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' 3'41" 5r I “v. - ,,... . lvfa",-;I “1‘41? 3. ' ‘ J gr. 1-. .~ ‘V U 0 ¢.:)-' \ 135:7.“ .‘ . ' I, 13/; I." [Sf-4:54 $19.“; v (sky; ‘L‘fl ' h .. w-se'” 4b,... _, "T I“ I," . C‘W‘xw'. ,hé’fi ”‘3 “n “W “1?; «wt ("4' 'd‘v'si“;3:{5‘J‘I’i‘m‘w‘2-zvt’ ‘ ' . V "‘7’ .I ‘ ' o V l |: r "h”: ‘1‘ 'q‘ I ' u'n‘ -“ “ “£3 hf’zg.:i-~ .. w.» n xvi-m3 . “I g _, ’ 7%}. ¢¢§$¢€WLF¢ i m "Mm?" 1-11 ' ."ra .- " '1" 4i ' "3“ I t‘. ILA" :1 '1 II - I fixw . {av W, ~ '. "I” . ‘ A I. .. .5313?! '. . “’3'. 5 h: ' " . . -.'-‘v "C x "13?“ ' X' I. -n.‘ n I‘ . ‘.- a» v u #1.... A) Egrr :. . "I'F'kl" . 3.. x I" (-"~? 1 s' hunt-d. 1." {Midi-icon: ._,;__,-:; ,. nth-141‘ g! 'I.'v--u' This is to certify that the thesis entitled Influence of Initial Freezing Temperature 0n Freezing Characteristics of Food presented by Uzoma Godwill Nwankwo has been accepted towards fulfillment of the requirements for M. S . degree in Agricultural Engineering W Major professor Date é/zfl’s’ Bu 44’. W144.) 0-7639 MS U is an Waive Action/Equal Opportunity Institution IV1ESI_J RETURNING MATERIALS: Piece in book drop to ”saunas remove this checkout from “- your record. FINES will be charged if‘book is returned after the date stamped beiow. INFLUENCE OF INITIAL FREEZING TEMPERATURE ON FREEZING CHARACTERISTICS OF FOOD BY Uzoma _- —--\ Godwill Nwankwo A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering 1985 ABSTRACT INFLUENCE OF INITIAL FREEZING TEMPERATURE ON FREEZING CHARACTERISTICS OF FOOD BY Uzoma Godwill Nwankwo The increasing use of freezing as a food preserva- tive method and the growing demand for ready-to-serve frozen foods have saddled the frozen food industry with responsibilities of finding new and improved freezing processes. Research has shown that formation of ice crystals in foods during freezing and the subsequent melt- ing of the ice during thawing before the product is used is an inefficient use of time and energy. Any steps that can partially or completely eliminate ice crystal forma- tion during freezing could improve the freezing process and the frozen food products. The objectives of this investigation were to use computer simulation models to predict how thermal proper- ties of food, and freezing time are affected by the depression of initial freezing temperature. The results illustrate that thermal prOperties change significantly and that freezing times can be Uzoma Godwill Nwankwo reduced considerably by depressing the initial freezing temperature of the food. Frozen food quality can be enhanced as a result of reduced ice crystal formation. This is especially important during storage and distribu- tion of frozen food during which temperature fluctuations cause alternate thawing and freezing of the product that could lead to loss of product texture as well as other quality characteristics. The result also shows that by substantially depressing initial freezing temperature, food products can be held at temperatures as low as -18°C with minimal ice formation. This would result in energy saving during freezing and reduce, if not completely eliminate, thawing time. To Daddy and Dada-~you have always believed in me. ii ACKNOWLEDGMENTS My heartfelt and sincere gratitude to Professor Dennis R. Heldman, Professor of Food Engineering for suggestion of this research tepic, and his continuous support, interest, fatherly guidance and encouragement throughout the period of this study. My indebtedness to him can never be repaid. My sincere appreciation also to members of my committee: Dr. James F. Steffe, of the Department of Agricultural Engineering, and Dr. Mark Uebersax, of Food Science Department for their willingness to serve on my committee, as well as their suggestions and help during this study. To my wonderful wife, Adrena, special thanks for keeping me happy and well fed, and for all those encourag- ing words. Last, but not least, I wish to thank my sponsors: the University of Maiduguri, Maiduguri, Nigeria, who provided me this opportunity. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . LIST OF FIGURES O O O O O O O O O NOMENCLATURE . . . . . . . . . . . INTRODUCTION . . . . . . . . . OBJECT IVES C O O O O O O O 0 LITERATURE REVIEW . . . . . . . Chapter I. II. III. 3.1 3.2 3.3 3.4 IV. Freezing Point Depression . . . Unfrozen Water Fraction Prediction Model . . . . . . . Thermal Properties Prediction Food Freezing Models . . . . THEORETICAL CONSIDERATIONS . . . . O O mxlONU‘l-bUJNI-J hbkkbbbb O O O O O Freezing Temperature Depression . Unfrozen Water . . . . . . Product Density . . . . . . Thermal Conductivity . . . . Enthalpy . . . . Apparent Specific Heat . . . . Thermal Diffusivity . . . . . Heat Transfer During Freezing . 4.8.1 Assumptions . . . . 4.8.2 General Heat Conduction Equa- tion I I O O O O 0 Initial and Boundary Conditions . Freezing Time Criteria . . . . iv Model Page vi vii Chapter V. EXPERIMENTAL CONSIDERATIONS . . . . . 5.1 5.2 5.3 VI. RESUL 6.1 6.2 6.3 6. 