He...” h we... “1.!— ‘11.“ 1:... IR“. nKU fl}! 1.1“ Ibul“ II My“ FDA urn m h!- ”v. u fiv «raw. flu. Mk I»! .,....r.. 192”» NH 3 1mm nun \) E... .| i v u .3 n ‘ B r. ‘ :u rm .. 1V! x Gun... a.» mu“ gnu... 5L“ M‘szb 6:0 v... . ,P“‘ F «Lin i... ‘L .- u..: b “an” J‘s-5.1"“: .4... “1.. .4. .. L B R A R Y M icing-an Staif Umvcrsi t .f’ I Mme ‘ - 2\ twat-\"A This is to certify that the thesis entitled DIFFUSION 0F SUGAR IN CUCUMBERS AND P l CKLES presented by BRUCE DELBERT EDER has been accepted towards fulfillment of the requirements for M degree in .EaadJcience /7/Zu€e 333% Major professor Date May 18, 1270 ‘ 0-169 ABSTRACT DIFFUSION OF SUGAR IN CUCUMBERS AND PICKLES BY Bruce Delbert Eder Sweet pickles are manufactured from cucumbers or de- salted salt stock pickles by diffusion of sugar from sugar solutions. Because the sweetening process is time consum- ing, there is considerable interest in speeding up the process. This project was undertaken to elucidate sugar diffusion characteristics of cucumbers and salt stock pickles and to investigate methods of increasing the rate of sugar diffusion into these products. This necessarily involved development of methods for measurement of the diffusion rates. Model diffusion systems were developed to measure rel- ative diffusion rates through the product tissue. The dif- fusion rates in various parts of the product were determined by measuring the rate of uptake of C14 labeled sugar by sections of cucumbers or pickles from a large volume of solu~ tion. The sugar concentration was monitored using a liquid scintillation counting technique. Diffusion rates in the fleshy tissue were expressed as diffusion coefficients while Bruce Delbert Eder the rates through the skin were expressed by permeability coefficients. Labeling studies revealed that sucrose was inverted by endogeneous enzymes in both cucumber and pickle tissues. Therefore, because it was difficult to quantitate the dif- fusion of sucrose in these tissues, the diffusion experi- ments were conducted using glucose as the diffusant sugar. The diffusion rates in various parts of the cucumber flesh were found to be nearly the same (D at 25C2r2.6x10'6 cmz/sec for 3 cm diameter cucumbers, variety SMR-lS). The skin offered a significant resistance to diffusion; the permeability coefficient of the skin (25C) was found to be approximately 1.7x10"5cm/sec. The diffusion rates in the different varieties tested (SMR-lS and a mixture of SMR-S8 and Pioneer) were not significantly different. In contrast, the diffusion coefficients (25C) in various parts of the salt stock pickle flesh (variety SMR-SB and Pioneer mixed) were different. The diffusion rate of 'GcmZ/sec) was glucose in parenchymatous tissue (D216.2x10 much higher than that in the seed cavity area (03:4.0x10'6 cmz/sec). The permeability coefficient of the skin (25C) was approximately 8.0x10'5cm/sec. Thus, the salt stock pickles were found to be much more permeable than the cucum- bers. The parenchymatous tissue of salt stock pickles offered little resistance to diffusion (D for glucose in water equals 6.9x10-6cm2/sec). Bruce Delbert Eder The diffusion rates in cucumbers were found to be highly temperature dependent. The diffusion rates through the skin were most sensitive to changes in temperature. lllllllll! ulllll DIFFUSION OF SUGAR IN CUCUMBERS AND PICKLES BY Bruce Delbert Eder A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Food Science 1970 one--- W—a To my wife and children ACKNOWLEDGEMENTS I want to express my sincere appreciation to my major professor, Dr. J. R. Brunner, for his guidance and encour- agement throughout this study and during preparation of this manuscript. I would also like to thank my guidance committee, Dr. D. Heldman, Dr. D. K. Anderson, and Dr. P. Markakis for their suggestions in preparation of this thesis and for their advice throughout the study. Thanks go to Dr. B. S. Schweigert, Chairman of the Food Science Department, for supplying the assistantship which partially supported this program. The author wishes to acknowledge Florasynth Inc. and General Foods for supply— ing fellowships through the IFT in support of this research. I also wish to thank Mr. G. Staby for his help in operation of the radiochromatographic scanner. Special appreciation goes to my wife for her patience and support throughout this program and for her assistance in preparation of this manuscript. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE INTRODUCTION LITERATURE REVIEW Diffusion and Transport in Biological Systems Diffusion in Cucumbers and Pickles MATERIALS AND METHODS Chromatography of Sugars Incubated with Cucumbers and Pickles Measurement of Diffusion Rates Calculation of Sugar Concentration and Recovery from Chromatograms Calculation of Diffusion Coefficients RESULTS AND DISCUSSION Density and Moisture Content of Test Sections Recovery of Glucose from Sample Sections by Homogenization Chromatography of Sugars Exposed to Cucumbers and Pickles Diffusion in Cucumbers and Salt Stock Pickles SUMMARY AND CONCLUSIONS LIST OF REFERENCES iv Page vi viii ix 16 16 18 35 38 51 51 53 55 65 93 99 APPENDIX I Derivation of the Equation Used to Calculate the Average Thickness of a Hollow Cylinder Wall and Error Analysis Derivation of Finite Difference Equations (Crank-Nicolson Method) APPENDIX II 102 102 104 115 Table Table Table Table Table Table Table Table Table Table Table 10. 11. LIST OF TABLES Densities and Total Solids (Moisture Content) Data for Test Sections of Cucumbers Density and Total Solids (Moisture Content) Data for Test Sections of Salt Stock Pickles (2.5 Cm Size) Recovery of Sugar from Sample Sections as a Function of the Homogenization Conditions Per Cent Inversion of Sucrose Incubated with Cucumbers and Salt Stock Pickles Recovery of C14 Labeled Sugars From Radiochromatograms of Sugars Incubated with Cucumbers and Salt Stock Pickles Diffusion Coefficients of Glucose (25C) in Cucumber Tissue as a Function of Storage Time at 25C in Aqueous Solution Results of Diffusion Experiments of C14 Glucose in Hollow Sections of Cucumbers Showing Correlation Coefficients of the Diffusion Coefficients (D) with Time (t) and with the Concentration Ratio (Y) Diffusion Rates of Glucose (25C) in Different Areas of Cucumbers (3 Cm Diameter) Diffusion Rates of Glucose (25C) in Cucumbers as a Function of Storage Time in Air at 4C Diffusion Rates of Glucose (25C) in Different Sizes of Cucumbers (Variety sun-15) " Diffusion Rates of Glucose in Cucumbers (2 Cm Diameter) vi at Different Temperatures Page 52 53 55 62 64 67 77 79 81 Table Table Table Table Table Table Table 12. 13. 14. 15. 16. 17. Diffusion Rates of C14 Labeled Glucose (3.4xlO"6 Per Cent) and Cl4 Labeled Glucose Plus One Per Cent Glucose in Cucumbers (SMR-lS) at 25C Comparison of Diffusion Rates (25C) of Glucose in Different Areas of Cucumbers and Salt Stock Pickles Diffusion Rates of Glucose (25C) in Salt Stock Pickles as a Function of Storage Time in Salt Brine (12.9 per cent) at 4C Diffusion Rates of Glucose (25C) in Salt Stock Pickles as a Function of Storage Time in Solution at 25C Comparison of Diffusion Rates of Glucose (25C) Determined Using Hollow Cylinders with those Determined Using Cylinders of 2.5 Cm Diameter Salt Stock Pickles Comparison of Diffusion Rates Calculated Using the Crank-Nicolson Finite Difference Method to those Calculated from the Exact Solution for Diffusion in a Cylinder A-l.Results of Diffusion Rate Experiments for C14 Glucose in Cucumbers and Pickles vii 81 84 85 86 87 92 115 .AJ‘Lll (kill (Ill I'll/I‘ll. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1. LIST OF FIGURES Cylindrical sections of cucumbers. A. Whole section with skin. B. Peeled section. C. Seed cavity section. Hollow cylindrical sections of cucumbers. A. Section with skin intact. B. Peeled section. C. Smaller section of parenchymatous tissue. Unassembled sample cells and samples Assembled sample cells containing samples Diffusion test system consisting of the diffusion test chamber, a temperature regulating water_bath, and a circulating pump Inside view of the diffusion test chamber showing the stainless steel coil and positioned samples Block diagram for computing D and K in cylinders and hollow cylinders from experimental data using the exact solutions for infinite cylinders and slabs, respectively Block diagram for computing D and K in composite cylinders using the Crank-‘ Nicolson finite difference method Recovery of glucose from sample sections as a function of homogenization time and Omnimixer shaft speed Profile of chromatograms obtained for incubation of C sucrose with cucumbers (36 hr) viii Page 25 25 28 28 31 31 46 49 55 56 Figure Figure Figure Figure Figure Figure Figure 11. 12. 13. 14. 15. 16. 17. Profile of chromatograms obtained for in- cubation of Cl4 sucrose, and l and 10 per cent sucrose with cucumbers (68 hr) Profiles of chromatograms obtained for incubation of C glucose with cucumbers (72 hr) Profiles of chrpmatograms obtained for incubation of C sucrose with salt stock pickles (30 hr) Profiles of chrimatograms obtained for incubation of C 4 glucose with salt stock pickles (30 hr) Diffusion coefficients of glucose (25C) in cucumbers as a function of storage time of the cucumber sections at 25C in aqueous solution (1% acetic acid and 1000 ppm potassium sorbate) Diffusion rate curves calculated for diffusion of glucose in 3 cm cucumbers with and without skin (D - 2.6::10"6 cm /sec, K - 1.7Sx10’5 cm/sec) Relationship of D, R, and Bi to cucumber size ix 57 58 59 60 68 75 78 ai' bi, Ci, d1 30(6): J1M) NOMENCLATURE recursion coefficients for finite difference equations area of interval i cm2 thickness of a hollow cylinder wall cm Biot number, E5 or 55. dimension- D D less concentration; Cav ' average concen- radioac- tration of solute In a body; Ccon' tivity concentration of sugar in radio- (dpm) chromatographic peaks of the control per solution; Ch, concentration of homo- unit genate; Cir concentration at node 1 time and time t; CI, concentration at node i and at time t + At; CO! initial con- centration; Cs, concentration of sugar in radiochromatographic peaks of the incubation solution; C , diffusion medium concentration diffusion coefficient (diffusivity); cmz/sec Do, initial diffusion coefficient: 5, average diffusion coefficient dilution factor Napierian base, 2.71828... height cm molar flux moles 3.773% Bessels functions of the first kind and of zero and first orders, respec- tively number of nodes in a cylinder permeability coefficient cm/sec (t) length modulus in finite difference equa- tions; = DAt/( Ar) modulus in finite difference equa- tion A—l9 modulus in finite difference equation A-23; 8 KAt/Ar mass; Mc' mass of sample new value of Bi level of significance variable radius radius; R0, outside radius; R1, inside radius percentage recovery of sugars from cucumbers or pickles percentage recovery of sugars from incubation solutions percentage total recovery of sugars standard deviation time; t , experimental time exp time increment t statistic volume; Vd' dilution volume; Vi' volume at 1. Vs, volume of solution moisture content distance increment of distance unaccomplished average concentration C .- C = C: _ cavg ; Ycalc' calculated 0 Y value; Yexp' experimental Y value xi C111 dimension- less dimension- less dimension- less 9 dimension- less cm SOC sec CC ml/g cm cm dimension- less root of the Bessel equation thickness of a membrane or of the cm skin Dt - 8 dimension- 52 less 3.1417.... density g/cc xii INTRODUCTION Sweet pickles may be produced from either fresh cucumbers or salt stock pickles; however whole sweet pickles are manufactured from salt stock pickles. Salt stock pickles are produced by fermentation of fresh cucumbers in salt brine maintained at 5-10 per cent salt. As the fermentation progresses to completion, salt is gradually added to the salting tank until the contents are brought to 14—16 per cent salt by weight. Under these conditions the pickles may be held for a year or more and processed when needed. When manufactured from fresh cucumbers, sweet pickles are made by adding both the cucumbers and the sweet liquor to the jar, followed by sealing and pastuerization. When sweet pickles are produced from salt stock pickles, desalted pickles are placed in tanks containing sugar solution and allowed to equilibrate. Then, additional sugar is added to the tanks and the contents are again allowed to equi- librate. This process is repeated until the desired sugar level is attained. Sucrose and glucose are used individ- ually or in combination for sweetening. The sugar diffusion process is time consuming, requir- ing several weeks to bring the product to the desired sugar concentration (20-25 per cent) and to restore the pickles 1 2 to their original volume. The maximum sugar concentration to which the cucumbers or salt stock pickles should be exposed is limited by shrinkage of the cucumbers resulting from osmotic pressure differences. Exposure to too high a sugar concentration will result in irreversible shrinkage. As the sugar diffusion rate is increased, the shrinkage is decreased. Consequently, there is considerable interest in developing methods to accelerate the diffusion process. This involves a better knowledge of the diffusion charac- teristics of the various parts of cucumbers and salt stock pickles. Although many processes involve diffusion of solutes in a foodstuff. little attempt has been made to describe the process rates in terms of transport properties such as diffusion coefficients. In homogeneous, isotropic systems, diffusion rates can be expressed by one parameter, the diffusion coefficient, which is independent of time and location. One of the objectives of this research was to develop methods of measuring and expressing the perme- ability of cucumber products in terms of diffusion coef- ficients. Since the diffusion rate of sugar in cucumbers cannot surpass that of sugar in water, it is also of interest to express the sugar diffusion rate in cucumbers in terms of a diffusion coefficient which can be compared to the diffusivity of sugar in water. It is generally believed that a salt stock pickles is more permeable than a fresh cucumber. In the production 3 of salt stock pickles, the cucumber is thought to undergo considerable cellular degradation during curing and fermen— tation. It would be of interest to know the magnitude of the diffusion resistance of cucumbers as compared to that of salt stock pickles. The specific objectives of this research were: 1. to develop a method of measuring sugar diffusion coefficients in cucumbers and salt stock pickles. 