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PERMEABILITIES TO LOW MOLECULAR VrEIGHT SOLUTES OF
COLLAGEN AlTD REGENERMED CELLULOSE
DIALYSIS MEMBRAJTES
by
CRAIG LEE DEARDEN, B.S. IN Ch.E.
A THESIS
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial FulfilLment of the Requirements for
the Degree of
MASTER OF SCIENCE IN CHEMICAL ENGINEERING
Approved
August, 1974
SOS*
Qop, 2
ACKNOINFLEDGEMENT
The author wishes to express his sincere appreciation to the
committee members, Dr. G. F. Meenaghan, Dr. L. J. O'Brien, and
Dr. W. J. Huffman, for their direction, interest, and confidence
during the course of this investigation.
Acknowledgement is given to Mr. Dean McKensie and Devro Inc.
for their supplying the collagen membrane for this study.
The author expresses his appreciation to Mr. Patrick Chui,
Ms. Mary Rodriques, and Mr. S. C. McHaney for their assistance
without which this study would not have been possible.
11
TABLE OF CONTENTS
ACKNOWLEDGEMENT ii
LIST OF TABLES v
LIST OF ILLUSTRATIONS vi
CHAPTER
I. Introduction 1
II. Literature Review and Theoretical Development . . 4
Liquid Side Mass Transfer Coefficient . . . . 8
Membrane Permeability 12
Overall Mass Transfer Coefficient 13
Dialysis Membranes 16
III. Equipment, Materials and Procedures 21
Equipment 21
Mass Transfer Cell 21
Analytical Tools 23
Materials 27
Membranes 27
Reagents 29
Procedures 30
Liquid Side Mass Transfer Coefficients . . 30
Overall Mass Transfer Coefficients . . . . 31
Tensile Strengths and Thicknesses of the Membranes 33
Phenomenological Data Analysis Procedure 34
iii
iv
IV. Discussion of Results 36
Boundary Layer Mass Transfer Coefficients . . 36
Mass Transfer Rates Through Cellulose and
Cellulose-Filled Collagen Membranes 42
Mass Transfer of Amino Acids Through
Cellulose-Filled Collagen Membranes ^8
Physical Properties of the Membranes 56
Application of Collagen Membranes
to Hemodialysis 61
V. Conclusions and Recommendations 63
Conclusions 63
Fiecommendations 64
LIST OF REFERENCES 66
APPENDIX 70
LIST OF TABLES
1. Membrane Processes 5
2. Physical Forms of Collagen Medical Products
and Their Clinical Application 18
3. Spectrophotometer Settings 25
4. Amino Acid Composition of Collagen 28
5. Summary of Phenomenological Coefficients
for Cellulose and Collagen 41 6. The Effect of the Solution of Hydration
on the Membrane Thickness 57
7. The Effect of the Solution of Hydration on Membrane Tensile Strength 59
8. Liquid Side Mass Transfer Coefficients
for Benzoic Acid 71
9. Overall Mass Transfer Coefficients 72
10. Tensile Strength of Regenerated Cellulose 74
11. Tensile Strength of Filled Collagen 75
12. Cellulose Membrane Thicknesses 76
13. Collagen Membrane Thicknesses 77
LIST OF ILLUSTRATIONS
1. Concentration Gradient Across a Semipermeable
Membrane 7
2. Dialysis Cell 22
3. System Flow Schematic 26
4. Liquid Side Mass Transfer Coefficient of Benzoic
Acid Versus the Reynolds Number 38 5. The Effect of Molecular Weight on the Overall
Mass Transfer Rate 43
6. Membrane Permeability Versus Molecular Weight 47
7. Molecular Weight Versus Effective Diffusivity 49
8. The Selective Effect of Collagen on Amino Acids . . . . 51
VI
CHAPTER I
INTRODUCTION
The maintenance of body fluid and electrolyte balances and the
excretion of metabolic toxins are among the primary functions of the
human kidney or renal system. During renal dysfunction, hemodialy
sis is clinically employed to simulate in vitro the functions of the
kidneys. In hemodialysis, blood is passed over a semipermeable mem
brane while a dialysate solution is allowed to flow on the opposing
side of the membrane. Mass transfer in the form of osmosis and
passive diffusion is then allowed to occur until the concentrations
of metabolic toxins in the blood fall to a level compatible with
life and the body fluid balances are adjusted to normalcy. The
dialysate solution is prepared to establish favorable concentration
gradients which allow the proper blood loss or gain by diffusion of
water, electrolytes, and toxins. Spent dialysate solution is dis
carded. The rate of diffusion and hence dialysis is dependent upon
the resistances to mass transfer, primarily the membrane permeabil
ity and boundary layer resistances.
Since the boundary layer resistances to mass transfer are prop
erties of fluid dynamics, the semipermeable membrane is the only
resistance that can be directly varied for a required fluid regime.
It is reasonable to expect that an increase in membrane permeability
or a decrease in membrane resistance could result in more efficient
hemodialysis leading to shorter dialysis times. Due to an insuffi
cient number of hemodialysis units, a shorter dialysis time would
allow more people to undergo dialysis per day. It appears, then,
that the development and application of a more permeable dialysis
membrane would ultimately lead to a decrease in the mortality rate
among people suffering a renal crisis.
Currently, the cellulosic membranes dominate all others in
their use in hemodialysis. Cellulose and cellulose acetate mem
branes are commercially available in tubing, sheet, or capillary
configurations. However, cellulosic membranes served only as a
standard for improvement during the course of this study. Collagen,
a naturally occuring protein, when cast as a membrane has been
found to be a superior dialysis membrane despite its poor tensile
properties (45). During hemodialysis it would be optimal for unde
sirable toxic components to diffuse quizkly from the blood through
the membrane and for desirable or essential components to pass
slowly or not at all from the blood. Collagen may exhibit such
mass transfer characteristics (46). Collagen has also been shown
to possess advantageous anticlotting and antigenic properties (41).
The purpose of this study was to estimate ija vitro: (1) the
effect of diffusing specie size on the permeabilities of regener
ated cellulose and cellulose-filled collagen membranes, (2) collagen
membrane selectivity to various amino acids, and (3) the effect of
hydration on the tensile strengths and thicknesses of regenerated
cellulose and cellulose-filled collagen membranes.
CHAPTER II
LITERATURE REVIEW AND THEORETICAL DEVELOPMENT
The generalized membrane is classically defined as an interface
between two regions of fluid. This interface must be a partial bar
rier to transport between the two fluid regions (13). Membranes may
be catagorized into three types according to the modes of transport
through them. Membranes that have structures that allow fluid
transport according to the normal equations of hydrodynamics are
called microporous membranes. Molecular diffusion membranes allow
mass transfer to only molecules that dissolve in the membranes.
Ultrafilter type membranes are intermediate between microporous
and molecular diffusion types. Their mode of transport is par
tially by a porous mechanism and also by membrane penetrant inter
actions. A summary of physical membrane processes is given in
Table 1 (13).
Three mechanisms of the method by which uncharged membranes
permit certain solutes to pass and reject others have been inves
tigated. However, no single theory is capable of general charac
terization of the transport process (23). The pore theory assumes
the membrane to be composed of a matrix of anastomosing channels
of varying diameter. Solutes whose molecular diameter is less than
the diameter of the pore are allowed to permeate but those whose
diameter is greater are rejected. The pore size distribution pro-
TABLE 1
MEMBRANE PROCESSES
i'rocess Membrane Primary flux
filtration ultrafiltration pressure permeation
pervaporation
dialysis diasolysis
piezodialysis forced flow electrophoresis
electrodialysis (conventional)
electroosmosis electrodecanta-tion
transport depletion
osmionosis osmosis
thermoosmosis
Soret effect Dufour effect
microporous ultrafilter solution-diffusion
solution-diffusion
ultrafilter ultrafilter
mosaic ion-exchange microporous and
ultrafilter alternate ion-exchange types alternate
any or none one ion-exchange type (usually cation-exchange)
one ion-exchange type (usually cation-exchange alternates with ultrafilter)
ion-exchange ultrafilter or solu
tion-diffusion ultrafilter or solu
tion-diffusion solution-diffusion solution-diffusion
solvent solvent selective transport of most mobile component
selective transport of most mobile component
small solutes selective transport of most mobile component
ionic solute solvent
ionic solute
solvent solvent
ionic solute
ionic solute
solvent
solute thermal
Source: J. Z. Friedlander, "Membranes," pp. 620-638 in Encyclopedia of Polymer Science and Technology, Vol. 8 , Mark, H. F. (ed.) Interscience Publishers. New York, N. Y. (1968).
vides a sieve effect (9, 13, 15, 44). Surface adsorption has
recently been investigated as a mechanism for membrane transport (3,
11, 40). The rate controlling step is not diffusion in either the
aqueous or membrane phase but considered to be the rate of solute
migration through the interface. The solubility mechanism, recently
fortified by Bonner, et al. (4), explains permeability on the basis
of solute solubility in the membrane phase. Solutes not soluble
in the membrane are rejected. Once in solution the rate of migra
tion of the solute is diffusion controlled (4, 12, 13).
The rate of solute transport through semipermeable membranes
has been considered to be controlled by the magnitude of the trans
port resistances (17, 22). The mass transfer resistances encoun
tered by permeators in an agitated diffusion cell include the bound
ary layer resistances on either side of the membrane and the mem
brane resistance to permeation regardless of the transport mecha
nism. The diffusion path for a mixed dialysis cell is shown in
Figure 1.
Mass transfer models have been used to describe mathematically
the transport rates through membranes and their boundary layers.
Neglecting osmotic and hydrostatic pressure effects. Smith, et al.
(43) have correlated mass transfer data using
N = KQ A (C^ - C2) (2-1)
where N = mass transfer rate (gm/sec)
K^ = overall mass transier coefficient 0
(cm/sec)
A = irrss t r ans fe r a r e a (cm )
Cji, ^2 ~ so lu te concen t ra t ion on s ides 3
1 and 2 in bulk phase (gm/cm ) .
Boundary Layer
Boundory Loyer
FIGURE 1 CONCENTRATION GRADIENT ACROSS A
SEMIPERMEABLE MEMBRANE
The reciprocal of the overall ma-ss transfer coefficient, K^,
represents the sura of the individual mass transfer resistances (23)
In transport phenomena, the reciprocal of a phenomenological coef
ficient represents a resistance (2). The overall coefficient may
8
then be expressed in terms of the sum of its parts as
_L_ = _1_ + JL + _L. (2-2) h \ ^S2 ^S2
where P = membrane permeability (cm/sec)
^ S l ' SJS2 ~ ^^*1^^*^ side coefficient on sides
1 and 2 (cm/sec).
Several investigators have used equation (2-1) in membrane studies
(41, 45). However, they reported the overall coefficient to be the
membrane permeability. As can be seen from (2-2), this is incorrect.
They did not account for the boundary layer contribution. Therefore,
to characterize the true phenomenological behavior of semipermeable
membranes, the boundary layer or liquid side mass transfer resis
tances must be known.
Liquid Side Mass Transfer Coefficient
Boundary layers develop when a mobile fluid phase contacts a
stationary phase. The fluid velocity at the phase interface
approaches the velocity of the stationary phase and progressively
increases back to the bulk phase velocity at some distance perpen
dicular to the plane of the interface (16). This distance is re
ferred to as the boundary layer thickness and its magnitude has
been shown to be inversely proportional to the bulk fluid velocity
(2, 24). The resistance to mass transfer offered by a boundary
layer is directly proportional to the thickness and hence, is
inversely proportional to the fluid velocity. In the Gilliland-
Sherwood form, Marangozis and Johnson (31) have reported the liquid
side mass transfer coefficients for an agitated chamber as
Kc^ I \ 0.33 / 2 \ 0.65
where k = liquid side mass transfer
coefficient (cm/sec)
1 = chamber diameter (cm)
2 D = molecular diffusivity (cm /sec)
y = fluid viscosity (gm/cm-sec)
3 p = fluid density (gm/cm )
r = stirrer speed (rpm)
d = stirrer dianeter (cm).
In similar studies, Johnson and Huang (21) have fit their liquid side
data to the form
k.„l /, \ 0.499 / ,2^\ 0.596 ,^ ,,
Using benzoic acid pellets in a stirred chamber. Smith, et al. (43)
found the liquid side coefficient for Reynolds numbers ranging from
8000 to 32000 to correlate as
k,^l , 0.33 / ,2 \ 0.567
-^=0.285^1 4 ^ . (2-5)
10
Smith, et al. (43) have shown that by monitoring the rate of
dissolution of a solid solute pellet in aqueous solution, the liquid
side coefficient can be estimated. By assuming that a saturated
solution exists at the liquid-solid interface, they developed the
following relationship betx een concentration and time:
( .at - Cl>t\ -hs At
3 where C = saturation concentration (gm/cm )
sat 3
C, = bulk fluid concentration (gm/cm )
k,, „ = liquid side coefficient (cm/sec)
2 A = mass transfer area (cm )
3 V = chamber volume (cm )
t = time (sec).
