A GENERALIZED THEORY OF SORPTTON
PHENOMENA IN BIOLOGICAL MATERIALS

Thesis for the Degree of Ph. D.
MMBHIGAN STATE UNIVERSITY
PATRTCK OBI NGODDY
196‘9

THEQS

This is to certify that the

thesis entitled

A GENERALIZED THEORY OF SORPTION
PHENOMENA IN BIOLOGICAL MATERIALS

presented by

Patrick Obi Ngoddy

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Agr. Eng.

4

 

Date sent. 241 1969

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ABSTRACT

A GENERALIZED THEORY OF SORPTION
PHENOMENA IN BIOLOGICAL MATERIALS

by Patrick Obi Ngoddy

A moisture sorption model unifying three basic concepts in
adsorption, is developed and verified for biological materials.

The B-E-T and Capillary Condensation theories are combined

 

into an integral isotherm equation for porous biological materials.
In order to solve the integral equation explicitly, it was necessary
to (a) characterize the physical structure of sorbing bio-materials
mathematically, and (b) to derive an explicit density function for
water sorbed by biological materials, utilizing the general frame-

work of the Adsorption Potential theory in conjunction with the

 

fundamental laws of thermodynamics.

Application of the moisture adsorption model to adsorption
data of a wide variety of biological materials led to the following
Specific conclusions:

1. The pores of biological materials are best modeled as inter-
connected spheroidal ink-bottles.

2. The numerical distribution of pore-size in biological materials
can be described by a power law type analytical function.

3. The density of water adsorbed by biological materials may be
expressed by the semi-theoretical relation:

= * °
p/po u (AHst/AHSt)

where p is the density associated with the isosteric heat of

8' o .. ‘
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5.

Patrick Obi Ngoddy

adsorption AHst’ po and AH;t are the corresponding density and
isosteric heat values at saturation, and u* is empirically
defined for the class of bio-materials of interest.

The three basic concepts in adsorption can be unified into an

isotherm equation of the type:

Ma = 355 (Zn-Kn)
l: and n are characteristic parameters of the physical structure
of the sorbent, and Z and A are functions of relative vapor
pressure.
A theory of sorption hysteresis based on the superposition
of the so-called "capillary condensation" hysteresis and

"swelling fatigue" hysteresis is deve10ped and used to predict

successfully the desorption isotherms of whole corn kernels.

Approvediqal Zfl/gZ/v/Zt COIL/1114

fiajor ProfessorJ(//.7 X4 ((7

Approved CUM M M

Department Chairman

A GENERALIZED THEORY OF SORPTION
PHENOMENA IN BIOLOGICAL MATERIALS

By

Patrick Obi Ngoddy

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1969

Ge/ 75?
4024-170

To

The pe0ple of BIAFRA

at their moment of supreme anguish.

ACKNOWLEDGEMENTS

It gives the author special pleasure to acknowledge with
sincere gratitude, the indispensable help and guidance of Dr.
Fredrick W. Bakker-Arkema (Agricultural Engineering), a truly
inspiring teacher and counselor, whose attitude and friendship
made this investigation a very rewarding and delightful experience.

The personal interest and unfailing assistance of Dr. J. B.
Kinsinger (Chairman, Chemistry) who guided the author's study of
the fascinating field of Surface Chemistry is gratefully acknowledged.

Sincere appreciation is extended to Dr. W. C. Bickert (Agri-
cultural Engineering) and Dr. D. R. Heldman (Agricultural Engineering
and Food Science) for serving on the author's guidance committee.

The author is indebted to Dr. Carl W. Hall (Chairman, Agri-
cultural Engineering) and his staff for making available the graduate
assistantship and for their kind c00peration and help in innumerable
other ways.

Grateful acknowledgement is due Dr. Bakker's group of graduate
students, David Farmer, Gonzalo Roa, John Rosenau, and David Thompson,
whose lively participation in several brain storming sessions helped
to crystallize many of the concepts presented in this dissertation.

It seems apprOpriate to recognize here, members of the author's
family: his parents, Mr. and Mrs. H. 0. Ngoddy and sister, Mrs. R.

A. Akudu, who were directly reSponsible for his early education; his
wife, Omogo, without whose understanding and kind support this study
could not have been accomplished.

Finally, thanks go to the members of the M.S.U. Glass ShOp for
their unfailing assistance, to Mr. M. P. Palnitkar for his assistance
in the laboratory experiments, and to Mrs. Nancy Hodge for her extra-
ordinary patience and diligence in typing the manuscript.

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2.4

2.5

 

TABLE OF CONTENTS

Page

INTRODUCTION . . . . . . . . . . . . . . 1
1.1 General Remarks . . . . . . . . . . . . 1
1.2 Hygrosc0pic Phenomena - Moisture Retention

and Transport in Bio-Materials . . . . . . . 2
1.3 The Nature of Adsorption . . . . . . . . 3
1.4 Adsorption as a Function of the Nature of the

Adsorbent O O O O O U C O O O O O O O 6
1.5 Adsorption as a Function of the Nature of the

Adsorbate - The Water Molecule . . . . . . . 8

1.6 Review of Related Literature . . . . . . . 12
1.7 Statement of the Problem . . . . . . . . . 28

1.8 Objectives . . . . . . . . . . . . . 29

THEORETICAL CONSIDERATIONS . . . . . . . . . . 30
2.1 Introduction . . . . . . . . . . . . . 30

2.2 A Physical Model of Water Adsorption by Bio-
Materials . . . . . . . . . . . . . 30

2.3 Generalization of Molecular and Capillary
Adsorbed Moisture . . . . . . . . . . . 33

(a) Pore Characterization - defining the size
of a pore filled by Molecular and Capillary
Adsorption . . . . . . . . . . . . 33
(b) The thickness of the adsorbed multi-layer . 37
(c) The Isotherm Equation . . . . . . . . 41
2.4 Adsorption Compression on Adsorbing Bio—Materials 48
2.5 Sorption Hysteresis in Biological Materials . . 68
(a) Capillary Condensation Hysteresis . . . . 69

(b) Swelling Fatigue Hysteresis . . . . . . 72

(c) Double Superposition of "Capillary Condensa-
tion" and "Swelling Fatigue" - Hysteresis . 76

i

III.

IV.

EXPERIMENTAL . . . . . . . . . . . . .

3.1 Product Preparation . . . . . . . . .

3.2 Determination of Equilibrium Moisture Content

(6)
(b)

The Adsorption System . . . . . .

Procedure . . . . . . . . . .

RESULTS AND DISCUSSION . . . . . . . . .

4.1 The Determination of Pore Structure from
Water Sorption Isotherms-Verification of a

Pore-

Size Distribution Function of the Power

Law Type for Bio—Materials . . . . . .

4.2 Experimental Results . . . . . . . .

4.3 Verification of the Derived Density Function
for Sorbed Water . . . . . . . . .

4.4 Verification of the Derived Isotherm Equation .

(a)
(b)

(C)

(d)

4.5 Verification of the Preposed "Capillary Condensa-

tion

Theoretical Isotherms . . . . . . .

n as the Primary Characteristic Parameter
of the Pore Structure . . . . . . .

g as the Secondary Characteristic Parameter
of the Pore Structure . . . . . .

Prediction of the Adsorption Isotherms of

Certain Biological Materials with the
Generalized Isotherm Equation . . .

- Swelling Fatigue" Superposition Theory of

Sorption Hysteresis in Biological Materials

(8)

SUMMARY AN
SUGGESTION
REFERENCES

APPENDIX

The Parameter, w, as a factor of pore
geometry . . . . . . . . . .

D CONCLUSIONS . . . . . . . . .

S FOR FURTHER STUDY . . . . . . .

ii

Page
77
77
78
78

83

85

85

103

111
119

119

121

122

123

144

147

149
152
154

163

 

Table

2.1

2,2

2.3

2.4

4.1.1

A.l.2(a)

4.1.2(b)

4.1.3

4.2.1
4.2.2

4-3.1

4.3.2

 

 

Table

2.1

2.2

2.3

2.4

4.1.1

4.1.2(a)

4.1.2(b)

4.1.3

4.2.1

4.2.2

4.3.1

4.3.2

LIST OF TABLES

The function u* as determined from the
empirical data of Stamm (1938) for
spruce wood at 25°C. . . . . . . . .

The reduced function p/po = u*(AH/AH°) for
corn. Based on the desorption isotherm (4°C)
data of Rodriguez-Arias (1956) . . . . . . .

The reduced function p/p = u*(AH/AH°) for
cotton. Based on the Adsorption isotherm
(10°C) data of Urquhart and Williams (1924) . .

The reduced function p/p = u*(AH/AH°) (for pre-
cooked freeze-dried beef powder based on adsorp-
tion isotherm (10°C) data obtained in the experi-
mental part of this study . . . . . . . .

Illustrative input data for pre-cooked beef
powder freeze-dried at 105°F platen temperature

Pore-Size Distribution for pre-cooked beef powder
freeze-dried at 105°F platen temperature. Calcu—
lated by a modified Cranston and Inkley scheme,
and based on 10°C isotherm. A cylindrical pore
model is assumed . . . . . . . . . .

Pore-Size Distribution for pre-cooked beef powder
freeze-dried at 105°F platen temperature. Calcu-
lated by a modified Cranston and Inkley scheme,
and based on 10°C isotherm. A spheroidal Ink
Bottle pore model is assumed . . . . . . .

Summary of the characteristic parameters of pore
structure as determined from pore-size distri-

bution plots . . . . . . . . . . . .

Equilibrium moisture content of pre-cooked

freeze-dried ground beef. Adsorption . . . .

Equilibrium moisture content of pre-cooked

freeze-dried ground beef. Desorption . . . .

Verification table for the density of sorbed
water. Product = Pre-cooked freeze-dried

beef powder . . . . . . . . . . . . .

Verification table for the density of sorbed

water. Product = Soda boiled cotton . . . . .

iii

Page

60

61

63

64

89

91

94

101

104

106

114

115

 

Tab‘

L‘

‘\
0

£\

L\

Table

4.3.3

4.3.4

4.4.1

4.2.2

4.4.3

4.4.4

4.4.5

4.5.1

4.5.2

Verification table for the density of sorbed

water. Product = spruce wood . . . .

Verification table for the density of sorbed

water. Product = Corn kernels .

Calculation of Theoretical Isotherm

(a) Pre-cooked freeze-dried beef powder (10°C)
(b) Pre-cooked freeze-dried beef powder (37.7°C)

Calculation of Theoretical Isotherm
(a) Raw freeze-dried beef slices (10°C)
(b) Raw freeze—dried beef slices (40°C)

Calculation of Theoretical Isotherm
(a) Whole Corn Kernels (22°C). . . .
(b) Whole Corn Kernels (50°C). . . .

Calculation of Theoretical Isotherm
(a) Wood (20°C) . . . . . . . .
(b) Wood (60°C) . . . . . . . .

Calculation of Theoretical Isotherm
(a) Cotton (10°C). . . . . .
(b) Cotton (30°C). . . . .

Work sheet for the calculation of desorption

isotherm of whole corn kernels at 22°C .

Work sheet for the calculation of desorption

isotherm of whole corn kernels at 50°C .

iv

Page

116

117

125
126

128
129

131
132

134
135

137
138

145

146

Figure
1.1

2.1

2.2

2.3

204

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12
3.1

3.2

LIST OF FIGURES

Molecule of water vapor . . . . . . . .

Pictorial representation of an organic tissue
as a random network of small, irregular pores.
The diameter of one pore is magnified for

illustrative purposes . . . . . . . .

Cylindrical capillary of radius, R, showing the

"inside annular tube", r, which may be given
either by the Cohan or Kelvin equation . . .
Capillary menisci

(a) Hemispherical (Kelvin) meniscus . . .
(b) Cylindrical (Cohan) meniscus . . . .
Packing of adsorbed molecules in successive

molecular layers, to illustrate variable
stacking configurations . . . . . . . .

Plot of X/X against P/P . . . . . . .
m o

Shapes of capillaries
(a) Cylindrical capillary . . . . . . .
(b) Interconnected spheroidal "Ink Bottles" .

Polanyi's iSOpotential contours . . . .

Diagram showing two thermodynamically identical

systems
(a) Contains an active sorbent . . . . .

(b) Contains the pure sorbent . . . . .

(0/00) and (AH/AH°) plotted against X/Xm . .

Plot of the function u* as obtained from the
empirical data of Stamm (1938) . . . . .

Generalized plot of the function, u*, as obtained

from the empirical data of Stamm (1938) . .
Plot of p/po against X/XL . . . . .
Adsorption apparatus . . . . . . . .

Diagram showing the principle of the Cahn-
Electrobalance . . . . . . . . . .

Page

10

31

36

36
36

38

40

44
44

49

53
53

65

65

66
67

79

80

Figure

3.3

4.1

4.3

4.4

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

(a) Close-up view of the electrobalance
in the vacuum line . . . . . . . . . .

(b) Close-up view of the Control unit and
recorder outside the experimental chamber . .

Pore-Size distribution plot for corn at 4°C,
15.5°C, 30°C isotherm temperatures . . . . . .

Pore-Size distribution plot for cotton at 10°C,
20°C and 30°C isotherm temperatures . . . . .

Pore-Size distribution plot for wood at 20°C,
60°C and 100°C isotherm temperatures . . . . .

Pore-Size distribution plots for pre-cooked
freeze-dried beef powder at 10°C, 22.2°C and
37.7°C isotherm temperatures . . . . . . .

Adsorption isotherms for pre-cooked freeze-dried
beef pOWder (10°C ' 37.7°C) o o O o o o o o

Desorption isotherms for pre-cooked freeze-dried
beef powder (10°C - 37.7°C) . . . . . . .

(a) Plot showing sorption hysteresis in pre-
cooked freeze-dried beef powder (10°C and 37.7°C)

(b) Plot showing sorption hysteresis in pre-cooked
freeze-dried beef powder (22.2°C) . . . . .

Vertical section of stretched out specific
surface, showing the multi-layer matrix . . . .

(a) Theoretical isotherms, Pm 0.008 PO . . .

(b) Theoretical isotherms, Pm = 0.1 PC . . . .
Comparison of experimental adsorption isotherms

with calculated isotherms for pre-cooked freeze-
dried beef powder . . . . . . . . . .

Comparison of experimental adsorption isotherms
with calculated isotherms for raw-freeze-dried
beef slices . . . . . . . . . . . .

Comparison of experimental adsorption isotherms
with calculated isotherms for corn . . . . . .

Comparison of experimental adsorption isotherms
with calculated isotherms for wood . . . . . .

Comparison of experimental adsorption isotherms
with calculated isotherms for cotton . . . . .

Comparison of experimental desorption isotherms
with calculated isotherms for corn . . . . .

vi

Page

81

81

97

98

99

100

105

107

108

109

112

120
120

127

130

133

136

139

148

d.b.

AH .Q

st st

AH, A'H

AH ,A'H
m m

AF

12

L(R)

NOTATION

average diameter of
sorbate molecule, [A°]

factor of pore geometry
dry basis
[calories]

enthalpy,

isosteric heat of
sorption, [calories/mole}

enthalpy change,
[calories/gram]

molar enthalpy change

elastic modulus, [psi]
surface free energy
bulk modulus, [psi]

pore-size distribution
parameter, eqn. (2.3.16)

parameter defined by eqn.
(4.1.5)

length of pore of radius,
R, [A°]

molecular weights of
sorbate, [grams]

specific adsorbed
mass [grams]

specific desorbed mass

equilibrium moisture
content, [M.C.d.b.]

moisture content
number of sorbate

molecules, number of
adsorbed multi—layers

vii

AS,A'S

AS ,A'S
m m

vapor pressure
vapor pressure
corresponding to the

mono-layer capacity

adsorption vapor
pressure

desorption vapor
pressure

swelling pressure,
eqn. (2.5.5)

pore radius, [A°]

universal gas constant

radius of curvature
of meniscus during
adsorption

radius of curvature
of meniscus during

desorption [A°]

generalized radius of
meniscus [A°]

Cohan radius [A°]

Kelvin radius [A°]

parameter defined by
eqn. (4.1.4)

entrOpy, [Cal./degreej

o 2
B-E-T surface area (A )

molar entrOpy,
[Ca1./degree-mole]

entrOpy change

molar entrOpy change

N(R)

12

Av

number of pores of
size, R

used as a subscript or
superscript implies

saturation conditions

specific adsorbed volume,
eqn. (2.3.13) [cc]

volume, [cc]

molar volume [cc]

volume of pore of radius,
R

finite volumetric step

volumetric hydro-
expansion

in chapter I used as
relative vapor pressure;

elsewhere used as specific

adsorbed mass

Specific adsorbed mass at
the mono-layer capacity

specific adsorbed mass
at hygrosc0pic or fiber
saturation point

variable, defined by eqn.
(2.3.23)

identification for a com-
ponent of a thermodynamic
system

coefficient of linear
hydro—expansion,

% M.C.d.b.]

surface tension of con-
densed sorbate

normal stress, [psi]

viii

[in./in.

12

U)

if“

m (R)

w2(R)

surface area pores
associated with a
given adsorption

step. [(A°) 3

temperature, [°K]

thickness of adsorbed
multi-layer, [A° J

variable defined by
eqn. (2.3.25)

variable defined by
eqn. (2.3.26)

factor of pore geometry

empirical constant
of the Halsey eqn.
(2.3.5)

average thickness of
a single layer of
adsorbed molecules,

[A°]

pore-size distribu-
tion parameter, eqn.
(2.3.16)

constant defined by
eqn. (2.3.24)

primary characteristic
parameter of pore struc-
ture, eqn. (2.3.20)

secondary character-
istic parameter of

pore structure, eqn.
(2.3.19)

function of pore
geometry

pore-size distribution
function

Poisson's ratio

luv-936.331.» I. IRWIN“; 1» ..

 

area which one adsorbed V,¢
molecule will occupy

(A°)2 n

density of sorbed water,
[Ema/0C0]

8
function defined by
eqn. (2.4.26)
chemical potential K

ix

contact angle

two dimensional
spreading pressure
in chapter I

fraction of surface
covered by mono-
molecular adsorption

factor of pore
geometry

I. INTRODUCTION

1.1 General Remarks

 

The growing socio-economic importance of food and fibres
together with the complexity of the technology for their production,
handling, processing, preservation and distribution has brought about
a new and exacting emphasis among Agricultural and Food Engineers
on the study of BIOLOGICAL MATERIALS SCIENCE. The primary intent
of this endeavor is to provide higher quality food and fibre pro-
ducts more economically, while the methodology consists of the basic
engineering approach to problems involving the handling and proces-
sing of biological materials. It appears logical that the physical
laws governing the response of biological materials to handling and
processing must be well understood in order to Optimize harvesting,
processing, handling and storage systems.

It is now well established that the moisture content of bio-
logical products exerts a profound influence on their mechanical,
physical, chemical and enzymic behaviors. This recognition has, in
the main, provided the stimulus for the large body of scientific work
to be found on the subject. The work, with a few limited exceptions,
has been primarily experimental in nature. In consequence, while
there exists a vast quantity of confirmatory data on the relation-
ship between biological products and those molecules of water asso-
ciated with them, the exact nature of water in food, and even such
fundamental datum as the adsorption isotherm itself has not been

satisfactorily derived from a theoretical point of view. The cause

of the sigmoid shape of this isotherm and of the distinctively
pronounced hysteresis effect must still be considered the subject
of conjecture rather than the consequence of an established theory.
A large body of isolated data still exists which to a greater or
lesser degree has refused to fit any general treatment. Yet, in
view of its all pervading influence, it is fundamental to the
entire field of biological materials science, that the laws which
underlie the hygroscOpic phenomena in biological materials be well
understood and particularly that these laws be rendered more con-
cise, understandable and useable by expressing them in more pre-
cise mathematical terms. The present work was undertaken in the
attempt to weld some of the data - old and new - into a coordinated

theory of adsorption for biological materials.

1.2 Hygroscopic Phenomena — Moisture Retention
and TranSport in Bio-Materials

Kuprianoff (1958) in his treatment of bound water in foods,
suggests that moisture in biological materials may exist as free
moisture, chemically bound moisture, and adsorbed moisture. In
appraising the relative contribution of each type of moisture to the
overall role of water in food, he concludes that adsorbed moisture is
by far the most important category of water in biological products.
The extensive body of scientific work on the subject amply vindicates
the point that problems associated with moisture transport and reten-
tion in bio-materials above the so-called hygrosc0pic point lend
themselves to relatively easy theoretical treatment. The hygrosc0pic

point has been defined by Lewis (1921) as the product moisture content

correSponding to a relative humidity of 100 percent. Above this
point, the physical laws of osmosis, liquid diffusion and capil-
larity govern the motion or retention of water molecules in organic
materials. These laws are coordinated with remarkable success
(Lykov, 1955) into a moisture transport theory. This theory has
been variously used (Van Arsdel and Capley, 1963) to reconstruct
the so-called constant rate drying period which characterizes the
drying of biological materials above the hygrosc0pic point.

In contradistinction with the so-called "free or removable"
moisture discussed above, water associated with biological products
below the hygrosc0pic point is subject to intense surface phenomena
called sorption. This explains why this type of moisture is called
adsorbed moisture. The removal of this type of moisture is character-
ized by a falling rate drying period. The motion and retention of
adsorbed moisture in biological materials have eluded rigorous
theoretical treatment for a long time. The reason why this is the

case will become apparent in subsequent discussions in this chapter.

1.3 The Nature of Adsorption

 

When a solid is exposed to a vapor at a definite pressure, the
concentration of the vapor is higher at the solid - vapor interface
than it is in the vaporous atmosphere. This is the phenomenon called
adsorption. It is not confined to solid-gas interface alone, but
must be expected to occur in gas-liquid, liquid-solid, liquid-liquid
and even in some circumstances solid-solid interactions (Brunauer,

1945).

Adsorption is described phenomenologically in terms of an
empirical general adsorption function, m = f(P,T) where m is the
amount of vapor adsorbed under conditions specified by pressure P
and temperature T. As a matter of experimental convenience the
relationship between the amount of vapor adsorbed by a solid and
vapor pressure is often represented by the so-called moisture
equilibrium isotherm, m = fT(P). This isothermal function can be
generated for several temperatures making it thereby possible for
alternative plots of the data as isobars, m = fp(T) or isosteres,

P = fm(T) to be develOped. While a major body of experimental and
theoretical work has focused on the precise reconstruction of the

adsorption isotherm, the isostere has almost exclusively been used
in conjunction with the Clausius-Clapeyron equation to obtain the

heats of adsorption.

