Optimal catalyst activity distribution for effectiveness factor, productivity and selectivity maximization in generalized nonisothermal reacting systems with arbitrary kinetics

Science, Vol. 47, No. 3, pp. 615-621, Chrmical Enginming F’ainted in Great Britain.

ooos-2509p2 $5.00 + 0.00 Q 1991 Pergamon Rau pie

1992.

OPTIMAL CATALYST ACTIVITY DISTRIBUTION FOR EFFECTIVENESS FACTOR, PRODUCTIVITY AND SELECTIVITY MAXIMIZATION IN GENERALIZED NONISOTHERMAL REACTING SYSTEMS WITH ARBITRARY KINETICS JINBIAO YE and QUAN YUAN+ Dalian Institute of Chemical Physics, Academia Sinica, 161 Zhongshan Road, Dalian, China (Received

4 July

1989; accepted

for publication

18 June

1991)

AliutracG-Uptimal catalystactivity distributionfor effectivenessfactor, productivityand selectivitymaximization in nonisothermal pellets has been determined for generalized reacting systems with arbitrary kinetics. It has been shown that optimal catalyst activity distributionis a Dirac 6(x) function in all cases. paper). Therefore, the conclusion that this Dirac 6(x) function distribution is the optimal activity distribution is not mathematically rigorous. In the present paper, a new method for solving the problem of optimal catalyst activity distribution for generalized cases is reported. It is shown that the 6(x) function does represent optimal activity distribution for maximizing effectiveness factor, productivity and selectivity for generalized reacting systems with arbitrary kinetics for nonisothermal catalyst pellets. The method is to be described for a single reaction in detail in Section 2 and then it will be applied to a general case in Section 3.

1. LNTRODUCIION

Performance of large catalyst pellets can be better or worse than that of micropellets, because of transport phenomena. The fact that catalyst pellets with nonuniform activity distribution can exhibit higher activity and selectivity has been proved (Shadman-Yazdi and Petersen, 1972; Corbett and Luss, 1974, Becker and Wei, 1977; Dadyburjor, 1982,1985; Cukierman et al., 1983). The problem of optimal catalyst activity distribution for effectiveness factor maximization has been analyzed for biomolecular Langmuir kinetics in pellets with/without internal temperature gradient (Morbidelli et al., 1982, 1985; Wu et al., 1984). More recently, the problem of effectiveness factor maximization in single reactions was solved for arbitrary kinetics with/without external concentration and temperature gradients (Chemburkar et al., 1987; Vayenas and Pavlou, 1987a). It was shown that effectiveness factor was maximized by a Dirac 6(x) function distribution. In generalized reacting systems, the optimization problem with activity distribution has been studied only for specific cases. Selectivity maximization has been given for isothermal pellets with no external gradients, for parallel reactions with arbitrary kinetics and also for consecutive reactions with arbitrary kinetics for the main reaction and positive-order kinetics for undesired reactions by Vayenas and Pavlou (1987b). They have also investigated nonisothermal parallel and consecutive reaction systems (Vayenas and Pavlou 1988; Vayenas et cd., 1989). The conclusion is that selectivity is maximized by a Dirac S(x)type activity distribution too. For more generalized reacting systems, the optimization problem was studied by Wu et al. (1990). A perturbation method was applied to find the optimal activity distribution maximized effectiveness factor, selectivity and productivity, but there does not exist a minimal perturbation about the Dirac delta function (it is cited in this

2. SINGLE

REACITON

( >

-$-&92 = f$Za(x)F(C). In the above equation, activity must satisfy the following equation

distribution

a(x)

1 s0

a(x)x”dx

and the corresponding are

= l/(n + 1)

boundary

dC -_=O dx C=l

conditions

at

x-0

at

x=1

(2) of eq. (1)

(4)

where @=c F(C)

= exp

[(

y

l-

R%(C) I? 0 1 1 + 80 E

+Author to whom comspondenaz

SY!3TEMS

Consider the reaction A + B with arbitrary kinetics under the assumption of negligible external mass and heat transfer. The steady-state mass balance equation for reactant A is

y=RT,.

should be addressed. 615

-

c)

)IrW/rtc

616

JINBIAOYE and QUANYUAN

The effectiveness factor q can be expressed as

= (n + 1)

a(x) F (C)x” dx.