2. to determine the relative sugar diffusion resist- ances of various areas (i.e., skin, seed cavity and paren- chymatous tissues) in cucumbers and salt stock pickles. 3. to determine the relative sugar diffusion resist- ances in salt stock pickles as compared to fresh cucumbers. 4. to study factors affecting sugar diffusion rates in cucumbers and salt stock pickles (i.e., diffusion temperature, sugar concentration, type of sugar, and size of product). 5. to find practical methods for increasing the rate of sugar diffusion into the product. LITERATURE REVIEW Diffusion and Transport in Biological Systems Mechanisms of Transport A considerable amount of work has been done on the study of mass transport in biological systems and numerous reviews on the subject have appeared in recent years (Tuwiner, 1962; Bunch and Kallsen, 1969). Tuwiner reviewed the findings on mass transport of ions and molecules through plasma membranes and listed five transport mechanisms. These are: (1) mass flow through pores; (2) diffusion which involves no specific structural relationship between the membrane and permeating material; (3) facilitated diffusion which involves a specific structural relationship between the membrane and diffusing substance; (4) active transport which involves a specific structural relationship between the membrane and the permeating species and a supply of energy from metabolism; and (5) through pincocytosis, an engulfing or invaginating mechanism of the cell membrane which can occur in plant cells. Although, in living cells, much of the transport through membranes cannot be accounted for by diffusion or other physical forces alone, many researchers have applied diffusion equations to quantitate the transport of ions 4 5 and molecules (Jacobs, 1933; Jacobs, 1967; Jacobs and Stewart, 1932; Tuwiner, 1962; Rogers and Perkins, 1968; Bunch and Kallsen, 1969). Other workers have used kinetic analysis based on a ”carrier" model, in which a carrier transports the solute across the cell membrane, to quan- titate the transport (Moore and Schowsky, 1969; Tuwiner, 1962). In nonliving membranes the transfer of molecules is by diffusion and is fundamentally no different from diffu— sion in liquids (Tuwiner, 1962). In theory the same laws apply. Structurally, some membranes have been shown to possess properties corresponding to a solid phase dispersed in a liquid phase while other membranes have been shown to possess properties of a solid solution. Expression of Diffusion Rates The driving force for diffusion is associated with a gradient of chemical potential which in the case of liquids is a function measuring the potential energy of a molecule of solute. These quantities depend upon the temperature and solution composition. Although the true driving force for diffusion (a gradient of chemical potential) is expressed in terms of activities, the diffusion equations are usually expressed in terms of concentration gradients. If J is the molar flux of solute (moles/cmz/sec), it is related to the concentration gradient, fig, by: = _ dC J DE? (1) where D is the diffusion coefficient in units of square centimeters per second and is defined as the amount of material which is transferred in unit time through a unit area under the influence of a unit concentration gradient. This is an expression of Fick's First Law. The gradient of J, dJ/dx, represents the time rate of change of the concentration, dC/dt: dC d2C = D— (2) at" dx2 This is Fick's Second Law. The equation is applicable when the diffusion coefficient is constant and when diffusion is one-dimensional. With appropriate boundary conditions this differential equation has been solved for many cases encoun- tered in diffusion (Jacobs, 1967; Crank, 1956; Tuwiner, 1962; Newman, 1931). If a membrane is sufficiently thin the concentration difference (Cm - C) across the membrane will be nearly linear at any instant and the concentration gradient, g§. can be expressed as BEEF—E! where is the membrane thickness. Then Fick's Law (eq. 1) can be expressed as J = mm”; C) (3) It is frequently convenient to combine D and ¢f and some- times a partition coefficient into a single permeability coefficient, K. For the case where J' is unknown and the 7 partition coefficient is unity, then, if we let K = D/J . eq. 4 becomes J = K(Ca;' C) (4) The permeability coefficient, K, is then defined as the amount of material which in unit time will cross a unit area of the membrane with a unit concentration difference across the membrane and has units of centimeter per second. This constant suffices to define the relative rates of diffusion of different substances across a membrane. But to obtain absolute values of these rates or even to obtain the same numerical values with different membranes, it is necessary to employ true diffusion coefficients. On the other hand, to obtain a true diffusion coefficient for membranes requires a knowledge of the partition coeffi- cient between aqueous and non-aqueous phases, and of the membrane thickness (which is most difficult to measure with accuracy). The permeability coefficient has been used to express diffusion rates through individual cell membranes (Jacobs, 1967). Methods of Measurement of Diffusion Rates Jacobs (1967) and Tuwiner (1962) describe methods of applying diffusion laws to transport in biological systems and membranes. It is not within the scOpe of this review to describe the numerous methods which have been used to quantitate transport in cells and membranes, but a few 8 will be reviewed which illustrate a variety of approaches. Collander and Barlund (1933), as cited by Jacobs (1967), studied diffusion into cells of the plant Chara. Their results indicated the applicability of Fick's Law to cell permeability. Their work justified the common simplify- ing assumption that the delay in diffusion is very slight in other regions when compared to that across the cell membrane; therefore, the solute could be considered to be uniformly distributed throughout the entire cell volume at any given time. Jacobs (1933), Jacobs (1967), and Jacobs and Stewart (1932) measured the permeability of Arabacia eggs to solute and water by observing volume changes in the eggs when exposed to hypertonic solutions. The eggs would at first shrink and then recover to their original volume. At the point of minimum volume(g!.- 0), the equations describing dt the transport of water and solute could be simplified and solved to obtain an approximate value of the permeability coefficient for the solute when the volume and area were considered constants. By solving the equations numerically using the Runge-Kutta method, the inexact assumption of constant area and volume was eliminated and the permeability of water as well as the solute could be calculated. Bunch and Kallsen (1969) studied the rate of intra— cellular diffusion of water, urea, and glycerol in the giant barnacle Balanus nubilus. The experiments were per- formed by exposing one end of the long muscle fiber to a 9 solution containing radioactively labeled solute and then determining the distribution of label after exposure for a certain time. Using the error function solution of the diffusion equation, they calculated diffusion coefficients and found that they were not different from those reported for dilute solutions. They concluded that one may reason- ably assume that the rates of intracellular diffusion are equal to those in dilute solution. Diffusion in Cucumbers and Pickles Measurement of Desalting Rates Several investigators have studied the diffusion of salt and sugar into salt stock pickles, but none have inves- tigated diffusion in fresh cucumbers. In some of the earliest work, Switzer et a1. (1939) measured the diffusion rate of salt from salt stock pickles. In the batch experi- ments they placed equal weights of pickles and water in jars and measured the rate of increase of salt (NaCl) in the solution by refractive index. A standard curve of salt concentration (which was determined chemically) versus the refractive index of the original brine at various dilutions was used to convert the refractive index readings to salt concentrations. During these experiments the content of the jars were agitated only when sampled; thus it is possible that stratification of the salt solution took place in the jars. Also, these same investigators measured the desalting rate in running water. The salt content remaining 10 in pickles at various times was determined by squeezing the juice from several pickles and measuring the salt by chemical determination or by salometer. Pflug et a1. (1967) performed similar desalting exper- ments. In the batch or equalization experiments, the contents of the experimental tanks were mixed by bubbling air through them. Experiments designed to measure the extent of desalting in running water were carried out in the same manner as the experiments of Switzer et a1. (1939). Diffusion coefficients were calculated by assuming that the geometry of the pickle approximated an infinite cylinder, that the diffusion of salt in the pickle is one-dimensional and that the pickle is homogeneous and isotropic. The diffusion coefficients were calculated from the equaliza- tion data using the charts of Crank (1956) for diffusion from an infinite cylinder into a stirred solution of limited volume. The data from the running water studies were ana- lyzed by plotting the logarithm of the difference between the average salt concentration in the pickle and the con- centration in the desalting solution (in this case zero) versus time. The diffusion coefficients were calculated from the slope of the straight line asymptotes of the resulting curves. In the batch experiments Switzer et a1. (1939) deter- mined diffusion rates at four different temperatures (38F, 68F, 98F, and 138F) and found that the rate of desalting increased with temperature. They concluded that the 11 difference between the diffusion rate at 38F and that at 68F was greater than the difference between the diffusion rate at 68F and the rates observed at the higher temper- atures. However, their data were difficult to interpret. The similar experiments, Pflug et a1. (1967) determined diffusion coefficients at 66F, 120F, and 160F and found that the diffusivity increased proportionally with in- creasing temperature. At 160F the diffusivity was found to be more than two and one half times that at 66F. Pflug et a1. (1967) determined diffusion coefficients on four sizes of cucumbers (3/4, 7/8, 1 1/8, and 1 3/8 inch diameter) in both equalization and flowing water experiments. They found that the average diffusion coef- ficients calculated from the equalization tests were con- sistantly larger for the smaller sized cucumber; however, the diffusivities calculated for each size were highly variable. These values increased markedly with time. Sometimes the increase was two to three fold over the range of experimental times observed. Possibly, the assump- tion that the salt stock pickle is homogeneous with respect to diffusion rate is not valid; the skin may not be as permeable to salt as the flesh. Also, if the contents of the tank were not agitated properly, there would be some resistance to diffusion at the surface of the pickles. Both of these factors would lead to the kind of results obtained, namely that the observed diffusion coefficient would in- crease with time. 12 The diffusion coefficients calculated from the flowing water experiments showed a slight tendency to increase with the cucumber size. The plots of the logarithm of the con- centration difference versus time appeared to yield straight line asymptotes; however, the intercepts of the asymptotes with the abscissa at zero time were generally higher than would be expected for a homogeneous system. These higher intercepts are observed when there is a finite surface resistance such as would be the case if the skin were less permeable than the flesh. As mentioned earlier, Pflug et a1. (1967) assumed that the shape of the pickle was close to an infinite cylinder for ease in calculating the diffusion coefficients. Gen- erally the ratio of length to diameter of cucumbers ranges from 2.5:1 to 3:1. For an isotropic medium, Kopelman (1966) showed that this assumption would lead to a 5-7 per cent error in the diffusion coefficient. If the diffusivity in the longitudinal direction was greater than that in the radial direction, the error would even be larger. Switzer et a1. (1939) also studied the effect of size on desalting rates in running water. From their data it was difficult to determine if the relative diffusion rates of the pickles varied with size. The only conclusion that could be made was that the salt diffused out of the smaller pickles faster. Pflug et a1. (1967) observed that the diffusion coef- ficients for the smaller size pickles (3/4 inches) were not l3 much different than the diffusivity of salt in water while the diffusion coefficients for the larger pickles were 20-40 per cent less than that of salt in water. The dif- fusion coefficients evaluated from flowing water and equal- ization tests did not appear to be different. Measurement of Sweetening Rates Fabian and Switzer (1940) studied the processing of sweet pickles and determined rates of sugar uptake by processed (desalted salt stock) pickles. Since pickles shrink when subjected to high sugar concentrations, both the volume change of the pickle and the change in sugar concentration of sweet liquor were measured as functions of time. To start the experiments pickles and sweet liquor containing sugar, vinegar, and salt were added to jars. The volume of pickles was measured at each sampling time by measuring the volume of drained liquor. At the same time the Brix of the liquor was determined. After 45 and 87 hours, additional sugar was added to bring the sugar to the desired levels. Readings of volume and Brix were taken throughout the experiment. The data were analyzed by plot- ting the percentage of the original pickle volume versus time and the percentage change in sugar concentration of the liquor versus time. These same researchers determined the rates of