It is customary to divide adsorption into two broad classes,
namely physisorption and chemisorption. Physical adsorption as the
former is alternately called, takes place relatively rapidly, and,
apart from hysteresis, the process is reversible. It is further
supposed that adsorption is induced by the same type of relatively
nonspecific intermolecular forces responsible for condensation
phenomena. Thus, in physical adsorption, the heat of adsorption
should be in the range of the heats of condensation. Physical
adsorption is important only for gases below their critical tempera-
tures, i.e., for vapors (Adamson, 1967). Chemisorption, on the other
hand, may occur below or above the critical temperature of the sor-

bate. It may be slow or rapid. It is qualitatively distinguishable

from physisorption in that the chemical specificity is higher and
the energy of adsorption so large to suggest the presence of an
activation energy necessary for full chemical bonding. While the
probability of chemisorption playing at least some role has been
considered by some investigators (Rodriquez-Arias, 1956), it is
now generally agreed that is is adsorption of the van der Waal's
type which is almost wholly in effect in the adsorption of water
vapor by biological products.

Even though no definitive concensus is discernible from the
literature on the state of the solid-adsorbate complex, certain
directive conclusions can be drawn regarding moisture retention in
biological materials by a careful comparison of available experimental
data and existing adsorption theory. While such conclusions are at
best tentative and must be updated as the frontiers of uncertainty
in surface phenomena shrink, they remain essential if our present
modelistic conceptualization of adsorption in biological materials is
to continue its steady advance. Bearing this in mind, observations
regarding moisture retention by biological materials can be summarized
as follows: their predominant sigmoid shape together with their pro-
nounced hysteretic behavior distinguish the moisture equilibrium iso-
therms of biological products as belonging to the type II (sometimes
type IV) classification of the Brunauer type. If the combined con-
sequences of (a) the B-E-T theory of multimolecular adsorption, (b)
Polanyi's adsorption potential theory, and (c) Zsigmondy's capillary
condensation theory are now admitted into the picture, it becomes
rather obvious that moisture is held in organic materials by inter-

molecular and capillary adsorption forces.

...-ss.|. hilt . 1.1.. fill) 441%

Molecular adsorption occurs when the water molecules adhere
to the pore walls as result of the divergent force field at the
surface of the cell walls. When moisture is adsorbed by organic
materials, it is compressed due to strong intermolecular forces
of attraction. The adsorption compression is highest at low rela-
tive humidities, so that there is a net decrease in the volume of
the water-solid aggregate. As layers of water molecules on the
cell wall increase with rising vapor pressure, the force of attrac-
tion is decreased and the resultant adsorbate compression is corres-
pondingly diminished. This simplified picture of molecular adsorp-
tion is essentially in harmony with the "hydrogen bonding" theory
of adsorption as discussed by Ward (1962).

Capillary adsorption occurs when voids in the cellular struc—
ture of a material are of a size large enough to hold water in liquid
form, under reduced vapor pressure, by the forces of surface tension.
The tensions originating in the capillary water are transmitted to
the capillary walls and there produce stresses within the cell walls.
These stresses combined with swelling stresses culminate in what
Hammerle (1968) termed the HYDRO-STRESS in biological materials.

It will be shown later in this work that the hydro-stress can be
used as a basis for the quantitative estimation of hysteresis in

biological materials.

1.4 Adsorption as a Function of the
Nature of the Adsorbent

 

The quantity of the vapor adsorbed by a given weight of adsor-
bent varies greatly from one adsorbent to another. Since, as has been

amply discussed in the preceding sections, adsorption is a surface

phenomena, this variation must be due to the active Operation of

at least two factors: the area of the interface and the specific
adsorbing prOperties of the substance per unit area. Because the
present interest lies in the class of organic plant or animal
materials, it must be quickly recognized that these are character-
ized by a great complexity and heterogeneity in physical structure.
Each being an assemblage of strongly hydrated high-molecular-weight
compounds, mostly belonging to the classes called proteins, carbo-
hydrates, and lipids. Complete quantitative representation of even
a single one of these systems is not possible. Consequently, the
average or combined behavior of a large number of individual units
or pieces must be dealt with in practice.

Bearing this limitation in mind, both plant and animal
tissues can be considered on the microsc0pic level as being funda-
mentally cellular in their natural state. These fundamental building
blocks are interposed with a complex labyrinth of passageways making
them microsc0pically intricate porous bodies (Babbitt, 1942). While
it is impossible to estimate from a priori considerations the poten-
tial surface area that plays a part in adsorption phenomena, the
present state of the art has made possible the rough estimation of
the relative pore size distributions in such porous complexes. If
this tool is used with caution, it is now entirely within the realms
of possibility to characterize the physical structure of biological
materials mathematically in terms of a pore-size distribution func-

tion and an idealized pore geometry. It is the thesis of this study

that if the structure of an adsorbing biological material can be
so mathematically characterized, its equilibrium isotherm shape
can be predicted accordingly. The details of this structural

characterization will be pursued in a subsequent chapter.

1.5 Adsorption as a Function of the Nature

 

of the Adsorbate - the Water Molecule

 

As has become evident from the foregoing discussions, the
adsorbability of water vapor by organic tissues must depend in a
very high degree upon the nature of water itself. While within
the context of a lumped or averaged treatment of the subject, water
must necessarily be conceived as an ubiquitous and fundamental part
of the structure of biological products (Kuprianoff, 1958; Barkas,
1953), cognizance must here be taken of the fact that scientists
and engineers often find it convenient to assume a double super-
position effect in defining the various properties of water-binding
biological products (see for example Hammerle, 1968). It becomes
therefore expedient to consider at this point, the nature and pro-
perties of the water molecule.

Water is, in many respects, a unique compound. Chemical
reactions and physical interactions in which it participates on the
molecular scale influence every gross characteristic of organic
materials. Except for a slight natural ionization which leads to
the formation of minute amounts of hydrogen and hydroxyl ions, pure

water consists of molecules made up of two atoms of hydrogen and

one atom of oxygen. These molecules may be aggregated by weak

forces into quasi-crystalline combinations whose size and form
depend upon the physical conditions prevailing at the time.

Thermochemical data show that the heat of formation of
gaseous water from hydrogen and oxygen gases is 57.8 K Ca1./Mole.
Since the heats of dissociation of hydrogen and oxygen gases are
103.4 and 118.2 K Cal./Mole., reSpectively, the bond energy for
each O-H bond is one-half of the sum:

57.8 + 103.4 + 59.1 or 110.2 K Ca1./Mole.
The chemical bonds are formed by completion of two electron pairs,
each of the two unpaired electrons of the oxygen atom associating
with the electron of a hydrogen atom. Four of the remaining six
electrons of oxygen are much farther apart from the nucleus than
the other two. These four plus the four electrons involved in the
chemical bonds tend to form four pairs that are as far apart as
they can be while still attracted by the oxygen nucleus. They form
the corners of an imaginary tetrahedron.

The water molecule is schematically represented in Figure 1.1.
The distance of the boundary shell from the center of the atom is
equal to the so-called van der Waal's radius. The equivalent diameter
of the molecule as determined from the density of ice is about 3.2 A.
The cross-sectional area of the water molecule is given either as
14.8 (A)? corresponding to its chemisorption as O-H groups or 10.6
(A)2, corresponding to physical adsorption as H20 molecule along
its longest axis (Gregg and Sing, 1967). These are important quan-
tities which will be used extensively later in this work.

Water vapor exhibits phenomena which can be most satisfactorily

explained by assuming that the molecules involved therein contain

 

0' = 104° 31'

Figure 1.1. Molecule of water vapor.

10

 

rigid, or nearly rigid, electrical dipoles. The distance between
the dipoles being significantly less than the diameter of the
molecule. The magnitude of these moments can be obtained from a
study of the dielectric constants (Dorsey, 1940). The unique
prOperties of water with respect to molecules of similar size, such
as NH3, H23 and CH3OH, have been attributed (Bernal and Fowler,
1933) not only to its dipole character, but also to the geometrical
structure of the molecule. In the present context, the later char-
acteristic together with the polarizability of the water molecule
(i.e., the distortion in the molecular charge distribution and the
resultant distortion of molecular geometry due to external force
fields) may in fact be more important than the dipole qualities
(Adamson, 1967).

In the vapor, the molecules are separated by relatively
great distances and are moving at high velocities. The trans-
lational energy of these molecules is so great that the van der
Waal forces are inadequate to hold them together when they collide.
Thus the vapor expands and exerts pressure in conformity with the
kinetic theory of gases and in the first approximation behave in
accordance with the ideal gas laws.

In the liquid the molecules are held in intimate contact
with one another by combined intermolecular forces. Each molecule
in liquid water occupies a volume of 29.7(A)3 which indicates a
porosity between molecules of about 36.7% (Matz, 1965). The nature

of the bonding forces is not well understood. Some contribution is

probably made by the weak forces of the van der Waal's type (with

11

bond energies of about 5 K Cal./Mole.) and must be, to a consider-
able extent, nondirectional, since the molecules are free to move
in the liquid. The magnitude of the attractive forces between the
molecules must be relatively large, since the vapor pressure of the
liquid is negligible compared to the pressure which is implied by
the gas laws.

In organic products hydrogen bonding exists between water
molecules, between water and small and large molecules and ions,
and as the direct link between these latter micromolecular components
(Ward, 1962). There is, therefore, competition between the compon-
ents for the available hydrogen bonding sites. The rate at which
water may break or promote particular intermolecular hydrogen bonds,
and the extent to which this can be carried out, together with the
converse process of displacement of water from bonding sites, affect
all practical dehydration and rehydration processes. Coulson (1959)
reduced the intermolecular forces to (a) electrostatic interactions
(b) diSpersion forces (c) repulsive forces and (d) delocalization
interaction. Under the combined influence of these forces the
water molecule in its adsorbed state appears to have undergone a
definite thermodynamic transformation (Ewing and Spurway, 1930).
While there exists no satisfactory theory of the liquid state, the
detailed treatment of the adsorbate-adsorbent complex is even

farther away.

1.6 Review of Related Literature

 

In view of the extensiveness of the subject, it is not

intended here to give complete coverage of the voluminous literature

12

on the general subject of gas adsorption. Instead, the principal
models or theories which have been advanced to account for the
sorptive behavior of biological materials or from which such theories
are directly or indirectly derivable, will be taken up partly for
their own sakes and partly as a means of introducing a new general-
ized model to be presented in a later chapter.

In order to expedite the task in this section, it is necessary
to outline some criteria against which the works to be examined can
be evaluated. As was previously mentioned, the adsorption isotherm
is not only the most convenient form in which to obtain and plot
experimental data, it is also the form in which theoretical treat-
ments are most easily develOped. The first demand of a theory of
adsorption then, is that it gives an experimentally correct adsorp-
tion isotherm. Adamson (1967), however, points out that this is a
necessary but insufficient test of the validity of the premises
underlying a theory. Since quite differently based models have been
found to yield equations which are experimentally indistinguishable
and even algebraically identical, he suggests that data on the heats
and possibly entrOpies of adsorption be used as a more discriminating
test of an adsorption model. The characteristic hysteresis observed
in adsorption by porous systems provides a third criterion of
evaluation in the case of porous biological materials. This demands
that the hysteresis phenomenon be both qualitatively and quantita-

tively derivable from a given adsorption model.

13

Labuza (1968) in a recent review of the subject, points
out that theoretical treatment of sorption has primarily come
under three modelistic frameworks, namely:

1. The Kinetic concept of Langmuir;

2. Polanyi's Adsorption potential theory;
and 3. Zsigmondy's capillary condensation theory.

The Kinetic Concept: Langmuir (1918) preposed the classical kinetic

 

model of adsorption based on his belief that adsorption was in
essence, induced by unbalanced chemical forces on the surface of
crystals leading to a unimolecular layer of the adsorbate. Assuming
that (a) adsorbed molecules are localized, (b) colliding adsorbate
molecules are reflected elastically, and (c) the heat of adsorption
for every adsorbate molecule striking the bare surface of a solid
adsorbent is the same and equal to the heat of vaporization,
Langmuir equated the rate of evaporation to the rate of condensa-
tion at the surface of the adsorbing solid under conditions of
dynamic equilibrium. The resultant isotherm equation is of the form:

P/v = 1/b vm + P/vm [1.1]
where v is the volume of gas adsorbed isothermally at vapor pressure
P; vm is the volume adsorbed at the monolayer point; and b is a con-
stant of adsorption dependent on both the heat of adsorption as well
as the isotherm temperature.

In Spite of the limitations inherent in its simplifying
assumptions, the Langmuir isotherm equation is perhaps the most
important single equation in the field of adsorption, serving in

many cases as the starting point in the derivation of other equations.

14

Type I adsorption isotherms are best interpreted in terms of
Langmuir's equation and accordingly, are sometimes referred to
as Langmuir isotherms.

While Langmuir's pioneering adsorption model may be con-
sidered a perfectly acceptable one at very low P/Po values, its
greatest merit lies in the fact that it forms the basis of the more
universally accepted B-E-T theory of multimolecular adsorption.
This theory has, with varying degrees of success, been used to
account for the occurence of types II, III, IV and even V adsorp-
tion isotherms of the Brunauer classification. Brunauer, Emmett
and Teller (1938) extended Langmuirs kinetical approach to multi-
layer adsorption. The basic assumptions in the B-E-T model are:
(a) the Langmuir equation applies to each layer, (b) for the first
layer the heat of adsorption, q, may take a unique value, whereas
for all succeeding layers, it is equal to qv, the heat of vaporiza-
tion of the liquid adsorbate and (c) evaporation and condensation
can occur only from or on exposed surfaces. A detailed balancing
of the forward and reverse rates of adsorption on a surface where
the number of adsorbed layers is restricted to n by walls of pores
having finite diameters leads to the equation:

1

_ vm C X . 1 - (n+1) Xn + n Xn+ [1 2}
1 ‘ X 1 + (C-l) x - c x“+1

 

where X is the relative vapor pressure, P/Po; v is total volume
adsorbed at the measured pressure, P; vm is the volume adsorbed in
the monolayer; and C is a constant related exponentially to the

heats of adsorption and liquifaction of the adsorbate. Relation

15

[1.2] is the so-called B-E-T 3-parameters equation. It reduces
to the Langmuir relationship when n f 1. Under conditions in
which the surface is free and adsorption is not limited [n = m],
it reduces to the more familiar 2-parameters equation:

X 1 + (C-l! X [1.3]

v(1-X) : v C v C
m m

I

from which values of vm and C are obtainable from straight line
plots of adsorption data in the low-pressure region. In a subse-
quent extension of the theory by Brunauer, Deming, Deming and
Teller (1940), a much more elaborate equation was obtained to
account for isotherms of Types IV and V. The multi-layer theory
thus became the first unification of physical—adsorption concepts
which applied to the complete isotherm from the monomolecular
region through the multi-layer and capillary condensation regions
to the saturation pressure. The B-E-T relationship works very
satisfactorily in the relative vapor pressure region from 0.05 to
0.3 (Adamson, 1967), and is extensively used as a general method
for the determination of surface areas of adsorbents.

The considerable success of the B-E-T equation stimulated
investigators to consider modifications that would give a better
fit to type II isotherms. Among these modifications are those of
Pickett (1945), Huttig (1948), McMillan and Teller (1951), Clampitt
and German (1958) and Dellyes (1963). A detailed summary of these

modifications is contained in a book by Young and Crowell (1962).

16

In spite of its considerable success, the assumptions of
the B-E-T theory are not realistic. It does not give the correct
variation of either the heat or entropy of adsorption with the amount
of gas adsorbed. While the B-E-T assumption correctly reflects the
approach of the adsorbed film to bulk liquid prOperties as P
approaches Po’ its assumption of localized multilayers is not con-
sistent, leading to erroneous configurational entrepy values (Hill,
1960). Related to this is a catastrOphe that Cassel (1944) has
pointed out, namely, that the B—E-T model predicts infinite adsorp-
tion at saturation.

In view of the fact that these same general criticisms apply
to the Huttig equation and indeed to the various modifications of

/

the B-E-T equation, the Opinion of Gorter and Frederikse (1949) that
"the kinetical B-E-T theory gives a simple and valuable first pic-
ture of the phenomenon of adsorption, but it seems difficult to
correct its obvious short comings without destroying the simplicity
which perhaps constitutes its chief attraction", seems very appro-
priate. Halsey (1950) concludes that the B-E-T and related iso-
therms should be regarded as merely convenient algebraic tools for
locating the point "B", which, he feels, marks the monolayer stage,
Since it is the point of most rapid change in the affinity of the
solid for the adsorbate.

The Potential Model: At approximately the same time that Langmuir

 

develOped his monomolecular theory, Polanyi (1914, 1916, 1920) formu-

lated an entirely different concept known as the potential theory.

17

This concept recognizes the existence of multi-layer adsorption,
and considers that there is a pontential field at the surface

of a solid into which adsorbate molecules fall. The adsorbed
layer thus resembles the atmosphere of a planet, being most com-
pressed at the surface of the solid and decreases in density
outward.

Polanyi defined the adsorption potential at a point on the
solid, as the work done by adsorption forces in bringing an adsor-
bate molecule from the vapor phase to that point. This work is
conceived as a work of compression, and mathematically is given by

the so-called hydrostatic equation:

6 = I V dp [1-4]

where e is the adsorption potential at a point where the density of
the adsorbed film is p; p0 is the density of the gas phase; and V

is the molar volume of the adsorbate. It is fundamental to Polanyi's
theory that the adsorption potential at any given point is charac-
teristic of the adsorbent alone and is temperature independent as
well as unaffected by the presence of foreign molecules. It is

hence possible to map out the entire adsorption space into a number
of equipotential surfaces, the nature of which would be character-
istic of the adsorbent.

On qualitative grounds, the potential theory appears to be
fundamentally correct. It accounts for the empirical fact that
systems at the same values of RgT 1n (P/Po) are in essentially
corresponding states (Brunauer, 1945) and that the multi-layer

approaches the bulk liquid in prOperties as P approaches P0.

18

However, in specific treatments, functions must be assumed to
represent the equation of state in the adsorbed phase. Since
the adsorbent - adsorbate complex still remains a major thermo-
dynamic puzzle, functions used can only be approximate. From
this point of view, the original formalism of Polanyi its subse-
quent modifications by Frenkel (1946), Halsey (1948), McMillan and
Teller (1951) and Hill (1952) together with the adjoint so-called
two-dimensional film theory of Harkins-Jura (1944), must be
regarded as still somewhat primitive in Specific applications.
Capillary Condensation: It has long been recognized that the vapor
pressure over the miniscus of a liquid contained in a narrow capil-
lary is lower than the vapor pressure of the free liquid at the
same temperature, provided that the liquid wets the capillary and
forms a concave meniscus. In other words, a vapor is liable to
condense in a capillary at a lower pressure than it would on a
plane surface. This phenomenon is called capillary condensation.
The quantitative relationship between the vapor pressure, P,
over a liquid confined in a capillary and the corresponding satura-
tion vapor pressure in a free space at the same temperature was

given by Lord Kelvin (1871) in the form:

 

2 O V cos a) [1 5]
r _

1n (P/PO) =
where o is the surface tension of the liquid; V is the molecular
volume of the adsorbate in its liquid state; r is the radius of the
cylindrical capillary; Rg is the universal gas constant; T is tem-

perature in degrees Kelvin and w is the contact angle.

19

Zsigmondy (1911) and later Foster (1932) applied the
capillary condensation theory to relationships between adsorp-
tion and pore structure in porous adsorbents. They argued that in
porous structures, the same relationship between vapor pressure
and meniscus radius exists as in the case of ordinary cylindrical
capillaries. As the equilibrium pressure is increased in an adsor-
tion experiment, condensation occurs in successively larger pores.
This rationalization leads directly to the calculation of pore-size
from adsorption isotherm data employing the Kelvin equation [1.5].
The principal assumption is that the adsorbate exists as a con-
densed liquid in the pores of the adsorbent and has preperties
characteristic of the bulk liquid phase.

Zsigmondy (1911) attributed the hysteresis phenomenon
characteristic of adsorption in porous systems, to contact angle
hysteresis due to impurities. While this may well be true in some
cases, it fails to account for systems having retraceable closed
hysteresis 100ps. The schemes of Barret et a1. (1951), Pierce
(1953) and Cranston and Inkley (1957) for calculating pore-size
distributions from adsorption isotherms represent refinements of
Zsigmondy's original theory. In Spite of the warning by Everett
(1958) that the bundle of capillaries model can be outrageously
wrong for real porous systems, these methods constitute in many cases,
the only available estimation of the real pore structure. The
"ink bottle" concepts of hysteresis as advanced by Kraemer (1931),
McBain (1935) and Cohan (1938, 1944) remain to date, the most
plausible qualitative description of the phenomenon. These too,

are founded on the capillary condensation theory of Zsigmondy.

20

In conclusion, while the applicability of the Kelvin
equation and the assumed bulk liquid prOpertieS of the adsorbed
film remain subjects of controversy, most investigators (see
Brunauer 1945, Adamson 1967) seem to agree that capillary condensa-
tion plays some role in physical adsorption. In general, it is
believed to come into play approximately in the region of hysteresis.

The Polarization Theory: In the interest of completeness, the

 

Polarization theory of de Boer and Zwikker (1929), and Bradley (1936)
is briefly taken up here. This theory explains the adsorption of
non-polar molecules on ionic adsorbents by assuming that the upper-
most layer of the adsorbent induces dipoles in the first layer of

the adsorbed molecules, which in turn induce dipoles in the next
layer, and so on until several layers are built op. Brunauer (1945)
severely criticized the polarization theory on the ground that the
effect was not large enough and consequently the theory has largely
been ignored. Even though some recent work by Bewig and Zisman
(1964) and Benson and King (1965) suggest that this neglect may be.
either mistaken or premature, Hill (1952) points out that there
exists no satisfactory theory of the liquid state, even for monatomic
liquids, and the detailed treatment of a liquid in a combination

electrical and dispersion long-range force field is still far away.

SecondaryfiTheories

 

The models to be taken up in this section of the review, are
designated secondary in so far as they are in varying degrees sub-

ordinate to one or in some cases a combination of the primary models

21

previously discussed. In the context of the present interest,
these theories are especially relevant because each one of them
has focussed on water vapor as the adsorbate and either polymeric
or organic solids as the adsorbent.

Although as early as 1882 Mfiller had preposed an equation
to predict the adsorption of water vapor by textile fibres, his
equation turned out to be quite valueless (Swan and Urquhart, 1927)
because his arbitrary assumptions - (a) a linear relationship
between the amount of vapor adsorbed and relative vapor pressure
and (b) zero adsorption at boiling point of water - were unsound.