4 VII. CONCL VIII. SUGGE APPENDIX BIBLIOGRAPHY Predicted Initial Freezing Tempera— tures . . . . . . . Simulation of Thermal Properties . . Simulation of Temperature Distribu- tion During Freezing . . . . . . TS AND DISCUSSIONS . . . . . . Influence of Reduced Initial Water Content on Product Thermal Proper- ties . . . . . . . . . . . 6.1.1 Initial Freezing Temperature Depression . . . . . . . 6.1.2 Unfrozen Water . . . . . 6.1.3 Density . . . . . . . . 6.1.4 Thermal Conductivity . . . 6.1.5 Apparent Specific Heat . . . 6.1.6 Enthalpy . . . . . . . Influence of Freezing Point Depres- sion on Product During Freezing . . 6.2.1 Temperature History . . . .‘ 6.2.2 Freezing Time . . . . . . 6.2.3 Refrigeration Requirement . . 6.2.4 Storage and Quality (Micro- bial) Considerations . . . 6.2.5 Freeze-Thaw Process . . . Justification of Simulation Approach to Food Freezing . . . Possible Industrial Application . . USIONS . . . . . . . . . . STIONS FOR FURTHER STUDY . . . . Page 24 26 26 26 29 29 29 31 34 36 39 41 43 43 53 57 60 60 61 62 64 66 67 73 Figure 4.1 4.2 6.3 6.8 LIST OF FIGURES Diagram of the two component homogenous three-dimensional dispersion system . . Schematic illustration of procedure used to predict thermal conductivity of frozen products . . . . . . . . . . . Influence of composition on initial freez- ing temperature of grape juice .. . . . Predicted unfrozen water fraction versus temperature for different initial freezing temperatures of grape juice . . . . . Predicted density as a function of tem- perature for different initial freezing temperatures of grape juice . . . . . Thermal conductivity as a function of initial freezing temperature for grape juice . . . . . . . . . . . Predicted thermal conductivity as a function of temperature for different initial freezing temperatures of grape juice . . . . . . . . . . . . Predicted apparent specific heat versus temperature for different initial freez- ing temperatures of grape juice . . . . Predicted enthalpy as a function of tem- perature for different initial freezing temperatures of grape juice . . . . . Predicted center temperature history for grape juice product thickness of 2 cm and h of 40 W/mzK . . . . . . . . . . vi Page 14 15 30 32 35 37 38 40 42 44 Figure 6.11 6.12 Predicted center temperature history for gra e juice thickness of 4 cm and h = 40 W/mK o o o o o o o o o o o 9 Temperature history at center for grape juice thickness of 6 cm and h of 40 W/m2 K o o o o o o o o o o o 0 Predicted surface and center temperatures for initial freezing temperature of -1.86 C h = 100 W/m K and thickness = 6 cm for grape juice . . . . . . . . . Predicted surface and center temperature for initial freezing temperature of -9.24C h = 100 W/mZK and thickness = 6 cm for grape juice . . . . . . . . . . Predicted surface temperature for grape juice thickness of 6 cm and h = 100 WImzh o o o o o o o o o o o 0 Predicted enthalpy for grape juice with different initial percent unfrozen water . Predicted temperature history for grape juice thickness of 2 cm and h = 60 W/mZK . Predicted temperature history for grape juice thickness of 4 cm and h = 100 W/m2 K . . . . . . . . . . Temperature history at center for grape juice thickness of 4 cm and h of 100 WImZK . . . . . . . . . . . Predicted temperature history for grape juice thickness of 6 cm and h = 60 W/mZK . . . . . . . . . . Predicted center temperature history for grape juice thickness of 6 cm and h of 100 W/m2 K . . . . . . . . . . . vii Page 45 46 48 49 51 59 68 69 70 71 72 Table 5.1 LIST OF TABLES Page Physical PrOperties of Unconcentrated Grape Juice Utilized in Simulation . . . 25 Predicted Freezing Time for Grape Juice Thickness of 2 cm . . . . . . . . . 54 Predicted Freezing Times for Grape Juice Thickness of 4 cm . . . . . . . . . 55 Predicted Freezing Times for Grape Juice Thickness of 6 cm . . . . . . . . . 56 Predicted Enthalpy Value for Different Initial Freezing Temperatures . . . . . 58 viii Cpa Cpi Cps pr HI H1 H3 Hw IW kc kd MS MW (T) (T) (T) NOMENCLATURE Apparent specific heat of frozen product (kJ/kg C) Specific heat of ice (kJ/kg C) Specific heat of product solutes (kJ/kg C) Specific heat of unfrozen water (kJ/kg C) Total heat content or enthalpy of product (kJ/kg) Sensible heat contribution for ice (kJ/kg) Latent heat contribution (kJ/kg) Sensible heat contribution for product solutes kJ/kg Sensible heat contribution for unfrozen water kJ/kg Initial water fraction Convective heat transfer coefficient, W/mZC Thermal conductivity of product W/m C Thermal conductivity of continuous phase W/m C Thermal conductivity of discontinuous phase W/m C Molal latent heat of fusion, kJ/mole Product‘thiCkness, cm" Volume fraction of solutes Molecular weight of solutes Molecular weight of water mi ms mw TAO TF Tf Ti a(T) pl 93 ow Xw Mass fraction of ice Mass fraction of solutes Mass fraction of water in product Universal gas constant, kJ/mole k Freezing temperature of water, C Experimental Initial Freezing Temperature of Product, C Predicted initial freezing temperature of product C Initial temperature of product C