sugar penetration with and without acetic acid added. When no acid was used, 100 ppm Dowicide B (a sodium salt of 2,4,5 l4 trichlorophenol) was added to prevent spoilage. They found the diffusion was fastest when no acid was added; however their results are subject to question since phenol com- pounds are known to interact with semipermeable membranes and, thus, increase their permeability. Fabian and Switzer also found that the manner in which acetic acid was added affected the sweetening rate. The acetic acid was introduced in three ways: (1) by mixing pickles which had been equilibrated to 4 per cent acetic acid with sugar solution which contained no acid; (2) by mixing pickles which had been equilibrated to 2 per cent acetic acid with sugar solution which contained the remain- der of the acid; (3) by mixing pickles equilibrated to 0 per cent acetic acid with sugar solution containing all the acid. Regardless of the method of addition, the final equilibrated acetic acid level was 2.15 per cent. From the results obtained, they concluded that the sugar pene- tration rate was fastest when all the acid was added to the sugar solution. But, because the acetic acid as well as the sugar contributed to the Brix reading of the liquor and because the Brix reading of all the liquors were the same, they were actually measuring the diffusion of varying amounts of sugar and acetic acid in combination. When testing the sweetening rates of dextrose and sucrose, and dextrose and sucrose in combinations, they found that dex- trose penetrated faster than sucrose alone. In their procedure, Fabian and Switzer made no mention 15 of the sweetening temperature or if the temperature was controlled. Since the diffusion rate is highly temperature dependent, small differences in temperature could lead to substantial differences in the observed diffusion rate. Anatomy of the Cucumber The cucumber (Cucumis sativus L.) is a member of the cucurbitaceae family (gourd family). The ovary of the fruit is usually tricarpellate, but occasionally the ovary may be four-celled (Hayward, 1938). The fleshy tissue of the cucumber consists of large parenchyma cells with those in the seed cavity area being larger than those in the sur- rounding parenchymatous tissue. The cucumber tissue contains intercellularly entrapped air; for cucumber variety SMR-58, Fellers (1964) found approximately 7 per cent air (v/v) in the seed cavity area and 10 per cent air (v/v) in the surrounding parenchymatous tissue. The surface of the cucumber contains a single cell thickness of epider- mal cells covered with cutin on the outer most surface which is almost impervious to water (Hayward, 1938). The cucumber surface is also dotted with spines through which the cucum- bers respire. MATERIALS AND METHODS At the beginning of this study, the diffusion rate data for sucrose in cucumbers was difficult to duplicate. Therefore, it was decided to determine the fate of sucrose during its exposure to cucumbers and salt stock pickles. Chromatography of Sugars Incubated with Cucumbers and Pickles Descending paper chromatography was performed to deter- mine if sucrose and glucose, the sugars used in the diffu- sion tests, were changed or broken down during diffusion into cucumbers and pickles. Weighed slices of cucumbers or pickles were placed into a known volume of aqueous solution containing the Cl4 uniformly labeled sugar, 1 or 2 per cent acetic acid, and 1000 ppm potassium sorbate. The contents were allowed to incubate for several days at room temper- ature. However, before incubation was initiated, 1 ml of the diffusion medium was withdrawn and placed in a test tube to serve as a control which was sampled at the same time as the cucumbers and incubation solution. At pre- selected times, cucumber sections were removed from the solution, rinsed with distilled water, allowed to drain several minutes, and homogenized with a measured volume of 16 17 water in a Sorvall Omnimixer. The homogenate was filtered and 10,ql portions of the filtrate, incubation solution, and control were spotted 1 inch apart on Whatman No. 1 paper (18 inches in length). Aliquots of 10,ul of the sugar standards, glucose, fructose, and sucrose, (2 per cent w/v) were also spotted on the same paper. The solvent system employed to develop the chroma- tograms, n-butanol: acetic acid: water (4:1:5 v/v/v), has been shown to give good separation of glucose, fructose, and sucrose (Brock e; :1" 1955). The paper was developed until the solvent front was several inches from the lower edge of the paper (about 12 hr) and then dried at room temperature for 1 hr. To achieve good separation, the paper was developed a second time in the same manner. After again drying the paper, the areas containing the standards were sprayed with 0.1N HCl and heated several minutes at 110C to hydrolyze the sucrose. These areas were sprayed with benzidine reagent (0.5 g benzidine, 200 m1 glacial acetic acid, and 80 ml absolute alcohol) and placed in the oven at 110C for 5 min. The sugar standards appeared as yellowish-brown spots. Strips (1 inch wide) containing radioactive spots of each sample were cut from the chromatogram and in the most cases locations of the radioactive spots were determined using a Packard Model 7201 Radiochromatogram Scanner. How- ever, in some experiments, the distribution of label on the 18 strip was determined by cutting the strip into 1 cm long segments, placing each of these segments in scintillation vials along with 10 ml of dioxane scintillation cocktail (6 g PPO, .2 g POPOP, and 100 g napthalene per liter p-dioxane; Rapkin, 1963) and counting the radioactivity using a Packard Tri-Carb Scintillation Spectrometer. Chase and Rabinowitz (1967) state that the radioactivity of the spots can be determined in this manner without extracting the labeled compounds from the paper. Wang (1959) also showed that the orientation of the paper strip to the photo- tubes of the spectrometer does not affect the efficiency of counting C14 labeled compounds. The recoveries of radioactive sugars from the peaks of the chromatograms which had been scanned were measured by cutting the areas of the strips containing the peaks into 2 cm segments and determining their radioactivities by liquid scintillation counting as described previously. From the total radioactivity of the spots on each strip, the per cent recovery of radioactive sugars from the cucumber sample and incubation solution could be calculated (see ”Calculation of Recoveries of Sugars from Chromatographic Peaks" for details). Measurement of Diffusion Rates Design of Experiments On a cellular level, diffusion through plant tissues is complicated. The solute must pass through cell walls, l9 membranes and into the cytoplasm, and is thought to pass through various compartments within the cell. For purposes of describing the diffusion rates in cucumber tissue, the diffusion in each part of the tissue was described by an overall effective diffusion coefficient, D. Since the size of the tissue is large compared to the cell size, the solute will diffuse at a uniform rate through a given area of the tissue even though the diffusion rate through each part of a cell is not the same. The diffusion rate through the cucumber or pickle skin was described by a permeability coefficient, K. Since the skin is not uniform (the outer surface of the cucumber con- tains spines which are broken off during commercial handling leaving small openings or pores in the skin) and because its thickness is difficult to measure, it was difficult to deter- mine diffusion coefficients for the skin. Because the thick- ness of the skin was very small (0.01 mm), the skin was considered to be infinitely thin, and, therefore, its diffu- sion rate was expressed by K. The prime objective in the develOpment of the experi- mental diffusion method was to obtain comparative diffu- sion and permeability coefficients for cucumbers and pickles but not necessarily to duplicate industrial conditions. Under industrial processing conditions the diffusion takes place from solutions of high sugar concentration where the product first shrinks to a minimum volume followed by par- tial recovery of the volume lost. Analytical description 20 of this process is complicated. Initially, there is bulk transport of water out of the product as well as diffusion of sugar inward. After the minimum volume is reached, both sugar and water are transported into the product. Therefore to determine diffusion or permeability coeffi- cients in this system, the simultaneous transport of sugar and water as well as the dependence of the sugar diffusion coefficient on the sugar concentration would have to be considered. Because this system is complicated and because we were mainly interested in obtaining relative values of the diffusion and permeability coefficients, the experi- ments were conducted in a model diffusion system. In this system the problem of shrinkage due to osmotic pressure differences was eliminated by using C14 labeled sugar which could be detected in low concentration as the diffusant. Use of radioactivitely labeled sugars had several advantages: (1) the amount of radioactivity could be measured essentially without interference from other soluble solids including sugars in the product; (2) because the amount of labeled sugar required to obtain measurable radio- activity was low, the diffusion coefficients could be measured at essentially infinite dilution where concentra- tion dependence of the diffusion coefficient did not have to be considered; (3) isosmotic conditions could be main- tained. Experiments were designed to measure the average in- crease of sugar concentration in cucumber tissue sections as 21 a function of the time of diffusion from a large volume of sugar solution. The volume of solution was large enough compared to that of the tissue sections that its sugar con- centration was essentially constant during an experiment. Initially, it was planned to use a limited amount of solution where the decrease in concentration of the solu- tion could be measured because it is easier to sample the solution than to sample the cucumber. But difficulty was encountered in setting up experiments where the ratio of cucumber to solution volume was high enough to permit an accurate calculation of diffusion coefficients from the experimental measurements. As the ratio of cucumber to solution volume decreased, the overall change in concen- tration of the solution was less. In turn, the calculated value of the diffusion coefficient became more sensitive to errors in the measured solution concentration. In the case where an essentially infinite volume of solution with respect to the cucumber volume was used, and where the cucumber was sampled, the overall change in concentration of the cucumber was maximized (from zero to the solution concentration) and the calculated value of the diffusion coefficient was least sensitive to measurement errors. Since the cucumber shape is approximately cylindrical, it was decided to determine rates of radial diffusion in the cucumber. The cucumber consists of several distinct areas: the seed cavity, the parenchymatous tissue and the skin. The diffusion rates in these areas were determined 22 by using whole or hollow cylindrical sections of the cucumber. In experiments where whole cylindrical sections were used, the diffusion rate was first tested in cylindrical sections of the seed cavity tissue. Assuming that these sections of the seed cavity tissue were homogeneous with respect to diffusion, the diffusion coefficient of the tissue was calculated. Then the diffusion rates were tested in larger cylindrical sections of the cucumber con- taining both the seed cavity tissue and the surrounding parenchymatous tissue. Knowing the diffusion coefficient of the seed cavity tissue, the diffusion coefficient in the surrounding parenchymatous tissue was calculated, assuming that it was homogeneous. Finally, with known diffusion coefficients in the seed cavity and parenchyma- tous tissue, the permeability coefficient through the skin could be determined by measuring the diffusion rates in whole sections of the cucumber with the skin left intact. When hollow cylindrical sections were used for deter- mining diffusion rates, the sections consisted of only one kind of tissue, except when sections containing the skin were tested. But in this case, the skin was too thin to obtain a homogeneous section. When hollow cylindrical sections other than those containing the skin was tested, the diffusion coefficient was calculated assuming that the section was homogeneous with respect to diffusion. The permeability coefficient of the skin was determined by 23 testing hollow sections containing the skin and par- enchymatous tissue and by using the previously determined diffusion coefficient of the parenchymatous tissue to calculate the permeability of the skin. Raw Material Cucumbers. The cucumbers used for most of the experiments were of the variety SMR-lS, a common pickling cucumber variety, which were obtained from the MSU Horticulture plots. The cucumbers were hand picked and the spines removed (in industry the spines are broken off during handling and trans- portation) to prevent damage to the fruit during storage and transport. The cucumbers were washed with water and stored at 4C (39F) until tested (less than 3 days). Cucum- bers of a mixture of SMR-58 and Pioneer varieties were obtained from Heifetz Pickling Company, Eaton Rapids, Michigan. Salt stockpickles. Salt stock pickles (varieties SMR-58 and Pioneer mixed) were supplied by Heifetz Pickling Company. The pickles were washed and stored at 4C (39F) in salt brine made up to the same concentration as the original salt stock brine. The average NaCl concentration of the pickles as obtained from the company was 12.9 per cent (w/w). Before testing, the pickles were desalted in running water (24 hrs) to less than 0.2 per cent salt. 24 Sample Preparation Before samples were cut from the cucumbers and pickles, the stock was sorted to remove any irregularly shaped or damaged fruit and then sorted according to size. The diam- eter of each fruit was measured with vernier calipers by measuring the distance from the base to the apex of the lobe in longitudinal center of the fruit. The product was sized within *0.2 cm of the desired diameter. Cutting of sections. Two parallel cross-sectional cuts were made through the cucumbers to obtain sections of the desired length (1.5 cm - 2 cm). Then tissue samples were prepared in the form of cylinders or hollow cylinders (see Figures 1 and 2). Cylindrical seed cavity sections were cut centrally from the whole sections using cork borers. Cylindrical sections containing seed cavity and parenchymatous tissue were prepared by peeling the skin from whole sections. Dif- fusion through the skin was tested using intact whole sections. Hollow cylindrical sections of seed cavity tissue were prepared from whole sections by making concentric cuts through the seed cavity with cork borers to give a wall thickness of between 0.25 and 0.3 cm. Hollow cylindrical sections of parenchymatous tissue were prepared by cutting out the inside of peeled sections with cork borers to leave a 0.25 to 0.4 cm thick ring of parenchymatous tissue. To prepare hollow cylinders containing the skin, the inside of 25 I y ‘ HF .77 I; B C A Figure l. Cylindrical sections of cucumbers. A. Whole section with skin. B. Peeled section. C. Seed cavity section. Figure 2. Hollow cylindrical sections of cucumbers. A. Section with skin intact. B. Peeled section. C. Small- er section of parenchymatous tissue. 