Optimal Response of Amaerobic Digesters to Inhibitors Entering with the Feed

A serious problem occasionally arising in anaerobic digester operation is reactor failure (resulting in lengthy and costly shutdowns) due to inhibiting factors such as chloroform, ammonia and oxygen entering with the feed. What is essentially happening is that as the inhibitor concentration builds up in the reactor, the specific growth rate of the methanogenic bacteria drops and when this drops below the dilution rate (inverse of residence time) the methanogens wash out. Lowering the dilution rate to below the ultimate value of the specific growth rate can save the digester, but the question is how should this be lowered. In attempting to answer the above question we used a simple model for the anaerobic digestion process and employed Pontryagin´s maximum principle to determine the control law that maximizes the total of methane produced over a time period of several time constants, assuming that it is known that an inhibitor enters with the feed at the initial time. The optimization problem is singular with the optimal dilution rate being initially ally on a bound (whether this is upper or lower depends on the initial conditions) until a singular arc is reached. By using the optimal steady-state values of the state variables as final values of the singular arc, the arc was determined using backwards integration. It turns out that although the singular arc is defined by a complex expression of the state and input variables, it is approximated extremely well by a linear relationship between dilution rate and methane rate.