(430e) Process Operability Analysis of High Dimensional Systems

Conventionally the plant and control designs are done sequentially, but a preferred approach would be to address control issues early in the process design. Operability analysis is one of the bridges that link process design and control. The Operability index framework introduced by Vinson and Georgakis^[1] laid a good foundation for this purpose. This methodology can be used to design new processes, and compare competing designs. It can also be used in the debottlenecking and assessment studies of existing processes. In this framework, the designer evaluates the input to output and output to input maps through elaborate simulation or design calculations.

Subramanian and Georgakis^[2] applied OI methodology to primarily two dimensional nonlinear systems using one dimensional boundary marching algorithms with heuristics to map the interior domains. In the present work, high dimensional continuation method is used for computing the entire solution manifold of algebraic systems. This approach systematically computes the entire region of interest. The power of this method is demonstrated by revisiting some of the processes studied in [2]. An open source software library called Multifario ^[3] is used for exploring the high dimensional input, output and/or disturbance spaces. It computes the solution manifold as a set of overlapping neighbourhoods of polytopes. The computation proceeds by establishing the boundary points and computing an additional polytope that is in the area of interest. This procedure is repeated until the entire solution space of interest is obtained.

The ideal Continuous Stirred Tank Reactor (CSTR) system is considered with a single first order reaction of type taking place. The system parameters are taken from [2]. The coolant flow-rate and the volume of the reactor are taken as inputs and, the exit concentration and the temperature of the reactor are considered as outputs. The disturbance variables are the feed concentration and the feed temperature. The high dimensional computational procedure was validated by reproducing the input to output and output to input maps that were reported in [2]. Interestingly, the overall Desired Input Space (DIS) that involved two inputs, two outputs and two disturbance variables was computed in one full calculation run and it matched well with the previously reported result that was approximated as a union of finite set of maps. Later, the method was applied to a CSTR with series reaction system. Here the new method identified the output maps from a given available input space along with the multiple steady-states.. The results from the new approach matched well with the previously reported results.

Having studied square systems that have the same number of inputs and outputs, a non-square system where one of the output variables is specified to be interval controlled rather than set-point controlled. (e.g. reactor temperature). The additional degree of freedom made available with such specifications, can be exploited to lower the input requirements. Competing design (input) options can be compared with a design objective function, for example that is least away (measured by norm) from the nominal operating point or that requires minimum reactor volume, etc. From the results computed from Multifario for the first order system, we select a group of points that approximate the minimal DIS through heuristics. The results reasonably compare with the theoretical calculation. We are pursuing to fine tune the approach to improve the accuracy.

The effects of key parameters in the software that control the accuracy and speed of computations were studied and will be reported. It has to be pointed out that these calculations are computationally intensive as higher dimensional manifolds of interest are explored fully. At present, we have been able to apply this approach for systems of containing upto eight variables within a few hours of computational time.

References :

^[1] D.R. Vinson, C. Georgakis, J. Process Control, 10, 185, 2000.

^[2] Subramanian, S. and Georgakis, C., Chemical Engineering Science, 56, 5111–5130, 2001.

^[3] M.E. Henderson, Int. J. Bifurcat Chaos, 12, 451, 2002.