26 whole sections was cut out with cork borers leaving a 0.25 - 0.4 cm thick ring of skin and parenchymatous tissue. Dimensions of samples. The average radius of a cylindrical section was calculated from measurements of its length with calipers and weight. Knowing the density,pv, and assuming the section to be a cylinder, then R. M . (5) where L is the length and M is the weight of a section. This method of determining the average radius was used rather than direct measurement because the peeled and whole sections were only approximately cylindrical and, therefore, the length and weight could be measured more accurately than the diam- eter . The average wall thickness of a hollow section was determined from measurements of the length, outside diameter, and weight of the section. The length of each section was measured in three places and averaged. The average outside diameter was determined by measuring the distance from each base to the apex of each lobe (a total of three measurements for each section) and then averaging the readings. The average wall thickness, A, was determined as follows (see Appendix I for details of derivation): A=R°- Jigf-M/pwx. . (6) where Ro is the average outside radius. The per cent error in the calculated wall thickness is approximately the same 27 as the per cent error in the measurements of radius, length, and weight (see Appendix I). This method of calculation is more accurate than direct measurement because the wall thickness of the sections was not uniform and the diameter, length, and weight can be measured much more accurately than the thickness. Deaeration of cucumber sections. To eliminate variability due to air loss during the experiments, the cucumber sections were deaerated prior to testing and infiltrated with the test solution (minus the sugar). The cucumber sections were placed in the solution contained in a desiccator jar and held submerged by a perforated porcelain plate. A vacuum of 28 inch Hg was pulled on the jar and maintained until no fur- ther air loss was observed (as evidenced by air bubbles). For whole cylindrical sections this required 6 - 15 hr while for the hollow sections this required only 1 - 2 hr. During this time the solution and samples equilibrated osmotically. Then the vacuum was slowly released allowing the air spaces to be filled with the solution. Fellers (1964) observed from photomicrographs that cucumber sections could be infil- trated with water without causing damage to the tissue. Assembly in sample cell. To prevent diffusion of sugar through the ends and to hold the sample in place during test- ing, the sections were placed in specially designed sample cells (see Figures 3 and 4). Each cell consisted of two plexiglass plates (3/32 inch thick) and four stainless steel bolts (1/4 inch by 1 1/2 inch) and nuts. Four symetrically 28 M. )H’ 9.9 99 mi) eons. Figure 3. Unassembled sample cells and samples Figure 4. Assembled sample cells containing samples 29 spaced holes (5/16 inch) were drilled in the two plates. The bolts were placed through the holes of the top plate and cemented to it with epoxy resin. A.small plexiglass piece was bonded to the center of the top plate to serve as a handle in placement of the sample cell unit. High-vacuum grease was used to seal the ends of the sample section. The grease was smeared in a thin layer on the side of the plate adjacent to the tissue section. The sample was positioned as shown with the ends pressed against the grease. The nuts were tightened until just snug enough to hold the sample in place. Difquion Test Medium The aqueous solution used for testing diffusion rates contained one per cent acetic acid (v/v), 1000 ppm potassium sorbate (w/v) and C14 radioactive sugar (uniformly labeled). Sucrose was used in the first tests conducted. But, during the course of the experiments we found that sucrose was inverted; therefore, glucose was used in all subsequent dif- fusion tests. All diffusion rate data reported in this thesis are for glucose. Acetic acid is used by the pickling industry in the manufacture of sweet pickles and it along with sorbate were used in the experiments to prevent bac- terial growth. When cucumbers were incubated in the test solution, no bacterial growth was observed even for incuba- tion times exceeding one month. 30 Diffusion Test Chamber The diffusion test chamber containing the diffusion medium was provided with means of temperature control and for continuous mixing of the diffusion medium. The test chamber and control equipment are shown in Figures 5 and 6. The chamber (7 3/8 by 8 3/4 by 5 1/4 inch, inside dimensions) was constructed of53/32 inch plexiglass and was reinforced with 1/4 inch plexiglass strips at the corners. The tOp consisted of a 1/8 inch glass plate (8 1/2 by 11 inch) which rested on a 1/2 inch wide rim of 3/32 inch plexiglass bonded around the top edge of the chamber. The tOp of the chamber could be sealed air tight to the rim using stopcock grease spread around the rim. The entire chamber was insulated with 1 1/4 inch Styrafoam. The temperature was controlled by pumping water from a regulated water bath through a stainless steel coil located in the chamber. The water bath consisted of a cooling coil and a Bronwill Thermoregulator and Circulator.1 The temper- atures could be regulated from 15C (using tap water as a coolent) to 90C. The temperature of the medium in the chamber was controlled within at least I 0.2C. A Manostat Pump2 was connected to openings in the front of the chamber to continually mix the solution during testing. The pumping rate could be regulated from 0 to 2 liters/min. 1 Bronwill Scientific Div., Will Corp., Rochester, New York 2 Manostat Corp., New York 31 Figure 5. Diffusion test system consisting of the diffusion test chamber, a temperature regulating water bath, and a circulating pump Figure 6. Inside view of the diffusion test chamber showing the stainless steel coil and positioned samples 7 . I III II II I ' II‘ 32 Approximately 1.5 to 2 liters of solution was used in the experiments and flow rates of between 1.5 and 2.0 liters/min. were maintained. Sampling and Determination of Radioactivity To determine the amount of radioactive sugar which had diffused in the sample, the sample was homogenized, the homogenate filtered, and radioactivity of the filtrate counted using liquid scintillation. After removal from the diffusion chamber, the sample cell unit was rinsed with dis- tilled water to remove adherring radioactive solution and allowed to drain several minutes. Then the sample was removed from the sample cell, cut into pieces, placed in the mixing chamber along with a measured volume of water and homogenized in a Sorvall Omnimixerl for a minute or more at 16,000 RPM. After homogenization, the homogenate was filtered through Whatman No. 1 paper and a 0.5 ml aliquot of the filtrate was pipetted into a liquid scintillation vial. After all the samples of an experiment were taken, 15 m1 of dioxane scintillation cocktail were added to the vials and the radioactivity counted on a Packard Tri-Carb Liquid Scintillation Counter (5.5 per cent gain; window setting 50 - 1000). The observed counts per unit time were cor- rected to disintegrations per unit time by external standard— ization (Rapkin, 1963). 1 Ivan Sorvall, Inc., Norwalk, Connecticut I 33 Procedure for a Single Test For routine testing the raw material was first cut into the desired sections. When cucumbers were tested, they were deaerated under vacuum and infiltrated with the solution minus the radioactive sugar. When sections of salt stock pickles were tested, they were equilibrated over night in the solution. Samples were weighed and appropriate measurements made to determine their dimen- sions before they were positioned in the sample cell. The assembled cells, containing the sections (5 samples per test), were placed upright in the diffusion chamber which contained 1.5 to 2 liters of diffusion medium. At pre- selected times, samples were removed and the average radio— activity per unit volume of sample was determined. The concentration of the diffusion medium was monitored through— out the test period. Physical and Compositional Measurements Density. The average densities of the sections of cucumbers and pickles used in the diffusion tests were measured by water displacement (A.O.A.C., 1960a). A small amount of triton x - 100 was dissolved in the water to eliminate the formation of air pockets on the surface of the tissue section. Total solids. The total solids or moisture content of the sections of cucumbers and pickles was measured by vacuum drying at room temperature (A.O.A.C., 1960b). 34 Salt concentration. To determine the original salt con- centration in salt stock pickles, several pickles were first mascerated in an Osterizer Blender. The mascerated tissue was strained through several layers of cheese cloth and the expressed pickle juice was clarified by filtering several times through Whatman No. 2 filter paper. The salt concentration of the clarified pickle juice was measured by titration with 0.1N AgNO3, using dichloro-fluorescein as the indicator (Richardson and Switzer, 1939). Measurement of Recovery of Glucose from Sample Sections as Affected by Homogenization Conditions The effect of the homogenization conditions on the con- centration of labeled sugar recovered from sample sections was investigated to determine the Optimum homogenization conditions. Whole cucumber sections, 3 cm in diameter and 1.5 cm in length, were placed in the diffusion medium. After allowing the sections to equilibrate with the medium (84 hr), the sections were withdrawn from the medium and homogenized with 15 ml of water at the following conditions: 1, 1 1/2, and 2 min at 16000 RPM; 1, 2, and 4 min at 8000 RPM after initially blending for l min at 16000 RPM. In all cases the Omnimixer speed was gradually brought from 0 to 16000 RPM in 30 seconds and then the samples were homogenized for the indicated times. After sampling and determining the radioactivity of the homogenate, the concen- tration of labeled sugar was calculated. The ratio of the 35 observed concentration of the section to that of the solu- tion was a measure of the recovery of sugar from the cu- cumber sections and of the completeness of mixing the tissue and the water. Measurement of the Change in Diffusivities in Cucumber Tissues with Time Cylindrical sections of peeled and seed cavity tissue were placed in 0.5 liters of the diffusion medium (minus the sugar) and kept at 25C in an incubator. At preselected incubation times, three samples of each section type were withdrawn from the incubator. Then the diffusion coefficient of C14 glucose in each of these sections was measured. Dif- fusion coefficients were measured by placing the incubated tissue sections in the diffusion chamber and determining the average concentration of each section after 5 hr diffusion time. In this short time, the increase in diffusivity of the samples during measurement of their diffusion coeffi- cients was relatively small and was regarded as negligible. Calculation of Sugar Concentration and Recovery From Chromatograms Calculation of the Average Sugar Concentration in a Sample Section To calculate the average concentration of radioactive sugar in a sample, C (dpm per unit volume), the dilution avg of the sample with water before homogenization had to be 36 taken into account. Assuming that the amount of free water in the cucumbers corresponds to its moisture content (see "Physical and Compositional Measurements"), then w V Cavg 8 Ch ( CMC + d) (7) We“. where Ch is the concentration of radioactive sugar in the homogenate (dpm per unit volume), MO is the mass of the cucumber or pickle sample (9), W0 is the water content (ml water per g cucumber or pickle), and Vd is the dilution volume (ml) . Calculation of Recoveries of Sugars From Chromatographic Peaks From known volumes and known radioactivities of the chro- matographic peaks obtained for samples of cucumber homogenate, incubation solution and control solution, the recoveries of radioactive sugars were calculated assuming the incubation solution and cucumbers had reached equilibrium. Recovery of sugars for the incubation solution were calculated by taking into account the dilution due to the water in the cucumbers. The dilution factor, Df, was calculated by: vs + Mowc V8 (8) Df. where V8 is the volume of solution incubated with the cu- cumbers (ml), MO is the mass of cucumbers incubated (g): and Wc is the water content of the cucumbers (m1 of water/ 37 g cucumber). The per cent recovery of sugar for the incu- bation solution is: (9) where CS is the total radioactivity recovered from chroma- tographic peaks obtained for the solution (dpm per unit volume) and Ccon is the total radioactivity of the peak obtained for the control solution in the same units. The average concentration of sugars recovered from the chromatographic peaks of cucumber and pickle samples, Cavg' was calculated by considering dilution of the sample before homogenation (see'Calculation of the Average Sugar Concen- tration in a Sample Sectiofi). Then the per cent recovery of labeled sugars from the cucumbers, % Rc' could be calculated in the same manner as that for the incubation solution: i.e., % RC = Cavstx 100 , (10) Ccon The total recovery of labeled sugar from the chromatographic peaks of both the solution and cucumbers were calculated by adding the total radioactivity of the sugars obtained from the chromatograms of the solution and the total radioactivity of the sugars obtained from chromatograms of the cucumber and dividing by the total radioactivity of the sugars originally in the solution: C V + C gM W % Rt = s 3V cav G ex 100. (11’ 3 con 38 Calculation of Diffusion Coefficients The diffusion rates through the fleshy tissue of the cucumbers or pickles were described by diffusion coeffi- cients (D), whereas the diffusion rates through the skin were expressed by permeability coefficients (K) corre- sponding to finite surface resistances. The biot number, Bi, was used to describe the relative resistance of the flesh to the skin and was expressed as Bi = g5 or EB . (12) depending on whether the section geometry was approximated as an infinite slab (in which A is the half thickness) or as an infinite cylinder (in which R is the radius). Cylindrical Sections The diffusion coefficients of the cylindrical flesh sections (sections not containing skin) were first calcu- lated using the exact solution to the diffusion equations by assuming that each entire section was homogeneous with respect to diffusion. If the diffusion coefficients calcu- lated for different sections of the tissue (peeled versus seed cavity) were observed to be different, then the Crank - Nicolson finite difference technique was used to calculate the diffusion coefficients of the tissues having different diffusion rates. 