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Sufficient conditions for uniqueness of the steady state
Pergamon Press.
Chemical Engineering Science, 1968, Vol. 23, p. 1501,
Printed in Great Britain.
Sufficient conditions for uniqueness of the steady state (First received
26 January
1968; in revisedform 29 January
THE STRIKINGLY
powerful condition for uniqueness of the steady state for diffusion and reaction given by Luss and Amundson [ l] rests on a theorem concerning the direction of change induced in an eigenvalue by a change in the weighting function. The reference given by them [2] only proves this for a positive weighting function, but this function may have a change of sign in the comparison equation to which this theorem is applied. Since the theorem is so useful in this kind of analvsis it mav be worth emohasizina that it is true even when the weighting function changes sign in the interval. Mason [3] showed that the first positive eigenvalue of the equation
z r$g>+ _(
XPp(r)v
= 0,
osrs1,
subject to suitable homogeneous boundary conditions (such as v( 1) = 0, v’(0) = 0) satisfied the following minimum problem. Among all functions w(r) satisfying the same boundary conditions and for which H(w)
=~,%‘p(r)d(r)dr=
1,
H*(v) =
1968)
,dPp*(r)v”(r)
dr 5 H(v) = 1.
If, however, we denote the value of H*(v) by a2 and take w = v/a, then H*(w) = 1 and w satisfies the boundary conditions. Then A* s jO’r”‘[w’(r)ledr
< u2 l
rm[w’(r)12dr,
for the first inequality is a property of the eigenvalue and the second a consequence of d being greater than 1. But aw = v so that the last integral is just A; it follows that A* G A. An interesting feature of the case where p(r) changes sign is that there are two sets of eigenvalues, a positive set diverging to +m whose lowest member is determined by the variational principle given above and a set of negative eigenvalues diverging - - to -Q) (141P. 237). The eiaenfunctions corresuonding to the least positive and greatestnegative eigenvalues are of one sign and the variational principle for the latter is 0 2X=-J;
rm[o’(r)12dr +-;
r’“[w’(r)]“dr,
where w(r) is any function satisfying the boundary conditions and the normalization
the first eigenfunction v(r) is such that H(w) =J’; r’“p(r)d(r)dr=-1. A = Jo1r*[v’(r)12dr
G ,d r’“[w’(r)]*dr.
This is true even though the function p(r) does change sign. Since the boundary conditions are homogeneous this represents a normalization of the trial functions, but in addition it tells us that w(r) is sufficiently large in the part of the interval where p(r) is positive that it makes the whole integral positive. Consider a similar equation with the same boundary conditions, but a different weighting function p*(r), which is never less than p(r), and let the lowest eigenvalue be A*. This eigenvahre satisfies a similar inequality. Since p*(r) 2 p(r), then H*(w) ---H(w) = J; r”(p*(r)
-p(r)}w’(r)
This integral is again a normalization, but it implies that the trial functions, and so also the eigenfunction, is sufficiently dominant in the part of the interval where p(r) is negative that it makes the integral negative. A proof similar to that given above shows that if v(r) satisfies (rm v’)’ + A* p*(r) P v = 0 and the same boundary conditions, then if p@) G p(r), the first negative eigenvahre is increased, i.e. x * 3 A. It would be worthwhile to study more fully the relationship between the sufficient and the exact conditions for uniqueness to determine how precise the sufficient condition may be. Such a comprehensive study is in progress for the nonisothermal first order reaction in a slab.
dr 2 0
RUTHERFORD
for any trial function w. The first eigenfunction v(r) of the first problem is not a candidate for the second minimization since
Department of Chemical Engineering University of Minnesota Minneapolis, Minn. 55455, USA.
REFERENCES [1] LUSS D. and AMUNDSON N. R., Chem. Engng Sci. 1967 22 253. [2] COURANT R. and HILBERT D., Methods of MathematicalPhysics, Vol. 1. Interscience 1953. [3] MASON M., Trans.Am. math. Sot. 19067 337. [4] INCE E. L., Ordinary Di&rential Equations. Longmans, Green 1926; Dover Edition, 1956. C.E.S.Vd a No,12--o
1501
ARIS
Chemical Engineering Science, 1968, Vol. 23, p. 1501,
Printed in Great Britain.
