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delphia: W. B. Saunders 1969.

Notation

a

ai

A

A

sphere radius i = 1, 2, ... stoichiometric coef­ficients in (4.4.1) various sphere radii intro­duced in (2.6.17) function of time introduced in (3.2.10) i = 1, 2,. set of constants in­troduced in (4.8.11) constant vector in the direc­tion of the average concen­tration gradient in (2.6.3a) magnitude of A in (2.6.4), also used as reactant species in (4.4.3)

Ap normalizing constant for the eigenfunction (3.3.1)

Ai reactants in (4.4.1) Aq quencher species [Aq] quencher concentration [A.] j<IJ(pd mean concentration of A.

at x + PI if x is in i [Aq]2 j(2)(PI ,P2) d3 PI d3 P2 mean number

of quencher pairs with two different quencher particles located respectively in the volume elements d3 PI and d3 P2 at PI and at P2 relative to a ran­domly selected point x in ..yo.

d(q, u) function of q and u on a1/', introduced in (1.6.2)

d., .rd., .rd •• , d •• , d.. vector and scalar derivatives of d introduced in equations (1.6.5) through (1.6.9)

b parameter in (4.3.10) bi i = 1,2, ... stoichiometric coef­

ficient in (4.4.1) hi i = 1,2, ... set of constants in­

troduced in (4.8.11) B product species in (4.4.3) Bi i = 1,2, ... product species in

(4.4.1)

B*

C(x)

excited species in the quench­ing problem solute concentration at x dimensionless concentration defined by (4.4.67) trial function defined for a spherical catalyst pellet to satisfy inequality (4.5.7)

< c > volume average of excited species concentration defined by (3.5.3)

Co, CL concentrations at the ends x = 0 and x = L of a long slab

C,

Co + c,

reactant concentration in the bulk phase far from the cata­lyst pellet concentration at sphere center in (2.5.5a) saturated concentration of solute initial concentration of solute defined by (3.2.3) total concentration in a binary mixture of A and B defined by (3.3.9)

CA , cA " CB , cB; i = 1,2, ... reactant A, Ai and product B, Bi concentrations in (4.4.2)

CA;f' CB;f i = 1, 2, ... concentration of species Ai and Bi far from the catalyst pellet in (4.4.8)

d3 x, d3 P infinitesmal elements of vol-

dfJjdn

ume infinitesmal element of surface elements of solid angle dimensionless volume element introduced in (4.8.6) dimensionless element of sur­face area, defined by (4.8.5) magnitude of the concentra­tion (0 = c) gradient in an

Notation 101

arbitrary catalyst pellet, de­fined by (4.5.11)

D, D(x), Do local diffusion coefficients, D also used in place of De for the effective diffusion coeffi­cient in section 4.5

De effective diffusion coefficient D" Da, Db diffusion coefficients for the

various phases of section 2.5 DeA " DeB, effective diffusion coefficients

of species Ai and Bi E defined by (4.4.22) f(x) arbitrary function of position fp(x) eigenfunction for (4.4.57) F(u) defined by (4.2.8) Fa(P) defined by (2.6.3b) ~G(()) defined by (1.3.14) F(u) defined by (4.6.10) Fa, (p,), Fa,(p,) defined by (2.6.18) .?(q. u) variational functional defined

by (1.6.2) g(x) stochastic function of pore

structure defined by (1.3.2) G(p, p') (G(p,p') = (g(x)g(x+ p)g(x + p'))

three-point correlation Go magnitude of the concentra­

tion gradient, defined by (4.5.11) ~(q) variational functional (1.6.18)

also used as variational lower bound functional for a cata­lyst pellet (4.2.15)

qjmax maximum value of the varia­tional functional qj(q)