39 Exact solution for diffusion in a cylinder. The exact equations were applied by assuming the overall mass trans- fer to be one-dimensional. The differential equation for diffusion in an infinite cylinder is (Crank, 1967) where C is concentration, t is time and r the radius. For the flesh tissue sections, we have the following conditions: C a q” , r = R, tZO, C = C0 ,0‘ Cavg is the unaccomplished concentration Cm-Co 40 ration, Y. The )8n's are the non-zero roots of the trans- cendental equation 309‘) = 0 (15) and their values are given by Abramowitz and Stegun (1964). For cases where there is a surface resistance due to the skin, the surface boundary condition is _%§.=K(c., ~C),r=R,t.?-0 and the other boundary conditions are the same. The solu— tion for these conditions is (Newman, 1931): a; -’8n “2 4Bi2 e R Y - 2 2 2 (16) “,1 An (Bi +£n) The .6n's are the roots of .6 J, (,8) - 31 J00!) = o (17) and Bi 8 §§° Finite difference solution for diffusion in a composite cylinder. Because it is difficult to apply an exact solution to the calculation of diffusion coefficients in a solid body made up of dissimilar materials, a finite difference tech- nique was used to calculate the diffusion coefficients in different areas of the cylindrical sections. The finite difference technique was applied by first performing mass balances at increments of distance (Ar) and over time intervals (At) to obtain the finite difference equations. These equations could then be solved by a direct or an 41 iterative method (see Smith, 1965 and Ozisik, 1968). The Crank - Nicolson method of finite differences was used because it is stable for all values of m (Ozisik, 1968), where DAt m = ————z (18) (Ar) The parameter, m, arises from the finite difference equa- tions (see Appendix I) and, in the explicit forward finite difference method, its value must be Sl/Z for stability of the solution. Since no limit is imposed on m in the Crank ~ Nicolson method, there is no limit on At; and, therefore, the computational time can be reduced in compar- ison with the explicit method. The truncation error for the Crank — Nicolson method is also the smallest (i.e. 0(0):)2 + 0(At)2 of the other methods (the forward and backward difference methods). On the other hand, the equa- tions for the Crank - Nicolson method must be solved simul- taneously, which is more involved computationally than solving each equation individually as in the case of the explicit method. The Gaussian elimination method described by Ozisik (1968) was used to directly solve the simultaneous equations which were written in tridiagonal form; that is, when the equations are written in matrix form, the coefficients of the matrix on the left side will be zero everywhere except on the main diagonal and on the two diagonals parallel to it on either side. For derivation of the finite difference equations and determination of the matrix coefficients, see 42 Appendix 1. After iterating the equations to obtain the concen- tration for each node at the experimental time,dt, the average concentration of the cylinder was found by numer- ically integrating over the entire cylinder volume. The flow diagram of the entire computational procedure is shown in Figure 8. The finite difference technique was checked in two ways. Its results were compared to those of the exact solu- tion for an infinite cylinder and, also, the radius and time increments used in the finite difference equations were halved until the solution was found to converge. Hollow Cylindrical Sections Crank (1967) shows uptake curves for hollow cylinders with both the inside and outside exposed to the same con— stant concentration. These curves show that if the ratio of the outside to the inside radius (Re/R1) is less than 10, the results are very close to those obtained by assum- ing the geometry to be an infinite slab. In our experiments the hollow sections were exposed to the solution on only the outside, but the largest ratio of Ro/Ri for the flesh sections was 1.75 while that for the sections containing skin, which offers some resistance to mass transfer, was 1.1. These ratios are small enough to enable us to calculate the diffusion coefficients within a negligible error by assum- in the hollow sections to be of an infinite slab configuration. 43 For the flesh sections, the outside surface concen- tration is the same as that of the diffusion medium and there is no mass transferred through the inside surface (§§.= 0). The differential equation for a slab is: 2C a 23 3C 5E Ji' (DEE) (l9) and the boundary conditions then are: C=C‘°,x=A,t?.O, C=C030 :3ocx a ea.oo on: .Hwo m wow "Odour .uHmu Fosam> M ofiommdxg I. M mo :owueaooamo 46 a mo cOMDMHoono A) mnmam new a e HA0 now a .ono axe» .OHMU Awoumo Hmucoamuomxo comm v I ‘ .ado>wuooonou .mnmau one muoocwaho ouacwmcw How mcoHucHom uoexo on» mean: sumo Hmucofiwuomxo Eoum muoocwaho soaaos one nuoocaaao cw M one o mcauomfiou Haw Ecumuwo xooam .n ouomwm 47 OZ 2m vemz one 33 * ”0'0 4 H HO no» 2282 one axed . Renew... «m2 5an s ham 0 Sean .u 3.82 :16 a now oz :3 no. uwmz no A32" so: .015 I as + u u a - .5137 unoJT ..o.usoo. .5 «names 48 N OZ mmoo.ow|mm OZ ww.+ a u a _ A .Hm .oo can .aan o How «ma .oo on: .Hao a now new .oamu c+m.~ ..mx Q. F _ ..o.ueoo..~ masons 49 Figure 8. Block diagram for computing D and K in composite cylinders using the Crank-Nicolson finite difference method. (:Read experimental data‘:) (: Read values of At and Ar ‘:) lCalc. the experimental J Cavg' and then Yexp‘ I Assume a value for D or K] % is <9 [C - 0 at all nodesJ [ta-t: At J‘——"@ [Ca1c. recursion coefficients for I the center node from eq. A-12 l Calc. recursion coefficients at nodes within the cyl. from eq. A-16 and at the interfaces of different materials from eq. A-19 Calc. recursion coefficients'for the node adjacent to the boundary: for no surface resistance use eq. A-20; for surface resistance due to the skin use coefficients of eq. A-23 m Figure 8. (cont'd.) so Gaussian elimination to solve the tri- lCalc. the unknown concentration (C+) using diagonal matrix of recursion coefficients Calc. the average concentration in the cyl. by numerical integra- tion. Determine Y calc' < 0.002? J i (:_ Print values of D or K :) D (or K) - - 1 Y D (or K) exp ) - 1 Ycalc RESULTS AND DISCUSSION Density and Moisture Content of Test Sections The results of density and moisture content measure- ments for test sections of cucumbers and salt stock pickles are shown in Tables 1 and 2, respectively. Because diffu- sion rates were determined in several sizes of cucumbers, it was necessary to measure the densities and moisture contents of sections from the different sizes of cucumbers. The data in Table 1 show that the densities and moisture contents of the sections did not vary much with the size of the cucumbers. Because the densities of the sections from the 2.5 and 3 cm cucumbers were found to be nearly the same, the densities of the sections from the 2 cm cucumbers were assumed to be the same as those of sections from the 2.5 cm cucumbers. The densities of the sections with skin intact were slightly higher than those of seed cavity and parenchymatous tissue; the densities of the seed cavity and parenchymatous sections were the same. The total solids content of the parenchymatous tissue was somewhat higher than that of the seed cavity tissue while the total solids content of the sections containing skin was consistantly higher than that of the sections of parenchymatous tissue. 51 52 Table l. Densities and Total Solids (Moisture Content) Data for Test Sections of Cucumbers Cucumber Section Density Total Moisture diameter description solids content (cm) (gm/CC) (%) (ml/9) Hollow cylinders Seed cavity 1.012 2.49 0.975 Parenchymatous 1.012 2.74 0.973 With skin 1.014 3.87 0.961 3 Whole cylinders Seed cavity a 1.012 2.49 0.975 Peeled b 1.012 2.62 0.974 With skin c 1.013 3.18 0.968 Hollow cylinders 2 1/2 Flesh 1.010 2.26 0.977 With skin 1.012 3.94 0.961 Hollow cylinders 2 Flesh 1.010 d 2.54 0.975 With skin 1.012 d 3.51 0.955 a These values were taken to be the same as the hollow cylinders of seed cavity tissue b These values were obtained by averaging those obtained for hollow cylinders of seed cavity and parenchymatous tissue c These values were obtained by averaging those obtained for hollow cylinders of seed cavity and for hollow cylinders with skin d These densities were taken to be the same as those found for 2 and 1/2 cm cucumbers 53 Table 2. Density and Total Solids (Moisture Content) Data for Test Sections of Salt Stock Pickles (2.5 Cm Size) Section Density Total Water description solids content (g/cc) (%) (ml/g) Hollow cylinders Parenchymatous 1.010 a 1.42 0.986 With skin 1.011 b 2.10 0.979 Whole cylinders Seed cavity 1.010 1.30 0.987 Peeled 1.010 a 1.46 0.985 With skin 1.011 1.79 0.982 a The density was taken to be the same as that observed for cylinders of seed cavity tissue b The density was taken to be the same as that found for cylinders with skin Recovery of Glucose from Sample Sections by Homogenization The effect of the homogenization conditions on recovery of glucose from cucumber sections was investigated. Table 3 and Figure 9 show that homogenization for l min at 16000 RPM was sufficient to recover 97 per cent of the sugar, (Cavg/qn - 0.971). No significant improvement in recovery was observed for the longer homogenization times and higher Omnimixer speeds. Although in this experiment the sample sections were allowed to equilibrate with the solution for 84 hr, it is still possible that the average concentration of sugar in the sections did not reach that of the solution. This may account for the value of C /Cn being consistantly avg less than one. 54 Homogenization time (min) at 16000 RPM (43-4J-) 1.0 1.5 2.0 i k 1.00” 0.98 / av B can 0.94__ 0.92‘_ 0.9% : t : : 0 1 2 3 4 Homogenization time (min) at 8000 RPM after homogenizing for l min at 16000 RPM t<>-{>1 Figure 9. Recovery of glucose from sample sections as a function of homogenization time and Omnimixer shaft speed 55 Table 3. Recovery of Sugar from Sample Sections asa Function of the Homogenization Conditions e o o o a Homogenization conditions Time at Time at 16000 RPM 8000 RPM caV /c,, b . . 9 (min) (min) c_ 1.0 0.971 10.007 1.5 0.993 10.014 2.0 0.949 t0.005 1.0 1.0 0.975 i0.005 1.0 2.0 0.970 i0.003 1.0 4.0 0.979 i0.007 Average Cavg/qno = 0.973 a Omnimixer was gradually brought up to full speed (16000 RPM) in 30 sec b from mean Chromatography of Sugars Exposed to Cucumbers and Pickles Average of duplicate samples showing deviation Figures 10 - 14 show profiles of chromatograms obtained for incubation of sugars with cucumbers and salt stock pickles. The peaks of the radiochromatograms were identified by com- parison to the chromatographic spots of known sugars. The results show that sucrose was inverted when incubated with either cucumbers or salt stock pickles (see Figures 10, 11 and 14). In both cases, no inversion of sucrose occurred in the control solutions; radiochromatograms of the control AHA on. muons—coon . use b wmo Spouse mo 3.393 3 o a new: omouoon :0 no seduced—0c.“ Hon pondeueo man u I ll. I III. |vllrl I'll-l l I III I l {I l l .l [I t l I I l i I It [ I U ( . l I u I l Ill. 