Multiple steadystates

In a linear system, the phenomenon of multiple steadystates cannot occur. It is the nonlinearity of the process, the exponential temperature dependence of the reaction rate, that can lead to more than one steadystate. 

With multiple steadystates, the process outputs can be different with the same process inputs. The reverse of this can also occur. This interesting possibility, called input multiplicity, can occur in some nonlinear systems. In this situation we have the same process outputs, but with different process inputs. For example, we could have the same reactor temperature and concentration but with different values of feed flow rate and cooling water flow rate. 

Figure 6.9d shows the Qg and Qn for another reactor which has some very interesting features. Now there are three intersections of the curves. This means that there can be three different temperatures that are steadystates. For exactly the same feed conditions and parameter values, the reactor could settle out at three different temperatures fj, and %. This phenomenon of multiple... 

The result is the most useful of all the Laplace transformations. It says that the operation of differentiation in the time domain is replaced by multiplication by s in the Laplace domain, minus an initial condition. This is where perturbation variables become so useful. If the initial condition is the steadystate operating level, all the initial conditions like are equal to zero. Then simple multiplication by s is equivalent to differentiation. An ideal derivative unit or a perfect differentiator can be represented in block-diagram form as shown in Fig. 9.3. 

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