Coordinates in the rectangular plane Coordinates in the rectangular plane Coordinates in the rectangular plane Thermal diffusivity of product, mzls Density of product at any temperature, kg/m3 Density of ice, kg/m3 Density of product solutes, kg/m3 Density of frozen water, kg/m3 Mole Fraction INTRODUCTION The dependence of the thermal properties of foods at both frozen and unfrozen states on composition has been recognized for considerable periods of time. Siebel (1892) had suggested a definite relationship between percentage of water in food and ”latent heat" during the freezing process. The amount of water in a food product is the single most dominant factor in establishing the magnitude of thermal prOperties for a food. These prOperties, which include density, thermal conductivity, specific heat, enthalpy and thermal diffusivity are essential for the following reasons: 1. Optimization of freezing equipment design by establishing accurate refrigeration requirements and/or freezing times 2. Prediction of product quality changes during temperature equilibration and frozen storage 3. Correlation of freezing parameters with quality parameters of the product Traditional freezing methods have relied on crystal- lization of water in the food product to reduce the amount 1 of free water available in the system; thus inhibiting ' mdcrobialgrowth and enzyme activity that cause food spoilage. However, unlike pure water, this crystalliza- tion process does not occur at a particular temperature in food products. As the temperature of the food product is lowered, ice formation begins when the product tempera- ture is equal to its initial freezing temperature. The ice crystallization changes the solute concentration in the product and reduces the amount of water available for freezing and depressing the freezing point of the product. This becomes a progressive situation as explained by Staph (1949). Since the thermal prOperties of the product are directly dependent on the state of water in the product (Heldman, 1982), these properties would continuously change as the freezing process progresses. Heldman (1970a) pro- posed a model for predicting the amount of unfrozen water in a food product during freezing using the freezing point depression equation for an ideal binary solution and has used the predicted values in the simulation of product thermal preperties during freezing (Heldman, 1982). It is obvious that depression of the initial freezing temperature is a major characteristic of food freezing. The magnitude of this depression varies with product and composition or more specifically, the water content as illustrated by Riedel (1951; 1956; 1957b). This depression in initial freezing point is due to increasing concentration of the product and results in significant changes in the product prOperties. The purpose of this study is to investigate the effect of lowering initial unfrozen water content and the resulting initial freezing point depression on thermal properties, refrigeration requirements, temperature his- tories, freezing times, and other freezing parameters of the product using computer simulation. were: CHAPTER II OBJECTIVES The principal objectives of this investigation To use a computer program for simulation of thermOphysical properties during the food freezing process. To determine the infuence of the depressed freezing temperature on thermophysical properties of frozen foods. To predict the influence of modified thermOphysical properties on mass average temperatures, freezing times, and refrigera- tion requirements for the food freezing process. To analyze potential benefits that might accrue during freezing, equilibration, and storage of frozen foods, as a result of modified initial water content and thermo- physical properties of the food product. CHAPTER III LITERATURE REVIEW 3.1 Freezing Point Depression One unique characteristic of food freezing is the depression of the initial freezing temperature. The magnitude of this depression has been shown in experi- mental research by Riedel (1951; 1956; 1957) to be a func- tion of the product composition especially the water content. Other researchers (Fennema and Powrie, 1964; Heldman, 1966; Hohner and Heldman, 1970) have suggested predicting the magnitude of the freezing temperature depression using the freezing point depression equation for an ideal binary system. 3.2 Unfrozen Water Fraction Prediction Model The importance of the percentage of unfrozen water in a food product during freezing in the prediction of thermal properties has been established by Heldman (1982). The accuracy of prediction of thermal properties, there- fore, depends not only on the prediction models used, but to an appreciable extent on the accuracy of prediction of the percent frozen (or unfrozen) water in the product as freezing progresses. Heldman (1974a) proposed a model 5 for predicting the unfrozen water in a product as a func- tion of temperature at sub-freezing temperatures utiliz- ing the freezing point depression equation for an ideal binary system. 