Chapter 8 Professional Reference Shelf

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R8.2 Steady-State Bifurcation Analysis

	In reactor dynamics it is particularly important to find out if multiple stationary points exist or if sustained oscillations can arise. Bifurcation analysis is aimed at locating the set of parameter values for which multiple steady states will occur.¹ We apply bifurcation analysis to learn whether or not multiple steady states are possible. A bifurcation point is a point at which two branches of a curve coalesce as a parameter is varied. Consider the function , in which x is a scalar variable and is a parameter. Figure CD8-1 shows curves (AB, BC, and BD) for which

		(CD8-1)

	is satisfied. We see that as we start to increase TIME GIF along AB, there is only one value of x for a given that will satisfy Equation (CD8-1). However, as we continue to increase along AB, we reach a bifurcation point * beyond which there are two values of x that satisfy Equation (CD8-1) for a given value of . Consequently, we analyze our system of equations to learn if a bifurcation point exists that denotes multiple solutions. There is another condition that is necessary for a bifurcation point to exist. If we were to move an incremental amount away from the bifurcation point but still remain on BC or BD, we would have

		(CD8-2)

	Figure CD8-1 Bifurcation point.

	Expanding Equation (CD8-2) in a Taylor series, it can be shown that at the bifurcation point

		(CD8-3)

	If Equations (CD8-1) and (CD8-3) are satisfied, there will be a set of parameter values for which we will have multiple steady states (MSS). We shall continue the discussion of the first-order reaction taking place in a CSTR to illustrate bifurcation analysis. A slight rearrangement of a combination of Equations (8-68) and (8-69) from the energy balance gives

		(CD8-4)

	which is of the form

		(CD8-5)

	where and are positive constants.

	Similarly, with some minor manipulation, the mole balance on an isothermal CSTR can be put in similar forms,

		(CD8-6)

	or

		(CD8-7)
	We observe that both the CSTR energy and mole balances are of the form

		(CD8-8)
	If F(y) is a monotonically increasing function as shown in Figure CD8-2, the derivative of the function with respect to y will never be zero, that is,



	Upon differentiating Equation (CD8-8), we have	(CD8-9)


	and we see that dF/dy can never be zero if the maximum value of the derivative of G is less than . Thus the sufficient condition for uniqueness is

Uniqueness		(CD8-10)

	When Equation (CD8-10) is satisfied there will be no multiple steady states. However, if max , we do not know at this point whether or not multiple solutions exist and we must carry the analysis further. Figure CD8-2 MSS with a parabolic function. Figure CD8-3 F(y ) curve common to chemically reacting systems. We now apply the condition for a bifurcation point, Equations (CD8-1) and (CD8-3), to the CSTR balances, that is, Equation (CD8-8). If multiple steady states exist, there will be a bifurcation point, y*, at which the following conditions are satisfied:

Conditions for multiple steady states		(CD8-11) (CD8-12)


	Figure CD8-2 shows that a parabola satisfies both conditions (1) and (2), that is, Equations (CD8-11) and (CD8-12), respectively. Consequently, we see that there will be a set of parameter values for which multiple solutions will exist, as demonstrated by the dashed-line parabola. Figure CD8-3 shows the shape of a typical curve in reaction systems with multiple steady states.