Do you lie awake at night wondering what the source of process dynamics is? Do you wonder why temperature and composition controllers tend to oscillate at low production rates and low levels? Are you perplexed about why some controllers need a gain of 0.2 and others need a gain of 20? Do you wonder why level controllers can have a controller gain of 100 without oscillating? This tip can help prevent another sleepless night, but if that is not enough, click this link to download Appendix F – “First Principle Process Gains, Deadtimes, and Time Constants” has the essential knowledge to help you fall asleep right away, especially if differential equations make you drowsy. At your next BBQ, amaze friends and relatives by explaining the effect of volumes and flows on process dynamics. Just provide recliners or hammocks as you lull them to sleep.
As a result of programming simulations to study unit operations, I learned how to set up the ordinary differential equations (ODE) for the material and energy balances. I did not have to solve them. For simulations I just needed to numerically integrate them. For real-time simulations, I could use the simplest method. For Appendix F, I just had to put the ODE in the form for self-regulating, integrating, and runaway processes. A runaway response develops when the heat from an exothermic reaction exceeds the heat removal rate. An integrating response of temperature and composition occurs when there is no discharge flow to let higher or lower temperatures or concentrations out of the vessel. Gas pressure has an integrating response when changes in pressure do not result in much of a change in vent flow. Level has an integrating response because the change in pump flow with level is negligible.
If the discharge valve is closed, the level can only go up. If there is no reaction or vaporization in a batch, composition from feed addition can only go up. If the reaction is not reversible and there are no side reactions, the product concentration in the batch from reactant addition can only go up. If an acid is added that is not consumed, the batch pH can only go down. If a base is added that is not consumed, the batch pH can only go up. Composition control loops for these one-sided responses (single direction) need to use the slope of the batch profile as the process control variable where the slope setpoint goes toward zero at the end of the batch.
The near-integrating process gain for slow continuous processes is equal to the true integrating process gain for batch processes, supporting the idea that slow continuous processes can be effectively treated as near-integrators.
Concept: Ordinary differential equations can be set up to solve for the process gain and process time constant. The process deadtime comes from thermal lags, mixing delays, and volumes in series.
Details: The integrating process gain for liquid level is inversely proportional to the product of the area and the density. The integrating process gain for gas pressure is proportional to the absolute temperature and is inversely proportional to volume. The integrating process gain for temperature for vessel temperature control by manipulation of jacket temperature is proportional to the product of the overall heat transfer coefficient and area and is inversely proportional to the product of the liquid mass and heat capacity. The integrating process gain for composition control of vessel concentration is proportional to feed concentration and inversely proportional to liquid mass. The turnover time for a well-mixed volume is the liquid mass divided by the summation of the mass flows from agitation and recirculation. The controller gain is inversely proportional to the product of the integrating process gain and deadtime. For well-mixed vessels the controller gain can be quite large (e.g., 10 to 50); the Lambda factor can be quite small (e.g., 0.1 to 0.02) because the turnover time is so small (e.g., 5 to 20 sec), the integrating process gain is so slow (e.g., 0.0001 to 0.00002%/sec per %), and process time constant is so large (e.g., 200 to 1000 sec). For inline temperature and composition control, the process time constant is negligible. The process gain for these plug flow volumes is inversely proportional to throughput flow. The process deadtime is the residence time (liquid mass divided by total mass throughput flow). Plug flow volumes by definition have no back mixing and no axial mixing, but may have radial mixing.
Watch-outs: For a volume to be well-mixed, there must not be any stagnant areas and the liquid volume must be completely back mixed from turbulence or agitation. The open loop response includes automation system dynamics in addition to process dynamics. The total loop deadtime is the summation of the final control element deadtime (e.g., pure deadtime from pre-stroke, deadband, resolution, and threshold sensitivity limits and equivalent deadtime from slew and ramp rates), process deadtime, transportation delay to sensor, 1½ times the analyzer cycle time, ½ of PID execution time, and the equivalent deadtime from sensor lags, transmitter damping, and signal filters. For the large process time constants in vessels and columns, 100% of all the automation system lags become equivalent deadtime. The open loop gain is the product of the final control element gain (e.g., the slope of the installed characteristic of a control valve in flow units per % signal), process gain in engineering units, and measurement gain (100% divided by the measurement span in engineering units). The open loop gain must be dimensionless.
Exceptions: The equations in Appendix F do not take into account a change in phase. The equations for mass transfer can be set up similarly to heat transfer except that the driving force is a difference in concentrations or pressures (component vapor pressure minus vessel pressure) rather than temperatures, and there is a mass transfer coefficient instead of a heat transfer coefficient. The loss of heat from a liquid through evaporation must be included in the energy balance. Crystallization can be treated as a reaction but population balances should be added to model the number of crystals in crystal size classes.
Insight: Relatively simple material and energy balances can be developed to provide first principle relationships for understanding the effects of equipment design and operating conditions on process dynamics.
Rule of Thumb: Work with the process engineers to set up the ordinary differential equations in a simplified form that enables the estimation of process dynamics.