# Support analysis for a semi-batch reactor control.

1. INTRODUCTION

The chromium sludge is processed in a chemical reactor by an exothermic chemical reaction with chrome sulphate acid (Kolomaznik, 1996). During this reaction a considerable quantity of heat is developing so that a control of the reaction is necessary. In order to investigate main properties of the real process, a mathematical model of the chemical reactor was derived based on Fig.1 (Macku, 2005).

2. SEMI-BATCH REACTOR

2.1 Mathematical model

Under usual simplifications, based on the mass and heat balance, the following 4 nonlinear ordinary differential equations can be derived (Macku, 2005):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The first equation expresses the total mass balance of the chemical solution in the reactor. The symbol [[??].sub.FK] [kg.[s.sup.-1]] expresses the mass flow of the entering chromium sludge, [a.sub.FK] (t)[-] denotes the mass concentration of the chromium sludge in the reactor and m(t)[kg] describes weight of the reaction components in the system. k [[s.sup.-1]] is the reaction rate constant expressed by the Arrhenius equation (2) where A [[s.sup.-1]], E [J.[mol.sup.-1]] and R [J.[mol.sup.-1].[K.sup.-1]] are pre-exponential factor, activation energy and gas constant respectively.

k = [Ae.sup.E/RT(t) (2)

[FIGURE 1 OMITTED]

The third and fourth equations describe the enthalpy balance. The individual symbols used above mean: [c.sub.FK] [J.[kg.sup.-1].[K.sup.-1]]--chromium sludge specific heat capacity, [c.sub.R] [J.[kg.sup.-1].[K.sup.-1]]--specific heat capacity of the reactor content, [T.sub.FK] [K]--chromium sludge temperature, [DELTA][H.sub.r] [J.[kg.sup.-1]]--reaction heat, K [J.[m.sup.-2].[K.sup.-1].[s.sup.-1]]--conduction coefficient, S [[m.sup.2]]--heat transfer surface, T(t) [K]--temperature of reaction components in the reactor, [T.sub.v] (t) [K]--temperature of a coolant in the reactor double wall, [[??].sub.v] [kg.[s.sup.-1]]--coolant mass flow, [c.sub.v] [J.[kg.sup.-1].[K.sup.-1]] coolant specific heat capacity, [T.sub.vp] [K]--input coolant temperature, [m.sub.vR][kg]--coolant mass weight in the reactor double wall.

3. CONTROL THEORY POINT OF VIEW

From the systems theory point of view the reactor has four input signals [[??].sub.FK] (t), [[??].sub.v] (t), [T.sub.FK] (t) and [T.sub.vp] (t), four state variables m(t), [a.sub.FK] (t), T(t), [T.sub.v] (t) and one output signal to be controlled given by the temperature inside the reactor T (t). Hence, it can be generally seen as a Multi Input--Multi Output (MIMO) system of 4th order. In addition it possesses strongly nonlinear behaviour. Practically, the only manipulated variables are input flow rates of the chromium sludge [[??].sub.FK] (t) and of the coolant [[??].sub.v] (t). Therefore, input temperatures of the filter cake [T.sub.FK] (t) and of the coolant [T.sub.vp] (t) can be alternatively seen as disturbances. For further analysis, the reactor model described by the system of differential equations (1) is transformed into a linear time-variant (LTV) system.

3.1 Linear model

Having generally a nonlinear model defined by a system of formulas

x'(t ) = f [t, x (t), u (t)] (3)

where x (t) defines a vector of state-variables [[x.sub.i] (t) [x.sub.2] (t) ... [x.sub.n] (t)], u(t) vector of input variables [[u.sub.i] (t) [u.sub.2] (t) ... [u.sub.m] (t)] and f is a nonlinear vector function [[f.sub.1] [f.sub.2] ... [f.sub.n]], then the linear model in a given operating (steady-state) point ([u.sup.s], [y.sup.s]) can be generally obtained using formulae with constant matrices A, B. As the reactor embodies astatic behaviour, it is not possible to compute the matrices in a chosen (steady-state) operating point. However, the linearization can be performed generally, resulting in a time-variant system:

x'(t) = A (t) x (t) + B (t)u (t) (4)

where the matrices A(t), B(t) are no longer constant but time-dependent. Using the formulas above, the originally nonlinear model of the reactor has been transformed into a linear time-variant model.

Generally, output from a linear system with matrices C, D is defined as:

y (t) = Cx (t) + Du (t) (5)

3.2 Transfer function

As the reactor analyzed in this contribution is astatatic, the linearized model is time-dependent. Then the transfer function (t.f.) is also time-dependent:

[??](s, t) = C (s[I.sub.n] - A[(t)).sup.-1] B(t) (6)

where s is the complex Laplace variable and [I.sub.n] is the n -by- n identity matrix.

