On minimal models of process systems

Minimal representations are known to have no redundant elements, and are therefore of great importance. Based on the notions of performance and size indices and measures for process systems, the paper proposes conditions for a process model being minimal in a set of functionally equivalent models with respect to a size norm. Generalized versions of known procedures to obtain minimal process models for a given modelling goal, model reduction based on sensitivity analysis and incremental model building are proposed and discussed. The notions and procedures are illustrated and compared on a simple example, that of a simple nonlinear fermentation process with different modelling goals and on a case study of a heat exchanger modelling. (C) 2004 Elsevier Ltd. All rights reserved.

Existence and uniqueness of Q-processes with a given finite μ-invariant measure

Let Q be a stable and conservative Q-matrix over a countable state space S consisting of an irreducible class C and a single absorbing state 0 that is accessible from C. Suppose that Q admits a finite mu-subinvariant measure in on C. We derive necessary and sufficient conditions for there to exist a Q-process for which m is mu-invariant on C, as well as a necessary condition for the uniqueness of such a process.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

A novel wastewater treatment process: Simultaneous nitrification, denitrification and phosphorus removal

Simultaneous nitrification and denitrification (SND) via the nitrite pathway and anaerobic-anoxic enhanced biological phosphorus removal (EBPR) are two processes that can significantly reduce the COD demand for nitrogen and phosphorus removal. The combination of these two processes has the potential of achieving simultaneous nitrogen and phosphorus removal with a minimal requirement for COD. A lab-scale sequencing batch reactor (SBR) was operated in alternating anaerobic-aerobic mode with a low dissolved oxygen concentration (DO, 0.5 mg/L) during the aerobic period, and was demonstrated to accomplish nitrification, denitrification and phosphorus removal. Under anaerobic conditions, COD was taken up and converted to polyhydroxyalkanoates (PHA), accompanied with phosphorus release. In the subsequent aerobic stage, PHA was oxidized and phosphorus was taken up to less than 0.5 mg/L at the end of the cycle. Ammonia was also oxidised during the aerobic period, but without accumulation of nitrite or nitrate in the system, indicating the occurrence of simultaneous nitrification and denitrification. However, off-gas analysis found that the final denitrification product was mainly nitrous oxide (N2O) not N-2. Further experimental results demonstrated that nitrogen removal was via nitrite, not nitrate. These experiments also showed that denitrifying glycogen.-accumulating organisms rather than denitrifying polyphosphate-accumulating organisms were responsible for the denitrification activity.