~(q) variational lower bound func-

c1Jmax

h h(p)

tional for a molecular sieve catalyst, defined by (4.6.17) maximum value of the varia­tional functional ~ (q) length of a cylinder introduced as part of trial function (2.4.2) value of h(p) that minimizes variational functional variation of the trial h (p)

about ho(p) h, hi i = 0, 1 defined by (2.4.20) ha(P' n, n') d3 p d2 n d2 n' probability that

two points on the pore wall which can see one another

hz

h*(x)

H(p)

H, H*(e) .yt(x,q,u)

lies within d2 n at the first point and within d2 n' about n' at the second defined by (4.4.42) defined in (3.7.12) and (3.7.13), an improved form of this trial function is defined by (3.8.7) function of p only defined m (2.4.7) defined by (4.5.17) trialfunction defined by(3.6.12) function of x, q, and u defined in Y· introduced in (1.6.2)

.K., Yf., J'l •• , .11' •• , .11'.. vector and sca-lar derivatives of Yf intro­duced in (1.6.5) through (1.6.9) unit vector across a slab in the positive x-direction

I (V . q) defined by (4.2.16) Io(a p), I dap) zeroth and first-order modi­

i(n· q) j(x) J

fied Bessel functions defined by (4.6.18) flux vector mean flux, net rate at which molecules pass through a unit total cross-section of a slab of suspended solid

J magnitude of the mean flux J "(u) variational functional defined

by (1.6.14), also used as a varia­tional upper bound functional for a catalyst pellet (4.2.1:;)

"min minimum value of the varia­tional functional" (u)

j(u) variational upper bound func­tional for a molecular sieve catalyst, defined by (4.6.9)

j min minimum of the variational functional j(u)

k, k, first-order reaction rate con-stant per unit volume intro­duced in (4.5.2) and (4.3.1)

kB Boltzmann's constant k, coefficient for external mass

k

transfer effective first-order reaction rate constant, defined by (3.5.4) first-order surface reaction rate constant

have a relative position vector k; p lying in d3 p and that the

rate constant in the numerator of the Langmuir-Hinshelwood rate law (4.4.9) unit normal n to the pore wall

102 Notation

Kp defined by (2.4.14) K constant in the denominator

of the Langmuir-Hinshelwood kinetic rate law (4.4.9)

K (u) variational functional defined by (3.4.1)

K(x, x') d2 x' probability that a molecule emitted from the pore wall at x will make its next wall col­lision within d2 x' at x:

K A , adsorption coefficient for spe­cies Ai radius of a circle passing through all the vertices of a triangle with sides p, p', and (p' - p) defined by (1.3.13)

I" 12 , 13 largest, second largest, and smallest of the sides of a tri­angle with sides p, p', and (p' - p), introduced in (1.3.12)

Iz defined by (4.4.45) L/2 ratio of volume of r to ex­

ternal surface area of a -Y', for a slab this IS one-half the thickness

!£' defined by (4.4.28) m adjustable constant in (2.6.3a) ma (ma = 44>/ s) average pore di-

ameter mo , mp adjustable constant in (2.6.18) M defined by (4.2.28) At sphere of volume V n density of sphere centers in a

random bed of solid spheres n, n', n(x) unit normals to a surface no, np densities of sphere centers for

the radii ao , ap, ... respectively introduced in (2.6.17)

N mass Biot number defined by (4.2.7)

Na number of sphere centers in a volume V introduced in (1.3.7)

Ni mass Biot number at one of i = 1,2,3, ... , .# data points

N,p modified mass Biot number, defined by (4.7.12)

.k· number of known data points for a molecular sieve catalyst pellet

P(e) de probability that a point chosen at random in the void region lies at a distance between 0

