SINGLE PARTICLE CONVECTIVE MOISTURE LOSSES FROM HORTICULTURAL PRODUCTS 1N STORAGE Dissertation for the Degree of Ph. D. MlCHIGAN STATE UNIVERSHY LUIS G. ViLLA 1973 . 17:,“ State ’I‘L’llllll 135mm ‘ML . 2‘ .. , f O - ‘ ‘ o ' - ,- 'v’ z ' N a . "f ..“k ‘W ~.‘ ‘5' ‘_ , . ' ’ ' 5‘ ~'.M‘. , V ' ‘ . ’ 5‘ sr'fic eh PARTICLE *c‘ONVECTIVE ’1 ‘ , .1. , :_ .. ' ISTURE LOSSES FROM 'u “‘ 1'" -‘ HORTICULTURAL PRODUCTS IN STORAGE ‘f v: H 1: ' . resented b I ' ' fr... . a" ' I p y x ’4 'fl . ‘. t ;. ‘ ' f . .4. - :1 ;. ’t ' , Lugs G. Villa 5, .z b! ‘a ‘ . ' i: -.;y 3 ’ . .' ' ; '_ . _' . has been accepted towards fulfillment - 1" V ‘ " f the requirements for "7 -' 2‘3 3‘" - . ‘ JLdEgneé'm W211 v ‘__'_. Engineering "HE‘SIS aLIBR ”\ARY mm M W M ‘ ~ “S“:Sfi“, 3 1293 10 35 6049 5'33...» w I. . ‘ 3 - e b 5 SINGLE PARTICLE CONVECTIVE ‘1. _ ' , ,,, s " ' MOISTURE LOSSES FROM ._, 5 f“ HORTICULTURAL PRODUCTS IN STORAGE "if . , ~- g , presented by g g .‘ '. . \ ~- : ‘4‘ " u U I 2‘ ‘ :‘ 3’: 1‘" Luis G. Villa 3: ~. ., I t ‘ - ‘ '.i . A 7 -‘ g ‘z _ - has been accepted towards fulfillment _-, 4‘ ‘ - *‘ .. of the requirements for T: .17 .1 . r r: , I , Ph. D. degree 1n Agricultural ‘ ‘ 3' . ' Engineering 1; .-"’ 5.? .I" . “W. ".~., “4"‘z‘ r' ABSTRACT SINGLE PARTICLE CONVECTIVE MOISTURE LOSSES FROM HORTICULTURAL PRODUCTS IN STORAGE BY Luis G. Villa Many horticultural products are stored for extended. periods of time. Moisture losses during the storage life of these products is a major cause of deterioration. Moisture losses not only cause wilting and shriveling but also reduce salable tonnage. A good understanding of the mechanism governing the moisture loss process in individual particles will certainly help in developing systems for minimizing the losses. Besides, the prediction of the expected losses at different storage conditions can be useful in the design of storage facilities. A A semi-theoretical mathematical model was developed for predicting moisture losses from horticultural products in storage. The model is based on the distinctive behavior of different regions of the skin of a product with regard to the movement of the water vapor from the product to the Luis G. Villa environment. Free water, porous membrane and impervious regions were identified as the components of the skin of a horticultural product. For using the model, it is necessary to determine the surface area of the commodity, the convective heat and mass transfer coefficients and the vapor pressure deficit of the environment—product system. Besides, three so-called "skin parameters" have to be known. These parameters represent the fraction of the surface behaving as a free water surface, Y1, the fraction of the surface behaving as a porous membrane, Y2, and the resistance to water vapor movement through membrane like regions, r6. The model was used for studying the behavior of apples (Jonathan), potatoes (Manona), and sugar beets (US H20). From moisture loss data of individual particles the skin parameters of these products were determined. Values of Y1 = 0, and 72 = .01286 were obtained for the free water and porous membrane fraction parameters of Jonathan apples. In the case of Manona potatoes, values of Y1 = 0 and Y2 = .00890 were obtained. It was found that the diffusional resistance parameter, r6, is a linear function of the vapor pressure deficit in apples and potatoes. On the other hand, the behavior of sugar beets can be eXplained by assuming that the skin of the sugar beet is a combination of free water and impervious regions. It was found that about 43.6% (Y1 = .436) of the surface Luis G. Villa area of US H20 sugar beet is of a free water nature. Good agreement between predicted and experimental values were observed. The model can be applied for studying other horticultural crops. P Graphs for predicting moisture losses and storage times for apples, potatoes and sugar beets at various storage conditions were developed. The graphs can be effectively used in studying the individual effect of each variable in the moisture loss process from these products. Approved fl6&% / %’ Major Professor //_' 2 -23 Approved Department Chairman SINGLE PARTICLE CONVECTIVE MOISTURE LOSSES FROM HORTICULTURAL PRODUCTS IN STORAGE BY pixel Luis GfVVilla A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1973 to my wife Gloria ii ACKNOWLEDGMENTS The author wishes to express his sincere apprecia- tion to Dr. Fredrick W. Bakker-Arkema, a truly inspiring teacher and counselor whose attitude and friendship made his graduate work a very rewarding and delightful experience. Thankful acknowledgment is extended to Dr. J. V. Beck (Mechanical Engineering), Dr. D. Dewey (Horticulture) and Dr. L. Segerlind (Agricultural Engineering) for serving on the author's guidance committee. Thanks to Dr. Bakker's group of graduate students in helping to crystallize some of the concepts presented in this dissertation. The assistance of Mr. Jim Woodward in making some of the experimental equipment is sincerely appreciated. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . LIST OF APPENDICES . . . . . . . . . . LIST OF SYMBOLS . . . . . . . . . . . Chapter I. INTRODUCTION . . . . . . . . . . II. REVIEW OF LITERATURE - . . . . . . III. IV. 2.1 Convection Heat and Mass Transfer from Free Water Surfaces . . . . . 2.1.1 Convective Heat and Mass Transfer Coefficients . . 2.1.2 Surface Area of Single Particles of Agricultural Products . . 2.1.3 Driving Force for Evaporation 2.1.4 Some Properties of Air-Vapor Mixtures in the Range 32-100°F 2.2 Diffusion of Gases Through Membranes 2.3 Skin Nature of Horticultural Products 2. 3.1 Apples . . . . . . . . 2. 3.2 Potatoes . . . . . . . 2.3.3 Sugar Beets . . . . . . OBJECTIVES . . . . . . . . . . . THEORY . . . . . . . . . . . . 4.1 Model for Predicting Moisture Losses from Horticultural Products . . . 4.2 Estimation of Parameters . . . . 4.2.1 Linear Models . . . . . . 4.2.2 Nonlinear Models . . . . . 4.2.3 Comparison of Models . . . iv Page vi ix xi xii 18 28 30 32 34 35 43 46 47 47 56 56 60 61 Chapter I Page V. EXPERIMENTAL PROCEDURES . . . . . . 63 5.1 Comparison and Development of Formulas for Predicting Surface Area of Individual Apples, Potatoes and Sugar Beets ' . . . . . 64 5.2 Study of the Effect of the Shape of the Body on Moisture Losses from Apples, Potatoes and Sugar Beets . . . 65 5.3 Estimation of Skin Parameters of Jonathan Apples, Manona Potatoes and US H20 Sugar Beets . . . . . . . 68 VI. RESULTS AND DISCUSSION . . . . . . . . 71 6.1 Prediction of Surface Areas of Single Particles . . . . . . . . 71 6.1.1 Jonathan Apples . . . . . . 72 6.1.2 Manona Potatoes . . . . . . 6.1.3 US H20 Sugar Beets . . . . . 80 6.2 Study of the Effect of the Single Particle Shape on Moisture Losses . . 84 6.2.1 Jonathan Apples . . . . . . 85 6.2.2 Manona Potatoes . . . . . . 89 6.2.3 US H20 Sugar Beets . . . . . 92 6.3 Determination of Skin Parameters . . . 95 6.3.1 Jonathan Apples . . . . . . 95 6.3.2 Manona Potatoes . . . . . . 104 6.3.3 US H20 Sugar Beets . . . . . 110 6.4 Prediction of Moisture Losses . . . . 112 6.4.1 Jonathan Apples . . . . . . 115 6.4.2 Manona Potatoes . . . . . . 121 6.4.3 US H20 Sugar Beets . . . . . 126 VII. SUMMARY AND CONCLUSIONS . . . . . . . 132 VIII. SUGGESTIONS FOR FUTURE RESEARCH . . . . . 137 APPENDICES . . . . . . . . . . . . . . 138 REFERENCES . . . . . . . . . . . . . . 221 77 ' LIST OF TABLES Table Page 1. Coefficients in equation (2.20) for various Reynolds numbers . . . . . . . . . . l3 2. Coefficients in equation (2.24) for various Prandtl numbers . . . . . . . . . . 15 3. Coefficients for equation (2.25) for various values of Schmidt number,‘§g. . . . . . . 16 4. Convective heat and mass transfer coefficients . for flow over conical surfaces . . . . . l7 5. Properties of air at atmospheric pressure in the range 32-100°F. . . . . . . . . 33 6. Published data on rate of moisture loss from apples . . . . . . . . . . . . . 40 7. Published data on rate of moisture loss from potatoes . . . . . . . . . . . . . 44 8. Formulas for estimating variances and standard deviations of Yi, b and b in the model _ o 1 Y1 ‘ bo+b1 Xi . . . . . . . . . . . s9 9. Estimated variances and standard deviations of the dependent variable and parameters of equation (6.1) . . . . . . . . . . . 74 10. Table for partition about the mean . V. . . . 74 11. Comparison of four models for prediction of surface areas of individual Jonathan apples . 75 12. Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.2) . . 79 13. Table for partition about the mean for the linear representation of equation (6.2) . . 79 vi Table ' Page 14. Comparison of models for predicting surface areas of individual Manona potatoes . . . . 80 15. Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equations (6.3), (6.4) and (6.5) . . . . . . . . . . 82 16. Table for partition about the mean for the linear representation of Equations (6.3), (6.4) and (605) o o o o o o o o o o o o 83 17. Comparison of weight-surface area relationships for predicting surface areas of individual US H20 sugar beets . . . . . . . . . 84 18. Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.7) . . 86 19. Table for partition about the mean for the linear representation of equation (6.7) . . 87 20. Comparison of equation (2.17) and (2.19) with equation (6.7) . . . . . . . . . . 87 21. Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.8) . 89 22. Table for partition about the mean for the linear representation of equation (6.8) . . . . . 90 23. Comparison of equations (2.17) and (2.28) with - equation (6.8) . . . . . . . . . . . 92 24. Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.9) . . 93 25. Table for partition about the mean for the linear representation of equation (6.9) . . 93 26. Comparison of equation (2.9) with equation (2.27) . . . . . . . . . . . . . 95 27. Skin parameters of Jonathan apples at 70°F. . 97 vii .Table i Page 28. Estimated variances and standard deviations of the dependent variable and parameters of equation (6.10) . . . . . . . . . . 100 29. Table for partition about the mean . . . . 102 30. Comparison of experimental versus predicted moisture losses from Jonathan apples . . . 103 31. Skin parameters of Manona potatoes at 70°F . . 105 32. Estimated variances and standard deviations of the dependent variable and parameters of equation (6.11) . . . . . . . . . . 108 33. Table for partition about the mean . . . . . 108 34. Comparison of experimental versus predicted moisture losses from Manona potatoes . . . 111 35. Comparison of predicted versus experimental results of moisture losses from US H20 sugar beets . . . . . . . . . . . . . . 114 viii Figure l. 10. 11. 12. 13. 'LIST OF FIGURES Comparison between convective mass transfer relationships for flow over spheres . . . . Comparison between convective mass transfer relationships for flat plates (equation 2.13) and for cylinders parallel to the air stream (equation 2.23) . . . . . . . Diagram of an Ellipse used as the Basic Curve for Generating a Model to Represent an Apple . Method of Measuring Model Parameters for Each Side of the Longitudinal Section of an Apple . WeightzHeight Ratio (W/H) Versus the Normalized Surface Area Parameter, o . . . . . . . Schematic Diagram of the Skin of a Horticultural PrOduct O O O O O O O O O O I O 0 Electrical Analogy to Represent the Behavior of the Skin of a Product with Regard to Moisture Losses . . . . - - - . - - Schematic Diagram of the Experimental Set-up Used in Moisture Losses Studies from Peeled Samples Weight-Surface Area Relationship for Individual Jonathan Apples . . . . . . . . . . Area Predicted by Moustafa's Model Versus the Experimental Planimeter Area . . . . . . Weight-Surface Area Relationship for Individual Manona Potatoes . . . . . . . . . . 'Weight-Surface Area Relationships for US H20 Sugar Beets . . . . . . . . . . . . Reynolds Versus Sherwood Numbers for Individual Jonathon Apples. . . . . . . . . . . ix Page 12 14 23 24 26 49 50 66 73 76 78 81 88 Figure 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Page Reynolds Versus Sherwood Number for Individual (Manona Potatoes . . . . . . . . . . 91 Reynolds Versus Sherwood Number for Individual Peeled US H20 Sugar Beets . . . . . . . 94 Environmental Relative Humidity Versus the {g Parameter for Jonathan Apples at 70°F . . . 98 VPD-rd Relationship for Jonathan Apples . . . 101 Relative Humidity of the Environmental Air Versus the {g Parameter for Manona Potatoes . 106 VPD-rd relationship for Manona Potatoes . . . 109 Frequency histogram and normal curve for the effective area parameter, Yl' in US H20 sugar beets . . . . . . . . . . . . 113 Predicted moisture losses from Jonathan apples . 116 Predicted allowable storage time for Jonathan apples. V = 1500 ft/hr . . . . . . . . 118 Predicted allowable storage time for Jonathan apples. V = 3000 ft/hr . . . . . . . . 119 Predicted moisture losses from Manona potatoes . 122 Predicted allowable storage time for Manona potatoes. V = 1500 ft/hr . . . . . . . 124 Predicted allowable storage time for Manona potatoes. V = 3000 ft/hr . . . . . . . 125 Predicted moisture losses from US H20 sugar beets O O O O O O O O O O O O O O 127 Predicted allowable storage time for US H20 sugar beets. V = 1500 ft/hr . . . . . . 128 Predicted allowable storage time for US H20 sugar beets. V = 3000 ft/hr . . . . . . 129 Appendix A. Conversion Table from British Units to SI units 0 O I O I O O O I O O B. ALEASQ Computer Routine . . . . . . C. Comparative Results of Four Models for Predicting Surface Areas of Individual Jonathan Apples . . . . . . . . D. COmparative Results of Two Methods for Predicting Surface Areas of Individual Manona Potatoes . . . . . . . . E. Comparative Results of Three Models for Predicting Surface Areas of US H20 Sugar Beets . F. Experimental Data of Moisture Losses from Individual Peeled Jonathan Apples . . G. Experimental Data of Moisture Losses from Individual Peeled Manona Potatoes . . H. Experimental Data of Moisture Losses from Individual Peeled US H20 Sugar Beets . 1. Experimental Data on Moisture Losses for Determining the Skin Parameters of Jonathan Apples . . . . . . . . J. Experimental Data on Moisture Losses for Determining a rd-VPD Relationship for Jonathan Apples . . . . . . . . K. Experimental Data on Moisture Losses for Determining Skin Parameters of Manona APotatoes . . . . . . . . . . L. Experimental Data on Moisture Losses for Determining a rB-VPD Relationship for Manona Potatoes . . . . . . . M. Experimental Data on Moisture Losses from LIST OF APPENDICES Peeled and Unpeeled US H20 Sugar Beets xi Page 139 142 145 154 165 170 172 174 176 186 194 204 212 » C1,C2,C3 d ,6 ,d3 LIST OF SYMBOLS ftz Surface area of a sphere having a diameter Area, surface area, equal to the height §_of an apple, ftz. Mass transfer driving force, defined by eqn. (2.53), dimensionless. Estimation of parameters 80 and 81, respectively Water vapor concentration, lbm/ft3 Air specific heat at constant pressure, BTU/lbm °F Water vapor concentration at the surface of the body, lbm/ft3. 3 Environmental water vapor concentration, lbm/ft Coefficients of equations for calculating the .convective heat and mass transfer coefficients. Specific gravity of apples, eqn(2.29), dimensionless. Equivalent diameter of a product ( = JA7W"), ft. Molecular diffusivity, ftZ/hr. Force units. Fraction of the surface area through which gaseous exchange occurs, eqn.(2.62), decimal. Convective mass transfer coefficient, defined lbm/hr-ftz. by eqn.(2.52), Height of an individual apple (Cooke's model), ft. Overall convective heat transfer coefficient, BTU/hr-ft2-°F. xii hl O h d app hfg 23" b t4 N é? 23' E: I: l w2 w3 woo WS Overall convective mass transfer coefficient, ft/hr. Overall convective mass transfer coefficient, lbm/ftz-hr-lbf/ftz. — Overall apparent convective mass transfer coefficient for an unpeeled product, lbm/ftZ—hr-lbf/ftz. Heat of vaporization, BTU/lbm. Local convective heat transfer coefficient at some point x, BTU/hr-ft2-°F. Thermal conductivity, BTU/hr-ft-°F. Units of length. Slant height of a sugar beet, ft. Units of mass. Molecular weight of water ( = 18.01), eqn.(2.48). Total mass flow rate, lbm/hr. Fraction of the total mass rate occuring through free water surface regions, eqn. (4.1), lbm/hr.’ Fraction of the total mass rate occuring through "membrane like" regions, eqn.(4.2), lbm/hr. Mass rate through impervious regions (=0). Environmental mass concentration defined by eqn.(2.54), dimensionless. Mass concentration at the surface defined by eqn.(2.54), dimensionless. xiii Nu Nu PL r6 Re Re x = Number of data points. = Nusselt number (=%§), dimensionless. = Local Nusselt number at some point x, dimensionless. = Percentage of moisture loss, eqns.(2.64) and (2.65), percentage. = Vapor pressure at the outside of a "membrane like" surface, lbf/ftz. = Prandlt number (=3), dimensionless. = Vapor pressure at the inside of a "membrane like" surface, lbf/ftz. = Saturated vapor pressure, eqn.(2.55), lbf/ftz. = Vapor preSsure, lbf/ftz. = Vapor pressure at the body surface, lbf/ftz. = Environmental vapor pressure, lbf/ftz. = "Resistivity" of a membrane to water vapor movement, dimensionless. = Skin parameter (=resistivity r times membrane thickness 6), ft. V00 x Reynolds number (= ), dimensionless. Reynolds number based on a x characteristic dimension, dimensionless. Rate of moisture loss, lbm/hr. Universal gas constant (=L544 ft—lb per (mole) (°R). xiv Sh St St s.d.(I) Voo Local radius of a body of revolution at some point x, ft. Regression coefficient for comparing two linear or nonlinear models, defined by eqn.(4.18), dimensionless. Regression coefficient for comparing eqn.(4.21) with eqn.(4.22), dimensionless. Overall Sherwood number (=hd x) dimensionless. wa Overall Stanton number (= h ), dimensionless. ~ pcP V0° Local Stanton number at some point x, dimension- less. Standard deviation of (I). Absolute temperature, °R. Surface temperature, °F or °R. Surface temperature calculated by eqn.(4.5),°R. Environmental temperature, °F or °R. Velocity component in the x direction, cgns. (2.3) (2.4) and (225), Et/hr. Velocity component in the yyiirection,e:gts.(2.3), (2.4) and (2.5), ft/hr. Variance of I. ‘Vapor pressure deficit; 1bf/ft2. Vapor pressure deficit (mm of Hg) x weeks of storage, eqn.(2.64) and (2.65). Environmental air velocity, ft/hr. XV a 80'851 811 8].- Y1 Y2 Width of an individual apple (Cooke's model),ft. Weight of the individual particle, lbm; Characteristic dimension of a body, ft. Cartesian coordinate. Independent variable of a given model. Cartesian coordinate. Dependent variable of a given model. Thermal diffusivity, ftz/hr. Parameters of a given model. Fraction of the surface area that behaves as a free water surface, decimal. Fraction of the surface area that behaves as a porous membrane, decima1.. Membrane thickness, ft. Error in measurement 1. Air dynamic viscocity, lbm/hr-ft. Air density, lbm/ft3. Time units. Absolute humidty, lbm water/lbm dry air. Dummy variable, eqn.(2.40). Dummy variable, eqn.(2.40). . . . . . 