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 why some controllers need a lambda factor of 2 and others need a lambda factor of 0.02? Do you wonder why level controllers can have controller gains of 100 without oscillating?

This blog can help prevent another sleepless night. In fact "Appendix F - First Principle Process gains, Dead Times, 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.

If you just want to know why controller gains can be so high and lambda factors so low, you can go to the last paragraph.

When my Dad said it was time for the "Talk" when I became a teenager, I thought he meant the "Facts of Life." Instead he explained where process dynamics came from. This talk turned out to be much more useful because I wasn't going to learn about these "Facts" on the playground. 10 years later I was able write the equations in Appendix F.

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 was no outlet discharge flow to let higher or lower temperatures or concentrations out of the vessel. An integrating response of pressure occurs because changes in pressure did not result in much of a change in vent flow. An integrating response of level happens since 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 material and energy balance equations reveal the near-integrating process gain for slow continuous processes is equal to the true integrating process gain for batch processes supporting the idea that processes can be simply treated as near-integrators. The Control Talk Blog a couple of weeks ago showed that even dead time dominant processes could be treated as near-integrators for tuning if a dead time block was used in identifying the near-integrating process gain. You can see all of my blogs by going to the Control Talk Blog website

To summarize, ordinary differential equations can be set up in a form to solve for the process gain and process time constant. The process deadtime comes from thermal lags, mixing delays, and volumes in series. If you don't have the time to read Appendix F or you need to stay awake, you can go with primary results as follows:

- The integrating process gain for liquid level is inversely proportional to the product of the vessel area and liquid 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 vessel 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.
- For a volume to be well-mixed there must not be any stagnant areas and the volume must be completely back mixed from turbulence or agitation.
- For inline temperature and composition control, the process time constant is negligible due to plug flow.
- The process gain for these plug flow volumes is inversely proportional to throughput flow.
- The process dead time for plug flow is the residence time (mass divided by total mass throughput flow).
- Plug flow volumes may have radial mixing but there is no appreciable back mixing; no axial mixing.

The equations in Appendix F do not include a change in phase. Mass transfer can be set up similar to heat transfer except the driving force is a difference in concentrations rather than temperatures and there is a mass transfer coefficient instead of a heat transfer coefficient. The loss of heat in teh liquid from evaporation must be included in the energy balance. Crystallization can be treated as a reaction but population balances should be added to the model to quantity of number of crystals for various crystal sizes.

The controller gain is inversely proportional to the product of the integrating process gain and deadtime. For well mixed vessels the temperature controller gain can be quite large (e.g. 10 to 50); lambda factor can be quite small (e.g. 0.1 to 0.02); because the turnover time is so small (e.g. 5-20 sec) and the integrating process gain is so slow (0.0001 to 0.00002 %/sec per %); process time constant is so large (e.g. 200 - 1000 sec). The best controller gain for a bioreactor temperature loop was 80; integrating process gain was 0.000008 %/sec per %. For level control, the deadtime is smaller and the integrating process gain can be slower leading to an incredibly large controller gain.