Catalyst activity distribution function a(x) must also satisfy a(x) > 0, thus leading to the activity distribution function set K, expressed as 2 O}

(H

+

SC

$x”u)dx.

+

(13)

0

(5)

K = (a(x):a(x)

s 1

X

(6)

and

Assume that a*(x) is the optimal activity distribution function which maximizes the effectiveness factor, eq. (13). Then any other distribution a(x) neighboring u*(x) can be expressed as u(x) = a*(x) + Sa.

(14)

Since a(x) and u*(x) must satisfy eq. (2), 1

K=MuN,uQ

(7)

where M, N, Q are function sets which are expressed as M = {u(x): a(x) is finite for any x>

s0

[u*(x) + 8u]x”dx a*(x)x”dx

It is clear that the effectiveness factor must be maximized by one of the catalyst activity distributions. If there does not exist an optimal activity distribution, an activity distribution a,(x) will be found which makes r~[a, (x)] B r~[u(x)] for any catalyst activity distribution u(x). That means the effectiveness factor is infinite in K. In fact, it is impossible.

2.1. No optimal activity distribution in M u N Apply the maximal principle to find the optimal distribution in M u N. The reason that the maximal principle can be applied in set N is given in Appendix A. 2.1.1. Necessary condition for optimulity. Let dC/dx = u, and rewrite eq. (1) as dC -_=1( dx

(8)

= r#?o(x)x”F(C)

(9)

with the boundary conditions

s0

6ax’dx

(17)

Thus C(x) and u(x) resulting from u(x) can be expressed as C(x) = c*(x)

+ K(x)

(18)

u(x) = u*(x) + &(x).

(19)

NOW substitute the boundary condition into eqs (18) and (19) leading to X(1)

= 0,

&J(O) = 0.

(20)

By introducing the following condition

(21) (221 4 (0) = 0,

A,(l)

= 0

(23)

and substituting eqs (lS)-(ZO) into eq. (13), the variation of effectiveness factor, eq. (13), becomes

'i3H -&dX 0

I aa

(24)

which defines the influence of 6a on ?I. The necessary condition for o* to be optimal is

at

x=0

(10)

61 6 0.

C=l

at

x=1.

(11)

By introducing eq. (25) into eq. (24), the necessary condition for optimality is

H = a(x)x”F(C)

operator

+ Air.4+ &@u(x)x”F(c).

(12)

Then the effectiveness factor given by eq. (5) becomes

1)

= 0.

1(= 0

Define the Hamiltonian

=(n+

(16)

1

611= (n + 1)

1)

= l/(n + 1).

I 0

Substitution of eq. (16) into eq. (15) leads to

Q = {a(x): a(x) is Dirac 6(x) function}.

Pl=(n+

(15)

I

N = {a(x): a(x) is finite except for some x, but there exist other x to satisfy a(x) 3 O}

&(x”U)

= l/(n + 1)

i[H-i. s0

s

1u -

jH-A,g-

&+za(x)PF(C)]

A&(x94)] dx

dx

s

law -6adx

0

aa

(25)

< 0.

(26)

We can prove (see Appendix B) that the necessary and sufficient condition which satisfies eq. (26) under the constraint of eq. (17) is i3H 1 = -constant. aa 9

(27)

2.1.2. No optimal actiuity distribution. The optimal activity distribution can be obtained from eq. (I) with

Optimal catalyst activity distribution its boundary conditions and eqs (21)-(23), (12) and (27). On substituting eq. (12) into eqs (21), (22) and (27), they become, respectively d& -= dJc

-

(1 + n,~z)a*(x)x*F’(c*)

(28)

617

and we therefore consider the optimal activity distribution to be a Dirac S(x) function, and Sa must satisfy a’(x) + 6a > 0. W) By introducing a*(x) = b(x comes 6a > 0

(1 + A,#P)F(C*)

= constant.

(30)

Substitution of eq. (29) into eq. (28) leads to $&

(

92

= (1 + n,@)a*(x)F’(C*).

>

Multiply eq. (31) with Y(dC*/dx) rate to obtain xnd&dC* -dx dx

(31)

=

XE[O,Z)

u (X, 11.