2 K1nemat1c Viscoc1ty of air, ft /hr. xvi I. INTRODUCTION Loss of water from harvested horticultural crops is a major cause of deterioration in storage. Moisture losses not only cause wilting and shriveling but also reduces salable tonnage. Wilting and shriveling caused by water losses seriously damages the appearance of produce and thus affects consumer appeal. Many fruits and vegetables will appear shriveled or wilted after water loss of only a small percentage of their original weight (3 to 5%). Besides, the loss in produce weight as a result of water 'loss becomes a direct economical loss. Although the different variables affecting moisture loss from horticultural products has been recognized for quite some time, insufficient use had been made of the physical transport phenomena theory in investigating the problem. The effect of variables such as the airflow of the environmental air on moisture losses has been insuffi— ciently studied. Lack of information was found with regard to the behavior of the skin of agricultural pro- ducts under different environmental conditions. A brtter understanding of the mechanism of the moisture loss process will certainly help in developing systems for minimizing moisture losses. The main objective of this research was to integrate the different variables affecting the moisture loss process into a mathematical model. 'This approach allows a systematic study of the individual effect of each variable. The developed model will be used in studying the behavior of individual Jonathan apples, Manona potatoes and US H20 sugar beets with regard to moisture losses. Graphs for the prediction of moisture losses and allow- able storage times at different storage conditions will also be developed. II. REVIEW OF LITERATURE The investigation of the process of moisture transfer from a moist body can be divided for purposes of analysis into two parts: (i) the transfer process occur- ring within the product, and (ii) the interaction of the product surface with the environment. The first of these processes occurs as a result of capillary flow and dif- fusion caused by a difference in concentration. On the other hand, the factors controlling the rate of transfer of water vapor from the moist body to its surroundings are associated with convection of the vaporized water away from the body. Van Arsdel (1963) stated that the factors that determine the rate of movement of water within the body can be regarded as independent of the external conditions. The phenomenon of moisture losses from perishable agricultural products can be analyzed at a macroscopic level as a process controlled by the rate at which moisture moves through the skin of the product and is carried away from the surface by convection. The following review of literature examines the theory behind the evaporation from a free water surface, the diffusion through membranes, and the effect of the skin of the product on the loss of moisture. 3 2.1. Convection Heat and Mass Transfer from Free Water Surfaces In the study ofevaporation of water from free water surfaces,macrosc0pic energy and mass balances lead to the following governing equations: Mw hfg = hA (T0° - T3) (2.1) and Mw = hd A (Cws - Cwm) (2.2) The convective heat and mass transfer coefficients, h and hd' are essentially aerodynamic properties of the system whereas the temperature and mass driving forces are thermodynamic properties. A study of the variables defined in equations (2.1)and (2.2) follows. Relationships for calculating the convective heat and mass transfer coefficients for flow over different geometrical shapes (to which agricultural products can be approximated) are analyzed first. The prediction of surface areas is studied Specifically for those products studied in this research. An analysis of the different unit systems used for expressing the driving force is given next. Finally, a tabulation is made of some properties of air and air-vapor mixtures in the range of temperatures used in this study. 2.1.1; Convection Heat and Mass Transfer CoeffICIents A considerable amount of research has been carried out on the prediction of the convective heat and mass transfer coefficients for flow over single bodies of various geometrical shapes. From principles of the boundary layer theory analytical solutions have been obtained for a limited number of situations. Regret- tably, it has not always been possible to obtain analytical solutions, especially for situations where separation of flow occurs. Experimental methods have been used in such cases. I Analytical solutions of the boundary layer for flow over external surfaces include assumptions sucheas: thermal and velocity boundary layers which develop along the surface of the body are not influenced by the develop- ment of boundary layers on any adjacent surface; all body forces are negligible so that the fluid is forced over the body by some external means unrelated to the tempera- ture field in the fluid. The solutions are also based on an idealization of constant fluid pr0perties, unaffected by temperature. Under the preceding assumptions the applicable differential equations of the boundary layer for a two dimensional situation are (Kays, 1966): Continuity .%% + %% = 0 (2.3) Bu Bu 32u Momentum u§§ +v3§ = v——§ (2.4) 8y 8T 3T 32u Energy 5; + V3; = a;_§ (2.5) Y Adding the proper boundary conditions, equations (2.4), and (2.5) have been solved for a limited number of situations.’ In cases where analytical solutions have not been obtained, dimensional analysis has been effectively used. Relationships in terms of dimensionless numbers have been developed for predicting the convective heat and mass transfer coefficients from experimental data. In general, the following relationships relate the different variables that affect the convective heat and mass transfer coefficients: clReczPrC3 (2.6) Nu I I' . Sh = c'lRec 2PrC 3 (2.7) Agricultural products can be approximated by shapes such as flat surfaces, spheres, cylinder, axisym- metrical bodies, cones or prolate spheroids. A discussion of the working formulas for the convective heat and mass transfer coefficients of flow over those geometrical shapes follows. a. Flow Over Flat Surfaces.--An analytical solution can be obtained if the following assumptions are made: laminar flow, incompressible and steady flow, no pressure variations in the direction perpendicular to the plate, constant properties of fluid, and negligible viscous shear in the direction perpendicular to the plate. The solution leads to the relationship (Holman, 1972): 1/3 1/2 Nu = .332 Pr Re (2.8) x x I for the local convective heat transfer coefficient. An overall value for E can be written by inte- grating over the length of the plate: h = = 2hx (2.9) or Nu = .664 131:1/3Rex:£2 , (2.10) Re < 5 x 105 Holman (1972) presents a semi-empirical formula for calculating the convective heat transfer coefficient -over a length L of a plate when both the laminar and the turbulent layers are present. A value of 5 x 105 for the Reynolds number is assumed for transition from laminar to turbulent flow. The resulting relationship is: 1/3 Nu = Pr (.036 Rex;§ - 836) (2.11) Re'5 5 x 105 Powell and Griffith (1935) and Powell (1940) obtained experimental data on evaporation rates of water from saturated surfaces into air for various geometrically shaped bodies. They expressed the evaporation data in terms of the quantities (ul) and el/(Pw - Pa), where u = Air velocity 1 = Characteristic dimension e = Rate of evaporation per unit area Pw = Vapor pressure at saturated surface Pa = Vapor pressure in the air stream. If Powell's data are rearranged in terms of dimensionless numbers, the following relationships hold for a flat plate: Sh = .091 Re'69 (2.12) 1000 < Re < 20,000 streamlined edge Sh = .046 Re'76 (2.13) 20,000 < Re < 400,000 streamlined edge Sh = .56 Re'50 (2.14) 2000 < Re < 20,000 non-streamlined edge Powell's data are valid for only a Schmidt number of .6. The mass and heat transfer equations for flow over flat plates without streamlined leading edges, agree well with each other in both the laminar and turbulent regions. The mass transfer data for flow over flat plates with streamlined leading edges fall approximately 30 percent below the mass transfer data for flow over non-streamlined flat plates in the laminar region, and fall approximately 5 percent below the mass transfer data for flow over non-streamlined flat plates in the turbulent region (Boelter et al., 1965). b. Flow Over Spherical Surfaces.--Flow over spheres typified the problem of separation of flow over external surfaces. Heat and mass transfer coefficients diminish from the forward stagnation point to the point of separation of the laminar boundary layer and then increase. When the transition to a turbulent boundary layer is followed by separation, the two points are indicated by sudden trends of increasing hx in the region 90 to 120 degrees from stagnation (Bennet;_and Myers, 1962). Extensive data have been taken to determine the convective heat and mass transfer coefficient for flow over spheres. McAdams (1954), on the basis of the results of a number of investigators, recommends the use of 10 the following equation for calculating the convective heat transfer coefficient: Nu = 0.33 Re'6 (2.15) 1000 < Re < 50,000 An analytical solution for Reynolds number approaching zero gives, Nu = 2 (2.16) Re+ 0 One of the earliest analyses of mass transport from spheres is contained in the work of Frossling (1938). In addition to a rather complete theoretical analysis, as well as some experimental work involving macrosc0pic transport to an air stream from drops of different hydro- carbons, a relationship for calculating the overall mass transfer coefficient was obtained: 1/3 1/2 Sh = 2.0 + .6 SC Re (2.17) The constant 2.0 in this equation corresponds to the analytical value for Sherwood number when Reynolds number tends to zero: Sh = 2.0 (2.18) 11 Powell's evaporation data for airflow over spherical surface results in the following equation: Sh = .29 Re'59 (2.19) 600 < Re < 46,000 Figure 1 shows a comparison between equations (2.17) and (2.19) in the Reynolds number range from 600 to 10,000. The formulas agree to within 1 18%. c. Flow Over Cylindrical Surfaces.-—The most extensive data for flow over cylinders are those of Hilpert (1933). He considered flow of air normal to cylinders at various diameters. The results presented in terms of the Nusselt and Reynolds numbers based on the cylinder diameter are: Nu = c1 ReC2 (2.20) The coefficients c1 and c2 from Hilpert's data are given in Table 1. For evaporation from cylinders normal to the air- stream, Powell's data lead to the relationships: Sh = .24 Re'59 (2.21) 100 < Re < 2000 Sh = .17 Re'64 (2.22) 2000 < Re < 40,000 12 .coflumuwm maflammnum mamum> sump m.HHm30m .mwumnmm um>o 30am MOM mmflnmcoeumawn uwmwcmuu mums m>fluom>coo cowsumn :omflummsooul.a musmfim AIWNH 000.0 000.m 000.v 000.“ 0 u - - fl 0 00. n um 1 0m s.ms 0.0m «.00 000.0H m.sa v.0m m.m0 000.m v.ns 0.s0 0.00 000.0 Aaohumsem masammmumc 0m om + 0.~ u hm hm N.SH n.vv 0.em 000.5 «\H m\H + 1 00 «.ms 0.Hv m.0e 000.0 s.es m.sm v.vq 000.m . . . . “mums m.HH030mc s 0H H am N mm 000 v 0m.0m 0n. n cm 0.0 p.0m m.~m 000.m . 1 00 m.m 0.0m m.mm 000.m 0.0 I H.0H H.SH 000.H m.ms- e.vs 0.NH 000 0 Isa.m0 10H.NV mm L 00 .mmflo .cvm .cwm . Em em 13 TABLE l.--Coefficients in equation (2.20) for various Reynolds numbers. Flow normal to a circular Cylinder (Kays, 1966). Re g1 c2 1-4 .391 .330 4-40 .821 .385 40-4,000 .615 .466 4,000-40,000 .174 .618 40,000-250,000 .0239 .805 In the case of cylinders parallel to the air- stream, the following relationship developed from Powell data holds: 8 Sh = .029 Re’ (2.23) The order of agreement over an extended range of Reynolds number (5,000 to 200,000) of equations (2.13) and (2.23) is shown in Figure 2. Differences of less than 10 percent were observed. d. Flow Over Axisymmetrical Bodies.--Smith and Spalding (1958) proposed an approximate solution for the case of flow over axisymmetric bodies. The solution is restricted to the laminar constant-surface temperature problem and to situations where separation of flow does not occur. The solution leads to the relationship: 14 .mume m.aamzom .Amm.m soeumsqmv Hudson was map on Hwaamumm mumocfiawo How one Ama.m coflumsqmv mmumam umam How mmflanOAHMHoH Hmmmsmuu mums m>fluom>coo cwm3umn consummEoonl.m mesmem T Om OH X mH OH m 0 0 0 q 4 l4 0 .d o m m OOH OO.NI m.hom N.mmw 000.com 0m.Hl O.v®v m.vmv OO0.0®H ON.HI m.ONv N.mHv 000.00H Om.OI 0.55m m.mbm OO0.0VH m we. I m.mmm H.Nmm OOO~ONH OON J 4. mo. 0.60N 0.00N OOO.om 0v.N m.bHN m.NNN OOOSOB O®.N m.©mH m.HhH 000.0m Om.v N.OHH v.mHH 000.0m Om.m ¢.®v m.om OO0.0H OOm Oh.OH m.®N m.mN OOO.m w Amm.NV AMH.NV mm .mman .va .cvm Sm Sm 00¢ 15 1/2 C2 c u Rx(Vmp) = 1 Stx (2.24) c 1/2 X 3‘3 2 [fo(va) Rx dx] The coefficients c1, c2 and c3, are given in Table 2, for variOus Prandtl numbers. In order to find the average convective heat transfer coefficient equation (2.9) can be used. TABLE 2.--Coefficients in equation (2.24) for various Prandtl numbers. Heat transfer to the laminar- constant property boundary layer (Kays, l966).* oPr C1 c2 C3 .7 .418 .435 1.87 .8 .384 .450 1.90 1.0 .332 .475 1.95 5.0 .117 .595 2.19 10.0 .073 .685 2.37 = constant By analogy, the local convective mass transfer coefficient gx, defined in equation (2.52) can be computed by Rx(Vm0) 2 x dx] (2.25) 16 The coefficients c1, c2 and c3, as computed by Spalding and Chi (1963) are functions of the Schmidt number, §Er and the driving force, B, and are given in Table 3. As before, equation (2.9) is used to calculate the overall mass transfer coefficient. TABLE 3.-—Coefficients for equation (2.25) for various values of Schmidt number, §E° Laminar constant- property boundary layer (Kays, 1966). Sc B c1 c2 c3 .7 -.9 1.850 .050 “1.10 -.6 .812 .150 1.30 0.0 .418 .435 1.87 1.0 .244 .650 2.30 3.0 .136 1.150 3.30 9.0 .060 1.900 4.80 1.0 -.9 1.430 .150 1.30 -.6 .633 .250 1.50 0.0 .332 .475 1.95 1.0 .200 .650 2.30 3.0 .113 1.000 3.00 9.0 .052 1.450 3.90 5.0 -.9 .431 .450 1.90 -.6 .205 .500 2.00 0.0 .117 .595 2.19 1.9 .073 .650 2.30 3.0 .045 .750 ' 2.50 9.0 .023 .900 - 2.80 5.0 -.9 1.037(Sc)_§;§ .90 2.8 -.6 .568(Sc)_2/3 .90 2.8 0.0 .339(SC)_2/3 .90 2.8 1.0 .230(Sc)_2/3 .90 2.8 3.0 .145(Sc)_2/3 .90 2.8 9.0 .077(Sc) .90 2.8 17 e. Flow Over Conical Surfaces.-—Luikov (1965) obtained experimental relationships for calculating the convective heat transfer coefficient of flow over bodies of a conical shape. Miranov (1962) reported relation- ships of convective mass transfer coefficients over cones during porous cooling. Table 4 presents the reported equations for each tested condition. TABLE 4.--Convective heat and mass transfer coefficients for flow over conical surfaces. Orientation Characteristic Relationship Reference to Airflow D1mens1on ”'1: 1 / Nu=.128Re'65 (2.26) Luikov (Metal bod ) (1965) 7 1 Y ‘9' Sh=.l6lRe'67 (2.27) Miranov ' (Porous Cooling) (1962) f. Flow Over Prolate Spheroids.--Lochiel and Caderbank (1964) developed relationships for calculating the convective mass transfer coefficient of flow over prolate spheroids by using results of flow over spheres: Sh ‘ 1/3 2 1/2 —P—S- = [3 (“Jul/2 [ 2E ”’3 ’ (2.28) Shs 3 E(1-E2)1/2 + Sin"1(1-1=;2)1/2 where l8 Shps = Sherwood number for prolate spheroids ShS = Sherwood number for spheres [Equation (2.17) or (2.19)] __ Ln[(l+e)/(leé)] - 2e J —{ 2} Ln[(l+e)/(l-e)] - 2e/(l-e ) e = (1-E2)1/2 til ll "Eccentricity" or width to height ratio. 2.1.2 Surface Area of Single Particles of Agricultural Products Formulas for the prediction of surface area of individual product are empirical in nature. A number of researchers have determined the surface area of the product by peeling the commodity in narrow strips and taking the planimeter sum of the tracings as the surface area. Others have used a method consisting of coating the product surface with spherical beads and then corre- late the weight of those beads with the surface area. In each case the value obtained was assumed to be the actual area of the commodity and it was used to obtain relationships between the surface area and product parameters such as the area of traverse cross section, traverse diameter, axial or longitudinal diameter, correla- tion with the geometric volume and correlation with the geometric volume and correlation with the weight. A review of the literature on prediction of surface area of individual apples, potatoes and sugar beets follows. 19 a. Apples.--Some researchers have calculated the surface area of an apple on the assumption it is a sphere. Magness et_al. (1926) used measurements of the circum- ference, while Hamilton (1929) used caliper measurements of the diameter of tagged apples. Gunther (1948), Baten and Marshall (1943), Chapman gt_al. (1934) and Smith (1926) measured the traverse and vertical axis and used the average of these diameters to calculate the area. Chapman §t_al. (1934) measured the volume displaced by an apple to calculate the diameter of a sphere of that volume and then used the diameter to compute the surface area. Barnes (1929) considered an apple as a cardioid and used the formula: A = .3095 3W (2.29) Barnes (1929) reported that the surface areas computed by equation (2.29) did not differ over 5% from his best "unstated" mechanical measurement.. Baten and Marshall (1943) compared several methods to predict surface areas of apples and other fruits. They found that the traverse diameter, i.e., perpendicular to the core, gave the best predictions for unpicked fruits, while the relation of surface area to weight gave the best predictions of surface area for picked apples. The following weight-surface area relationships were reported: 20 For Delicious apples A = .045993 + .40635WO (2.30) For Jonathan apples A = .044701 + .42840WO (2.31) For McIntosh apples A = .049458 + .40635WO (2.32) For Stayman Winesap apples A = .058472 + .35280WO (2.33) Frechette and Zahradnik (1965) compared a linear and a second degree polynomial weight-surface area relationship for McIntosh apples. The following two regression equations were compared: A .054306 + .34650WO (2.34) and 3’ ll .155764 + .71820(WO - .28983) - .6044(W02 -.08675) . (2.35) 21 Although the second degree polynomial gave the best fit curve for the data, the two regression equations differ insignificantly. The researchers pointed out that the linear equation can be used with confidence because it gave a correlation coefficient of .975 and a maximum error of 2.9% from the mean. Frechette and Zahradnik (1965) also developed an empirical equation for the surface area of equivalent spheres based on the average density of the McIntosh apples tested: .667 A = .36069(Wo) (2.36) The values of this curve differ a maximum of 3.86% from the best fit curve for the 84 McIntosh apples that were teSted for the relationship of surface area based on weight. Recent studies have idealized several fruits as bodies of revolution having some characteristic parameters. Moustafa (1971) idealized the apple fruit as an ellipse whose coordinate axes were translated and rotated to create a shape similar to one half of an apple cross section. The elliptical shape was then rotated 360 degrees about its new major axis. The surface area resulting from the rotation of the upper part of the ellipse through an angle of Zn around the x-axis will be 22 2 Sin 8 d0 (2.37) y II n 2n f0 R where R = a2c Sin (0-2) 'D/Ia4c28in2(0-z) - [b2+(a2-b2)Sin2(0-z)][czaz-azbz] h2 + (az-b2)Sin2(0-z) (2.38) a, b, c and z are the characteristic parameters. Their definition is given in Figure 3. Using numerical approximations, equation (2.37) takes the form: n 2 . A = Zn 2 R. Sin 0. (A0) (2.39) i=1 1 1 a, b, c and z are measured from a longitudinal section through the longitudinal axis of symmetry of the fruit. Figure 4 shows the method of measuring model parameters. Moustafa's results Show close agreement between experimental and predicted surface areas for apples with differences of less than 10 percent. No information of the variety used for testing the model is given. Moustafa pointed out that small errors in measuring the model parameters can result in a large error in the theoretical Ellipse in Cartesian a2 b Coordinates if x = x1 §=Yl-C ble2 + a2(yl - c)2 = azb2 or in polar coordinates b2r2C052¢-+a2r28in2¢-2a2chin¢ = a2(b2-c2) where x1 = rCos¢ Y1 = rSin¢ rotating througn an angle 2, b2R2Cosz(0ez)- 2a2cRSin(0-z) = a2(b2-c2) Figure 3.