The liquid side coefficient can be estimated by evaluating the slope
of a semilogarithmic plot of (2-6).
Using dimensional analysis. Smith, et al. (43), have shown the
liquid side mass transfer coefficient for any solute at a constant
Reynolds number can be determined from a single experimental measure
ment of a reference k^^ by
where i = ith component
r = reference component.
11
Molecular diffusivities for aqueous biochemical solutions
have been studied by Longsworth (28, 29). Longsworth also demon
strated that molecular weight was a justifiable means of predicting
molecular size and hence the molecular diffusivity. To within 4.7%
his data were fit to the following expressions:
D = 10.772 X 10"^/(M°-^•^•^-1.893) (2-8)
2 where D = molecular diffusivity (cm /sec)
M - molecular weight.
The work of Longsworth has indicated that the diffusivities of
amino acids and polypeptides may have appreciably higher molecular
diffusivities than indicated by their size (28). The Wilke-Chang
equation,
D = 7.4 X 10"^ r((|{M)°-^ T/yV^°-^J (2-9)
where j» = association parameter of the solvent
M = molecular weight of solvent
T = absolute temperature ( K)
y = viscosity of solvent (gm/cm-sec)
V, = molal volume of solute at its normal
3 boiling point (cm /g-mole),
has been used to correlate the diffusivity of glycine to within 1%
of experimental results (42). For estimating the molal volume, the
group contribution methods of Le Bas and Schroeder were used (42).
12
As indicated by Longsworth, the prediction of some amino acid and
polypeptide diffusivities on the basis of molecular size may be
erroneous. His work therefore appears to limit the range of appli
cation of the Wilke-Chang equation.
Membrane Perncability
Assuming equilibiium conditions exist at the membrane inter
faces, the permeability to a solute as a function of its interfacial
membrane concentration can be expressed as
N = lm * <C^M - S M ^ (2-10)
where k = m^einbrane phase mass t r a n s f e r m ^
coefficient (cm/sec)
^IM' ^2M ~ i' t ^ f l l membrane concentration
3 of solute on sides 1 and 2 (gm/cm ).
By paralleling this system with a gas-liquid equilibrium system, it
can be seen that the interfacial solute concentration in the mem
brane phase is related to the interfacial solute concentration in
the fluid phase by equilibrium constraints. Using a distribution
coefficient which relates the solute concentration in both phases
at equilibrium, the interfacial membrane concentrations for dilute
solutions can be expressed as
and
13
S K = ' ILS/? (2-11)
SM = SLS/5 (2-12)
where 0^ , C^ _ = interfacial boundary layer concen-
3 trations on sides 1 and 2 (gm/cm )
C = distribution coefficient, mole
fraction solute in the fluid phase
divided by the mole fraction solute
in polymer phase.
By substituting equations (2-11) and (2-12) into (2-10), the rate
expression for permeating solutes in terms of the membrane coeffi
cient is as follov;s:
where P = membrane permeability,
k /C(cm/sec). m
Overall Mass Transfer Coefficient
Experimentally, the membrane permeability cannot be evaluated
from (2-13) due to problems inherent in measuring the interfacial
solute concentrations. The membrane permeability is however func
tionally related to the overall mass transfer coefficient by equa
tion (2-2). Assuming that k ,, and k n ^^ commensurate, Individ-
14
ual rate expressions can be written for both sides of the membrane
as
^ = " S * (1 - SLS^ (2-1*>
and
where N^, N. = mass trnasfer rates on sides
1 and 2 (gm/sec)
C,, C« = solute concentration in bulk 0
phases on sides 1 and 2
3 (gm/cm ).
Expressing equations (2-14) and (2-15) in terms of their liquid
side concentrations and substituting into (2-13) yields
k A / N N
Upon achieving a pseudo-steady state, N- must equal N_. Using this
constraint, (2-16) may be re-written as
^ " | i 7 k ; j T i 7 i r ] ^ ( S - V - (2-17)
The bracketed term in (2-17) upon inspection is identically the
overall mass transfer coefficient defined by (2-2). Therefore
(2-17) is equivalent to (2-1).
Kaufman and Leonard (23) have modified equation (2-17) using
material balances to yield a correlation from which the overall
15
coefficient can be estimated from experimental data. Using chambers
of equal volume, the following correlation was obtained:
/(Cj - C ) \ -2K A t
" 1( ):) --^ 3
where V = chamber volume (cm )
subscript t = any time t
subscript 0 = time zero
Kaufman and Leonard (22, 23) demonstrated that (2-18) fails to
account for one significant phenomena, filtration of water due to
an osmotic gradient. By correcting for osmotic flow, equation
(2-18) can now be written as
(C. - C ) \ -2K_ A t R T k. A
' ' ' <^h^>oj ^ ~^ ^ ^'' ^ ''^" '"'' where k-. = filtration coefficient
(cm/atm-sec)
T = absolute temperature (**K)
R = gas constant, 0.08205
(atm-liter/mole^K)
C,, C^ = concentration of solute in fluid
phase (moles/ liter).
It can be seen that for very dilute solutions, equation (2-19)
reduces to (2-18).
16
By evaluating the slope of equation (2-18) using a semiloga
rithmic plot, the overall coefficient can be estimated. If, how
ever, the plot is not linear, osmotic effects are not negligible
and therefore, must be analyzed. Once the overall coefficient is
known, the membrane permeability can be calculated using
P = Q ^ S (2-20)
" ^LS " ^h
One of the first groups to investigate true membrane permeabil
ity was Smith, et al. (43). Smith and co-workers have reported
permeabilities to sodium chloride for various types of cellulosic
dialysis membranes and varying dialysis conditions.
Dialysis Membranes
Cellulosic membranes originally developed for the meat packing
industry were first used for sausage casings (9). The casings were
developed to withstand considerable pressure, to be as thin and
uniform in thickness as possible, and to be free of pin hole defects.
By coincidence, the same characteristics are required by dialysis.
As a result, cellulosic membranes have evolved into good dialysis
and hemodialysis membranes. The high strength of cellulose pre
cludes membrane rupture due to low blood velocities and high shear
rates. The thickness and uniformity of cellulose membranes allows
17
relatively high permeability to low molecular weight solutes. The
mass transfer characteristics of cellulose are well explored (7, 14,
25, 27, 32, 43).
Chvapil, et al. (8) have recently reviewed the medical and
surgical applications of collagen. A summary of the physical forms
of collagen medical products is given in Table 2 (8). Since 1957
collagen has been used in dialysis and hemodialysis and found to be
superior to cellulosic membranes (26).
Collagen constitutes a majority of the protein component of the
connective tissue of animals. One of the basic requirements for
admission of a protein to the collagen class is an elevated hydroxy-
proline content (19). The collagen molecule is composed of three
strands wound into a tight triple helix with small non-helical pep
tide groups on the collagen molecule which account for many of its
biological properties (45). Being composed of acidic and basic
eimino acid groups, the molecule contains both positively and nega
tively charged sites (8).
In an attempt to develop a more efficient hemodialysis mem
brane, Rubin, et al. (41) investigated thin films cast from calf
skin collagen. From their studies it was found that an anticlot
ting surface could be formed by heparinizing the collagen film.
The heparin is believed to bond electrostatically to the positively
charged collagen sites. Collagen, a naturally occuring protein in
TABLE 2
PHYSICAL FORMS OF COLLAGEN MEDICAL PRODUCTS
AND THEIR CLINICAL APPLICATION
18
Form of Collagen
Solution
Gel
Flour
Fibers
Film, membrane, tape
Sponge, felt
Tubing
Application
Plasma expander
Vehicle for drug delivery system
Vitreous body
Hemostatic agent
Suture material, weaving blood vessels Valve prosthesis
Corneal replacement Hemodialysis, artificial kidneys Membrane oxygenators Wound dressing Hernia repair Patches (aneurysm, bladder)
Wound dressing Surgical tampons
Vessel prostheses Reconstruction surgery of hollow organs
(esophagus, trachea)
Source: M. Chvapil, Kronenthai, and W. Van Winkle, "Medical and Surgical Application of Collagen," Int. Rev. Connective Tissue Research 6: 1-61 (1973).
19
man, was also found to be non-reactive with antibodies native to
collagen.
Unlike cellulosic membranes, collagen dialysis membranes have
been found to have poor tensile characteristics (26, 41, 45, 46).
Several effective means of cross-linking the collagen molecule to
improve its tensile strength have been developed. Miyata, et al.
(33) have found that short, controlled ultraviolet irradiation in
nitrogen is a valuable method to introduce cross-links in collagen.
Aldehydes, particularly formaldehyde, are the simplest bifunctional
compounds which have been used for collagen cross-linking (8).
Bovcc and Cater (5) were also successful in cross-linking with di-
isocyanates, cyanuric chloride compounds, and difluordinitrodiphenyl
sulphone. Chvapil and co-workers have reported that cross-linking
can be simulated Ix^ vitro by tanning agents such as chromium salts
(8). Supporting the membrane on physical supports has been employed
by Stenzel (45) and Higly (20) as a means of reducing the danger of
rupture during membrane use.
Stenzel, et al. (45) have reported collagen membranes to have
transport properties superior to cuprophane. Dry, ultraviolet
irradiated collagen was found to be 19% more permeable than cupro
phane for the compounds studied. Verjr little other work on the
mass transfer properties of collagen has been done.
The possible occurence of collagen membrane selectivity to
amino acids was first noted by Stenzel, et al. (46) in 1969. The
20
selective effect was not noted for the same amino acids dialyzed
through cellophane coils. According to Abel and co-workers (1),
the rate of recovery of patients dialyzed during acute renal fail
ure was significantly higher when they were maintained on a high
L-amino acid diet. Selectivity to amino acids would have a similar
effect by preventing their loss during dialysis. The selective
effect noted by Stenzel was only observed and not quantified or
further investigated.
CHAPTER III
EQUIPMENT, MATERIALS AND PROCEDURES
Equipment
Mass Transfer Cell
In choosing a working model from which to study the phenomeno
logical behavior of semipermeable membranes, several criteria had
to be satisfied. First, the diffusion cell needed to be easy to
work with and mathematically defined. Secondly, the mass transfer
area needed to be constant and easily specified. Lastly, the model
needed to be previously studied to provide quantitative insurance
that experimental data agreed satisfactorily with published values.
A prototype was constructed from the specifications of the
diffusion cell of Smith, et al. (43). The model was constructed
from plexiglas tubing with an inside diameter of 6.1 cm and chamber
volumes of 235 ml. The impeller shaft was made from 0.635 cm out
side diameter, 316 stainless steel tubing. The impeller blades
were arranged in a paddle configuration and constructed from plexi
glas pieces 5.68 cm long, 0.635 cm wide, and 0.159 cm wide. A V
Nalgene polyethylene fitting served as a seal between the impeller
shaft and chamber wall. Stainless steel tubing, 0.159 cm I.D., was
used for the sampling probes. Gaskets were made of 0.159 cm thick
Teflon sheets. Details of the prototype are shown in Figure 2.
21
22
CM
111 Q: 3 O
u.
UJ
o
<n (o >-_ j
<
o
o a.
23
Several assumptions concerning the operation of the apparatus
were made: (1) no concentration gradient existed in the bulk fluid
phase because of sufficient chamber agitation, (2) the liquid side
mass transfer coefficients on either side of the membrane were
assumed equal, and (3) the effects of hydrostatic and osmotic pres
sures were negligible.
The prototype was operated at ambient conditions. Impeller
speeds vjere syncronous and controlled using a gear reducer coupled
with a 0.25 hp electric motor. Fluid leakage was occasionally noted
during operation at the plastic fittings and flanges. The plastic
fittings may be replaced by rotating mechanical seals to eliminate
fluid loss around the impeller shaft entrances. Flange leakage was
probably due to nonuniform torque on the flange bolts. A staggered
o-ring type seal could be installed in the flanges to eliminate this
type of fluid leakage and also maintain a constant membrane tension.
Analytical Tools
The change in solute concentration in the chambers was monitored
by spectrophotometric techniques. A Beckman ACTA CIII spectrophoto
meter was utilized. Since one of the special features of the ACTA
was a concentration mode, calibration curves were not necessary.
By standardizing a high and a low concentration sample, the ACTA
linearized all points between the limits. This was found to be
correct by checking points in between the limits with known concen-
24
trations of urea solutions. Distilled, deionized water was used
as a reference.
Continuous monitoring was made possible by use of flow through
cells. The flow scheme of the apparatus is depicted in Figure 3.