P,

pte)

q(x) q(p,n,n')

q. q*(xi - x) qo(x)

iii

r(c)

r,

r'

and 0 + de from the closest point on the interface pressure at the two ends of a slab x = 0 and x = L respec­tively probability of finding at least one sphere center within a shell of radii (0 + a) to (E + a) + dE. probability that the volume v in a random bed of spheres contains no sphere centers surface average of the concen­tration gradient magnitude, defined by (4.5.13) probability that two points chosen at random on the sur­faces of two different spheres are exposed and can see one another, given by (1.4.3) trial flux vector field arbitrary function defined in (2.9.3) defined by (4.4.31) defined by (3.7.13) trial flux that maximizes the variational functional variation of the trial flux q about qo dimensionless trial flux, de­fined by (4.8.3) i = 1,2, ... ,IV· dimensionless flux distribution for one of the known data points trial flux in excess of maxi­mizing value from section 1.6 defined by (4.4.23) and (4.4.24) defined by (4.4.34) radius of a cylinder reaction rate per unit volume of a homogeneous catalyst pellet radii of the smaller and larger of two concentric circles reaction rate r(c) evaluated at cf radius of the sphere with the slime volume as the cubic cell of section 3.2 radial vector for spherical co­ordinates first derivative of the reaction rate r(c)

I' (C)

,-I

R

R(c) R(c) R. R*

!:I' Sip)

!:I' t T :Y u(x) uo(x)

Notation

inverse of the reaction rate function ric)

reaction rate on the catalyst internal active surface per unit of surface reaction rate ric) evaluated at Cf first derivative of the reaction rate ric) inverse of the surface reaction rate function ric) derivative of the inverse func­tion r- I

derivative of the inverse func­tion 1'-1 radius of a sphere whose vol­ume is fixed equal to that of the cylinder in section 4.5 defined by (4.2.29) defined by (4.6.36) constant defined by (3.4.15) radius of a spherical region drawn concentric with each spherical quencher molecule, defined in (3.7.13) region in l' pore wall surface per unit total volume catalytically active surface area per unit total volume of molecular sieve catalyst pellet void-solid interface (S(p) = (g(x)g(x + p)), two­point correlation outer wall of the cubic cell, introduced in (3.2.4) intersection of planes x = 0 and x = L with the void re­gions of a slab outside surface area of 1', area of surface 81' total internal catalytically ac­tive area within a molecular sieve catalyst pellet, area of 81/ arbitrary surface in '/I"

time temperature defined by (4.4.35) trial concentration scalar field trial concentration that mini­mIzes the variational func­tional

u'(x)

ii

v

v V

Vie)

Vie) v"

X, x' Xi

i,x

y y

z

103

variation of the trial concen­tration u about Uo trial fl uct uation aboutthe mean concentration, defined in (2.3.7) dimensionless concentration, defined by (4.8.2) i = 1,2, ... AI' dimensionless concentration distribution for one of .JV known data points trial scalar field in excess of minimizing function, used in section 1.6 finite volume from which sphere centers are excluded, introduced in (1.3.7) mean molecular speed total volume volume enclosed by a surface l:e of constant concentration c = e within a catalyst pellet, introduced in (4.5.10) derivative of Vie) large volume excluding a sphere of radius a at the origin volume of a composite sphere of radius rb , introduced In

(2.5.1) volume of a sphere of radius a

arbitrary region of volume V subvolume of the total volume l' volume of the region Q of all points lying within a distance a of either end of the vector p. given by (1.3.10) volume of the region Q' of all points lying at a distance a or less from one or more of the three vertices of the triangle with sides p, p', and (p' - p), given by (1.3.12) defined by (4.4.37) length coordinate across the slab position vectors position of the i'''sphere center dimensionless position vector and length coordinates, de­fined by (4.8.4) variable of integration in (3.8.3) defined by (4.4.36) defined by (4.4.40)

104 Notation

0, 1, 2 subscripts on variational functionals refer to forms zeroth, first, and second order in the variation of the trial function

fJ

fJ(x)

Y

Yo

magnitude of the vector "­mean concentration gradient (l1c), also (t/lL - t/lo)i./L in (2.8.4) arbitrary constants in (2.5.6) and (2.5.7) set of A'" positive numbers Ii'= 1 (Xi = 1 introduced III

