Modeling The Probabilities in Flow Systems

Deflnitions. The basic elements of Markov chains in flow systems are the system, the state space, the initial state vector and the one-step transition probability matrix. The system is a fluid element. The state of the system is the concentration of the fluid element in the reactor, assumed perfectly mixed. The state space is the set of all states that a fluid element can occupy, where a fluid element is occupying a state if it is in the state, i.e. at some concentration. For the flow system depicted in Figs.(4-1) and (4-la), the state space SS, which is the set of all states a system can occupy, is designated by 

Finally, the movement of a fluid element from state Cj to state Ck is the transition between the states. 

The initial state vector given by Eq.(2-22), corresponding to Figs.(4-1) and (4-la), reads  

Z+1 designates the number of states, i.e. Z perfectly mixed reactors in the flow system as well as the tracer collector designated by As shown later, the probabilities Si(0) may be replaced by the initial concentration of the fluid elements in each state, i.e. Cj(0) and S(0) will contain all initial concentrations of the fluid elements. The one-step transition probability matrix is given by Eqs.(2-16) and (2-20) whereas pjk represent the probability that a fluid element at Cj will change into Ck in one step, pjj represent the probability that a fluid element will remain unchanged in concentration within one step. 

In the following, general expressions are derived for the transition probabilities corresponding to two general flow configurations. The latter can be reduced to numerous systems encountered in Chemical Engineering elaborated below. 

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