The Smith Equation: It was not until 1947 that a partially suc-

 

cessful treatment of water sorption by polymers was formulated by
Smith who recognized the existence of two principal classes of
sorbed moisture: (a) bound moisture, (ab), held on the adsorbent
surface by intermolecular forces in excess of forces responsible
for condensation and (b) normally condensed moisture, (ac). He
assumed that the relationship between ab and P/Po can be approxi-
mated by the Langmuir equation. While accepting the multi-layer
concept of Brunauer, Emmett and Teller as the structural framework
of ac and further adepting the B-E-T position that the multi-layer
is essentially in the same thermodynamic state as the bulk liquid,

he derived the following expression for ac:

ac = - a’ 1n (l-P/PO) [1.6]

and summed ab and ac to obtain:

_ = u ’ -
a - ab + ac ab a 1n (1 P/PO) [1.71

22

where a, is defined as the adsorbed mass per gram of the adsorbent
in a unimolecular layer of normally condensed moisture, and a is
the total specific adsorbed mass. In order to arrive at equation
[1.6], Smith assumed that at any stage of sorption, the completed
fraction of a, is a measure of the total potential evaporating
surface - a perfectly valid assumption. However, his subsequent
assumption that this fraction, (9), is equal to P/Po is thermo-
dynamically unsound. While it is true that in the assumed thermo-
dynamic state of ac, both 9 and P/Po represent some measure of the
escaping tendency of the vapor molecules, it can be shown that they
do not reduce to an identity. All that can be deduced from thermo-
dynamics is that: 9 = 8(P/PO). Invoking (a) Lewis and Randall's
representation (Lewis and Randall, 1961, p. 147) of fugacity as the
molal free energy and (b) Cibb's definition of the surface free
energy as the work done against surface tension forces by the
Spreading two - dimensional pressure on the adsorbing surface
(Gregg and Sing, 1967, p. 234), the following approximation can be

written:
n 9 = AF = -R T ln (P/P ) [1.8j
g o

where 9 is taken as a measure of the surface on which the spreading
pressure, (H), is active. Equation [1.8] suggests that Smith's
fundamental simplification cannot be thermodynamically justified.
It must therefore be concluded that even though the Smith equation
has been found in certain cases (for example Becker and Sallans,

1955) to fit experimental isotherm data remarkably well in the

23

region 0.5 S P/Po S 0.95, his exponential representation of the
multi-layer region is basically empirical. That empirical func-
tions can be selected to fit a wide variety of isotherms has been
demonstrated by Strohman et al. (1967) and Do Sup Chung et a1.
(1967), both of these works will be elaborated upon later in this
section.

Other weaknesseS'of the Smith equation include (a) its
inability to give the correct variation of either the heat or
entrepy of adsorption, (b) its inability to account for hysteresis
and (c) the fact that temperature dependence is not inherent in
the formulation.

The Henderson Equilibrium Equation: Perhaps the best known and

 

most widely used equation for predicting the equilibrium moisture
content of biological materials is the semi-empirical equation of
Henderson (1952). Using the Gibb's adsorption equation as a
starting point, Henderson derived an isotherm equation which can
account for the temperature dependence of the experimental curve.

His equation is of the form:

kTMan

(1 - P/P.) = e' [1.9]

where T is absolute temperature in degrees Rankine; and k and H are
empirical constants. When apprOpriate values of the parameters, k
and H, are available, the Henderson equation or its modifications
by Day and Nelson (1965) and Thompson et a1. (1967) have been found
to fit isotherm data for cereal grains fairly well (see Rodrigues-

Arias, 1956). However, the Henderson equation has been found to be

24

quite inadequate for certain biological products (see for example
Pichler 1957, Bakker-Arkema 1961, Day and Nelson 1965). One
deficiency of the Henderson equation is that it is totally based
on thermodynamics, it is therefore not founded on a model and
gives no information about the nature of the adsorbent or its
surface.

The Young-Nelson Hysteresis Equation

 

A creditable effort to construct a theory of adsorption for
biological materials to reflect their basic cellular nature was
made by Young and Nelson (1967). These investigators considered
the cell as the ultimate basis of adsorption and recognized the
existence of three modes of adsorbed moisture: (a) the unimole-
cular-bound moisture of Langmuir, (b) the normally condensed or
multiple-layer moisture of Brunauer, Emmett and Teller; and (c)
an adsorbed moisture which results from a diffusional passage of
moisture into the inner cell and which on account of the irreversi-
bility of the diffusion process is responsible for the occurence
of hysteresis.

The development of a representative expression for the first
category of moisture is kinetical; being in essence identical to
the B-E-T process. Their develOpment of an expression for the
normally condensed moisture, with only slight differences, parallels
the Smith method of attack and implies the same basic assumptions.
It is therefore not surprising that they arrived at an equation
which has the prOperties of a combined B-E-T-Smith equation
(Strohman et al., 1967) including the advantages of both but impeded

by the fundamental empiricism of the Smith equation.

25

Although these authors by their perceptive introduction

of the "absorbed moisture" have deve10ped a comparatively straight
forward quantitative representation of hysteresis, the simplifi-
cations and reasoning leading to their explicit expression for the
absorbed moisture effectively destroyed their model. If the ulti-
mate cell of a biological product is taken as the basis of sorption,
it appears logical that moisture transfer across the semi-permeable
cellular wall can take place only as a result of osmosis. From
thermodynamic considerations, the osmotic pressure necessary to in-
duce such transfer is given by: (see Guggenheim, 1967, pp. 183-184):

RT .

__a_
H _ V1 1n (Pl/P) [1.101

where H is osmotic pressure in atmospheres, Rg (the Universal gas
constant) = 82.06 cc.atm. °K-1mole-1; V1 is the volume of adsorbed
moisture in cc.; and T is absolute temperature in degrees Kelvin.
For the 30°C isotherm, the values of H correSponding to the relative
vapor pressure values of 0.1, 0.3, 0.8 and 1.0 are respectively:

1.6 x 106, 4.6 x 105, 3.46 x 104 and 0.0 atmospheres. The values of
V1 corresponding to the above P/Po values are obtained by conversion
of the adsorption data for corn. The conversion constants will be
developed later in this work. The specific values of V1 used are
0.0351, 0.0654 and 0.1590 cc. Since the osmotic pressure must be
exceeded or at least balanced for transfer across the cell wall to
take place, it appears clear that except at saturation, infinite

layers of adsorbed molecules will be necessary to create a suffi-

ciently large hydrostatic force. Moisture transfer across the membrane

26

cannot therefore be justified prior to saturation. On the other
hand, with a predicted zero osmotic pressure at saturation, all
the normally condensed moisture on the surface would be expected
to diffuse into the cell. In summary, our present line of
reasoning allows for no adsorbed moisture except at saturation,
at which point the normally condensed moisture begins to diffuse
into the cell. This moisture passage should continue until the
cell fills to its maximum absorptive capacity. The Young-Nelson
picture of the absorbed moisture does not therefore agree well
with thermodynamic reasoning. It must, however, be stressed that
in spite of its apparent inability to stand up to its theoretical
implications, the Young-Nelson model of hysteresis represents the
best quantitative effort yet to solve explicitly the problem of
hysteresis in water adsorption by biological materials.

The ChungePfost Eqpation: The general framework of the potential

 

theory was utilized by Chung and Pfost (1967) to develop an iso-
therm equation for cereal grains and their derivatives. The

equation is of the form:

_'_‘.‘__ -
1n (P/Po) — RgT exp( BM) [1.11]

where M is the specific adsorbed mass and the parameters A and B

are product and temperature dependent empirical constants. Like

the earlier but algebraically identical relation of Dubinin and co-
workers (1955, 1965), the Chung and Pfost equation is semi-empirical
being based on an over-simplified equation of state for the adsorbed
film. While specific instances of success in the use of such equa-
tions can be cited, their wider usage cannot be justified on
theoretical grounds.

27

The Empirical Eguation of Strohman and Yoerger: Strohman and
Yoerger (1967) using the Othmer linear plots as a starting point
developed a purely empirical isotherm equation for corn. Since
they had no working model, their method must be considered as
essentially a curve fitting scheme which offers practically no
descriptive picture of the phenomenon. While such schemes may
become useful in secondary applications such as drying for example,

they throw no light on the basic process of adsorption.

1.7 Statement of the Problem

 

While the three basic theories of adsorption and their
subordinate equations have failed in their individual capacities
to produce a generally satisfactory mathematical formulation of the
sorptive response of biological materials when exposed to a vaporous
atmosphere of water, their apparent complementary character indicates,
as has been suggested by KUhn (1964), that the fundamental concepts
may be successfully combined into a unified model of adsorption.

_As a broad statement of the problem, it is attempted to con—
struct an isotherm equation for biological materials based primarily
on the B-E—T and capillary condensation theories and somewhat in-
directly on the potential theory.

This integrated theory will be justified if the porous nature
of a given biological material can be considered fundamental to its
sorptive behavior. In the light of the best available knowledge
in the area of sorption, it is reasonable to consider the adsorption

process by biological products as resulting from mono and multi-layer

28

phenomena up to the incidence of hysteresis; the subsequent
isotherm progress into the region of hysteresis is attributed
to both multi-molecular adsorption as well as capillary con-

densation on the porous solid.

1.8 Objectives

 

The study reported in this dissertation was undertaken

for the following specific objectives:

1. To construct, utilizing the B-E-T, and capillary
condensation theories together with a working model
of their pore structure, a moisture adsorption model
for biological materials.

2. To derive, using the general framework of the potential
theory in conjunction with the fundamental laws of
thermodynamics, an explicit density function for adsorbed
moisture. This function is then integrated into the
moisture adsorption model to obtain a generalized iso-
therm equation for porous biological materials.

3. To verify the validity of the prOposed model by showing
that:

(i) The heat of adsorption and its variation with the

 

amount of vapor adsorbed are inherent in the
formulation.

(ii) Sorption hysteresi§_is both qualitatively and

 

quantitatively expressible within its governing
framework;

(iii) The derived isotherm equation is experimentally

 

correct.

29

II. THEORETICAL CONSIDERATIONS

2.1 Introduction

 

In this chapter the framework of a generalized model of
adsorption is formed. The B-E-T and Capillary Condensation theories
are combined into an integral isotherm equation accounting for mole—
cular and capillary adsorbed moisture in porous biological materials.
In order to solve the integral equation explicitly, it became neces-
sary to:

(a) characterize the pore structure of adsorbing

biological products mathematically;

(b) define an explicit density function to satisfy

the fundamental postulates of the adsorption

potential theory as advanced by Polanyi.
As a necessary self-consistency test for this line of attack, a con—
cept of sorption hysteresis is constructed within the generalized
framework, thereby making possible the quantitative estimation of
hysteresis in swelling biological materials.

2.2 A Physical Model of Water
Adsorption by Bio-materials

 

 

The physical picture advanced here is illustrated in Figure
2.1 which shows an organic tissue in a vaporous atmosphere of, for
instance, water at a specified pressure and temperature. The tissue
is visualized as a random network of small pores, the diameter of one
of which has been magnified for illustrative purposes. The pores are
interstices between ill-fitting cellular building blocks which make
up the tissue. Although these intercellular passageways certainly

are not cylindrical, a useful picture can be made in which they are

30

07:301W .>E<JJ.Q<O n
ISJNO..ZIKNL.Z. 0N.u:2mv(.)_N

 

.mmmoausa

m>fiumuuwsfiflw new pmwmwcwwe ma open one we umuoEme wLH .mouoa hogawwuuw
.Hfimem mo xuosuoc Eopcmu m on msmmflu omcmwuo no mo cofiumucmmmuaou HMHHOuoLm .H.N muowwm

MCDFQOZ ommmomod
62.30...» $144453
KSJNO_2thhZ. omr=20<2

 

 

mm><4 1:332
ommmomo< mo mmmzxofk
«
manic V
mo z.>.._mv.
L

 

31

Jim‘s... hi“ ,. Um, . I wflé

treated as cylinders, parallelepipes (slitted pores) or inter-
connected spheroidal "ink bottles", with rough walls and inter-
secting with other pores.

The sorption of water vapor by the tissue can now be thought
of as a process by which the pores are isothermally filled or emptied
of water vapor under the influence of surface forces active on the
pore walls. While this simplified picture classifies the tissue as
essentially a porous body, it does not preclude its basic organic
and biological nature. The position is taken here that unique modes
of behavior such as respiration, and ”water-active” sites which
characterize biological materials in their natural state, exert a
tremendous influence on the nature and distribution of surface forces.
In consequence, it is logical to assume that at least some Of the
"live processes" find commensurate expression in the energy and
entrOpy configurations associated with the sorptive process. The
point is stressed here that in the gross treatment of physisorption
in biological materials, the separate and detailed consideration of
associated life phenomena may become superfluous since such processes
are already inherent in energy or entrOpy terms.

As a micrOporous body, the pore filling process proceeds due
to the Operation of two mechanisms: (a) molecular adsorption, and
(b) capillary condensation. At any stage in the process, its adsorp-
tive capacity is measured by the adsorptively utilizable volume of
the constituent pores. The practical objective Of the physical model
being developed is to characterize these pores sufficiently in terms

of size, and size distribution to make the computation of the afore-

mentioned volume possible.

32

2.3 Generalization of Molecular and
Qgpillaryyédsorbed Moisture

2.3 a. Pore Characterization — defining the size f.g pore filled

 

 

by Molecular and Capillary Adsorption:

In its strictest sense, the term "capillary condensation" is
applied to the particular adsorption mechanism described by the Kelvin
equation. In this mechanism the equilibrium vapor pressure of a liquid
in a cylindrical capillary is reduced below its saturation value by
negative hydrostatic pressure arising from tensile components of curved
surfaces of tension of the liquid, i.e. from the menisci. The reduc-
tion in pressure is related to the radius of curvature of the menisci
by the Kelvin equation (1.5). If complete wetting is assumed to occur
so that q>is equal to zero, equation (1.5) can be reduced to the simpler

form:

 

_ 2 0'6
rk ‘ RgT 1n(Po/P) [2°3’1]

where rk is the radius of curvature of a hemispherical meniscus.
Although eQUation (2.3.1) is basically descriptive of such
microsCOpic phenomena as capillary rise and depression, Zsigmondy (1911)
and later Foster (1932) applied it to relationships between adsorption
and pore size in micrOporous adsorbents. Their principal assumption
was that the adsorbate behaved as a single-component liquid in a poten-
tial field inside the micrOporous capillary of the adsorbent and pos-
sessed prOperties characteristic of the bulk liquid phase. In view
of the undefined thermodynamic state of the adsorbent-adsorbate complex,

this rationalization remains open to question. However, investigators

consider that the Kelvin equation can be used with reliability for the

33

calculation of pore size along the desorption path of the isotherm
(Flood, 1967, p. 55). This is particularly true in the case Of H20
where adsorbent contraction accompanying adsorption hysteresis is
indicative of (a) the presence of a hemispherical meniscus, and (b)
the active Operation of a negative hydrostic pressure - both of
which are consistent with the Kelvin mechanism.

Wheeler (1945) preposed an improved theory which took into
account the effects of multi-layer adsorption. Two new assumptions
were made. First, it is assumed that at any point on the desorption
branch of the isotherm, all pores larger than a certain radius R, are
covered with an adsorbed multi-layer of thickness t, whereas all
pores smaller than R are filled by capillary condensation. Secondly,
since all unfilled pore walls have an adsorbed layer of thickness
t, the radius of the meniscus in a filled pore, where it joins a
larger pore, is assumed not to be the pore radius R, but a smaller
radius (R-t). This means in effect that under conditions of capil-
lary adsorption it is not the pore of the true physical radius R
which is important, but rather, a pore whose radius has been effec-
tively diminished by the thickness of an adsorbed multi-layer.
Wheeler thus argued that the Kelvin equation applied not to the actual
pore radius but to the effective radius Of the "inside annular tube”
left after multi-layer adsorption has taken place. The maximum true
pore radius R, which will be filled by both molecular adsorption
and capillary condensation at a relative vapor pressure P/PO is thus
given by:

= 2.3.2
R t + rk 1 1

34

where t the thickness of adsorbed multi-layer, yet to be defined.

r the Kelvin radius defined by equation (2.3.1). The

k
summed radius is shown diagrammatically in Figure (2.2).

Because of its Kelvin component, the Wheeler equation (2.3.2)
is applicable only to the desorption branch of the isotherm. How-
ever, Cohan (1938) and Coelingh (1938), working independently, con-
cluded that capillary condensation occurred along both branches of
the isotherm 100p, and explained hysteresis in terms Of the different
shape of the meniscus during adsorption and desorption. Along the
desorption branch the Kelvin mechanism is assumed - the meniscus
being hemispherical - but along the adsorption branch, the meniscus
assumes a cylindrical shape, the pore being Open at both ends (see
Figure 2.3). Cohan by (a) specializing the Young-Laplace equation
for the case of a cylindrical meniscus (see Gregg and Sing, 1967, p.
149), and (b) employing the same basic thermodynamic argument as
Kelvin, arrived at the equation:

(IV

C = R T 1n(P /P) [2.3.3]
g 0

 

where rC = the radius of curvature of a cylindrical meniscus (the
Cohan radius).
The Wheeler equation (2.3.2) can be now generalized into the
form:
R = r + t [2.3.4]
where r = rc for adsorption; and r = rk for desorption.
Having already defined r in terms of the variable, P/PO, the thickness

t needs to be similarly defined in order to complete the characteriza—

tion of R.

35

 

Figure 2.2.

 

 

Figure 2.3. Capillary menisci:

 

or Kelvin equation.

:\

 

36

1

I’III’I’I”III77III’IIIrf77r7'II'll/rilllrrifiirr

 

i

 

 

Cylindrical capillary of radius R, showing the "inside

annular tube" r which may be given either by the Cohan

\1

(a) Hemispherical (Kelvin) meniscus.

(b)

Cylindrical (Cohan) meniscus.

 

 

-*-

 

 

2.3 b. The thickness pf the adsorbed multi-layer:

 

 

Halsey (1948) employed the concept of COOperative adsorption
to refine the B-E-T theory. This concept implies that adsorbate
molecules influence each other by horizontal interaction during
the adsorption process. He derived the following semi—empirical

relation for the adsorbed multi-layer thickness:

2
Qst S

RgT 1n(Po/P) ]

 

cw) = T [ [2.3.5]

where T - average thickness of a single layer of adsorbed molecules
Q = the isosteric heat of adsorption
Q = an empirical constant dependent on the adsorbate, the
adsorbent and possibly temperature.
The most direct empirical way of estimating t, is to determine
by the standard B-E-T procedure the monolayer capacity Xm of the
adsorbate. The adsorption X at any pressure can then be converted into

the thickness of the film, by employing the relation.

cw) = (i—‘-) . 'r [2.3.6]
In
or t(A°) = n T [2.3.7]

where n is the number of molecular layers adsorbed. It needs to be
emphasized that the value of T is not necessarily the same as the
molecular diameter, d of the molecules. The relationship between the
two depends on (a) the mode of stacking of successive molecular layers
in the adsorbed film as illustrated by the variable configurations

postulated in Figure 2.4, and (b) the degree of adsorbate compression

37

   
        

            
  

    

0969
s‘s‘e‘s‘o’e’e’4‘4
O. 3.0.0.0331...
(a )

   

    

 

 

Figure 2.4. Packing of adsorbed molecules in successive molecular

layers, to illustrate variable stacking configurations.

38

within the adsorbed film. Two different values for nitrogen of
3.5(A°) (prOposed by Lippens, Linsen and deBoer, 1964) and 4.3(A°)
(prOposed by Shull, 1948) are clearly illustrative of the state of
controversy existing on the subject. For the adsorbed water mole-
cule, an even greater degree of uncertainty must be entertained
with regard to the value of T because of the high degree of compres-
sion associated with water adsorption (see Babbitt, 1942; Stamm,
1938).

In Figure 2.5, X/Xm is plotted against P/Po for a number of
biological products. Superposed on this plot is a curve described

by the modified Halsey equation:

1 75 I
= ° 2. .
mm [W3 E 383
Qst
where the quantity (E1?) in the Halsey relationship is approximated
8

by the constant 1.75. This theoretical curve fits the empirical X/Xm
values fairly well. The individual points are however, scattered
around the composite curve. This means that some amount of uncer-
tainty exists in the value of X/Xm defined by equation (2.3.8) for a
given sample. If this degree of uncertainty is considered within
limits of acceptable error, then t is defined by the Halsey equation

in the form:

1.75

t(A°) = 3.2[31—12713]

[2.3.9]

where for the water molecule, T is taken to be approximately equal

to d for lack of a better approximation.

39

 

g THEORETICAL
CURVE (eqn- 2'3-8)

 

 

J L 1 J 1 J 1 L a L +

0
0.0 0.: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 LG
P/P.

Figure 2.5. Plot of )(/)(m against P/I’O.

a , wood 20°C isotherm. A, cotton 10°C isotherm. 0 , cotton 20°C
isotherm. A, cotton 30°C isotherm. X, corn 4°C isotherm.
[3] , Ground freeze-dried beef 10°C isotherm. I . ound :meze-driml

hoof 22.2”": isotherm.

40

Equation (2.3.9) in a slightly different form was used by

Shull (1948), Wheeler (1955), Dollimore and Heal (1964), and
Viswanathan and Sastri (1967) for the calculation of the adsorbed
multi-layer. It is adapted in the present study in preference to
the B-E-T 3-parameters equation from which n = X/Xm is derivable,
for a number of reasons. It is consistent with the concept of
mobile adsorbed film. It allows for the variation of heats of
adsorbtion with amount adsorbed. It has a form convenient for

usage in a larger context.

2.3 c. The Isotherm Equation:

 

With R fully defined as a function of relative vapor pressure,
the equation for the adsorption isotherm is deve10ped as follows:

The Wheeler equation (2.3.4) specifies the relative vapor
pressure at which condensation will occur in a capillary of a given
size. Such condensation will also occur on the adsorbed layers of
water molecules already covering the concave surfaces of a porous
system. If such condensation leads to a significant reduction in the
radius of curvature of a concave surface or Open ended capillary,
condensation will continue on this surface as long as the radius of
curvature continues to reduce. In this discussion, a pore will be
defined as any void region in a porous material which is partially
or completely filled by the combined consequence of molecular and
capillary adsorbed moisture. Also the pore radius will be considered
to mean the radius of curvature of the surface on which the above
described filling-process started to occur. This radius is defined

by the generalized Wheeler equation (2.3.4).

