Responses to Forcing

The response to a step input takes time. The figure shows a first-order response where the rate of change is proportional to the driving force.

Response to a pulse starts out the same as for a step, but the response may not be complete when the forcing changes.The figure shows that the response starts down instantly when the forcing heads in the other direction.

Response to ramp forcing has some initial adjustment but becomesa ramp itself and is parallel to the forcing ramp. The horizontal distance between them after the transients have died out is equal to the time constant.

Response to sinusoidal forcing may again shows some initial adjustment before it also becomes sinusoidal. Note that the amplitude may be different and the peaks may not line up.

We may be interested in how microorganisms respond to a step in nutrient concentration. Switching to a different input pipe will not generate a step in nutrient concentration in the vessel, but sudden dumping of more nutrient may work well. Consumption of nutrient may distort the step, but we could compensate by increased feeding. Since environmental systems have response times of seconds, minutes, or even days, a step that is complete in a few seconds may be perfectly acceptable for analysis based on an idealized input. In other words, the input may be close enough to an ideal case when the time constants are slow relative to the time for the input.

A step down for some nutrient or condition in a vessel can be nearly impossible. We can reduce hydrogen ions to low numbers by adding caustic suddenly, but other complications may result from the salt and heat that are formed. Removing a substrate such as sugar cannot be sudden unless we use a specific adsorbent or reactant in large excess. On the other hand, consumption of sugar by cells will lower its concentration, but not in stepwise fashion.

A ramp up in concentration in the vessel would not be too difficult togenerate for an inert material; assuming perfect mixing the flow rate for agiven set of concentrations could be based on simple mass balances. An ingredient that is consumed or evolved by chemical or biochemical reactions in the vessel would best be converted to a ramp function with the use of acomputer and sensors for concentration. The feed rate would be based on mass balances that consider the reactions. A ramp down might be possible for an ingredient that is being consumed rapidly.