3.3 Thermal PrOperties Prediction Models VOlumes of literature dealing with experimentally determined values of thermal prOperties exist. Riedel (1951: 1956; 1957) used calorimetric methods to determine thermodynamic properties of specific food products. Experi- mentally measured thermal property data and empirical relationships have been compiled by Woodams and Nowrey (1968), Reidy (1968), and Morely (1972). However, much of the published thermal properties data as compared by Reidy and Rippen (1971) show considerable variations in values for same product. They concluded that these appar- ent discrepancies in experimental values are due to inherent instrumental errors, methods of measurements, and unattainable assumptions and conditions imposed by the heat transfer models utilized. To overcome problems associated with experimental determination of thermal prOperties of foods, several authors have proposed the use of models for prediction of thermal pr0perties. Hsieh et a1. (1977) proposed a model for predicting density of a freezing product. Long (1955) and Lentz (1961) used modified forms of the Maxwell (1904) equation to predict thermal conductivities for several frozen products. KOpelman (1966) and Mascheroni et a1. (1977) have deveIOped mathematical models for predicting thermal conductivity for foods. Kopelman's model has been successfully utilized by Heldman and Gorby (1975) and Hsieh et a1. (1977) to predict thermal conductivities of food during freezing. Models for prediction of enthalpy of frozen products have been proposed by Heldman and Singh (1981), Levy (1979), Larkin et al. (1983), and Schwartzberg (1976). Heldman (1982), Lescano (1973), and Schwartzberg (1976) have pro- posed mathematical models for the prediction of specific heat capacities for frozen foods. This apparent heat capacity could then be used to predict or simulate thermal diffusivity of the product during freezing (Heldman, 1982). 3.4 Food Freezing Models Food freezing rates have been predicted using both analytical and numerical techniques. Golovkin et a1. (1973), Hayakawa and Bakal (1974), Komori and Harai (1974), and Plank (1941) have used the analytical approach and constant thermophysical properties to solve food freezing problems for infinite slab, infinite cylinder and Sphere. Cleland and Earle (1977), Mascheroni and Calvelo (1982), and Nagaoka et a1. (1955) have all prOposed modified forms of Plank's original equation to account for the change in properties. All these methods have either assumed constant thermophysical prOperties during freezing or utilized average values of properties within ranges of tempera- ture. Numerical techniques became increasingly useful in solving heat transfer problems in food freezing with the rapid development of digital computers. Many researchers, such as Albasiniy (1956), Charvarria (1978), Charm (1971), Comini and Bonacina (1974), Cordel and Webb (1972), Cullwick and Earle (1973), Fleming (1973), Gorby (1974), Heldman (1974b), Heldman and Gorby (1975b), Lescano (1973), and Purwadaria (1980) have used computer freezing simula- tions based on finite difference techniques to predict freezing times for foods. Lescano and Heldman (1973) used the Crank-Nickolson finite difference equation to solve the heat transfer problem during freezing of a slab of codfish and predict the temperature history during freezing. Heldman and Gorby (1975b) developed a model to predict freezing times for spherical geometric food products using computer simu- lations based on finite difference techniques. This was modified by Hsieh (1976) and applied to fruits and vege- tables. Purwadaria (1980) developed a finite element model using computer simulation to solve heat transfer problems involving phase change for elliptical and trapezoidal geometries. Commonly used finite difference schemes have been compared by Cleland and Earle (1983) while different computer simulations related to food freezing process have been compared by Heldman (1974b). CHAPTER IV THEORETICAL CONSIDERATIONS 4.1 Freezing Temperature Depression Unlike pure water, food substances have no single freezing temperature, but will freeze over a range of tem- peratures. Woolrich et a1. (1933) established that the presence of salts, sugars, and fats considerably depressed freezing temperatures of foods by considerable magnitudes. During the freezing process, ice formation in the product begins when the temperature of the product equals its initial freezing point temperature. The remaining unfrozen product has a higher concentration resulting in a lower freezing temperature. The process of increased ice forma- tion, increased concentration of the unfrozen product, and resultant depression of freezing temperature during freezing is progressive. The depression, as a result of reduction in unfrozen water in the product, causes corre- sponding changes in thermal properties. Heldman (1974a) utilized the freezing point depres- sion equation in predicting the relationship between