--Diagram of an Ellipse used as the Basic Curve for Generating a Model to Represent an Apple. Axis x and y were Translated to x1, Y1! and then Rotated to x, y. The Major Portion of the Ellipse was Then Rotated 360 Degrees about the x-axis, Generating the Model. Moustafa (1971). 24 Figure 4.--Method of Measuring Model Parameters for Each Side of the Longitudinal Section of an Apple. Moustafa Model. 25 predicted area due to the high power to which the parameter is computed. A mathematical model of the apple using bi- sphercal coordinates was developed by Cooke and Rand (1969). The width W and height H are the descriptive parameters. The surface area A,of the apple can be computed from: 2 0° 2n ‘Sinedtp dn A = 2a fn=o fw=o (Cosh n-Cos 0) (2.40) where 0 = cos-1 (3- - 1). (2.41) a = E—S—i-l—e- (2.42) Equation (2.40) may be normalized by dividing by the surface area (As) of a sphere having a diameter equal to the height H of the apple: A 2 = [sin(3+ (n - 0) cos 0] = 0 (2.43) 4“ (8/2) or A = 0 A (2.44) 26 The function 0 defined in equation (2.44) increases monotonically with W/H and is equal to unity for W/H=1. A plot of W/H versus a is presented in Figure 5. 'W/H + Figure 5.--Weight:Height Ratio (W/H) Versus the Normalized Surface Area Parameter, 0. Equation (2.44). t b. Potatoes.--Maurer and Eaton (1971) described a method for predicting surface areas of potatoes by , measuring the major and minor axis of potato tubers. It was assumed that the shape of the pctato is closer to the geometric form of a prolate spheroid than to other geometrical forms. The equation for a surface area of a prolate spheroid is: 27 2 . -l _ nb ndb Sin Ec where d is the major axis, b the minor axis and 02-b2 RC = b is the eccentricity. Because of differences in the traverse sections of individual tubers, the minor axis was estimated by adding the value of the width (b1) and thickness (b2) and dividing by 2 to obtain the minor axis. In terms of a potato tuber, equation (2.45) becomes: n(bl + b2) [(b1 + b2) Ecc + 2d Sin-lEccl‘ A = 8Ecc (2°46) where ECC = Maurer and Eaton (1971) applied equation (2.46) to tubers of several varieties including Red Pontiac, Kennebec and Warba. Although there was close agreement between experimental and predicted results some varieties did not correlate as well as others. The errors resulted mostly from tubers with a flat side. 28 c. Sugar Beets.--Sandera and Suecova (1954) suggested the following weight-surface area relationship for sugar beets _ 2/3 The constant C is a function of the _£EEEEE_ (E) o diameter D ratio. For the variety for which equation (2.47) was developed the value of (g) ranged between 4 to 6 and Co between 6.83 and 7.17. 2.1.3 Driving Force for Evaporation Units of the energy driving force defined in equation (2.1) are temperature. Three different sets of units are used to express the mass driving force: a. In terms of concentration units. C in equation (2.2) can be expressed in [ML-3] units. In this case hd will be given in [L0Hl] units. b. In terms of vapor pressure units. Assuming water vapor as an ideal gas: (Cws " C...) = if" ‘7‘“ " “:‘jj’ (2°48) Substituting (2.48) into (2.2) 29 ° — 31 _!§.- .22 Mw — hd A R0 (TS T...) (2.49) Since TS and Tm are absolute temperatures and close in value, equation (2.49) can be approximated by: = ' - where . _ M h d ’ hd ROT (2'51) Units of the mass transfer coefficient h'd, defined in equation (2.51), are [Me'l F-ll. c- In terms of dimensionless units. Equation (2.2) can also be written as (Kays, 1966): MW = gAB (2.52) where mvw m m S B = ~¥fi-~:ET , dimensionless . (2.53) we and _ _ l mwi — l 1 + m (2.54) 30 where Mass of water) w. = Absolute humidity (Mass dry air 1 Units of the mass transfer coefficient g, defined in equation (2.52), are [Me-1 L-Z]. Although dimensionless units are more convenient from an engineering stand point, pressure units have most often been used to present data on moisture losses from agricultural products. The term "vapor pressure deficit" (VPD) is employed to describe the driving force for the moisture losses phenomenon. 2.1.4 Some Properties of Air-Vapor Mixtures in the Range 32-100‘? For completeness purposes, tabulation of those properties of air and air-vapor mixture important in the evaporation process is presented in this section. The listing of such properties is limited to the 32-100°F temperature range. Equations describing properties were preferred whenever they were available in the literature. Linear interpolation was used in some cases; a. mgfituration Pressure Line of Air-Vapor Mixtures.-- From Brooker (1970): 31 A1 + A2 + A3T2 + A4T:3 + AST4 Ln(P /144 A ) = e (2.55) sat o 2 AGT - A7T 491.69 : T(°R) : 959.69 where, Ab = 0.3206182232000000 D 04 A.4 = 0.2153211916363544 D -04 A1 = -0.2740552583614256 D 05 (A5 = -0.4620266568199822 D -08 .A2 = 0.5418960763289505 D 02 ‘A6 = 0.2416127209874000 D 01 .A3 = -0.4513703841126545 D-Ol A7 = 0.1215465167060546 D -02 . b. Latent Heat of Vaporization, hfg.--Brooker (1967), using Keenan and Keyes data, developed the following linear regression curve for the latent heat of vaporiza- tion: hfg = 1075.8965 - 0.56983 (T - 459.69) (2.56) 491.69 :,T (°R) : 609.69 c. Absolute Humidity.—-Assuming that the air and . water vapor are ideal gases the well known‘psychrometric expression for the absolute humidity can be derived: .6219 Pv w = 5...:7§_ (2.57) atm V 459.69 3 T (°R) : 959.69- 32 d. Air Density.-- (Patm - Pv) = ""'5_3. 3'_5 T (2 ° 53) '0 Equation (2.58) is the ideal gas law and needs no explanation. e. Additional Aerodynamic and Thermodynamic Properties of Air in the Range 32—100°F.--Table 5 shows a tabulation of additional aerodynamic and thermodynamic air properties which affect the evaporation process. Holman (1972) and Perry (1963) were used as the source of information. Linear interpolation was used whenever the exact value at a given temperature was not found. Throughout the present research the units are expressed in the British system. In Appendix A a conver- sion table from British units to SI units is presented. 2.2 Diffusion of Gases Through Membranes Fick's law of diffusion may be applied to describe the movement of water vapor through porous membranes. In terms of molecular flux such an equation is: M = -D A —— (2.59) When equation (2.59) is used to describe the movement of water vapor through a membrane, a parameter 33 000. 040.0 004. 000. 4000. 040. 040. 4040. 0040. 000 400. 000.0 004. 000. 0000. 040. 000. 0040. 0040. 00 000. 000.0 004. 400. 0000. 040. 000. 0040. 0040. 00 000. 000.0 404. 040.! 0000. 040. 000. 0040. 0440. 00 000. 000. 004. 400. 0000. 040. 000. 0040. 0440. 00 000. 000. 404. 040. 0000. 040.. 000. 0440. 0440. 04 000. 400. 004. 000. 0400. 040. 000. 4440. 0440. 04 000. 000. 004. 000. 4400. 040. .044. 4040. .4040. 00 400. 000. 404. 004. 0400. 040. 040. 0040. 4040. 00 000. 000. 404. 004. 4400. 040. 000. 0040. 0040. 00 000. 000. 004. 004. 0400. 040. 000. 0440. 4040. 00 400. 000. 004. 004. 0400. 040. 000. 4040. 0040. 04 000. 000. 004. 004. 0400. 040. 000. 0040. 0040. 04 000. 400. 404. 004. 0000. 040. 000. 0000. 4040. 00 000. 400. 004. 004. 4000. 040. 000. 0000. 0040. 00 he! MW 0.00.00 0. 800 .ww 10W E 000802 000 000802 000 000 040 000 800 000 .0504 upwanom .mDMMHQ possum .mfimmaa .onvsou “mom zuamoom0> muflmsmo hywmoomw> seagumaoz anemone assumne ownwommm owuufimsflm baseman .moooalmm 00:00 0:» 20 musmmmum owumsmmosum um 000 mo mmflgummoumln.m mqmee 34 which characterizes the behavior of the specific - membrane must be included. This parameter (r) is a measure of "resistivity” to the flow of water vapor through the membrane. Equation (2.59) can be transformed in: (2.60) If a constant gradient concentration and iso- thermal diffusion are assumed, equation (2.60) becomes: M = - ——— A -——-(P - P ) (2.61) Equation (2.61) may be used to describe the mechanism of water vapor movement through a porous membrane. 2.3 Skin Nature of Horticultural Products The nature of the skin of a commodity as a barrier to the evaporation process limits the independent use of equation (2.1) and (2.2) or equation (2.61) for describing the phenomenon of moisture loss from horticultural products. The thickness and nature of the protective coating is highly variable. Mushrooms behave as free water sur- faces (Fockens, 1967). Carrots and sugar beets have less protective coating than apples or pears and consequently loose water faster. Tomatoes have a relative impermeable 35 skin and lose almost no water at all. Differences have also been observed in varieties of the same commodity (Lutz, gp_al., 1968). In order to predict the behavior of a specific commodity with respect to moisture losses the nature of its skin must be known. Due to the individuality in skin composition, only the three products covered in the present study will be considered in this section. Information on the characteristics of the skin of specific products with regard to moisture losses is scarce.' Most of the available literature on this tOpic is related to fruits. 2.3.1 Apples There have been several conflicting reports con- cerning the avenue of gas exchange in fruits. In general, three different routes have been proposed (1) pedical opening of floral end (Brooks, 1937); (2) lenticels or stomata (Haberlandt, 1971); and (3) the cuticle (Smith, 1954). Burg and Burg (1965) found that diffusion of gases through the pedicel opening or floral end was important in some commodities. When this pathway was sealed off the rate of gas evoluted declined in tomatoes and green peppers but no measurable response was observed in apples. 36 Clement (1935) studied the morphology and physiology of lenticels in apples. Conclusions of his work are the following: 1. Lenticels may be open or closed depending on the character of the hypodermal cells. The cells may be cutinized or suberized and then rendered closed to the free movement of gases or liquids. Lenticels may also be closed when the stomata, associated with the lenticel, is closed over by means of the epidermal cuticle. Lenticels may be open when the hypodermal cells are unmodified or when the modified cells have been torn apart. 2. The total number of lenticels per apple is characteristic of the apple variety. The number may range from 450 to 800 in the case of Winesap apples and from 1500 to 2500 in the case of Spitzenburger apples. 3. The number of lenticels per apple varies depending on the amount of water available to the plant during the early development of the apples. This reaction is varietal rather than general. The Winesap apple when given more water produced more lenticels per apple than when grown with less water. The Delicious apples when given more water actually produced fewer lenticels even though the apples were larger. 4. Lenticels are closed by processes which favor dehydration of the outer tissues of the apple. While the apples are still unmature, they respond more completely to such treatment than do mature apples. After apples have been in storage for 6-8 weeks, they respond only after prolonged treatments. A 5. Carbon dioxide gas within the apple escapes with equal speed whether the apple has many or few open lenticels. Some studies have shown that about 99.5% of the surface of the apples is impervious to water vapor (Fockens, 1967). The presence of wax in the skin of the apple is probably the cause of this behavior. Smock (1950) observed that if an apple cuticle is separated from its adjacent tissues and is them immersed in warm ether, the ether solution fraction, referred as "wax," amounts to some 50 percent of the total cuticle matter. Horrocks (1964) compared the permeability to water vapor of a disk of apple skin with and without wax. Results showed a very dramatic increase in permeability whenever the wax was removed from the skin by soaking in two successive solu— tions of hot chloroform. Burg and Burg (1965) suggested that regardless of the nature of the pathways for gas exchange in apples (unless it directly or indirectly involves one or more enzymatic steps) it is highly likely that the process will be governed by Fick's law of diffusion: 38 . D M = ——— fA —— (2.62) Fockens (1967) proposed a model to predict moisture losses from agricultural products: M = A{YiR - hd l ' l ——+Y [( )( )1} (P -P ) (2.63) w o T 2 RO T +- r6 Fockens tested his model on beds of apples. The vapor pressure deficit was calculated assuming a vapor pressure at the apple surface equal to the saturated vapor pressure at the environmental dry bulb temperature. Values of Y1 = 0 and Y2 = 1/250 for apples were reported. Fockens also found that the so called"coefficient of diffusional resistance," r6, is an inverse function of the relative humidity of the surrounding air. Wilkinson (1965) noticed a similar increase in permeability to water vapor of the apple skin when the relative humidity of the surrounding air was increased. Lentz and Rooke (1964) observed a non-linear response to vapor pressure deficit changes in eight different variety of apples. When the vapor pressure deficit was decreased, i.e., increasing the relative humidity, the moisture loss per unit of vapor pressure deficit increased. Although the trend was the same for all varieties, the quantitative response was varietal dependent. 39 A considerable amount of data was found in the literature for the moisture loss from apples. Most of the reported experiments were conducted in beds of products. The information is generally presented in terms of mass loss per unit time per unit weight per unit of vapor pressure deficit. Vapor pressure deficit is always calculated at the environmental dry bulb temperature. Table 6 summarized the published moisture loss data for several varieties of apples. 2.3.2 Potatoes Smith (1968) described the skin of the potato as a layer of corky periderm 6 to 10 cells deep acting as a protective area over the surface of the tuber. Small lenticels-like structures occur over the surface of the tuber. These develop in the tissue under the stomata and are initiated in the young tuber when it still has an epidermis. Periderm thickness varies considerably among varieties. Cultural conditions also influence periderm thickness. Burton (1966) stated that "water is lost from the tubers by evaporation, there being no regulation mechanism, and the rate of loss of any particular sample of potatoes being proportional to the water vapor pressure deficit between the tuber and the surrounding air." 4O 00 x 004.00 000.4 000004 400000 00000 ”“00 x 000.40 000000 0 000. 4.40 000000 060 0-00 x 044.0 000.0 .0400 0 00 x 000.4 000.0 00 0 0cm 000000000 0000 0H00 0 000.0 000.0 00 0000 000000 .008400 wuo0 x 040.0 000. 000 00 40000000 0000040 040000 00000 0:00 0 000.4 000 04-40 044. 0.00 00000000 .00000000 0000 .0400 00 x 000.0 000.0 0.00 00000 00-0 00000 0000 0000 100000 000 WH00 x 000.0 000000 0 000.4 0.00 00000000 . .0400 00 00000 000000 0000 00 0000 00 00000000 000 000000 00080 0-00 x 000.00 000000 0 000. 4.40 400000 0003 000 0044 0.00 x 400.0 000000 0 004.0 0.00 .40000000 0>000000 00 000000 00 000000 0000000 000000 on 0000000 00 x 040.0 000.0 00000000000 00 0000000 0-00 x 040.0 000.0 0000 00 .00 40 00000 A0000V 00080 WH00 x 004.0 0400 4 040.0 00 0.4000000 00 .MII lunlmammm AH .. mxumEom moocmuomom MQH 0009 uM\QH mo >00000> 00003 00 m0 .@509 coaumuso om> 0000 00000002 no 0000 .mmammm 800m mmoH 00500008 mo mums so some pmpmHHndelt.o mqmde 41 0-00 x 040.0 000.0 0000 00800008 . 0-00 x 440.0 000.0 «0 0000 ms» vomuum 0-00 x 000.0 044.0 000 000 008\00 000 00 0-00 x 000.0 000.0 400 00 8000 40000000 000 040000 00000 0-00 x 000.0 000 04-40 000. 0.00 00000002 cummu mucuwfl 000 40 000 00 000 000 0004000 0-00 x 044.0 000 0 400.0 0.44 0.44 00 0000 0000800 000000000 0-00 x 000.0 000 00-0 000.4 0.40 .0000 00 0000 0-00 x 000.0 . 000.0 000 400000000000 000000 0-00 x 400.0 040.0 000 000 008400 000 0-00 x 000.0 000.0 00 00 8000 40000000 000 040000 00000 0-00 x 000.0 000 04-40 400. 00 00000002 0-00 x 400.0 044.0 000000 00003 0-00 x 000.00 0400 0-0 000.0 4.40 00000000 0-00 x 400.00 044.0 000000 00003 0-00 x 000.00 0400 0-0 000.0 4.40 008000 0-00 x 000.00 400.0 0.44 .umwu muomwfi 000 04 000 00 000 0004000 000 0.44 0000 0000800 000000000 0-00 x 000.00 00000 0 000.4 0.40 mica x 000.40 mph-N msowowama 000000 00003 0-00 x 000.40 0400 0-0 000.0 4.40 000000 000 .Mmm nun-00mm0 AH 0009 mxumsmm mmoamummmm 00003 AA mo NHM\QH mo 400000> 000.0 m fig .9058 000A .u Hun 00000002 00 0000 .UmaqfluaOUIl.o mamas 42 Ammmav uuuqum mnoa a mpm.m mwm.a nozoamaamm can :maad mica x m~¢.n mxmms ma bow. m.mmumm zoaamw mica x mmm.m m>>.m muoa x ~H¢.HH Nom.a Avmmav magma mica x mmo.va mus mnuvm mmo. mm zowcmm .umwu muomwn mun mv uou mm wom van an .muvmav mica x mnm.¢ man an «Ho.o 0.55 mobs um cam: muamamm xmumwamwm ouoa a mm>.v mun mm omm.¢ m.vm ouoa x mam.m Hom.~ muoa x vuo.oa mve.fl mcflammuu mica x mvH.mH mam. .wcmHmH Avmmav spawn muoa x m¢~.va omm. mm mwonm mnoa x ~m~.m Hom.~ muoa x vbv.m omm.a maoa x mmm.HH mnm.a mloa x moo é.” mmo. mucunuuflmo Avomav magma muoa x mmw.ma mun mnuvm Hon. mm vmm um .mll Inglmammm QH mxnmaom mmocmummmm mad umme uM\nH he mumflum> umum3 AH mo _ om> .msms mmoq coflumuao mudumwoz mo mumm .uoscflucoouu.o mqm¢a 43 Butchbaker (1970) suggested that equation (2.61) expressing Fick's law of diffusion could be used to pre- dict moisture losses in potatoes. The difficulty in measuring the thickness of the potato tuber skin was con- sidered as the limitation for using Fick's law for calculating the moiSture losses. Schippers (1971) reported two different moisture loss relationships obtained for the same variety of potatoes (Katahdin) in two consecutive years: 1968: PL =.676vw + 1.40 (2.64) 1969: PL ==.874Vw + .61 (2.65) Limited data were found in the literature for moisture losses from potatoes. Table 7 summarizes the reported data for several varieties of potatoes. 2.3.3 Sugar Beets Very little useful information about the behavior of the sugar beet skin in regard to moisture losses was found in the literature. Khelemski and Zhadan (1964) found empirical relationships between the velocity of the surrounding air and the mass transfer coefficient for individual beets. The relationships are given in terms of an apparent con- vective mass transfer coefficient for sugar beets, hd anp When such relationships were expressed in terms of British units the following equations were obtained: 44 no as -oa x voa.a 0» m m .muM\mnH u «penua\umum3 OH a one. no mooumuom How mmoa mums mmmnm>m cm mm>wm “woody souudm .mmeuoaum> uommam can cavamumu you NHM\maa Isnuoumuom mo usa0m\umpm3 mo ussom m OH x «mm.m mo mmoa mums mmmumbm cm Umuuomou Aanmav mummmwnum once osa.m tango awe emuumz OH mov.m GOHSU .mmoumuom mloa mmb.m possess manna m on o no cmumamcoo on moahfiflm “use .mmmuouw 0H Hmv.m woman «women Awmxa and mmoa mica mov.am cannons ommamm unmamsc mama wanes o- HON mm wooalom can OH bNm.