disturbances.
Once a process is disturbed, compensation is required. Compensation can occur as part of a manual or automatic control system. Manual process control occurs when operators notice that a process variable has deviated from the desired setpoint and they manipulate a controlled variable (e.g., valve opening or pump speed) to try to return the process variable to the setpoint. Under manual control, operators determine the timing, duration, magnitude, and direction of the required adjustment based on their experiences with and knowledge of the process (see Figure 7.1 for an example of information flow in a process).
Automatic process control involves using various hydraulic, pneumatic, electromechanical, and electronic equipment to emulate manual controls (Figure 7.2). The operator establishes the objective, but automatic process control equipment executes the required adjustments.
The goal (i.e., control objective or objective function) of any control scheme must be defined in terms of measurable output variables. It may be as simple as “minimize effluent biochemical oxygen demand (BOD) of the wastewater” or as complex as “minimize sludge disposal and energy costs while keeping effluent BOD below effluent permit requirements”. If some objectives (e.g., cost) conflict with others (e.g., effluent quality and reuse), engineers typically can assign weights to each objective or use some as constraints (e.g., permit requirements).
FIGURE 7.1 Information flow in a process.
FIGURE 7.2 An example of information flow in feedback process control.
Often, the objective will be stated in terms of keeping one output variable as close to a desired value (i.e., setpoint) as possible. One example of a setpoint is “maintain dissolved oxygen level close to 2 mg/L”. In other instances, the objective will be to minimize one parameter (e.g., energy use).
2.0 CONTROL THEORY
The better a process’s behavior and inputs are understood and measured, the more its results can be controlled. However, a process’s performance can be improved even if its behavior is only approximately known (i.e., a mathematical model of it has been created). Process models are typically used to develop a process control strategy, and the task of determining which model and parameters to use is called process identification.
Although complex, detailed models are available for many processes. Process control strategy designers typically only need a simplified model: steady-state first order, first-order lag, or second-order lag (and, perhaps, a dead-time [transport] lag).
In a steady-state process, the rate of change of all measurable quantities is zero. Any process is considered steady state if its inputs change more slowly than its dynamics.
The model for a steady-state first-order process is expressed algebraically, typically as a simple proportional relationship. The proportionality constant is called process gain (KP). If a process’s input is X and its output is Y, then
For example, the sludge production in an activated sludge process may be proportional to the BOD load for a certain range of operating conditions. The process gain is the observed yield (kilograms [pounds] of sludge produced per kilogram [pound] of BOD removed), as follows:
If a process has more complex kinetics, however, this model is inadequate. When input conditions change in such processes, the outputs adjust gradually to steady state, not instantaneously. Mathematical models of dynamic systems may include differential equations.
The simplest type of dynamic process is a first-order lag process, which is characterized by capacity and resistance. One example of a first-order process is a completely mixed, constant-volume (V) tank in which some nonreactive substance (C), such as table salt, has been added to the water (Q) (Figure 7.3). When a first-order process’s input changes, the process approaches a new steady state, at first rapidly and then more slowly. This type of response is called exponential decay.
The dynamics of a first-order process can be described by two parameters: process gain and process time constant (TP). The process gain of a first-order process (once it reaches steady state) is defined as the ratio of output changes to input changes. In this instance, the gain is 1 (i.e., 1 unit change in input concentration results in 1 unit change in output concentration). Chemical reactions could change this gain.
FIGURE 7.3 A first-order system and the response curve for a change in input concentration from 0.0 to 1.0.
The process time constant in this example is equal to the process’s detention time (V/Q). The higher the flowrate, the faster the output responds to changes in influent concentration. In the presence of a chemical reaction that produces salt, the dynamics would slow and TP would increase.
Numerically, TP equals the time required to achieve 62.3% of the difference between the initial and final conditions. After three TP intervals, a first-order process will have completed about 95% of the time needed to achieve steady state. (In an activated sludge process, the TP for changes in biomass equals the sludge age.)
Many processes have more complex behavior than first-order processes. Processes with two capacities and resistances, such as two first-order processes connected in series, are called second-order processes (Figure 7.4). The response of such a process to an incremental change in input would resemble a sigmoidal (S-shaped) curve. The process would be described by process gain and two time constants.
Another type of second-order process is characterized by capacity, momentum, and, typically, resistance. One common example of this is a shock absorber.
Any second-order process can oscillate around the steady-state value. For example, if someone pushes down on a car fender, the spring will return the car to the original position. If the shock absorber is not providing sufficient resistance (i.e., the process is “underdamped”), momentum will cause the car to overshoot its original position. The car will then vibrate up and down with decreasing amplitude until it settles to the steady-state position. If the shock absorber is providing enough resistance (i.e., the process is “critically damped”), the car will not overshoot its original position (Figure 7.4).
FIGURE 7.4 An example of a second-order system and a response curve for a change in input concentration from 0.0 to 1.0.
Another common factor in a process’s dynamic response is dead time (transportation lag). Suppose a first-order process is connected to a 305-m-long (1000-ft-long), 0.15-m-diameter (6-in.-diameter) pipe with a volume of almost 5700 L (1500 gal). If the flow through the pipe is 6 L/s (100 gpm), its discharge would be representative of what
