψ_{i} (k)$ or $ψ_{i} (j) = z_{2}^{r} ψ_{i} (k)$ If the answer is yes, insert either $z_{1}^{r - s} ψ_{i} (k)$ or $z_{2}^{r - s} ψ_{i} (k)$ at an appropriate position according to the descending order of power products in $ψ_{i}$ for $s = 1,, r 1$ . Let $n_{i} = n_{i} + (r 1)$ , and return to Step 3. In the case that the answer is no, check whether $r < \deg ψ (j) - 2$ . If yes, repeat Step 6. Otherwise, proceed to Step 7.

Step 7: Check whether 1 2 $\deg_{z_{1}} ψ (j) \geq \deg_{z_{2}} ψ (j)$ . If yes, insert $z_{1}^{- 1} ψ_{i} (k)$ , while if no, insert $z_{2}^{- 1} ψ_{i} (k)$ , into $ψ_{i}$ according to the descending order of power products. $n_{i} = n_{i + 1}$ . Return to Step 3.

Once $ψ_{1}, \dots, ψ_{l}$ are constructed by Algorithm 1, $ψ = diag ψ \dots ψ$ , can be readily obtained. $ψ$ is now of size nl, where $n = \sum_{i = 1}^{l} n_{i}$ . Note that n will be smaller than $n'$ if some power products are absent in the polynomial entries of $F (z_{1}, z_{2})$ . This will be illustrated by a non-trivial example in the next section. Next, express $D (z_{1}, z_{2})$ and $N (z_{1}, z_{2})$ in the form $D (z_{1}, z_{2}) = D_{HT} ψ$ , $N (z_{1}, z_{2}) = N_{HT} ψ$ ,

where $D_{HT} = (\begin{matrix} D_{11} & \dots & D_{1 l} \\ ⋮ & ⋱ & ⋮ \\ D_{l 1} & \dots & D_{ll} \end{matrix})$ , $N_{HT} = (\begin{matrix} N_{11} & \dots & N_{1 l} \\ ⋮ & ⋱ & ⋮ \\ N_{m 1} & \dots & N_{ml} \end{matrix})$ where $D_{i_{j}} \in R^{1 n_{j}}$ and $D_{i_{j}} \in R^{1 n_{j}}$ are row vectors whose entries are the coefficients of the $(i, j) -$ indexed polynomial in $D (z_{1}, z_{2})$ and $N (z_{1}, z_{2})$ , respectively. The system matrices $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , and C can then be constructed by the following algorithm.

Algorithm 2 (Xu et al., 2005)

Step 1: Introduce $n_{i} * n_{i}$ matrices $A_{10}^{(i)}$ and $A_{20}^{(i)}$ , $i = 1, \dots, l$ which are determined in the following way. Initially set all of the entries of $A_{10}^{(i)}$ and $A_{20}^{(i)}$ to zero. For $k = 1, \dots, n_{i}$ , let $A_{20}^{(i)} (k, h_{k}) = 1$ only if there exists certain $h_{k}$ such that $ψ_{i} (k) = z_{1} ψ (h_{k})$ , and let $A_{20}^{(i)} (k, m_{k}) = 1$ only if condition (10)does not hold and there exists certain $m_{k}$ ( $k < m_{k} \leq n_{i}$ ) such that $ψ_{i} (k) = z_{2} ψ_{i} (m_{k})$ . It is obvious that $A_{10}^{(i)}$ and $A_{20}^{(i)}$ are upper triangular matrices and have at most only one entry equal to 1 in the same row of $A_{10}^{(i)}$ and $A_{20}^{(i)}$ .

Step 2: For $k = 1, 2$ , construct column vector $B_{k}^{(i)} \in R^{ni}$ , $i = 1, \dots, l$ , by initially setting all of the entries of $B_{k}^{(i)}$ to zero. If there exists some h such that $ψ_{i} (h) = z_{k}$ , change the $h - th$ entry of $B_{k}^{(i)}$ to 1.

Step 3: Construct the $n * n$ matrices $A_{10}$ , $A_{20}$ and the $n l$ matrices $B_{1}$ , $B_{2}$ as follows. $A_{k 0} = diag {A_{10}^{(1)}, A_{10}^{(2)}, \dots, A_{10}^{(l)}}$ , $B_{k} = diag {B_{10}^{(1)}, B_{10}^{(2)}, \dots, B_{10}^{(l)}}$ , $k = 1, 2$ .

