Composition, pH and temperature loops that largely determine product quality have a hidden factor that affects the loop linearity and particularly the ability to perform well at low production rates. Here we detail the factor, the consequences and the solutions for continuous unit operations and fed-batch reactions.
Composition, pH and temperature of continuous operations and fed-batch reactions are a function of the ratio of the manipulated rate of change of material or energy to a reference rate of change of material or energy. Most often it is a ratio of flows but the ratio can involve energy (e.g., exothermic reaction heat release or extruder specific energy consumption) or speed (e.g., conveyor speed or sheet line speed). Here we will focus on the reference flow being a feed flow and the manipulated flow being either a reagent flow for pH, reactant flow for reactor composition, a reflux flow for distillation column temperature, or a coolant flow for reactor or heat exchanger temperature. The dependency of these key process variables (composition, pH and temperature) on a ratio is a fundamental consequence seen in the ordinary differential equations (ODE) for the material and energy balances with the help of a charge balance for pH. Since ODE put in a form useful for process control is something we rarely have seen even in university courses, the factor is typically not considered in the process design, expectations for process turndown and the capability of Model Predictive Control (MPC) whose models are based on the manipulation of a flow rather than a ratio.
The effect of the manipulated variable on the process variable is best seen in a plot of the process variable versus the ratio of the manipulated flow to feed flow. We will start with a more detailed look at a continuous pH loop to illustrate better how to compute the open loop self-regulating process gain.
For pH, this ratio is the X axis (abscissa) of the titration curve. The X axis of a titration curve from the lab is often just given in terms of milliliters of reagent added. This axis needs to be divided by the sample volume in milliliters. If the reagent concentration used in the pH control system is the same as was used in the lab, this ratio is the ratio of volumetric reagent flow divided by the volumetric feed flow (e.g., each flow in liters per minute) so that the ratio is dimensionless. Often the lab reagent concentration is more dilute for safety reasons and to make the titration easier (less sensitive to each drop of reagent added). In these cases, the reagent flow must be multiplied by the ratio of the lab reagent normality to the pH control system reagent normality.
The slope of this curve is the more obvious part of the process gain. This slope must be multiplied by the hidden factor so that the engineering units of the process gain are (delta pH)/ (delta reagent flow). The slope of the titration curve is (delta pH)/(delta ratio) where the delta ratio is (delta reagent flow)/( feed flow). If we multiply the slope by the hidden factor of 1/(feed flow), we end up with the process gain being (delta pH)/delta reagent flow). If we multiply this process gain by the valve gain that is the slope of the installed valve characteristic (delta reagent flow)/(delta % PID output) and the process variable measurement gain that is simply (100% PID input)/(pH scale span), we end up with a dimensionless open loop gain. Since the PID algorithm works with % signal, the open loop gain must accordingly be (delta % PID input)/(delta % PID output) which is dimensionless since the % units cancel out. If the pH loop manipulates a reagent flow setpoint, instead of a valve gain we simply have (reagent flow span)/(100% PID output) for the manipulated variable gain. In the literature, this open loop self-regulating process gain is often mistakenly called the process gain eliminating the recognition that the gain depends upon the manipulated variable gain (e.g., valve gain or reagent flow span), measurement gain (e.g., pH span) and whether the process is self-regulating or integrating. For pure batch operations and for level and gas pressure loops, the open loop response ramps at a rate determined by the open loop integrating process gain that has units of (delta % PID input/sec)/(delta % PID output) giving units of 1/sec. For more details on these terms that are often misunderstood see the 8/24/2015 Control Talk Blog “Understanding Terminology to Advance Yourself and the Profession”
For reactor composition control where the reactant A feed flow is manipulated to maintain the reaction stoichiometry, the delta ratio for the slope is set based on the stoichiometric coefficient and is subsequently (delta reactant A feed flow)/(reactant B feed flow). The hidden factor to make the open loop gain dimensionless is the inverse of the reactant B flow.
For column temperature control where the reflux flow is manipulated, the delta ratio is (delta reflux flow)/(column feed flow) and as you probably figured out the hidden factor is the inverse of the column feed flow. For heat exchanger temperature control where the coolant flow is manipulated, the delta ratio is (delta coolant flow/(exchanger feed flow) and the hidden factor is the inverse of the exchanger feed flow. For reactor jacket temperature control where the makeup coolant flow is manipulated, the delta ratio is (delta coolant makeup flow)/(jacket feed flow) and the hidden factor is the inverse of the jacket feed flow. Here the dramatic increase in open loop gain at low production rates is particularly detrimental. The nonlinearity introduced by the hidden factor can be eliminated by a constant jacket flow where the coolant return flow equals the coolant makeup flow from pressure control and the recirculation flow is constant. Unfortunately, the engineers doing the process design may not recognize the hidden factor or its consequences.
While theoretically, this nonlinearity introduced by the hidden factor can be compensated by a valve’s theoretical equal percentage flow characteristic whose slope is proportional to flow, the installed flow characteristic is rarely the theoretical characteristic due to the effect of a changing valve pressure drop and due to the inherent flow characteristic not matching the theoretical characteristic. Also, often the composition, pH and temperature PID manipulates a flow loop setpoint rather than a valve to better compensate for valve response problems and to enable flow feedforward. The valve gain is best made as linear as possible as noted in the 10/202015 Control Talk Blog “The Unexpected Benefits of Signal Characterizers”. Finally, for well mixed volumes, the increase in process time constant (open loop time constant) with residence time cancels out the effect of the increase in open loop gain for a decrease in feed since the PID gain is proportional to the open loop time constant divided by the open loop gain.
For volumes with little to no back mixing (essentially plug flow), the process time constant from the residence time is negligible relative to the deadtime or heat transfer time constant. This is the case for gas reactors, extruders, sheet lines, static mixers, heat exchangers, coils, and jackets. Unfortunately, the increase in residence time from the decrease in feed rate shows up as increase in transportation delay of composition, pH or temperature from the volume inlet to the volume outlet. The combination of an increase in open loop gain and increase in deadtime can easily cause controllers to go unstable. This is often the case for jacket temperature control when total jacket flow is manipulated rather than the recommended makeup coolant flow with a constant jacket recirculation flow.
The fact that the loops important for product quality in continuous operations and fed-batch reactors have a process variable dependent upon a ratio of flows is the reason why ratio control should be used with the ability for the operator to see the actual ratio and go totally on ratio control for startup and measurements not being representative of the process as noted in the 5/30/2015 Control Talk Blog “Essential Feedforward Control and Operator Interface Tips”. For plug flow systems, a correction by the PID controller of the ratio setpoint (effectively a feedforward multiplier) provides the needed compensation for the hidden factor. For well mixed volumes, a correction by the PID controller that is a bias to the manipulated flow (effectively a feedforward summer) provides a correction independent of the hidden factor effectively compensating for errors in the flow measurements. A feedforward summer is less sensitive to feedforward scaling and flow measurement problems especially at low production rates.
For a concise presentation of the concepts and details on the effects of the hidden factor on PID controllers see my ISA book Good Tuning: A Pocket Guide, 4th Edition. If you are up for a more comprehensive view, see my Momentum Press book Tuning and Control Loop Performance - 4th Edition.