(41)

It. is clear that 6a = 0 is the unique condition which satisfies eq. (17) under the constraint of eq. (41), that is, the perturbation about the Dirac 6(x) function vanishes. Suppose that activity distribution a(x) in Q is cy=, Al 8(x - xi). Substitute this into eqs (l), (2) and (5), and the following equations result, namely

and then integ-

*G+,

- G = i: cpF(C,)A&

&-cl -

= ,d12d2C* s 0 x dx

a) into eq. (40), it be-

xk

(42)

,=1

dx

+ A,4Z).*(X)x”~dx.

(32)

On combining eqs (l), (30) and (32), it follows that --

dx = 0.

(33)

Solving eq. (33), we obtain dL2 dC = -0 dx dx

Equation (34) implies that either

dh

--0

XE

dx

(9

i=l

The effectiveness factor rl is maximized under the constraints of eqs (42) and (43) by applying the Lagrange multiplier method to obtain optimal AI. xi. The Lagrange multiplier operator is defined as

J = (n + 1)

XE[O,l].

E Aix$F(C,).

q = (n + 1)

I

+h

x;

i=l

co,11

+ a 5 A,x; - -n+l1

2 A&F(C,)

i=l

ci+l

xi+l

i=l

-

CJ

-

xi

-

4*AjxfF(Cj)

i j=i

>

1 .

(45)

The necessary conditions for maximizing J are

or dC* -= dx

0

d&

-=O

xaCO,xrf

XE

dx

Lx,, 13-

(36) (37)

Substitution of either eq. (35) into eq. (31) or eqs @6) and (37) into eqs (1) and (31) yields (1 + A, ~“)a*(X)F’(C*) In general, (1 + A,+z) fore,

= 0.

(48) aJ

(33)

-_=O

aa

is not equal to zero, and there-

a*(x)F’(C*)

= 0.

aJ a

=0

(i=

1,2,.

. . , Al)_

(39)

(50)

Since F’(C*) = 0 holds only for some C*, a*(x) is equal to zero except for these C*. It appears that a*(x) does not exist in the function set A4 u N. The optimal activity distribution must therefore exist in Q.

Substitute eq. (45) into eq. (46) to obtain

2.2. Optimal activity distribution in Q The maximal principle could not be applied to find optimal activity distribution in Q. Since the maximal principle is based on perturbation about optimal activity distribution a*(x), Sa must be a minimal value,

Substitute eq. (45) into eq. (47) and then subtract eq. (51), to get

F(C,)(n

+ 1-

5 A) + a = 0 k=i

(i= 1,2, . . . , IV).

ng,x;- l

-

ci

xi+1 -

xi

C I+1

+ B&

ci*l (xxi+1 -

-

ci x,)2

(51)

JINWAOYE and QUAN YUAN

618 ci -

B1-lxl-l(x*

- c,-, _ xi_1)2

D, =

0

(i = 1, 2. . . . , N).

(52)

D=

(i = 1,2, . . . , N).

(53)

Substitution of eq. (53) into eq. (51) leads to F(C,) = constant such that the effectiveness factor is

max F(C). D
(56)

optimal activity distribution existing in Q can thus be expressed as =

6(x -

a)

(57)

(n + l)xl’

Of course, the optimal activity location X is where F(C) reaches its maximum value. The determination of this optimal location is omitted since it is a trivial case, and any other standard optimization technique can be utilized.

3.

---

0

0

0

0

0

o-q-

1

(‘53)

d,

.a-

1

(i = 1,2, . _ . , I)

(W

G(C)=Dxa,xF(C)

(55)

The

a*(x)

D,

(65)

with the boundary conditions

rl = WC,) %MX=

0

0 0

. .

Di = Del/De1 (54)

... 3-a

0

. .

By inductive inference for eq. (52), then pi = 0

0 D2

0 0

dC -=O dx

atx=O

(66)

c=c,

atx=l

(67)

where the kinetic function F(C) includes actual effect of temperature on reaction rate because the steadystate heat balance is V2T=

-

b2&)

9 &M,(C). i=1

(68)

Thus the temperature distribution in pellet can be represented explicitly by the concentration distribution of C. Either the effectiveness factor or productivity can be expressed as the object function: 1

GENERALIZED

J = (n + 1)

SYSTEMS

REACTING

Consider the generalized reacting system Cc+%

= 0

i=

1,2,.

. . ,I

j=

1,2,.

