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12DOI: 10.12731/2227-930X-2016-1-27-37
ANALYTICAL SYNTHESIS OF CHEMICAL REACTOR CONTROL SYSTEM
Labutin A., Nevinitsyn V.
The problem of the analytical synthesis of the synergetic control system of chemical reactor for the realization of a complex series-parallel exothermal reaction has been solved. The synthesis of control principles is performed using the analytical design method of aggregated regulators. Synthesized nonlinear control system solves the problem of stabilization of the concentration of target component at the exit of reactor and also enables one to automatically transfer to new production using the equipment.
Keywords: analytical synthesis; control system; chemical reactor; synergetic control theory; computer simulation.
Introduction
The fundamental concept of the design of contemporary flow processes is the concept of the design of cybernetically organized chemical engineering processes and systems, which was rationalized by Academician V.V. Kafarov [1].
According to this concept, during the step of the design of chemical production, which is related to the conversion of initial substances to final products, the problem of optimal synthesis of reactor junction and problem of synthesis of the process control algorithms is solved; and, at the step of consumption, the sub-problem of the organization of optimal functioning of object under the effect of parametrical and signal disturbances [1-5].
Despite the large number of the works related to the automation and control of chemical reactors [6-9], the problem of synthesizing
control systems that provide the maintenance of optimal modes of their work remains completely unsolved. This is related to the principal feature of chemical reactors as control objects, namely, manifold, nonlinearity, and multicoupling.
The solution of this situation is to develop a physical theory of control, in particular synergetic control theory, the principal features of which were formulated in [10-12].
The use of synergism ideas in the problems of control assumes the development and realization of the directed target self-organization of object-regulator dissipative nonlinear systems. Furthermore, the aim of the motion of system is formulated as the desired invariant manifold in phase space of object, which acts as a target attractor [12].
In general, the problem of synergetic synthesis of the control system is formulated as follows: the control principle, u = (up..., um)T, should be determined as the function of state variables of object u1(x1,..., xj, ..., um(x1,..., xn), which transforms the representative point (RP) of system in phase space from the random initial state to the environment of the given invariant manifolds (x1,..., xn) = 0, S = 1,..., m and subsequent motion along the intersection of manifolds to somewhat stationary point or to somewhat dynamic mode. In the given equations, n is the dimensionality of state vector and m is the number of external controls. On the path of motion, the minimum of the criterion of optimality of system (J) should be attained and its stability should be ensured as follows:
co
W
T^Vs + vl
dr. (1)
The motion of RP in phase space follows the functional equation Tsîfs+Mfs= 0, S = 1,..., m, (2)
where TS is time constant. This is the equation of stable critical point, which gives minimum to the optimizing functional (1). The condition of asymptotic stability of system generally has the form TS>0.
The effectiveness of the method of analytical design of control algorithms by nonlinear objects with the use of synergetic principle (the method of analytical design of aggregated regulators (ADAR)) is given in [13-16].
In this work, the problem of synthesizing the effective control algorithms of the chemical reactor in the realization of the complex series-parallel reaction is stated. The synthesized control system should provide the stabilization of the concentration of target component of chemical reaction on the exit of device under the action of disturbances on object, as well as the transfer of object from one mode of work to another (switch), namely, change of its productivity with the retention of the required quality of target component.
Description of object and statement of control problem
A chemical reactor is a capacity-type device equipped with a mechanical stirrer (Fig. 1). The device functions in isothermal mode. The multistep series-parallel reaction is carried out in the reactor as follows:
A + B—^Pj, A + Pj—A + P2— where A and B are initial reagents; Pp P2 and P3 are products of reaction; kp k2, and are rate constants of steps. The key component is P2 substance. Initial reagents A and B are given to the device by separate flows.
Fig. 1. Flow scheme of chemical reactor
In Fig. 1, the following designations are given: jef, x™ are the concentrations of initial reagents; u1, u2 is the use of initial reagents; u is the use of reaction mixture on the exit from device; x1, x2, x3, and x4 are the concentrations of components A, B, P1, and P2 in reactor; and V = x5 is the volume of reaction mixture in device.