'l‘ l {I III. I ‘ Figure 11. Profile of chromatograms obtained for incubation of C sucrose, and l and 10 per cent sucrose with cucumbers (68 hr) \i{('|'.."lll.lll'!'l[lllltlil Figure 13. :toffiesm of chruatogra- obtained for incubation sucrose with salt stock pickles (30 hr) Figure 14. Profiles of chromatograms obtained for incubation of C 4 glucose with salt stock pickles (30 hr) 61 solutions show only one peak which was identified as sucrose. These results indicate that the inversion was enzymatic. If the observed inversion of sucrose had resulted from acid hydrolysis, the sucrose in the control solutions should have also been inverted; the acidity and composition of control solution were the same as that of the incubation solution. A consistantly higher percentage of invert sugar was found in the homogenate than in the incubation solution (see Table 4), indicating that the sugar became inverted within the cucumber or pickle and then diffused into the solution. It was of interest to determine in which parts of the cucumber the inversion occurred. Incubation tests were performed on the following areas of 3 cm diameter cucumbers: cylindrical sections of seed cavity tissue (1.8 cm diameter); hollow cylindrical sections of paren- chymatous tissue (2.4 cm outside diameter and 1.8 cm inside diameter) and hollow cylinders including the skin (3.0 cm outside diameter and 2.4 cm inside diameter). The results obtained using these areas of tissue were the same as those shown in Figure 9. In each case sucrose became inverted. The inversion was quantitatively the same for each of the areas tested. Most fruits contain invertases so it was not unexpected to find invertase activity in cucumbers. But, it was sur- prising to find that the salt stock pickles also possessed invertase activity. During curing in salt brines, the 62 Table 4. Per Cent Inversion of Sucrose Incubated with Cucumbers and Salt Stock Pickles Description of Total area Area under Amount of experiment under peaks glucose and inversion' fructose peaks (cmz) (cmz) (*5) C14 sucrose incubated with cucumbers (36 hr) Homogenate 0.80 0.72 90.0 Incubation solution 1.43 1.06 74.1 C14 sucrose incubated with salt stock pickles (30 hr) Homogenate 0.71 0.59 83.1 Incubation solution 1.21 0.87 71.8 Total radio- Radioactivity activity of of glucose and peaks fructose peaks (cpm) (cpm) C14 sucrose + 1% sucrose incu- bated with cucumbers (68 hr) Homogenate 2299 1554 67.6 C14 sucrose + 10% sucrose incu- bated with cucumbers (68 hr) Homogenate 2180 1640 75.2 pickles are exposed to acid and other fermentation by- products as well as to high salt (NaCl) concentration (13 - 16 per cent). Yeast invertases have been shown to be unstable at pH's less than 4.0 (White 22 cl, 1964); the 63 pH of the salt stock pickles as obtained from the manu- facturer was 3.8. The pH range for optimum activity of invertases has been found to be 4.5 - 5.5 (Biochemists Handbook, 1961). In our diffusion experiments, the pH of the cucumbers, pickles, and solution was 3.5. Because the concentration of C14 labeled sucrose was low (<0.001 per cent), it was decided to determine if a significant amount of inversion occurred in higher concen- trations of sucrose. Figure 11 shows that inversion also occurred at concentrations of 1 per cent and 10 per cent sucrose. A somewhat greater percentage of the sucrose was inverted in the 10 per cent sucrose solution (75.2 per cent in 68 hr) than was inverted in the 1 per cent sucrose solution (67.6 per cent in 68 hr). But, in each of these cases, the relative rate of inversion was lower than that found when C14 labeled sucrose was the only sugar added to the solution (90.0 per cent in 36 hr; see Table 4). Each of the chromatograms obtained when glucose was incubated with cucumbers or salt stock pickles showed only one peak which was identified as glucose (see Figures 5 and 14). All the radioactive glucose which was initially present in the solution was accounted for in this spot (Table 5). Table 5. 64 Recovery of C14 Labeled Sugars from Radiochromat- ograms of Sugars Incubated with Cucumbers and Salt Stock Pickles Description of Solution Total Recovery RT a incubation sampled radioactivity of sugars experiment of spots (dpm) (%) (%) Control for tests 1,11, and III 4279. - - Homogenate I 1748. '96.8 14 Incubation 103.5 C sucrose medium I 3716. 104.6 incubated with cucum- Homogenate II 1704. 98.0 bers (36 hr) Incubation medium II 3773. 104.6 Homogenate III 1938. 105.9 Incubation 102.3 medium III 3572. 101.6 C14glucose Control 1278. - - incubated with cucum- Incubation bers (72 hr) medium 913. 100.4 - C14 sucrose Control 5220. — - incubated with salt Homogenate 1925. 107.8 stock pickles 109.3 (30 hr) Incubation medium 4155. 110.0 C14 glucose Control 4584. - - incubated with salt Homogenate 1663. 100.8 stock pickles 102.2 (30 hr) Incubation medium 3324. 102.7 a Total recovery of sugars incubation medium Likewise, all the label initially present in the sucrose solu- from the homogenate and tions was accounted for in the chomatographic spots of 65 sucrose, glucose and fructose (Table 5). It was there- fore concluded that these sugars were broken down no further and that they were not metabolized. Diffusion in Cucumbers and Salt Stock Pickles Table A-1 of the Appendix II shows the results for all the diffusion experiments. Throughout the discussion, tables are drawn from this data to illustrate pertinent points. Diffusion in Cucumbers Diffusion in cylindrical sections. In tests 48 - 57 (see Table A-l), diffusion rates were determined using cylin- drical sections of cucumbers. The diffusion coefficient of each seed cavity or peeled section was calculated by assuming the section to be homogeneous. The permeability coefficient of each skin section was calculated using the average value of the diffusion coefficient determined for the peeled sections. In test 48, the experimental value of the diffusion coefficient at 25 hr was believed to be in error because it was not within i 23d of the average; therefore this value was discarded. In test 50, the value of Bi calculated for 60 hr was indeterminate (00); there— fore, this point was discarded. The diffusion coefficient of the peeled sections was noted to increase with time and, correspondingly, to become larger as the value of the concentration ratio Y decreased 66 (see tests 49, 52, and 56 in Table A—l); the value of Y ranges from 1 initially before any diffusion has taken place to 0 at equilibrium. In each experiment, the diffu- sion coefficients were determined over a range of times or Y values to detect any change in the diffusion coef- ficient. This required experimental times of as long as 60 hr when using the cylindrical sections. The noted tendency of the diffusion coefficient of the peeled sec- tions to become larger at longer times or, correspond- ingly, for smaller Y values indicated that either the diffusion coefficient of sugar in the tissue actually increased with time or that the peeled section of tissue was not homogeneous which would lead to erroneous values of the calculated diffusion coefficient. If the inner tissue in a section was more permeable than the outer, the value of the diffusion coefficient calculated assuming the section to be homogeneous would become larger as more solute was transferred into the inner tissue (as the Y value became smaller). Because the average diffusion coefficient of the seed cavity did not appear to be different from that of the peeled tissue, the apparent change in the diffusion coef- ficient of the peeled tissue was not attributed to non- homogeneity. Therefore, the change in the diffusion coef- ficient of the tissue with the time of exposure to the dif- fusion medium (minus the sugar) was investigated. The results of this investigation on cucumber tissue are shown in Table 6 and Figure 15. The diffusion coefficients of 67 Table 6. Diffusion Coefficients of Glucose (25C) in Cucumber Tissue as a Function of Storage Time at 25C in Aqueous Solution Test Storage Seed Cavity tissue Parenchymatous tissue no. time _ t 6 _ _ a _ + 6 _ _ a (hr) (D sd)x10 D/Do (D - sd)x10 D/Do (cmz/sec) (cmz/sec) 58A,59A 0 2.13 10.081 1.0 1.60 1’0.205 1.0 58B,59B 24 2.62 10.207 1.23 2.08 30.297 1.30 58C,59C 70 3.78 10.162 1.77 3.49 i0.227 2.18 a Do refers to the average initial diffusion coefficient b One per cent acetic acid and 1000 ppm potassium sorbate both the parenchymatous and seed cavity tissues increased markedly with time. The diffusion coefficient of the paren- chymatous tissue increased faster than that of the seed cavity tissue; in 70 hr, the diffusion coefficient of the parenchymatous tissue increased by approximately 100 per cent while that of the seed cavity increased by approximately 75 per cent. These changes in diffusion rates were probably caused by enzymatic breakdown of the cell structure but may have been partially caused by acid hydrolysis of the mem- brane. Diffusion in hollow cylindrical sections. Because of the large change in the diffusion rate of glucose in the cucum- ber with time, the remainder of the diffusion tests on cucumber tissue were performed using hollow cylinders. 68 2.53? A Parenchymatous 2 . 0.. 0 Seed Cavity 0:" U) 1.54 1.02. l J I I J. V T 0 20 40 60 80 Storage time (hr) Figure 15. Diffusion coefficients of glucose (25C) in cucumbers as a function of storage time of the cucumber sections at 25C in aqueous solution (1% acetic acid and 1000 ppm potassium sorbate) 69 These sections were thinner than the cylindrical sections; therefore, the diffusion rate experiments using hollow cylinders could be performed in a much shorter time and could be conducted over a fairly wide range of Y values. Typical results of diffusion experiments in hollow sections of cucumbers are shown in tests 65 to 67 and 70A to 70C (Table A-l). The data for the diffusion coeffi- cients of parenchymatous tissue show a tendency for the diffusion coefficient to increase with time. To determine whether this increase was significant, correlation coef- ficients of the diffusivities (D) with both time and Y were calculated (Table 7). The results show that there was no significant correlation for the parenchymatous tissue when the tests (66 and 70C) were considered individually. When the tests results were combined, the correlations of D with t and D with Y were both found to be significant at p = 0.05. No significant correlation of D with t or Y was found for the results of tests on seed cavity tissue (tests 65 and 70A) or on the skin (tests 67 and 70C). Because the tests on hollow cylinders were conducted for such a short time (200 min), it was not considered likely that the observed increase in the diffusivity of the parenchymatous tissue was due to an actual increase in the diffusivity of the tissue; after 3 hr storage time in solution, the diffu— sion coefficient of the parenchymatous tissue was observed to increase by only 5 per cent (Figure 15). The observed increase in diffusivity could have resulted from the outer 70 .1 - I..." ' .i‘il'Illl’l‘t I'll. n'l so>os mo. n e um ucmowmscmsm « ee.~ ee~.e eem ee.~ emm.e ees eee.e- mee.e sm.~ ese.e ems mes em.~ eem.e ee ee.~ eee.e es .sme.e- .eee.e msoussssosonso ee.~ ~em.e eem me.m eme.e ees mme.en eee.e ee.~ see.e ems ee se.~ eem.e ee em.~ see.e ee me.~ es~.e eem ee.~ ee~.e ees ses.e- sse.e ee.~ eem.e ems «es ee.m eme.e ee ee.~ eme.e ee auw>mo comm m~.~ me~.e ee~ m~.~ emm.e ees ~ee.e- ~e~.e- se.~ eem.e ems me ee.~ eee.e ee e~.~ sse.e ee 1s.eees lu.eee4 1s.eesn lu.eesu luemxmsoe less. mumou oocsbaoo Mom umou some wom boa x a w weds .oc umoa coma osmmsa mucosowmmooo cesumHousou “we asuom eosussueoocoo 0:» cuss one Ase ease cuss xoe mucoso Ismoou scemomuso on» no mucos0fimmoou cosuososuoo ocHSOem muonaooou mo mcowuoom Basso: cw omoooao vHU mo mucoEwummxm cowmomwea mo madamom .h OHQMB 71 vv.a Nmm.o com mv.a Nmm.o ovm mmH.o Hmm.o| mm.H ooh.o omH Uon . mv.a owe.o ONH mm.a mmm.o om Nmm.o+ mom.ot nflxm mm.H voo.o oom mm.a mmm.o oem mmv.o+ mmm.ol hm.H mmh.o own no ~v.~ meh.o oma m~.N Hem.o ow is.eeeh la.oees is.eess la.eess losm\~sue lasso mummy pocmnfioo now one» some you boa x o w mafia .0: once was» oommfla mucowosmmooo :0wumsosuoo A.U.ucoov h OHQMB 72 parenchymatous tissue being more resistant to diffusion than the inner tissue. Comparison of diffusion rates in various areas of cucum- REEE- The results of the diffusion experiments on differ- ent areas of cucumbers are shown in Table 8. For each set of tests the average diffusivities of the seed cavity and parenchymatous tissue were not significantly differ- ent. Therefore, it was concluded that the fleshy tissue of the cucumber (excluding the skin) could be considered homogeneous with respect to diffusion. Perhaps the observed differences between sets of diffusion tests were due to inherent differences in the cucumbers which were harvested at different times. The results in Table 8 (see also Table A-1) show that the diffusion coefficients obtained for hollow cylindrical sections were generally higher than those obtained for the cylindrical sections. Again, this was probably due to inherent differences in the cucumbers. A comparison of the results obtained using hollow cylinders with those obtained for cylinders will be made subsequently. It should also be noted that there was no apparent difference between the diffusivities of the different varieties of cucumbers tested (SMR—lS and a mixture of SMR-58 and Pioneer). However, the permeability coeffi- cients of the skin for the mixture of SMR-58 and Pioneer varieties were much more variable (sd = 0.83) than those of variety SMR-lS (sd = 0.17 to 0.50). The mixture of 73 vhH.0H mm.a Hmm.oH oo.~ mmH.oH hh.~ mm\ma\m Uonndon mHImZm oev.ou mm.s ov~.oH em.~ mm~.oH vv.~ mw\os\m hmumm mHImZm msoocssho zossom ooxse Hoocowm . one ewe es em.s ses.es ee.~ ~em.es e~.~ ee\m\e metee emumZm sm~.ofl em.s mm~.ow mv.~ mms.ofi ha.~ mw\h~\m mmusm mHImZm mom.0H mm.~ mam.ow mv.m N-.ow mv.~ mm\wa\m omnmv msnmzm muoocwaho csxm msoumawnocmsmm hus>mo comm omumo>ume .oc auowum> mcosuoom Aoom\aov Aoomxoeov muonasooo umoa woneooou mo 6 o sumo hnuoaoou .mes .2 m s we .ees .2 m s m: AHoDOEmsQ 80 my muoneoooo mo mmosd acouommwo as Acmmv omoosaw mo moumm cOsmswwwo .m manna 74 SMR-58 and Pioneer varieties was obtained from a pickling company while the cucumber of variety SMR-lS were hand picked by the author. The cucumbers obtained from the company were somewhat damaged as a result of rough han- dling during harvesting, transportation, and grading. This variable amount of damage may account for the large variability observed for K. The thickness of the skin,¢{, was observed micro- sc0pica11y to be 0.01 mm. For K = 1.7 x 10'5cm/sec and.JSO.001 cm, the corresponding diffusion coefficient of the skin, D, is approximately: D = K6'= 1.7 x 10'5cm/sec x 0.001cm = 1.7 x 10'8cm2/sec. This diffusion coefficient is comparatively much smaller than that of the fleshy tissue where D = 2.6 x 10‘6cm2/sec. The contribution of the skin to resistance of diffusion in the cucumber is evident in Figure 16. This illustra— tion shows diffusion rate curves as calculated from the experimentally observed values of D and K. These curves show that the overall diffusion rate in cucumbers if com- posed only of fleshy tissue would be much greater than that of cucumbers with skin and of the same size. To reach an unaccomplished concentration ratio of Y = 0.12, which indi- cates that the cucumber has taken up 88 per cent of the sugar ultimately transferred, required 72 hr for cucumbers of flesh alone but required 96 hr for those having skin. Thus, it can be seen that the skin offers a significant 75 .omn\eomuesxme.s u s .oonxmsoenesxo.