One head of the double headed peristaltic pump was used to transport
bulk fluid phase solution from one chamber of the dialysis cell to
the spectrophotometer and back into the same chamber. The second
head of the purp was used to transport fluid at a commensurate rate
from the other chamber through a dummy loop and back into the same
chamber. The dummy loop served to balance the fluid displacement
from the chambers and insure that the mixing due to sampling was
identical on both sides of the membrane. The sampling loops were
each made from 0.159 cm I.D. Tygon tubing and had 9 ml volumes. The
fluid flow rate was 20 ml/min through each of the loops.
Only one chamber could be monitored during a run. The concen
tration in the other chamber was calculated from a material balance.
Periodically, at the end of a run the sampling loop was changed so
that the concentration could be determined in the other chamber.
Invariably, the experimentally estimated concentration agreed with
the calculated value thus closing the material balance.
A summary of the spectrophotometric settings is given in
Table 3. The end result of an experiKient was a recorder strip of
the change in concentration with time. From these, all phenomeno
logical coefficients were determined.
TABLE 3
SPECTROPHOTOMETER SETTINGS
25
Solute
Aniline
Bacitracin
Benzoic Acid
Creatinine
Glycine
L-arginine
L-cystine
L-hydroxyproline
L-leucine
L-phenylalanine
Riboflavin
Urea
Concentration
mg%
10
10
10
10
50
10
10
10
25
50
1
50
Wavelength
nm
245
220
245
254
201
204
201
200
204
224
245
201
Slit Width
mm
0.37
0.39
0.30
0.32
0.98
0.99
0.98
0.94
0.81
0.42
0.30
0.98
26
o <
LiJ I O
ro CO
Ul
a: ID e> L.
^ o _J u. 2 llJ K-co >-en
27
Tensile strength measurements were performed using an Instron
Model TM-S testing instrument. Membrane thicknesses were measured
using a Mitutoyo micrometer. This instrument had a range from 0 to
2.54 cm and accuracy to within 0.0003 cm. All weight measurements
were made using a single pan, high sensitivity, analytical Mettler
balance. All solutions were made with distilled, deionized water.
Materials
Membranes
Standard regerxerated cellulose dialysis tubing obtained from
Fisher Scientific Co. was considered to be representative of the
cellulosic membrane family and was therefore studied. The flat
width of the tubing was 38 mm. The tubing was sectioned into 16 cm
segments, split along one of the seams, and pulled open into flat
sheets. Before insertion into the diffusion cell the sheets were
thoroughly washed and hydrated with distilled, deionized water.
Collagen m.embranes were secured locally from Devro Inc. Little
information was supplied by Devro Inc. due to the proprietory nature
of their process. Devro Inc. produces collagen tubing for sausage
casing. Their membrane is made from collagen solubilized from
bovine skins. The flat width of the membrane is 43.5 mm. During
the manufacturing process the collagen tubing is coated with a thin
film of mineral oil to lubricate the surfaces. However, a special
non-oiled batch was prepared fcr this investigation. An amino acid
analysis was m.ade on the Devro Inc. membrane. Table 4 summarizes
28
TABLE 4
AMINO ACID COMPOSITION OF COLLAGEN
Amino Acid
Reported
Aspartic Acid
Threonine
Serine
Proline
Glycine
Alanine
Valine
Methionine
Leucine
Isoleucine
Phenylalanine
Hydroxyproline
Devro
y
Inc. Collagen
moles/1000
43
16
39
687
175
81
21
6
22
18
11
9.9%
Calf Skin Collagen
y moles/1000
52
19
37
118
344
119
24
5
14
31
19
12.8%
Source: Shetlar, M. R.: Department of Biochemistry, Texas Tech University School of Medicine, Lubbock, Texas,
29
the results for the amino acids analyzed and compares this composi
tion to calf and reptilian collagen. The discrepancy in the glycine
and proline contents is probably due to Devro Inc.'s solubilization
technique from which the collagen is extracted from the animal hide.
The collagen membrane was reported to contain 1% by weight cellulose
as paper added as a polymer filler (30).
Before use, the filled collagen membrane was sectioned into
16 cm segments, slit along one of the seams, and pulled open into
sheets. The segments were then washed and hydrated. During hydra
tion it was observed that an ultraviolet absorbing organic compound
was leached from the membrane. It is known that glycerine was used
at some point in the manufacturing process (30). Due to the solu
bility of glycerine in water, it is possible that the unknown com
pound may be glycerine. Identification of this unknown should be
undertaken if future work with the Devro Inc. membrane is considered.
Reagents
All solutes used in this study were A.C.S. reagent grade. The
primary requirement for all solutes was that they all be ultraviolet
absorbing.
Benzoic acid was chosen to investigate the liquid side mass
transfer resistances due to its low solubility in water, 340 mg% at
25*'C. Benzoic acid pellets the size of the chamber mouth were made
by melting benzoic acid crystals in a beaker, pouring the molten
liquid into a siliconized pellet die, and pressing the die with a
30
hydraulic press. A force of 1.72 x 10^ dynes/cm (25,000 psi) was
exerted by the press until the die cooled to room temperature. The
pellets were carefully removed and polished with //600 fine grit using
standard methods. For estimating the effect of molecular weight on
membrane permeability, compounds were chosen whose molecular weights
varied from 60 to 1411. Clinically, urea and creatinine are the
most commonly monitored compounds during renal dysfunction. As a
result, they were selected for study. Bacitracin has a large molec
ular weight, 1411, and accordingly was studied. Aniline, benzoic
acid, and riboflavin were studied because of their intermediate
molecular weights, 93, 122, and 376, respectively.
The amino acids studied were selected on the basis of collagen
chemistry. L-cystine was studied because of its absence from the
collagen molecule. L-leucine and L-phenylalanine were studied
because they are both essential amino acids that are contained in
collagen. L-arginine and glycine were studied because they repre
sent non-essential amino acids present in the collagen molecule.
L-hydroxyproline was studied because of its importance in differ
entiating between collagen proteins and non-collagen proteins.
Procedures
Liquid Side Mass Transfer Coefficients
In determining the liquid side mass transfer coefficient only
one chamber of the diffusion cell was utilized. A polished benzoic
acid pellet was placed across the mouth of the chamber. The pellet
31
was fixed into position by an aluminum backing plate which was
bolted directly to the chamber flange. The ACTA was standardized
and calibrated using a saturated benzoic acid solution, 340 mg%, for
the high concentration reference and distilled, deionized water for
the low concentration reference. All standard spectrophotometric
procedures were followed (38). The chart time drive was set at
4.17 in/min. Instead of distilled water, a 22.4 mg% benzoic acid
solution was charged to the chamber and sampling loop to retard the
rate of dissolution. In sequence, the sampling loop pump was start
ed, the motor driving the impeller was turned on, and the chart
drive was started. Each run was allowed to proceed for 30 minutes
before reversal of the above procedure was initiated. Prior to re
use, the benzoic acid pellets were polished with //600 fine grit to
normalize the surface area. Runs were made for impeller speeds of
19, 36, 46, and 64 rpm which correspond to Reynolds numbers of 1890,
3550, 4530, and 6340. Each run was performed in duplicate. Table 8
in the Appendix summarizes the results of each repetition.
Overall Mass Transfer Coefficients
In estimating the overall mass transfer coefficient, a semi
permeable membrane was placed between the chambers. Regenerated
cellulose and cellulose-filled collagen membranes were randomly
selected at different lengths along their respective tubings. After
splitting and hydrating the segments, a membrane was draped over the
mouth of one of the chambers. The other chamber was then placed and
32
aligned over the first chamber. The comers of the membrane which
extended beyond the flanges were used to stretch and seat the mem
brane. When all surface wrinkles were smoothed, the membrane was
anchored by bolting together the two flanges.
All solutions were made using standard techniques. The spec
trophotometric procedure was the same as above except for the use
of different solutions. Table 3 summarizes the solution types and
concentrations used during the course of this study. A constant
impeller speed of 47 rpm vras used in all overall coefficient estima
tions. This speed was selected because it provided adequate mixing
and was low enough to preclude the incipience of membrane flutter.
The procedure used for all overall coefficient estimations was
rigidly follov/ed throughout the course of the study. The low con
centration chamber was continuously monitored using the sampling
loop. By infusion with the peristaltic pump, the sampling and dummy
loops were initially filled with distilled, deionized water and the
solute solution, respectively. The loops were then attached to
their chambers. The solution side chamber was sealed while the
water side was charged with distilled, deionized water and purged
of air bubbles. Sealing the solution side prevented the hydro
static pressure on the water side from bulging the membrane and
allowing the infusion of too great a volume of water. Upon sealing
the water side chamber and opening the other, the solution to be
studied was charged. After purging the air bubbles and sealing.
33
the peristaltic pump was started, the filled diffusion apparatus was
placed into its frame, and the impellers were started. To allow the
incipience of a pseudo-steady state, a ten minute latency period was
observed prior to starting the recorder. Experiments were allowed
to proceed at least one and one-half hours before termination.
Upon termination the recorder and impellers were stopped, the
recirculation loops disconnected and purged with distilled, deion
ized water, and the diffusion cell emptied. Both chambers were then
water rinsed three times. The third rinse was allowed to remain in
the cell for twenty minutes in an attempt to clean the membrane.
Each membrane was used twice before replacement.
To obtain credibility in the data, repetitions of the runs on
each solution were made in no fewer than triplicate. As many as
ten repetitions were made on some solutes. The repetition sequence
of experimentation is given in Table 9 which is found in the Appen
dix,
Tensile Strengths and Thicknesses of the Membranes
Samples from both membranes were randomly selected and cut in
2,54 cm X 2.54 cm squares. The squares were then hydrated in one
of four aqueous solutions, namely, distilled, deionized water,
200 mg% urea, 10 mg% creatinine, and 10% mg% sucrose. A square was
then aligned parallel, perpendicular, or oblique to the membrane
seam in the jaws of the instrument. The tensile strength for each
square was then estimated for one of the three alignment positions
34
using the American Society for Testing and Materials* standard
method for breaking fabrics. No fewer than ten repetitions were
made. The data are summarized in Tables 10 and 11 of the Appendix.
Membrane thicknesses were estimated in triplicate for twelve
random positions using standard micrometer procedures. The hydrated
squares were surface dried with blotting paper to remove the surface
film prior to measuring the thickness. Care was used not to stretch
or fold the Lier brane during the micrometer readings. Dry thicknesses
for both types of membranes were measured to estimate the effect of
hydration on thickness. All thickness data is presented in Tables
12 and 13 found in the Appendix.
Phenomenological Data Analysis Procedure
The recorder output data from both the liquid side and overall
mass transfer studies were treated in similar fashion. From the
recorder strips, fifteen concentration-time points were extracted.
The concentration values were used to calculate the concentration
ratios on the left hand side of equations (2-6) and (2-18), respec
tively. The fifteen concentration ratios and their corresponding
times in minutes were then used as input data for a computerized
regression analysis (36). A copy of the program and sample output
are shown in the Appendix. The regression analysis yielded the
slopes of equation (2-8) or (2-18) and determined the statistical
significance of the data. From the slopes, the overall or liquid
side coefficients were calculated.
35
The raw data obtained during the phenomenological investigations
were too voluminous to be included in this text. They are, however,
available in computer card form from the Department of Chemical
Engineering, Texas Tech University, Lubbock, Texas.
CHAPTER IV
DISCUSSION OF RESULTS
The study of the mass transfer rates through semipermeable
membranes was performed in four phases:
(1) the estimation of boundary layer mass transfer coeffi
cient,
(2) the estimation of mass transfer rates of non-amino acids
through cellulose hemodialysis membranes and cellulose-
filled collagen membranes,
(3) the estimation of mass transfer rates of amino acids
through cellulose-filled collagen membranes, and
(4) the estimation of the physical properties of cellulose-
filled collagen membranes and cellulose membranes.
Each of these phases are discussed separately below.
Boundary Layer Mass Transfer Coefficients
The liquid side mass transfer coefficient for benzoic acid was
evaluated from the slope of the equation
^ sat " S\\ " .S
"' ( t^> ; ) - -^"^ • The slope, -k A/V, was calculated by performing a first order
polynomial regression on the data of each experiment. An F test
36
37
was performed to check statistically the linearity of the data.
A correlation coefficient of 0.99 for all benzoic acid experiments
was obtained (47).
For benzoic acid the liquid side mass transfer coefficient
varied from 0.80 x 10 cm/sec to 1.63 x 10 cm/sec as the mixing
Reynolds number increased from 1890 to 6340, The liquid side mass
transfer coefficient is presented in Figure 4 as a function of the
mixing Reynolds number. The work of Smith, et al. (43) is also
presented in Figure 4 to provide a comparison of results.
