^{(1)}, Éva Mihálykó-Orbán

^{(2)}, Csaba Mihálykó

^{(3*)}

(1) Department of Mathematics and Computing, University of Pannonia, Egyetem u. 10, Veszprém, H-8200, HUNGARY

(2) Department of Mathematics and Computing, University of Pannonia, Egyetem u. 10, Veszprém, H-8200, HUNGARY

(3) Department of Mathematics and Computing, University of Pannonia, Egyetem u. 10, Veszprém, H-8200, HUNGARY

(*) Corresponding Author

### Optimal Design and Operation of Buffer Tanks Under Stochastic Conditions

#### Abstract

Uncertainties often appear during the operation of chemical systems. The amortization of machinery, mistakes of operators, and changes in the quality of raw materials are all stochastic events. Safety regulations demand the elimination of random mistakes and the reliable operation of production units. However, the control and maintenance of batch and semi-continuous processes has always been difficult. In this paper, a way of preventing malfunctions in batch and semi-continuous processes is presented by using appropriately designed buffer tanks. A stochastic model was investigated in which batch and continuous subsystems were linked by an intermediate storage tank. The main concern was the reliability of the system. Reliable operation was neither defined as the exhaustion of raw materials nor the excessive accumulation of them. The processing system was investigated over finite time intervals. The counting processes that describe the random batch-input and random batch-output processes are supposed to be independent homogeneous Poisson processes with different rates. By introducing a function that describes the material in storage, reliable operation is defined as when this function satisfies two inequalities for a time interval of any duration. The reliability can be defined as the probability that these inequalities are valid. The first two variables of the function are the initial buffer as well as tank capacities and the third one is the endpoint of the time interval. By applying probabilistic methods, an integral equation is capable of proving the reliability. Nevertheless, its analytical solution cannot be determined, hence the values according to a Monte Carlo simulation are approximated. Although an analytical solution cannot be provided, based on our previous research, when random outputs were not allowed, an approximate function of the reliability was constructed. The parameters were determined by the least squares method. By applying this function, an approximate link could be identified between the necessary initial buffer and tank capacities that belong to a reliability level. As a given reliability level can be guaranteed by many pairs of initial buffer and tank capacities, economic investigations were conducted to help determine the optimal initial buffer and tank capacities from an economical point of view. A stochastic profit function was defined, and its expectation was maximized numerically.

#### Keywords

intermediate storage; stochastic modelling; batch system control; Monte Carlo simulation; economic optimization