Step 4: It is easy to see that $(I - A_{10} z_{1} - A_{20} z_{2}) ψ = (B_{1} z_{1} + B_{2} z_{2})$ and it can be shown, in the same way of Bisiacco et al. (1989), that ${I - (A_{10} + B_{1} D_{HT}) z_{1} - (A_{20} + B_{2} D_{HT}) z_{2}}^{- 1} \cdot (B_{1} z_{1} + B_{2} z_{2}) = ψ D_{R}^{- 1} (z_{1}, z_{2})$ ,

Therefore, we have

N_{R} (z_{1}, z_{2}) D_{R}^{- 1} (z_{1}, z_{2}) = N_{H T} ψ D_{R}^{- 1} (z_{1}, z_{2}) = N_{H T} {I - (A_{10} + B_{1} D_{H T}) z_{1} - (A_{20} + B_{2} D_{H T}) z_{2}}^{- 1} \cdot (B_{1} z_{1} + B_{2} z_{2})

(Xu et al., 2005)

2D modelling of a rotary drum filter

Calculation of the transfer matrix

The state vector is

F_{x} = (\begin{matrix} F_{x 11} & F_{x 12} \\ F_{x 21} & F_{x 22} \end{matrix})

(37)

with

\begin{array}{l} F_{x 11} = \frac{\partial f_{1}}{\partial h} = - \frac{{\bar{w}}_{f}}{θ} \\ F_{x 12} = \frac{\partial f_{1}}{\partial q_{A F}} = - \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) . θ} \\ F_{x 21} = \frac{\partial f_{2}}{\partial h} = α + (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) {\bar{w}}_{f}^{2}, \\ F_{x 22} = \frac{\partial f_{2}}{\partial q_{A F}} = 0 \end{array}

The command vector is

B_{u} = (\begin{matrix} B_{u 11} & B_{u 12} \\ B_{u 21} & B_{u 22} \end{matrix})

(38)

with

\begin{array}{l} B_{u 11} = \frac{\partial f_{1}}{\partial q_{b}} = \frac{2 . ρ_{g}}{(R_{1}^{2} - R_{2}^{2}) θ} \\ B_{u 12} = \frac{\partial f_{1}}{\partial w_{f}} = - \frac{\bar{h}}{θ} \\ B_{u 21} = \frac{\partial f_{2}}{\partial q_{b}} = 0 \\ B_{u 22} = 2 (ρ_{s} - ρ_{l} + μ_{l}) β_{2} (ϑ + \frac{μ}{ρ}) {\bar{w}}_{f} \\ + 2 (\frac{ρ_{s} - ρ_{l} + μ_{l}}{ρ}) (μ \cdot β_{2} - S) {\bar{w}}_{f} \cdot h \end{array}

At first, we calculate the determinant of $\begin{matrix} F_{x} \\ p . I_{2} - F_{x} = (\begin{matrix} p & 0 \\ 0 & p \end{matrix}) - (\begin{matrix} F_{x 11} & F_{x 12} \\ F_{x 21} & F_{x 22} \end{matrix}) \end{matrix}$

$det (p . I_{2} - F_{x}) = p^{2} + \frac{{\bar{w}}_{f}}{θ} . p + \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) . θ} . F_{x 21}$ Using the previous determinant, we calculate the adjugate matrix of $F_{x}$ to obtain the final transfer matrix $M (t, \frac{1}{θ})$

\begin{matrix} G_{X} (com (p . I_{2} - F_{x})^{T}) . B_{u} = (\begin{matrix} p . B_{u 11} & p . B_{u 12} \\ F_{x 21} \cdot B_{u 11} & F_{x 21} \cdot B_{u 12} + (p - F_{x 11}) B_{u 22} \end{matrix}) \\ M (t, \frac{1}{θ}) = (\begin{matrix} M_{1} & M_{2} \\ M_{3} & M_{4} \end{matrix}) \end{matrix}

(39)

with

M_{1} (t, \frac{1}{θ}) = (p^{2} + \frac{{\bar{w}}_{f}}{θ} . p + \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) \cdot θ} . F_{x 21})

(40)

M_{3} (t, \frac{1}{θ}) = \frac{F_{x 21} \cdot B_{u 11}}{p^{2} + \frac{{\bar{w}}_{f}}{θ} . p + \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) \cdot θ} . F_{x 21}}

(41)