41

If a porous system of organic tissue has a given distribu-
tion or pore radii, one can say that at a specified vapor pressure,
pores of radius equal to or less than the value determined by the
Wheeler equation (2.3.4) will be completely filled. That is, if

the volume of a pore in a porous system is expressed by the function

vR = Cpl(R) [2.3.10]

and if the number of pores of radii between R and R + dr is given by
dN = m2(R)dR [2.3.11]
then, the volumes of all pores with radii between R and R + dR is:

d vR = deN = m1(R) x m2(R) dR [2.3.12]

Thus, the sum of the volumes of all pores whose radii are equal to or

less than R is given by the integral:

R
Va = JR“. [ 01m X 202m) 1 cm [2.3.13J

where Rm, the lower limit of integration is used in recognition of
the so-called "molecular sieves" effect which assumes that adsorbed
molecules in the first approximation, cannot penetrate into pores
smaller than their own diameter, 2Rm.

Equation (2.3.13) is essentially an integral isotherm equation
defining the adsorptive volume of the porous adsorbent. In order to
solve this integral equation explicitly, it is necessary to obtain
explicit analytic functions for (a) the pore geometry, w1(R) and (b)

the pore size distribution, w2(R).

42

de Boer (1958) has shown that a wide variety of geometric
models are possible. However, only two broad classes of these
geometric shapes are consistent with the assumptions of the present
deve10pment. These geometric forms represented diagrammatically
in Figure 2.6 are:

(i) The Cylindrical type pore model for which:

1T L(R) R2 [2.3.14]

c0101)

where L(R) variable length of pore.
(ii) The Interconnected Spheroidal "Ink Bottle"

pore model for which:

cp1(R) = 5:;- rr R3 [2.3.15]

Wheeler (1945) suggested and Shull (1948) demonstrated that
pore size distributions may be represented by simple analytical
relationships of:

(i) The Maxwellian type for which:

 

.3

@2(R) = A exp [-R/Ro]

W

0

(ii) The Gaussian type for which:

2
w2(R) = A exp [- 82( - 1) J

wlw

o

where A, B and R0 are constants. Gregg and
Sing (1967) have further suggested the possible
utilization of:

(iii) The Log:Normal distribution for which:

 

m2(R) = A exp [- 82(1n R/RO-1)2]

43

(a)

g (b)

Figure 2.6. Shapes of capillaries.
(a) Cylindrical capillary

(b) Interconnected spheroidal "Ink Bottles".

44

No closed form solution to equation (2.3.13) is possible
when any one of the above analytical functions is used to define
m2(R), the worst of them is mathematically intractable, the best
leads to unwieldy end results. In the attempt to overcome this
difficulty, Foster (1948) and Kuhn (1964) recommended a power

function distribution of pore radii for which:

cp2(R) = KlRY [2.3.10]

where the symbol y is an exponent dependent on the product. Equa-
tion (2.3.16) is essentially of an exponential character, and for
the most simple cases, it reduces to a simple power function in
which y can assume constant values, being positive or negative,
integral or fractional.

A detailed computation of the pore-size distribution of
several biological products was undertaken using a modified Cranston
and Inkley (1957) scheme. The results - to be presented in chapter
4 - show that if the spheroidal "ink bottle” geometric model is
assumed, m2(R) [as defined in equation (2.3.11)] is well described
by the power law equation (2.3.16). In view of its central impor-
tance to the deve10pment presented, the above mentioned computational
scheme together with the results need to be considered here in detail.
However, in order to maintain continuity in the line of thought, it
is desirable to postpone this side detail for consideration in a
later section.

Combining the "ink bottle” pore geometry with the power func-

tion distribution of pore radii in equation (2.3.13) yields:

45

R
v = -% U K I R3+Y dR [2.3.17]

Equation (2.3.17) integrates to:

= ,3 n_ n
va n ( R Rm ) [2.3.18]
where
5:5-TTK [2319]
.3 3 1 o.
and
n = 4 + Y [2.3.20]

Equation (2.3.18) is a simplified geometric expression of the
adsorptively utilizable volume of a porous adsorbent. The exponent
n can have positive or negative, integral or fractional values.
Substitution of the generalized Wheeler equation (2.3.4) into
equation (2.3.18) and using the Cohan equation (2.3.3) together with

the specialized Halsey equation (2.3.9) yields:

 

 

l, _
_ E 1.75 3 0 v n
Va — n { [3'2 (1n Po/P) + RgT 1n(Po/Pm)J
L -— n
- 1.75 7 0 V 1
“ L3°2 (In P /P) + R T 1n(P /P ):J } [2°3°21J
o g o m

where Pm is the vapor pressure corresponding to the adsorbed mono-
layer or point "B" of the isotherm.

Equation (2.3.21) is an isotherm equation relating the specific
adsorbed volume of the adsorbent, Va’ with relative vapor pressure.
It shows strange deviations from experience in that it gives zero
adsorption, Va = O at P = Pm, and negative values of Va at pressures,

P < P5. This means that isotherms calculated with equation (2.3.21)

46

must start not at P/Po = 0, but at a low relative vapor pressure
Pm/P0 > 0. This irregularity stems from the fact that the choice

of the lower limit of integration, Rm, in equation (2.3.13) while
conforming with the specifications of the "molecular sieves"

effect, overlooks the contribution to Va due to the partial filling
of the first mono-layer in the vapor pressure range 0 S P S Pm'

Yet, the choice of Rm is substantiated in the now accepted argument
(Wheeler, 1955) that the Kelvin or Cohan radius is inapplicable in
the low pressure range 0 S P < Pm. In order to correct for this
irregularity, a shift of the reference axes of the isotherm plot was

performed, thereby reducing equation (2.3.21) to the form:

=5 n_ n
Va n (Z X ) [2.3.22]
where
g _
1 75 0 V
Z = 3.2 (—L———) -————————- [2.3.23]
1n X1 RgT ln x1
)(=32(.1_'7_§_)%+_G_V__ [2324]
° ln x2 RgT ln X2 ° '
X1 = (P() + Pm)/(P + Pm) [2.3.25]
x2 = (P0 + Pm)/Pm [2.3.20]

Equation (2.3.22) is a volumetric isotherm equation defining
the specific adsorbed volume due to molecular and capillary phenomena
within the intermicellular capillaries of the tissue. It is reduceable
to its gravimetric equivalent by a process which converts the specific
adsorbed volume to specific adsorbed mass. This can be accomplished

by introducing the apprOpriate density term into equation (2.3.22)

47

to obtain the terminal relation:

Ma . 0% (2” - A") [2.3.26]
where, in view of the demonstrated variable compressibility of the
adsorbed water "film" (Katz, 1933; Stamm and Seborg, 1935; and Stamm,
1938), p is an undefined function which is dependent on the magnitude
of the intermolecular forces, the surface-impressed pressure and
possibly the isotherm temperature. Polanyi's Adsorption Potential
theory describes the adsorbed multi-layer as resembling the atmos-
phere of a planet with the highest compression at the surface of the
solid and the density falling off outwards (see Figure 2.7). Babbitt
(1942) insisted that any theory of adsorption must account for this
adsorption compression. Stamm and Seborg (1935) demonstrated that
actual adsorption compression values can be obtained, and that
adsorption compression extends to the fiber saturation point. A den-
sity function for the adsorbed water film reflecting these points of
view needs now be formulated to augment the derived isotherm relation
(2.3.26).

2.4 Adsorption Compression on
Adsorbing Bio-Materials

Even before Polanyi (1914) first expounded his now famous
theory of isopotential contours within the adsorbed film, a large
number of investigators had preoccupied themselves with detailed
theoretical and experimental study of the adsorbed phase. Among
sudh early investigators are Rose (1849), Jungh (1865) and Parks (1902)

who speculated that adsorbed water molecules on the surface of a

48

 

/ ___ _~ / —-—— "" "‘ —' ABSORPTION
/ ’ ‘ \ ./ SPACE

/ ”——

‘g’ ‘—' .-—_

\ un—
/
/ /
// ,_ \ // (ttWWW Hm ADSORBENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.7. Polanyi's isopotential contours. The space between
each pair of equipotential surfaces corresponds to

a different degree of adsorbate compression.

49

solid have undergone a definite thermodynamic change in state and
are held against the surface by a force which is expressible in
terms of pressure and compressibility. Calorimetric measurements
of the heats of adsorption undertaken by Lamb and Coolidge (1920),
Patrick and Grimm (1921) and later supported by the extensive
thermodynamic calculations of Harkins and Ewing (1921), estimated
values of the compressive force in the range of 37,000 to 100,000
atmospheres. From specific volume determinations for cotton in
different liquid and gaseous media, Davidson (1927) calculated
values of the average compression of sorbed water of approximately
2000 atmospheres on the basis that compression takes place entirely
in the water and that the adsorbent is essentially incompressible.
Katz (1933) has shown that a qualitative pr0portionality exists
between the heat of swelling of binary aqueous systems and the
corresponding volume change. Gibson (1934) gives the relationship
between the specific compression of water, the internal or intrinsic
pressure change, and the externally applied pressure for aqueous
solutions of certain sulfates.

Measurements are available giving the contraction in volume
as a function of moisture content for cellulosic materials. Thus,
Stamm and Hansen (1937) employed the Gibson compressibility relation-
ship for water to obtain specific volume contractions for wood, pulp
and cotton. Filby and Maass (1932) calculated the apparent density of
water adsorbed on cotton cellulose as a function of moisture content.
They concluded that below the 3% moisture level an apparent density
of 2.6 gm./cc. is obtained. The apparent density value falls off

with increasing moisture content, until above the 9% moisture level

50

where the adsorbed water appears to have its normal bulk liquid
density. This result was contradicted by the work of Stamm and
Seborg (1935) who found that the density of water originally
adsorbed is 1.3, and demonstrated that adsorption compression
extends to the fiber saturation point. Ewing and Spurway (1930)
obtained apparent density values for water adsorbed on silica gel.
Their values extend all the way from 1.03 at 1% moisture content

to 0.54 at 7% moisture level. Morrison and McIntosh (1945)

obtained the apparent density of water adsorbed on four different
charcoals using the helium displacement method. Their results which
exhibited considerable inconsistency, showed a notable difference in
apparent density values of water between the adsorption and desorp-
tion branches. In another study, Tuck, McIntosh and Maass (1947)
investigating the density of various adsorbates on charcoal, con-
cluded that in view of the uncertainties which they observed, the
utility of apparent density values of adsorbates in checking con-
cepts in physical adsorption is limited.

From the foregoing review, it is apparent that while the
broad concept of adsorption compression seems to be well established
among investigators, considerable uncertainty exists regarding
compressibility and apparent density values of adsorbates. This
uncertainty must account, at least in part, for the seeming lack of
interest on the part of investigators to come up with a descriptive
density function for the adsorbed phase. The compressibility rela-
tion of Gibson notwithstanding, there exists, to the knowledge of

the author, only two such equations. The first is the density

51

relation of Stamm (1938), which expresses adsorbate density

in terms of the "true" and "apparent" specific gravities of

the sorbent. The second is a computational scheme using the
Lowry and Olmsted (1927) modification of Polanyi's theory to
generate the so-called characteristic curves from empirical
isotherm data (see Brunauer, 1945, pp. 101-104). The character-
istic curves are then integrated numerically to obtain the density
values. Because of the difficulty of obtaining accurate Specific
gravity values for porous substances, the Stamm relation does not
lend itself for usage here. The Polanyi characteristic curves
scheme on the other hand, is only as good as the simplified equa-
tion of state which must be assumed for the adsorbed phase. For
their work, Lowry and Olmsted used a modified form of the van der
Waals two-dimensional equation of state given by van Laar (1924).
Even though Lowry and Olmsted (1927) and De Vries (1935) appear
to have successfully used the method to obtain reasonable density
values for C0 adsorbed on charcoal, its direct application to

2
H20 systems does not seem to be as yet possible since the van Laar
constants for the water molecule have not been determined.
Because of the urgent need for an easily applied adsorbate

density relation, a derivation of such a function will now be

presented.

2.4 a. g Density Function for Sorbed Water:

 

 

Consider a closed vessel (Figure 2.8) of volume V containing
the sorbent and a fixed number of molecules, n, of the sorbate at

the temperature T and pressure P. If Vm denotes the molar volume

52

 

 

 

 

 

 

;\\\\\\x\\\\xx\ x;\\\\\\\\ \\x\' x \\x

K \

k \

K N

N \

\

x
N
\
x
x
x
\

 

 

T 111111111
I
I

     
   
 

Ansoasms T
SYSTEM
p

n

SIDE

PURE n CYLINDER

ADSORBATE

m

 

\ \\\\ ;\\\\x\\\\\\\;;\\ x\\\\\\\ \\\xx

7r/ir/friI/r/Ir/lr/I/7r4

 

 

 

Figure 2.8.

 

iii/TITIIIII/II

(a) (b)

Diagram showing two thermodynamically identical systems.
(a) Contains an active sorbent

(h) Contains the pure sorbate

53

of the gaseous sorbate and the volume of the sorbent is assumed

to be negligible, then:
a
V = (n — n ) Vm [2.4.1]

a
where n is the adsorbed excess due to sorption over and above the
amount of gas which would be contained in the same volume, at the
same pressure and temperature, in the absence of the sorbent.

From.Maxwe11's relation:
(bS/BP)T = -(0V/BT)P [2.4.2]

combining equations (2.4.1) with (2.4.2) gives:

- (aS/BP)T = (BVIBT)P
a a
= (n-n )(avm/aT)P-(0n /0T)Pvm [2.4-3]

also 8 3?

(as/62)., = - (as/an )T(507)T
consequently

a _ a
(as/an )T - (BS/0P)T (BF/5n )T

a a a a
- (n-n )(BVm/BT)P(aP/dn )T - Vm(dn /dT)P(5P/dn )T
a a
= - 5 2. .
(n n )(de/GT)P( Plan )T + Vm(5P/8T)na [ 4 4]
Now compare the system described above with another system
(shown in Figure 2.8.b) consisting of a vessel of the same volume V
containing the same gas at the same temperature and pressure, but
without any sorbent. If primed symbols are adOpted to represent
quantities relating to this second system when they may differ

from those relating to the first system, then:

54

n, = n - na [2.4.5]

where n, denotes the number of molecules of pure gaseous sorbate
remaining in the vessel after the amount n3 has been lost iso—
thermally through the lateral cylinder. Moreover, since both
systems are in equilibrium and the gas is in identical condition,
the chemical potentials are the same, i.e.

U = u [2.4.6]

If the molar entrOpy change in the first system per unit change in

na resulting from a decrease in the pressure, is denoted by ASm, then
AS = (aS/ana)
m T
a a
= _ 2.4.
(n n )(avm/aT)P(6n )T + vm(aP/aT) a [ 7]

n
If the molar entrOpy change in the second system corresponding to

the same decrease in pressure is further denoted by A’Sm, then,

I

I a .
= 2. . '
A Sm n (avm/aT)P(0P/0n )T [ 4 8J

Subtracting equation (2.4.8) from equation (2.4.7) and using

equation (2.4.5) gives:

— i = a — a : [
Asm A Sm (BS/5n )T (as/0n )T vm(0P/0T)na [2.4.9]

Multiplying equation (2.4.9) by T gives:

T as - T A’ s = T(5P/8T) v [2.4.10]
m m nam

55

It follows immediately from equation (2.4.6) that:
A u = A’ u [2.4.11]

From the fundamental equations of thermodynamics (see Guggenheim,

1967, p. 24),

(1 G.
.g—i‘. (“a/T) = - -—- .. £5 a: - 357 [2.4.12]
T T

It follows immediately from equation (2.4.12) that:

(1. CI.
r A s + A n = A a“ [2.4.13]

Subtracting equation (2.4.11) from equation (2.4.10) and using

equation (2.4.13) yields:

A Hm - A Hm = T(3P/8T)navm [2.4.14]

The left hand side of equation (2.4.14) may be taken to
represent the equilibrium molar enthalpy (heat) of sorption. It
is the heat which must be supplied per unit change in n8 resulting
from decrease of pressure under isothermal quasi-equilibrium
conditions, minus the heat that must be supplied to the second
system when the pressure is identically reduced by the same
amount. It needs to be emphasized that every quantity occuring
in equation (2.4.14) is experimentally determinable without the
use of approximations or extraneous assumptions. Thus the quantity
A’sm is a molar work function for a real gas, and AHm and (op/or) a

n
are obtainable from sorption isosteres.

56

If the sorbed mass in the sorbent-sorbate system is

denoted by M; and the corresponding volume by Va then,

V
a

a
n vm a (Ma/M) vIn [2.4.15]

also p Ma/Va = M/Vm [2.4.16]

where p is the density of the sorbate and M.is its gram molecular
weight. Multiplying equation (2.4.14) by n8 and using equations

(2.4.15) and (2.4.16) gives:

'H
a I
i- (AHm- 7] 11m) = '1‘(51>/a'r)uava [2.4.17]

so that:
T(BP/BT)na

= -—-—--—-- 2 ° .
p (AH-A’H) [ '4 18"

Equation (2.4.18) can be expressed in terms of a standard density,

p0, such that:

o I o BP/bT) a
= AH -A H ( n z

where the superscript "0" denotes saturation values, and p0 is the
density of the sorbate at saturation pressure.

The differential terms of equation (2.4.19) can be defined
by invoking the Clausius-Clapeyron equation without any loss of

accuracy. Thus:

(3.; .. 211.123 [2.4.20]
na RgT

57

(92) = AH°P0
dTno RgTz

combining equations (2.4.20) and (2.4.21) yields:

(BF/5T)na = 92. P/P )
(BF/5T)n0 AH°( o

 

Substituting equation (2.4.22) in equation (2.4.19) gives:

P AH°-A’H° . Ag

9/0. ' '130‘W) no

or

9/90 = u* (AH/AH°)

AH°-A’H°

*3
where u (P/Po) (Zfi:37§")

for pure water vapor A'H° i 0
consequently

P AH°

“ (————7')
Po AH-A H

[1*

[2.4.21]

[2.4.22]

[2.4.23]

[2.4.24]

[2.4.25]

[2.4.26]

Equation (2.4.24) states that the adsorbate density is a

direct function of the isosteric heat of sorption which in turn

is a measure of the intermolecular force of attraction.

result appears to support the earlier qualitative observation of

Katz (1933), that a prOportionality exists between the heat of

swelling of a binary aqueous system and the correSponding volume

change. Since a purely theoretical determination of the function

H* is not possible without simplification of the work function

A'H, the decision was made to obtain an estimate of H* using the

58

empirical data of Stamm (1938). It seems reasonable, judging

from the nature of the terms present in equation (2.4.26), that u*
will not vary appreciably from one biological material to another.
Thus, while the quantity (AH/AH°) - the ratio of the heat of
adsorption at any P/Po value to that at saturation - characterizes
the specific character of the individual substance, the deter-
ministic function u* classifies it as a class or sub-class.

The work sheet used for the calculation of u* from the
empirical data of Stamm (1938) is shown in Table 2.1. Column 1 shows
the standard relative vapor pressure and column 2 gives the corres-
ponding specific adsorbed mass read from the isotherm. The figures
given in column 3 are the mono-layer capacities determined by the
standard B-E-T procedure. Columns 4 and 5 show the numbers of
adsorbed layers of water molecules. Column 6 is simply a reduced
form of column 5, obtained by dividing each value of column 5 by
the number of adsorbed molecular layers at saturation. Column
7 contains the isosteric heat values for spruce wood taken from
Figure A.l of the appendix. Column 8 shows the average AH values
in terms of molecular layers, obtained by careful averaging of the
relevant values of column 7. Column 9 expresses the average iso-
steric heat values as a ratio of the isosteric heat value at satura-
tion. Column 10 is a tabulation of the density values obtained
from Stamm (1938), and columns 11 and 12 are reduced forms of
column 10. Column 13 is obtained by dividing terms in column 12 by
the corresponding terms of column 9 in accordance with equation

(2.4.24).

