Reference ID: MET-DC1C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The determination of the process gain (K) and time constant (tau) is a fundamental requirement in process control engineering. These parameters define the First-Order Plus Dead Time (FOPDT) model, which serves as the primary empirical representation of system dynamics. By characterizing how a process variable responds to a step change in an input variable, engineers can effectively tune control loops, ensure system stability, and predict transient behavior in industrial applications such as surge tank management, thermal regulation, and flow control.
Methodology & Formulas
The calculation relies on the analysis of a step response test. The process gain represents the ratio of the change in output to the change in input, while the time constant represents the temporal duration required for the system to reach 63.2% of its total response. The following table outlines the validity criteria and operational thresholds required to ensure the accuracy of the model.
Parameter
Constraint/Condition
Minimum Valve Percentage
\( MIN\_VALVE\_PCT \)
Maximum Valve Percentage
\( MAX\_VALVE\_PCT \)
Minimum Delta Input
\( MIN\_DELTA\_INPUT \)
The mathematical derivation of the model parameters is defined by the following algebraic expressions:
To ensure the validity of the model, the system must operate within the linear range defined by the valve limits, and the input change must be significant enough to avoid division by zero errors during the gain calculation.
To calculate the time constant (tau) from a first-order process response, follow these steps:
Apply a step change in the controller output.
Record the process variable until it reaches a new steady state.
Identify the time required for the process variable to reach 63.2 percent of the total change from the initial value to the final steady-state value.
Subtract the initial time of the step change from this timestamp to obtain the time constant.
Process gain (Kp) represents the sensitivity of the process output to changes in the input. You can determine it using the following calculation:
Measure the initial steady-state value of the process variable.
Measure the final steady-state value after the process has settled following a step change.
Calculate the change in the process variable (delta PV).
Calculate the change in the controller output (delta CO).
Divide the change in the process variable by the change in the controller output (Kp = delta PV / delta CO).
Establishing a true steady state is essential for accurate parameter identification for the following reasons:
It provides a stable baseline, ensuring that the observed change is solely due to the input step rather than external disturbances.
It allows for a precise calculation of the total magnitude of the process response.
It prevents errors in the time constant calculation that occur if the process is still drifting from previous load changes.
Dead time, or transport delay, must be identified separately from the time constant to ensure effective controller tuning:
Observe the time elapsed between the moment the step change is applied and the moment the process variable first begins to show a measurable response.
Record this duration as the dead time (theta).
Ensure that the time constant calculation begins only after the dead time has elapsed, rather than from the moment the step change was initiated.
Worked Example: Tank Level Step Test for FOPDT Model
In a process control application, a step test is performed on a vertical cylindrical surge tank to empirically determine its dynamic response characteristics for PID tuning. The tank level is monitored after a step change in the inlet flow rate.
Knowns:
Initial tank level: 40.000 %
Final tank level: 60.000 %
Initial inlet flow: 50.000 L/min
Final inlet flow: 60.000 L/min
Time at start of step input, \( t_{start} \): 0.000 s
Time when level reaches 63.2% of total change, \( t_{63.2\%} \): 120.000 s
Calculate the process gain, \( K \):
\( K = \frac{\Delta \text{Output}}{\Delta \text{Input}} = \frac{20.000\%}{10.000 \text{ L/min}} = 2.000 \% \text{ per L/min} \)