._,I

with arbitrary kinetics and without external mass and heat transfer. The steady-state balance equations for

I0

a(x)x*G,(C)dx

(69)

In the above equation, when i = r, and if the rth component is a reactant, then J represents the effectiveness factor; whereas when i = p. and if the pih component is a product, then, J represents productivity. The global selectivity can be expressed as

A, are V2C = 42a(x)G(C)

i mu sj. I I=1

S

I-

D.,(dCp/Wix=i Der(dCrlWlx=,

=

I

1

o

4x)x” G, 03 dx

s

’a(x)x”

*

(70)

G,(C) clx

0

3.1. .The maximal object function in M v N Apply the maximal principle to find the optimal activity distribution in M u ZV.

dG -=

Let

dx

Define the Hamiltonian

“‘* operator

Ef = u(x)x”G~(C) +

C &us k=l

r+1

91 F,(C)

F(C) =

gz Fz(C) 8sPs(c) ... [ BFF,_(C)

81 =

ri(C*)/r,

1

(Co)

(71) For effectiveness factor, productivity and selectivity, the subscript i is r, p and s, respectively. For the object function J and selectivity S, the necessary conditions for optimality are respectively 1 dH

--

A-? au

= constant

(72)

Optimal catalyst activity distribution

$g

W, - WpGrW

c

= constant

II

(73)

we can obtain S,,,=

where 1 w, =

I0

a*(x)x”G,(C)

dx

L,,G,(C)

= constant

s[6;J.

(84)

The conclusion is that the maximal effectiveness factor, productivity or selectivity is either J_,= J[G(x - 1)/n + 1] and S, = S[G(x - 1)/n + l] or they do not exist in h4 u N.

(i = r, p)

that is G,(C) + 4’ 1

619

(74)

3.2. Optimal activity distribution

in Q

k=l

Suppose that activity distribution a(x) in Q is f

I+1 G,(C)w,-

Gr(C)w,

+

"'r@

i=l

1 &zkGk(C) k=l

[

W,2 = constant.

We apply the same method as in Section 3.1 to find the optimal activity distribution, and we obtain A,S(x

Ii

(75)

The adjoint equations can be expressed as

-

xi)_

J max= G,(C)

x=xL $I I i

wijgj

(k = I,29 . . + 3 We (85)

I+1

d’d,iz

1

+

For object function S, we have

i=l

f-q”+)

=

>

c

-

@AZi~

1 (k

k

i=1

(76)

~(w,/w.)

1,2,.

=

..,I+1)

at x = i

+a=0 .X=X% 1,2,

(78)

(k = 1,2, . . . , Z + 1). (79)

For object Function J, a*(x) can be solved according to balance equation (58) with boundary conditions (66) and (67), adjoint equation (76) with boundary conditions (78) and (79), and eq. (74). Let

2 AixIG(Cj)

wk =

(k = p. r).

Multiply eq. (86) with A,x; k = 1 to k = N, leading to

Multiply eq. (76) with xn(dC:/dx) and sum from 1 to k = I. After integration and combination with eqs (58) and (74), we obtain

and then sum from

a = 0.

(87)

Substitution of eq. (87) into eq. (86) leads to [ mrG,(C)

-

F&G,(C)][,=,,

= 0

(k =

I0

nx” - ’P(x) dx = 0.

(88)

CG,(WG(C)I 1x=x,,

P(x) = 0.

(81)

(90) (91) such that the optimal activity distribution for maximizing effectiveness factor, productivity or selectivity in Q can be expressed as

Integration of eqs (58) and (74) gives

I

1

0

a* (x)Gi(C)x’ dx +

-

I

If1

c x”d,,+ t=i

dC*

a*(x)

i

6(x - 2)

(n + 1)x+-

(92)

0

= G,(C,)/n

+ 1).

(82)

0

Substitution of eqs (79) and (81) into eq. (82) leads to i wijgj = J [ ‘:;;)I. (83) I j=1 By applying the same method to object function S, J allax= G,(C,)

_

4. CONCLUDING

‘P(x)x”dx

(8%

and

(80)

It follows that

152, - . . , N).