59

 

 

 

000.H 000.H 000.~ can 000.0 m 0
000.~ 00~.H «mm 00.0 : 00m. 00.~

~00.H ~00.~ 000.H «0m nmw. o %
m~0.~ cad.“ «0m 0m.m : mum. m0.0

q~0.~ c~0.~ 000.H «0m saw. m *
«m0.H and.“ own 0m.¢ : mum. 00.0

mm0.H N¢0.H 000.H mmm Hum. a *
0m0.H ~m~.~ 00m 0m.m : mca. 00.0

moo.a ~00.H 0N0.fl 00m 0N0. m ¢
050.H «RH.H mom 0n.~ : mma. 0n.0
000.~ m~.~ «00 0N.N : mag. 00.0

m¢0.H H0~.H mm0.H mac mmm. N e
000.H ¢0~.~ 0H0 00.H : m00. 0n.0
OHH.H m-.~ 000 00.H : 000. 0d.0
0-.~ ¢MN.H one m~.H : moo. 0m.0
00.0 mmH.H qm~.a 0¢~.H 00«.H 000 #00 qu. H % 00.H : Ono. 0N.0
00H.“ «NN.H 0am on. : mac. 0~.0
05d.a omm.H 055 cm. : 0mH0. no.0
00H.H 00m.H .. 00.0

0m0.
o. o . mm a... a a o
«.1 So So .0 mm mum mo ax; mamas x? x m3
ma NH HH 0H m 0 h 0 m a N H

.oomm um 0003 moauam MOw Awmmflv EEmum we oumv HmoHuHan msu Boom omcHEhouom mm *3 cofiuocsm mLH .H.N mHQMH

60

10.. 93.5.}.1 I55 I» I'rwWhHUW‘

 

 

 

 

000 000.0 : 000. 00.H

000.H 00.H 000.H mom 00.H q %
mom 000.0 2 000. 00.0
000 HNH.0 : ONN. 00.0

mq0.H mmo.H 000.H 000 05.0 m t
000 00¢.N : MHN. 00.0
0H0 00H.N : 00H. 05.0

00H.H 000.H 000.H 000 00.0 N *
0N0 0N0.H : 00H. 00.0
#00 ~00.H : 00H. 00.0
000 05¢.H : 0NH. 00.0
005 HNN.H : 0HH. 00.0
005 050.H : 000. 0N.0

mmN.H 0m. NNm.H «0m 0N.0 H e
NHO H05. : 000. 0H.0

~00.
E
oo\a 43 .mm mm ma ”\ax mowaoa sx\x ax x om\m
1| x\x
HH 0H m 0 . n 0 m o m N H
.Aomoav moaoa soowaoooa Mo
mumw Auoqv EumSuOmH COHuauOmow msu co vmmmm .cuoo pom -

O .
... one 43 n Q\o ooaooaou ooooooo och

.N.N massa

61

 

000.H 000.H 00.H 00m 000.H n 0
000 NNN.0 : 0N. 00.H

~00.H n00.H 00.H 00m n00. 0 0
0N0.H 0N0.H H00.H H00 N00 0H5. m 0 mom.0 : 00H. 00.0

000.H mmo.H mHo.H 0mm Hum. 0 0
000 500.0 : NMH. 00.0

N00.H 000.H 000.H 5H0 0N0. m 0
0N0 nn.N : 000. 00.0
000 0N.N : 000. 0n.0

HNH.H n00.H Nno.H 000 m0N. N 0
0N0 . 000.H : 000. 00.0
000 0N0.H : 000. 00.0
N00 Nnm.H : 000. 00.0
000 mH.H : H00. 00.0

00H.H 00. 00H.H 005 NOH. H 0
000 000. : N00. 0N.0
0H5 000. : 0N0. 0H.0
on“ 00. : 5H0. 00.0

5000.
a
030 «1 oWW mm ma WWW” muwmma ax\x Ex x om\m
HH 0H 0 0 n 0 m 0 m N H

 

HoooHV EuosuomH coHuaMOmwm

ago so wmmmm

.oooooo toe x.:<\:<v 4: u oo\o coaoooou eoooooo may .m.~ oaooa

.Aomaav mamasaaz ego oooooso: uo some

62

000.H 000.H 000.H 000 00.H 0 0

63

 

 

 

000 000.0 : 000. 00.H
N00.H 000.H 000.H 0N0 000. 0 0
0H0 000.0 : 000. 00.0
0NH.H 0N0.H 000.H N00 000 000. 0 0 00H.0 : 000. 00.0
00H.H 000.H 0HH.H 000 000. 0 0
000 000.N : 00N. 00.0
000 000.0 0 000. 00.0
0H0.H 000.H 00N.H 000 000. N 0
000 000.H : 00H. 00.0
000 000.0 z 000. 00.0
000 H00.H : NNH. 00.0
000 000.H ; 000. 00.0
0N0 000.H : 000. 00.0
000.H 00. 000.H 0N0 00H. H 0
000 000. z 000. 00.0
000 00. =. 000. 00.0
H00.
5
oa\q «1 omd ma :0 “\Hx muome Bx\x 8% x om\m
mm 1: x\x 0
HH 0H 0 0 0 0 0 0 0 N H

 

.00500 meu 00 uuma HmucoEHomaxm m5» cH cmchuno mumo HoooHV EnonuomH coHuauomvm

o
co 0mmm0 .umwBom 0m00 meuw mummuw wmxoouumua now Aomq\zqv «3 n o\o coHuoc30 woosvou mLH .0.N mHan

Superposed plots of p/po (column 12, Table 2.1), and
AH/AH° (column 9, Table 2.1), against the number of adsorbed
molecular layers (column 5, Table 2.1), are shown in Figure 2.9.
The function, U*: as generated from Table 1 (column 13) is plotted
in Figure 2.10 as a function of the number of adsorbed molecular
layers. In anticipation of its wider usage in this work, the
modifying function u* is plotted in a more generalized form in
Figure 2.11. Using Figure 2.11 as the invariant function H* for
the class of biological products of interest, Tables 2.2, 2.3 and
2.4 are developed as typical work sheets for the calculation of
the reduced density function, p/po. In view of their close identity
to Table 2.1, the detailed description of these tables is not
given. Density curves for water adsorbed on spruce wood, freeze-
dried beef powder, corn kernels and cotton based on these tables
are plotted in Figure 2.12. The plots represent a wide variety of
biological species - cellulosic materials, animal tissue, seeds
and textile fibres.

In conclusion, it may be noted that the nature and composi—
tion of equation (2.4.23) can be derived from purely dimensional
considerations. While the present definition of U* may appear sus-
pect because it is based on just one set of empirical data, it will
be shown later - from independent calculations in chapter four —
that the density values are acceptable first estimates. It will be
desirable, however, to revise the function U“ to reflect a broader
spectrum of empirical data as more experimental density values for

water sorbed on biological products become available.

64

 

 

 

 

1.16 EMPIRICAL DATA 0F STAMM
(1938)
go
‘\
9L
Figure 2.9. (*—-), reduced density of water, adsorbed on wood as a function
of adsorbed layers of water molecules. (---), isosteric heat

ratio, AH/AH° as a function adsorbed layers of water molecules.

1.05
1.04
1.03
1.02
2L 1.01
1.00

 

 

Figure 2.10. Plot of the function u* as obtained from the empirical data
of Stamm (1938).

65

1.05-

I.O41-

1.03—

l.02 -

3. 1.01“-

I.OO -

 

 

.sm’ 1 i l l l l l 1 J. l l ,g’_
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x x .
_E_ = x/xL (REDUCED NO 0F 5)

32'; Xu. ADSORBED LAYER

Figure 2.11. Generalized plot of the function, 0* as obtained from the
empirical data of Stamm (1938) for Spruce wood.

XL = Moisture content at hygroscopic point.

66

P II; = p." 1AH/AH°)

Figure 2.12.

1.36
1.34

SPRUCEWOOD AT 25’0
STAMM (19381

"32 u GROUND FREEZE DRIED BEEF
1.30 0 ~ AT IO’C

0 CORN AT 4’0 (Rodgriqusz
1-28 Arias 1956)

1.26
1.24
1.22
1.20
1.18

1.14
1.12
1.10
1.08
1.06
1.04
1.03
1.02

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

 

A COTTON AT 10°C (Urquharta
Williams 1924)

1 l 1 1 1 1 1 I 1 1 +

r”

X/Xm

Plot of reduced density function against dimensionless number

of adsorbed layers.

XL = Moisture content at hygrosc0pic point.

67

2.5 Sorption Hysteresis in
Biological Materials

 

 

A fundamental characteristic of sorption isotherms of
biological materials is their dependence on the previous sorption
history. Thus, the adsorption isotherm obtained by placing a dry
biological material in atmospheres of increasing relative vapor
pressures of water in the words of Ewing (1881) ”lags behind” the
desorption isotherm obtained by placing the wet product in atmos—
pheres of decreasing relative humidity. This phenomenon, of general
occurence for biological materials, is called hysteresis. It
executes a closed 100p - the hysteresis loop - between the upper
and lower bounds of relative humidity, 0 S P/Po S 1. In some sys-
tems the size of the hysteresis 100p depends upon the speed and
frequency with which the 100p is traversed. EEEE,QXPerim9“E§a§£?

carried out very slowly, or are repeatedua sufficient numbet.9f

v»v‘d‘n'fiv‘b“w"e - e V ~-'9-‘ "‘-
‘—

times, hysteresis may disappear. Time-dependent hysteresis which

h... AA- _. _.
'7 w-w-n'fva-a ,~-, t-,.t..-\._.pu \HHJ‘Q‘H" .

has been attributed to relaxation phenomena (Alfrey and Doty, 1945;
ter Haar, 1950), raised some serious doubts among investigators as

to the reality of the phenomenon (McGavack and Patrick, 1920; Patrick
1929). However, the excruciating and extensive work of Allmand and
collaborators (1929), Urquhart (1929), Urquhart and Eckersall (1930),
Lambert and Foster (1932), Foster (1934), Burrage (1934) and Rao
(1941) appear to have established the phenomenon as real and funda-
mental. In consequence, it has long become an important self-

consistency rule for any given theory of adsorption that sorption

hysteresis be inherently explicable within its governing framework.

68

In pursuing this practice in this section from posterior
considerations, it seems possible, taking the facts as a whole,
to think of the sorptive behavior of biological materials as a
double superposition effect. On the one hand, adsorption hysteresis
in bio-materials must be linked with condensation and evaporation
processes in irregular voids - leading logically to the so-called
capillary condensation theories (Everett, 1967). On the other hand,
as "organogels" (Urquhart, 1929; Katz, 1933; Rao, 1941), adsorbed
water molecules influence such physical properties of biological
products as strength, elasticity, rigidity, swelling and evolution
of heat. This water—induced change leads to what will be called

for lack of a better term - "swelling fatigue" hysteresis.

2.5 a. Capillagy Condensation Hysteresis:

 

While Zsigmondy's earliest ”impurity - contact - angle"
hysteresis theory is inadmissible as a general explanation of
"permanent hysteresis" in bio-materials, it does give a creditable
qualitative account of time - dependent hysteresis in these systems.

If the soluble components of adsorbing biological products
are considered to be "impurities" present on the adsorbing insol-
uble surfaces, then the alternate desolving and deposition of
such soluble matter with adsorption and desorption, reSpectively,
can conceivably lead to varying wetting patterns in both processes.
Suppose now that each desolving-deposition cycle is associated with
a measurable translocation of the solute particles, then the contact-
angle hysteresis induced by such translocatable impurity can con-

ceivably be even more pronounced.

69

Foster's (1932) "Open-pore" or "delayed meniscus" theory
of adsorption hysteresis is based on a consideration of the mechanism
of meniscus formation in the adsorption process and of meniscus
disappearance in desorption. These mechanisms are supposed to differ.
On the other hand, the so-called "ink bottle" or "bottle neck" theory
of hysteresis advanced separately by Kraemer (1931) and McBain (1935)
depends on the geometrical irregularity of the pores. In essence,
both theories attribute capillary condensation hysteresis in
physical adsorption by rigid non-swelling adsorbents to t0pographical
factors. On qualitative grounds, both theories are compatible with
the general physical model which has been prOposed. They are conse-
quently each capable of making a possible contribution to adsorption
hysteresis in biological materials. It appears necessary, as a
realistic approach to capillary condensation hysteresis in these
products, to combine the essential features of Foster's theory and
of the ink bottle theory. Katz (1949), in a detailed theoretical
discussion of permanent hysteresis seems to have been the first to
prOpose this point.

The open-pore theory was applied to the simple case of an
Open-ended cylindrical capillary by Cohan (1938). This development
is often regarded as a limited quantitative expression of Foster's
ideas. However, in a later paper, Cohan (1944) showed conclusively
that his considerations are also applicable to closed capillaries
possessing constrictions. For the present discussion therefore,
the Cohan theory of capillary condensation is adOpted with modifica-
tions as an essential combination of the Open-pore and ink bottle

theories in the first approximation.

70

This theory, simultaneously deve10ped independently by
Cohan (1938) and Coelingh (1938), explained hysteresis in terms of
the different shape of the meniscus during adsorption and desorption.
Along the adsorption path, the meniscus assumes a cylindrical shape
with a radius of curvature defined by the equation:

Ra = t + O V/RgT 1n(PO/Pa) [2.5.1]

Along the desorption branch on the other hand, the Kelvin mechanism
is assumed - the meniscus being hemispherical with a radius of curva-
ture:

Rd = t + 2 0 V/RgT 1n(Po/Pd) [2.5.2]

Comparing equations (2.5.1) and (2.5.2) under conditions of perfect
wetting, yields the well known Cohan relation for Open ended tubular
capillaries:

2 _
(Pa/Po) = (Pd/Po) [2.5.3]

From the detailed consideration of the shapes of capillaries by
de Boer (1958), equation (2.5.3) is easily generalized to the form:

(.1)
(Pa/Po) = (Pd/Po) [2.5.4]

where the exponent, w, is a parameter specifying the pore geometry.
Considering the different shapes of interconnected "ink bottles" possi—
ble in the porous structure under consideration, w lies in the range

m 2 1 (de Boer, 1958; Linsen and van den Heuvel, 1967). While the
exact value of w must depend on the particular product, it suffices

to point out that its predicted domain in the range above 1 ensures
that the adsorption path consistently lags behind the desorption

branch.

71

 

2.5 b. "Swelling Fatigue" Hysteresis

Since biological materials are elastic organogels which
swell and contract with adsorption and desorption of water vapor,
respectively, the purely tOpographical explanation Of hysteresis
given above is necessarily incomplete. In addition to the geo-
metrical factors discussed, there exists a contribution to hystere-
sis due to water-induced changes in the properties of the swelling
adsorbent.

Urquhart (1929), Katz (1933), White and Eyring (1947) recog-
nizing the importance of these water-induced changes attributed
hysteresis in elastic gels such as celluloses to the free secondary-
valence bonds of the hydroxyl groups of the cellulose molecules.

In the original water-soaked condition, the free hydroxyl groups
are practically all satisfied by water. When the cellulose is
dried, these hydroxyl groups are freed and, as a result of the
shrinking, pairs are drawn together so that the individual groups
mutually satisfy each other. Upon adsorption, part of the hydroxyl
groups that have mutually satisfied each other are not free for
water adsorption, thus giving reduced adsorption. RaO (1941)
explained hysteresis in some cereal grains in terms of the water-
induced changes in their elastic prOperties. Chung and Pfost
(1967) acknowledged these changes in their molecular shrinkage—
crack concept of hysteresis in cereal grains. Henderson (1969)

in a recent paper on the subject, stressed the importance of unique

changes associated with moisture adsorption by biological materials.

72

All these accounts attributing hysteresis in biological
products to water-induced physico-chemical changes are basically
qualitative descriptions of the phenomenon. In order to make
them expressible quantitatively certain approximations are mandatory.
Thus it appears logical to consider that, in spite of their separate
physical and chemical origins, adsorbed moisture—induced changes
are ultimately mirrored in such physical properties as elasticity,
strength and swelling. Such generally irreversible changes can be
collectively classified as fatigue - hence the term "swelling
fatigue" hysteresis.

An approximate thermodynamic relationship exists between the
pressure of swelling and the vapor pressure of a partially swollen
substance. This relationship has been given by Katz (1933) and
White and Eyring (1947) as:

R T

= .8.
PS Mva 1n (Po/P) [2.5.5]

where Va is the specific adsorbed volume corresponding to the rela-
tive vapor pressure P/Po. The pressure, Ps’ is reSponsible for the
so-called hydro-stress in biological materials (Hammerle, 1968). To
correlate dimensional changes with the changes in stress resulting

from adsorption the relation:

OH = BO E (Ma- MO) [2.5.6]
was prOposed by Hammerle (1968) by assuming Hooke's law and isotrOpic
prOpagation of stress within the adsorbing bio-material. If it is

further assumed that the deviatoric stress is negligible and the

the swelling pressure, PS, is a mean normal stress, then equations

73

(2.5.5) and (2.5.6) can be combined to yield:
R T 1n (Po/P)

MVa 8001a“ M0)

E [2.5.7]

 

The well known relationship (Jastrzebski, 1959) between the modulus

of elasticity E, and K, the bulk modulus is:
x a E/3(1-2\2) [2.5.8]

Substituting equation (2.5.8) in equation (2.5.7) gives:

 

R In (P /P)
K3'3152 )av em M) [2'5‘93
V a O a” 0
Using the definition Of K as:
Av _
g —— 20 o
R PSI V [ 5 10]

s
in equation (2.5.9) and rearranging terms yields:
3(1-2 v) Ps M V8 V8 80(Ma- Mo)

Av = R T In (P I?) [2.5.11]
g 0

 

Substituting equation (2.5.5) in (2.5.11) yields:
Av = 3(1-20) vs 00 (M8- M0) [2.5.121

Equation (2.5.12) gives the volumetric hydro-expansion of an
adsorbing bio-material as a function of moisture content. The
volume V8, is defined not as the bulk volume of the sorbent, but
instead as the effective volume filled in a finite adsorption step
from the relative vapor pressure Pl/Po to Pz/Po, where P1 corres-

ponds to the Wheeler radius R and P to the radius R2 (R2 > R1).

1 2
The total volume adsorptively filled during this step is:
R2 -
v12 =.- J‘ vr dR [25.131
R1

74

It is the implication of the capillary condensation theory that:

v = UR L(R) [2.5.14]

12

where the capillary is assumed to be cylindrical with mean radius
‘R = % (R1+ R2), and length, L(R).

Making the simplified assumption that for each finite adsorp-
tion step the associated hydro-expansion Av is restricted within

the corresponding volume v it is justifiable to consider for

12’

suitably small steps that:

vS = v12 [2.5.15]
Thus, equation (2.5.12) can now be rewritten as:
Av = 3(1-2v) v12 80 (Ma- Mo) [2.5.16]

This volumetric expansion means, in effect, that a pore of mean

radius R changes its volumetric capacity from v in adsorption to

12

v12 + Av in desorption. The volumetric hydro-expansion, as defined
above, depends on the mechanical prOperties of the product. Since
these prOperties exhibit water-induced changes of physico-chemical
origins, their collective contribution to permanent hysteresis is

well represented in Av. For purposes of usage, Av is converted to

its gravimetric equivalent by introducing the apprOpriate density

term, D, so that:

_ AX = _
AMd - p 3(1 2v) v12 Bo (Ma- Mo)/p [2.5.17]

75

If the bulk modulus, K of a substance is known as a function Of
moisture content, it can be readily shown that equation (2.5.17)

can be written in the form:

A M R T 1n(Po/P) gig

. 2. .
d va M K p [ 5 17a]

 

2.5 c. Double Superposition £5 "Capillary Condensatiog: - and -
"Swelligg Fatigue" - Hysteresis

USing the Boltzman superposition principle that the effect of

 

the sum of causes equals the sum of effects of each of the causes,
total hysteresis may now be approximated by the following additive
procedure:

(1) Compute for each Pa/PO value, the specific desorbed mass,

Md, using the relationship:

M = M + A M
a

0 [2.5.18]

d

(ii) Compute, using the Cohan relationship, the desorption
relative vapor pressure, Pd/Po, corresponding to each Md

value, such that:

w
Pd/PO = (Pa/Po)

where W is empirically determined for each product.
(iii) Plot Md obtained in accordance with equation (2.5.18) against
the corresponding Pd/Po value to Obtain the desorption

isotherm.

This scheme will be tested in chapter four for biological
products whose bulk moduli have been empirically determined as a

function of moisture content.

76

III EXPERIMENTAL

The adsorption and desorption equilibrium moisture isotherms
of pre-cooked freeze-dehydrated beef were determined gravimetrically
by exposing the product to a temperature-controlled free water sur-
face under vacuum. The sorptive weight change was automatically and

continuously recorded by s Cahn electrobalance.

3.1 Product Preparation

Low fat commercial grade beef was secured in the form Of
approximately one-inch cubes. The initial fat content was approxi-
mately 9%. The product was cooked in a forced-convection-air oven
at 300°F for a period of 30 minutes. The product cubes were placed
in aluminum trays and covered during the cooking process.

The cooked beef cubes were subsequently frozen in aluminum
foil trays at 20°F prior to freeze-drying. The freeze-drying was
done in a commercial freeze-drier at a platen temperature of 105°F
with a pressure of less than 1 mm of mercury. Both the platen
temperature and pressure were held constant during the freeze-drying
Operation. Approximately 15 to 20 hours were required to dry the
product to less than 2% moisture content on the dry basis (d.b.).

After drying, the product was ground in a Fritz-Patrick mill
using a 0.063-inch screen. The resulting powder was immediately
placed in sealed bottles which were stored in a desiccator at -20°F.
Portions of the product to be used for equilibrium moisture content
determinations were equilibrated to the isotherm temperature prior

to the run.

77

The moisture content of the freeze-dried beef following the
freeze-drying Operation and prior to preserve-storage was determined
in a heated-air oven in accordance with a procedure prescribed by
Triebold and Aurand (1963). In this method an oven temperature of
80°C was used for a time duration of 16 hours.

3.2 Determination of Equilibrium
Moisture Content

 

 

3.2 a. The Adsogption System:

 

 

Equilibrium moisture content of the pre-cooked freeze-dried
beef powder was determined by observing the weight change of the
product sample when exposed to a constant water vapor pressure under
vacuum. Following the attainment of equilibrium at a specified
condition, the vapor pressure was adjusted to a different value and
the new equilibrium point determined. In this way it was possible
to obtain adsorption and desorption isotherms in one continuous
experimental process on the same product sample.

The basic component of the system was a Cahn RG automatic
electrobalance. This highly sensitive instrument (effective sensi-
tivity of 0.2% of full scale) was enclosed in a Cahn glass vacuum
bottle with hangdown glass tubes for product exposure; see Figures
3.1 and 3.3. The principle of Operation of the balance is illus-
trated schematically in Figure 3.2. The instrument in essence,
converts the sorptive weight change of the sample into an electrical
signal by the use of a beam position sensing apparatus shown in the
diagram. This signal is amplified in a 2-stage servo amplifier and

then employed to drive a strip chart recorder. The electrobalance

78

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80

81

 

Figure 3.38 Close up view of the Cahn automatic
electrobalance in the vacuum line.

 

Figure 3.3b Control unit and recorder outside the
experimental chamber.

works on the null-balance principle. Thus, an accurate and con-
tinuous recording of product weight change can be obtained in the
entire sorptive range. The control unit (see Figure 3.1) functions
as a standardization and calibration device.

Figure 3.1 is a schematic diagram of the vacuum line. The
glass chamber containing the‘balance was connected at one end to a
13 mm inside diameter main vacuum line, and at the other end to a
pure water vapor source. The main line was connected through a
vapor trap to a mechanical-diffusion pump combination capable of
pumping the system down to a pressure of l x 10'3 torr in fifteen
minutes. The vapor trap was a glass trap with external reservoir.
In its position at the inlet to the pump, the trap served two pur-
poses: first, it acted as a baffle to the molecules of the pumping
fluid back-streaming from the diffusion pump, and secondly, it pre-
vented the possible contamination of the pumping fluid by condensable
vapors emanating from within the vacuum line itself.

The vapor source consisted of a temperature-controlled dis-
tilled water reservoir; F of Figure 3.1. The temperature of the
reservoir was controlled to within 0.05°C using a constant tempera-
ture laboratory bath with an effective Operating range of -30°C to
+71°C. Thus, desired relative pressure values were easily attained
by precise temperature control of the free water surface in the
reservoir. As a check on steam table predictions, pressure measure-
ments were made in the system by means of three over-lapping and

independent vacuum guages, namely: (i) a mercury u-tube manometer

82

for measurements in the pressure range from S to 760 torr, (ii) a
Virtis McLeod guage for the range 0.5 to 10 torr, and (iii) a
standard type McLeod guage for the pressure range from 1 x 10"3 to
0.5 torr.

All connections in the vacuum line incorporated hyvac step-
cocks in such a manner that the isolation of any desired component
of the system was easily achieved. The entire apparatus, with the
exception of the control unit and the recorder, was enclosed in a
5 ft. x 4 ft. x 9 ft. controlled temperature cubical. Figure 3.3(a)
is a close-up view of the balance in the vacuum line, and Figure
3.3(b) shows the control unit and recorder outside the experimental
chamber. A continuous record of temperature was monitored by capper-
constantan thermocouples from (a) the immediate vacuum around the
test sample, (b) several locations in the cubical, and (c) the vapor-

regulating bath.