Since the system generally has one output T (t) and 4 inputs [[??].sub.FK] (t), [[??].sub.v] (t), [T.sub.FK] (t), [T.sub.vp] (t), the resultant t.f. is a vector of the 1-by-4 size:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The first term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] describes the relation between the temperature inside the reactor T(t) and the input flow rate of the chromium sludge [[??].sub.FK] (t). The other terms describes the relations between the temperature and the variables [[??].sub.v] (t), [T.sub.FK] (t), [T.sub.vp] (t) respectively. As stated earlier, the only practically manipulated variables are [[??].sub.FK] (t) and [[??].sub.v] (t). The transfer function for [[??].sub.FK] (t) has this general form (using (5) and (6)):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

From the equation presented above, it can be seen that the relation between T (t) and [[??].sub.FK] (t) is generally integrative. At present, the only practically manipulated variable is [[??].sub.FK] (t), therefore, the further investigation is focused on the analysis of the transfer function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.3 Transfer function coefficients range

In order to determine the range of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] coefficients, a series of simulation experiments were performed in the MATLAB/Simulink environment. Some of the coefficients are very small and consequently they could be possibly neglected for the control system design.

3.4 Poles and zeros

Given the range of coefficients, it is possible to compute also the range of poles and zeros of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Results are summarized in Table 1.

Zeros at (or very close to) the origin indicate derivative behaviour whereas poles at the same position signalize integrative properties. The table shows that one pole ([p.sub.1]) is directly at the origin resulting in integrative behaviour of the temperature T(t) with respect to [[??].sub.FK] (t). If the poles are located in the left part of the complex plane (their real parts are negative), the system is stable. From this point of view the table shows that generally the system embodies also instability. In addition, when the poles are complex (they also have imaginary parts), it indicates oscillatory behaviour. As revealed by the table, in some conditions the system may embody oscillatory behaviour, however absolute values of complex parts of the poles are relatively small which shows that this effect is not so significant. From the results, it can be also deduced that the system possesses non-minimum phase (NMP) behaviour some of the zeros may become positive (unstable). Generally, NMP-systems are more difficult to control.

4. CONCLUSION

Havig the approximate uncertainty intervals of the coefficients, it suggests using the robust control approach (Morari, 1989). As parameters of the linearized model change, an alternative idea could also be the usage of adaptive control strategies (Astrom, 1989). Another possible approach which proved to be successful is the predictive control (Samek, 2007).

5. ACKNOWLEDGEMENTS

The work was supported by the Grant Agency of the Czech Republic under the grant no. 102/07/P148 and by the Ministry of Education of the Czech Republic under the grant no. MSM 7088352102.

6. REFERENCES

Kolomaznik, K.; Mladek, M.; Langmaier, F.; Tay-lor, M.; Diefendrof, E.J.; Marmer, W.N. & Tribula, E. (1996). CR Patent 280655

Macku, L. (2005). Modeling of tanning salts regeneration process, Proceedings of the 15th Int. Conf. Process Control 2005 (High Tatras, Slovakia, Jun. 7-10), Bratislava: Slovak University of Technology, 127/1-127/4. 2005

Morari, M., Zafirou, E. (1989). Robust Process Control. Prentice Hall, Englewood Cliffs, New Jersey. 1989

Samek, D., Macku, L. (2007). Simulation of model predictive control of semi-batch reactor. Proc. Int. Symp. on Systems Theory SINTES 13 (Craiova, Romania, Oct. 18-20). Craiova: University of Craiova, 180-185. 2007
```Tab. 1. Range of Poles and Zeros

[z.sub.i],
[p.sub.i] Real min. Real max.

[z.sub.1] 0.0185 0.1601
[z.sub.2] -0.0062 -0.0062
[z.sub.3] 1.973 x [10.sup.-5] 0.0016
[p.sub.1] 0 0
[p.sub.2] -0.0226 -0.0059
[p.sub.3] -0.0067 0.0021
[p.sub.4] -0.0018 8.464 x [10.sup.-4]

[z.sub.i],
[p.sub.i] Imag. min. Imag. max.

[z.sub.1] 0 0
[z.sub.2] 0 0
[z.sub.3] 0 0
[p.sub.1] 0 0
[p.sub.2] 0 5.590 x [10.sup.-4]
[p.sub.3] -5.590 x [10.sup.-4] 9.108 x [10.sup.-4]
[p.sub.4] -9.108 x [10.sup.-4] 0
```