It is clear that

k =

X

(86)

where

S =

x”P(x) +

. . . , N)

i=l

atx=O

0

dx

AZk = 0

r,zG,(C)

(k =

(77)

with the following boundary conditions

dA2k_

G,(C) pp W,z

&(1)[~

2+1 +

(k = 1,2, . . . , I + 1)

REMARKS

The optimal catalyst activity distribution in a symmetric porous pellet of any geometry which is not subject to poisoning, maximizing the effectiveness factor, or the global yield, or the global selectivity has been investigated analytically. The conclusion is that the optimal distribution is always a Dirac 6(x) function regardless of the complexity of reaction network

620

JINBIAO YE and QUAN YUAN

and generality of reaction kinetics. Of course, the determination of the optimal distribution consists of identifying the optimal one among all possible Dirac G(x)-type distributions, that is, the optimal location of catalyst, 2, needs to be identified. The problem is particularly simple because the balance equation reduces to an algebraic system for the Dirac 6(x) distribution. It can thus be solved by any standard optimization technique. NOTATION

a C :

Di D

D, F F G G 9 H Z J .T K A4 N L r R S t 11 x

activity distribution function dimensionless concentration, c/cl0 concentration I-dimensional dimensionless concentration vector D,llD,i vector defined by cq. (63) effective diffusivity dimensionless kinetics expression, r(cl, cl, . . .

Cl, W(c,cl,

c20,

- - . czo,

to)

vector defined by eq. (61) vector defined by eq. (65) element in the vector defined by eq. (65) defined by eq. (62) Hamiltonian operator number of components in the reacting system object function number of reactions in the system function set defined by eq. (6) function set defined by eq. (7) function set defined by eq. (7) integer characteristic of pellet geometry Dirac 6(x) function set reaction rate characteristic pellet dimension selectivity temperature concentration gradient dimensionless distance from the center of the pellet

Greek letters number of Lagrange multiplier Prater number of the reaction, or number of ; Lagrange

Becker, E. R. and Wei, J., 1977, Nonuniform distribution of catalyst on supports. J. Catal. 46, 372-381. Chemburkar, R. M., Morbidelli, M. and Varma, A., 1987, Optimal catalyst activity profiles in the pellets--VII. The case of arbitrary reaction kinetics with finite external heat and mass transport resistances. Chem. Engng Sci. 42, 2621-2632. Corbett, W. E. and Luss, D., 1974. The influence of nonuniform catalytic activity & the ~&focmance of a single sphericalpellet. Chern. Engng Sci. 29, 1473-1483. Cukierman,A. L., Laborde, M. A. and Lemcoff, N. O., 1983, Optimum activity distribution in a catalyst pellet for a complex reaction. Chem. Engng Sci. 38, 1977-1982. Dadyburjor, D. B., 1982, Distribution for maximum activity of a composite catalyst. A.2.Ch.E. J. 28, 7-728. Dadybujor, D. B., 1985, Selectivity over unifunctional multicomDonent catalyst with nonuniform distribution of 24, 16-27. components. ZEC &dam. Morbidelli, M., Servida, A. and Varma, A., 1982, Optimal catalyst activity profiles-I. The case of negligible external mass transfer resistance. ZEC Fundum. 21. 278-284. Morbidelli, M., Servida, M., Carra, S. and Karma, A., 1985, Optimal catalyst activity profiles in pellets-TIT. The nonisothermal case with negligible external transport limitations. ZEC Fundam. 24, 116-125. Shadman-Yazdi, F. and Petersen, E. E., 1972, Changing catalyst performance by varying the distributionof active catalyst within porous supports. Chem. Engng Sci. 27, 227-237. Vayenas, C. G. and Pavlou, S., 1987a, Optimal catalyst activity distribution and generalized effectiveness factor in pellets: single reaction with arbitrary kinetics. Chem. Engng Sci. 42,2633-2645. Vayenas, C. G. and PavIou, S., 1987b. Optimal catalyst distribution for selectivity maximization in pellets: parallel Chem. Engng Sci. 42, and consecutive reactions. 1655-1666. Vayenas, C. G. and Pavlou, S.. 1988. *timal catalyst distribution for selectivity maximiza&on in nonisothermal pellets: the case of parallel reactions. Chem. Engng Sci. 43, 2729-2740. Vayenas, C. G., Pavloy S. and Pappas, A. D., 1989. Optimal catalyst distribution for selecti+iiymaximization in nonisothermal pellets: the case of consecutive reactions. Chem. Engng Sci. 44, 113-145. Wu, H., Yuan, Q. and Zhu, B. L., 1984, Optimal catalyst activity Drofiles in nonisothermal Dellets. J. chem. Znd.