unfrozen water fraction and temperature. The equation is given by: 10 11 1 1 _ [éf - TA5:] - 1n Xw (4.1) molal latent heat of fusion wu« where L 71 II universal gas constant TAO = freezing temperature of water TF = product freezing temperature Xw mole fraction of water in product If the mole fraction of water in the product is known, the product's freezing temperature can be calculated. '4.2 Unfrozen Water The state of water in the product exerts a strong influence on the prOperties of a frozen food. Heldman (1974a) noted that the relationship between frozen water fraction and temperatures may be the most basic character- istic of a frozen food needed in freezing design computa- tion. This statement underscores the importance and need for a reliable model for predicting the state of water in a product during freezing. Heldman (1974a) proposed a model utilizing the freezing point depression equation (Equation 4.1). By using the relationship: 12 mw/Mw mw/Mw + msst Xw = (4.2) where mw = mass fraction of water in product Mw = molecular weight of water ms = mass fraction of solutes MS molecular weight of solutes The computed value of Xw, the effective molecular weight of product solutes (Ms) can be computed. This molecular weight of solutes is then assumed constant throughout the freezing process and used in computing the mass fraction of water at different freezing-temperatues of the product. Heldman (1974a) has shown that this procedure gives reason- ably reliable prediction of unfrozen water content as a function of temperature. 4.3 Product Density Hsieh et al. (1977) have shown that the effect of freezing on product desnity can be predicted using the equation: 231+“ +m—I (4.3) 1— — o ' ow 08 pl where p density of product at any temperature during freezing pw = density of unfrozen water 03 = density of product solutes cl = density of ice 13 The use of the above equation requires knowledge of the solute density which is not known in most cases. Another limitation to this model is the fact that the equation lumps all the solutes and neglects the fact that the product may contain different solutes with different densitites. The first difficulty can be overcome by using published density value of product (p) at known composition and temperature to calculate the density of the product solutes (ps). This value can then be used as a constant value in subsequent calculations of product density (0) during freezing. 4.4 Thermal Conductivity The large difference between the thermal conductiv- ity of ice and water makes prediction of the thermal con- ductivity of a product during freezing very complex and difficult (Heldman, 1982). A modified form of the thermal conductivity model proposed by Kopelman (1967) has been successfully utilized by Heldman and Gorby (1975a) to predict the thermal con- ductivity of different types of foods. Using Kopelman's two component-three dimensional dispersion model, the thermal conductivity of an isotropic material can be pre- dicted by the equation: 14 Figure 4.1. Diagram of the two component homogenous , three-dimensional dispersion system. A. Natural random state B. The rearrangement of the components SOURCE: Kopelman, 1967.. 15 _ l ‘ 1 " Q __ with Q = M [1 - kd/kc] (4.5) where k thermal conductivity of product volume fraction of solutes 3 II kc thermal conductivity of continuous phase kd = thermal conductivity of discontinuous phase Since a frozen food product is a three-phase system, ice, water, and solids, Heldman and Gorby (1975a) modi- fied Kopelman's model by first considering two of the three product phases and reducing the system to two phases for a second use of the expression (Figure 4.2). Choice of the continuous and discontinuous phases is also impor- tant. In this investigation, water is considered the con- tinuous phase when evaluating the ice and water system, while water-ice mixture is considered the continuous phase when dealing with ice-water and product solids system as suggested by Heldman and Gorby (1975a). The thermal conductivity of product solids was found additively from the thermal conductivity and mass fraction of each component that made up the product solids. 16 Thermal Thermal Thermal Conductivity Conductivity Conductivity of Water of of kc ice Solids kd kd Thermal Conductivity water-ice mixture kc Thermal Conductivity of Frozen Product Figure 4.2.--Schematic illustration of procedure used to predict thermal conductivity of frozen products (Heldman and Gorby, 1975a). 17 4.5 Enthalpy The total heat content of a frozen food product is the sum of the sensible and latent heats of the product. The total heat content or enthalpy can be expressed, as proposed by Heldman (1982), by: H=Hs+Hw+H1+HI (4.6) where Hs, hw, and HI are sensible heat contributions for product solutes, unfrozen water and ice, respectively H1 is the latent heat contribution Using -40°C as a reference for H = 0 and considering the temperature dependence of the sensible heat contribution, Heldman (1982) proposed integrating the change in heat content for the different phases of the product over appropriate temperature limits. Ti Ti :13 II ms - Cps J dT + mw pr J dT Tf + J mw(T) pr(T) dT + mw(T) L -40 Tf + I mi(T) Cpi(T) dT (4.7) -40 18 where ms = mass fraction of solutes Cps = specific heat of product solutes mw = mass fraction of unfrozen water pr = specific heat of water mi = mass fraction of ice specific heat ice Cpi Using this model, Heldman (1982) predicted values of enthalpy for sweet cherries that showed reasonable agree- ment with experimental data obtained by Riedel (1951). 