First Principal Process Gains, Dead Times, and Time Constants

First principal relationships can define process cause and effects that can lead to improved controller tuning and performance by the selection of better tuning rules and process variables for scheduling of tuning settings. It also affects the choice of control valve trim and the feedforward design. The understanding of these relationships does not require a degree in chemical engineering but presumes just some understanding of common terms (e.g. heat transfer coefficient and area), relationships (e.g. ideal gas law), and physical concepts (e.g. conservation of mass and energy).

Equations have been developed from first principal relationships for the process gains, dead times, and time constants of volumes with various degrees of mixing. The results show that for well mixed volumes with negligible injection delays, the effect of flow cancels out for the controller gain if one of the following methods is used: Lambda self-regulating rule where Lambda is set equal to the dead time, or the reaction curve method. The effect of flow also cancels out for the reset time besides the controller gain if the process is treated as a “near integrator” and the Lambda integrating tuning rule is used. This is because the flow rate cancels out in the computation of the ratio of process gain to time constant that is the “near integrator” gain. This ratio and “near integrator gain” are inversely proportional to the process holdup mass (e.g. liquid mass). However, for temperature control the effect of changes in liquid mass cancels out because a change in level increases the heat transfer surface area covered. Several authors have mistakenly tried to schedule controller tuning based on liquid level for reactor temperature control. One author has reported being bewildered by its failure. This is not the case for gas pressure control. The equations show that liquid level has a profound effect on the process integrating gain for vessel pressure control because it changes the vapor space volume without any competing effect. To summarize, the integrator gain for composition and gas pressure is inversely proportional to liquid level (liquid mass). For temperature, the effect of level cancels out unless the level is above or below the heat transfer surface area, which is unusual but can occur at the beginning or end of a batch when coils instead of a jacket is used for heat transfer. For temperature, the integrator gain is nearly always proportional to the overall heat transfer coefficient that is a function of mixing, process composition, and fouling or frosting.

The equations also show that if the transport delay for flow injection is large compared to the time constant, which does occur for reagent injection in dip tubes for pH control), then the controller gain will be proportional to flow. Note that pH control is a class of concentration control.

For the control of temperature and concentration in a pipe, the process dead time and process gain are both inversely proportional to flow and the process time constant is essentially zero, which makes the actuator, sensor, transmitter, or signal filter time lag the largest time constant in the loop. Thus, the largest automation system lag determines the dead time to time constant ratio. For a static mixer, there is some mixing, and the process time constant is inversely proportional to flow but is usually quite small compared to other lags in the loop. The controller gain is generally proportional to flow for both cases.

Finally, the above has implications so far as whether a flow feedforward multiplier or summer and whether a linear or equal percentage trim should be used. A flow feedforward multiplier and equal percentage trim, which both have a gain proportional to flow, can help compensate for a process gain that is inversely proportional to flow provided the process time constant is not also inversely proportional to flow. This is generally the case for temperature and concentration control of essentially plug flow volumes (pipelines, static mixers, and heat exchangers). For well mixed volumes, feedforward summers and an installed linear characteristic for valves is generally best. For control valves this corresponds to a linear trim when the available pressure drop that is much larger than the system pressure drop or critical pressure drop so the installed flow characteristic is close to the inherent flow characteristic.

The results are also useful for determining the dead time to time constant ratio, which has a profound effect on the tuning factors used and the performance of dead time compensation, which has been discussed in the category of controller tuning and control performance on this blog site. A copy of Advanced Application Note 4 that summaries and derives these equations is available from me.