Engng-(China) 4, 283-290. Wu, H.. Brunovska. A.. Morbidelli. M. and Varma. A.. 1990.

Optimal catalyst ac&ity profiles in pellets-VIIi. denerai nonisothermal reacting system with arbitrary kinetics. Chem. Engng Sci. 45, 1855-1862. APPENDIX A

multiplier

dimensionless

activation

energy

i;:

Dirac

3

matrix defined by eq. (60) Thiele modulus { = R[r(c,,,

6(x)

REFERENCES

Function set N can also be described as

function

N = {a(x): a(x) is infinite for

czo,

. . , cIo,

~oV~,~~~ol~‘~~

Subscripts bulk condition 0 ith component in reaction system or ith value i kth component in reaction system or kth value jth reaction in reaction system or jth value i r reactant product P max maximum value of object function * optimal value

X1* %,

- - I XN7a(x) is finite for the other x)_

Without loss of generality, we can assume that x, satisfies x1 < x1 <

x: s xi-

a(x)x- dn = ci

. . . < XN (i = 1,2,

(Al) . . . , N).

ew

It is clear that a(x) in set N is a combination of Dirac delta function set and function set M. Of course, it consists of a function that satisfies e, = 0, too. But such a function could be omitted from a practical viewpoint. Thus, activity distribution a(~) in set N can be expressed as

a(x) = a,(x)

+

2 &6(x

i=1

- Xi)

(A3)

Optimal where a,(x) Let

621

catalyst activity distribution APPENDIX 3

is finite for any x. %(x)

= al(x)

x E Cxi* xi

f

xe cx,, xi+1

C(x) = G(x)

(A4)

11

WV

3.

Substitution of eqs (A4) and (A5) into eq. (1) and eq. (2) gives, respectively

Su#icient condition When (aH/da)x -” = constant, the left-hand side of eq. (26) becomes

s

I

IaH

--Sodx=gf 0 ao

+ &

where x0

=

0

X.4+1

= l/(n + 1)

1

=

The corresponding

1

5 dx

dx

Ix=x,

at x=x,

d&_ dx

d& .i--i dx

I +

dx

=o

2

6

s

x2--
6 2

X,-t2dX
0

SA x;

(B3)

1

(A12)

x-x,

(i = 1, 2,

. . . ,N-

-h.,U~ = 0.

1)

(aH/a+-n 3.constant, dH aax

#J2W’(C,-1) (A13) (Al4)

SAS. (B4)

.X=x,

Suppose that which make

at x = xi

cc1 + A,.,-

IX=0

I

6

2

-Es

x=x,

%.?_

1

6

where SA is a minimar value. Substitution of eq_ (B3) into eq. (26) leads to

(All)

boundary conditions are = &.,-1(x,)

x,--
~,,,!P)~:w~‘(c:) (i = 0, I, . . . , N).

Li

SA -

Xl+stxCL

= (1 +

The corresponding

o
0--

(AlO)

Applying the maximal principle to find the optimal distribution o:(x) in x E [xl, xi+, 3 gives the adjoint equation

k&!i!p)

6a=

(A9)

+ S,JTG-,Wl.

0

x;

(A@

boundary conditions of eq. (A6) are

_ G-, _-

032)

which means that eq. (26) is satisfied.

dx

-

ci = ci_l(x,)

0

aa

Necessary condition Since 6a is an arbitrary perturbation about a*(x), we can let

a&c) F(Ci)x”

+ S,F(CiW

(A7)

(Bf)

s = LaH -Sadx

o

1

x^6adx.

Substitution of eq. (17) into eq. (Bl) leads to

646) a,(x)x”dx

s0

-

I

1111

two

points x,,

x2

8H 1

x=31, ’ aaF

IX=X*

can be found. By introducing eq. (B5) in eq. (84) and supposing A r 0, we have -&dx

> 0

W)

The necessary condition is (1 + L,+,+2)F(C:)

= constant.

(A15)

It appears that the solution of eqs (All)-(Al5) is the same as that of eqs (28)+3(I). Thus, the maximal principle can be applied in set N.

which contradicts eq. (26). Therefore, the necessary condition for which effectiveness factor is maximized by a*(x) is aw 1 --= aa xR

constant.

(B7)