~ u no essn usoeuss one sass muoneoooo E0 m as omoossm mo chmommwo Hem oouoasosoo mo>uoo mums cowmommso .ma ounces lune mess em se me ee es en es ss e r b P d 7 db P 7 mo.o All/Ir ee.e es.e s\\\\l.xeousme assn usages: «.0 xesusme essn Bus3|\\\\< s v.0 Y .1 u m.o o.H 76 resistance to diffusion. Thepermeability of cucumbers as influenced by_storage time in air at 4C. Generally, several days were required to complete a series of diffusion experiments. In many cases, it was desirable that the cucumbers used in these tests be harvested on the same day. Therefore the cucum- bers were stored for several days in air at 4C. Because the cucumbers were subject to bacterial and enzymatic action, the effect of storage time on their permeability was investigated. The results which are shown in Table 9 indicate that there was no significant change in permea- bility of the cucumbers after three days of storage. After ten days, the permeability coefficient of the skin increased greatly while the diffusivity of the flesh did not change significantly. At this time, the skin was observed to be somewhat slippery due to microbial growth and the observed increase in permeability was attributed to bacterial breakdown of the skin. For all other diffu- sion experiments, the cucumbers were stored less than three days. 77 Table 9. Diffusion Rates of Glucose (25C) in Cucumbers as a Function of Storage Time in Air at 4C -' ‘—_—7 (B‘i sd)x106 (F i sd)x105 Cucum- 2 ber di— Storage (cm /sec) (cm/sec) ameter Test time 'Séed cavity Parenchyma- Skin (cm) no. (dcys) tous 65 - 67 0 2.44 $0.265 2.67 i0.240 1.89 i'o.440 3 70A-70C 3 2.77 1-’0.183 2.60 10.331 1.55 ’50.174 69A-69B 1.75 - 2.84 i0.370 1.46 10.155* 73A-73B 10 - 3.18 -0.167 6.11 -3.78 * difference significant at p = 0.05 Diffusion rates in different sizes of cucumbers. Cucumbers of different sizes were tested to determine the influence of size on their diffusion characteristics. The cucumbers tested were all harvested the same day. Because the diffu- sion coefficients of different areas of the flesh in 3 cm cucumbers were found to be the same, it was assumed that the flesh of the smaller cucumbers was also uniform throughout. Accordingly, the diffusion rates in the flesh of the smaller cucumbers were determined on only one area of the flesh which included both the seed cavity and parenchymatous tissue. The results of the study on different sizes of cucumbers are shown in Table 10 and Figure 17. The diffu- sion coefficient of the 2 cm diameter cucumber flesh was found to be significantly greater (p = 0.01) than that of the 2.5 cm cucumbers. The average diffusion coefficient of the 2.5 cm cucumbers was not different from that of the 3 cm 78 Hz—v—I Ia N1 m. ml' muwm Hmnfisoso ou Hm can .m .m mo mwnmcowumHmm .ha gunman EU .mm9m24HQ mumZDUDU o.m m.N o.N flu- . d “m Lt? 3’ lo I 0 <1 OBS/Zara ' t-O-O-I 901 X g 79 Ho.o u m as unmowmwcmwm n Eocmoum mo mmmummn m mm.m epa.ofl mm.H mm~.ow mo.m mow..m mam.60% of K). The large variation of K observed for the cucumbers obtained commercially may have resulted from damage to the surface of the cucumbers during harvesting, transportation, and grading. The surface of cucumbers contains a variable number of 89 spines which leave openings in the skin when broken off. The non-uniformity of the skin could, in part, account for the observed variability in K. One reason for the higher variability in K compared to that of D is that the average D value used in calculation of K for a section could have been different from the actual D value of the section. This would result in an error in the calculated value of K. The higher variability of K compared to that of D also partially resulted from the calculated value of K being more sensitive to small differences in the concentration ratio Y. For a cylinder, if Ya:0.5, then a l per cent error in the experimental value of Y causes a 2.5 per cent error in the evaluated D; while, if Bi is in the range of 5-10 and Y2’0.5, then a l per cent error in the experimental Y value causes a 4 per cent error in the evaluated K. For a slab, if Ys'0.5, then a l per cent error in the experimental value of Y causes a 1.5 per cent error in the evaluated D; while if Bi is in the range of 1-2 and Yc!0.5, then a l per cent error in the experimental Y value causes a 2 per cent error in the evaluated K. This also indicates that the coefficients obtained using hollow cylinders (which were assumed to be of infinite slab configuration) should be less variable than those found when cylindrical sections were used. Accordingly, it was found that the K values obtained using hollow cylinders were generally considerably less variable than those obtained using cylinders (Table A-l). In general, the D values 90 obtained using hollow cylinders were also somewhat less variable than those obtained using cylinders. Another source of error in the evaluated coefficients is in determining the dimensions of a section. This error is larger for a hollow section where the per cent error in the evaluated wall thickness is approximately the same as the per cent error in measurement of its radius, length, and weight. For a cylinder, the per cent error in the eval— uated radius is half of the per cent error in measurement of the length and weight of a section. The cucumber was assumed to be cylindrical for calculation of diffusion rates in cylindrical sections and for-calculation of the average thickness of the hollow cylindrical sections. Because the cucumber shape deviated somewhat from cylindrical, a slight error was introduced by this assumption. Accuracy of Diffusion Coefficients Calculated Using the Finite Difference Method Because the diffusion coefficients of the seed cavity and parenchymatous tissue in salt stock pickles were found to be different, the diffusion rates in salt stock pickles were calculated using the Crank-Nicolson finite difference technique. This calculation method was checked by halving the radial and time increments until the results obtained were within 1 per cent. The accuracy of the method was also checked by apply- ing it to cases which could also be solved using an exact 91 solution and comparing the results of the two methods. The diffusion and permeability coefficients which were calculated from experimental data using both the finite difference and exact solutions are shown in Table 17. The coefficients were calculated assuming the cucumber tissue (not including the skin) to be homogenous. When the finite difference method was employed, the value of the radius for each section had to be rounded-off to obtain a whole number of radial increments (ibr). The diffusion coefficients calculated for test 52 (peeled sections) using the finite difference method were nearly the same as those calculated from the exact solution. The largest percentage deviation was 3.1 and this difference resulted from rounding-off the value of the radius. When the same radius was used in both methods the results were within 0.5 per cent. Similarly, the values of the permeability coefficients of the skin calculated using the two methods were fairly close. When the same radii were used in the two methods, the differ- ence in the K values obtained was less than 1 per cent. However, at the longer experimental times, small differ- ences between the radii used in the calculation led to large differences in the K values. The largest difference in K values calculated using the two methods was 10.4 per cent. The overall average K (1.62x10-5cm/sec) and D values -6 (2.17x10 cmZ/sec) calculated using the finite difference method were very close to those obtained using the exact so- -6 lution (1.64x10-5cm/sec and 2.18x10 cmz/sec, respectively). 92 o.m+ va.c+ ch.a mc.a Nc.a cc.a comm v.cau H~.o| ac.H cm.H ~o.~ mm.H ochw m.c+ Ho.c+ mm.a cc.H mm.a cc.H coma N.H+ Nc.c+ mc.H mc.a mc.H vv.a coma m.an mc.c| co.H mm.a mc.H cm.a ccc Amc Aucsoamc Aommwfioc “any .ommNEoc AEUV “sway mod x M coma coauoom mafia msmwmm mod x M NO msflcmm acme cOwusHom uomxm Aces ma n u<.fio mc.c u qu iwummxm Eoum mucoummwwo cowusaom comaoowZIxsmuu coflMSHom uooxm nexm sues nooeuoom Hooeuoceamo . mm some hHhh n mod x_m mn.m u cod x m ~.H+ mc.c+ mv.~ mc.a cc.~ es.e ccam v.c+ ac.c+ cm.~ mv.a mm.~ mv.a coma m.c+ ac.c+ om.a mc.H om.a mv.a com o.~| mo.c| cm.a cv.H mm.H H¢.H ccc H.mn hc.c| ca.~ cw.a hH.~ N¢.H com 2: 3.828 Sofimfie 83 Bonxeeoe Es 35 ca x a com: cowuoom mas» o mawcmm cod x a mo msflcmm acme ooeuoeom uooxm Aces me u u<.so mo.o u nee -auooxm Eoum oocmummwflo somusaom comHOOflZIxcmuu cowusaom uooxm ncoeuoom dooeuooeaso oedema . mm unoa sootheso o as toenaumeo MOM :oHusHom uooxm on» scum cousasoamu muons on cacao: oocoumumwo muesam somaoowZIxcmuo on» mcwmo woumHsUHmo moumm cowmsmuwo mo camaummeoo .hH OHQMB SUMMARY AND CONCLUSIONS Methods of measurement of sugar diffusion rates in cucumbers and salt stock pickles were developed. In the first experiments, diffusion rates of sucrose in cucumbers were measured; however, the results of these experiments could not be consistantly duplicated. Therefore, the pos- sibility that sucrose was inverted and metabolized when exposed to cucumbers and pickles was investigated. Radio- chromatographic profiles showed that sucrose was, indeed, inverted when exposed to either cucumbers or salt stock pickles. The inversion was attributed to endogeneous invertase. The possibility that sucrose was inverted by acid hydrolysis was discounted because the sucrose in the control solutions did not become inverted. The inversion of sucrose was found to be both qualitatively and quan- tatively the same in different parts of the cucumber. Radiochromatograms of C14 glucose incubated with cucum- bers or salt stock pickles showed that all the glucose orig- inally present in the incubation solution was recovered as glucose. Similarly, in each case all the sucrose originally present in the incubation solution was recovered as sucrose, glucose, or fructose. These observations led to the conclu- sion that sugars were not broken down further than to glucose 93 94 or fructose and, therefore, not metabolized. The remainder of the diffusion tests were performed using glucose as the diffusant. Diffusion rates in cucumbers or pickles were measured by determining the rate of uptake of Cl4 glucose by sections of cucumbers or pickles from a large volume of solution which could be considered infinite. Liquid scintillation counting was used to measure the Cl4 glucose concentration. The diffusion rates through the tissue were expressed by diffusion coefficients and those through the skin were expressed by permeability coefficients corresponding to surface resistances. The first diffusion experiments were performed using cylinders of cucumber tissue. Results of these experiments showed that the diffusion coefficients of the parenchyma- tous tissue increased with the experimental time; the experi- ments were run for as long as 60 hr. Further experimen- tation revealed that the diffusion coefficients of both the seed cavity and parenchymatous tissue increased with time of exposure to the solution. After storage for 20 hr in the diffusion medium (minus the diffusant sugar) at 25C, the observed diffusion coefficients of the seed cavity and parenchymatous tissues increased by 75 and 100 per cent, respectively. To minimize the experimental diffusion time and, thus, minimize the change in the diffusion coefficients of the tissue, subsequent experiments were conducted using hollow 95 cylindrical sections. Because the wall thickness of the hollow sections (0.25 — 0.40 cm) was much less than the radius (1 - 1.5 cm) of the cylindrical sections, the rate of uptake of sugar by the hollow sections was faster. Thus, the experimental times could be reduced considerably (.v 3 - 5 hr).by using the hollow sections. Diffusion and permeability coefficients of cylindrical sections having homogenous fleshy tissue were calculated using the exact solutions for one-dimensional diffusion in an infinite cylinder. When the tissue of the cylin- drical sections consisted of several areas having different permeabilities, the Crank-Nicolson finite difference tech- nique as applied to diffusion in composite infinite cylin- ders was employed. The ratio of the outside radius to the inside radius of the hollow cylinders was relatively small (xv 1.5); therefore, the diffusion and permeability coef- ficients in hollow cylinders were calculated using the equations for diffusion in an infinite slab. Use of the two types of sections in the measurement of diffusion rates were discussed. The diffusion coefficients of glucose in cucumbers determined using hollow cylinders (‘é’2.6x10'6cm2/sec) were somewhat higher than those deter- mined using whole cylinders ( £32.3x10'6cm2/sec). The differences between the tests conducted with cylinders and hollow cylinders were attributed to inherent differences in the cucumbers themselves; the cucumbers were harvested and tested at different times of the season. Comparison of 96 diffusion rates determined using cylindrical sections of salt stock pickles with those determined using hollow cylindrical sections of salt stock pickles obtained from the same batch showed no significant difference between the rates. The results of experiments performed using hollow cylindrical sections of both cucumbers and salt stock pickles were generally less variable than those conducted using cylindrical sections. The diffusion coefficients found for the parenchymatous and seed cavity tissues of cucumbers were not significantly different (Dcy2.6x10'5cm2/sec for 3 cm diameter cucumbers). Therefore, it was concluded that the fleshy tissue of the cucumbers tested was essentially homogenous with respect to diffusion. The cucumber skin was found to offer a signi- ficant resistance to diffusion (K131.7x10'5cm/sec). The diffusion and permeability coefficients of the different varieties of cucumbers tested SMR-IS and a mixture of SMR-SB and Pioneer) were nearly the same. The D value of the cucumber tissue was found to de- crease with the cucumber size. When sizes of 2.0, 2.5, and 3.0 cm diameter were tested, the D values ranged from approximately 3.8x10"6 to 2.7x10'6cm2/sec. The permeability coefficients for the skin of the various sizes of cucumbers were not significantly different, but the Biot number increased markedly with size. This indicated that the skin offered proportionally more resistance to diffusion in the smaller cucumbers than in the larger ones. 97 Although only two different temperatures were used (25 and 40C), the diffusion rates in cucumbers were found to be strongly temperature dependent. The effect was most marked for the skin. The tissue of the salt stock pickles tested (varieties SMR-lS and Pioneer mixed) was found to be non-homogeneous. The diffusion coefficients of the seed cavity tissue was about 4.0x10'6cm2/sec while that of the parenchymatous tissue was approximately 6.2x10’6cm2/sec. The parenchym- atous tissue offered little resistance to diffusion; the 'Gcmz/sec. The diffusivity of glucose in water is 6.9x10 permeabilities of the tissue in salt stock pickles were con- siderably higher than that found for fresh cucumbers of the same variety ( 5’2.3x10'5cm2/sec). Correspondingly, the K value of the skin found for salt stock pickles ( 2’8x10'5cm/sec) was nearly 6 times that found for cucumbers ( c’l.4x10'5cm/sec). The skin of salt stock pickles offered proportionally less resistance (Bifl'20 for the 2.5 cm size) than that of the cucumbers (Bic’7 for the 2.5 cm size). The increase in permeability of cucumbers during curing to pro- duce salt stock pickles was attributed to enzymatic break- down of the tissue from endogeneous and bacterial enzymes and to acid hydrolysis of membranes. The standard deviation of observed diffusion coefficients was usually within 7-10 per cent of the average value, while that for the skin of the hand picked cucumbers (SMR-lS) was approximately 20 per cent. For the cucumbers obtained 98 commercially the per cent sd of the average K was much higher ( >60 per cent); this large variation in K was attributed to damage of the product surface during handling. A major portion of the variation in the observed D values was attri- buted to inherent differences among cucumbers. The larger variability in K when compared to D was accounted for by non-uniformity of the cucumber skin and greater sensitivity of K to errors in measurement. The diffusion and permea- bility coefficients of replicate tests were found to be in good agreement. 