§;£__2, Procedure

For the adsorption test, about 100 milligrams of the sample
weneplaced in the aluminum sample container, and the entire system
evacuated to l x 10"3 torr in order to establish the 0% relative
humidity reference with which the sample was allowed to attain
initial equilibrium. After the sample had attained the isotherm
(cubical) temperature, the stapcock S4 of Figure 3.1 was Opened to
expose the sample to a pre-conditioned free water surface in the
reservoir, F. The adsorption isotherm was established by adjusting

the vapor pressure of the free water surface to pregressively higher

83

values corresponding to higher relative humidities and allowing

time for equilibration at each level. »A minimum of ten experimental
moisture equilibrium points were employed to establish an isotherm.
Following the establishment of equilibrium at 100% relative humidity
the vapor pressure was progressively reduced - step by step - to

determine the desorption isotherm of the same sample.

84

IV RESULTS AND DISCUSSION

4.1. The Determination of Pore Structure from
Water Sorption Isotherms - Verification of a
Pore-Size Distribution Function of the
Power Law Type for Bio-materials

The crux of the isotherm equation deve10ped in chapter II,
is the power law type distribution of pore radii which was postu-
lated. It is therefore of central importance to the considerations
to be undertaken in the present chapter, that the basis of this
hypothesis be explored and verified in detail.

The determination of pore-size distributions of catalysts
from sorption isotherms is a well-established technique (Lester,
1967). In this technique (See for example Gregg and Sing, 1967,
pp. 160-172), the Kelvin equation is applied directly to the desorp-
tion branch of the isotherm; the results so obtained give a qualita-
tive picture of pore structure. Of the various refinements which
have been prOposed (Carman, 1951; Barrett, Joyner, and Halender,
1951; Harvey, 1943; Dollimore and Real, 1964; Viswanathan and
Sastri, 1967), the approach of Cranston and and Inkley (1957) is of
particular interest because: (a) it is more exact than any of the
other methods, (b) it may be applied either to the adsorption or
desorption branch of the isotherm and (c) the total Specific sur-
face area estimated by this method can be compared with the corres-
ponding B-E-T values and the differences used as an evaluation of
the physical assumptions regarding the pore geometry.

The original scheme of Cranston and Inkley is generalized
to handle pores of shape other than a cylindrical geometry as

follows:

85

Consider a finite adsorption step from relative vapor pressure
Pi/Po to Pi+l/Po with condensation occuring in pores with radius

varying between R and R (P P R > R1). If the step in

1 1+1 1 ’ 1+1

relative vapor pressure is kept small, the above described pores

-—- 31+ R1+1
have an average radius of R1 8 -——§-—-— . At the same time,

smaller pores are already filled, while in larger pores the thickness

>
1+1

of adsorbed layers on their walls increases from ti to ti+l° The

total volume of water adsorbed in the step, i = l is shown in

appendix B to be:

 

 

R2
vl2 = IRI vR dR
71.
IR2 (R-tl)
= . f . v dR
N.
R 1 R
°° R-t v
12 . f . a
+ (cz-cl) IRZ a 2 R as [4.1.1]

where t12 = (t1 + t2)/2

VR = total volume of pores in the range dR

K, fl and f2 are shape factors such that:

{a a 2, fl
n a 3, f1 = 3/4, f2 - 3 for a spheroidal geometry
R-t
and the quantity C-i-l) is a correction factor for the curvature

= 1, f2 = 2 for a cylindrical geometry

of the meniscus.

Assuming that V is suitably constant over the range R1 to

R

R2 and letting this value of VR be specified as V 2, equation

1
(4.1.1) becomes:

86

 

 

 

+ (t - t ) . f . dR
2 l 2 R2 2R2
Rearranging equation (4.1.2) gives:
m R-t
12
v =r (v -k F . v .dR)
12 12 12 12 “R2 2R2 R

R2 K n
where r12 = (Rz-R1)/f1 . [R1 [(R-tl) /R ] dR

U

l/f1

[(R-t1)/R]K

:51

—.

 

for a suitably small difference, dR = (RZ-Rl)

and R12 = 2f2(t2-t1)

[4.1.2]

[4.1.3]

[4.1.4]

[4.1.5]

For computational purposes, the integral term of equation (4.1.3) is

replaced by a summation term of all increaments of radii from R2 to

the largest possible pore radius, so that:

Rmax R-t

 

12
= r { v - k L ( .
12 12 12 12 R2 2R2 R

V

Pore-size distribution is defined in terms of:

v . AR) ]

[4.1.6]

(a) Pore-Length L(R) in the case of a cylindrical geometry

 

where:
-2
L(R) = VIZ/W R
and

cpz(R) = m

dR

87

[4.1.7]

[4.1.8]

Also, because V H R2L(R)

12
and $12 = 2 H R L(R)
therefore
812 = 2 VIZ/R [4.1.9]

or (b) Pore—Number N(R) for spheroidal "ink bottles" where:

3

4 ..
N(R) = val-5 n 3 [4.1.10]
42m) = anon/an - [4.1.113

and 512 = 3 VIZ/R [4.1.12]

Cummulative specific surface areas computed using equations (4.1.9)
and (4.1.12) can be compared with corresponding B-E-T surfaces -
thus, affording a convenient criterion of geometrical selectivity.
The complete calculation is performed on a CDC 3600 computer
using a modified Cranston and Inkley (1957) method. In this scheme
only 12 to 15 experimental isotherm.points are read in as shown in
the illustrative input tabulation of Table 4.1.1. The 40 or more
points usually needed are synthesized by the use of the Aitken-
Neville modified Lagrange iterated interpolation scheme (this scheme
is discussed by Moursund and Duris, 1967, p. 135). After this point,
the calculation of pore-size distributions follows closely the method
of Cranston and Inkley (1957). The essential differences being that
the "cylindrical" formulations of Cranston and Inkley are replaced by
their generalized counterparts as developed earlier in this section.
This generalization makes it possible to assume any desired geometri-
cal model for a pore. Two geometrical models - cylindrical pores

and spheroidal ink-bottles - are investigated for comparative reasons.

88

Table 4.1.1.

Illustrative input data for beef powder freeze-dried at 105°F platen

temperature.

Isotherm.temperature - 10°C

 

 

P/Po M; 8 mass va 0 adsorbed Pore Radius (R)
2 for PIPo
adsorbed (gm/gm) (cc/gm) (Ac)

0.000 0.000
0.100 .070 0.0515 6.742
0.200 .095 0.0699 9.635
0.300 .110 0.0825 12.463
0.400 .122 0.0927 15.617
0.500 .135 0.1026 19.680
0.600 .155 0.1178 25.624
0.700 .191 0.1546 34.563
0.800 .260 0.2234 52.743
0.900 .380 0.3375 106.054
0.950 .450 0.4240 214.982

.963 300.000
1.000 .505 0.505

89

It turns out that in all cases investigated (see Tables 4.1.2(a)
and 4.1.2(b)), the cummulative specific surface areas calculated
on the model of interconnected spheroidal ink-bottle geometry com-
pares better with the B-E-T surfaces, than the surfaces bases on
the cylindrical pore model. It is concluded on the basis of this
finding, that the pores of biological materials are best modeled for
theoretical treatments as interconnected spheroidal ink-bottles.

Figures 4.1 through 4.4 are log-log plots of pore-size dis-
tributions of several biological materials calculated on the basis
of the spheroidal ink-bottle pore model. The remarkable linearity
of these plots indicates that a strong tendency exists in bio-
materials toward a power law type distribution function for pore-
size. Thus, the power law equation (2.3.16) when expressed in its
logarithmic form:

In m2 = In K + y In (R) [4.1.13]

1
yields an empirical straight line with a SIOpe y, and an intercept
of magnitude 1n K1.

In Table 4.1.3, the parameters Y and K1, as determined from
the pore-size distribution plots, are summarized. These values were
obtained from least square straight line approximations of the data
plotted in figures 4.1 through 4.4.

In spite of the elaborate refinements of the Cranston and

Inkley method, there exists (as with the other methods) no generally

accepted accuracy criteria for evaluating pore-size distributions so

90

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1n [$2(R) = dN(R)/dRJ

97

 

'7
' :L: o
«r-
dl- o
q- o
4..
J-
43 D
O
‘ .A o
' f
o o o A
t o
' A
4. 44
i A0
4; ° .Jf‘
t

)

0

6
-—--+- -+-»+~H+++n—
b. n
D»o
a
n
T

-:L
(3

A6 Figure 4.1 Pore-size distribution
plot for corn at the tempera-
tures: 4°C, 15.5°C, 30°C.

Q3 Based on the desorption data of
Rodriguez-Arias (1956).

w~+-+~+~+ H-H+
m 0

'érl o oo 4‘
.A
on
.13 A °
T 9°“
+
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£5.

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nnnt: nAn-rivn 1.- in [A 0‘)

' i1.‘.j.fll|.fv: :~
1

:¢2(R) = dN(R)/dRJ

In

98

6. 54.
—+ W'WWW‘HW
D
O
U o D 0
a
pm
0 ‘9
ON» a
p o

0
l8 l2;
0° .3

n o
30 C
A A-—-
A
no °
‘ 0
H3
0
° Figure 4.2. Pore-size distribution
‘P plot for cotton at isotherm
9
A. temperatures: 10°C, 20°C,
a
Q 30°C. Based on the adsorption
A
o a data of Urquart and Williams (1924).
‘II A o
l0 '1: 5,4“
I: ”.A
‘D a
-- 8'0
#- Ago. 0
o
.4 O
n
3 4

 

’IZ. .. ----.. . - - .. +4-} 1 .4
no 33+H+1 r ffiHH—r + 1344 s '

PORE RADIUS — 1n iR(A°)J

I67

-m..-+n_+..+—+_T_f-m

H3

-—+++1+'r++----+-—'r—~t—+~++—t+r—

52.
~ +~+~+4+++H9——- +

+ ~+—+—++—++H—--+~

5.

Ié'gl-H—Hw 1
I03

99

u
o
a
a
o
A
u
n
n o
o
A
A
“A
o
8
a
20°C
A. ° . °-_-
0 a
60°C
a o n—
n
u 0
100°C
0 _
A
A
8° A
A .0A Figure 4.3. Pore-size distribution
0A plot for wood at isotherm
0
temperatures: 20°C, 60°C, 100°C.
0 Based on the adsorption data of
an OA Lewis (1921).
9e. A
J’
nI:
0A
A D
o
a A
%o 0
4% 8
0°
0 A
+- i—vwggHH-aw w» +H+l I s . i 4
l l() L()

PORE RADIUS - 1n LR(A°)J

742m) = dN(R)/dR]

b—

1n

IO

4.
-r ‘llllll

- +~++~H+++--~ +-

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+ -+-++++H— ~

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.9-

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a...

5.;
N

6

0
l
+

100

go
A A
o
o
D O
o£4
a”
o
o
A 10°C
o __
A o
n A
22.2°C
a _
° 0
.A
o 37. C
Av——-°7o
G
.A
o
n
0 A Figure 4.4. Pore-size distribution
a
n A plot for pre-cooked beef powder
0 freeze-dried at 105°F platen
a An temperature. 10°C isotherm,
oo
n on 22.2°C isotherm, 37.7°C isotherm.
°o 0 Based on adsorption isotherms
0
AA obtained in the present study.
A
6'
o
o
48.
a A A
'5; .A
o A
a
a?

m9

D

~I+H+H+~+ -—--+ +
LO

T
.L

fiW—f

|PORE RADIUS - 1n IR(A° I)J

Table 4.1.3 Summary of the characteristic parameters of pore structure
as determined from pore-size distribution plots.

 

 

 

 

 

PRODUCT Temp Y K1 0 §
from from = 4 + Y = 4/3 0K1
°C pore-size pore-size EQN. EQN
distribution distribution '
plots plots (2.3.20) (2.3.19)
4.0 -4.636 .335 -0.636 1.404
CORN 15.5 -5.015 .237 -1.015 5.182
30.0 -5.059 .853 -1.059 7.763
10.0 -4.394 .049 o0.394 0.207
COTTON 20.0 -4.560 .214 -0.560 0.897
30.0 -4.603 .238 -0.603 0.995
20.0 ~4.416 .0934 -0.416 0.391
WOOD 60.0 ~4.555 .2053 -0.555 0.860
100.00 -5.202 .448 -1.202 18.634
PRE-COOKED 10.0 -4.344 .119 -0.344 0.497
FREEZE-DRIED
BEEF 22.2 ~4.599 .487 -0.599 2.039
POWDER
37.7 -4.876 0.756 -0.876 3.167

 

101

calculated (Everett, 1958). It seems apprOpriate that until such
criteria are established so that the magnitude of the various
quantities involved can be ascertained in absolute terms, pore-size
distribution plots of the type represented in figures 4.1 through

4.4 should be considered as having qualitative but not absolute
significance. Thus, the pore-size distribution parameters - Y and

K1 - of Table 4.1.3, cannot be expected to be quantitatively accurate.
Nevertheless, the exponential nature of the pore-size distribution
function,‘Pz(R), is conclusively established from a qualitative stand-
point, because of the well defined linear tendency of the log-log
plots. In addition, the following observations regarding the para-

meters y and K are qualitatively discernible from Table 4.1.3. :

l
1. As characteristic parameters for the pore structure

of the sorbent, Y and K are unique for each sorbent

1
system.

2. Both Y and K are clearly temperature sensitive. This

1
implies that the pore structure of the sorbent changes
with isotherm temperature. While it is as yet premature
to determine from a priori considerations, the exact
nature of this change, it appears reasonable, in view of
the effects of temperature on swelling forces, that the
pores could undergo a certain amount of deformation with
changing isotherm temperature. Although a purely tem-

perature-induced expansion or contraction cannot reasonably

be expected to alter the pore structure appreciably,

102

Hammerle (1968) did show that the so-callcd hydro-
expansion in bio—materials is considerably temperature
sensitive.

In closure, it has been shown in this section that (a) the
pores of biological materials are best modeled as inter-connected
Spheroidal ink-bottles; (b) the pore-size distributions of these
products are well described by an analytical function of the power
law type; (c) while the present state of the art does not allow for
precise quantification of the pore-size distribution parameters,
useful insights have been gained into their qualitative characters.
These results have important application in the theoretical considera-

tion of moisture transport in biological materials.

4.2 Experimental Results

Adsorption and desorption data for pre-cooked freeze-dried
beef powder obtained in this investigation are summarized in Tables
4.2.1 and 4.2.2, respectively. The specific adsorbed mass is
recorded on the dry basis (d.b.). Each value represents an average
of three independent tests. The isotherm data are reproduced in
Figures 4.5 and 4.6.

The plots show the traditional sigmoid shape expected of
porous water-binding biological materials. They are temperature
dependent, showing decreasing adsorption with rising isotherm tem-
perature. A distinctively pronounced hysteretic behavior is apparent

in figures 4.7(a) and (b). While the desorption curves show a

103

 

< A D S O R P T I O N ;>

 

 

 

 

in (d.b.)

P/Po ‘1919 22.2°C 37.7°C
.05 .047 .03 .0136
.10 .070 .045 .0209
.20 .095 .062 .0309
.30 .110 .075 .0391
.40 .122 .087 .0500
.50 .135 .100 .0591
.60 .155 .125 .0791
.70 .191 .163 .102
.80 .260 .230 .141
.90 .380 .33 .200
.95 .450 .405 .241
-98 .049 .465 .27

1-00 .505 .490 .286

Table 4.2.1. Equilibrium moisture content of pre-cooked freeze-dried ground
beef.
platen temperature = 105°F

100-150 milligrams

9.8%

sample size

Fat content

104

105

 

 

 

 

.50-
.45.
Io’c
40* 22 2 C
2 .35.
6
'2 .30.
E
2 377.6
8
.25L
UJ
(I
:3
5', 20+
5
2
.154
.10;
.05»

 

 

Figure 4.5. Adsorption isotherms for pre-cooked hoof powdvr

freeze-dried at 105°F platen temporaturv. (10”C - 37.7““)

 

 

 

< D E S O R P T I O N -“—>

 

 

in (d.b.)
P/Po '19:g 22.2°C 37.7°C
0.05 .065 .04 .0427
0.10 .088 .057 .0500
0.20 .110 .075 .0546
0.30 .123 .085 .0682
0.40 .137 .097 .0846
0.50 .156 .113 .109
0.60 .191 .143 .138
0.70 .260 .200 .175
0.80 .425 .280 .213
0.90 .487 .465 .254
0.95 .498 .4755 .273
1.00 .505 .490 .286

 

Table 4.2.2.

Equilibrium.moisture content of pre-cooked freeze-dried ground beef.

platen temperature = 105°F
sample size = 100-150 milligrams
fat content = 9.8%

106

107

 

.s..?

 

 

 

 

.451-
.40-
a
15 .351-
10°C
’5
u: 30% 22.2°C
*z'
o 37.7 c
t)
25+
111
a:
.2
6 .20”
2
.I5»
.I04
.05

 

 

0.0 .l .2 .3 .4 .5 .6 .7 .8 .9 1.0
PIP.

Figure 4.6. Desorption isotherms for pro-cooked hoof powder freeze-

dried at 105°F platen temperature. (10°C - 37.7°C)

108

 

o——— Adsorption

.5()r

 

°-—- Desorption

(db)

CONTENT

MOISTURE

 

 

00.1.2.3; 5.8.7.8910
PH;

Figure 4.7(a). Plot showing sorption hysteresis in pru-conkod

freeze-dried beef powder.

109

 

 

.50~
u/
q’
.454 I,
I
I
.40- I
.2 I
‘-’ I . I
3 .35? desorpnon—jf
i
1— i I
5 odsovplion
8 ‘ 1’
g /
w .25;- /
g I
t- . /
'fl . /
2 i /
. /
/ '
ISI- 15/
g /
/ .
; ,1
101' In/ °
I ’,n" W
2 .°"
? f// .
05;. J
}/
II

 

 

Figure 4.7(b). Plot showing sorption hysteresis in pre-cooked freeze-
. dried beef powder (22.2°C isotherm)

flattening tendency near saturation vapor pressure - suggesting a
type IV classification of the Brunauer type - the adsorption curves
are strictly sigmoid-type II, with the "B" point fairly well defined.
These salient features sUpport - as do most sorption isotherms
of biological materials - the micrOporous assumption so fundamental
to the deve10pment here presented. While a number of biological
materials will be used to test the theory as a means of demonstrating
its SCOpe, the data presented in this section will in the main, be

employed to test the generalized model.

110

4.3 Verification of the Derived
Density Function for Sorbed Water
It is assumed in the interest of simplicity that the B-E-T

surface area (S ) of a given sorbent can be stretched out into

B-E-T
an equivalent flat surface. The mono-layer and subsequent multi-
molecular layers are assumed to form a matrix, see Figure 4.8. The
individual molecules file into vertical columns. The molecules of
each column are forced by virtue of their presence in that column

to conform to the cross-sectional area (oh, [A°]2) of the first (mono-
layer) molecule of that column. The area, om, which one adsorbed
molecule would occupy in the completed mono-layer depends among

other things, on (a) the sorbate surface tension and (b) the intensity
of the sorbent-sorbate interaction represented by the so-called
spreading pressure at the surface. Because of the recognized com-
pressive effect of this surface force together with its fundamental
polarizability, the adsorbed water molecule is visualized in Figure
4.8 not as a spheroid, but as an ellipsoid, the degree of ”flattening”
of which is prOportional to the gross compressive force acting on

the molecule. According to the potential theory, this flattening

is expected to be highest in the mono-layer and diminishes progres-
sively outward in the succeeding layers.

With the assumptions enunciated above, the multi-layer thick-

ness t, can be defined as:

t(A°) = va/sBET [4.3.1]

111

I

T
t
_I_

 

Figure 4.8.

 

To

 

 

1 I T2 TI <7'2"“(1'0
3‘1

specific surface strvtvhnd our

 

 

 

 

 

Vertical section of stretched out specific surface, showing

the multi-layer matrix. Each adsorbed water molecule is
visualized as a flattened elipsoid reflecting Ihv degree of com—
pression of the molecule as a result of intermolecular forces

of attraction and polarizability effects. Each layer of
molecules is subject to a different but uniform compression.

The first layer being the most compressed and the tOp laver

the least compressed.

112

The average thickness of a single layer of adsorbed molecules, T,

previously defined by equation (2.3.7), becomes:

 

va Ma
T = t/n = = ————-—' [4-3-2]
“ “SBET p “SBET

where n is the number molecular layers adsorbed and p is the mean
density corresponding to n. In accordance with equation (4.3.2),
the average thickness at saturation vapor pressure, To is given by:

M

a
T (A°) =-—————-— [4.3.3]
0 ponSBET

Combining equations (4.3.2) and (4.3.3) yields:

T
E a: ;2. L4.3.4j
00 11

Equation (4.3.4) implies that the density ratio and the
thickness ratio should be approximately equal. This furnishes an
approximate verification of the density function derived, tabulated
and graphed earlier in chapter II. In Tables 4.3.1 through 4.3.4,
the above verification is performed for a number of biological
products. A comparison of columns 5 and 9 of each table shows a
remarkable agreement between the ratios.

Three things need to be pointed out immediately regarding
Tables 4.3.1 through 4.3.4. First, the saturation density, p0, is
assumed to be 1 cc/gm, corresponding to the normal bulk liquid
density of water. Secondly, since the B-E-T surface area, SBET’

determined from H20 isotherms is known to vary with temperature

113

Table 4.3.1.

Isotherm temperature

10°C S

Product = Pre-cooked freeze-dried beef powder.
2

Verification table for the density of sorbed water.

 

 

1 2 3 * 5 6 7 8 9
17/130 Ma _;5_ p/Do Va t Tn :2
m ' Mtge) ,, :61 a :5 = t/n Tn
m = .091 p 5881‘

.05 .047 .52
.10 .070 .769

1.360 .0669 2.079 2.079 1.351
.20 .095 1.044
.30 .110 1.210
.40 .122 1.341
.50 .135 1.484
.60 .155 1.703

1.316 .1380 4.320 2.160 1.300
.70 .191 2.10

1.164 .2340 7.200 2.400 1.170
.80 .260 2.86

1.126 .3230 10.000 2.500 1.124
.90 .380 4.18
.95 .450 4.95 1.062 .4280 13.250 2.650 1.060

1.00 .505 5.55 1.000 .5050 16.860 2.810 1.000

 

* In these calculations, Po

114

1 cc/gm., the normal density of bulk water

Table 4.3.2.

Verification table for the density of sorbed water.