From boundary layer theory it has been shown that the hydro-
dynamic boundary layer thickness over a flat plate is inversely
proportional to the fluid velocity (2, 24). As the boundary layer
thickness decreases with increasing fluid velocity, the diffusion
rate increases due to a shorter diffusion path (18). Experimen
tally this effect was manifested as an increase in k with
increasing Reynolds numbers. The pattern exhibited by k in Fig
ure 4 is therefore to be expected. The liquid side mass transfer
coefficients reported by Smith (43) and shown in Figure 4 are not
significantly different, as calculated by a standard t test, from
the liquid side mass transfer coefficients calculated in this study.
Smith's benzoic acid data for Reynolds numbers between 8000 and
32000 have been mathematically formulated by regression analysis as
K3 = 0.285 . 3 - " C - "
38
"^ 5 .00
o
•P O
o
2.50
1.50
t; 1.00 c o
<D 0 .50
*.j 0 .30
O Smi th , et al.
A Experimental
1000
1
2500 5000
Reynolds Number
10000
FIGURE 4
LIQUID SIDE MASS TRANSFER COEFFICIENT
OF BENZOIC ACID VERSUS THE REYNOLDS
NUMBER
39
where N = Stanton number, k 1/D
N = Schmidt number, y/pD
2 N = Reynolds number, r d p/y.
The liquid side coefficients of Smith's in Figure 4 were obtained
by solving equation (3-1) for k using Reynolds numbers from 1890
to 6340. This assumed that the results of Smith and co-workers
could be linearly extrapolated down to the Reynolds number range
investigated. Extrapolation would not, however, be valid for Rey
nolds numbers low enough to result in insufficient mixing causing
the occurence of a concentration profile in the bulk fluid phase.
Nonhomogeneity of the concentration in the bulk phase would manifest
itself in nonlinear behavior of the data. However, for the Reynolds
numbers studied, 1890, 3550, 4530, and 6340, the data were statis
tically linear. Therefore, any effects due to insufficient mixing
were negligible. This justifies the extrapolation of Smith's
results to the Reynolds number range studied.
According to Figure 4, the mass transfer operation of Smith's
cell and the prototype cell studied here are identical. Therefore
the behavior of the prototype cell can be predicted by using the
results of Smith, et al, (43) for the laminar boundary layer Rey
nolds numbers ranging from 1890 to 32000.
As mentioned in Chapter III, a primary reason for selecting
benzoic acid as the reference compound was its relatively low solu
bility in aqueous solutions, 340 mg% (25''C). This should insure
40
a constant mass transfer area over the duration of an experiment,
0,25 to 1 hour. However, upon termination of an experiment, the
surface was observed and found to contain striations approximately
0,5 mm wide and 2 to 3 cm long caused by surface erosion. This was
unvaryingly noted for each pellet after every run. Polishing the
pellet after each use re-established the original surface area and
prevented a compounded erosion effect. A time variation in surface
area would manifest itself in a deviation from linearity of equation
(2-6). As indicated by the F test, the data were indeed linear.
Therefore, the observed change in surface area did not affect the
mass transfer rates.
The liquid side mass transfer coefficients at a Reynolds num-
_3 ber of 6340 for each compound studied ranged from 1.44 x 10 cm/sec
-3 to 0.371 X 10 cm/sec. These values are listed in Table 5. Using
benzoic acid as the reference, the liquid side coefficients of the
other compounds were estimated using the following equation:
0.667
•isi = "^Sr 57) . (2-7)
-5 2 The diffusivity of benzoic acid was found to be 1.14 x 10 cm /sec
from the work of Chang (6). The time averaged benzoic acid concen
tration was used in estimating the value of D^, D^'s were found in
literature or estimated from the Wilke-Chang equation (28, 29, 42).
41
in
w PQ < c-^
CO H ^ W M U M ft. Pt4 W o c_> H J < o M
o 1-4 O 2
w s o z w sc p pti
o
z w o a h-3 O CJ
Q
§ W c/D O hJ ^ H J • J [X]
u c^ o PM
Q (U
CM
0)
W)
o u
0) .H iH •H
o> o o
+4 >^ vO
o
r i H
» O
+A <r vO
! 1
M3 CT.
• o
+J^ o i n
r o
-J un i n
o
i n i H
• O
U i n
i n
i n o o
r>. ro O
><r r-i
• o
+cl vO
m
rH CN
•Q rH i n
i H
ON - ^
• o +J o i n
-
-d-O
p
^ i H
m
o CM
tH
<U C
•H
• H C <:
r o Q
+^ CO
i H
r ro O
C •H U cd M •u
d PQ
<t-o o
+J, c» CNJ
vO i H
i H
QJ C
•H C
•H 4J
S u •_5
•<f O
Q +J 0^
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-3-00
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c •H JJ CO
a 1
H-J
m o o t l 00
•<f
>d--d-
r-i
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0)
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0)
o
d
0)
CM O
4 CM
iH CM
O O O Q O
in ON r^ 1/
00
o W rH •
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vO O O r-i KO r-i
v£> CM iH ,-1
? m o
I +=i +J 0 CM CM
P +4
vO
p +d
00
CM ro rH
i n o o +1 vO
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+ 1 vO
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P +«1 o
O
P 14 vO
CM O
o tl
CO
CM CM
O CM
r^ vo <t •<*• ro rH CO <J-
(U
c •H rH •H C <
C •H O Ctf U U •H O CO
CQ
<U C
•H C
•H +J cd (U V4
o
0) fl
•H 4J CO > . a 1
H4
cd 0) M p
42
Mass Transfer Rates Through Cellulose and
Cellulose-Filled Collagen Membranes
The overall mass transfer coefficient, K- , for all compounds
was evaluated from the slope of the equation
(V^)o)°~^^ •
The slope, -2K A/V, was estimated by performing a first order
polynomial regression analysis on the data of each experiment. An
F test was performed to test statistically the linearity of the
data, A correlation coefficient of 0.99 resulted for all experi
ments. The assumptions under which the mass transfer rates were
estimated therefore were valid.
The effect of molecular weight on mass transfer rates through
cellulose hemodialysis and cellulose-filled collagen membranes was
investigated. The overall mass transfer coefficient was found to
-4 / -4 / vary from 4,87 x 10 cm/sec to 1.61 x 10 cm/sec for collagen and
-4 -4
from 4,32 x 10 cm/sec to 1.36 x 10 cm/sec for cellulose for
molecular weights increasing from 60 to 1411. Graphical represen
tation of this effect is presented in Figure 5 for both membranes.
For the conditions of constant boundary layer and membrane
thickness, constant mass transfer area, and the assumption of non-
selectivity between the solute and membrane phases, the overall
diffusion rate is inversely proportional to the molecular radius of
43
• •
c 0) o» « _J
o "5 — a>
O
•
c 0)
o
o o
o
o a> k.
ID
a> c
W M M
C <
CVJ
0}
c c n 0) ^ o FO
«> C
*-
u 1
_J
^
c u o
itr
u o
CD
lO
1 1
o o o CVJ CO
<
o o o
o o lO
o o
o
._
igh
Q>
^
k -
cula
V
o S
a> 3
O 0)
If)
Ul Q: 3 O
U-
< (T
HE
O
VE
»-
z o h-X o UJ ^
a: < _ j
o Ul
MO
L
u.
E
EF
FE
CT
0
X
UJ
< en
cr, UJ U-
w z < a:
o o ro
(09S/UJ0) ._OIX fUa/OIJJSOQ J9J.SUDJ± SSD/V IIDJdAQ
44
the diffusing species (18, 28). As noted previously, the molecular
weight is a good indication of the size of a molecule (28). Figure 5
demonstrates that for both membranes the overall diffusion rate
decreased with increasing molecular weight. Both curves indicate
that the diffusion rate asymptotically approaches zero and infinity
as the molecular weight approaches infinity and zero, respectively.
For the compounds studied, the results presented in Figure 5 vari-
fied the work of Stenzel and co-workers (45) for pure collagen.
Due to a malfunction in the air conditioning system of the
laboratory, the ambient temperature was noted to be 32**C, 7*'C above
normal, on the day that bacitracin-cellulose mass transfer rates
were determined. This increase in temperature represented a 2%
increase in the absolute temperature. Since the liquid side dif
fusivities according to popular theories of liquid diffusion are
directly proportional to the temperature, the increased temperature
would lead to an increased value for the liquid side mass transfer
coefficient (2, 9), The membrane diffusivity, a directly propor
tional function of temperature, would also be elevated (2), As a
result, the overall mass transfer coefficient would be elevated.
However, Figure 5 indicates that the overall coefficient of bacitra
cin followed the expected trend (28, 45), Since this variation in
temperature had no apparent effect on the results, the use of a
nonregulated temperature system was justified. However, for large
molecules the effect of temperature is not as pronounced as it
45
would be for small molecules. Had the temperature anomalies occured
during the study of low molecular weight compounds, the calculated
effect would have been approximately three times more distinguish
able. Therefore, it is recommended that constant temperature regu
lation be employed for future work to avoid error due to temperature
variations.
Figure 5 also illustrates that the overall mass transfer coef
ficients of the cellulose-filled collagen membranes were an average
46% greater than those of the cellulose membranes for the compounds
studied. This is in agreement with the work done on pure collagen
by Stenzel, et al, (45).
The overall mass transfer coefficient, K_, is functionally
related to the liquid side coefficient and the membrane permeability
as follows:
0 m LS
Since the liquid side coefficients for both membranes are the same,
the difference between the two membranes' overall coefficient must
then be due to the differences in membrane permeability, P . Mem-m
brane permeabilities were calculated from equation (3-2), The mem-
-4 brane permeability was found to vary from 15,1 x 10 cm/sec to
-4 -4 3.64 x 10 cm/sec for collagen and from 10.8 x 10 cm/sec to
-4 2.62 x 10 cm/sec for cellulose for the compounds studied. The
effect of molecular weight on membrane permeability is presented
46
in Figure 6. The results indicate that the diffusion rate through
the membrane was inversely proportional to the molecular weight.
This trend was expected (45).
As shown in Figure 6, the membrane permeability to bacitracin
is grossly inconsistent with the trend of the data. However, the
overall coefficient for bacitracin was consistent with the molecular
weight trend. Since the permeability was calculated directly from
experimental values of K- and k ^, the value of the liquid side
coefficient may be in question. From the mass transfer aspect
bacitracin (molecular weight 1411) is a relatively unstudied poly
peptide. As a result the molecular diffusivity, required in calcu
lation of k values, was estimated using the Wilke-Chang equation,
D = 7.4 x 10"^ r(cj)M) T/y V^°*^1 . (2-9)
The molal volume needed by the Wilke-Chang equation was estimated
using the group contribution methods of Le Bas and Schroeder (42).
-5 2 Both methods yielded a molecular diffusivity of 0.180 x 10 cm /sec
for bacitracin in dilute aqueous solution. The work of Longsworth
(28) has shown that polypeptides have higher diffusivities than
indicated by their molal volumes. Therefore, the Wilke-Chang
equation may not be valid for polypeptides. Molecular diffusivities
for creatinine and cystine were estimated by the Wilke-Chang equa
tion. These compounds are not polypeptides and, as indicated by
Figure 6, their permeabilities follow the trend.
47
• •
c <u o» 9>
c a> o» o
o O
U. 1
o
o
llu
a> O
o
k .
13
a> il
in
c <
<1) c
c
ea
t k-
o
a> c
v>
o 1
_l
c u o
cit
o OD
• o — CVJ ro ^ in O O o CM
o o o
X o Ul
a: <
O UJ
o o lO
JZ
en a> ^
k.
o 3 O a> o S
Ul
o
a>
o CO
O O
CO
CO Q :
U l >
m < UJ
Q: U l CL
UJ z <
m
Ul
o
CD CM GO
(09S/UJ0) ofx AimqDdujJdtj auojquuayij
48
The results presented in Figures 5 and 6 are slightly mislead
ing. If both membranes were of the same thickness, the results
reported above would be directly comparable. However, it was found
that the hydrated thickness of the collagen membrane was i00,4±10,4
-4 X 10 cm and the thickness of the cellulose hemodialysis membrane
was 57.1±1.1 x 10~ cm. Therefore, the diffusion path of the colla
gen membranes was approximately 1.8 times that of the cellulose
membranes. For the compounds studied, the effective diffusivities
-5 2 -5 2 for collagen ranged from 1,51 x 10 cm /sec to 0.370 x 10 cm /sec
-5 2 -5 2 and for cellulose from 0.617 x 10 cm /sec to 0.149 x 10 cm /sec.
The average increase in the effective diffusivity of the cellulose-
filled collagen membrane was 2.6 times the effective membrane diffu
sivity of cellulose. The effective diffusivities are presented as
a function of molecular weight in Figure 7. The results of this
phase of the investigation are summarized in Table 5.
This phase of the study clearly indicates that the cellulose-
filled collagen membranes are superior from the aspect of relative
diffusion rates to clinically used hemodialysis membranes. Similar
findings are reported by Stenzel (45) for pure collagen membranes.
