Bifurcation diagram adiabatic case

The shape of the bifurcation diagram can easily be predicted from the simple heat generation-heat removal diagram in Figure 4 (A-2). For simplicity we assume the adiabatic case with yf as a bifurcation parameter. If we plot y versus the bifurcation parameter yf we obtain the hysteresis curve shown in Figure 5 (A-2). 

It is important to note that while the selection of the bifurcation variable does not affect the maximal number of steady-state solutions, it affects the number and type of bifurcation diagrams. For example, if we selected the coolant or feed temperature as the bifurcation variable then Eq. (18) would be the universal unfolding for both the adiabatic and the cooled case and no isolas would exist [1 2 ]. 

The shape of the bifureation diagram ean be easily eoneluded from this simple heat generation heat removal diagram. Let us take, for simplicity, the adiabatic case with 7 as a bifureation parameter. If we plot 7 versus the bifurcation parameter 7, we obtain the S-shaped hysteresis eurve shown... 

The increase in increases the slope of the removal line because the slope is given by (1 -I- K ). If a bifurcation diagram is drawn for this non-adiabatic case with as the bifurcation parameter and the jacket cooling temperature is the temperature of the middle steady state we get the pitchfork-type bifurcation diagram shown in Figure 7.8. 

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