 

 

Product Soda boiled cotton (Urquhart and Williams, 1924)
Isotherm temperature = 10°C. SBET = 126(m2)
1 2 3 4 * 5 6 7 8 9
P”. Ma 3’} n 9’9. V. t Tn ;2
m n
g .036 a ”(g-3°) va/SBET
.05 .017 .48
.10 .023 .644
.20 .032 .896
1 1.220 .0295 2.300 2.300 1.217
.30 .041 1.15
.40 .049 1.372
.50 .058 1.624
.60 .068 1.904
2 1.120 .0642 5.000 2.500 1.120
.70 .080 2.240
.80 .099 2.970
3 1.089 .0990 7.800 2.600 1.080
.90 .132 3.697
4 1.035 .1390 10.800 2.700 1.037
.95 .164 4.593
5 1.008 .1780 13.715 2.743 1.02
1.00 .240 6.722
6 1.000 .2400 16.980 2.830 1.00

 

* In these calculations

po

‘3 1 cc/gm., the normal

115

density of bulk water.

Table 4.3.3.

Verification table for the density of sorbed water.

 

 

Product = Spruce wood (Stamm, 1938)
Isotherm temperature = 20°C. SBET = 177(m2).
1 2 3 4 * 5 6 7 8 9
1”? “a 3’} n 9’9. a t n :9
m = “*(950) = M /0 _ Tn
Km 2 .05 A“ a v /s - t/n
a BET

.05 .013 .26
.10 .025 .50
.20 .025 .99

1 1.15 .0434 2.45 2.45 .14
.30 .063 1.26
.40 .080 1.6
.50 .095 1.9

2 1.10 .0900 5.2 2.6 .08
.60 .113 2.26
.70 .135 2.7

3 1.067 .1400 7.98 2.66 .05
.80 .165 3.3

4 1.042 .1910 10.8 2.70 .04
.90 .225 4.5

5 1.035 .2410
.95 .275 5.5

6 1.000 .3000 16.8 2.8 .00

1.00 .340 6.8
7 1.000 .3400 19.6 2.8 .00

 

* In these calculations, 00

a 1 cc/gm., the normal density of bulk water.

116

Table 4.3.4. Verification table for the density of sorbed water.

Product = Corn kernels (Rodriguez-Arias, 1956).

 

 

Isotherm temperature = 4°C. SBET = 281(m2).
l 2 3 4 * 5 6 7 8 9
"P. M. 3.;— n 9’92“ V. t T. _:_9_
m = “*(Zfi-O) . M [p -= t/n n
Xm=.0865 a
va/SBET

.10 .065 .751

1.295 .0667 2.37 2.37 1.31
.20 .0925 1.07
.30 .1100 1.271
.40 .1275 1.473
.50 .1425 1.647
.60 .1625 1.878

1.145 .1510 5.370 2.685 1.150
.70 .1825 2.109
.80 .2125 2.456

1.043 .2480 9.00 3.00 1.02
.90 .2700 3.121
.95 .3000 3.468

1.000 .3460 12.4 3.1 1.00

1.00 .3500 4.046

* In these calculations, Po

117

211 cc/gm., the normal density of

bulk water.

(because the mono-layer capacity, vm, varies with isotherm tempera-
ture), the absolute Specific surface area of the sorbent is not

known. Even though it is customary to use a mean Specific surface
based on several isotherms, this average value is questionable when
wide variations in vm values occur with isotherm temperature.

Because of the doubt in the SBET value used, the t (column 7) and

Tn (column 8) values of these tables do not have absolute but do

have relative significance. Consequently, the ratio TO/Tn (column 9),

is unaffected by the degree of accuracy of S Finally, the veri-

BET’
fication performed is insufficient as a test of the validity of the
derived density function. At best, it only adds a certain margin of
confidence to the derived function. The absolute test for the den-
sity relation can be done experimentally by comparing gravimetric
sorption data with the corresponding volumetric data for the sorbate,
water, on the same sorbent system. There exists in current litera-

ture a glaring lack of sufficient data along these lines and thus

such a test is not possible at the present time.

118

4.4 Verification of the Derived
Isotherm Equation

 

 

4.4 a. Theoretical Isotherms

 

 

For illustrative purposes, two sets of isotherms calculated
with the volumetric equation (2.3.22) are presented in Figure 4.9.
The Specific adsorbed volume, Va’ is plotted in its reduced form by
dividing each calculated isotherm point by the corresponding calcu-
lated adsorbed volume at P = 0.98 Po' This reduction of all isotherm
heights to l or 100% allows for a better comparison of characteristic
forms of adsorption curves and facilitates the eventual matching of
theoretical with experimental isotherms. The adsorbed volume at a
relative humidity of 98% is chosen as a relatively well defined
point. In contrast, considerable doubt still exists among investiga-
tors (see for example, Bangham and Sever, 1925) as to whether equili-
brium is ever truly obtained at saturation. As a result, the adsorbed
volume at 100% relative humidity is not well defined. In these plots,
the volumetric isotherm expression is chosen in preference to its
gravimetric equivalent because it lends itself well for consideration
on a general basis. The isotherm temperature used for the calcula-
tion of Figure 4.9 is 10°C. Two sets of theoretical curves are pre-
sented to illustrate the effect of the Pm value on the isotherm shape.

The curves are obviously very characteristic showing (a) well
defined initial "knees" at moderately positive values of n [the
knees are clearly better defined at lower values of Pm, see Figures
4.9 a and 4.9 b], (b) certain linear parts, and (c) distinctive

individual starting angles of the isotherms. All regular types

119

 

 

.0 E “O E
momwm u oooa u e mAH.oV u a momma n oooa u e mamoo.v u a
.ANN.m.NV .cem cuss .ANN.m.~V .cem suns
“63350.30 mechanic: Hmoauowooafl. .Anvoé 35m: woumasoamo mEmnuOmH Hmoeuwuomne .Amvmé muawwm
.68\a
. o... m. m. N m. n. w. . N. e.
o o . . _ a . . u . .

 

 

   
     

038

 

 

 

 

 

0.0209029“?an

":0.

<2 a e “:“Q'Q s e 9:

 

 

[95 "WM-707w ow mamas HUHJSIOW ason

 

(D

[‘5 '"d’djow/ 0w mamas 3301570717 03700038

120

of isotherms of the Brunauer classification are obtained merely

by the variation of n. This finding is significant in that it
vindicates the leading premise of the generalized theory that the
three basic concepts in adsorption are complementary and thus can

be unified into a self-contained isotherm equation. The present
results must also be viewed in their broader context as strengthening
considerably the significance of a purely power law distribution

concept for the pore-size.

4.4 b. .3 as the Primary Characteristic Parameter of the Pore Structure

 

 

 

As the primary shape criterion for the set of theoretical
isotherms represented in Figure 4.9, n must be viewed as the character-
istic value for the representation of the quantitative behavior of a
porous sorbent at any relative vapor pressure. In the range n < -l,
the theoretical isotherms are clearly of the Langmuir-type. Thus,
this range of n values may be associated with materials characterized
by exceedingly narrow pores, probably not more than two molecular
diameters in width. These are the so-called micrOpores. Sigmoid
type II isotherms are defined in the range -1 S 0 S + 1. Accordingly,
pores so characterized fall within the so-called intermediate and
lower macrOporous range (Brunauer, 1945; Gregg and Sing, 1967). n
values in the range n > + 1 define type III isotherms and in conse-
quence may be associated with large pores of the upper macrOporous
range. It is well recognized that the isotherms of most biological
products are sigmoid. Consequently, it would seem logical that the
main focus of the present investigation should be restricted to the

intermediate range, -1 S n S + l.

121

As defined by equation (2.3.20),

n = 4 + Y [4.4.1]
Ordinarily, this relationship would have provided a convenient com-
putational formula for 0. However, it has been previously argued
that the pore-size distribution parameter, Y, is not defined with
sufficient quantitative accuracy to warrant this usage. This short-
coming notwithstanding, the definition of n as a function of Y in
equation (4.4.1) provides a preliminary insight into its qualitative
character. It is unique and specific for each sorbent system. It
is temperature sensitive, assuming progressively changing values as
the isotherm temperature increases [see table 4.1.3]. As a conse-
quence of the previously discussed relationship between 0 and the
characteristic isotherm types, it is apparent that an appreciable
temperature-induced change does take place in the sorbent pore struc-
ture. Even though the exact nature of this change cannot be specified
at this point, it is obvious that in applying the derived isotherm
equation to specific products, 0 must not only be defined for each

product, but needs also to be defined for each temperature.

4.4 c §_2_ a Secondary Characteristic Parameter g; the Pore Structure
According to equation (2.3.19),

. 4
7; == '5 TI Kl [4.4.2]

g is thus a quantity combining the pore-size distribution parameter,

K , with the factor of pore geometry, (% W). Since 5 is dependent

1

on K its value is unique for each sorbent system and it is also

1’

temperature sensitive (see Table 4.1.3). Because K1 is not accurately

122

defined from a quantitative standpoint, the exact value of 5 cannot
be determined from equation (4.4.2). A progression from smaller

to larger values of 5 is evident as the isotherm temperature is
increased (Table 4.1.3). It would seem therefore, that a decreasing
n value combined with an increasing § value implies a change in the
pore structure. This change means that the effective size of each
associated pore is affected by a combined hydro and thermo expansion

or contraction of the solid component of the sorbing tissue.

4.4 d. Prediction of the Adsorption Isotherms of Certain Biological

 

 

 

 

Materials with the Generalized Isotherm Equation

Since it has not been possible to determine reliable values
of n and g from a priori considerations, these quantities will be
defined empirically to satisfy qualitative criteria stipulated from
earlier theoretical considerations. To determine n for a Specific
product and temperature, the empirical isotherm in its reduced volu-
metric form is superposed on a set of theoretical isotherms such as
shown in Figure 4.9. The best matching n and Pm values are thus
selected. Based on these values of n and Pm, the function [Ejfi—fiflj
is defined for each relative vapor pressure. Combining the approxi-
mate density term, pl, with the above function, the best fitting
average 5 value, E, is defined. It is now possible to define for
each P/Po value, the first approximation of the specific adsorbed

. ._ Zn.- AT]
mass, defined by p1.§ [__0_—_

J. The discrepancy between this approxi—
mate value and its empirical counterpart is minimized by the careful

correction of the density term. This gives rise to a new set of

density values, designated as oz.

123

In Tables 4.4.1 through 4.4.5 the relevant parameters for
(a) Pre-cooked freeze-dried beef powder (10°C and 37.7°C)
(b) Raw freeze-dried beef slices (10°C and 40°C)
(c) Whole corn kernels (22°C and 50°C)
(d) Wood cellulose (20°C and 60°C) and

(e) Soda boiled cotton (10°C and 30°C), are shown.
zn- in
0

tion possible for the product and temperature under consideration.

The quantity E p2[ J in each table represents the best predic-
Predicted isotherms based on these tables are represented in Figures
4.10 through 4.14. These plots together with the standard error
estimates shown on each table are, for most of the points, within
limits of the magnitude of error ordinarily associated with adsorp-
tion experiments. The error estimates are also within the limits of
variability to be expected on account of the highly Specific character
of biological materials.

The Corrected Density,pzi In rationalizing the corrected density
term, p2, it should be kept in mind that the density function derived
in this work, on which p1 is based, is acceptable only as a first
estimate. In so far as the isosteric heat values determined by the
combined usage of the Clausius-Clapeyron equation and the SIOpes of
the so-called adsorption isostere plots are, at best, crude estimates
of the actual quantities (Ross and Olivier, 1964; Gregg and Sing,
1967; Adamson, 1967), the ratio, (AH/AH°), of the derived density
function is only as good as the isosteric heat values used. The
relatively small correction factors needed to up-grade O1 to 02, in
most cases, are justified on the basis of the associated isosteric

heat difficiencies discussed above. Moreover, p2 is seen to conform

124

Table 4.4.1(a).

Calculation of theoretical isotherm.

Product

Isotherm temperature = 10°C

pre-cooked freeze~dried beef powder

 

 

 

 

[211-110]
._ 0 0
Ma 9 ‘E 2 : 3.01 p SDZEZ ”x
P/Po (Expt1.) 1 .454 m ' 2 Error
.1 .070 1.360 " .0957 1.612 .070 .000
.2 .095 1.360 " .1450 1.451 .095 .000
.3 .110 1.316 " .1870 1.302 .110 .000
.4 .122 1.316 " .2290 1.160 .132 -.010
.5 .135 1.316 " .2750 1.160 .144 -.009
.6 .155 1.164 " .3260 1.160 .171 -.016
.7 .191 1.164 " .3900 1.160 .205 -.014
.8 .260 1.164 " .4790 1.160 .252 -.008
.9 .380 1.126 " .6350 1.160 .334 +.056
95 .450 1.062 " .7990 1.160 .420 +.030
.98 .483 1.00 " 1.0400 1.023 .483 .000

 

Standard Error Estimate = i.0064

125

Table 4.4.l(b).

Calculation of theoretical isotherm.

 

 

 

 

Product = pre-cooked freeze-dried beef powder
Isotherm temperature = 37.7°C
zn_xn
E n J
7 .. 0 0
Ma 9 g '0. -= 0.2 50.,[2 -1 J
P/Po (Expt1.) 1 81.26 PM ~ 0.1 02 Error
.1 .0209 1.360 .0089 1.844 .020 +.009
.2 .0309 1.360 .0165 1.532 .031 .000
.3 .0391 1.316 .0240 1.292 .039 +.000
.4 .0500 1.316 .0319 1.248 .049 +.001
.5 .0591 1.316 .0407 1.248 .063 -.004
.6 .0791 1.164 .0512 1.248 .079 .000
.7 .1020 1.164 .0648 1.248 .100 +.002
.8 .1410 1.164 .0844 1.248 .132 +.009
.9 .2000 1.126 .1210 1.248 .190 +.01
.95 .2410 1.062 .1620 1.248 .254 -.013
.98 .2860 1.000 .2260 1.000 .284 +.002

 

Standard Error Estimate - i.0064

126

127

 

 

 

.50 1 T I I 1 I l I T
A EXPERIMENTAL IO' 8
.454 ..
9 EXPERIMENTAL 37.720
.40L 5
.357 '1
3
3
.30~ a
"2' THEORETICAL Is0TI-IERM 9
111 7) 8 0.1 I
5 P . 0I - I
_ m , .. q
8 '25 6 . .454 [I
m
s /
I- .20- A p n
‘3 I
0 I
2 A I
'5”‘ I q
2. 0
- /
A /
.IO~ ._ ff 4
I
55/ THEORETICAL
2. [‘4
It ISOTHERM
.(753’ ‘,a0" 17 a (3.2: "
I”
.r‘)’ F;.‘ Ctl
,I" E . l.383

 

.00 1 l 1 1 l 1 1 l
.l .2 .3 .4 .5 .6 .7 .8 .9 LC

P/P.

Figure 4.10. Comparison of experimental adsorption isotherms with
calculated isotherms for pre-cooked freeze-dried

beef powder.

Table 4.4.2(a).

Calculation of theoretical isotherm.

 

 

 

 

Product = raw freeze-dried beef slices
(Data taken from Saravacos, 1965)
Isotherm temperature 10°C.
[Zn-1"]

11 E = n - 2n- 1”

a p n = 0.01 §pz[ n J
P/Po (Expt1.) 1 .1047 Pm = 0.10 92 Error
.1 .030 1.360 " .214 1.382 .030 .000
.2 .060 1.360 " .390 1.382 .055 +.005
.3 .080 1.316 " .556 1.382 .080 .000
.4 .100 1.316 .724 1.351 .101 -.001
.5 .120 1.316 .906 1.294 .121 -.001
.6 .142 1.164 1.110 1.241 .143 -.001
.7 .170 1.164 1.370 1.204 .172 -.002
.8 .200 1.164 1.710 1.132 .202 -.002
.9 .250 1.126 2.280 1.064 .253 -.003
.95 ‘.310 1.062 2.860 1.050 .313 -.003
.98 .375 1.00 " 3.630 1.00 .38 -.005

 

Standard Error Estimate = 10.0027

128

Table 4.4.2(b). Calculation of theoretical isotherm.

raw freeze-dried beef slices
(Saravacos, 1965)

Product =

Isotherm temperature = 40°C

 

 

 

M
‘E [2:Eflih (Theorztical)

M 0 —. Zn_xn

a = 0 - .04 =§Pz[ 0 J
P/Po (Ethl') p1 .136 P111 3 '10 p2 Error
.1 .027 1.36 .126 1.600 .027 .000
.2 .050 1.36 .230 1.600 .049 +.001
.3 .065 1.316 .329 1.520 .066 -.001
.4 .080 1.316 .430 1.380 .080 .000
.5 .090 1.316 .539 1.270 .092 -.002
.6 .100 1.164 .666 1.145 .103 -.003
.7 .120 1.164 .821 1.145 .127 -.007
.8 .155 1.164 1.030 1.145 .160 -.005
.9 .215 1.126 1.400 1.145 .217 -.002
.95 .250 1.062 1.770 1.076 .258 -.008
.98 .300 1.000 2.280 1.000 .310 -.010

 

Standard Error Estimate

129

= $0.

0048

 

 

 

 

40 l l 1 I I l | l I
9 EXPERIMENTAL IO° C
} (Saravacos,
.35 - A EXPERIMENTAL 40°C 1965)
.30 e 4.
A THEORETICAL ISOTHERM L
- ——>
"f 97 8 .0I I
3 pm . .l I
,_ '25 _ 6 = .I047 f”
2 /
m
g I‘
O .20 '- / ‘4
0 I
In /
5 .l5 - f“ 4
53 /
g xe‘
/’
_ ,A _
.IO ’4’
,A"u-THEORETICAL
. //A’ ISOTHE RM
L . 4
1355 3’45 I" ,cyq
'/ 8": J
6 - .I360
13c, 1 1 1 l J l I l J
.l .2 .3 .4 .5 .6 .7 .8 .9 LG
P/Po

Figure 4.11.

Comparison of experimental adsorption isotherms
with calculated adsorption isotherms for raw

freeze-dried beef in slices.

130

Table 4.4.3(a). Calculation of theoretical isotherm.

 

 

 

 

 

Product = whole corn kernels (Chung and Pfost, 1967)
Isotherm temperature = 22°C
M g [7‘7“an
a n - 0 x0
= n = -. 1 §pz[z .3 J
P/PO (Expt1.) 91 .01415 Pm = .008 92 Error
.1 .055 1.295 .0300 1.380 .058 -.003
.2 .072 1.295 .0436 1.215 .074 -.002
.3 .090 1.145 .0548 1.215 .094 -.004
.4 .105 1.145 .0654 1.188 .109 -.004
.5 .118 1.145 .0763 1.129 .121 -.003
.6 .132 1.145 .0882 1.104 .137 -.005
.7 .150 1.043 .1020 1.104 .159 -.004
.8 .182 1.043 .1200 1.104 .187 -.005
.9 .232 1.000 .1490 1.104 .232 0
.95 .266 1.000 .1790 1.081 .273 -.007
.98 .286 1.000 .2080 1.000 .294 -.008
Standard Error Estimate = £0.0046

131

Table 4.4.3(b). Calculation of theoretical isotherm.
Product = whole corn kernels (Chung & Pfost, 1967)

Isotherm temperature = 50°C

 

 

 

 

M p " [211'an

a 1 5 n ._ 27140

n =- - .08 502E771
P/Po (Expt1.) =.0184 Pm * .008 92 Error
.1 .050 2.08 1.400 .053 -.003
.2 .068 3.04 1.240 .068 .000
.3 .080 3.83 1.146 .080 .000
.4 .092 4.58 1.106 .092 .000
.5 .105 5.36 1.106 .108 -.003
.6 .120 6.21 1.106 .126 -.006
.7 .141 7.22 1.106 .145 -.004
.8 .179 8.54 1.106 .173 +.006
.9 .230 10.70 1.106 .216 +.024
95 .260 12.60 1.106 .255 +.005
98 .287 15.20 1.000 .279 -.001

Standard Error Estimate = 10.0079

132

 

 

 

 

-30 I r I I r T I I I
.23- 0 EXPERIMENTAL 22°C
} (Chung & Pfost, 2
A EXPERIMENTAL 50°C 1967) I
.26- p-
.24— I _
"I
.22— f —
., I
4120- / 4
g THEORETICAL ISOTHERM /
F ,l 8r- ” "0.1 A .—
a PMI .008 fl
'5 .I8e 6 - .OI42 / a
o ‘/
6)
MP /° —
I: /
:5 _ I
*- .12 /A "
Q
o ’ 1’
2 4'0 _. y/ —I
A /
.081- .. ,fi’ THEORETICAL ISOTHERM ‘
/°’ 1; . -.08
'06 - a / P” I .008 _
6 - .OI84
.04 - I —
.02 ’ ..
l I I l I I l I I
.I .2 .3 .4 .5 .6 .7 .8 .9 10

Figure 4.12.

Comparison of experimental adsorption isotherms with
calculated isotherms for corn (based on the empirical

data of Chung and Pfost, 1967).

15')

9 v .5)...

.l

Table 4.4.4(a). Calculation of theoretical isotherm.
Product = wood (Lewis, 1921)

Isotherm temperature = 20°C

 

 

 

.. [2";an
1“a E 1’1 ' - . 10 E D 2 [£212.93]

P/PO (Expt1.) Pl .0172 PIn - 0.10 92 Error
.1 .0250 1.153 1.33 1.22 .027 -.002
.2 .0495 1.153 2.40 1.22 .050 .000
.3 .0630 1.101 3.38 1.123 .065 -.002
.4 .0800 1.101 4.36 1.094 .081 -.001
.5 .0950 1.101 5.39 1.050 .097 -.002
.6 .1130 1.067 6.53 1.050 .116 -.003
.7 .1350 1.067 7.90 1.030 .139 -.004
.8 .1650 1.042 9.68 1.030 .171 -.006
.9 .2250 1.024 12.50 1.030 .221 +.004
.95 .2750 1.0065 15.10 1.030 .267 +.008
.98 .3080 1.000 18.30 1.000 .314 -.006

 

Standard Error Estimate 8 $0.00415

134

Calculation of theoretical isotherm.

Product = wood (Lewis, 1921)

Table 4.4.4(b).