Mass Transfer of Amino Acids Through
Cellulose-Filled Collagen Membranes
As previously mentioned, preliminary investigations of pure
collagen as a hemodialysis membrane have indicated that collagen may
49
2 0
o q> «)
o
•o I
o
^ 10 -
(O
o «>
Uj
4 0
I
Legend-
• C-F Collagen
0 Cellulose
1 Urea
2 Aniline
3 Creatinine
4 L-cy stin e
± 00 2 0 0
Solute Molecular Weight
4 or
2 6 0
FIGURE 7
MOLECULAR WEIGHT VERSUS EFFECTIVE
FUSIV ITY
D I F -
-„,r, -rj*^ j r /A
50
exhibit selective transport to amino acids (46). That is, for simi
larly sized molecules the diffusion rate of an amino acid molecule
through collagen membranes would be slower than the diffusion rate
of a non-amino acid molecule.
The m.embrane permeability to the amino acids studied ranged from
6.79 X 10 cm/sec to 2.17 x lO""" cm/sec. The results of the amino
acid permeability studies are presented in Figure 8. All phenomeno
logical coefficients were calculated as in the previously presented
phases. All mass transfer assumptions were statistically validated
by the use of the F test with a correlation coefficient of 0,99
resulting for each experiment.
As compared to the membrane permeability to non-amino acid
compounds of similar size, the average decrease in permeability to
glycine, L-leucine, and L-arginine was 35%. A standard t test was
used to evaluate the significance of the membrane selectivity to
these three amino acids. A 0.99 correlation coefficient was
obtained indicating the significance of the difference. The phenom
enon of selectivity would be advantageous in hemodialysis since the
quantity of glycine, L-leucine, and L-arginine lost during dialysis
would be minimized. Physiologically this would, of course, be
beneficial (1),
Glycine, L-leucine, and L-arginine, as do all amino acids, have
one or more carboxyl groups contributing a negative charge (34).
Collagen has both positively and negatively charged sites within its
51
16
o
Q)
I
O
12 -
o 8 -
0}
c:
40 100 2 0 0 Solute Molecular Weight
5 0 0 1000
Legend
1 2 3 4
5 6
Urea Glycine Aniline Creatinine Benzoic acid L- 1 e u ci n e
7 8 9
10 I I
L- hydroxyproline L- phenylalanine L- arginine L- cystine Riboflavin
FIGURE 8
THE SELECTIVE EFFECT OF COLLAGEN ON AMINO
ACIDS
52
molecular structure (35). In an attempt to explain the selective
behavior of the three amino acids, an organic acid, benzoic acid,
was studied. However, as shown in Figure 8, no selectivity to ben
zoic acid was observed. The cause of selectivity due singly to the
carboxyl group may possibly be eliminated.
The pore theory of semipermeability has long been a classic
means of rationalizing how uncharged membranes permit the passage
of certain solutes and reject the passage of others. The membrane
is considered to be a matrix composed of anastomosing pores (9).
The membrane allows the passage of solutes whose molecular diameter
is less than the diameter of the pore. The membrane contains a
distribution of pore diameters and therefore acts like a molecular
sieve. The selectivity of collagen to L-leucine, L-arginine, and
glycine cannot be rationalized by the pore theory. Some other mem
brane phenomena must therefore be responsible for this effect.
Amino acids are grouped according to their basic or acidic
tendencies (35), The amine functional group serves as a Lewis
base while the carboxyl functional group acts as a Lewis acid (34),
Amino acids whose molecular constituency includes one amine and one
carboxyl group are termed neutral amino acids. Those that are com
posed of more amine groups than carboxyl groups are called basic
amino acids. Those that are composed of more carboxyl groups than
amine groups are named acidic amino acids.
53
In solution the nitrogen of the functional group can furnish
its pair of non-bonding electrons to an electrophile which results
in a positive charge on the amine group (34). In polar solvents the
carboxyl group has the ability to form a carboxylate anion upon
losing a proton (34). Therefore, regardless of the classification,
an amino acid will carry a charge when in aqueous solution.
It therefore appears plausible that membrane-solute charge
interactions could be responsible for collagen membrane selectivity.
Charge repulsion and electrostatic bonding between the charged mem
brane sites and the charged functional groups of amino acids seem
the most likely interactions to occur. However, on this basis
selectivity should be observed for not only all amino acids but also
all charged solutes. This is contrary to the results. Urea with
its two amine groups and benzoic acid with its carboxyl group show
no evident selectivity. Selectivity due to charge interactions may
in fact occur to some degree, however, these interactions fail to
predict consistently the results of this study and, therefore,
appear not to be the controlling factor in collagen membrane selec
tivity.
Aside from the charge postulations, collagen membrane selec
tivity could possibly be explained on the basis of polymer solution
thermod3mamics. That is, the degree of selectivity would be gov
erned by the equilibrium phase distribution between the solute,
solvent, and polymer phases. If the solute was insoluble or only
54
partially soluble in the polymer membrane phase, the diffusion rate
would be significantly decreased. If the solute was completely
soluble in the membrane, the mass transfer would be diffusion con
trolled. Since collagen is essentially a poljmierized amino acid
chain, the solubility of free amino acids in the membrane would be
expected to be decreased. The greater the membrane composition of
an amino acid, the less soluble a free amino acid may be in the mem
brane phase. That is, a smaller quantity of amino acid would be
required to form a saturated solution. Based upon the polymer
thermodynamics of Flory and Huggins (10), a theoretical model has
been developed to predict the phase distribution for the ternary
system. The development of this theory is given in the Appendix.
The quantitative application of the theory was not feasible due to
the lack of required parameters, namely, solvent interaction para
meters, polymer segments/molecule ratio, and specific volume of the
collagen molecule.
Qualitatively the theory appears to present a plausible cause
for collagen's selectivity to some amino acids. L-cystine was
absent from the composition of the cellulose-filled collagen mem
brane. Therefore, the sorption of free cystine into the polymer
phase would not be hindered by intra-membrane cystine and no selec
tivity should be seen. Experimentaily, cystine was found to behave
as a non-amino acid compound. No selectivity was observed. Glycine
was found to represent 18% of the total amino acid content of the
55
membrane. The collagen membrane was found to be selective to gly
cine. The membrane permeability to glycine was decreased by 25%,
This indicated that the equilibrium conditions were such that fur
ther solubility of glycine in the polymer phase was restricted.
Therefore, the diffusion rate was decreased. A similar pattern was
noted for L-arginine and L-leucine. The membrane permeability to
L-arginine and L-leucine was decreased by 42% and 30%, respectively.
L-phenylalanine exhibited this effect to a lesser degree. Its per
meability was decreased by only 11%. However, the concentration of
L-phenylalanine in the membrane phase was only 10%, L-hydroxypro
line did not exhibit a decreased permeability. However, the L-hy
droxyproline concentration was estimated to be 9.9% in the amino
acid analysis of the cellulose-filled collagen membrane as shown in
Table 4. To be consistent with the thermodynamic model, the lack
of selectivity to L-hydroxyproline must mean that the membrane is
far from saturated with L-hydroxyproline. That is, even though
the membrane is composed of 9.9% L-hydroxyproline, the solution will
absorb much more L-hydroxyproline before saturation conditions are
approached. Permeability studies should be initiated on the balance
of the amino acids contained in the collagen membrane as well as
other neutral, basic, and acidic amino acids not present in the
collagen molecule. Glucosamine and galactosamine, amino acid sugars,
should also be studied.
56
In summary, the cellulose-filled collagen membrane was found
to be selective to glycine, L-leucine, and L-arginine. For these
amino acids, the permeabilities were 35% less than expected. The
results of this phase of the study indicate that the pore mechanism
and charge interaction mechanisms inadequately explain selectivity,
and a polymer solubility theory may represent the diffusion mecha
nism more consistently.
Physical Properties of the Membranes
A comparative study was made on cellulose hemodialysis mem
branes and cellulose-filled collagen membranes to observe the effect
different solutions of hydration had upon the membrane thickness and
tensile strength.
The results of the membrane thickness study are summarized in
Table 6. The dry thicknesses of the cellulose and collagen mem-
-4 -4 branes were 26.0±1,6 x 10 cm and 43.0±2.5 x 10 cm, respectively.
For solutions studied, the average hydrated thickness for the cellu-
-4 lose membranes was 57.1±1.1 x 10 cm and the average hydrated
-4 thickness for the collagen membranes was 100.4±10»4 x 10 cm.
Hydration resulted in an increased membrane thickness of 2,2 times
the unwetted thickness for cellulose and 2.3 times for collagen.
However, the hydrated membrane thickness was not significantly
affected by the solute present in the solution of hydration.
TABLE 6
THE EFFECT OF THE SOLUTION OF HYDRATION
ON THE MEMBRANE THICKNESS
57
Solution
of Hydration
None
Distilled Water
200 mg% Urea
10 mg% Creatinine
10 mg% Sucrose
Average
-4 Membrane Thickness (10 cm)
Filled Collagen
43,0±2.5
106,3±10.5
94.9±11.7
97.1±11.4
103.5±8,0
100,4±10.4
Cellulose
26.0±1.6
56.6±1.5
57.2±1.2
57.3±1.3
57.1±1.4
57.1±1.1
58
The membrane thickness of collagen is a function of the pH of
the wetting solution (8). The pH for all phases of this investiga
tion ranged from 6.0 to 6.2. However, the pH of blood encountered
during hemodialysis may vary from 7,2 to 7.6. Therefore, the effect
of blood pH on collagen membrane thickness should be estimated in
future studies.
The results of the tensile strength studies are summarized in
Table 7. The tensile strength was estimated in three membrane
positions: parallel, perpendicular, and oblique to the membrane
seam. For cellulose, the average tensile strengths for parallel,
5 2 perpendicular, and oblique positions were 4.59±0,50 x 10 dynes/cm ,
5 2 S 2
3.74±0.61 X 10 dynes/cm , and 2.97±0,57 x 10 dynes/cm , respec
tively. For collagen, the average tensile strengths for the three
positions were 0.85 0,13 x 10^ dynes/cm^, 0,83 0.14 x 10^ dynes/cm^, 5 2
and 0.85 0.15 x 10 dynes/cm . The cellulose membranes had tensile
strengths 5.4, 4.5, and 3.5 times greater than the collagen mem
branes for the three respective positions. As in the membrane
thickness study, the solute present in the solution of hydration
had no significant effect upon the tensile strengths of the mem
branes.
The results presented on pure collagen by Kon (26) and Sten
zel (45) indicate that the poor mechanical properties of collagen
represent a limiting restriction to hemodialysis application. Dur
ing the course of this study, an occasional defect in the cellulose-
59
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60
filled collagen membrane was noted. Upon observation, the membrane
defect appeared much the same as does peeling necrosed human epider
mis several days following a prolonged infrared exposure. Such a
defect would be vulnerable to rupture. Within experimental limits,
no significant change in the mass transfer rates was noted.
Due to the poor tensile properties of the collagen membranes,
a means of improving its tensile strength needs to be evaluated. The
membrane should be physically supported or cross-linked to improve
its mechanical strength. The membrane could be physically supported
on cellulose triacetate or wax extended polyethylene porous sup
ports (20). Investigations should then be undertaken to obtain
the effect of the supports of the effective mass transfer area and
resulting mass transfer rates. Membrane cross-linking could be
achieved by physical or chemical methods. Physically, ultraviolet
or gamma ray exposure forms cross-links within the polymeric struc
ture (5). Chemically, chromium salts, polyvalent cations, and
aldehydes are used to form cross-links with the collagen molecule
(6, 8). The effect of chemical cross-linking on membrane permeabil
ity should be investigated. Preliminary work has shown that cross-
linking pure collagen resulted in a 27% decrease in membrane per
meability to small molecules (41).
61
Application of Collagen Membranes
To Hemodialysis
The primary advantage to the use of cellulose-filled collagen
membranes would be their relatively low resistance to mass transfer.
The resistance offered by cellulose membranes used for clinical
dialysis was an average 46% greater than the resistance of the col
lagen membranes.
The membrane resistance in hemodialysis constitutes a large
portion of the overall resistance to mass transfer. Using the
method of Volk and Zaltzman (48), the blood side resistance to mass
transfer was estimated for a capillary flow and flat plate hemo-
dialyzer. The dialysate resistance was assumed to be small and was
therefore neglected. The collagen membrane resistance to urea was
the least of all compounds studied. The resistance to L-arginine
was the greatest. For a capillary flow dialyzer, the membrane
offered 62% and 89% of the total mass transfer resistance to urea
and L-arginine, respectively. For a flat plate dialyzer, the mem
brane offered 39% and 70% of the total resistance to urea and L-
arginine transport. Therefore, the membrane resistance is the most
important controllable factor in maximizing the efficiency of hemo
dialysis. By minimizing the membrane resistance, the diffusion
rate will be greatly increased and hence dialysis time or dialyzer
area and size will be decreased. Collagen appears better suited
to dialysis than cellulose from the mass transfer aspect.