¢ GmHDU m.om um awn» masoaoa ouoa ~H~.s amazon: aflcamumx un condo mums mmoumuom on .mummu saw an umma mm: as sme.oa mango emu oH mo 30am new as “flames magma wmoa mom.mm oosnmm oqumm tenuous omnmacmx um Imumm IHEI.UO&. DH mxumsmm mmosmummmm w mo .uomumno Hmumz AH mm .mEmB mamsmm mpmflum> mmoq musumwoz mo mumm .mmoumuom scum mmoa musumfioa mo mums so memo pogmflandmlu.h mqmde 45 V .6 ——L .000528 (11,808) = .000294 still air (2.66) (2.67) III. OBJECTIVES The objectives of this study were: 1. Development of a model for predicting moisture losses from horticultural products in storage. Comparison and development of relationships for predicting surface areas of individual Jonathan apples, Manona potatoes and US H20 sugar beets. Study of the effect of the shape of the individual particles on moisture losses from apples, potatoes and sugar beets. Determination of the skin parameters that affect the rate of moisture losses in Jonathan apples, Manona potatoes and US H20 sugar beets. Preparation of prediction graphs of moisture losses from Jonathan apples, Manona potatoes and US H20 sugar beets. 46 IV . THEORY 4.1 Model for Predicting MoiSture Losses from Horticultural Products Weight losses of horticultural products result as the combination of the respiration and evaporation (moisture losses) processes. Weight losses due to respiration are of negligible nature in comparison with losses due to evaporation. For that reason the contribu- tion of respiration to the weight loss process will be neglected in this study. Moisture losses from horticultural products is basically a mass transfer process, and the theory of physical transport phenomena may thus be used to study the problem. The agricultural products studied in this investi- gation Contain between 80 to 95 percent of water by weight. Some of this water is lost during storage by evaporation. The moisture movement within the product is rapid enough to maintain during normal storage conditions a saturated condition just below the skin. Therefore, no concentra- tion gradient exists within such products. This "lumped concentration capacity" assumption is transient in nature, but considering that the maximum allowable moisture loss from the product is relatively small (from 3 to 10 percent 47 48 in a 6 to 9 month period), a constant rate of moisture loss during the storage time can be expected as long as the storage conditions do not change. The steady state condition has been observed by most of the researchers after the first few days of storage time. The problem of moisture losses from horticultural products can be analysed as a process controlled by the rate at which water vapor moves through the skin of the product and is carried away from the surface by convection. In general, the skin of a commodity may be con- sidered as a combination of zones that present a distinctive "resistance" to the movement of water vapor (Figure 6). Certain areas of the surface may behave as free water surfaces; equation (2.1) describes their contri- bution in the moisture transfer process. Other regions of the surface may behave as porous membranes and their contribution to the loss of moisture can be predicted by Fick's equation of diffusion. Finally, zones of the skin may be impervious to water vapor. The overall behavior of a product with regard to moisture losses is given by the relative magnitude of these areas and by the value of the parameters affecting the governing equations. In the following analysis, an electrical analogy is used to model the skin of an agricultural product. Figure 7 shows the equivalent electric circuit that represents the different paths water vapor may follow in 49 .Amhmav .Hm um Hamnouwz.soum pmumop< .mmanmummm> can muwdnh mEom cw canon msflnmboo hxmz amusumz 0 ma maowuzo one .s3onm mum Homm> swung mo mousom cosEoo .pOSUonm amusuasowuuom n no sflxm gnu mo Emummflo oaumfimnomll.m musmfim Homm> nouns nufl3 cmuMHSHMm woman Ham HMHSHHmUHmusfi smasofluso smxoun Hmoeucma Ho upmsoum 50 . R T M1 I Mh o A w le r6 R T R.o T ”2 Mfrs 12:“ “M. 3412.} M 3 - 0 + w pf A’, 1 MW ‘41 II: P P ws w0° Governing Equations: -M h Y A . d l _ Mwl _ ( Rb T ) (Pws wa) ° -( i 1 )'(P -‘P ) w2 : r6 Ro T Ro T ws , ww + M Dwa Y2 A M hd Y2 A w3 0 Mw = wl + Mw2 + Mw3 ~ hAT - 1337.8416 a T = w - O s hA 56983 fig Figure 7.-—Electrical Analogy to Represent the (4.1) (4.2) (4.3) (4.4) (4.5) Behavior of the Skin of a Product with Regard to Moisture Losses. 51 leaving the product. The governing equations are also given. The first path corresponds to that region of the skin that behaves as a free water surface. The governing equation [see equation (2.50) and (2.51)1 is them: = (M Egsyl A) (P - P ) (4.1) R0 T ws w0° Mwl The second path corresponds to the portion of the total area behaving as a porous membrane. Two resistances . in series, the one for the diffusion through the membrane and the one for the convection process, characterize this path. The describing equation for moisture losses through this path is then: - _ 1 sz‘iran'r RT 0 + o , M Dwa Y2 A M hd Y2 A ’°.(Pws - Pwm’ (4.2) Finally, the third path is an Open circuit corresponding to impermeable regions of the_skin: W3 = O (4.3) The overall mass losses from a product can be obtained by adding these partial losses: 52 An analysis of the variables and parameters of the skin model described by equations (4.1), (4.2), (4.3), and (4.4) follows. The convective mass transfer coefficient, hd' is not a constant but is a function of position on the surface. Regrettably, the variation of hd with position is known for only those cases where separation of flow does not occur. An overall value for the convective mass transfer coefficient has to be used. 4 Relationships for calculating the mass transfer coefficient of flow over different geometrical shapes were discussed previously. Agricultural products can only be approximated by the following shapes: apples by spheres, potatoes by prolate spheroids and sugar beets by\ggne§. Due to the variability in shape of individual samples of the same commodity, a study of accuracy of such approximation with regard to the prediction of hd seems appropriate. The effect of shape of the product on moisture losses can be isolated by studying the behavior of peeled samples. 1f the skin of the different products under 'investigatnon is removed, the behavior of the individual samples with regard to moisture losses is governed by the shape of the product under similar environmental condi- tions. This 1s true if the study is performed duriqg an apprOpriate short period of time, where an approximately 53 constant rate of moisture loss is observed, and a steady state assumption is justified. The study of moisture losses from peeled samples will also give comparison values to measure the effective- ness of the product as a barrier for the migration of moisture from the product. Study of moisture loss from peeled samples of those products under investigation is one objective of the present research. The surface area determination is empirical in nature. Equations for prediction of the surface area apply to the variety of the commodity they were developed for. This is particularly true in the case of weight-surface area relationships. More general formulas are needed. However, the goodness of the prediction may be more important than its generality at least for certain type of studies. 0n the other hand, due to the fact that products are stored in beds, weight-surface area relation- ships could be more useful when knowledge of the behavior of individual bodies is going to be applied to practical situations.‘ Weight-surface area relationships for Jonathan \ apples, Manona potatoes, and US H20 sugar beets will be \ developed in the present research. Comparative studies of different formulas will be conducted whenever it seems appropriate. 54 The driving force (voltage in the electrical analogy) needs especial consideration. In general, regardless of the nature of the product, the assumption of a vapor pressure at the surface equal to the saturated vapor pressure at the environmental dry bulb temperature has been used in the literature for calculating the driving force for the mass loss proceSs. Although this approximation could be good enough for products with highly impervious skins, i.e., apples, it is misleading for products that behave as free water surfaces. To determine the driving force affecting the evaporation process, the temperature at the surface, Ts' must be known. Equation (2.1) can be used to obtain an estimate of an average value of the temperature at the surface. If the formula for hfg given by equation (2.56) is substituted into equation (2.1) the following relation- ship is obtained: ~ h AT - 1337.8414 s » W 0 TS = h A - .56983 MW ‘ R) (4‘5) In calculating the driving force for evaporation the vapor pressure at the surface can be assumed to be equal to the saturated vapor pressure at TS. Parameters Y1, Y2, 3 and g characterize in general the behavior of the skin with regard to moisture losses. From these four parameters only three are independent; r and 6 cannot be simultaneously determined by using the 55 described model. The product £§_will be considered as one parameter from now on. The individual behavior of a product may allow simplifications into the model. For some products the fraction of the total area that behaves as a free water surface might be negligible in comparison with the "membrane like" path. In such case Y1 = 0 and the effect of the environmental air velocity will be a minor variable in the moisture loss process. On the other hand, in some products the porous membrane path might be considered negligible in comparison with the free water path. In the latter case the parameter 71 represents an "effective area" of moisture loss. Finally the skin of some products may be considered approximately impervious to moisture migration. Two different problems may be considered with regard to the model: the "inverse" problem of estimating the parameters from mass loss data and the "classical" one of calculating moisture losses when the parameters are known. Both problems will be studied.’ Techniques for predicting the parameters of the models discussed above are outlined in the next section. The statistics associated with the prediction is also discussed. 56 4.2 Estimation of Parameters The models discussed in the previous section fall into two categories: linear or nonlinear with respect to the parameters. Weight-surface area relationship may be expressed by models which are linear, or by models that can be linearized. Dimensionless relationships for calculating the convective mass transfer coefficients are also equa- tions that can be linearized. On the other hand, the model for predicting the moisture loss from horticultural products (equation 4.4) is a nonlinear model with respect to the parameters, Y1: Y2 and £6. In analysing the problem of estimating the parameters of those models described above, the concepts and notation presented by Beck (1973) are used. 4.2.1 Linear Models Three models will be considered in determining weight-surface area relationships for individual products. A linear model, Yi = 80 + lei + 8i (4.6) and the "intrinsically" nonlinear models, ._ 31 Yi — BO Xi + Ei (4.7) and 57 _ Xi Y. — 80(81) + e. (4.8) J. 1. Equation (4.7) is also the basic model for determining the convective mass transfer coefficient, h d' Equations (4.7) and (4.8) can be linearized by taking logarithms: = I Ln Yi Ln(BO) + 81 Ln(xi) + e i (4.9) ._ I Ln Yi — Ln(BO) + Ln 81 X1 + e i (4.10) Equations (4.6), (4.9) and (4.10) may be similarly analyzed with regard to the determination of parameters 80 and 81. If the criteria of minimization of the sum of squares is used, the following set of equations allows the determination of estimates for 80 and 81: n i=1 (xi-x)(Yi-Y) n — 2 z (xi-x) i=1 and h0 = Y - blx (4.12) where, 58 b0 and bl = estimates of parameters 80 and 81, respectively _. 1 n X = I!- 2 X1 (4.13) 1=l _ 1 n Y = E )3 Y1 (4.14) i=1 n = number of data points. The predicted regression value of Yi is denoted by Y.: Yi = be + lei (4.15) The residual ei is the measured value of Yi minus the predicted value or, ei = Yi - Yi (ei # Si) (4.16) Equations for estimating the variance and standard deviations for Yi' b0 and b1 are summarized in Table 8. Parameter confidence intervals can be calculated from: b. - s.d.(b ) t (n-p) < B. < b, + s.d.(b.)t (n-p) l 1 l 1 l 1—d/2 l-d/Z (4.17) 59 TABLE 8.--Formulas for estimating variances and standard deviations of Yi' b0 and b1 in the model Estimated I Estimated Variances standard deviations Var (I) s.d. (I) 1 n ~ 2 Y. ——— Z (Y.-Y.) VVar(Y.) i n-2 ._ 1 1 1 1—1 n 2 Var(Yi) 2 (xi) b0 i=1 /Var(bo) n X (X.-f)2 1 1:1 Var(Yi) b JVar(b ) l n _.2 l 2 (Xi-X) i=1 where n = number of data points p = number of parameters tl_a/2(n-p) = value from a t-distribution-table for (n-p) degrees of freedom and (l-d/Z) ' probability. 60 4.2.2 Nonlinear Models The dependent variable, MW, of the model for predicting moisture losses (equation 4.4) is linear with respect to Y1 and Y2 but is nonlinear with respect to £§° The problem of predicting these parameters is then of a nonlinear nature. Iterative techniques must be used for predicting the parameters. Meeter and Woolfe (1968) developed a computer routine called GAUSHAUS which estimates parameters enter- ing nonlinearly into a mathematical model. In GAUSHAUS the estimates of each iteration are obtained by a method developed by Marquardt (1963) which combines the Gauss (Taylor series) method and the method of steepest descent. The user must provide a main program to read input data from cards or tape and to initialize certain constants. A subroutine that determines values of the model for a choice of parameter values transmitted to it by GAUSHAUS must also be provided by the user. Output from GAUSHAUS is a printed report which includes a description of the problem, a summary of each iteration relating to the precision of the estimates and possible to the adequacy of the mathematical model. The skin parameters of horticultural products can be estimated by applying the GAUSHAUS routine to the moisture loss data. '61 4.2.3 Comparison of Models In general, if two or more linear or nonlinear models apply to the same data, a coefficient, R2, can be used to compare the effectiveness of the models in repro- ducing the experimental data. This coefficient is defined as: 2 213 e12 EA ei where EB ei2 = sum of square residuals for model B EA ei2 = sum of square residuals for model A, X e. > 2 e. (4.19) Because of condition (4.19), an examination of (4.18) leads to 0 i R2 i l, where R2 = 0 corresponds to the models being nearly equally effective, and R2 = 1 corresponds to model B being much better than model A. For simple linear models such as (4.6), (4.9) and (4.10) the following expression for R12 is frequently used: R = _ (4.20) 62 The value of R12 defined by equation (4.20) implies the comparison of Y- ~ ? model A A 1 = Bo + Ei,AYi = = b0 (4.21) . Y._ ~__ w1th model B B 1 — 80 + lei + €i,BYi - bo + blxi (4.22). As before, a value of R12 + 0 corresponds to model A (equation 4.21) being equally effective as model B (equation 4.22). If R12 + 1, model B is much better than model A. With regard to model A (equation 4.21) and model B (equation 4.22), an F-test can be used to obtain a measure of. howmuch the additional term (81) has improved the prediction. If F is small, then the two parameter model does not significantly improve the fit compared to the one parameter model. Finally, the computed F value can be statistically bounded by comparing the value with a tabulated F1_a (l, n-2) value. If the calculated value F exceeds the tabulated value, the probability that the hypothesis HO: B1 = 0 is false is d. If the calculated P value is less than the tabulated one, the null hypothesis is rejected; that is, it may be that 81 = 0. V. EXPERIMENTAL PROCEDURES To achieve the objectives, a group of experiments were conducted during Fall, Winter and Spring seasons of 1972-1973 at the Agricultural Engineering Processing Laboratory of Michigan State University. A description of how the products were handled and the apparatus and procedures used is given below. Before any weight loss test was performed, the products were handled in a somewhat different manner. Mature Jonathan apples from the MSU Horticultural Farm at Grand Rapids were hand picked from four different trees. They were packed in plastic bags, placed on carton boxes, and immediately stored at 36°F. Potatoes of Manona variety were picked at Stanton, Michigan after they were machine harvested. They were placed in mesh bags and stored at 65°F, 90-100% relative humidity, for two weeks. Immediately after the suberiza- tion process they were stored at 36°F. Sugar beets of US H20 variety were hand dug from a MSU field in East Lansing. They were tOpped at the base, placed in plastic bags and stored at 36°F. In general, the moisture loss tests consisted of measuring the weight loss history of individual peeled or 63 64 unpeeled samples placed in a test chamber in which the temperature, relative humidity and airflow of the air were controlled. Specifics about the procedure and apparatus used during each set of experiments follows. 5.1 Comparison and Development of Formulas for Predicting Surface Areas of Individual Apple377 Potatoes and Sugar Beets Experimental surface areas of apples, potatoes, and sugar beets were obtained by peeling each individual sample in narrow strips and calculating the planimeter sum of the tracings. Shape parameters were measured in each sample to allow comparison of the different methods for predicting the surface area of individual particles. The weight of the unpeeled sample was taken for the development of surface area-weight relationships of each product. In the case of apples, before each sample was peeled a longitudinal section was made through the longitudinal axis of symmetry of the fruit. The section was then drawn on paper, and the parameters for predicting surface areas were measured. Parameters a, b, c, and z for the Moustafa model were measured according to the method described in Figure 4. Width and height of each sample were measured for the prediction of surface areas using Cooke's bispherical model. Weights of the unpeeled samples were taken to the nearest .001 gram. 65 Comparison of a weight-surface area relationship with Maurer's prolate spheroid model for potatoes was performed. In order to predict the surface area of each potato sample by Maurer's model, sections were made through the major and minor axis of each sample. These - 1' b2 and d (equation 2.46) were measured. Weights of the unpeeled sections were drawn on paper, and parameters b samples were obtained to the nearest .001 gram. In the case of sugar beets, three weight-surface area relationships were compared with each other. Weights of the unpeeled samples were obtained to the nearest .01 gram. 5.2 Study of the Effect of the Shape of the Body on Moisture Losses from Apples, Potatoes and Sugar Beets A set of experiments was conducted to study the effect of the product shape on moisture losses from peeled samples. Tests consisted of measuring the weight loss from individual peeled samples placed in a test chamber during a one hour period. Temperature, relative humidity and airflow of the environmental air were kept constant during each experiment. Figure 8 shows the experimental set-up for this study. The test chamber consisted of a box of two feet in length and one foot square cross sectional area. The box was insulated with one inch expanded polystyrene. 