4.6 Apparent Specific Heat Heldman (1982) noted that the specific heat of a frozen product includes the latent heat portion in the enthalpy. The folloWing equation was proposed for predict- ing apparent specific heat: _ dH Cpa(T) — dT (4.8) From Equation (4.8), it is quite evident that apparent specific heat for a frozen food will vary with temperature. Since most of the water in the product is frozen within 10°C below the initial freezing point tem- perature, temperature increments for computation of apparent specific heat in this range should be quite small so as not lose the real impact of temperature on apparent specific heat. 19 4.7 Thermal Diffusivity As with most other thermal properties, thermal diffusivity of a frozen food will vary significantly with temperature and can be predicted from the equation k(T) MT) = 0(T)Cpa(T) (4.9) The thermal diffusivity of the product is a very important prOperty involved in the solution of transient heat-conduction equation for temperature history during freezing. Equation (4.9) shows that variation of thermal conductivity, density, and apparent specific heat with temperature will influence thermal diffusivity. 4.8 Heat Transfer During Freezigg Thermal properties of products and their magnitudes are of limited value unless they could be put to practical use in the prediction of freezing times. Heat Transfer during freezing is a complicated process due to the phase change that occurs in the product and variable thermal properties which are functions of product temperature. Calculations of heat transfer rates during freezing can be accomplished using either analytical or numerical methods. Analytical methods assume constant thermal properties; an assumption that is not realistic with biological or food products. Numerical methods can incor- porate variable thermal properties and provide more 20 reliabile and accurate heat transfer problem solutions during freezing (Heldman, 1974b). 4.8.1 Assumptions The following assumptions have been made in using a numerical method of solution of the heat transfer problem: 1. Initial temperature of product is uniform and constant. Product thermal properties are constant for small temperature ranges above the initial freezing point. Overall product composition remains constant throughout the freezing process. Freezing medium temperature is uniform and constant. Heat transfer from freezing medium to product is by convection only. Heat transfer within the product is by conduction only. Heat flow in product is one-dimensional. Mass transfer between product surface and freezing medium and/or within product is negligible. Surface heat transfer coefficients (h) are constant during the freezing process. 4.8.2 General Heat Conduction Equation Fourier's differential equation for the temperature history in a solid body (assuming only conduction) derived from conservation of energy for an infinitesimal element with no internal heat generation and assuming 3-dimensional heat flow is given by 5 (xx 6t) 5 (k at) 6 (kzdt) 5T 3;.__3;__— + 6? ——%§—— + 3; -—dE—— = Cpp XE (4.10) for a homogeneous infinite slab. Since Cpa, p, and k are functions of temperature during freezing of food products, then considering heat flow in only one direction, Equation (4.10) can be rewritten as: k(T) a? = a; ‘4-11’ or 1 (ST ___ g a(T) 3? 6X (4.12) Cp(T)p(T) .. 1 where k(T) - a(T) (4.13) 4.9 Initial and Boundary Conditions A. Initial Condition: The product is initially at uniform temperature 22 at t = 0 (4.14) B. Boundary Conditions 0T _ _ 1. E - 0 at X - 0 (4.15) x=0 2. At the convection surface (5r _ kx (T) E? + be (T-Tf) - o t _>_ o ' (41.6) 4.10 Freezing Time Criteria There is a lack of consistency in the definition of freezing time. This does not detract from the fact that the time required to accomplish a given freezing process in a food product is of considerable importance to design of freezing systems. One commonly used approach is to determine the time for the center of the product to reach an arbitrarily selected temperature. This method becomes especially use- ful when finite difference techniques are used to predict temperature history in a freezing product because the time for product center to freeze any desired temperature can be easily obtained (Cleland and Earle, 1977). Another approach is to define freezing time in terms of the time required to decrease the heat content 23 (enthalpy) of the entire product from some initial value to the equivalent of the product