Flow control with nonlinear controller

Estimating the inventory of reactants and anticipating their dynamic effects is fundamental in the design and control of chemical plants. The occurrence of nonlinear phenomena is often interrelated with the method of controlling the makeup of fresh reactants [8]. There are two methods for controlling the component inventory in a plant. By self-regulation the fresh reactant is set on flow control at a value given by the desired production rate. No attempt is made to measure or... 

The performance of PID controllers suffers from several limitations disturbances, analyzer deadtime, process nonlinearity, constraints, windup, abrupt startup of a loop, and flow control over a wide operating range. This section addresses approaches that have been developed to improve the performance of PID controllers with respect to this set of control problems. 

The Redwood Mass Flow Controllers (MFCs) were able to control mass flow, but with some limitations. In particular, great care had to be exercised when calibrating the MFCs since the calibration curve sometimes exhibited an inflection point in the flow rate versus set point response. A few MFCs did have linear calibration curves, but many exhibited this nonlinearity to varying extents. This was probably due to the flow measurement and control techniques of these MFCs, which operated by controlling the pressure drop across an orifice. Discussions with Redwood staff indicated that adjusting the internal flow control and measmement parameters for these controllers required a considerable amoimt of work. 

In many situations, a smooth bottom product flow is desired. Some means of achieving this have been discussed in guideline 11 in Sec. 16.6. For maximum smoothing, the bottom product should be flow-controlled, with the set point adjusted by a cascade level (or in scheme 16.4e, temperature) controller (68). Eliminating the bottom baffle can also reduce bottom flow fluctuations (258). A nonlinear level controller (below) can also help. 

Finite element analysis (FEM) has become a popular method for numerical simulation of flow through dies. One of the benefits of EEM is that it can handle non-linear fluids well. A newer numerical technique gaining popularity is boundary element analysis (BEM) Three-dimensional flow analysis with BEM can handle complex flow geometries well however, BEM at this point is not as good as FEM in handling nonlinear fluids. Less detailed analyses often use control volume analysis to reduce the computational effort. The different numerical techniques will be discussed in more detail in Chapter 12. 

Cascaded flow controllers can reduce the effect of any upstream pressure variations or changes in control valve characteristics resulting from nonlinearities or from fouling, as noted in Step II.A.3. Ratio control between wi and W2 can maintain the desired stoichiometric ratio of reactants approximately constant, despite changes in production rate or feed composition. Finally, cascade control can help deal with disturbances introduced by intentional changes in production rate W4, as is discussed next. 

Build a simulation in HYSYS using water with a flow of 20 kmol h at 15°C and 1 atm as the oifly conponent and the Peng-Robinson equation of state as the fluid property package. Use the default tank volume of 2 m and specify liquid flow control on the Liquid Valve page of the tank unit operation. Calculate the flow out of the tank using Equation W2.1, which describes a linear valve, and then Equation W2.2, which describes a nonlinear valve. In both cases, the outlet flow rate is a function of the liquid head on the tank oifly. 

Adaptive Control. An adaptive control strategy is one in which the controller characteristics, ie, the algorithm or the control parameters within it, are automatically adjusted for changes in the dynamic characteristics of the process itself (34). The incentives for an adaptive control strategy generally arise from two factors common in many process plants (/) the process and portions thereof are really nonlinear and (2) the process state, environment, and equipment s performance all vary over time. Because of these factors, the process gain and process time constants vary with process conditions, eg, flow rates and temperatures, and over time. Often such variations do not cause an unacceptable problem. In some instances, however, these variations do cause deterioration in control performance, and the controllers need to be retuned for the different conditions. 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

Figure 7.4c shows a AP transmitter used with an orifice plate as a flow transmitter. The pressure drop over the orifice plate (the sensor) is converted into a control signal. Suppose the orifice plate is sized to give a pressure drop of 100 in H2O at a process flow rate of 2000 kg/h. The AP transmitter converts inches of HjO into milliamperes, and its gain is 16 mA/100 in HiO. However, we really want flow rate, not orifice-plate pressure drop. Since AP is proportional to the square of the flow rate, there is a nonlinear relationship between flow rate F and the transmitter output signal ... 

Chemical reactors intended for use in different processes differ in size, geometry and design. Nevertheless, a number of common features allows to classify them in a systematic way [3], [4], [9]. Aspects such as, flow pattern of the reaction mixture, conditions of heat transfer in the reactor, mode of operation, variation in the process variables with time and constructional features, can be considered. This work deals with the classification according to the flow pattern of the reaction mixture, the conditions of heat transfer and the mode of operation. The main purpose is to show the utility of a Continuous Stirred Tank Reactor (CSTR) both from the point of view of control design and the study of nonlinear phenomena. 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

The interference process in this collinear approach is, however, different from the interference realized by mixing the local oscillator and the CARS field on a beam splitter. Interference takes place in the sample, which, in the presence of multiple frequencies, mediates the transfer of energy between the beams that participate in the nonlinear process. The local oscillator mixes with the anti-Stokes polarization in the focal volume, and is thus coherently coupled with the pump and Stokes beams in the sample through the third-order polarization of the material. In other words, the material s polarization, and its ability to radiate, is directly controlled in this collinear interferometric scheme. Under these conditions, energy from the local oscillator may flow to the pump and Stokes fields, and vice versa. For instance, when the local oscillator field is rout of phase with the pump/Stokes-induced anti-Stokes polarization in the focal interaction volume, complete depletion of the local oscillator may occur. The energy of the local oscillator field is not redistributed in terms... 

Whereas the operation of batch reactors is intrinsically unsteady, the continuous reactors, as any open system, allow for at least one reacting steady-state. Thus, the control problem consists in approaching the design steady-state with a proper startup procedure and in maintaining it, irrespective of the unavoidable changes in the operating conditions (typically, flow rate and composition of the feed streams) and/or of the possible failures of the control devices. When the reaction scheme is complex enough, the continuous reactors behave as a nonlinear dynamic system and show a complex dynamic behavior. In particular, the steady-state operation can be hindered by limit cycles, which can result in a marked decrease of the reactor performance. The analysis of the above problem is outside the purpose of the present text ... 

These three nonlinear ordinary differential equations will be used to simulate the dynamic performance of the CSTR. The openloop behavior applies when no controllers are used. In this case the flowrate of the cooling water is held constant. With closedloop behavior, a temperature controller is installed that manipulates cooling water flow to maintain reactor temperature. 

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