M_{2} (t, \frac{1}{θ}) = \frac{p . B_{u 12}}{p^{2} + \frac{{\bar{w}}_{f}}{θ} . p + \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) \cdot θ} . F_{x 21}}

(42)

M_{4} (t, \frac{1}{θ}) = \frac{F_{x 21} \cdot B_{u 12} + (p - F_{x 11}) B_{u 22}}{p^{2} + \frac{{\bar{w}}_{f}}{θ} . p + \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) \cdot θ} . F_{x 21}}

(43)

We have $t = z_{1}$ and $\frac{1}{θ} = z_{2}$

M (z_{1}, z_{2}) = (\begin{matrix} \frac{b_{1} z_{1} z_{2}}{z_{1}^{2} + a_{1} z_{1} z_{2} + a_{11} z_{2}} & \frac{b_{2} z_{1} z_{2}}{z_{1}^{2} + a_{2} z_{1} z_{2} + a_{22} z_{2}} \\ \frac{b_{3} z_{2}}{z_{1}^{2} + a_{3} z_{1} z_{2} + a_{33} z_{2}} & \frac{b_{4} + b_{44} z_{2} + b_{444} z_{1}}{z_{1}^{2} + a_{4} z_{1} z_{2} + a_{44} z_{2}} \end{matrix})

with

\begin{matrix} b_{1} = B_{u 11} \\ b_{2} = B_{u 12} \\ b_{3} = F_{x 21} . B_{u 11} \\ b_{4} = F_{x 21} . B_{u 12} \\ b_{44} = F_{x 11} . B_{u 22} \\ b_{444} = B_{u 22} \\ a_{1} = a_{2} = a_{3} = a_{4} = \frac{{\bar{w}}_{f}}{θ} \\ a_{11} = a_{22} = a_{33} = a_{44} = \frac{2 m_{ρ}}{(R_{1}^{2} - R_{2}^{2}) . θ} F_{x 21} \end{matrix}

From the transfer matrix we can formulate the following matrices

\begin{matrix} D_{R} (z_{1}, z_{2}) = (\begin{matrix} z_{1}^{2} + a_{11} z_{1} z_{2} + a_{12} z_{2} & 0 \\ 0 & z_{1}^{2} + a_{11} z_{1} z_{2} + a_{12} z_{2} \end{matrix}) \\ N_{R} (z_{1}, z_{2}) = (\begin{matrix} b_{11} z_{1} z_{2} & b_{12} z_{1} z_{2} \\ b_{21} z_{2} & b 0 + b_{22} z_{2} + b_{222} z_{1} \end{matrix}) \end{matrix}

F (z_{1}, z_{2}) = (\begin{matrix} N (z_{1}, z_{2}) \\ D (z_{1}, z_{2}) \end{matrix}) = (\begin{matrix} b_{11} z_{1} z_{2} & b_{21} z_{2} \\ b_{12} z_{1} z_{2} & b 0 + b_{22} z_{2} + b_{222} z_{1} \\ z_{1}^{2} + a_{11} z_{1} z_{2} + a_{12} z_{2} & 0 \\ 0 & z_{1}^{2} + a_{11} z_{1} z_{2} + a_{12} z_{2} \end{matrix})

Thus the column degrees of columns 1 and 2 in $F (z_{1}, z_{2})$ are $k_{1} = 1$ and $k_{2} = 2$ , and the power products occurring in the polynomial entries of the columns with non zero coefficients are ${z_{2}, z_{1}}$ and ${z_{2}, z_{1}, z_{1}^{2}, z_{2} z_{1}}$ respectively. The initial column vectors $ψ_{1}$ and $ψ_{2}$ satisfying condition (1).

$ψ_{1} = {(\begin{matrix} z_{1} z_{2} & z_{2} & z_{1} \end{matrix})}^{T}$ ; $ψ_{2} = {(\begin{matrix} z_{1} z_{2} & z_{1}^{2} & z_{2} & z_{1} \end{matrix})}^{T}$

Apply now Algorithm 1, $ψ_{1}$ and $ψ_{2}$ should satisfy conditions (2) and (3).