62
The selectivity exhibited by colla.gen was found to decrease
the membrane permeability to glycine, L-leucine, and L-arginine by
35%. The selectivity of collagen to glycine, L-leucine, and L-ar-
ginme would be of importance in nutritionally maintaining chronic
renal patients. The application of collagen to hemodialysis would
therefore minimize the loss of amino acids.
One of the most demanding requireiaents of the membrane in dia
lysis is high mechanical strength. The shear stress exerted by the
fluids at low dialysis flow rates may be great enough to rupture a
weak or defective membrane. The tensile strength of cellulose
hemodialysis membranes was found to be no less than 3.5 times great
er than the cellulose-filled collagen membranes. This represents
a serious limitation to the dialysis application of cellulose-filled
collagen membranes. Therefore, the collagen membrane or membrane
configuration must be modified to improve its mechanical strength
before its utilization is feasible.
Regardless of the favorable mass transfer results obtained by
this study, a clinical evaluation must be undertaken to optimize
the cellulose-filled collagen membrane for hemodialysis application.
Such an evaluation would, of course. Involve blood dialysis rather
than single component dialysis. From clinical studies solute-sol
ute interactions, solute-membrane interactions, and blood-membrane
interactions could be evaluated.
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
1. For non-amino acid compounds, whose molecular weights varied
from 60 to 1411, cellulose-filled collagen membranes exhibited a
46% greater membrane permeability than did regenerated cellulose
hemodialysis membranes.
2. The average decrease in membrane permeability of cellulose-
filled collagen to three amino acids, namely glycine, L-leucine,
and L-arginine, was 35% as compared to non-amino acid compounds of
similar size.
3. With regard to applications in hemodialysis;
a) the semipermeable collagen membrane would offer from 39%
to 86% of the total mass transfer resistance in parallel
plate and hollow fiber dialyzers for the compounds studied,
b) cellulose-filled collagen membranes would not be suited for
clinical dialysis unless a suitable support structure was
included. The tensile strength of the cellulose-filled
collagen membrane was only 20% as great as regenerated
cellulose hemodialysis membranes.
63
64
Recommendations
During the course of this study certain limitations and areas for
improvement were observed. Each of these recommendations are noted
separately below.
1. The diffusion apparatus should be improved by:
a) use of a constant temperature jacket to eliminate the effects
of ambient temperature variations,
b) use of rotating mechanical seals at the impeller shaft inlets
to the chambers thus eliminating leakage and reducing main
tenance,
c) use of a staggered o-ring seal in the flanges to reduce
gasket leakage and insure proper membrane seating.
2. Due to the poor stress properties of the cellulose-filled colla
gen membrane, the membrane should be physically supported or cross-
linked to improve its mechanical strength thus alleviating the pos
sibility of stress trauma or deformation. The membrane could be
physically supported on cellulose triacetate or wax extended poly
ethylene porous supports. Investigations should then be undertaken
to determine the effect of the supports on the effective mass trans
fer area and resulting mass transfer rates. Membrane cross-linking
could be achieved by physical or chemical methods. Physically,
ultraviolet or gamma exposure forms cross-links within the membrane
structure. Chemically, chromium salts, pol3r7alent cations, and
65
aldehydes are used to form cross-links with the collagen molecule.
The effect of chemical cross-linking on membrane permeability
should be investigated,
3. The unknown compound which was extracted from the cellulose-
filled collagen membrane during initial membrane hydration should
be identified.
4. All of the amino acids present in the collagen molecule should
be investigated as membrane permeators. From this, the function
ality between permeability and collagen's amino acid composition
may be determined in an attempt to explain selectivity.
5. Amino acid sugars, namely D-glucosamine and D-galactosamine, and
other substituted amino related compounds should be studied as per
meators. By modifying the carboxyl group, the effect of charge on
collagen membrane permeability can be determined.
6. The changes in mass transfer resistances affected by collagen
membrane swelling due to pH variations should be determined.
7. The cellulose-filled collagen membrane should be clinically
evaluated to optimize the membrane for actual dialysis service.
LIST OF REFERENCES
1. Abel, R. M., Beck, C. H., Abbott, W. M., Ryan, J. A., Barnett, G. 0. and Fischer, J. E.: "Acute Renal Failure: Treatment with Intravenous Amino Acids and Glucose." New Eng. J. Med. 288: 695-699 (1973).
2. Bird, R. B., Stewart, W. E. and Lightfoot, E. N.: Transport Phenomena. pp. 513-515, 502-505. John Wiley and Sons, Inc., New York, N. Y. (1960).
3. Blank, M.: "Monolayer Permeability and the Properties of Natural Membranes . " J. Phys. Chem. 66 : 1911-1918 (1962).
4. Bonner, D. C , Cabasso, I., Dearden, C. L., Huffman, W. J. and Smith, J. E.: "Prediction of Solute Rejection by Water-Permeable Membranes." (Work to be published, 1974).
5. Bowes, J. H. and Cater, C, W,: "Crosslinking of Collagen," J. Appl. Chem. : 296-304 (1965).
6. Chang, S. Y.: "Determination of Diffusion Coefficient in Aqueous Solutions," pp. 2, 31, 41, 45. M.S. Thesis, Library, M.I.T. (1949).
7. Chang, T. M. S.: Artificial Cells, pp. 146-176. Charles C. Thomas, Publisher, Springfield, 111. (1972).
8. Chvapil, M. , Kronenthal, R. and Van Winkle, W.: "Medical and Surgical Applications of Collagen." Int. Rev. Connect. Tissue Res. i6: 1-61 (1973).
9. Craig, L. C : "Dialysis," pp. 824-857 in Encyclopedia of Polymer Science and Technology, Vol. 4_, Mark, H. F. (ed.) Interscience Publishers. New York, N. Y. (1966).
10. Flory, P, J.: Principles of Polymer Chemistry, pp. 548-554, Cornell University Press, Ithaca, N, Y. (1953),
11. Flynn, G, L. and Roseman, T. J.: "Membrane Diffusion II." J. Pharm. Sci. : 1788-1796 (1971).
12. Flynn, G. L. and Smith, R. W.: "Membrane Diffusion III." J. Pharm. Sci. 61: 61-66 (1972).
66
67
13. Friedlander, J. z.: "Membranes," pp. 620-638 in Encyclopedia of Polymer Science and Technology, Vol. 8, Mark, J. F. (ed.) Interscience Publishers. New York, N. Y. (1968).
14. Ginzburg, B. Z. and Katchalsky, A.: "The Frictional Coefficients of the Flows and Non-Electrolytes Through Artificial Membranes." J. Gen. Physiol. 47: 403-418 (1963).
15. Gliozzi, A., Morchio, R. and Ciferri, A.: "Transport Properties of Collagen Membranes." J. Phys. Chem. 73: 3063-3070 (1969). ~
16. Graham, R. R.: "Limited Range Moclel for the Dehydrogenation of Cyclohexane," M. S. Thesis, Library, Texas Tech University, Lubbock, Texas (1967).
17. Grimsrud, L. and Boff, A. C : "Velocity and Concentration Profiles for Laminar Flow of A Newtonian Fluid in a Dialyzer." Chem. Eng. Progr. Symposium Ser. No. 66 §1} 20-31 (1966).
18. Guyton, A. C : Textbook of Medical Physiology, pp. 473-474. W. B. Saunders Co., Philadelphia, Pa. (1971).
19. Harrington, W. F.: "Collagen," pp. 1-16 in Encyclopedia of Polymer Science and Technology, Vol. 4, Mark, H. R. (ed.) Interscience Publishers. New York, N. Y. (1966).
20. Higley, W. S.: "Collagen Membranes for Reverse Osmosis." United States Patent No. 3,644,202 (1972).
21. Johnson, A. I. and Huang, C. J.: *'Mass Transfer Studies in an Agitated Vessel." A. I. Ch. E. Journal 2: 412-419 (1956).
22. Kaufmann, T. G. and Leonard, E. F.: "Mechanism of Interfacial Mass Transfer in Membrane Transport." A. I. Ch. E. Journal L4: 421-475 (1968).
23. Kaufmann, T, G, and Leonard, E. F.: Studies of Intramembrane Transport: A Phenomenological Approach." A. I. Ch. E. Journal 2A' 110-117 (1968).
24. Kays, W. M.: Connective Heat and Mass Transfer, pp. 79-85, McGraw-Hill Book Co,, New York, N, Y. (1966).
25. Kedem, 0. and Katchalsky, A.: "Thermodynamic Analysis of the Permeability of Biological Membranes to Non-Electrolytes." Biochem. Biophys. Acta : 229-246 (1958),
68
26. Kon, T., Mrava, G. L., Weber. D. C- and Nose, Y.: "Collagen Membrane for Hemodialysis." J. Biomed. Mater. Res. 4_: 13-23 (1970).
27. Lane, J.A. and Riggle, J. W.: "Dialysis." Chem. Eng. Progr. Symposium Ser. No. 24 55: 1Z7-143 (1959).
28. Longsworth, L. G.: "Diffusion M«.i3;surements, at 1°C, of Aqueous Solutions of Amino Acids, Peptides, and Sugars." J. Amer, Chem. Soc. 7^: 4155-4159 (1952),
29. Longsworth, L. G,: "Diffusion Measurements, at 25* 0, of Aqueous Solutions of Amino Acids, Peptides, and Sugars." J. Amer. Chem. Soc. 75.: 5705-5709 (1953),
30. McKensie, D.: Comments of the Mamtufacturing of Collagen Tubing, Personal Communication to C. L. Dearden, Lubbock, Tx. (1973).
31. Marangozis, J. and Johnson, A. I.: "Mass Transfer With and Without Chemical Reaction." Can. J. Chem. Eng. 39; 152-158 (1961).
32. Marshall, R. D. and Storrow, J. A.: "Dialysis of Coustic Soda Solutions." Ind. Eng. Chem,. 43: 2934-2942 (1951).
33. Miyata, T., Sohde, T., Rubin, A. L. and Stenzel, K. H.: "Effects of Ultraviolet Irradiation on Native and Telopep-tide-Poor Collagen." Biochiit. Biophys. Acta 229; 672-680 (1971).
34. Morrison, R. T. and Boyd, R. N.r Organic Chemistry, pp. 579, 592. Allyn and Bacon, Inc.^ Boston, Mass. (1973).
35. Orten, J. M. and Neuhaus, 0. W.: Biochemistry, pp. 57-63. C. V. Mosby Co., St. Louis, ¥io, (1970).
36. "Polynomial Regression." System/360 Scientific Subroutine Package GH20-0205-4. Interoational Business Machines, White Plains, N. Y. (1968).
37. Prausnitz, J. M.: Molecular Thermodynamics of Fluid-Phase Equilibria, pp. 181-214, 3a5--404. Prentice-Hall, Inc., Englewood Cliffs, N. J. (1969).
38. Price, A.: "Acta CIII UV-Visible Spectrophotometer." Instruction Manual No. 015-082250- Beckman Instruments, Fuller-ton, Ca. (1972).
69
39. Reid, R. C. and Sherwood, T. K.: The Properties of Gases and Liquids, pp. 86-89, 5/8-560. McGraw-Hill Book Co., New York, N. Y. (1966).
40. Rosano, H. L., Duby, P. and Schulman, J. H.: "Mechanism of the Selective Flux of Salts and Water Migration Through Nonaqueous Liquid Membranes." J. Phys. Chem. : 1704-1708 (1961).
41. Rubin, A. L., Riggio, R. R., Nachman, R. C , Schwortz, G. H. , Miyata, T. and Stenzel, K. H.: "Collagen Materials in Dialysis and Implantation." Trans. Amer. Soc. Artif. Intern. Organs j^: 169-175 (1968).
42. Sherwood, T. K., Brian, P. L. T. and Fisher, R. E.: "Desalination by Reverse Osmosis." Industrial and Engineering Chemistry Fundamentals 6: 2-12 (1967).
43. Smith, K, A., Colton, C, K., Merrill, E. W. and Evans, L, B.: "Convective Transport in a Batch Dialyzer: Determination of True Membrane Permeability from a Single Measurement." Chem, Eng. Progr. Symposium Ser. No. 84 64_: 45-58 (1968).
44. Snell, F,, Wolken, J. and Iverson, G, J,; Physical Principles of Biological Membranes, pp. 137-141, Gordon and Breach Science Publishers, New York, N, Y, (1968).
45. Stenzel, K. H,, Rubin, A. C , Yamayoshi, W,, Miyata, T,, Suzuki, T., Sohde, T. and Nishizawa, M.: "Optimization of Collagen Dialysis Membranes." Trans. Amer. Soc. Artif. Intern. Organs 17_: 293-298 (1971).
