Stability in the Operating Point

The component balance and energy balance are highly interactive. When the temperature increases, the rate of reaction increases according to an Arrhenins equation. However, the increased conversion due to the temperature rise will slow down the rate of reaction. The stability should therefore be considered for an equiUbrinm point, taking into account the interaction between the temperature T and conversion C. The other input variables are assmned to be constant F, , Ct, 7, and Tc. 

In Chapter 5 is was stated that a linear system based on deviation variables, is asymptotically stable if the roots of the characteristic eqnation of an inpnt-output relationship have negative real parts. The roots of the characteristic eqnation correspond to the poles of the transfer function or the eigen valnes of the homogenons part of the differential equation. Therefore, it is necessary to derive the characteristic linear input-output relationship for the process in an operating point. Because only the homogenous part of the differential equation is of interest, the inputs of the differential equations can be ignored. Next, fiom the roots of the characteristic equation, conditions for stability can be formulated. The following three steps will be performed  

The state equation for the temperatuie is given by Eqn. (7.10). When Tc and V are constant, linearization gives  

Step 2 Substitution in order to obtain the characteristic equation. 

The substitution to arrive at the characteristic system equation can be done as follows. Suppose the system can be described by the following state equations  

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