The mathematical model of chemical reactor at constant temperature of reaction mixture and variable degree (volume) has the following form:
-— Aj H----j -— A2 H----, -— A3--y
dx xs x5 dx x5 x5 dx x5 (3)
dx4 ux4 dx5
dx xs dx
where R1 = -k1x1x2 - k2x1x3 - k3x1x4, R2 = -k1x1x2, R3 = k1x1x2 - k2x1x3, R4 K3x1x4 is
the rate of reaction on components.
The problem of control of chemical reactor is formulated as follows: one should synthesize the control principle, which provides the transfer of device from one production, G = xAv, to new production, G = j^u, and stabilization of the concentration of target component at the given degree x4 under the action of disturbances. The change of the output flow with time from u to u can proceed by any principle, including in a stepwise manner.
The transfer to new production G, with the maintenance of the given concentration xA, is made possible by the selection of the particular value of the average time of reaction mixture in device, x. At the given G, x„, u, the required mean time can be achieved by the change in the mixture volume in the device.
The flow of the initial reagent u2 at the input to the device is suggested as the control effect for the volume regulation. In addition, one should also choose the control for stabilizing the concentration x4 at the given degree under the action of disturbances. The analysis of the structure of equations of mathematical model of reactor (3) shows that
variables xl and x3 may act as the internal controls and the direct external effect can be performed only on xl by the change of the consumption of initial reagent u at the input to reactor. Thus, the control channels of the concentration of the target component and volume of the mixture in the device are represented as follows: ul -> x1-> xA, u2 ->x5, where
Because the mathematical model of object (3) contains two external controlling effects ux = u and u2 = u2, we use the ADAR method on the basis of parallel-series combination of invariant manifolds [12]. The procedure for synthesizing the control principle involves the following. At the first step, the invariant manifolds are considered as shown below:
which determines the given relationships between phase coordinates of object, which in turn reflects the specificity of control object and requirements of designer to system. The control principle u = (up u2)T is synthesized so as to perform the transition of representative point of system in phase space from arbitrary initial position to the intersection of manifolds, ..., x5) = 0.
Let us introduce two aggregated macrovariables to consideration, the first of which determines the relationship of xl with controlled variable x4 and the second reflects the technological requirement to the volume of reaction system as follows:
where v1(x4) is somewhat function, which should be determined at subsequent procedure of synthesis. Macrovariables (4) should follow the solution of principal functional equation of ADAR method (2).
Let us introduce the macrovariables ^ and XV2 of Eq. (4) to functional equation (2) for the synthesis of control principle, u = (u1, u2)T. As a result, we obtain the following equations:
Synthesis of control principles via ADAR
\|/s(jc1,...,:r5) = 0, S = 1, 2,
V! =*1+v1(x4), v|/2 =x5-x5,
(4)
cbq 5Vj dx4
dx
dx4 dx
dx,
+ x, + v1=0, T2-± + x5-dx
0
Due to the equations of object (3), these relationships have the following form:
T,
^ i |
/ _ \ -
ux4 + xl +vt =0,
V *5 , _
(5)
r2(Ml + w2 -u)+x5 -x5 = 0, where u is the new set value of the consumption at the exit from reactor, which determines the required production.
We obtain the following relationships for the control principle from Eq. (5):
u _ Oci + vi)Jts *ix5 + uxt <5vj (i?4x5 - x4u)
Txm 1\xi
dx.
(6)
(X5~X5
Controls uj and u2 transfer the RP of the system in the phase space to the intersection of manifolds, ^ = 0 and XV2 = 0, where the relationships xj = -Vj, x5 = x5 are realized and the compression of phase space is realized, i.e., a decrease occurs in the dimensionality of the system of equations (3). The equations of decomposed system with the assumption of relationships xj = -vt and u = u have the following form:
dx2 _ £ u2x2 dx 2 x.
ux.
dx,
ux, dx,
dx
dx
where R2 = kjvjx2, R3 = -kjvjx2 + k2vtx3, R4 = -k2vtx3 + k3vtx4.