66 electric hygromet r temperature recorder variable air intake speed humidity AMINCO- fan sensor AIRE unit ‘— insulated chamber 12" i 5 6 24" # Thermocouple No. \lO‘U‘lanNH —w Measurement of dry bulb temp. wet bulb temp. environ. temp. surface temp. surface temp. surface temp. environ. temp. Figure 8.--Schematic Diagram of the Experimental Set-up Used in Moisture Losses Studies from Peeled Samples. 67 Temperatures were measured with 20 gage copper- constantan thermocouples placed at the locations shown in Figure 8. A texas Instrument* potentiometer was used for continuous monitoring of the temperature. Tempera- tures were recorded to the nearest .5°F. Relative humidity was measured with a series of Hygrodynamics** humidity sensing elements. An electric hygrometer was used to monitor the sensor readings. The sensor's accuracy was checked continuously with a dry-wet thermocouple set (thermocouples l and 2). Air velocity was measured with a calibrated hot- wire anemometer. Thermocouples were placed on the surface of the sample at three different locations. In order to condition the environmental air, an Aminco-Aire*** unit was used. This unit is able to control the dry bulb temperature and relative humidity of the air to within 1 .75°F and .5 R.H., respectively. A variable speed fan controlled the airflow in the environmental chamber. The procedure described below was followed during each test: * Texas Instruments Incorporated, Houston, Texas. ** Hygrodynamics, Inc., Silver Springs, Maryland. *** . . American Instrument Company, S11ver Springs, Maryland. 68 a. The sample was removed from storage (36°F) and placed in the chamber for a 18-24 hrs period under test conditions. b. The sample was peeled and the resulting narrow strips of skin were recorded on paper to determine the surface area. c. Holes were drilled through the samples in order to place thermocouples for measuring the temperature at the surface. d. Initial weight of the sample was obtained. The scale was located immediately adjacent to the test chamber‘ to reduce difficulty with handling during the weighing procedure. Weights were obtained to the nearest .001 gram in the case of apples and potatoes and .01 gram in the case of sugar beets. e. The sample was placed in the test chamber and after one hour of exposure to the conditions, a final - weight was obtained. f. Characteristic shape parameters of the sample were recorded. 5.3 Estimation of Skin Parameters of Jonathan Apples, Manona Potatoes and US H20 Sugar Beets A set of tests was conducted to determine the parameters which characterize the behavior of Jonathan apples, Manona potatoes and US H20 sugar beets with regard to moisture losses. 69 The tests consisted of measuring the weight loss history of individual samples placed in a test box during a certain period of time. The duration of each experiment and the frequency of weight measurements varied with the product being tested. In the case of apples and potatoes a 110-120 hrs period was used, with periodical weight measurements every 24 hrs. In the case of sugar beets, a 12-18 hrs period was used, with periodical weight measurements every two hrs. The test chamber consisted of an 8 foot long and one foot square cross sectional area insulated box. The temperature, relative humidity and airflow of the environmental air were measured with the same type of instruments described previously for the weight loss tests of the peeled samples. Aminco-Aire units were also used for conditioning the environmental air. In this set of experiments, however, the test chamber was arranged in a closed circuit with the conditioning Aminco unit. BecauSe variation in the behavior of individual samples was expected, several samples were placed in the test chamber during each individual test. In the case of sugar beets, four samples evenly distributed in the chamber were studied simultaneously. In the case of apples and potatoes, eight samples were studied simultaneously. 70 The following steps were followed during each individual test: a. Samples were removed from storage (36°F) and placed in the chamber for a 18-24 hrs period under test conditions. b. The initial weight of each sample was recorded. Weights were obtained to the nearest .001 gram in the case of apples and potatoes, and to the nearest .01 gram in the case of sugar beets. c. Periodic weights of each individual sample were obtained. d. Samples were peeled and the resultant narrow strips of skin were drawn on paper to determine the surface area. e. Peeled samples were replaced in the chamber and periodic weights of the peeled samples were taken. f. Characteristic shape parameters of the samples were recorded. VI. RESULTS AND DISCUSSION 6.1 Prediction 9f Surface Areas of Single Particles Weight-surface area relationships were developed for each one of the products being researched. A computer routine, ALEASQ, was written to estimate the parameters 80 and 81 of the linear model A = 80 + Ble. ALEASQ also linearizes models A = B'0(Wb)8'l and A a B'O(B'1)Wo and givesan estimation of parameters 8'0 and 8'1. The routine finally gives a statistical analysis of the estimated parameters. A listed of ALEASQ is given in Appendix B. I Comparison of the deve10ped relationships was made with some of the models described in the literature. In the case of Jonathan apples, Baten's weight- surface area relationship, Cooke's two—parameters, and Moustafa's four parameters model were compared with the linear relationship developed from planimeter data. For Manona potatoes, the prolate spheroid Maurer's model was compared with the developed weight-surface area relationship. Sandera's formula for predicting surface area of sugar beets was developed for a different variety than the one used in this study. For this reason no attempt was made to COHDRIC the experimental weight—surface area 71 72 relationship with Sandera's equation. Instead, the weight- planimeter area data were applied to those models described by equations (4.6), (4.7) and (4.8). A comparison of the prediction of surface area by these models was made. Results and discussion on the developed relation- ships and model comparisons for each individual product are presented below. 6.1.1 Jonathan Apples The linear least square analysis on the planimeter data of Jonathan apples gave the following weight-surface ' area relationship for individual apples: A = .04164 + .4359 W6 (6.1) A plot of the planimeter surface area data and of the linear relationship (equation 6.1) is presented in Figure 9. The statistical analysis associated with equation (6.1) is given in Tables 9 and 10. A high regression coefficient, R12 = .98811, and a calculated F value (F 4655.42) much larger than the tabulated F value (F.01 = 7.10) characterizes statistically the prediction of equation (6.1). The 95% intervals of confidence for b0 and b1 are: .03780 < bo < .04548 and .4231 < bl < .4487. Predicted surface areas by equation (6.1) and by the relationships developed by Baten, Cooke and Moustafa are summarized in Appendix C. 73 .24 P .23 — I' .21 .20 .19 I r—U .18 4 1 .17 .16 P .15 - Surface Area, A (ftz) .14 )- .13 h .12.- .ll - .104 wglnlnlnlnlLLlnljlnlLLJL .20 .22 .24 .26 .28 .30 .32 .34 ‘.36 .38 .40 .42 “hymns WE CUnfi anne:9u~WerfiurSurfimxaAmealkflatflmusup:fln:Inihfidhal Jenathan.Apples. Equation (6.1). 74 TABLE 9.--Estimated variances and standard deviations of the dependent variable and parameters of equation (6.1). Estimated Variance of I Estimated Standard 95% interval of confidence for b 95% interval of confidence for b I Deviation of I Var (I) s.d. (I) A .0000038 .00195 bO .0000037 .00192 b1 .0000408 .00639 Coefficient R12 = .98811 .03780 < b < .04548 0 .4231 <-b < .4487 1 TABLE lO.--Tab1e for partition about the mean. Equation (6.1). Source of Sum of D. of F Mean F Variation Squares Squares Re51duals .0002 56 .000003 Fcalculated 4655.42 Deviation Between Line and Mean .0178 1 .0178 F.01 table 7.10 TOTAL. .0180 57 75 Table 11 presents a comparison of the models. The linear relationship developed by Baten gives comparative results to those of equation (6.1) (coefficient R2 = .17). On the other hand, equation (6.1) gives better prediction than Cooke's or Moustafa's models (R2 = .83 and R2 = .99, respectively). It is interesting to notice, however, that the predicted surface areas by Cooke's model are within 1 6% of the experimental ones. This model could be useful for studies where the weighing procedure is difficult or not possible, i.e., in preharvesting studies. TABLE 11.--Comparison of four models for prediction of surface areas of individual Jonathan apples. Model Residuals Rigigiils R2 Linear experimental Equation (6.1) .0000 .0002 .00 Baten .0479 .0003 .17 Cooke -.l696 .0013 .83 Moustafa 1.3364 .0648 .99 Moustafa's model showed poor prediction of surface areas of Jonathan apples. It was observed that the residuals increase as the size of the sample increases, Figure 10. It may indicate that the deviation from the 76 .moud umuosflcmam Housmswnmmxm one momHm> Hope: m.@mmumsoz mo empowooum mmH¢II.oa shaman mm. mm. mm. mm. vN. mm. mm. Hm. 0N. ma. ma... ha. ma. ma. 3”. ma. NH. I .- (I - I )- I - I - I - I q 4+ I .- . q I - I J- I) 1“ ‘ u (I d J - . (A- O C" (n l‘ \0 Ln V m N N H H H H H H H H o o o o o o o o o H 4 N .mm. (243) 981v SOEJIHS lenemruetd 77 assumed elliptical shape is larger for big than for small samples. 6.1.2 Manona Potatoes The least square analysis on the planimeter data of 120 Manona potatoes gave the following weight-surface area relationship for individual potatoes: A = .3018 wo'6638 (6.2) Figure 11 shows a plot of the planimeter surface area and of the relationship expressed by equation (6.2). The statistical analysis on equation (6.2) is given in Tables 12 and 13. A high R12 coefficient (R12 = .97888) was obtained. The calculated F value is much higher than the tabulated one at 99% confidence limit. The 95% intervals and b are: .2967 < b < .3069 and of confidence for b0 1 0 .6461 < bl < .6816. Predicted surface areas by Maurer's model and by equation (6.2) are presented in Appendix D. Maurer's model showed poor prediction of the surface area of Manona potatoes in comparison with equation (6.2). A R2 coefficient of .97 was obtained (see Table 14). 78 . .i~.ev nonsense .mooumuom moose: Hmsofl>wosH How mwcmcofluwamm mend mesmnsmluamwmzui.aa mucmflm ,Asnflv a .unmsmz we. 8. mm. mm. am. am. cm. as. we. 3. me. 3. mm. mm. em. mm. om. i Ti...q....dilu...¢-4J..d.mJ-1.quqd NH. . ..MH S .. m o x. as. .4 1 e O .00. a o i . 0 ma coon . N . i . a coo-0* .3 m. ¢5WI¢. lvhn—Ho V I I . \Tw ...$ U as. .uc o O. O 1 ad. 0 o . o _ now. 0 .3 seem. n 4 . Hm. mmmm. 1 79 TABLE 12.--Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.2). Estimated Variance of I Estimated Standard I Deviation of I S.d¢ (I)I” Ln(A) .01575 Ln(bo) .00858 bl .00898 Coefficient R 95% interval of 95% interval of 2 1 = .97888. confidence for b : 0 confidence for b : l '.2967 < b < .3069. O ‘.6461 noucH oocooncou wmm on you Ho>noucH oocooncoo wmm «Hm pcoHonmooo AHQVQH no HQ mo coHDMH>oQ osmoconm ooumEHumm Hooch no on we coHHMH>oQ onoocoum oouoEHumm asch no a mo COHumH>oo onoocmum ooDoEHumm Aeneas no He no oUCMHHm> ooumEHumm Honch no on we OOCMHHm> Umflmfifiumm Ase an no a mo oUGMHHm> ooumfiHumm Am.mv coHDmsqm 1e.sc consumes Am.ov GOHpmoUM .Am.mv ocm A¢.mv .Am.mv mcoHuooqo mo coHumucomonmoH HoocHH one we muouofimnmm one oHQMHnm> usoocomoo ecu mo mGOHDMH>oo onmocmum ocm moUCMHHm> ooumsHumMIl.mH mamme 83 HHvo.H m.m no HHem.H e.o mmmm. m.o gases onmm.H onmm.H m.m coo: mmNm.H H mmNm.H «.0 one oCHH coosuom name. name. m.m :OHuoH>oQ mo.Hm~.H mHoo. memo. m.o mo.n om.hom hHoo. om MHHH. v.o om.vem mooo. Nmmo. m.o mHooonom Ho m .Heo.m onmcqm . . monocom coHuoHnm> m coo: m me a we saw coHumoom monsom . 3.3 one 3.8 .Am.mv mCOHuocom mo GOHpoucomonon HoosHH one now some on» scone COHDHDHMQ now oHomenl.mH momma 84 Results of the prediction of the surface area by equations (6.3), (6.4) and (6.5) are summarized in Appendix E. Equation (6.5) shows to be more accurate in predicting the surface area of individual sugar beets (see Table 17). However, the comparison in accuracy between equation (6.4) and (6.5) gives a relative small R2 value (R2 = .25). The simple linear relationship expressed by equation (6.3) gives the poorest prediction of the three compared models. TABLE l7.--Comparison of weight-surface area relationships for predicting surface areas of individual US H20 sugar beets. Model Residuals Square Residuals R2 Equation 6.3 .0000 .83858 .96 Equation 6.4 -37.6695 .03635 .25 Equation 6.5 -.0266 .02715 .00 6.2 Study of the Effect of the Single Particle Shape on Moisture Losses The study of the effect of the shape of the individual particle on moisture losses was performed by measuring the mass loss of individual peeled particles at different environmental conditions. The results are expressed in terms of dimensionless numbers. 85 The vapor pressure deficit was calculated as the difference between the saturated pressure at the surface (at the recorded temperature) and the environmental air vapor pressure. Surface areas of the individual particles were obtained by the planimeter technique. The diameter of a sphere with a surface area equal to the planimeter area was used as the character- istic dimension of apples and potatoes. In the case of the individual sugar beet, the slanted height of the beet was used as its characteristic dimension. The ALEASQ routine was used to calculate the coefficients Bo and 81 of the following model: Sh = 80 Re81 (6.6) Comparison of the experimental relationships for peeled apples, potatoes and sugar beets was made with those relationships described in the literature for spheres, prolate spheroids and cones, respectively. Results and discussion on the developed relation- ships and comparisons for each individual product are presented below. 6.2.1 Jonathan Apples Appendix F summarizes the results of experimental data for peeled Jonathan apples in the 2,000-10,000 86 Reynolds number range. Analysis of such data leads to the following relationship: Sh = .539 Re'504 (6.7) Tables 18 and 19 summarize the statistical analysis of equation (6.7). A high regression coeffi- cient (R12 = .91369) and a calculated F value larger than the tabulated P value at a 99% confidence limit are obtained. The 95% confidence intervals for b0 and b1 < 1.376 and .395 < b < .613. are: .2110 < b0 1 TABLE 18.—-Estimated variances and standard deviations of the dependent variable and parameters of the Alinear representation of equation (6.7). Estimated Standard I Estimatedvzirtihce of I Deviation of I s.d. (I) Ln(Sh) .00925 .09620 Ln(bo) .17719 .42094 bl .00240 .04903 Coefficient R12 = .91369 95% interval of confidence for b0: .2110 < to < 1.376. 95% interval of confidence for b : .395 < b < .613. 1 l 87 TABLE 19.-—Tab1e for partition about the mean for the linear representation of equation (6.7). Source of Sum of D of F Mean F Variation Squares ° Squares Re51duals .0925 10 .0093 Fcalculated 105.86 Deviation Between Line and Mean .9797 1 .9797 F;Ol table 10.04 TOTAL 1.0723 11 Equation (6.7) was compared with the Frossling and Powell relationships for flow over spheres [equations (2.17) and (2.19), respectively]. The experimental data fall closer to the Frossling equation than to the Powell equation (see Figure 13). When a residual comparison of equation (6.7) with equations (2.17) and (2.19) was made (Table 20), R2 values of .34 and .75 were obtained, respectively. TABLE 20.--Comparison of equation (2.17) and (2.19) with equation (6.7). Model Residuals Square Residuals R2 Equation 6.7 - 2.1989 142.19498 .00 Equation 2.17 -24.82239 217.11945 .34 Equation 2.19 65.23221 585.71629 .75 88 monmd conuoc0b HosoH>HocH.n0m muonsoz ooosnonm msmuo> moHocmoMIn.mH ouson o Dom oooNH ooooH ooom oooo ooov ooom o u] u -‘. q - d d L. A J - d o a 1 cm 5.2 .8888 o . .om .om m. + o.m n cm m mm I :\ 1 0% 5m AHmucoEHHomxmv vom.om mmm. u cm _ . 0 SEE monotone 0 o H 0 mm. m am pm r.. . l om \\. m l AN\F\ " Q 1 cm 89 6.2.2 Manona Potatoes Appendix G summarizes the results of experimental data for peeled Manona potatoes in the 2000 to 10,000 Reynolds number range. Analysis of such a data led to the following relationship: Sh = .344 Re'539 (6.8) In Tables 21 and 22 the statistical analysis of equation (6.8) is presented. A regression coefficient of .91421, and a calculated F value larger than the tabulated F value at the 99% confidence limit are obtained. The 95%| confidence intervals for b0 and b1 are: .126 < b < .933, 0 and .423 < bl < .655. TABLE 21.-~Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (6.8). Estimated Standard Estimated Varlance of I Deviation of I Var (I) s.d. (I) Ln(Sh) .01084 .10412 Ln(bo) .20058 .44786 bl .00273 .05220 Coefficient R12 = .91421 95% interval of confidence for b : .126 < b < .933. 0 0 95% interval of confidence for bl: .423 < bl < .655. 90 TABLE 22.--Tab1e for partition about the mean for the linear representation of equation (6.8). Source of Sum of Mean Variation Squares D' Of F' Square F Re31duals_ .1084 10 .0108 Fcalculated 106.56 Deviation Between Line and Mean 1.1551 1 1.1551 F.01 table 10.04 TOTAL 1.2635 11 Equation (6.8) was compared with the Frossling equation for flow over spheres (equation 2.17) and with the Lochiel relationship of flow over prolate spheroids [equations (2.28) and (2.17)]. All the experimental data fall below the curve expressed by equation (2.17) (see Figure 14). Improvement of the prediction was obtained when the Lochiel relationship [equation (2.28)] was added to equation (2.17). Table 23 shows a comparison of residuals of equation (6.8) with equations (2.17) and (2.28). Equation (2.28) showed to be equally effective as equation (6.8) in predicting the experimental data (R2 = .100). 91 . . .mmoumuom macaw: HmsoH>HwGH How Hwnfidz voo3umsm momum> moaocmmmll.va musmflm meH oooma cocoa ooom ooom ooov ooom o 1 . _. l _ . _ ._ _ . _ . o 1 ON Aamucmeflummxmv mmm.mm vwm N am 1 l ow .nm Ana.~s coflumswm O I O "u 1 mm. m m + o N gm 1 cm Am~.mv cam A>H.NV mGOAumsqm . Gowuowwmum Hmwnooq 4 <\5. H on Hmucmfiflummxm o 1 cm 92 TABLE 23.