46. Stenzel, K, H., Sullivan, J, F,, Miyata, T. and Rubin, R. C : "Collagen Dialysis Membranes: Initial Clinical Evaluation." Trans. Amer. Soc. Artif. Intern. Organs 15: 114-117 (1969).
47. Volk, W,: Applied Statistics for Engineers, pp, 159-166, 271-273. McGraw-Hill Book Co., New York, N. Y. (1968).
48. Wolf, L. and Zaltzman, S.: "Optimum Geometry for Artificial Kidney Dialyzers." Chem. Eng. Progr. Symposium Ser. No. 84 64: 104-111 (1968).
APPENDIX
70
TABLE 8
LIQUID SIDE MASS TRANSFER COEFFICIENTS
FOR BENZOIC ACID
71
Run No.
1
2
3
4
5
6
7
8
Impeller Speed
(rpm)
19
19
36
36
46
46
64
64
Reynolds No.
1890
1890
3550
3550
4530
4530
6340
6340
\s (cm/sec)
8.08x10"^
7,78x10""'
1.17x10"^
1.17x10""
1.26x10"-
1,28x10"^
1,64x10"^
1.61x10""
72
TABLE 9
OVERALL MASS TRANSFER COEFFICIENTS
Run No.
7 8 9
10 11 12 16 17 18 19 20 21 22 23 24 25 26 12-C 13-C 14-C 15-C 20-C 21-C 22-C 23-C 24-C 25-C 30 31 32 33 34 35 36 37 38 39 40
S o l u t e
Urea II
II
B a c i t r a c i n II
II
C r e a t i n i n e II
II
II
A n i l i n e II
II
II
L - c y s t i n e II
II
C r e a t i n i n e II
II
II
B a c i t r a c i n II
II
II
11
11
A n i l i n e t l
It
II
L - p h e n y l a l a n i n e II
II
It
L - c y s t i n e It
t l
Membrane
C e l l u l o s e II
II
II
II
II
II
II
It
t l
II
t l
t t
II
t l
t l
It
Col lagen It
It
It
It
It
It
It
It
It
tt
t l
»»
It
It
It
tt
It
It
It
t t
K (10 ^ cm/sec)
4 .36 4 . 3 1 4 .29 1.64 1.71 1,46 2,00 2.16 2.04 2 . 1 1 2.75 2.75 2 .53 2 .65 1.49 1,65 1,68 2 ,73 2,82 2 ,94 2,82 1.43 1.54 1.33 1.16 1.22 1.50 2.97 3 . 1 1 3.12 3,09 2,34 2 ,02 2 ,05 1,90 2 , 0 1 2 ,05 1,91
TABLE 9—Continued
73
Run No.
41 49 50 55 56 57 58 59 60 61 62 64 65 68 69 70 74 75 76 77 78 79 80 81
CM
CO
00
00 84 85 86 87 88 89 91 92 93 94 95 96 97
So lu t e
L - c y s t i n e L - a r g i n i n e
It
Urea II
II
It
II
II
II
It
It
It
Glycine It
It
L-hydroxypro l i ne It
II
II
L - l e u c i n e tl
t l
t l
Glyc ine It
L - l e u c i n e II
A n i l i n e R i b o f l a v i n
It
It
L - a r g i n i n e tl
II
It
Benzoic Acid II
It
Membrane
Col lagen It
It
It
It
II
It
II
It
It
It
It
t l
t l
It
It
It
II
It
It
It
It
tt
It
It
tt
tt
It
It
tt
It
It
11
II
It
It
It
tl
tl
K (10~^ cm/sec)
1,88 1,47 1.54 4 ,89 4 ,74 5,17 4 .77 4 .86 4 .63 4 .86 5.05 4 .87 4 .87 3 .21 3,04 3,06 2,54 2 ,38 2 .25 2 ,25 1.97 1.94 1,89 1.89 3 .33 3,32 1.99 2,02 3 ,23 1,76 1,57 1,47 1,44 1,54 1,45 1.32 2 .74 2 .87 2 ,98
74
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78
Phase Equilibria Model
It is assumed that two equilibria systems are encountered in
dialysis: (1) equilibrium between the solute and solvent in the
bulk fluid phase and (2) equilibrium between the solute, solvent,
and poljrmer in the membrane phase. It is believed that equilibrium
constraints in the membrane phase may be responsible for the selec
tive effect exhibited by collagen to several amino acids.
At equilibrium, the polymer and bulk fluid phases may be related,
assuming that the membrane is insoluble in the solvent, as
f l = f^ (A-1)
and
f3 = f3 (A-2)
where f = fugacity
a = liquid phase
3 = poljnner phase
1 = solute
3 = solvent.
From the definition of chemical potential,
U^ - y^ E RT In (Y^ X^) (A-3)
79
where y = chemical potential
R = gas constant
T = absolute temperature
Y = activity coefficient
X = mole fraction
i = ith component of
equilibrium system
= standard state.
Equation (A-3) can be rewritten as
@ RT In (Y^ X^) = RT In (- l (A-4)
f. or X = - ^ (A-5)
Now the solute-solvent equilibriimi will be considered. At
equilibrium.
a ^1 =
^1
M (A-6)
where f° = standard state fugacity
arbitrarily chosen as the
pure subcooled liquid at the
temperature under its own
saturation pressure.
80
This standard state must be evaluated. By assuming that the solid
has a low solubility in the liquid phase and using Margules'
approach, the activity coefficient can be expressed as
^" X. sat - " ( - 1. sat)' (- )
where A = constant
sat = saturation.
Considering saturation conditions,
,-s a ex L ^1 = ^1. sat ^1. sat 1 (*-8«)
or
\ , sat = -^ T • (A-8b) 1, sat 1
By substituting (A-8) into (A-7), the constant term in (A-7) can
now be expressed as
(A-9)
To evaluate (A-9), the saturation concentration of the solid in the
s L liquid phase and f /f for the solid must be known. Prausnitz (37)
s L has correlated the f /f ratio in terms independent of the solvent
as follows:
81
AC /T - T\ AC
where Ah = enthalpy of fusion
R = gas constant
T = subcooled liquid temperature
T = triple point temperature
Ac = heat capacity of liquid
minus solid phases,
OL
Once A has been determined, the activity coefficient can be
expressed as
f" Y^ = = exp [ A ^ (1 - X ^ ) n (A-11)
where f = f* ,
To evaluate the activity coefficient for the solvent, the iso-
baric, isothermal Gibbs-Duhem equation is applied,
d In Y. a ^ " 3 X^ = X^ ^ (A-12) 1 C6 3 a • d X^ d X^
Equation (A-12) upon integration and rearrangement yields
Y^ = il = exp (A" (X^)2). (A-13)
' ^ ^
82
Having evaluated the activity coefficients for the solute and
solvent in the bulk fluid phase, the three phase system may now be
investigated. Turning to Flory-Huggins theory (10), the chemical
potentials for each component of the ternary mixture can be estima
ted as follows:
V l - Vii = RT fin (|) + (1 - 0^) - (|>2 ( i/ 2 " 3 ''l/''3 "
(Xi2 2 + \ 3 *3><^2 ^3^ - 23 (^l/^2>*2*3] ^^"^^^
^2 - 12 = ^ [ m ^2 + (1 - *2> - *1 <^2/^l> - 3 ^^2/^3^ ^
(X21 1 - X23 3X^1 3) - Xi3 (r2/ri)*i*3] (A-15)
and
y^ - 13 = RT fin (|)3 + (1 - 4)3) - i>^ (1 3/ 1) - *2 ''3/''2 "*"
(X31 1 32 *2>(^1 ^2^ - 12 <-3/^l)^I*2] ^^'^'^
where 1 = solute
2 = pol3mier
3 = solvent
y = chemical potential
R = gas constant
T = absolute temperature
X = interaction parameter
(j) = volume fraction
r = number of structural units
in a given polymer molecule.
83
Each of the volume fractions, (J), can be expressed as follows:
X V (t) = ±—i (A-17) ^1 X V + X V + X V \^ -^f J
1 1 2 2 3 3
*2 = X, V^ / x ^ v , . X3 V3 <-->
and
X V <\>n = :^ r-^r—^-r^ (A-19) ^ ^1 ^1 - 2 ^2 •" S 3
where X = mole fraction
V = specific volume.
For very dilute solutions, cf) is approximately zero and <|)- plus
(j) approaches unity. Listed below are the assumptions:
""3 ""l -^ = -^ = 1 (A-20) ^1 3
r r -^ = — = 0 (A-21) ^2 ^2
^2 ~ " 2 — = — = large number (A-22) r r ^3 1
4, = 0 (A-23)
(j), + (j)„ = 1 . (A-24)
Substituting these assumptions into equations (A-14), (A-15), and
(A-16) yields
^1 " 1 = T Q ^ ^^ + 1 - <t>^ (r^/r^) + Xi2 2 ^ ^13 Q ^ " ^
^2 " 2 = RT [jLn ^^ + l - ^^ - ^^ {r^lr^) + X23 H (A-26)
and
y3 - y^ = RT [in 3 + 1 - (|) + X32 4)f] . (A-27)
However at swelling equilibrium, y equals y° and Flory (10)
expressed the solvent contribution as
- |ln (1 - ^l) + 4)2 + X32 ( 2 ] =
^ ( ^ - ^ n ^ ) ( ( ^ 2 > * ' ' - ^ ) ^^-2«)
where v_ = molar volume of solvent
- = specific volume of polymer
M = molecular weight per cross-c
linked unit
M = primary molecular weight
of the polymer,
, 33 By assuming that M /M approaches zero and ^' » <^J2, equation
(A-28) can be written as
- [in (1 - 6) + ^e + X32 W|)^] = » ^ ) - " . (A-29)
85
Now, by equating chemical potentials in the bulk fluid and
membrane phases, equations (A-25) and (A-27) can be written as
In Yi X^ = In 4)^ + 1 - 4, (r^/r3) + x^2 4 " 13 ^3 ^^"^^^
and
In Y3 X^ + In 4)^ + 1 - 4) + X32 (*2 ^ . (A-31)
By realizing that <^ approximately is equal to 1.0 - 4) , equation
(A-31) can be written as a function of 4) as
In Y3 X^ = In 4)3 + 1 - 4)3 + X32 (1 - <J>3) • (A-32)
It should now be obvious that by solving equations (A-29), (A-30),
and (A-32) the three volume fractions 4) , 4)o» and <p^ can be esti
mated from which the distribution coefficients can be calculated.
Before solving the above equations, several parameters must
first be estimated. The activity coefficients for the solute and
solvent in the bulk fluid phase can be calculated using equations
(A-11) and (A-13), respectively. Also required are the interaction
parameters, x» the specific volumes, v, and M , and v-.
Once all the parameters have been obtained, the equations can
be solved. One feasible means of solving these equations appears
to be;
B x3 (1) trial and error equation (A-29) for (^^ iterating on (p^t
o o (2) trial and error equation (A-32) for 4>3 iterating on 4)^,
86
6 S (3) substitute <p and 4)? into equation (A-30) and solve
o
algebraically for <p ,
By knowing the 4)*s, the mole fractions of solute and solvent in the
polymer phase can be calculated from equations (A-17), (A-18), and
(A-19),
Membrane behavior should now be predictable on the basis of
solution thermodynamics. From this model, the selectivity of col
lagen to glycine, L-arginine, and L-leucine may be explained.