enthalpy at the desired storage temperature. This method has been utilized by Heldman and Gorby (1975b) and provides the lowest possible residence time in the freezing medium for a given set of freezing parameters and for minimum refrigeration require- ment (Heldman and Gorby (1975b). In this investigation the first approach has been used and freezing time defined as the time required for the center temperature of product to reach -18°C (storage temperature). This approach is justified since only one initial product temperature was utilized in the simula- tion. CHAPTER V EXPERIMENTAL CONSIDERATIONS A computer program with the following subroutines were utilized: 1. Prediction of initial freezing temperature of product. Prediction of unfrozen water fraction, thermal conductivity, density, enthalpy, and apparent specific heat of product as a function of temperature for different initial freezing temperatures. Prediction of temperature distribution within a food product of infinite slab geometry during freezing. To operate the program, the following information is required: Initial water content of product. Product thermal properties above freezing. Experimental initial freezing temperature of product. Freezing medium temperature, 24 25 .Aoamonmm .oz Madam mama .moxo cfluumz can floso monocowv 30H>mmllm©oow owsqu mo mmfiuuomoum HcEHmsa "mousom ham.o xE\3 .Uocm um. nouns mo >ue>fluosccoo anemone mH.v U ox\nx «Doom up. Houc3 mo upon camwoomm mom.o 0 mx\bx Auoom up. and «0 you: oamwommm sqm.H o 6x\nx Aooom um. managesnonumo mo hams cemeomam mmm.~ o mx\nx AoooN um. one Go hams ohmuomdm ~H>.H o mxxhx Avoom uov :Hmuoum mo upon oamaomam m~.o mooa\m acoucoo and mm.qfi moos\m , mmumunsaonumo mo.m mconxm acmucoo uch mm.o macaxm acoucoo aflououm owoa mE\mx hufimcoc uoscoum HcfiuwcH mm.~i U Aha. mneucummamu mcwnmoum chcofifluomxm wma.vm noun 0 ooa\o~mlm acoucoo Hmuc3 HofiuwcH usac> mafia: huuomoum cowucH56Hm ca pmuflawua moasb mmmuw.cmucuucmocooco mo mwfluummonm Annamanmil.a.m mqmda 26 5. Convective heat transfer coefficient. 6. Thickness of product. 5.1 Predicted Initial FreezinggTemperatures Using Equation 4.1 and the experimental initial freezing temperature of product, the moledular weight of product solutes was computed. This value was used as an input to the computer simulation of initial freezing tem- perature while product initial water content was varied beween the normal water content of 84.12 percent and 50 percent. 5.2 Simulation of Thermal Properties Thermal properties of the food product above freez- ing were computed from knowledge of product composition, values of the thermal properties of the product components, and use of the models presented in the previous chapter. These properties were incorporated:h1the computer program to predict product thermal properties as functions of tem- perature with initial freezing temperature of product being depressed. 3 Simulation of Temperature istribution During Freezing .5... D A computer program to simulate the freezing processes of a product with infinite slab geometry was developed by Scott (1984). The program used the modified 27 Crank-Nicholson finite difference scheme to generate product properties needed and the predicted temperature history for a set of initial conditions. Additional information required to operate this program include: computing time increment criterion for termination of freezing number of nodes into which product will be divided The criterion for termination of freezing simulation was established as the center temperature of product equal to -18°C. The following parameters were chosen to evaluate: 1. 6. Initial freezing temperatures of -1.86°C, -2.45°C, -4.15°C, -6.33°C, and -9.24°C corresponding to initial water contents of 84.12%, 80%, 70%, 60%, and 50%, respectively. Initial product temperature of 25°C. Product thickness of 2, 4, and 6 cm. Freezing medium temperature of -40°C. Convective heat transfer coefficients of 40, 60, and 100 W/m K. Final product (center) temperature of -18°C. The physical properties of grape juice used for the simulation are listed in Table 5.1. The output information included: 1. Predicted initial freezing temperature. 28 Predicted unfrozen water, density, thermal conductivity, apparent specific heat, and enthalpy of food product as a function of temperature. Predicted temperature history for the entire freezing process. Predicted freezing time as determined by the criterion for termination of freezing. CHAPTER VI RESULTS AND DISCUSSIONS 6.1 Influence of Reduced Initial Water Content on Product Thermal Properties 6.1.1 Initial Freezing Temperature Depression Figure 6.1 presents the results of reduced initial water content on the predicted initial freezing tempera- ture of the product. At normal composition (84.12% water), the experimental initial freezing temperature of grape juice is -1.86°C. Starting with the freezing point depres- sion equation for an ideal binary solution and reducing the initial water content to 60%, the predicted initial freezing temperature of the product was reduced to -6.3°C. Further depression of initial freezing temperature to -9.2°C occurred at a composition of 50% water. This result is not surprising since several researchers (Deshpande et al., 1982; Huxsoll, 1982; Thijssen, 1969) have shown that an increase in the concentration of the product solids (decreasing the initial water content) results in depression of the initial freezing temperature of the product. 