--Comparison of equations (2.17) and (2.28) with equation (6.8). Model Residuals . Square Residuals R Equation 6.8 - 1.75610 157.94197 0.00 Equation 2.17 46.49266 336.62772 0.53 Equation 2.28 18.58314 175.17557 0.10 6.2.3 US H20 Sugar Beets Results of experimental data on moisture losses from peeled US H20 sugar beets in the 6,000-35,000 Reynolds number range is presented in Appendix H. The analysis of such data led to the relationship: Sh = .199 Re'634 (6.9) The statistical analysis of equation (6.9) is summarized in Tables 24 and 25. A regression coefficient, R12, of .94017, and a calculated F value larger than the tabulated F value at a 99% confidence limit are obtained. The 95% confidence intervals for b0 and b1 are .0662 < bO < .597, and .521 < b < .746. 1 Figure 15 shows a plot of the experimental data and of equation (6.9). The Miranov relationship for flow over cones [equation (2.27)] is also plotted in Figure 15. When the residuals of equation (6.9) were compared wirh residuals of equation (2.27) a R2 value of .74 was obtained (see Table 26). 93 TABLE 24.--Estimated variances and standard deviations of the dependent variable and parameters of the linear representation of equation (619). Estimated Variance of I Estimated Standard I Deviation of I Var (I) s.d. (I) Ln (Sh) .00861 .09281 Ln (b0) .24362 .49358 b1 .00256 .05057 Coefficient R12 = .94017 95% interval of confidence for b0: .0662 < b0 < .597 95% interval of confidence for b1: .521 <‘b1 < .746. TABLE 25.--Table for partition about the mean for the linear representation of equation (6.9). Source of Sum of Mean Variation. Squares D’ Of F' Square F Re51duals -0861 10 '0086 Fcalculated 157.14 Deviation Between Line and . Mean . 1-3535 1 1'3535 F.01 table 10.04 TOTAL 1.4396 11 94 .mumom Hmmsm owe me cmammm Hmsoa>acsH How Hmnfinz woo3ho£m msmum> mcaochmmlr. ma Gunman nwm ooomn oocom ooomN oooom ooomH cocoa ooom o T..ld......fi................o n ow n ow . om Aauusmadummxmu lemme man. u gm 0 sea gm wmm. n QNH 1 ova ism. N .nwme . omH A Ammo HSH..H am . hm omH \ com 95 TABLE 26.-~Comparison of equation (2.9) with equation (2.27). Model Residuals Square Residuald R2 Equation (6.9) -6.43599 1035.7098 0.00 Equation (2.27) 182.2309 4007.2051 0.74 6.3 Determination of Skin Parameters Moisture losses data of individual Jonathan apples, Manona potatoes and US H20 sugar beets were applied to the model described by equations (4.1), (4.2), (4.3), (4.4) and (4.5) for determining the skin parameters of each one of these products. The GAUSHAUS computer routine was used to determine the parameters in the case of apples and potatoes. In the case of sugar beets a simplification of the model was used. Sugar beet results are expressed in terms of an "effective surface area" for moisture migration. Results and discussion on the skin parameter values for each individual product are presented below. 6.3.1 Jonathan Apples Moisture losses data on 72 Jonathan apples were used for determining the skin parameters of this commodity. Tests were performed at an environmental temperature of 70°F. Three different airflows (3000, 6000 and 9000 Et/hr), and three different relative humidities (50, 62.5 and 75%) 96 were tested. Eight samples were used at each storage condition. Tabulation of the experimental data is pre- sented in Appendix I. A computer program was written for applying the experimental data and the model described by equations (4.1), (4.2), (4.3), (4.4) and (4.5) to the GAUSHAUS routine. The mass transfer coefficient, hd' was calculated by using equation (6.7). Equation (6.1) was used for calculating the surface area of the individual samples. In order to calculate the vapor pressure deficit equations (4.5) and (2.56) were used. ‘ The results gave a comparatively small value for the free water surface area fraction, Y1 (of a 10.6-10-?'l order of magnitude). Because of such a small effect of the parameter Y1 on the mass losses process and because a decreasing tendency of the Y1 parameters was observed during the iterative procedure, the model was simplified by making Y1 = 0. It was observed that a decrease in the sum of squares resulted when the simplified model was used. Table 27 shows the results of the skin parameter values when the experimental data and the simplified model (Y1 = C) were fed to the GAUSHAUS routine. Average values of Y1 = 0, Y2 = .01286, and £9 = .01943 feet, were obtained. 97 TABLE 27.--Skin parameters of Jonathan apples at 70°F. fists? Y1 Y2 .. (Dafinaly (Dimensionless) (Dimensionless) (ft) .500 0 .01150 .02282 .625 0 .01385 .01920 .750 0 .01322 .01628 AVERAGES ' 0 .01286 .01943 The results show negligible free water regions in the apple skin. As a result the air velocity of the I environmental air has little effect on moisture losses from this commodity. The results also show that approximately 98.7% of the Jonathan apple skin is impervious to water vapor. Besides, it was found that the parameter £2 does depend on the relative humidity of the environmental air. Figure 16 shoWs the experimentally determined relationship between £6 and the relative humidity of the environmental air. It can be seen that rg has a higher value when the relative humidity is low and vice versa. So the diffusional resistance against moisture loss is higher when the relative humidity is low. This phenomenon has also been observed by Wilkinson (1965) and Fockens (1967). Fockens (1967) suggests that the cells of the skin change in shape at different environmental relative humidities. 98 .0230 - .0210 - .0180 ' r6 ' (ft) .0170- .0150 _ .0130 P QLALLLIL.IAJ;I.IAI.I .500 .626 .750 Relative Humidity (Decimal) Figure 16.--Environmental Relative Humidity Versus the {g Parameter for Jonathan Apples at 70°F. 99 The cells become flatter at lower relative humidities reducing the amount of intercellular spaces through which the water vapor moves from the product to the environ- mental air. This reduction in intercellular spaces originates an increase in the resistance to the movement of the water vapor through the "membrane like" skin. On the other hand, high environmental relative humidities result in round cells which originate a larger amount of intercellular spaces and a decrease in resistance to the movement of water vapor through the porous membrane skin. A study of the effect of vapor pressure deficit on the Eg parameter was performed. A 6-26 lb/ft2 VPD range was investigated by placing 56 individual samples in air atmospheres at different combinations of temperatures and relative humidities. Temperatures ranged from 50 to 80°F and relative humidities from 50 to 75%. The air velocity was kept at 6000 ft/hr. Eight samples were tested at each storage condition- In Appendix J the experimental data is presented. Equations (4.1), (4.2), (4.3), (4.4) and (4.5) were used for calculating the £2 parameter for each sample, at each VPD condition. Values of Y1 = 0 and Y2 = .0186 were used as the other two skin parameters. When VPD was plotted against ré, a decrease in r6 was observed whenever the VPD was increased and 100 vice versa. The following linear relationship was obtained when the VPD-r6 data was applied to the ALEASQ routine: r6 = .00770 + .00064(VPD) (6.10) Tables 28 and 29 summarize the statistics analysis of equation (6.10). A coefficient R12 = .70047 was obtained. The 95% intervals of confidence for the 1 are: .00584 < bo < .00956 and .00052 < b1 < .00076. The calculated P value is larger than the tabulated F parameters b0 and b 01 value. Figure 17 shows the experimental data and the relationship expressed by equation (6.10). Each data point in the graph represents an average value of eight samples. TABLE 28.--Estimated variances and standard deviations of the dependent variable and parameters of equation (6.10). Estimated Variance of I Estimated Standard I Deviation of I var (I) s.d. (I) r6 .000007 . .00262 b0 .000001 .00093 bl .000000 .00006 Coefficient R12 = .70047 95% interval of confidence for b0: .00584 < b0 < .00956. 95% interval of confidence for bl: .00052 < b1 < .00076 101 mm rmwammd casemGOb now mflnmcoflumamm mnlam>uu.ha madman om um N Tlumflv om> ma 0H Ammamfimm usmflm mo mmmnm>m may we £0fl33 mDHMPVou m mucmmmnmmu usflom Somme .1 J - a vwooo..+onnoo. o 03. 33 @H omo. omo. 102 TABLE 29.--Table for partition about the mean. Equation (6.10). Source of Sum of Mean . Variation Squares D' Of F' Square F Re31duals .0004 54 .0000 Fcalculated 126.28 Deviation Between Line and Mean .0009 l .0009 F.01 table 7.12 TOTAL .0013 55 The prediction of moisture losses was compared with the experimental data. An apparent mass transfer coeffi- cient for apples, h' , was used as the comparison d app criterion. Table 30 summarizes the results of the compari- son. It was observed that the predicted h' values d app were within i 10% of the experimental eight sample average values. The quantitative effect of the environmental air velocity on moisture losses can be observed in Table 30. Doubling the airflow resulted in an increase of the mass transfer coefficient for Jonathan apples of only 10 percent. This increase in the mass transfer coefficient is small compared with the effect of air velocity on moisture losses from free water surfaces. For free water 103 .mmamfimm unmam mo mmwnw>m map ma enam> somm* mm.> mica x mmm. mica x awn. cccm mm.mm com. on mm.m mica x new. mica x mmm. cccm mm.mm com. on mv.m mica x ch. mica x mum. cccm mommm com. on mc.mi mica x amm. mica x new. cccm mv.ma mmm. ch cw.cai mica x mmc.a mica x mam. cocm vv.ma mmm. ch mc.cai mica x maa.a mica x mcc.a cccm mv.ma mmm. cc ca.bi mica x mmc.a mica x mmm. cccm vc.ma cmh. cm mm.m mica x cwc.a mica x cmc.a cccm mc.ma mmm. cm mm.ai mica x mam. mica x mum. cccm mm.ma cmh. cc cm.ai mica x mNa.a mica x aaa.a cccm mm.ma cmc. cc ea.mi mica x mam.a mica x mma.a cccm mm.ma cmb. cc ma.h mica x mva.a mica x hNN.a cccm ma.m cmh. cm mc.MI mica x mnm.a mica x vmm.a cccm cm.m cmh. cm w WMMiNuMiun WMMiNuMinn u: mum Hmauqmumuuaa and and MW and amaMan M. amucmEaHmmxm cwpoapmnm 2> om> mam v.5 new 6.: «.mmammm umnumuoc EOHM mmmmoa musumaoe cmpoapmum mswum> amucmaaumexm mo cowaHMQEOUii.om mamme 104 surfaces the mass transfer coefficient is approximately proportional to the square root of the air velocity. 6.3.2 Manona Potatoes A behavior somewhat similar to the Jonathan apples was observed in Manona potatoes. An identical set of experiments was performed with 72 individual Manona potatoes for determining the skin parameters. Tabulation of the experimental data is presented in Appendix K. As in the case of apples, the model was simplified because a comparatively small value for Y1 was obtained. Table 31 shows the results of the skin parameter values when the experimental data and the simplified model (Y1 = 0) were applied to the GAUSHAUS routine. The convective mass transfer coefficient, h was calculated d' by using equation (6.8). The surface area of the samples was obtained experimentally by the planimeter technique. Equations (4.5) and (2.56) were used for calculating the vapor pressure deficit. Average values of Y1 = 0. Y2 = .00890 and £§_= .01143 feet were obtained. The results show negligible free water regions in the skin of Mibflnd potatoes. As a results, the air velocity of the environmental air has little effect on moisture losses from this commodity. The results also 105 TABLE 31.--Skin parameters of Manona potatoes at 70°F. Relative Y1 Y2 r6 Humidity (Dimensionless) (Dimensionless) (ft) Grated) .50 0 .008935 .01515 .625 0 .008867 .01086 .75 0 .008908 .00827 AVERAGES 0 .008900 .01143 show that approximately 99.1% of the Manona potato is impervious to water vapor. It was found that, as in the apples case, the value £6 for Manona potatoes ia a.function of the relative humidity of the environmental air. Figure 18 shows the experimental relationship between 3g and the relative humidity of the environmental air. Decreasing of the environmental relative humidity resulted in an increase of the resistance of the skin to water vapor migration. The effect of vapor pressure deficit on the £6 parameter was also studied for Manona potatoes. Fifty- six individual samples were placed at different storage conditions, covering a 6-26 VPD range. Temperatures ranged from 50 to 80°F and relative humidities from 50 to 75°F. The air velocity was kept at 6000 ft/hr. Eight Parameter r6 (ft) 106 T = 709F .0160 _ i- .0130 ' .0100 ” i- h .0070 b b {W ILIAJALA—lalnlmlnl .500 .625 .750. Relative Humidity (Decimal) Figure 18.--Relative Humidity of the Environmental Air Versus the £§ Parameter for Manona Potatoes. 107 samples were tested at each storage condition. In Appendix J the experimental data is presented. Equations (4.1), (4.2), (4.3), (4.4) and (4.5) were used for calculating the £6_parameter for each sample at each VPD condition. Values of Y1 = 0 and Y2 = .00890 were used as the other two skin parameters. When VPD was plotted against £6, a decrease in £6 was observed whenever the VPD was increased and vice versa. The following linear relationship was obtained when the VPD-r6 data was analysed with the ALEASQ routine. r6 = .00162 + .00052 (VPD) (6.11) Tables 32 and 33 summarize the statistic of equation (6.11). A coefficient R12 =-80595 was obtained. The 95% intervals of confidence for the parameters b0 and b are: .00050 < b < .00274 and .00046 < b < l 0 1 .00058. The calculated F value is larger than the tabulated F.01 value. Figure 19 shows the experimental data and the relationship expressed by equation (6.11). Each data point in the graph represents an average value of eight samples. The prediction of moisture losses from Manona potatoes was compared with the experimental data by using the apparent mass transfer coefficient as the 108 TABLE 32.-~Estimated variances and standard deviations of the dependent variable and parameters of equation (6.11). Estimated Standard Estimated Variance of I Deviation of I var (I) s.d. (I) r6 .000002 .00158 b0 .000000 .00056 b1 .000000 .00003 Coefficient R12 = .80595 95% interval of confidence for b0: .00050 < b0 < .00274. 95% interval of confidence for bl: .00046 < b1 < .00058 TABLE 33.-—Table for partition about the mean. Equation (6.11). Source of Sum of Mean Variation Squares D' 0f F‘ Square F Residuals .0001 54 .0000 Fcalculated 224.27 Deviation Between Line and Mean .0006 l .0006 F.01 table 7.12 TOTAL .0007 55 - -.—-—. .... -—-..—_—._.-— -- .oum—e— 109 .mmoumuom mcocmz How manmcoaumamn cniom>ii.ma musmam mum. Ammav cm> mm cm i ma ea m e 1a! . _ . _ . .a i _ . e l moo. l oao. O am>c «mooea.+ meaee. u we r o i. mae. emeee. II N >. 110 comparison criterion. Table 34 summarizes the results of the comparison. It was observed that the predicted h'd app values were within 6 9% of the experimental eight sample average values. The quantitative effect of the environmental air velocity can be observed in Table 34. Doubling the airflow resulted in an increase of about 15 percent in the apparent mass transfer coefficient for Manona potatoes. 6.3.3 US H20 Sugar Beets Data on moisture losses from 68 individual samples. of US H20 sugar beets were used for determining the skin of this commodity. A simplification of the model described by equations (4.1), (4.2), (4.30), (4.4) and (4.5) was per- formed. From preliminary research results and some evidences from the literature it was found that the skin of a sugar beet can be represented by free water and impervious regions only. As a result the parameter Y1 represents an "effective area" of moisture migration. In analysing the data the effective area was defined as: Effective _ Mass transfer coefficient of the sample unpeeled Area - Mass transfer coefficient of the sample peeled .mmamfimm unmam mo mcmnm>m ecu ma usaoe seem an 111 e.a mica x mmmee. mica x memee. eoem mm.mm com. on e.ai mien x emmee. mica x cemee. oeee ma.mm com. oe N.ei mica x emeae. mica x emeae. eeea ee.em com. on m.m mioa x Vbhmh. mioa x NmBMB. ooom Nv.ma mNo. on m.mi mioa x thmm. mioa x vmamm. ooom m¢.ma mNm. on a.vi mioa x momma. mioa x cameo. ooom hv.ma mNm. on m.m mica x mvmcm. mioa x «moam. ooom ho.ma omh. om m.hi mioa x ommmo.a mioa x hamao.a ooom mh.ma mNm. om N.Ni mioa x homvm. mioa x mmmNm. ooom vm.Na omb. on m.OI mloa x mmnwo.a mioa x wmmmo.a ooom mm.Na omh. on h.oi mica x aomma.a mica x ommma.a ooom mm.Na own. on N.h mioa x amamo.a mioa x mhmma.a ooom ha.m omh. om m.mi mioa x mmvmm.a mioa x mammN.a ooom hm.m omh. om wmmi avian wmmi uMiun um w wna N mna N H: N HeauemanMHe aha sea mm .wma Hmemmmo M. amucmaaummxm cmanomnm s> om> mam ©.£ mam ©.£ [1 All, lillliili IT '11" I. '11! 11!" i. I’luiill Z i...’ ' ,Iliiilili Il’iilll.’ Illliiililli. A 111.1 Iiili‘iilil «.mmoumuoe mcocmz Eonm mmmmOa mnsumaoe Umuoapmnm m5mum> amucmaanwmxe mo cowaHMQEOUii.vm mamee 112 In order to determine the effective area parameter, Yl' a set of weight loss experiments were conducted. The 1.8 - 9.6 1bf/ft2 VPD range was studied by placing 68 samples:h1air atmospheres at different combinations of temperatures and relative humidities. Temperatures ranged from 50 to 80°F and relative humidities from 50.0 to 87.5%. Airflows of 6000 and 9000 ft/hr were used. The experimental data is presented in Appendix M. Figure 20 shows a histogram prepared with the calculated effective area values. A Chi square test was | used for checking the hypothesis of a Gaussian behavior of the effective area parameter. The test proved that the normal hypothesis can be accepted. An average value of Y1 = .436 was obtained for the effective area parameter. This value is bounded by a .4181 - .4539 interval at a 95% probability confidence. A comparison of predicted versus experimental moisture losses was made. Table 35 shows that predicted values are within 1 15% of the experimental four sample average values. 6.4 Prediction of Moisture Losses The experimentally determined parameters were applied to the developed model for obtaining prediction graphs of moisture losses at various storage conditions. 20 15 >1 0 C‘. (D :3 3 10 H [u 5 o . / i w 113 Y1 = .4360 .4181 < Y1 < .4539 normal curve A A A A A .275 .325 .375 .425 .475 .525 .575 Class Interval Figure 20.-—Frequency histogram and normal curve for the effective area parameter, 71, in US H20 sugar beets. 114 .mmamamm snow mo monum>m map ma peace comm i h.mi mommaoo. mamcaoo. ooom mum. om N.aa vmamaoo. omhaaoo. ooom omh. om N.Na mmmmaoo. hNNwaoo. ooom omh. om N.Oi whvmaoo. mammaoo. ooom mhm. on e.H eemaaee. emeaaee. eeom. emu. em a.va mbmmaoo. mamvaoo. ooom omc. om o.ma ONmNaoo. Noomaoo. ooom omh. on N.a mwNONoo. aaOONoo. ooom 0mm. on m.mi mvaaoo. manmaoo. ooom omh. om m.Ni hmmmaoo. oomnaoo. ooom omh. om a.ol mONoaoo. vaNoaoo. ooom mNm. on o.m mthaoo. NNvNaoo. ooom mNm. on a.mi oomoaoo. mmmaaoo. ooom com. on N.wa mawmaoo. aommaoo. ooom com. on m mm“: henna mmmimueine u amaacmnmmmao EMa . Ena MW amEmmmo Mo amucmEanmmxm Umuoaownm 8> mew o and p .5 .3 ['0 l i. . .i'liil‘ 'l 11;. iii Illlllliill Ii.‘ , ii". liliii «.mummb umcsm cmm mo some mmmmOa musumaofi mo muadmmn amucmEaanxm msmnm> omuoaomne mo cowaummEOOii.mm mamde 115 Two different types of graphs are presented. The first type can be used for determining the rate of moisture loss per unit area (or per unit weight) at different storage conditions. The second type can be used for predicting the allowable storage time at different storage conditions and at various percentages of moisture losses. A discussion follows on the development and significance of these prediction graphs for each product studied in this investigation. 