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^

i H

-3-00

o

OJ

c •H JJ CO

a 1

H-J

m o o t l 00

•<f

>d--d-

r-i

u D

0)

O

0)

o

d

0)

CM O

4 CM

iH CM

O O O Q O

in ON r^ 1/

00

o W rH •

O

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

o o

+ 1 vO

O

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

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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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M CO

H

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c Ps T3

m o i H N—/"

. d •p

bO ti 0)

CO

0) ,—1 • H CO ti o H

o • H •M CO >-i

' d >^ ffi I H

o

o •H 4->

3 i H O

CO

ose

u o ti

CO

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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

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THE MASS T^ANSFcf^ C ' : ^ ^ F F I C I E M S .

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