So, for $ψ_{1}$ a new term has to be inserted into $ψ_{1} z_{1}^{2}$ , so, $ψ_{1} = {(\begin{matrix} z_{1} z_{2} & z_{1}^{2} & z_{2} & z_{1} \end{matrix})}^{T}$ . This new $ψ_{1}$ now satisfies conditions (2) and (3), therefore, we have

\begin{matrix} ψ = diag {ψ_{1} ψ_{2}} = \\ (\begin{matrix} z_{1} z_{2} & z_{1}^{2} & z_{2} & z_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & z_{1} z_{2} & z_{1}^{2} & z_{2} & z_{1} \end{matrix}) \end{matrix}

For $ψ$ constructed above, it is ready to calculate that $D (z_{1}, z_{2}) = D_{HT} ψ$ ; $N (z_{1}, z_{2}) = N_{HT} ψ$

\begin{matrix} D_{HT} = (\begin{matrix} 0 & 0 & a_{11} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{22} & 1 & 0 & 0 \end{matrix}) \\ N_{HT} = (\begin{matrix} 0 & 0 & b_{11} & 0 & 0 & 0 & b_{12} & 0 \\ 0 & 0 & b_{21} & 0 & 0 & 0 & b_{22} & b_{222} \end{matrix}) \end{matrix}

Since only $ψ_{1} (1) = z_{1} ψ_{1} (3)$ holds for $ψ_{1}$ , its follows from Algorithm 2 that

A_{10}^{(1)} = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), A_{20}^{(1)} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

Calculation of $A_{i}$ and $B_{i}$

On the other hand, for $ψ_{2} (1) = z_{2} ψ_{2} (4)$ , Algorithm 2 produces

A_{10}^{(2)} = (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), A_{20}^{(2)} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

It is easy to obtain

\begin{matrix} B_{1}^{(1)} = {(\begin{matrix} 0 & 0 & 1 & 0 \end{matrix})}^{T} \\ B_{2}^{(1)} = {(\begin{matrix} 0 & 0 & 0 & 1 \end{matrix})}^{T} \\ B_{1}^{(2)} = {(\begin{matrix} 0 & 0 & 1 & 0 \end{matrix})}^{T} \\ B_{2}^{(21)} = {(\begin{matrix} 0 & 0 & 0 & 1 \end{matrix})}^{T} \end{matrix}

Since $ψ_{1} (3) = z_{2}$ , $ψ_{1} (4) = z_{1}$ , $ψ_{2} (3) = z_{2}$ , $ψ_{2} (4) = z_{1}$

A_{10} = {A_{10}^{(1)}, A_{10}^{(2)}}

\begin{matrix} A_{10} = (\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}), A_{20} = 0 \\ B_{1} = {(\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix})}^{T}, \\ B_{2} = {(\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})}^{T} \\ A_{1} = A_{10} + B_{1} D_{HT} = (\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{11} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{22} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) \end{matrix}

A_{2} = A_{20} + B_{2} D_{HT} = (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{22} & 1 & 0 & 0 \end{matrix})

C = N_{HT} = {(\begin{matrix} 0 & 0 & b_{11} & 0 & 0 & 0 & b_{12} & 0 \\ 0 & 0 & b_{21} & 0 & 0 & 0 & b_{22} & b_{222} \end{matrix})}^{T}

Conclusion

In this research a two dimensional model was proposed to describe the dynamic behaviour of a rotary filter, which works in a permanent regime and it is dedicated to the aspiration of various types of acid: strong, medium, and weak. This method is considered among the most complex, multi-variable, and unstable systems on an industrial scale. Therefore, its study requires deep research in order to know all of the parameters that have an influence on the normal operation of the filter. Using the parameters that characterize the “AOUSTIN-UCEGO” filter, we find a two dimensional model for the filter. A constructive state-space realization procedure has been used for 2D systems which may produce an FM-II local state-space model.

Footnotes

Acknowledgements

This work was carried out in the Tunisian Chemical Group (TCG), SKHIRA factory in the unit of the phosphoric acid production.

Funding

This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia.

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1D and 2D modelling of a chemical process using the FM-II model: Rotary drum filter

Abstract

Keywords

Introduction

1D non-linear modelling of rotary drum filter

The rotary drum filter description

Formulation of the mathematical model

Calculation of the height of the cake equation ‘h’

Calculation of the flow rate of the phosphoric acid equation q AF

The 1D nonlinear model of the rotary drum filter

Linearization of the 1D model of the rotary drum filter

Simulation results and validation of the model

2D modelling based on FM-II method

Implementation of the FM-II method algorithms

2D modelling of a rotary drum filter

Calculation of the transfer matrix

Calculation of A i and B i

Conclusion

Footnotes

Acknowledgements

Funding

References

Calculation of the flow rate of the phosphoric acid equation $q_{AF}$

Calculation of $A_{i}$ and $B_{i}$