(4.8.7) parameter in trial concentra­tion (4.3.10) Langrangian multiplier intro­duced in (3.7.3) set of ft· positive numbers I','= 1 fJi = 1 introduced in (4.8.8) values of fJ that respectively minimize " and maximize C§,

listed in Table 4.4.3 cube root of diameter to height ratio for a cylinder and as a constant in (1.7.1) scalar function defined by (4.2.20) equilibrium distribution coef­ficient for the solute between the solid and void phases at the pore walls mean number of quencher particles in a sphere of radius a, defined by (3.8.3) the extremum value of the variational functional K (u), defined by (3.4.3)

Yl positive constant in (4.4.47) and (4.4.48)

Ymin defined by (4.4.59) Ymin defined by (4.4.65) [' [n variational functional for

Knudsen diffusion, defined by (2.8.1 )

[' mm' [' mf' [' f f mean and fluctuating parts

b b(p,)

of [' [~] defined by equations (2.8.6) through (2.8.8) parameter introduced in (1.6.3) Dirac delta function

i5 pp ' Kronecker delta 8Y- external surface of total vol­

ume Y'

AY(B)

f.(x)

~i

(J

B(q) K

external surface of a composite sphere of radius rb introduced in (2.5.2) external surface of the sub­volume j?' element of volume between two surfaces of constant con­centration c = Band c = (J + dB in an arbitrary catalyst pellet, introduced in (4.5.12) minimum distance from point x in j?' to the reactive interface aj?', introduced in (1.5.1) and (3.6.12) arbitrary constant in (2.5.6) variable of integration variable introduced in (3.7.18) effectiveness factor for a cata­lyst pellet effectiveness factor for a finite cylinder effectiveness factor for a mo­lecular sieve catalyst pellet i = 1, 2, ... ft· effectiveness factor at anyone of ft' known data points parameter used in spherical symmetrization to establish inequality (4.5.7) interior angles of a triangle with sides p, p', and (p' - p), introduced in (1.3.12) parameter with a value be­tween zero and one polar angle for spherical co­ordinates radial coordinate in a spheri­cal catalyst pellet selected so that the volume enclosed by the trial function 2'(0) in the sphere and the volume enclosed by the concentration value c = B in an arbitrary catalyst pellet are equal, defined by (4.5.10) scalar field defined by (1.6.17) [K = C fK] dimensionless num­ber for Langmuir-Hinshel­wood kinetics, defined by (4.4.18)

v

~ ~(X)

~o(X)

5(q)

II

P,p',p"

Pi

u

u*

ft

E(u)

Notation

adjustable parameter optimum value of A eigenvalue for (3.2.7) Thiele modulus defined by (4.2.6) modified Thiele modulus, de­fined by (4.7.12) Thiele modulus for a mo­lecular sieve catalyst, defined by (4.6.6) i = 1, 2, ... % Thiele modulus at one of % known data points defined by (4.4.26) non-negative constant in (4.4.47) and (4.4.48) variable used in (4.2.22) and (4.6.28) parameter in (4.3.6) and (4.4.19) trial function for Knudsen diffusion, introduced in (2.8.1) trial function that minimizes the variational functional rm variation of the trial function ~(x) about ~o(x) lower bound variational func­tional for diffusion controlled quenching, defined by (3.7.1) permeability, mean molecular flux per unit pressure gradient relative position vectors [Pi = (Xi - X)], relative posi­tion vector mean pore surface area which can be seen from a typical point on the void-solid inter­face, given by (1.4.5) uniform generation rate of excited species B* introduced in (3.5.1) dimensionless surface area for catalytically active surface, de­fined by (4.6.7) upper bound variational func­tional for diffusion, defined by (2.3.1) dissipation integral for a com­posite sphere, defined by (2.5.1) surface of constant concen­tration c = e, introduced in (4.5.12)

Y'(u)

x

"'(X)

"'p(X) "'2/3 (Yq) w w(x)

Q'