Sufficient conditions for uniqueness of the steady state (First received
26 January
1968; in revisedform 29 January
THE STRIKINGLY
powerful condition for uniqueness of the steady state for diffusion and reaction given by Luss and Amundson [ l] rests on a theorem concerning the direction of change induced in an eigenvalue by a change in the weighting function. The reference given by them [2] only proves this for a positive weighting function, but this function may have a change of sign in the comparison equation to which this theorem is applied. Since the theorem is so useful in this kind of analvsis it mav be worth emohasizina that it is true even when the weighting function changes sign in the interval. Mason [3] showed that the first positive eigenvalue of the equation
z r$g>+ _(
XPp(r)v
= 0,
osrs1,
subject to suitable homogeneous boundary conditions (such as v( 1) = 0, v’(0) = 0) satisfied the following minimum problem. Among all functions w(r) satisfying the same boundary conditions and for which H(w)
=~,%‘p(r)d(r)dr=
1,
H*(v) =
1968)
,dPp*(r)v”(r)
dr 5 H(v) = 1.
If, however, we denote the value of H*(v) by a2 and take w = v/a, then H*(w) = 1 and w satisfies the boundary conditions. Then A* s jO’r”‘[w’(r)ledr
< u2 l
rm[w’(r)12dr,
for the first inequality is a property of the eigenvalue and the second a consequence of d being greater than 1. But aw = v so that the last integral is just A; it follows that A* G A. An interesting feature of the case where p(r) changes sign is that there are two sets of eigenvalues, a positive set diverging to +m whose lowest member is determined by the variational principle given above and a set of negative eigenvalues diverging - - to -Q) (141P. 237). The eiaenfunctions corresuonding to the least positive and greatestnegative eigenvalues are of one sign and the variational principle for the latter is 0 2X=-J;
rm[o’(r)12dr +-;
r’“[w’(r)]“dr,
where w(r) is any function satisfying the boundary conditions and the normalization
the first eigenfunction v(r) is such that H(w) =J’; r’“p(r)d(r)dr=-1. A = Jo1r*[v’(r)12dr
G ,d r’“[w’(r)]*dr.
This is true even though the function p(r) does change sign. Since the boundary conditions are homogeneous this represents a normalization of the trial functions, but in addition it tells us that w(r) is sufficiently large in the part of the interval where p(r) is positive that it makes the whole integral positive. Consider a similar equation with the same boundary conditions, but a different weighting function p*(r), which is never less than p(r), and let the lowest eigenvalue be A*. This eigenvahre satisfies a similar inequality. Since p*(r) 2 p(r), then H*(w) ---H(w) = J; r”(p*(r)
-p(r)}w’(r)
This integral is again a normalization, but it implies that the trial functions, and so also the eigenfunction, is sufficiently dominant in the part of the interval where p(r) is negative that it makes the integral negative. A proof similar to that given above shows that if v(r) satisfies (rm v’)’ + A* p*(r) P v = 0 and the same boundary conditions, then if p@) G p(r), the first negative eigenvahre is increased, i.e. x * 3 A. It would be worthwhile to study more fully the relationship between the sufficient and the exact conditions for uniqueness to determine how precise the sufficient condition may be. Such a comprehensive study is in progress for the nonisothermal first order reaction in a slab.
dr 2 0
RUTHERFORD
for any trial function w. The first eigenfunction v(r) of the first problem is not a candidate for the second minimization since
Department of Chemical Engineering University of Minnesota Minneapolis, Minn. 55455, USA.
REFERENCES [1] LUSS D. and AMUNDSON N. R., Chem. Engng Sci. 1967 22 253. [2] COURANT R. and HILBERT D., Methods of MathematicalPhysics, Vol. 1. Interscience 1953. [3] MASON M., Trans.Am. math. Sot. 19067 337. [4] INCE E. L., Ordinary Di&rential Equations. Longmans, Green 1926; Dover Edition, 1956. C.E.S.Vd a No,12--o
1501
ARIS