87
C c c c c c c c c c c c c c c c c c c c c c c c c c c
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EP>'E\ 'SICN X( I I O O ) DIN':E\'SIf}N o n 100 ) 0 1 v<^NS inN; D { 6 6 ) D I M- \'S I ON ? ( i u ) , ^ ( 1 0 ) , 5 P ( 10 ) , T ( i 0 ) O r ' f . -'.S lOK' XE AC ( 1 1 ) ,STP( 11> tCE'EI 11 ) ,SUMS0( I I ) DIME'MS I ON i r S ( 1 U ) • I S A V ^ d l ) DIMENSION P(3UU)
1 F r j 7 M ^ T ( A 4 , A 2 , 1 5 , 1 2 , 1 1 ) 2 F r P ' ^ A T ( 2 F 6 . 0 ) 3 F O R M A T ( ' l * , 9 ( / ) , i a x , 2 5 H Pa tYNOMIAL R E G P E S 5 1 ^ ^ . • • ,
I A A , A ^ / ) 4 F0F>^^AT( / /18X, 'NU^^BEP OF OSSERVATI ONS • , 1 6 / / ) 5 F 0 P ' ^ A T ( / / 1 C X , • P O L Y N l ^ ' I A L KrGRESSICN OF D E G ? E F « , I 3 ) 6 F O P V A T C / / i a x , » INTEPCEPT" , E 2 0 , 7 ) 7 F 0 P - / A T ( / / 1 8 X , • (-E^^PESSin-vi CDE FF I C I EN IS • / 18X ,
1 ( 6 E 2 0 . 7 ) ) 8 P G H K A T ( 1 H 0 / Z 4 X , 2 4 H ^ \ ' A L Y E I S OF VARIANC ' r n P , I 4 ,
119H D EGk t fc POL Y^ OM U L / ) 9 FOPMAK1HU,18X,1QHSCURCE OF VARIAT I U N , 5 X ,
19H0EGREF GF,5X ,6HSU** CF , 5X , AH'^E A r i / 43X , ThFP E F D ^ ^ ' , 2 6X ,7HSQUAPES,3X,6HS0UAPc >
88
10 F r > & M A T ( / / l 1)
11 I^TKN'AT ( 18X
1 ^ 1 0 . 5 ) 12 FP^^v AT ( 18X 13 F'..R,v'AT ( 1 7H I 4 FnP 'MT (}.HC
ITHOBS r C , 11 JHY t STI^ '
lb ^U- ' -AT ( I HO 1 b FORMAT ( / / /
ISMVALUE, 11 ? ' 5 X , ! ^ ? o . 5 )
1 0 0 ^ > A r . ( 5 , ' , r WKITE ( o , 3 UPITP ( 6 , 4 L = r:- M
8X,«rMjE TO REGRESSION* , e x , 1 6 , F 1 4 . 5 , p* iC,5
, 2 5 H L E V I A T I ^ ^ i ABOUT REG. , I 6 , F 1 4 . 5 ,
, 7 H T 0 T A L , 1 6 X , I 6 , F 1 4 . 5 ) 0 NC IMP^OVt '^E\T ) / / 3 2 X , 18HTA6LE 0 = RE S I L " ) U A L S / / 1 8 X ,
3X,7HX VALUE,3X*7HY V A L U E , 4 X , ATE,4X,BHPF S i n i i A L / ) , 1 7 X , I 6 , ^ 1 3 . 5 , ' " I C . 5 , F 1 4 . 5 , F 1 2 , 5 ) 3?X , IHF , l2X,2CHI.vPR(:VEMENT IN T t l ^ M S / 3 0 X , X,17H:iF SUN CF S 0 U A R E S / / 2 4 X , F 1 3 , * ^ - ,
r.r?=300) P K , P R : , f J ,M ,NPLnT ) ?F- , D f 1
) M
1 = 1 , N
no
or. 110 J = L + I R F /i L ( b , 2 ) X X ( J)=ALOc;( CALL (n>ATA '^•1M = M • » - 1
SUM=0.0 NT = h:- l DO 200 1=1 I SAVE( I ) = I CALL OK^Ef* CALL ^MNV CALL MULT*-
1 S B , T , A N S ) WRITE ( 6 , 5 I F { A N S ( 7 ) ) SUMIP=ANS( I«=( SIJMIO) W^ ITE ( 6 , 1 GO Tn
IbO WKITE WP ITE WRITE WRITE SUM=ANS(4) W R I T F ( 6 , 1 0 N I = A N S ( 8 ) WRITE ( 6 , 1 WRITE ( 6 , 1 W R I T £ ( 6 , 16 C O E d ) =ANS
( I ) , X ( J ) X ( J ) )
( N , ^*, X , X B A P , STO , 0 , SUMSQ )
130
140
M
{ ;^M,r ,MM, I , I S A V E , D I , t - ) ( 01 , 1 , D E T , P , T }
( N , I , X B A P , S T D , S U M S Q , C I , t , I S A V E , r ^ ,
) I 1 4 0 , 1 3 0 , 1 3 0
4 ) - S U M 1 4 0 , 1 4 0 , 1 5 0 3)
210 ( 6 , 6 ( 6 , 7 ( 6 , 8 ( 6 , 9
) A N S ( l ) ) ( P ( J ) , J = 1 , I ) ) I )
) I , A N S ( 4 ) , A N S ( 6 l
1) N I , A N S ( 7 ) , A ^ v i S ( 9 ) 2 ) NT,SUMSO(vN') ) A M S ( 1 0 ) , S U M I P (3 )
89
160
200 210
23 0
240
250
300
DO 160 J = l , I C 0 E ( J + 1 ) = B ( J ) LA= I EC NTIMU^ I F ( N J L O T ) R j , 1 0 0 , 220 \P3=NH-N DM -iho 1 = 1 ,0 ^ P 3 = ^P3^-l ^ ( N P 3 ) = C 0 E { 1 ) L = I o n 230 J=J ,LA P (N P3) = P(M P 3 ) + X ( L ) ^ C C E ( J + 1 ) L = L + 0 rj2=N L = . ;'»' 0 0 240 1=1 P( I )=X( I ) N2 = N2^-1 L=L+1 i ( N 2 ) = X ( L )
( 6 , 3
,N
•? I T - ( 6 , 3 ) P c , P R l wEITE (6 , ^3 ) LA W P I T E ( 6 , 1 4 ) NP2 = N 0P3='1'»-N 00 250 1=1 ,N NP2 = NP2-H MP3 = NP3-^1 R E S I 0 = P ( N P 2 ) - P ( N P 3 ) W^MTE ( 6 , 1 5 ) I , P ( I ) ,P(NP2,3 , P ( N P 3 ) ,RESID CALL PLOTT ( L A , P , N , 3 , C , 0 ) GO TO l O u CONTINUE END
90
C
c c c c c c c c c c c c c c c c c c c c
T H I S SU^»5fUTINE HAS BE :^• DESIGNED TO PLOT THE
i r ;PUT DATA F^QM ABOVE. ""HE TIME AXIS IS THrT V F ^ ^ -
IC LE A X I S Ar:0 THE LOG CC ^XENTr^ AT I ON A ^ X I S IS ^ ^^
Hn^-IZHNTAL A X I S , THE (^EGRESSION CUr^VE IS ^LS^'
PLOTTED or: •[ n r SAME GRAPH. THE • - » CHAPAC^^^
' ^ E P K ^ S E N T S A N E X P E R I - ^ ' E N T A L P O I N T . THE • »»• CH/^P-
ACTEW KE PRESENTS A f'OINT CALCULATED EY REGf^ESSICN
A N A L Y S I S .
SUBcnUT I N ' PLOTT (NO, A , N, •^,, ML , N S ) 0 I '•"- NS I ON OUT ( b» > , YP R ( 6 ) , ANG( 9 ) , A ( : ) DATA RLArK ,ANG( 1 ) , A N G ( 2 ) / « • , • * - • , • ^ • /
1 Ff PM^T(« 1» , 6 ( / ) ) 2 FORMA^(« • , 1 7 X , E 3 . 0 , 2 X , 5 1 A 1 ) 3 FORMAT (1 H ) 7 F 0 P V A T ( « • , 2 2 X , 6 ( » . • , 9 X ) ) 8 FORVAT ( ' C , 1 3 X , 6 F 1 0 . 4 )
NL=46 MLL=NL I F ( N S ) 1 6 , 1 6 , 1 0
1 0 DO 15 I = 1 , N DO 14 J = I , N I F ( A ( I ) - A ( J ) ) 1 4 , 1 4 , 1 1
1 1 L = I - N L L = J - N 00 12 K = 1 , M L = L + N L L = L L + N F = A ( L ) A ( L ) = A ( L L )
12 A ( L L ) = F 14 CONTINUE 15 CONTINUE 16 T F ( N L L ) 2 U , 1 8 , 2 0 18 NLL=50 20 WPIT!=-(6,1)
91
?6 28
3u 40
45
50 55
60
7 0
80
84
86
9 0
X S r : L = ( A ( N ) - A ( 1) ) / ( P L T A K N L L - l ) )
Y'.iT w=A ( '"1 ) yMAX=YV IN M 2 = • -• M.
00 4J J=V1 ,M2 I E ( M J )-Y'^ IN ) 2 8 , 2 6 , 2 6 1 F( / ( J )-Yf A X ) 4 U , 4 0 , 3 0 y M I ^ = A ( J ) or TO 40 Y*'AX = /* ( J ) O"?' TINUE Y 3C '' L= ( YMAX-YM IN ) / 5 0 . 0 <! = A,(1 ) L = l nY = ^ ' - l 1 = 1 F = I - 1 XPp = XB•^R^XSC AL I F ( A(L )-XPR ) 5 C , 5 J , 7 . } DO 55 IX=1 , 5 1 0UT( T X ) = E.LAN;<
no 60 J=1,MY LL = l +J» N J P = ( ( A ( L L ) - Y ' U N ) / Y S C A L ) + 1 . 0
O U T ( J P ) = ANG( J ) CO^'TINUE WR T T E ( 6 , 2 ) X P P , ( O U T ( 1 7 ) , I Z = 1 , 5 1 ) L = L+1 GO TO 8U W P I T E ( 6 , 3 ) 1 = 1 + 1 I F ( I - N L L ) 4 5 , 8 4 , 8 6 XPP=A(N) GO TO 50 W P I T E ( 6 , 7 ) YPP ( 1 ) =YfMN OC 9 ) KN = 1 , 4 Y P R ( K N + 1 ) = Y P R ( K N ) + Y S C A L ^ 1 0 . 0
YPR(6)=YMAX Y P K I o ; = T ^!« A W R I T E ( 6 , 3 ) ( Y P R ( I P ) , I P = 1 ,6 RETURN
)
END
92
PCLYN'^MIAL REGRESSION . . . S A M P L E
NUMBER OF HBSERVA-riONS 15
POLYNOMIAL REGRESSION OF OEGRFE 1
INTERCEPT - 0 . 2 5 5 7 0 3 9 F - 0 3
REC^^ESSIOSJ COEFFICIENTS - 0 . 5 7 8 5 3 6 1 E - 0 2
ANALYSIS OF VARIANCE FOR 1 DEGREE POLYNOMIAL
SOURCE OF VARIAT ION DEGREE OF SUM OF MEAN FREEDOM SQUARES SQUARE
DUE TO REG'^ESSION I 0 . 0 4 0 6 4 0 . 0 4 0 6 4 DEVIAT ION ABOUT REG. 13 0 . 0 0 0 0 0 0 . 0 0 0 0 0 TOTAL 14 0 . 0 4 0 6 4
F IMPROVEMENT I N TERMS VALUE OF SUM OF SQUARES
248 78 3 . 9 3 750 0 . 0 4 0 6 4
PCLYNO^ K L R ^ - G K E S S I O M . . . S A K P L E
PCLYNOMIAL ^EG^f^SSIC\ OF DEGREE 1
93
TABLE OF RESIDUALS
CBS NO, X VALUE Y VALUE Y ESTIMATE RESIDUAL
0 . 0 0 . 0
3
4
5
6
7
8
9
10
11
12
13
14
i 5
2 . 0 8 0CJ - 0 , 0 1 2 0 7
4 , ^ 7 0 0 0 - 0 . 0 2 4 2 9
6.250C0 - 0 , 0 3 6 6 6
8 , 3 3 0 0 0 - 0 - 0 4 9 1 9
1 0 . 4 1 0 0 0 - 0 . 0 6 0 8 1
12.5CCO0 - 0 . 0 7 2 5 7
1 4 . 5 8 0 0 0 - 0 . 0 8 4 4 7
1 6 . 6 5 9 9 9 - 0 . 0 9 6 5 1
1 8 . 7 3 9 9 9 - U . 10870
2 0 . 8 1 9 9 9 - 0 . 1 1 9 9 1
2 2 . 8 9 9 9 9 - 0 . 1 3 2 3 9
2 4 . 9 8 9 9 9 - u . 1 4 5 0 3
2 7 . 0 6 9 9 9 - 0 . 1 5 6 6 5
2 9 . 1 5 9 9 9 - 0 . 1 6 9 6 0
0 .00026
0 .01229
0 .02438
0 . 0 3 6 4 1
•0 .04845
•0 .06048
0 .07257
•0 .08461
•0 .09664
•0 .10867
0 . 1 2 0 7 1
• 0 . 1 3 2 7 4
0 . 1 4 4 8 3
•0 .15687
•0 .16896
0 .00026
0 ,00022
0 .00009
- 0 . 0 0 0 2 5
- 0 . 0 0 0 7 4
- 0 . 0 0 0 3 3
0 .00000
0 . 0 0 0 1 4
0 . 0 0 0 1 3
- 0 . 0 0 0 0 3
0 . 0 0 0 8 0
0 . 0 0 0 3 5
- 0 . 0 0 0 1 9
0 . 0 0 0 2 1
- 0 . 0 0 0 6 5
94
0, #t
3,
5.
6.
a.
21.
23.
Time Versus Log-Concentration
Benzoic Acid
11.
13. #
15.
17.
19.
25. #
27.
29. i . . . . .
-0.1696 -0,1357 -0.1018 -0.0678 -0.0339 0.0