Life's a batch!

A large gap between the tip of the sheath and the bottom of the thermowell can introduce a disastrously large lag in the sensor’s response. If the excessive lag is in the secondary temperature measurement, it can cause the secondary loop to be too slow and even violate the cascade rule that it be five times faster than the primary loop. If it is in the primary temperature measurement, the reactor can move to undesirable operating regions before the control system sees it and become unstable if it becomes larger than the positive feedback time constant of an accelerating response of an exothermic reaction.

Profiles in Courage
Often operators or process engineers will find times to manually step up the feed rate. Sometimes this sequence is automated as part of the batch operation. A more effective method that has been shown to reduce batch cycle times by 25% uses a model predictive control (MPC) system to maximize the feed without violating constraints. Figure 2 shows that an MPC can look at the positions of jacket, condenser, and vent system valves and maximize a feed rate that prevents these valves from exceeding their maximum throttle positions for good temperature and pressure control. (Click the Download Now button at the end of this article to view a PowerPoint document with the figure mentioned here)

If the reaction rate is proportional to the product of the mole fractions of reactants, the optimum ratio of reactants can be computed. If Coriolis flow meters are used on the feeds, the concentration of reactant in each feed stream can often be computed from the density of the feed. The inventory of each reactant can be totalized and the mole fraction in the batch computed. The actual exponential ratio of mole concentrations in the reactor can become a controlled variable in an MPC that then manipulates the ratio factor for the feed rates to achieve an optimum target.  If the exponents of the mole concentrations in the reaction kinetics are not exactly known, a first principal model can be run and adapted online by a separate MPC to identify these parameters1. These models can also be run much faster than real time to predict the batch cycle time and yield.

When there is hold time in a batch phase to wait for a reaction to go to completion, it is usually conservatively set to make sure there is never a customer complaint. Process simulations can provide the ammunition and an adapted online process model can predict the proper hold time and the product concentration, which can also eliminate operator attention requests and wait times for lab sample results.

Ultimate Gain
Figure 3 shows the reactor temperature response if the primary loop was in manual or the secondary controller was in the local automatic instead of the cascade mode. (Click the Download Now button at the end of this article to view a PowerPoint document with the figure mentioned here) This is called the open loop response for the primary controller. All the responses accelerate and then ramp but the self-regulating response decelerates to a steady state, the integrating response continues ramping, and the runaway response further accelerates. If you look at the energy and material balances, there are few true integrating or runaway responses, but in the operating region many fed-batch reactors look like an integrating response because the lack of a discharge flow reduces process self-regulation. Often if there is a steady state it is beyond the limits of time or conditions for a batch phase.

The oscillations from valve deadband show up in the primary loop with a larger amplitude and period than those associated with a resolution equivalent to the offset from deadband. Previously, it was thought that deadband only caused sustained oscillations in integrating and runaway processes because the investigations centered on a single loop2. However, recent tests show the oscillations from deadband will not die out in a self-regulating process with a cascade loop because of the interaction between the primary and secondary controller. The effect of deadband can be minimized by tuning the controllers with a gain setting closer to the ultimate gain.

The ultimate gain is the controller gain that causes sustained equal amplitude oscillations. Unfortunately, deadband and resolution can also cause such oscillations. However, the oscillations from being close to the ultimate gain will decay and grow in amplitude, as the controller gain is lower and higher, respectively, than the ultimate gain. A controller gain exactly equal to the ultimate gain is unlikely, so an equal amplitude oscillation is almost always associated with valve deadband or resolution.

Normally, controller gains are less than one-fourth the ultimate gain to ensure stability despite a change in dynamics. However, low controller gains lead to more oscillations from non-ideal valve performance and can cause slow, nearly sustained oscillations for an integrating process and instability for a runaway process. Thus, for reactors, there is a gain window, where too low of a gain beside too high of a gain is a problem. Since the large process lag (open loop time constant) of reactors makes the ultimate gain larger than users realize, the more common problem is for reactor loops to be too close to or even below the low gain limit, which for a runaway is the inverse of the process gain. Since conventional wisdom is that the oscillations are caused by too high of a controller gain, the gain setting is decreased and the problem gets worse.

Equations 1a, 2a, and 3a show that the ultimate period as a multiple of the dead time (d) increases as you increase the process time lag also known as the open loop time constant (o)3. For an integrating and a runaway process, the multiple is greater than four. If this time constant (o) is taken as additional dead time, the multiple is exactly four for an integrating process. The multiple approaches infinity when the either the dead time or this time constant (o) approaches the positive feedback time constant for a runaway. In other words, if the slowness in the correction of the response indicated by the dead time or process time lag approaches or exceeds the speed at which the process is accelerating, the controller is unstable regardless of tuning and the window of allowable gains is closed.