The function Vj(x4) in the decomposed system (7) can be considered to be the internal control, under the action of which the motion of object (7) along the intersection of manifolds Y12 = 0 takes place. At the second step of procedure, the investigation of the expression for vj(x4) is performed. For this purpose, the aim of the motion of system
(7) is considered in the form of invariant manifold, which reflects the technological requirement to system as follows:
= x3 - x4 = 0. (8)
Macrovariable ¥3 corresponds to the solution of functional equation, r3\j/3 + \|/3 = 0, which has the following form in extended form with the assumption of Eq. (8) due to the model of decomposed system (7) shown below:
T.i
\
■ K^y^x^ I tC^V^X^
+ x4-x4 = 0. (9)
\,s y
The internal control is written as follows in accordance with Eq. (9): Vi=----W-. (10)
Final equation for the control principle u1 can be obtained by the introduction of v1 function (10) and its partial derivative dvl /dx4 to Eq. (6). The parameters of the adjustment of control laws, which affect the quality of the dynamics of the processes in object-regulator isolated system, are the time constants T1, T2 and T3. The conditions of asymptotic stability have the following form: T1>0, T2>0, T3>0.
Mathematical simulation and results
In order to verify the operation of the synthesized control law by the chemical reactor, the computer simulation of the object-regulator isolated system was performed. Properties of the control system, such as the ability (transfer) of the chemical reactor to switch from one mode of work to another (transfer to new production), disturbance invariance, covariance to the given actions, and the asymptotic stability of isolated system, were also studied.
The simulation was performed with the following technological and constructive parameters of object: V = x5 = 500 L, x™ =19.74 mol/L, x™ =10.93 mol/L, u1 = 1.5 L/min, u2 = 3.5 L/min, u = 5 L/ min, activation energy E1 = 60300 J/mol, preexponential multiplier of rate constant k1 k10 = 109860 L/(mol min), ratios of rate constants of
consecutive steps k2kx = 2, k3kl = 2.5, and the given concentration of target component xA = 0.54 mol/L. The parameters of adjustment of regulators are T1 = 20 min, T2 = 50 min, and T3 = 50 min.
In Figs. 2-4, the examples of transient control processes in the object-regulator isolated system are given with the transition from production G = x4u = 2.7 mol/min to production G = jc4ij = 2.16 mol/min by the change of stress. In Fig. 2, which demonstrates the change of regulated variables, the variant, when switching is performed in manual mode, is given.
Fig. 2. Change of controlled variable x4 (bold line) and the volume of mixture in device x5 (dashed line) with decrease in load u> by 20%: (1) control system
il , L/min
600 X, min
Fig. 3. Change in control action ul = u with decrease in load u by 20%: (1) control system and (2) manual mode
u2, L/min
5-
4 3 2 1 0
Fig. 4. Change in control action u2 = u2 with decrease in load u by 20%: (1) control system and (2) manual mode
As follows from Fig. 2, the major drawback of the manual mode of transfer to new production is the large over-regulation on the concentration of target component, which leads to the loss of product.
Conclusions
In this work, the problem of the analytical synthesis of the control law of chemical reactor with the transition from one production to another was solved via the methods of synergetic theory. Computer simulation of the object-regulator isolated system confirmed these properties of synthesized control system as the ability to switch chemical reactor from one mode of work to another (transfer to new production), disturbance invariance, covariance to the given actions, and asymptotic stability. These facts make synergetic control theory very promising applied to such complex, manifold, and nonlinear objects of chemical engineering as chemical reactors.
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DATA ABOUT THE AUTHORS Labutin Alexander
Ivanovo State University of Chemistry and Technology 7, Sheremetevskiy Avenue, Ivanovo, 153000, Russian Federation
Nevinitsyn Vladimir
Ivanovo State University of Chemistry and Technology 7, Sheremetevskiy Avenue, Ivanovo, 153000, Russian Federation
nevinitsyn@gmail. com