6.4.1 Jonathan Apples A graph for predicting moisture losses from Jonathan apples at various storage conditions is presented in Figure 21. In Figure 21 the vapor pressure deficit is plotted against the rate of moisture loss per unit area (or per unit weight). A second abscissa scale represents the equivalent relative humidity of the environmental air at 35°F. A third abscissa scale represents the equivalent environmental temperature at 90% relative humidity. The assumption of a surface saturated condition at a tempera— ture equal to the dry bulb temperature was used in drawing the relative humidity and temperature scales. A .33 pound (150 grams) apple was used for developing the graph. The nonlinear relationship between vapor pressure deficit and the rate of moisture losses can be observed 116 x10 x10- 15 p Jonathan Apples (Based on a 150 grm apple) 13 ' 12 ' \ I H [.4 I as I ) lbm hr-lb apple lbm hr-ft2 H c: I l 1 Rate of Moisture Loss/Weight ( Rate of Moisture Loss/Area ( O_L_ OalnlLlnlnLlljlnlnl o 1 2 3 4 5 6 7 8 9 VPD (lbf/ft2) L _l A l J l A l A l 100 90 80 7O 60 50 40 Storage Relative Humidity at 35°F (%) .- b b IL Inlnlml l l n i 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 21.--Predicted moisture losses from Jonathan apples. 30 15000 9000 6000 4500 3000 117 in Figure 21. The increase in skin resistance to water movement at higher vapor pressure deficits is responsible for such a behavior. The comparative effect of the different variables affecting the moisture loss process in Jonathan apples can be studied in Figure 21. Relative humidity is the most critical variable affecting moisture losses of Jonathan apples. A small decrease of the environmental relative humidity results in an important increase in the rate of moisture losses. For example, a reduction of 6% (from 93 to 87%) in the environmental relative humidity results in a doubling of the rate of moisture losses at 35°F. A similar doubling of the rate of moisture losses only occurs after an environmental temperature increase of about 20°F, i.e., from 35°F to 55°F, at 90%. On the other hand, doubling of moisture losses at 35°F and 90% only occurs after a ten times larger airflow (from 1500 to 15000 ft/hr) is used. Allowable storage time predictions for Jonathan apples at various percentages of moisture losses are presented in Figures 22 and 23. In Figure 22 an environmental air velocity of 1500 ft/hr (25 ft/min) was ‘used. In Figure 23 a 3000 ft/hr air velocity was used. Relative humidity and temperature scales were also drawn 118 500 400 " 300 5 Jonathan Apples . VI= 1500 ft/hr 200 L- b 0% 100 :- c: 90 7 " 80 :' a 'N) - '9 60 - g D m 50 " s *1 4U)“ 40 i- m D H {3, 3o - 5 H '2 20 L 10 E- 9 t' 8 L- 7 - 1% moisture 6 L losses 5 b r i I L A I A l j l 1 l L I O 2 4 6 2 8 10 12 VPD (lbf/ft ) 1 a L - l L l . I . 1 . L - l . g 100 90 80 70 60 50 40 30 20 Storage Relative Humidity at 35°F (%) % i.i . l . i‘ i i . i . 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 22.n-Predicted allowable storage time for Jonathan apples. V = 1500 ft/hr. 119 500 400 300 - Jonathan Apples i v = 3000 ft/hr 200 L 100 90 so 7 7o - 60 - so . 4O ' 3O - 20 h Allowable Storage Time (Days) 1% moisture losses mmaicnuoo ONrfi'er'v'v' b D D P . i . i . .l . .4 4 6 :2 8 10 12 VPD (lbf/ft ) A l a l A l A L A l A l A I L L 100 90 80 70 6O 50 40 30 20 Storage Relative Humidity at 35°F (%) l l L l L 1 L | . L I l I j 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 23.--Predicted allowable storage time for Jonathan apples. V = 3000 ft/hr. 120 in both figures. A .33 pound apple was used in calculating the predictions. The effect of the different variables on the storage time can be studied in Figure 22 and 23. It can be observed for example, that a storage relative humidity between 90 to 95% has to be maintained for keeping Jonathan apples for a six month period with a 3% maximum of moisture loss, using a 1500 ft/hr air velocity. If twice this velocity is used (3000 ft/hr), the storage time, under the same conditions, will be reduced in only 20 days. Changes in the storage relative humidity will dramatically affect the storage time. A 3% of moisture loss, for example, will occur in five months at a 90% relative humidity, 35°F of temperature, and 1500 ft/hr. If a 5% decrease in relative humidity takes place (from 90 to 85) the same amount of loss will occur in only 3 1/2 months at the same storage temperature and air velocity. On the other hand, a change of 5°F (from 35 to 40°F) during the storage of Jonathan apples (at 90% relative humidity and 1500 ft/hr of air velocity) will result in a reduction of only 25 days in the storage time. 121 6.4.2 Manona Potatoes Figures 24, 25 and 26 present the graphs for pre- dicting moisture losses and allowable storage times of Manona potatoes at various storage conditions. In Figure 24 the vapor pressure deficit is plotted against the rate of moisture loss per unit area (or per unit weight). A second abscissa scale represents the equivalent relative humidity of the environmental air at 40°F. A third abscissa scale represents the equivalent environmental temperature at 90% relative humidity. The assumption of a saturated surface at a temperature equal to the dry bulb temperature was used in drawing the relative humidity and temperature scales. A .397 of a pound (180 grams) potato was used for developing the graph. A similar analysis to the one for Jonathan apples can be performed on Figure 24. The nonlinear relationship between the vapor pressure deficit and the rate of moisture losses from Manona potatoes can be observed in this figure. Besides, as in the case of Jonathan apples, relative humidity is the most critical variable affecting the moisture loss process from this commodity. On the other hand, the environmental air velocity has compara- tively little effect on moisture losses. f potato’ lbm Rate of MOisture Loss/Weight (hr-lb 0 x10-5 5 _ .Z. 4'. 3L 2 L. 1'. .i ) lbm hr-ft Rate of Moisture Loss/Area ( 122 x10 157' Manona Potatoes 14* (Based on a 180 grm potato) 13" 12" H I-’ r H O I 8 Va A l A l A j A l A l L l A VPD (lbf/ftz) L A l l I # l A J A l A 1 100 90 80 7O 60 50 40 Storage Relative Humidity at 40°F (%) L lllA 1 L4 A l k L A 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 24.--Predicted moisture losses from Manona potatoes. J 0 1 2 3 4 5 6 7 8 9 10 11 15000 12000 9000 7500 6000 4500 3000 1500 ft hr 123 Allowable storage time predictions for Manona potatoes at various percentages of moisture loss are pre- sented in Figures 25 and 26. An air velocity of 1500 ft/hr was used in Figure 25. In Figure 26 a 3000 ft/hr air velocity was used. Relative humidity and temperature scales were drawn in both figures. A .394 pound potato was used in developing the graphs. The effect of the different variables on the storage time of Manona potatoes can be studied in Figures 25 and 26. It can be observed that, if an environmental relative humidity between 90 to 95% is maintained, Manona potatoes can be kept for eight months with a maximum of 3% in moisture losses, using a 1500 ft/hr air velocity. If twice this velocity is used (3000 ft/hr) the storage time, under the same conditions, will be reduced in 40 days. Changes in relative humidity will affect the storage time considerably. A 3% of moisture losses for example, will occur in seven months at a 90% relative humidity, 40°F of temperature and 1500 ft/hr. If a 5% decrease in relative humidity takes place (from 90 to 85%) the same amount of losses will occur in only 4 1/2 months. On the other hand, changes of 5°F (from 40 to 45°F) during the storage of Manona potatoes (at 90% relative humidity and 1500 ft/hr of air velocity) will result in a reduction of 45 days in the storage time. 124 Manona Potatoes 400 V = 1500 ft/hr 300 250 Fri'l 200 100 .. 90 :p 80 :- 70 - 6O .- 50 .. 4o — 30 - 20 — Allowable Storage Time (Days) 1% moisture H m 0" \lmLOO l . losses 0 2 4 6 2 8 10 12 VPD (lbf/ft ) I I A l A I A 0 A I l I 90 80 7O 60 50 40 Storage Relative Humidity at 40°F (%) I lAlAlA I A I l i A 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 25.—-Predicted allowable storage time for Manona potatoes. V = 1500 ft/hr. 125 500 q 400 - Manona Potatoes V = 3000 ft/hr 300 F 200 i- 100 90 80 70 60 I 'I'U'I'I'I 50 r V 40 V 30 20 .. Allowable Storage Time (Days) 10 1% moisture losses U'IOWQG)\D O <3 FJ\¢r'r'l 'l'l'l I p b l . l . l i l i __J 2 4 6 2 8 10 12 VPD (lbf/ft ) l n l l L A L L I 4 J A l 106 90 80 7o 60 50 40 Storage Relative Humidity a1 40°F (%) r, 11,.1 .. L 1 _L_ . l .1 l ,_1_.___... 35 45 55 65 75 85 Storage Temperature at 90% RH (°F) Figure 26.--Predicted allowable storage time for Manona potatoes. V a 3000 ft/hr. 126 6.4.3 US H20 Sugar Beets In Figures 27, 28 and 29 the graphs for predicting moisture losses and allowable storage times of US H20 sugar beets at various storage conditions are presented. In Figure 27 the vapor pressure deficit is plotted against the rate of moisture loss per unit area (or per unit weight). A second abscissa scale represents the equivalent relative humidity of the environmental air at 35°F. A third abscissa scale represents the equivalent environmental temperature at 90% relative humidity. The assumption of a saturated surface at a temperature equal to the wet bulb temperature was used in drawing the relative humidity and temperature scales. A 2.645 pounds (1200 grams) beet was used for develOping the graph. The effect of the different variables affecting the moisture loss process in US H20 sugar beets can be studied in Figure 27. Relative humidity is the most critical variable affecting moisture losses from US H20 sugar beets. A reduction of 5% (from 90 to 85%) in the environmental relative humidity at 35°F is equivalent, in terms of the increase in the rate of moisture losses, to a temperature increase of 20°F (from 35°F to 55°F), at 90% relative humidity. ) lbm hr-lb sugar beet Rate of Moisture Loss/Weight ( .127 x10"5 x10"s 12.000 500 ' US H20 Sugar Beets 15,000 (Based on a 1200 grm ’ 6,000 90 - ' 4,500 80 "' 300 - 7O - F 3,000. 60 - 50 i. 200 " 40 I- i- _5 #3100}. Pil 90 7 20 L}! 801' m 70f- g «50.. R 50- m L 10 ..£3 40 ' m 9 - g i 8 b-g 30- 7 h“§ 6 in- mi 0 20L. 5 i- 0 4.! 4 b'g I 3 i:- 10)- 9b 2 r- 8:- 7i- p 5'; 0 1 2 3 2 4 5 6 VPD (lbf/ft ) L A | A J A I A l A l A 1 A l 100 90 80 ‘70 60 50 40 30 Storage Relative Humidity at 35°F (%) i.i.i‘.i 35 55 75 95 Storage Temperature at 90% RH (%) 1!- Figure 27.--Predicted moisture losses from US H20 sugar beets. Allowable Storage Time (Days) 500 400 300 200 100 90 80 70 60 50 40 30 20 U1 0 -4 m mic .. 128 i US H20 Sugar Beets V = 1500 ft/hr E moisture : losses - 7% - 6% — 1% '5% r A l A l L .l O l 2 3 2 4 5 6 VPD (lbf/ft ) I L A l A I l l J I I l L l 100 90 80 70 6O 50 40 30 Storage Relative Humidity at 35°F (%) L I 1 l 1 l 1 .I 35 55 75 95 Storage Temperature at 90% RH (°F) Figure 28.—-Predicted allowable storage time for US H20 sugar beets. V = 1500 ft/hr. 129 400 - 300 200 US H20 Sugar Beets V = 3000 ft/hr 100 90 80 70 60 50 40 30 20 Allowable Storage Time (Days) 10% moisture losses Uld‘QCDkDO l'l'lfl'l'I O _l 2 3 2 4 5' 6 VPD (lbf/ft ) IA A I A I A I A I A I A I A I 100 90 80 7O 60 50 4O 30 Storage Relative Humidity at 35°F (%) i. i .__L . i 35 55 75 95 Storage Temperature Humidity at 90 RH (°F) u- Figure 29.--Predicted allowable storage time for US H20 sugar beets. V = 3000 ft/hr. 130 On the other hand, the air velocity has a more important effect on moisture losses from US H20 sugar beets than its effect on moisture losses from Jonathan apples or Manona potatoes. When the air velocity is doubled (from 1500 to 3000 ft/hr for example) at 35°F and 90% relative humidity the rate of moisture losses is increased in 45%. Allowable storage time predictions for US H20 sugar beets at various percentages of moisture losses are presented in Figures 28 and 29. In Figure 28 an environmental air velocity of 1500 ft/hr was used. A 3000 ft/hr air velocity was used in Figure 29. Relative humidity and temperature scales were also drawn in both figures. A 2.645 pound beet was used for develOping the graphs. The effect of the different variables on the storage time of US H20 sugar beets can be studied in Figures 28 and 29. It can be observed that, unless a very high relative humidity (from 95 to 100%) will be kept in the storage of this product, important moisture losses will occur during the storage time. For example, losses as high as 10% will occur in a two month period when the product is stored at 35°F, 90% relative humidity and a 1500 ft/hr air velocity is used. If the air velocity is doubled (300 ft/hr) at the given storage conditions (35°F and 90% relative humidity) the 10% moisture losses 131 will occur in only 40 days. Therefore in the storage of sugar beets it is very important to maintain a very high relative humidity and keep the airflow to a minimum. VII. SUMMARY AND CONCLUSIONS A semi-theoretical mathematical model for pre- dicting moisture losses from horticultural products in storage was developed. The model is based on the distinctive behavior of different regions of the skin of a product with regard to moisture losses. Free water, porous membrane, and impervious regions were identified as the components of the skin of a horticultural product. An electric analogy was used to present the model. Equations (4.1), (4.2), (4.3), (4.4) and (4.5) describe the model mathematically. For using the model, it is necessary to determine the surface area of the commodity, the convective heat and mass transfer coefficients, and the vapor pressure deficit of the environment-product system. Besides, the values of three so called "skin parameters" have to be known. These parameters represent the fraction of the surface behaving as a free water surface, the fraction of the surface behaving as a porous membrane and the resistance to water vapor movement through membrane like regions (parameters Y1, Y2 and r0, respectively). 132 133 The model was used for studying the behavior of Jonathan apples, Manona potatoes and US H20 sugar beets with regard to moisture losses. Weight—surface area relationships were developed for predicting surface areas in single Jonathan apples, Manona potatoes and US H20 sugar beets. Good correla- tions were found in all cases. The developed weight-surface area relationships for apples and potatoes were compared with some models from the literature. In the case of Jonathan apples, Baten's weight- surface area relationship, Cooke's two parameter model, and Moustafa's four parameter models were compared with the linear relationship developed from planimeter data. Baten's weight-surface area relationship gave comparative results to those of the developed relationship. Although the developed relationship gave better prediction of the surface area than Cooke's model, pre- dictions by Cooke's model were within 6 6% of the experimental ones. Cooke's model can be useful for studies where the weighting procedure is difficult or not possible, i.e., in preharvesting studies. Moustafa's model showed poor prediction of surface areas of Jonathan apples. It was observed that the devia- tion from the assumed elliptical shape is larger for big 0 than for small samples. 134 For Manona potatoes, Maurer's prolate spheroid model was compared with the developed weight-surface area relationship. Maurer's model showed poor prediction of the surface area of Manona potatoes in comparison with the developed weight-surface area relationship. Dimensionless relationships were developed for calculating the convective mass transfer coefficients of individual peeled Jonathan apples, Manona potatoes and US H20 sugar beets. Good agreement was found when these relationships were compared with equations described in the literature for flow over spheres, prolate spheroids and cones, respectively. Moisture loss data of individual Jonathan apples, Manona potatoes and US H20 sugar beets were applied to the model described by equations (4.1), (4.2), (4.3), (4.4) and (4.5) for determining the skin parameters of each one of these products. A nonlinear routine (GAUSHAUS) was used for pre- dicting the skin parameters of Jonathan apples and Manona potatoes. In the case of US H20 sugar beets a simplifica- tion of the model was used. Sugar beet results were expressed in terms of an "effective surface area" for moisture migration. The results show negligible free water regions in the Jonathan apple skin. It was also found that approxi- mately 98.7% of the Jonathan apple skin is impervious to 135 water vapor. Average values of Y1 = 0, Y2 = .1286 and r6 = .01943 ft, at 70°F, were obtained for the skin parameters of Jonathan apples. It was found that the parameter r6 is a function of the vapor pressure deficit. Equation (6.10) describes the r6 - VPD relationship for Jonathan apples. Due to the fact that free water regions are of negligible nature in the Jonathan apple skin, the velocity of the environmental air has comparatively little effect on moisture losses from this commodity.) It was observed that doubling the airflow resulted in only a,10% increase in moisture loss. This in comparison with free water surfaces where the mass transfer coefficient is approxi- mately proportional to the square root of the air velocity. A somewhat similar behavior to the Jonathan apples behavior was observed in Manona potatoes. As in the case of apples, negligible free water regions in the skin of the Manona potato were observed. It was found that approximately 99.1% of the Manona potato is impervious to water Vapor.- Average values of Y1 = 0, 72 = .00890 and r6 = .01143 feet were obtained for the skin parameters of this commodity. As in the case of apples, a linear relationship between the VPD and the parameter réhwas obtained for Manona potatoes. Equation (6.11) describes this relationship. 136 The environmental air velocity has little effect on moisture losses from Manona potatoes. Doubling the airflow resulted in a 15% increase in the loss of moisture from this commodity. The behavior of sugar beets with regard to moisture losses can be explained by assuming that the skin of the sugar beet is a combination of free water and impervious regions. It was found that about 43.6% (Y1 = .436) of the surface area of US H20 sugar beets if of a free water nature. Finally, graphs for predicting moisture losses and storage times were developed for Jonathan apples, Manona potatoes and US H20 sugar beets. The effect of each variable on the moisture loss process can be analysed by using the graphs. It is expected that the prediction graphs will be useful in designing storage facilities for these products. VIII. SUGGESTIONS FOR FUTURE RESEARCH Further studies should be made in the following areas: 1. Of practical importance is the study of the effect of low ranges of vapor pressure deficits (l to 5 lbf/ftz) on the £6 parameter. 2. Treatments for decreasing moisture losses should be evaluated. This may include studies of the effect of covering the skin with different types and amount of waxes. 3. Models for predicting moisture losses during the cooling process should be developed. Results from the present research may be useful for that purpose. 4. The developed model may be particularly useful in studying the suberization process. The effect of temperature on the skin resistance to moisture losses may be effectively studied through the analysis of the effect of the temperature on the skin parameters. 