8 The ultimate objective of this study was to find prac- tical methods of increasing the rate of sugar. uptake by cucumbers and pickles but only the increase in diffusion rate with temperature was investigated. However, a method for measurement of the diffusion rates was develOped and diffusion characteristics of cucumbers and salt stock pickles were further elucidated. LIST OF REFERENCES LIST OF REFERENCES Abramowitz, M. and Stegum, L.A., ed. 1964. "Handbook of Mathematical Functions.” 4th Ed. U.S. Government Printing Office, Washington, D.C. A.O.A.C. 1960a. Official Methods of Analysis. No. 20.003. 9th Ed. Assoc. Offic. Agr. Chemists, Washington, D.C. A.O.A.C. 1960b. Official Methods of Analysis. No. 20.009. 9th Ed. Assoc. Offic. Agr. Chemists, Washington, D.C. Batson, H.C. 1958. "An Introduction to Statistics in the Medical Sciences." Burgess Publishing Co., Minneapolis. Brock, R.J., Durrum, E.L., and Zweig, G. 1955. "A Manual of Paper Chromatography and Paper Electrophoresis." Academic Press Inc., New York. Bunch, W.H. and Kallsen, G. 1969. Rate of intracellular diffusion as measured in barnacle muscle. Science 164, 1178-1179. Chase, 6.0. and Rabinowitz, J.C. 1967. "Principles of Radioisotope Methodology," pp.303-304. Burgess Publishing Co., Minneapolis. Colowich, S.P. and Kaplan, N.O. 1955. "Methods In Enzy- mology." Academic Press Inc., New York. Crank, J. 1967. ”The Mathematics of Diffusion." Oxford University Press, Ely House, London. Fabian, F.W. and Switzer, R.G. 1940. Experimental work on processing and finishing of sweet pickles. III. Use of dextrose and sucrose in manufacture of sweet pickles. Proceedings of the Institute of Food Technology 1, 329-382. Fellers, P.J. 1964. The effect of several factors on white- ness of cucumbers pickle tissue. Ph.D. thesis, Michigan State University, East Lansing, Michigan. Jacobs, M.H. 1933. The simultaneous measurement of cell permeability to water and to dissolved substances. Journal of Cellular and Comparative Physiology 3, 427-444. 99 100 Jacobs, M.H. 1967. "Diffusion Processes,” reprinted from Ergebnisse der Biologie, 1935. Springer-Verlag Inc., New York. Jacobs, M. H. and Stewart, D.R. 1932. A simple method for the quantitative measurement of cell permeability. Journal g§_Ce11u1ar and Comparative Physiology 1, 71-82. Kopelman, I.J. 1966. Transient heat transfer and thermal prOperties of food systems. Ph.D. thesis, Michigan State University, East Lansing, Michigan. Long, C. ed. 1961. "Biochemist's Handbook." pp.224—225. D. Van Nostrand Co., Inc., Princeton. Moore, T.J. and Schlowsky, B. 1969. Effects of erthrocyte lipid and of glucose and galactose concentration on trans- port of sugars across a water butanol interface. Journal of Lipid Research 10, 216-219. Newman, A.B. 1931. The drying of porous solids; diffusion and surface emission equations. Transactions of the American Institute of Chemical Engineers 27, 203-220. Ozisik, N.M. 1968. Numerical solution of heat conduction problems. In "Boundary Value Problems of Heat Conduction." ed. Obert, E.F., pp.388-484. International Textbook Company, Scranton, Pennsylvania. Perry, J.H., ed. 1963. "Chemical Engineers' Handbook." 4th Ed. pp.l4-25. McGraw-Hill Book Co., New York. Pflug, I.J., Fellers, P.J. and Gurevitz, D. 1967. Diffusion rates in desalting of pickles. Food Technology 21, 90-94. Richardson, D.E. and Switzer, R.C. 1939. Method of testing acid and salt in pickles and pickle products. Fruit Products Journal 18, 292. Rogers, H.J. and Perkins, H.R. 1968. “Cell Walls and Mem- branes," pp.385-392. E and F.N. Spon Ltd., London. Smith, G.D. 1965. "Numerical Solution of Partial Differtial Equations with Exercises and Worked Solutions." Oxford University Press, London. Switzer, R.G., Richardson, D.E. and Fabian, F.W. 1939. Exper- imental work on processing and finishing pickles. I. Rate of diffusion of salt from pickles during the freshing process. Fruit Products Journal 18, 260-261, 281, 283. 101 Tuwiner, 5.8. 1962. "Diffusion and Membrane Technology." Reinhold Publishing Co., New York. Wang, C.H. and Jones, D.E. 1959. Biochem. Biophys. Res. Comm. 1, 203 as cited by Rapkin, E. 1963. Liquid scintil- lation measurement of radioactivity in heterogenous systems. Packard Technical Bulletin No. 5, Packard Instrument Company, Downers Grove, Illinois. White, A., Handler, P. and Smith, E.L. 1964. "Principles of Biochemistry," p.232. McGraw-Hill Book Co., New York. APPENDICES APPENDIX I Derivation of the Equation Used to Calculate the Average Thickness of a Hollow Cylinder Wall and Error Analysis It was of interest to determine the wall thickness of a hollow cylinder in terms of the outside radius, weight, density, and length of the cylinder. The volume of a hollow cylinder is terms of the in- side and outside radii, Ri and R0, reSpectively, is: M/p=7rR:-7erL, (A-l) where pis the density, M is the mass and L is the length. Solving for R, we obtain: 1 (A-Z) Let A be the wall thickness, then A = R - R, . o 1 Substituting (R0 - A) for R1 in eq. A-2 and solving for A, we obtain: M i p7rL ) (A-3) A=R -(R2- 0 O 102 103 It was also of interest to know the relative error in the value of A which would occur as a result of a measurement error in the value of R0. Although the outside radius, R0, of the sample sections was measured in three places and averaged, its value was the most uncertain of the parameters. In order to obtain an expres- sion of the error, the partial derivative of A is taken with reSpect to R: o NIH 8A 8R 0 (A-4) = 1 - R (R2 - M/anY O O MpL To obtain the relative error, 8A is divided by A and BBQ by RC: R aA/A o ( 2 - g) ———7—- = — 1 - - M - 8R0 R0 A RO(R0 /P7TL) (A 5) -1 From eq. A- 3, the term (R.O - M/p7rL) 2 is seen to equal 135-17; . o Substitution in eq. A- 5 yields: aA/A .-. .110. 1 - .31.... (A-6) 8R 7R A R - A ' o o o By rearrangement of the terms, we find that aA/A = _ 1 (A-7) 3R7R l-A/R o o o The relative error in the calculated value of A, caused by an error in measurement of R. , depends on the ratio of the wall thickness to the 0 outside radius (A/RO). In our experiments, the value of the A/R0 104 was usually less than 0. 2. Therefore, for a one per cent error in the measured value of R0, the calculated value of A was in error by l. 2 per cent or less. As is shown by negative error term in eq. A-7, a positive error in the value of RO will cause a negative error in the value of A. An expression relating the relative error in the calculated value of A for errors in the measurements of M and L was obtained in the same manner as that for R0. This expression is: 1 - 0.5 (g) BA/A : aA/A = 0 8M M aL/L (A-s) 1 - A R 0 When the value of A/Ro is 0. 2 or less, a one per cent error in the value of M or L will lead to an error of 1. l per cent or less in the value of the thickness, A. Derivations of Finite Difference Equations (Crank - Nicolson Method) In order to employ the Crank - Nicolson method, we take the average of the mass transferred at times t and t + At and equate it to the accumulated mass over the time interval, At. Calculation of the Center Concentration 105 Center node The mass transferred into and out of node 0 at time, t, is: IN OUT C-C C-C é-(DA% 1 0 —DA —-————O 1) Ar % Ar And the mass transferred at time, t + At, is: IN OUT 4. C _ C+ C+ — C4- 1 DA] 1 0 _. DA1 ——9———1— 2 2 Ar ‘2‘ Ar ’ where D is the diffusion coefficient; Ag is the area at the half inter- val; C0 and C1 are concentrations at r = 0 and r = Ar, reSpectively, + + and evaluated at time t; C 0 and C 1 refer to concentrations evaluated at time t + At. The accumulated mass is: C6-C0 M ), Ar where V% is the volume at the half interval. Combining these terms gives: 106 + C - C C - C C - C l lDA1 1 O-DA1 O 1 +— DA,1 0 2 '2' Ar 2 Ar 2 2 Ar + c; - 01' CO - cO - DA ) = V1 (——-—). (A-9) Ar 2 Ar Ar Ar 2 Since A1 = 2 7r —2— h and vi = 27! (—§—) h, they can be substituted 5 in eq. A-9. When terms are cancelled and the equation is divided through by 7th, we obtain: + 2 C - C + + _ (Ar) 0 0 D(C1 - C0) + D(C1 - C0) — 2 At . (A 10) Rearrangement gives: + + + 2_13_é.I_(c-c +C-C)=C-C. (A—ll) 2 l 0 1 0 0 0 (Ar) If we let m = DAtz and we arrange the unknown concentrations at (Ar) t + A t on the left hand side, we obtain: + —2m 01 + (2m+1)C; = 2m c1 + (1- 2m)Co. (A-12) The coefficients of the general recursion equation, + + + ‘ + + = - aiCi-l biCi cici+1 di ’ (A 13) are then: 107 a = 0, O b = 2m+l, 0 =-2m, Co d = (1-2m)C0-2mC . 1 Calculations of the Concentration at Nodes Within a Cylinder In the derivation of the equation for any node, i, within the cylinder, we use the same procedure: 1. e. , Cylinder node \ (C. - C.) (C. - C. ) 1 1—1 1 1 1+1 — DA, - DA. 2 ( 1";2' Ar 1+2 Ar ) I + + + + (C. - C. ) (C. - C. ) 1 DA 1'1 1 _ DA 1 1+1 2 i-% Ar i+§ Ar + Vi (Ci ‘ C1) = At . (A-14) where 108 A, = 2 7r(i--§)Arh, 1-‘2' , = 27r (i+-§-)Arh, 1+5 V = 2 771(Ar)2 h, Substitution of the areas and volumes in eq. A-l4 yields, after re- arrangement: + c. - c. = —1.- D“ [(21 - 1)c. - 41c. + (21+ 1)C. ] + 1 1 41 2 1—1 1 1+1 (Ar) + + + ' - - ‘ + ‘ + - [(21 1)Ci_1 41Ci (21 1)Ci+l ]) . (A 15) t Substitution of m = 2 and subsequent rearrangement with the un- (Ar) known concentrations at t + At on the left hand side gives: . + . + . + _ -m(21 - 1)Ci_1 + 41(1 + m)ci m(21 + 1)Ci+1 _ m(2i - 1)C1- + 41(1 - m)Ci + m(21+ 1)C1+ (A-16) l 1 ° The coefficients of the general equation (A—13) are then a = -m(2i - 1), b = 41(1 + m), c = -m(2i 4' 1) . d = m(Zi — l)C, + 4i(1 - m)C, + m(21+ DC, 1- 1 1+ 1 l ’ wherei = 1,2, ------------- k-l and k=R/Ar. 109 Calculations of the Concentrations at the Interface of two Different Materials 4-1 4' MI If we let kj be the number of increments of Ar. in material j and n], be the number of increments of Arj+1 in material 3' so that n. Arj+1 = kJArj = Rj’ then a mass balance around the interface i yields 1 (Ci-l ' Ci) (Ci ' Ci+1) _ D.A. 1 _ D. A 1 2 J 1-_2' Ar 3+1 1+5 A1“ + j j+l + + + + 1 (Ci-l - C ) (Ci - C1+1 ) — DNA 1 ' “ D. A. 1 = 2 J 1-5 Ar, 3+1 1+—2— Ar J j+l + (C1 - Ci) V- A—17 1 At ( ) where A. 1 = 2 77 (k.‘ %)Ar-h: 1‘2 J J l 110 Ar, Ar.+1 :: .. l + l 1! Vi 77 (k). 2) Arj (——Lz ) h + 7r (nj 2) Arj+1 ( 2 ) h. The letter 3' refers to the material number; Dj and Dl+1 are the diffusion coefficients of materials 3' and j+l reSpectively. Substi- tution of the areas and volume and multiplication of each side by 4 . Wh yields: ) + D 2k - l - - D 2 + 1 - j< J. ) = J J 1-1 1 3+1 J 1 1+1 + (2k - l)(Ar )2 + (2 + l)(Ar )2 -C—1—--—El-— (A-18) j j “j 3+1 ) At ° D.(2k. - 1)At Let m3 = J J 2 2 a 2k. - l)(Ar.) + (2n. + l)(Ar. ) ( J J J 1+1 + 3+1(2nj 1)At m"+1 2 2 3 (2k, - l)(Ar.) + (2n. + l)(Ar. ) J J J + J 1 Then eq. A-18 becomes after rearrangement with the unknown con- centrations on the left hand side: + + + _ v + 1 + l + 1 _ 1 = iji- 1 ( mj mj+1)Ci mj+1C1+1 I _ I + I _ 1 + I C . _ iji-l (mj mj+1 )Ci mj+1 1+1 (A 19) 111 The coefficients of the recursion eq. A-13 are: 1 Li b1 = 1+m3+m3+1, C1 = -m3+1, d1 = m3C1__1 — (m3 + m3+1 - 1)Ci + m3+1Ci+1 . Calculation of Concentration at the Flesh and Solution Interface At the flesh and solution interface, 8 , the surface concen- tration, CS, is assumed to be the same as the solution or medium concentration, Coo , since the solution is well stirred. Also since the solution volume is large compared to that of the cucumbers C00 is es sentially constant: Flesh- 5 olution inte rface C , Solution 00 The derivation procedure for the node s-l is the same as the 112 previously described. If we let k = s-l, then, + + - - + + = (2k 1)CS_2 4k(1 m)CS_ 1 - + - + + (2k 1)CS_2 4k(l m)CS_1 chng 1), DAt . . . where m = and C 18 the solutlon concentration. (Ar)2 °° efficients of eq. A-13 are: ak = -m(2k - l) , bk = 4k(m + 1) , ck = O , dk = m(2k - l)CS_2 + 4k(l - m)CS‘_1 Calculation of the Concentration at the Skin and Flesh Interface Skin— flesh interface C (X) /_—\k-1 (A-20) The co- + 2m(2k + 1)C . co If K is the permeability coefficient of the skin, then a mass balance around k yields: 113 (C C ) 1 k-l k — - - C + 2 (DAk-% Ar KAk(Ck 00)) 1 ( (C - c ) — D 1 k-l + _ 2 1%(“2— Ar - KAk (Ck (300)) — + (C - C ) l k k _ V _ 2 k Ar ’ (A 21) where A 1 = 2 1r (k — %) Arh , k-2 Ak = 2 7r k Ar h, Vk = 2 17 MA r)2 h . Substitution and multiplication by ;23 yields: (2k l)D(C - c ) - 2k KAr(C - c ) + (2k- 1)D(C+ c+) - k-l k k on k-l k + + 2 Ck ' Ck 2k KAr(C - C) = 2k(AI‘) ————- . (A-22) k 00 At . . . 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