Isotherm temperature = 60°C

 

 

 

 

M [221“]

a E _ 2 1’1_ x“

n - -o.o7 ngE n J
p/ro (Exptl.) 01 -.020 Pm - 0.10 Oz Error
.1 .020 1.153 .783 1.249 .020 .000
.2 .035 1.153 1.410 1.224 .035 .000
.3 .045 1.101 2.000 1.145 .046 -.001
.4 .060 1.101 2.590 1.145 .060 .000
.5 .070 1.101 3.210 1.145 .075 -.005
.6 .089 1.067 3.910 1.145 .091 -.002
.7 .110 1.067 4.750 1.145 .111 -.001
.8 .140 1.042 5.870 1.145 .136 +.004
.9 .188 1.024 7.670 1.145 .178 +.010
.95 .225 1.0065 9.390 1.145 .218 +.oo7
.98 .246 1.000 11.600 1.050 .245 +.001

 

Standard Error Estimate = £0.00423

135

136

 

 

 

 

.32 I I I r I I I l I
.30- A EXPERIMENTAL 20°C —
1 (Lewis, 1921)
.23- 9 EXPERIMENTAL 80°C 5
26I— _.
.24— f.
3.22 — 7 -
'0'
F_.2O~- / —
z
m ,8_ 9 _
F. .
z /
8 16‘ A / _.
' *' THEORETICAL ISOTHERM /
m . , _
g .14- 7) 0.1 A J _.
,_ Pu, . 0.! /
g f - .OI72 /
o .‘2 " I. "
S ' /
.IOP , fl/ 1
.08 P a / _
,o’
. M/ THEORETICAL
~06 ' ’90’ ISOTHERM i
“ ° lgr’ ‘0 "'-.(77
.o .— fl, Pm' mo _
4 /
- 0.02
.02 - ,6’ 5 ~
I l I J I I l I l
.I .2 .3 4 .5 .6 .7 .8 .9 IO

Figure 4.13.

PIP.

Comparison of experimental adsorption isotherms with
calculated adsorption isotherms for wood cellulose

(based on the data by Lewis, 1921).

Table 4.4.5(a). Calculation of theoretical isotherm.
Product = cotton (Urquhart and Williams, 1924)

Isotherm temperature = 10°C

 

 

 

M. [Zn-1n]~

° 3' —~ zn—I"

n . 0.10 €92E n J
P/PO (Expt1.) p1 =.199 Pm . 0.01 92 Error
.1 .023 1.19 .0957 1.243 .023 .000
.2 .032 1.19 .145 1.122 .032 .000
.3 .041 1.072 .187 1.122 .041 .000
.4 .049 1.072 .229 1.093 .049 .000
.5 .058 1.072 .275 . 1.078 .058 .000
.6 .068 1.045 .326 1.078 .070 -.002
.7 .080 1.045 .390 1.045 .081 -.001
.8 .099 1.045 .479 1.045 .099 .000
.9 .132 1.013 .635 1.043 .131 +.001
.95 .164 1.001 .799 1.032 .164 .000
.98 .209 1.000 1.04 1.014 .209 .000

 

Standard Error Estimate = 40.000738

137

Table 4.4.5(b). Calculation of theoretical isotherm.
Product = cotton (Urquhart and Williams, 1924)

Isotherm temperature = 30°C

 

 

 

M ‘E [Zn-1n] Mé(Theoretical)

a 0 _. 20-x”

n - 0.20 §DZ['—Er-J
P/Po (Expt1.) p1 =.799 PIn - 0.01 02 Error
.1 .018 1.19 .0935 1.388 .018 .000
.2 .027 1.19 .141 1.311 .027 .000
.3 .035 1.072 .183 1.311 .035 .000
.4 .044 1.072 .224 1.311 .044 .000
.5 .052 1.072 .268 1.273 .052 .000
.6 .061 1.045 .319 1.273 .062 -.001
.7 .073 1.045 .381 1.221 .073 .000
.8 .091 1.045 .469 1.221 .092 -.001
.9 .121 1.013 .621 1.178 .121 .000
95 .147 1.001 .783 1.091 .147 .000
.98 .163 1.000 1.020 1.00 .185 .022

 

Standard Error Estimate - +0.0066 (based on 11 pts)
8 £0.000447 (based on 10 pts)

138

 

 

 

 

 

'24 T j T I I 1 I I I
.224 9 EXPERIMENTAL IO°C 61
I (Urquhart and
.20— A EXPERIMENTAL ao'c Williams, 1924) -
3 l
. .Is- '.
3 I
I- .IGP °A..I
z I
m ‘I .
5 .I4 e f‘ -
o THEORETICAL ISOTHERM . ,
0 17 - 0J0
.l2 - J
B .10 I- E ' 0.197 o l A
a ,5
O .08 +- 0 / _.
2 ’A/
° /
.06 - o ,{5 THEORETICAL -
g I’A’ ISOTHERM
-°4 _ z’ a; - 0.20 —
0’, . Pm- O.OI
°°2 " , . 6 cor/99 "
l l l l I 1 l 1 l '
.l .2 .3 .4 .5 .6 .7 .8 .9 LG
P/ P,

Figure 4.14. Comparison of experimental adsorption isotherms with
calculated adsorption isotherms for soda-boiled cotton

(data taken from Urquhart & Williams, 1924).

139

 

 

 

in every case, with the essential implications of the adsorption
potential theory. In showing consistently increasing density values
with rising isotherm temperatures, p2 reflects the correct tempera-
ture dependence of the isosteric heat of adsorption as evident in

the well known relation (Ross and Olivier, 1964):

Ans: = qdiff + RgT

where qdiff is the differential heat of adsorption.

It appears reasonable, in the absence of conclusive empirical
density values, to accept the O2 values on their demonstrated quali-
tative merit.

Empirical_g and E Parameters: A summary of the empirically deter-
mined values of n and E for corn, cotton, wood, raw freeze-dried
beef slices and pre-cooked freeze-dried beef powder is presented

in Table 4.4.6. The values are clearly product and temperature
dependent. However, when compared with their counterpart semi-
theoretically determined n and E values (Table 4.1.3), drastic dif-
ferences in magnitude and character are evident. These differences
once again emphasize the fundamentally qualitative character of the
pore-size distribution parameters, Y and K1, as determined.

While temperature induced-changes in the pore-structure
were generally predictable from the figures of Table 4.1.3, certain
interpretations deduced from those figures still need to be revised
or finalized. Since the empirical values of n and E (Table 4.4.6)
show increasing trends from smaller to larger values with rising

isotherm temperatures, a progression from smaller to larger pores

140

Table 4.4.6.

Summary of empirical n, and E parameters.

 

 

 

 

 

PRODUCT Temperature 0 P111 5
°c
22.0 -0.10 .008 .01415
CORN
50.0 - .08 .008 .01840
10.0 0.10 0.01 .19900
COTTON
0.20 0.01 .79900
20.0 - .10 0.10 .01720
woos
" a 07 0. 10 a 02000
RAW 10.0 0.01 0.10 0.10470
FREEZE-DRI ED
BEEF SLICES 40.0 0.04 0.10 0.13600
FREEZE-DRIED
BEEF ’°”DER 37.7 0.2 0.10 1.26000

 

141

is indicated. .This expansion in the pores must be accompanied
by a concomitant contraction in the solid component of the porous
structure.

From tabulated Pm values (Table 4.4.6), it is at once obvious
that the Pm parameter as used in this study is quite different from
the actual pressure corresponding to the mono-layer capacity. Part
of this discrepancy can be traced back to the transformation of
coordinate axes which became necessary in the original formulation
in chapter II. It would appear that in the attempt to systematize
the plots of the derived isotherm equation, the actual physical
significance of Pm has been sacrificed.

Stability Considerations in the Sorbent Pore Structure: The standard
error estimate of i 0.021 calculated in Table 4.4.1 8 for pre-
cooked freeze-dried beef powder is by far the worst recorded. Since
this product happens to be the only biological material in its pow-
dered form tested, the comparatively poor error estimate raises a
question of structural stability in powdered adsorbents.

It would seem that the pore structure of a powdered adsorbent
is basically ill defined. For one thing, the pore sizes and pore-
size distributions associated with such sorbents must necessarily
change with the degree of sorbent compaction. It is also clear,
that both hydro and thermo induced changes in the sorbent pore
structure will be more pronounced during the adsorption process.
These trends appear to be well reflected not only by the compara-
tively poor error estimate of Table 4.4.1 a, but also by (a) the

change from D - 0.1 and 5 - .454 at 10°C (Table 4.4.1 8) to

142

 

‘0 = 0.2 and E = 1.26 at 37.7°C (Table 4.4.1 b), and (b) the accom-
panying change in Pm value from 0.01 at 10°C to 0.1 at 37.7°C.
Closure: It has been demonstrated in this section that the derived
isotherm equation can be used to reconstruct the adsorption isotherms
of biological materials with a fair degree Of accuracy. Even though
certain quantitative inadequacies of present methods made impossible
a priori determination of the structural parameters, 0 and E, it is
still possible to formalize the temperature dependence of these
parameters either on a general or specific basis. While no such
attempt has been made in the present study because of the limited
number of isotherm temperatures investigated, the necessary founda—
tions appear to have been laid to make such an effort a logical

extension of the present work.

143

4.5 Verification of the PrOposed
"Capillary Condensation - SwellingfiFatigue" Doublg
Superposition Theory of Sorption Hysteresis
in Biological Materials

 

 

As an example, the estimation of desorption isotherms from
adsorption data, employing the scheme outlined in section 2.5 c, is
implemented in Tables 4.5.1 and 4.5.2 for corn kernels at isotherm
temperatures of 22°C and 50°C. Corn is used because, to the knowledge
of the author, it is the only bio-material of which the bulk moduli
have been empirically determined for a wide range of moisture con-
tents (White, 1966).

The work sheets (Tables 4.5.1 and 4.5.2) are deveIOped as
follows: column 2 represents calculated or experimental adsorption
data; in this case the data of Chung and Pfost (1967) for whole
corn kernels were used. Column 3 gives the number of adsorbed multi-
layers, n = X/Xm. Columns 4 and 5 are the approximate density and
volumetric data. The terms of column 6 are obtained by linear inter—
polation Of the incremental volumes; in this case each incremental
volume is divided by a factor of 10- to give suitably small adsorp-
tion steps. Column 7 gives the bulk moduli data Obtained by extra-
polation of the data by White (1966). Columns 8 and 9 are calculated
in accordance with equations (2.5.17a) and (2.5.18), respectively.
Columns 10, 11 and 12 Show trial adsorption relative vapor pressure
values Obtained by the selective variation of the parameter, w, in
the generalized Cohan relation (eqn.(2.5.4)). In this case, m was
found to have the value 1.25. Thus the desorption relative vapor

pressure values calculated with the equation:

144

x no z.o o
Na> Ao\omo ca 9 m 5.5A n so 4

 

145

 

 

oo.a oo.a oo.H oooN. ooo. oo.o . oom. ooo.a ooN. o.”
who. do». How. Anna. Haoo. on.e omo. Nmm. ooo.a Now. o.
we». was. was. mood. naoo. oo.m omo. as“. neo.a and. o.
owe. «me. who. oNnH. ooNoo. om.o amo. mes. moo.a on“. a.
can. «on. an. mama. naoo. oa.a oao. maa. nea.a «ma. 6.
cos. nae. ooe. ooua. ouoo. an.“ ago. nod. nee.” wag. m.
«on. mom. «on. smog. aeoo. ma.» Mao. ooo. noa.a mod. 4.
oou. com. com oooo. oooo. oo.o mac. who. med.“ ooo. n.
«ma. no”. AH. Noao. Nooo. om.o mNo. moo. mo~._ mao. N.

mo. moo. mo. +ommo. nao «so. nou.a mno. a.

Neo.
Aom\omv ozo+mzu AoooHoMHszo Aoa xv c A.auome
m.aaom\mo ~.HAom\mv H.~Aom\ao 62 one one a NH. 6> a e 6: om\m

NH AA oa o o 4 a o m 4 m a a

 

.oomm u eunumwooEmu EomsuomH

Anomfi .umomm a mangov Hmcumx once u uosooum

 

.eConuomH cowomuomoo mo coauoaooaoo one cow ooozm xuoz .H.m.q mapmw

x no x a o
m u so 4

Nao Aa\oooca e a n.4H

 

 

oooo.a ooom. oooo. oo.e ooom. ooo.a o om. o.a
oooo.

moan. Nana. Naoo. om.o oomu. ooo.a ma. o.
oomo. m

ooma. oooa. oaoo. om.n oaaa. Mao.“ oaa. o.
oano.

ooqo. owed. muoo. om.o coma. moo.a moo”. a.
cane. N

ooNn. «ANA. naoo. oa.a oooa. msa.a NA. o.

. omao. ,6

come. mooa. oaoo. nm.~ oaoo. mea.a mod. n. “m
Roao. .

ooan. oooo. oeoo. ma.» mono. mea.a «mo. 4.
none.

oNNN. omoo. onoo. oo.o oooo. noa.a oo. n.
naao. a

onma. oeao. oooo. om.o ammo. moN.H woo. N.
omao.

oono. oomo. oomo. mow.” omo. a.
oomo.

Aoooa.ooapzv
Aooaxo Aoaxo a A.Auoxav
m~.HAom\mo u oo\oa Aozc+mzv 4624 x «Ho mo 6 e 6: om\a
oa o o a o m e n N A

 

Doom u musumuanou EnenuomH

Amooa .umme a wcssuv Hmcumx upon I uoswoum
.ECoLooma coHuauomoo we cowowaooamo one new noose xuos .N.m.¢ dance

_ 1.25
Pd/Po — (Pa/Po) [4.5.1]

were plotted against the corresponding specific desorbed mass Md
(column 9) to obtain the theoretical desorption curves (Figure 4.15).
Figure 4.15 Shows that the theoretical desorption curves fit

the experimental points remarkably well for the isotherm temperatures
investigated. While broad generalizations must await more extensive
application of the theory to other biological products and other iso-
therm temperatures, this finding indicates that the new approach pre-
sented here offers an initial framedwork for more detailed considera-

tion of the complex questions of hysteresis in bio-materials.

fi_._5__§. The parameter, _u)_, _a_s_ 3 factor of pore geometry

A convenient classification of sorption hysteresis based on
(a) the pressure range over which the main boundary 100p extends,
and (b) the steepness of the adsorption and desorption branches, has
been prOposed by de Boer (1958). Within the frame-work of this
broad classification, the parameter, m, can be employed to gain
more insight into the geometrical shape of the pores of sorbing
biological materials (Barrer st 31., 1956; de Boer, 1958; Linsen and
Van Den Heuvel, 1967).

Thus, for the corn kernel, equation (4.5.1) satisfies the

de Boer (1958) criterion:

(Pa/Po)2 < Pd/Po [4.5.21

Consequently, the pores are tubular capillaries with widened spheroidal
parts Open at both ends. This finding supports the earlier preference

in section 4.1 for the interconnected spheroidal ink bottle model.

147

 

.30 T I
0 EXPERIMENTAL 22.0

     
  
 
 
  

A EXPERIMENTAL 50°C

0)
(I
I

09
C)
I

THEORETICAL
L. ISOTHERM

(I

 

A’ _.

I
.IA’ ‘\r THEORETICAL
ISOTHERM

MOISTURE CONTENT (d. b.)
5

.05

 

 

 

.I .2 .3 .4 .5 .6 .7 .8 .9 IO
P/Po

Figure 4.15. Comparison of experimental desorption isotherms with
calculated desorption isotherms for corn (expl. data

taken from Chung and Pfost, 1967).

148

V. SUMMARY AND CONCLUSIONS

A generalized model of water sorption is developed and

verified for biological materials.

The B-E-T and Capillary condensation theories are combined

into an integral isotherm equation for porous biological materials.

In order to solve the integral equation explicitly, it became

necessary to:

(a) characterize analytically, the physical structure of sorbing

(b)

bio-materials in terms of a pore-size distribution function
and an idealized pore-geometry;

derive, utilizing the general framework of the potential
theory in conjunction with the fundamental laws of thermo-
dynamics, an explicit density function for water adsorbed by
biological materials. As an essential complement for this
method of attack, a new theory of sorption hysteresis is syn-
thesized within the generalized framework.

In consequence of a detailed application of the deve10ped

model to the sorption data of several biological products, the

following specific conclusions can be drawn.

1.

The pores of biological materials are best modeled for trans-
port processes in these systems as interconnected spheroidal

"ink-bottles".
The numerical distribution of pore-Size in bio-materials is

well described by a power law distribution function of the type:

_ Y
m2(R) — KlR

149

where the pore characterization parameters, K and Y are

1
both product and temperature dependent.

The density of water sorbed by biological materials can be
expressed, in the first approximation, as the semi-theoretical
relation:

p/po = 0* [Mist/Mist:l

where p is the density associated with the isosteric heat of

adsorption, AH po and Afizt are the corresponding density

st’
and isosteric heat values at saturation vapor pressure; and the
quantity, u*, is empirically defined for the class of biological
materials of interest.

The interconnected spheroidal ink-bottle pore model, the power

law distribution function and the density function are integrated

into an isotherm equation of the form:

a

M = 9715 [zn- in]

where n and g are the primary and secondary characteristic para-
meters of pore structure; and Z and 1 are functions of relative
vapor pressure.

The derived isotherm equation,
(a) reflects through its density term, the well established fact

that the heat of adsorption varies with moisture content,

(b) reconstructs the empirical adsorption isotherm and its

dependence on temperature reasonably well,
(c) yields most characteristic isotherm types principally through

the variation of the principal structural parameter, 0.

150

It thus appears conclusive that the three basic concepts
in adsorption, namely: (a) the Kinetic concept as exemplified
by the B-E-T theory of multi-molecular adsorption, (b) Polanyi's
adsorption potential theory, and (c) Zsigmondy's capillary con-
densation theory, play complementary and possibly overlapping
roles and can be unified into a coordinated theory of water
sorption by biological materials.
A new theory of sorption hysteresis based on the superposition
of (a) Capillary Condensation hysteresis, and (b) Swelling
Fatigue hysteresis, is developed in quantitative terms and has
been successfully employed to predict the desorption isotherms

of whole corn kernels.

151

VI. SUGGESTIONS FOR FURTHER STUDY

As a consequence of the present study, it is recommended that

further investigations should be undertaken in the following areas:

1.

W. The function derived in this study

for the density of water can be improved considerably if (a) more
precise isosteric heat values are measured and (b) the determina-
tive function, u*, is revised to reflect a wider spectrum of
biological materials. Furthermore, because of the almost complete
absence of reliable experimental data on the density of water
sorbed on biological materials, the derived density function is
yet to be conclusively verified.

The Structural Parameters, n and g. The dependence of n and g

on temperature needs to be formalized in order to make it possible
to extrapolate from a set of empirical TI and g values at one
isotherm temperature to another. If these parameters are deter-
mined for a sufficiently large number of biological materials
(over a wide enough temperature range) formalization on a general
basis would be possible. Furthermore, water-induced Changes in
biological materials should be studied in terms of the 0 and 5
parameters. In this case the structural parameters determined
from a desorption isotherm can be compared with those resulting
from the adsorption branch. This type of parametric study can
conceivably furnish additional information on the phenomenon of

hysteresis.

152

39

Mechanical PrOperties of Biological Materials. The bulk
modulus of most biological materials as a function of moisture
content and temperature has not been experimentally determined.
Therefore, it was not possible to test the "Capillary Condensa-
tion - Swelling Fatigue" superposition concept of sorption
hysteresis developed here in sufficient detail to establish its
general validity. Additional work should be done along these

lines.

153

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162

APPENDICES

163

APPENDIX A

Isosteric heat of adsorption plotted as a function of moisture

content for several materials.

164

AH” - (CAL./GRAM)

800

750

700

650

600

550

 

500

?\ o SODA- 3011.50 COTTON
X (from Urquhart and Williams, 1924)
\ A SPRUCE WOOD
\ (from Stamm, 1938)

> El CORN KERNELS (DISSPT'N).
\\ (from Rodriquez-Arias, 1956)
\x 0 CORN ADEPTN
\. (Chung & Pfost, 1967)
\
\ (Chung & Pfost, 1967)

 

llillLLLllllllllll11111111111

V

0.00 .04 .08 .I2 .16 .20 .24 .28

X-MOISTURE CONTENT (d.b.)

Figure A-l. The isosteric heat of adsorption as a function of

moisture content.

165

M., - (CAL. IGRAM)

 

 

 

\A\ 0 FIRE-COOKED FREEZE-DRIED BEEF POWDER (Adptn.)

\\ A Pas-000K150 sneeze-omen sees pownm wept...)
900—
soo~
700—
600—

550 I : -
0.00 0.10 0.20 0.30 0.40 0.50 0.60

X - MOISTURE CONTENT (d.b.)

Figure A-2. The isosteric heat of sorption as a function of moisture

content.

166

APPENDIX B

Deve10pment of the working equation of the Cranston and Inkley (1957)
method for the calculation of pore-size distribution from sorption
isotherms.

Let Vrér be the volume of pores having radii between r and
r + 0r, where 6r is small compared with r. Consider an adsorption
step from a relative pressure Pr such that the small pore in the

range is about to fill with condensate, to a pressure P ), such

(r+0r
that the largest pore in the range has just filled with condensate.
During this pressure change, pores in the range considered became
filled with condensate, smaller pores are already filled, while in
larger pores the thickness of the adsorbed layer on their walls in-
creases from tr to tr + 6t. The total volume of liquid water adsorbed
in this step is given by the volume of liquid condensed in pores

with average radius '? 8 (r1 + r2)/2, and the volume of water con-

densed in larger pores contributing to the thickness of the adsorbed

layer. Thus, assuming a cylindrical geometry,

2
v_r.5r ”(r - tr) Lt + c(tr - tfiér) 2:5 [A-l]

r-5r

where IL? total length of pores with mean radius ?,

ESP-0r = surface of pores covered with multi-layer thickness tr’

c = correction factor for the curved surface

The volume of pores with radius ? is given by:

-2
V? . U r 1‘? [A-Z]

167

so that L; = V1411 “f2 [A-3]
Also the capillary surface 31' is given by:
St = 2n rL? - 2%]? [A4]

If equations (A-3) and (A-4) are substituted in equation (A-1) and

the summation sign is replaced by an integral sign:

 

 

5"?)z m r-cr 2v?
VYOr .- -—-2-—- VYOr + 0t 1‘ "f- T (It [A-5]
‘? ?+0r
'i’ - t...
where c = ‘f t , and

V?6r is the total volume of pores in the range or considered.

In the limiting case, where 5r~0, equation (A-S) becomes:

(r-tr)2 ”0° ”#:4er
a A-6
vrdr 7— Vrdr + dt Jr r [ 1

where vr derived from experimental measurements, while r, tr, dr
and dt are all functions of pressure which can be evaluated. Thus,
Vr can in theory, be evaluated by applying equation (A-6) to the
experimental results. In practice, however, it is not convenient
to use the equation as it stands and it is preferable to integrate
it over small finite ranges of radii.

Consider a finite adsorption step from pressure P to pressure

1

P2, where P1 corresponds to the critical radius rl and P2 to radius

r2. The total volume of water adsorbed during this step is:

1'2
v12 a I vrdr
I1

2
r (r-t ) V (Zr-t -t )
2 l " r l 2 _
Vrdr + (cz-cl) Jr 2 [A 71

l r 2 r

«J~

1'

 

168

E ,IHIIw'liltll....| .11

«A 11111 1.1.! II 1.. 11.111111171111941 {in}!

1...!!51111311333 .. .

 

 

   

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