137 APPENDICES 138 mun; 139. APPENDIX A Convertion.Tab1e from.British Units to 81 Units Quantity British 81 multiply British Unit Unit by Obtain 81 Length foot metre 3.048x10'1 Area square square foot metre 9.290x10'2 volume cubic cubic foot metre 2.832x10'2 Time hour second 3.600x103 Velocity foot/ metre/ hour second 8.467x10'5 Mass pound kilo- mass gramme 4.536x10'1 Force pound Newton force 4.448 Energy BTU Joule (heat) 1.055x103 Pressure pound Newton/ force/ metre £c2 4.788x10 Temperature °F °C °C=5/9(°F-32) 140 APPENDIX B 141 APPENDIX B ALEASQ Computer Routine SUBROUTINE ALEASQCH,ND,X,Y,BD,81) DIMENSION XCIZD),Y‘iZD)9YH‘120),RES‘iZO),QSQ(120) GO TO (500,600,700)H 700 00 705 I319ND YCI)=ALOG(Y(I)) 705 CONTINUE GO TO 500 600 00 605 I=1gND ' XCI)=ALOG(X(I)) YCI)=ALOG(Y(I’) 605 CONTINUE 500 AN=ND 5X30.0 5'3000 DO 505 I=19ND SX=SX+X(I) SY=SY+Y(I) 505 CONTINUE XH=SXIQN YM=SYI0N SUHXSQ=000 SUHYSQ=000 SUHXY=0o0 DO 510 J=10ND SUHXSQ=SUHXSQ+(X(J)-XH)"2 SUMYSQ=SUMYSQ*(Y(J)-Y1)‘72 SUHXY=SUHXY+(X(J)-XH)*(Y(J)-YW) 31=SUHXYISUHXSQ 80=YM-31‘XM 510 CONTINUE 55530.0 SSR=0.0 SST=0.0 35x30.” REX=0.0 00 520 K=19ND YH(K)=BO+81‘X(K) RESCK)=Y(K)-YH(K) RSQCK)=RES(K)”2 SSE=SSE+RSQ(K) SSR=SSR+((YH(K)-YH)“2) SST=SST+((Y(K)-YM)“2) SSX=SSX+(X(K)“Z) 520 CONTINUE VYI=SSE/(AN-Z.) V80=VYI’SSX/(AN‘SUHXSQ) V31=VYTISUHXSQ SDYI=SQRT¢VYI) SOBO=SOQT(V80) SDBi=SORT(V91) RR=(81“Z)‘SUHXSOISUHYSQ F=SSQIVYT GO TO(5229523,710)H 710 80=EXP(801 81=EXP(31) 142 APPENDIX B ALEASQ Computer Routine 00 715 L=19ND Y(L)=EXP(Y(L)) YH(L)=EXP(YH(L)’ RESCL)=Y(L)-YH(L) RSQCL)=QES(L)”2 715 CONTINUE HRITE‘61,720)80,91 720 F0RHATK‘1‘915X,*HODEL YH(J)=80( 81 TO X(J))’,//,23X,’80= 9,F10.5. 1/,23X,'81= ‘gF10o5g/l) GO TO 528 523 BO=EXP(30) DO 524 L319ND XCL)=EXP(X(L)) YCL)=EXP(Y(L)) YHCL)=EXP(YH(L)) RESCL)=Y(L)-YH(L) RSQCL)=RES(L)“2 520 CONTINUE NR1TE‘619526180991 . 526 F0RHAT(‘1‘,15X,‘NOUEL YH(J)=B0(X(J) TO Bl)’,//,23X,‘80= ',510.5, 1/,23X,‘Bi= T’F1005,//) GO TO 528 522 HPITE‘619525780,81 525 FORHAT(‘1’,15X,‘MO0EL YH(J)=30+81(X(J))‘gl/923X,‘30= ’9F10o5, 1,123X,531= ‘eF10.5,//) 528 HRITE(61,530) 530 FORMATNSX,“l (J) ‘,5X,‘X(J)’y7X9‘Y(J)‘,5X,‘YH(J)*9QX, 1‘RES(J)‘.3X9*SQ RESCJ)‘,/) 00 540 J=19ND HRITE(619550)J,X(J)9Y(J),YH(J’,RES(J)9RSQ(J) 550 FORMAT(5X9I3g6X,F11.5,QF10.5) 540 CONTINUE HRITEC619570)VYI,VBOyVBi,SDYI,SOBD,SCBIgRR 570 FORHAT(5X,//,5X,‘VYI= ‘,F10.5,/,5X,’V80= ‘,F10.5,/,5X, i’VBi= ‘9F10.5,/,5X,’SDYI= .9F10059/p5x’T80803 ‘,F10.5,I,5X, 2T8081= T,F10059/ySX,TRR= ‘9F10059/I) NFQ=ND-2 NFLN=1 AVSR=SSEINFR AVSLH=SSRINFLN NFT=ND-1 HRITE‘519585) 585 FORHAT(15X,‘TABLE FOR PARTITION ABOUT THE MEAN‘9/ly5X, 1‘SOURCE’,9X,‘SUM 0F SQUARES’94X9‘0oOFoF.’,6X, Z‘HEAN SQ‘,7X,‘F‘,//) HRITE(61,590)SSE,NFR,AVSR,SSR9NFLHgAVSLN,F,SST,NFT 590 FORMAT(5X,'RESIDUALS‘,8X,F10o4,7X,IZ,9X,F10.h,/g 15X9’LINE AND HEAN‘,QX,F10o“,7X,12’9X,F10.“9510ghglg 25X,’TOTAL‘,12X,F10.4,7X,12) RETURN END 143 APPENDIX C 144 14s msnou. o.~ No. mm. cu.~ humma. omona. o.n mm. on. BH.~ nsmoa. n¢.~ nw.u mn¢u~. «mama. ¢~H~H. mnuom. m Hmong. N.a hm. om. NH.~ NCHHN. mmmeu. 0.0 nm. mm. wN.H wmohH. u¢.~ aw.~ amend. HHNNH. mwoua. mN¢mN. n HoHcN. m.m mm. mm. nu.~ qmuom. memofl. m.m Hm. Hm. m~.~ xm¢wofi. m¢.~ um.~ comma. omth. mmmha. wmoon. o owmma. 0.0 an. as. mo.~ caaqfi. mmw¢H. o.o ow. om. m~.~ ~0m¢fl. mu.~ mo.~ mnemfi. wfimqa. oqoma. moo¢~. m HooHN. m.m mm. ow. mm.“ mmwwa. unnoH. o.m Hm. mm. wH.H omwma. o¢.N mn.~ omaoa. momma. waooa. m¢~¢~. q osmou. ~.~ on. ma. ¢~.H . Acoofi. wmmNH. w.H on. us. oo.~ omqma. wm.~ «N.a mmmofl. BNNoH. onoofi. «nonw. m mummm. o.w mm. mm. oN.H I mecca. «csmfi. o.m «m. om.. o~.H oomua. wq.~ Nm.~ ommna. ONHNH. ommnfi. mmnmm. N monom. o.c mm. aw. mN.H wmoow. «mama. m.o mm. mm. ou.H «mama. s¢.~ mm.~ mumofi. NamoH. m¢aoa.. mmamu. H Amuwv . AmumV «can Amumv A umv m u< Loam A.mmnv Aawv ACHV Aawv A umV Acfiv Acwv mmu< mmu< AuumV AEDHV mwmum>< mmu< n u n m mwud m 3 Hmvoz .mmummm mwu< 03 02 #0602 ammumaoz Hmvoz mxooo :mumm ummawq umumawcmfim unwam3_ oHQEmm .mmana< cucumaoh Hmsvw>wcaH mo mmmu< mumwusm wcHuowvmum you mamvoz Mach mo wufismmm m>HumummEoo U anzmmm< «AAAN. o.o mm. .mw. ou.~ mmnuu. «coma. o.m on. on. om.A Ao¢AA. om.~ om.~ sauna. om¢mfi. oAaAH. ANANM. 0A monA. m.A An. as. eA.A omooA. ofi¢mfi. o.~ on. «a. “H.H ooumA. mn.~ «A.~ m¢mm~. ~¢Amfi. Hummfi. Agnew. nA momofi. o.¢ an. «A. ¢~.A oAqu. mmqu. o.n on. «a. m~.A mouofi. mq.~ Hm.~ momofi. mfiAoH. nomad. AmAwu. «A «om¢H. o.~ Hm. mA. cg.” . moaefi. oNAAH. m.m on. ma. H~.A H¢mqfi. mm.~ Ao.~ Aoomfi. .A¢o¢H. quqH. mmAqm. MA ocfimfl. o.q co. om. ¢A.H mHaHN. mmo¢~. o.H ¢m. ~¢. A~.A ommAH. A¢.N ¢¢.~ oonH. AoAmA. ANAAA. AAon. NA mAAmN. ~.m mm. mm. mm.A ,o ¢0Am~. anNN. m.~ «m. «a. A~.H mmNmH. No.N Ao.m mooofi. ouoofi. NAqu. oA¢mm. HAHN moo¢A. o.m Am. mA. ao.A NoomA. AmmHN. o.H Am. Ho.A -.H mooofl. ~m.~ N¢.N oAmoH. mumofi. oqwoA. «momm. oH “meow. o.¢ Am. Aa. on.H mamAN. «omcw. o.o cm. ea. mm.H omAmH. oA.~ oA.n mAAON. o¢oo~. NHmON. ommAm. w AmumV ANUHV mafia A “NV A umV mmu< :uwm A.wmnv Ach Acwv AGHV A uwv Aawv Acwv mmu< mmu< A umV AEnHV mmwum>¢ me< u u n m mmu¢ m 3 Hmvoz mmuwmm mmu< 03 oz H0602 ammumdgz amvoz axooo cmumm ummcwa umumEHcmHm uswfimz maaamm .mmama< cwnumcoh Hmsvw>wvaH mo mmoud mommusw wcauoavmum now mHmwoz usom mo muasmmm m>wumummaoo U anzmmm¢ Gamma. m.n an. an. oo.~ unmofi. «maafl. w.m o¢. mm. NN.H Hanna. mm.« mn.N chwa. nanoa. apnea. Noowu. mu omeN. «.0 ¢m. Hm. aN.H «anew. Hausa. c.~ mm. no. an.H modms. mo.~ mo.n onmma. «mama. mm¢ma. mau¢m. Nu Aquu. o.o an. «a. m~.~ mmHNN. «Nmmfi. o.m mm. mm. 0N.H wowNH. om.N cm.~ ammwfi. mm¢wa. Henna. onwum. Hm named. o.m mm. mm. MH.H mwoNH. «Hamfi. w.~ «m. um. o~.~ Henna. mm.~ ¢w.~ ¢nmo~. “wwofi. mnwoa. madam. ON mmn¢~. o.~ Hm. mm. NH.H a: mcHHN. nmmnu. w.m mm. mm. em.H mamofi. Hm.N Nm.~ wH¢NH. ¢mmna. omoufi. mmmom. m~.4 1 «anoa. ~.¢ mm. om. 0H.H hqmma. ammmH. o.m o¢. mm. mH.H mmmqa. mm.~ oo.~ owmma. onHmH. Hmm¢H. nqmmm. ma nnu¢a. o.H Hm. om. mH.H moomfi. mNNmH. m.m n¢. ¢m. «H.H mHHmH. 0N.N ¢n.~ nmqma. mumma. m¢~ma. comma. NH AuumV ANumV ”can AmumV ANumV umu¢ scum A.mmav Aafiv Acwv Anfiv AmumV Aswv Acwv mmu< mmu< AuumV AEAHV wwwum>< amu< N o A a wmu< m 3 Hmuox mmuwmm wmu< 03 oz Hove: wwmumsoz Have: mxoou :mumm Hammad umumawcmHm uswfim3 mHgEmm b .mm~&a< cucumcoh Hmsvw>wvcH mo mmmu< mummuam wcwuowvmum you mHmvoE “:03 mo muasmmm w>wumummsoo o Nanmmm< N¢wma. m.m #m. cm. «H.H canmfi. mmmnH. m.¢ m¢. mm. o~.~ eedma. mm.~ mh.~ mm¢mH. manna. manna. «nnmu. om ¢mmm~. o.H ¢m. «n. 00.3 mmwma. noo¢~. 0.0 mm. mu. HA.~ unwefi. Hm." wo.u oom¢H. om¢¢a. oun¢~. «mmmu. mu oemuu. ¢.H mm. mm. HN.H ; . oNHom. mfimna. ¢.¢ an. em. mH.H nmana. N¢.N ~m.~ waena. .mmmua. «Anna. nmwom. mu w¢mhu. H.@ mm. mm. w~.~ mqmmm. oomom. m.~ mm. mm. mm.H oomHN. wo.~ m~.m szNN. #ouwm. HmNNN. O¢mfiq. nu ummmu. o.m on. Co. Hm.H «Hemm. oaNmN. o.m mm. mm. mm.H quofi. Nn.N HH.m oo¢o~. «mqou. HHHON. «mmnm. eNAU 4. SEN. o3 Nm. 2. 8; 1 nooofi. mummfi. o.w om. mm. 0H.H nmomfi. o¢.~ mw.~ qmqnfl. mmmnfi. mumNH. oomom. mm oHHON. 0.0 mm. «m. 0N.H memou. mowou. m.o Hm. Ho. o~.H Homnfi. o¢.N do.~ mowsa. Hmnna. ¢mnna. NNHHM. ¢N AmumV ANumv mug» _ ANHMV ANumV wou< comm A.wunv Aafiv Aawv Acflv AmumV Acfiv Asfiv mmu< mmu< ANuMV AEAHV mwmum>< mmu< N o n m mmu< m 3 Home: mmuwom mmu< 03 m2 Hmvoszmmumzoz Hmvoz oxoou cmumm ummcwq umumesmHm unwwm3 mHaEmm .mmann< conum:OH Hmsww>fi©CH mo mmwu< momwusm wcfiuowvmum you mflmuoz usom mo muasmmm m>wumumaeoo u anzmmm< aaonu. o.n do. am. on." . AAnnu. anom~. A.~ em. n¢. an.“ QNAaA. no.~ mo.n «ANON. naHoN. ¢-o~. m¢Aon. An «ummA. m.n on. A». m_.~ . Aqoofi. ohmmfi. o.o an. «A. AA.A ANAmfi. ~n.~ AA.~ .Aw¢efi. named. oonoA. mnomu. em «NoAH. m.m co. “A. oA.A . mo~mfi. nanofi. m.~ Hm. N¢. AN.“ AnoeH. ~e.~ om.~ A¢mofi. Aqeofi.. «wmoA. AAAmN. mm ~o¢NH. o.~ mm. mm. oH.H Newoa. mmNoH. N.¢ A on. um. mawfi om¢mfi. mm.~ c~.~ onmH. «afima. nn¢ma. «ommm. en oésHN. n.o Ne. mm. o~.H “and". mNoHN. o.N «m. . mm. «N.a Hm¢wfl. s¢.~ «o.m o¢¢wfi. mnnwfi. onmma. oHowm. mm a, 4. HN¢mH. o.n mm. mm. nH.H 1; momma. manafi. m.~ «m. mm. MN.H enamfi. wm.~ ow.N momma. manoa. mnwoa. manna. um ¢mmw~.. o.m mm. mm. MN.H nam¢~. mmaoa. m.¢ on. Hm. «N.a 00¢oa. mm.N nw.~ wmnna. «coma. wm¢sH. o~¢om. Hm AN”: Auuuv «can AmumV ANUMV amu< scum A.mmnv Aggy Acfiv Aafiv ANuWV Aggy Aggy umu< mmu< ANUMV Aggfiv wwmum>< wmnfi u o n m mmu< m 3 Ammo: mmnwmm mmp<, 03 m2 gave: ammumsoz Hmwoz minno amumm pmwcwq HmumEHcmHm uanmB mfiaémm .mwfinn< cosuQcOH Haswfi>fivcH mo mmmu< mowwunm wcwuofikum you mHmvoz uaom mo muasmmm w>wumumano U Nanmmm< V madam. o.n me. an. -.H ocmmA. AAoeA. ¢.n me. A». «A.A ancAA. Aw." am.~ o~oAA. «naoA. «nHAA. nAuou. ¢¢ aaomA. e.o am. An. no.” mamAA. «Hmofi. c.m mm. «A. «a. oon¢~. NH.” ~A.~ e~m¢fi. oaaqfi. oomqfi. ooonm. mq ”coma. o.~ co. m». an.A . coca“. oAQNN. A.e um. am. An.“ AAAmH. oo.~ mm.~ ooAmA. qqomfi. mflmmfi. ofiunm. Ne mnn¢~. o.n me. an. n~.A Aoonm. Am¢m~. e.~ mm. mm. Am.A ANNAA. mo.~ oo.m ooqu. muqaa. m¢¢oA. «comm. Ha aouafi. A.m on. ma. A~.H NAAAH. «AmmA. o.¢ Am. mm. -.A oerH. A¢.N mw.~ mouAA. QNAAA. AHAAA. mmAAN. oq n. 5 Amamfi. o.e A¢. on. «H.A .. oNNNH. mm¢ofi. o.A n¢. MA. oo.H «Ammfi. n~.~ «m.~ ANAmA. ommmfi. moqma. «AoHN. mm mocHN. o.n on. ma. «Nufi «awed. oooufi. o.¢ m¢. AA. nH.A manna. H¢.~ «A.A mHooH. «Hana. maamfi. mmmou. mm .ANuwv Auumv «can ANumV ANHAV aou< comm A.wmnv Aafiv Aaflv AcHV Awumv Aafiv Aafiv mwu< mmu< A umV Aenflv mmmhm>< umu< N 0.. £ w me< m 3 Hmvoz mmuwmm mmu< 03 02 Hmvpz mmwumsoz H0602 axooo amumm ummcwa umumeficmfim ucwfim3. wHaEmm .mmHnn< caanOh HwavH>HvGH mo mmmu< mommusm wcHuoawoum uom.mamvoz usom mo muasmmm m>wumuwano o Nanmmm< «aauu. o.n an. on. ao.~ amend. mc¢¢u. ~.N an. on. o~.~ MHHmH. wN.N < umu< u u a. a mou< m 3 Hmvoz mmuwmm mmu< 03 m2 Hmvoz.mmmumsoz vaoz mxooc amumm ummcwg umumfiacmfim unmwm3_ mHaEmm .mmfian< casumCOh HmsvH>chH mo ammu< muwmusm mcfiuuavmum mom mamvoz usom mo mufismmm m>wumummsou o anzumm< ccaom. o.e an. ca. «N.a nncou. Aooaa. N.a an. an. ea.“ «Anna. nn.~ «m.~ meAH. moms”. NQMAA. «Owen. mm magma. 0.0 on. om. mN.~ MNNHN. Amman. o.n an. Aa. <~.~ «Humfi. ¢m.~ am.u momma. manA. Houwfi. omuum. Am «Amen. o.e Am. An. «N.a . w . mqmau. AHHHN. N.¢ mm. O¢. MN.H «Auofi. em.N ma.~ ouams. oaawa. wanH. mmwam. on @aan. w.u Nm. um. ON.H mmowH. muooa. o.~ on. Na. HN.H momma. w¢.~ Nm.N mecca. nnnofi. onoH. ooqwm. mm wma¢N. o.m mm. mm. .mN.H mmhum. oo¢HN. ¢.¢ mm. mm. mN.H mmnwfi. oo.~ Ho.m oonH. anoma. Nwmwa. onwam. «mnu 1. MMNHN. N.a co. Aw. «N.a . anNN. o¢e-. o.m mm. 0¢. m~.H momma. ~m.~ mo.m Amme. moqwfi. mHAwH. memm. mm monofi. H.@ mm. ow. nH.H «wmmA. No¢m~. A.N on. um. mH.H ¢mmm~. mm.~ 0A.~ «mme. manna. mwoma. Accom. Nm Ami Amumv .uuflm. ANumV Auuwv mmu< .nomo. A.mmnv Acwv Ac“ Acwv A umV Aswv Aafiv mmu< mmu< A umV AEQHV uwmum>¢ wou< N u n m mmu< m 3 Hmvoz mmnwmm mmu< 03 m2 vaoz ammumaoz vaoz axooo :mumm Hammad umumEHcmHm uswwm3 mHmEmm .mmanm< casumcoh Hmsvfi>wvaH mo mmou< mowmusm wafiuowvmum now mHmcoz much «0 mufismmm m>wumumaeou o anzmmm< APPENDIXD A 153 m¢ao~. «N.a as.~ wa.N ”anon. QNNON. smash. NH mace“. no.~ «N.a ua.~ mwcwu. omNcH. ausnc. AH ~N¢na. mo.N no.N nc.N naoau. NNNaa. nwmm¢. ea H¢¢wa. mH.N ao.n wo.n Hamsm. anHN. Nocfio. m sanofi. HN.N mn.N om.N ooaafi. ¢N¢aa. owaom. w mms¢~. mN.N mm.~ mm.~ nn¢o~. wHooH. wwuo¢. A omoHH. OA.H MN.N M@.N ANM¢H. mwméa. wmmmm. o «weefi. mm.~ mo.~ mo.~ wm¢mu. mecca. moqwm. m «Anna. sm.H m~.~ mn.~ AOQAH. momma. ooom¢. ¢ mo¢wa. 0H.N ¢m.~ nH.m mamma. «mama. ¢mmn¢. m Nooha. o~.~ ¢N.N mH.m nHomH. oomaa. nnw¢¢. N mwowfi. mH.~ HH.m mo.m mNHmH. mmmma. monom. A Asfiv Acflv Aafiv AAumv A “Av up An A Au.ov.cum A “My AEQHV mmu< me< manna mwx< mawua mflx< wcoA mmu< amuc 03 m2 . Hove: Human: scammmuwmm “mumaficmfim ucwflm3 maasmm .mmoumuom «scams. Hmsvw>HvCH mo mmmu< mommusm wcfiuowvmum How mwonumz 039 mo muasmmm m>Humumasoo Q xHszmm< 154 moooA. No.“ A».A nm.~ AooAA. mumofi. AmeAe. «A «mnefi. AA.A am.~ ~a.~ AnAoA. omemA. AAAwn. mu «QAAH. Ho.~ oo.~ mm.~ ANAAA. ANAeA. q¢nnn. NA ofinmfi. oA.~ «e.~ Aw.~ AAHAA. oAmoH. oAmN¢. AA oow¢fi. oo.~ mo.~ we.~ mmnoA. oqfiofi. awoam. ou «AAAH. o~.N A¢.N nA.~ nooAA. aAmoA. «Ammc. AA monH. Ao.~ A¢.~ An.“ mmmmA. Amaufi. «exam. ”A ~mm¢fi. mm.A AA.~ NA.~ mA¢eA. HonH. wAAoq. AA «AAmA. oA.~ nm.u ~m.~ oNAAH. mmmAH. o¢wq¢. 0A eNQAA. om.N oe.~ mH.m #«mou. NAAAH. mmoom. mH ofiwqfi. mo.~ mm.~ ow.~ AHmAH. moqAA. mmo¢¢. «A o¢o¢fl. mo.~ on.“ om.~ Aqaofi. NNAAH. «mmfiq. «A AaAV Aafiv Aafiv A uwv A~umv A. H. a AN..W.cvm Amumv Aenfiv mmu< mwx< manna mwx< mamuH mwx< wcoA mmu< mmu< 03 m2 Hmvoz amusmz cowmmmnwwm uwumaaamam unwwm3 mHaEmm .mmOumuom macaw: Hmscw>vaH mo mmmu< mommusm wcwuowkum you muosumz.o39 mo muasmmm m>wumummEoo Q xH92mmm< 155 ............. «Amos. mc.N cw.N «o.w «Hooa. onNoH. «Oman. mm om¢nfl. o~.~ mo.~ NA.~ w¢omd. oAmmA. mmAAn. mm NAAud. mo.~ «m.~ «A.A AAAmH. «mama. Afimmm. ¢n AAAoH. Ho.~ . AA.~ mfl.n ANoAA. ooAAA. oA-¢. mm “moufl. om.H‘ me.~ nm.~ . nmm¢g.: mo~¢fi. . AAANA. an mm~¢fi. mA.H «o.~ AA.N .mofimg. oom¢fi. Hounn. Am HAAmH. Ao.~ «A.~ om.~ omoAA. . NAAoH. wAmmq. on mmAqfi. «c.~ ~m.~ om.~ ofimmfi. cammfi. AeAAm. AN m¢m¢fi. ow.H oo.~ om.~ oANmA. «onA. Henna. mm «omofl. mo.~ #o.u NA.m A¢aofi. wamofi. «mmfiq. AA ANAAH. wA.A oo.~ wA.~ ofiw¢~. aoomfi. m¢~¢m. ow Aaomfi. AA.H mm.~ mo.~ Afloqfi. HANQH. oqnmm. mm Aafiv Acfiv Acfiv Auumv Auumv up An A A~.ov.acm Amumv Aanfiv mmu< me< manna mwx< mcmua mwx< wcoA mwufi mwu< 03 m2 Hmvox,umunmz scammmuwmm nonmawcmfim uswfim3 «Haemm .mmoumuom mcoamz Hmsve>wvcH mo ammu¢ mummusm wcwuuwvmum Mom avocumz 039 mo mufismmm m>wumumqsoo Q xHQZNmm< 156 no.u magma. «N.a ua.u Hénom. un¢o~. an-¢. aw «mung. wo.~ n¢.~ oc.u “amen. Hmmea. nmnen. s¢ wanna. HH.N n¢.N so.u camnd. on¢n~. «oHHn. o¢ on¢-. N~.N H5.N w~.m swans. Houma. s¢0m¢. n¢ ceaua. mN.N oA.N HA.m Nomha.. «Howa. N¢nm¢. «é «song. om.~ mm.N N~.n «Hema. muonu. Hmnwm. mq manna. 0A.N NN.N mn.~ HHAoH. h¢noa. mmwmm. N¢ «mama. mH.N N©.~ wA.N némcfi. Afiqoa. Nm¢o¢. H¢ soosfi. om.N wu.m Hm.N «Nova. omooa. «mmmm. ow mocha. mH.N Ho.N mo.m omwoa. nmona. oumfid. mm aoswa. no.N mm.N Nm.N ou~¢H. wmo¢a. «mHNm. mm NA¢AH. NM.N mA.N No.m ¢mmoH. ammoa. .«cmmm. Am Aafiv Acfiv ‘Aafiv Amumv Amuuv up #3 g.. 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N0.N ~0.N mA.m 000mg. 00002. 0A0m0. mofi .AcAv AaAV AcAV ANAAV AAAAV AA AA A AA.00.:VA A 000 AEAAV umu< 0Ax¢ mamu9 mwx< mauu9 0Hx< 000A umu< w u< 03 m2 20002.u0u002. scammmuwom “mumeHcmHm uswA03 020500 .mmouwuom 000002 Hmswfl>avcH mo 000u< momwuam wawuuwvmnm H00 0003002 039 mo 0020003 0>wumummEOU a Nanmmm¢ 163 APPENDIXE 164 APPENDIX E Comparative Results of Three Models for Predicting Surface Areas of US H20 Sugar Beets. Sample .Weight Planimeter Model 1* Model 2** Model 3*** N9 Wo Area Area Are Area (lbm) (£12) (£62) (ft ) (6:2) 1 1.34352 .41722 .40801 .40811 .42185 2 1.88386 .50653 .52833 .53023 .52279 3 1.63514 .48889 .47294 .47517 .47363 4 2.99162 .79639 .77500 .75859 .81161 ‘ 5 2.52784 .67549 .67173 .66582 .67511 6 2.35743 .60569 .63378 .63079 .63094 7 2.32315 .60201 .62615 .62367 .62241 8 2.63889 .73319 .69646 .68836 .70555 9 2.46168 .67986 .65700 .65229 .65761 10 2.40157 .66431 .64361 .63991 .64210 11 2.31647 .61375 .62466 .62228 .62077 12 2.29171 .63993 .61915 .61713 .61469 13 1.93739 .53868 .54025 .54186 .53402 14 2.35769 .65097 .63384 .63684 .63101 15 1.67765 .48771 .48241 .48470 .48169 * Model 1: A = .10883 + .22268wo ** Model 2: A = .32469 wo°77439 *** Model 3: A = .24745 (1.48743)W° 165 APPENDIX E Comparative Results of Three Models for Predicting Surface Areas of US H20 Sugar Beets Sample Weight Planimeter Model 1* Mbdel 2** Mbdel 3*** N9 W0 Area Area Area Area (lbm (2:2) (6:2) (6:2) (6:2) 16 2.06592 .53944 .56887 .56950 .56198 17 1.94264 .55090 .54142 .54300 .53514 18 2.24482 .56875 .60871 .60733 .60335 19 2.19184 .55799 .59691 .59620 .59080 20 2.09149 .57521 .57456 .57495 .56772 21 2.39599 .65000 .64237 .63876 .64068 22 2.20269 .61201 .59932 .59848 .59335 23 2.34364 .66431 .63049 .62772 .62725 24 2.20331 .58847 .59946 .59861 .59349 25 2.43541 .65993 .65115 .64689 .65078 26 2.25617 .57708 .61123 .60970 .60608 27 1.86590 .51542 .52433 .52631‘ .51908 28 2.20254 .56562 .59929 .59845 .59331 29 2.23902 .57049 .60741 .60611 .60197 30 1.82119 .49007 .51437 .51652 .50994 31 1.64308 .47187 .47471 .47695 .47513 32 1.75860 .48056 .50043 .50272 .49743 33 1.75313 .47222 .49922 .50151 .49635 166 APPENDIX E Comparative Results of Three Mbdels for Predicting Surface Areas of US H20 Sugar Beets Sample Weight Planimeter Mbdel 1* Model 2** Mbdel 3*** N9 Wb Are; Are? Are; Are; (lbm) (ft ) (£1: ) (ft ) (£1: ) 34 1.84039 .50000 .51865 .52073 .51385 35 2.14350 .55049 .58614 .58599 .57956 36 1.72405 .47938 .49274 .49505 .49065 37 1.63622 .45569 .47318 .47541 .47384' 38 1.69034 .46076 .48524 .48754 .48413‘ 39 1.50545 .45097 .44406 .44571 .44986 40 1.39464 .43090 .41939 .42009 .43050 41 1.55930 .46965 .45606 .45801 .45958 42 1.51658 .50278 .44654 .44826 .45185 43 1.62496 .47556 .47068 .47287 .47172 44 1.51318 .43299 .44579 .44748 .45124 45 2.58891 .72931 .58533 .67824 .69168 46 2.35388 .69986 .63299 .63005 .63006 47 1.80545 .50646 .51087 .51306 .50677 48 2.31349 .61444 .62400 .62167 .62003 49 1.71345 .49604 .49038 .49270 .48859 ' 50 1.51587 .47028 .44638 .44810 .45173 51 1.80245 .50403 .51020 .51240 .50616 167’ APPENDIX E Comparative Results of Three Models for Predicting Surface Areas of US 320 Sugar Beets Sample Weight Planimeter Model 1* Mbdel 2** ‘Model 3*** N9 Wo Area Area Area Area (lbm) (ftz) (ftz) (ftz) (fez) 52 1.55842 .44424 .45586 .45781 .45942 53 2.12191 .56229 .58134 .58141 .57462 54 1.77288 .50896 .50362 .50588 .50026 55 2.31966 .62174 .62537 .62295 .62155' 56 2.54444 .67750 .67543 .66920 .67958 57 1.81993 .52437 .51409 .51624 .50969 58 1.46953 .46194 .43607 .43745 .44349 59 1.51074 .45188 .44524 .44692 .45081 60 1.58706 .46701 .46224 .46431 .46468 61 1.45448 .46424 .43271 .43398 .44085 62 2.40946 .63847 .64573 .64154 .64411 63 2.37454 .63174 .63759 .63433 .63524 64 2.27798 .61437 .61609 .61426 .61135 65 1.58933 .47479 .46274 .46482 .46510 66 1.44365 .43632 .43030 .43148 .43896 67 2.26221 .60972 .61258 .61097 .60754 68 2.37059 .63938 .63671 .63351 .63425 168 APPENDIX F 169 AA.AA AAAA AAAoo. 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HHHHHHH. HHHHHHH. HHHHHHH5 HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHg HHHHH. HHH. HHH.H HHH. oH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH.H HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHaHHHe HHIH mm Has mnHuuuuLH ”HI HHHHV 33 wou< BAH BAH unaumz mou< “any .Hm> Hmsaomn ho mz H>HHHHHHH H.H HzH HHHH HHuHaHHHHH H HHH mm H HHHaHH 218 HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHg HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH . HHHH. HHHHHHH. HHHHHHH. HHHHHHg HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHg HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHoo. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH AHmaHuonv «mm. as Hh ”ml AmumV Ag muu< 89H ouaumz mou< AumV .Hm> Hmaaumn mo oz H>HHHHHHH H.H 32H HHHH HHHHaHHHHH H HHH mm H HHHaHH HOHuHHuwuouuwso nonwoum HQOHqucoo Haunoaaouw>cm .Huomm uwwsm 0N mm: vaomgab Ham HmHmmm Scum HomHoH musumHoz so mama kuamaHuomxm 2 Manmmm< 219 220 HHHHHHH. HHHHHHH.. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHH: HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH.H HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. HHHHHHH. HHHHHHH. HHHHHHHH HHHHH. HHH. HHH.H HHH. HH HH HHHHHHH. HHHHHHH. HHHHHH HHHH. 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