105

measured tortuosity factors, see Figure (2.6.1) tortuosity bound for a bed of spheres, given by (2.6.14) tortuosity bound for any iso­tropic suspension, given by (2.6.16) decay time, defined by (3.2.14) upper bound variational func­tional for diffusion controlled quenching, defined by (3.6.1) upper bound variational func­tional with added constraint, defined by (3.6.4) minimum value of Y(Uh given by (3.6.10) defined by (4.5.5) defined by (4.5.2) azimuth angle for spherical coordinates void fraction, a ratio of void volume to total volume inert volume to total volume in a molecular sieve catalyst dimensionless trial concentra­tion at slab edge, introduced in (4.3.6), (4.3.10), (4.4.19), and (4.4.32) values of X that respectively minimize" and maximize ~, see Table (4.4.3) rate at which molecules are reemitted from a unit element of pore wall surface located at x, introduced in (2.7.3) defined by (2.7.2) mean and fluctuating parts of the trial function for Knudsen diffusion, introduced in (2.8.5) eigenfunctions for (3.2.7) incomplete Gamma function Lagrangian multiplier in (3.6.4) Lagrangian multiplier in (4.6.17) dimensionless eigenvalue de­fined by (4.4.64) eigenvalue for (4.4.57) a region of all points lying within a distance a of either end of the vector P a region of all points lying at a distance a or less from one or

106

< ... >

< ... >'/'

Notation

more of the three vertices of the triangle with sides p, p', and (p' - p) V volume average defined by V; (1.3.1)

surface average over the void- Ilu - ell,

solid interface, defined by (2.9.2) gradient operator gradient operator in dimen­sionless coordinates x defined by (4.4.61)

Abramowitz, M. 56, 97 Amundson, N. R. 79, 83,

98 Anderson, H. R. 9,97 Anderson, N. 97 Aris, R. 59,65,79,85,91,

97,98,99 Arthurs, A. M. 13, 16, 61,

71,75,97

Becker, M. 97 Beran, M. J. 97 Berman, A. S. 97 Biot, M. A. 1,2,97 Brown, W. F. 97 Brumberger, H. 9,97 Bueche, A. M. 6, 97

Carman, P. 33,34,41,97 Coles, C. W. 75, 97 Coriell, S. R. 98 Courant, R. I, 97

Debye, P. 5, 6, 9, 97 Derjaguin, B. 41,97 Donnelly, R. J. 1,97

Finlayson, B. A. I, 97 Funk, P. 1,97

Author Index

Gould, S. H. 1,98 Gunn, D. J. 79,98

Ham, F. S. 43, 98 Hashin, Z. 98 Herman, R. 1,97 Hilbert, D. 1,97 Hill, R. 98

Jackson, J. L. 98 Jiittner, F. 59, 98

Keller, J. B. 98 Kohn, J. 79,99

Lanczos, C. 1,98 Lassettre, E. 41, 98 Lauwerier, H. A. 1,98 Luss, D. 79, 83, 98

deMarcus, W. C. 2,35, 36, 98

Noble, B. 13, 16,98 Noyes, R. M. 98

Ockham, William of

Polya, G. 80, 82, 98

Prager, S. 8, 11, 30,49,91, 98,99

Prigogine, 1. 1, 97

Reck, G. P. 11,49,98 Reck, R. A. 11,49,98 Rester, S. 65, 85, 91, 96, 98 Roberts, G. W. 69,99 Robinson, P. D. 97 Rotne, J. 98

Satterfield, C. N. 69,99 Schechter, R. S. 99 Scriven, L. E. 1, 97 Sewell, M. J. 13,98 Shtrikman, S. 98 Smoluchowski, M. 56, 99 Stegun, 1. A. 56, 97 Strieder, W. 8, 79, 99 Synge, J. L. 99 Szego, G. 80, 82, 98

Thiele, E. W. 59,99

Weissberg, H. L. 5, 6, 8, 30,99

Woodbury, G. W. 99

Young, L. C. 1,99

SUbject Index

Biot number 77 Bounds on solution derived from vana·

tional principle 75-78 Brownian motion 40

Calculus of variations, general remarks on 1,2

Catalysis, heterogeneous 59-96 passim

Diffusion coefficient, effective, best bounds for 26-30

- -, -, definition of 3, 4 - -, -, for a porous medium 2, 18-41

passim - -, -, variational formulation for 20-

22 Diffusion, Knudsen 2, 18, 35-41,69

Effectiveness factor, definition of 60 - -, discrete model 84, 89, 93 - -, first order reaction in slab, estimate

for large A 67,68 - -, first order reaction in slab, estimate

for small A 66 - -, Langmuir-Hinshelwood kinetics in

slab, estimates for large A 72-75 - -, Langmuir-Hinshelwood kinetics in

slab, estimates for small A 70-72 - -, lower bound 62-64 - -, upper bound 61-62 Entropy production 22 Euler-Lagrange equations 20, 25, 26, 51,

53,62,63,87,90 Experimental data, analysis of 91-95

Fick's Law 2,3, 18,21 Finite cylinder, effectiveness factor for 79

Heat conduction 2, 3, 18, 21 Heterogeneous catalysis 2 - -, discrete model, effectiveness factor,

lower bound 86-89 - -, - model, effectiveness factor, up­

per bound 85, 86

- -, - model, first order reaction in 89-91

- -, - model, variational principles for 83-92

- -, homogeneous model, effectiveness factor, lower bound 62-65

- -, - model, effectiveness factor, up­per bound 61,62

- -, - model, first order reaction in slab 65-68

- -, - model, Langmuir-Hinshelwood kinetics in slab 68-75

- -, non-isothermal particle 95, 96 Incomplete Gamma function 56 Integral equation for Knudsen diffusion

37,3S

Kinetics, first order 65-68 --, non-monotonic 95,96 Knudsen diffusion 8-10, IS, 69 - -, through a porous medium 35-41

Langmuir-Hinshelwood kinetics 59, 6S-75

Laplace's equation 2, 28, 48

Mean free path statistics 8-11

Nearest distance to surface II, 12, 51, 52, 56

Notational conventions \6, 17

Pellet shape, effect on effectiveness factor in heterogeneous catalysis 7S-83

Permeability, Knudsen, upper bound for 18,41

Pore diameter, average 9, 10, 36 Porous media, correlation functions for

4-S - -, diffusion in IS-41 passim - -, - in, isotropic 18-20,22-30,39-

41 - -, overlapping sphere model of 5-10 Porosity 2, 5, 19

Subject Index 109

Precipitation, diffusion limited 42-48 -, isolated particle approximation 46 -, spherical cell approximation 45, 46 -, upper bound on rate 46-48 -, variational principle for 47

Quenching 42,49-58 -, effective first order constant for 50 -, - first order constant for, lower bound

52-55 -, - first order constant for, upper bound

50-52 -, interference at high concentration 42,

49, 57 -, random distribution of particles 56-58

Reaction, diffusion limited 42 Random media, electrical properties of 2 - -, internal surface area 9, 40, 41 - -, correlation functions for 4-6, 19,

23-26, 30 - -, overlapping sphere model of 6-12 Random sphere model, bounds on effective

diffusion coefficient 30-35

- - -, with spheres of different sizes 34, 35

Schwartz'inequality 24, 82, 94 Suspensions, isotropic, bounds on diffusion

coefficient for 22-25 -, random, correlation functions for 4-8 Symmetrization process of Steiner 80, 81

Thiele modulus, discrete model 85 - -, homogeneous model 60, 65, 70, 77 Three-point correlation 6, 19, 23, 30 Tortuosity factor 33, 34 Two-point correlation 5, 19, 23-26, 30

Variational principles, complementary 13-16

Void fraction 5,7, 19,36 Void-point, surface statistics 11, 12 Volume average 4, 19

Zeolite 59

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