29 30 '12 - INITIAL FREEZING TEMPERATURE, C -24 L 1 1 i 20 40 60 80 100 INITIAL WATER FRACTION, % Figure 6.1. Influence of composition on initial freezing temperature of grape juice. 31 The implication of this result is that by extra- polating the curve in Figure 6.1, it would be possible to select a storage condition for product at sub-freezing temperature without freezing water. In this investiga- tion, this would be accomplished by reduction of the initial water content. For example, if it is desired to store product at -l7°C, a reduction of the initial water content to about 30% will depress the initial freez- ing temperature of the product to about -17°C, enabling the product to be stored at any temperature above -l7°C without formation of ice crystals. 6.1.2 Unfrozen Water The result in Figure 6.2 illustrate the influence of temperatrue on the predicted unfrozen water fraction for product with different initial water contents. With high initial water contents (84.12%, 80%, and 70%), a large proportion of the water freezes within a few degrees of the initial freezing temperature. For a composi- tion of 84.12% water, the percent unfrozen water decreases to about 28% between the initial freezing temperature of -1.86%:and -5°C, a reduction of almost 70% for a 3°C change in temperature. Water contents of 80% and 70% show the same effect, although to a lesser extent. This rapid decrease in unfrozen water fraction at temperatures just below the initial freezing temperature was noted by Heldman (1974). .oofi5n macho mo mousucummaou mcflnoouu Acwuaca ucmummmwo HON was» (chanson msmuo> c0wuocuu nouns conouuc: omuowooum .~.m ousmwm o .mmoeammEZme 32 ‘ «it. .I.I. ‘normovaa HEIVM nszoaaun 9 % Ste. 5.1.! a .2- .. i. are. ...... (.om «fie. III! .9»: 33 When the initial water content of the product is reduced to 50%, the unfrozen water percentage changes more gradually after freezing begins. Between the initial freezing temperatures of -9.2°C and -12.24°C, the unfrozen water percent changes only about 26% as compared to a 70% reduction to unfrozen water for the same 3°C change in temperature for a composition of 84.12% water. Although no experimental values were used to compare results of the current predictions, Heldman (1974) has shown that there is a good agreement between experimental values of unfrozen water and values predicted using the freezing point depression equation. The previous research has shown that a reasonable agreement can be obtained at even higher solids concentration of 50% or more between experi- mental and predicted values of percent unfrozen water as a function of temperature. The very gradual rate of ice formation immediately below the initial freezing temperature for low initial water content product (50% to 60%) can be attributed to the increase in the concentration of the noncrystallizing solutes. This results in increased viscosity of the solution and a decrease in the mobility of water molecules being crystallized (Fennema et al., 1973). Secondly, heat transfer during unagitated freezing is primarily by con- duction. Reduction in the initial water content results in a lower rate of heat transfer by conduction since the 34 solutes have lower thermal conductivity than water. This slow rate of heat conduction decreases the growth rate of ice crystals and the rate of freezing (Fennema et al., 1973). 6.1.3 Density The variation of density with temperature for product with different initial water contents is shown in Figure 6.3. By reducing the initial water content from 84.12% to 50%, the density of the unfrozen product increased from 1070 kg/m3 to 1278.5 kg/m3; an increase of approximately 20%. During freezing, the change in density is more rapid at temperatures just below the initial freezing temperature for the high moisture compositions. This is due to the fact that it is within this temperature range that most of the phase change occurs (Heldman, 1982). Predictions for compositions of 50% to 60% water show a much slower decrease in density as a result of the lower amounts of phase change. In general, the decrease in density was more pro- nounced for high moisture predictions (7% for initial freezing temperature of -l.86°C) and smaller, although still significant, for depressed freezing temperatures (5% for -6.3 and 4% for -9.2). 35 .ooasn mocha mo mousumuoosou mcwnoouw chuflca ucoummmwo How unsucummfiou mo coauocsm n no huwmcoo omuowomum U .flmaaflmmmzmfi BUDQOMQ N- m- «H: «m- on- _ . . . . . . . com I 0mm #Illlolllli 1| & .Lll|l+.nflil ~\\\. 1 oGOH x\\. a. llllllllll L i...l¢|\ \\.\.\\.I.III..I1.|I|) \.\\ IIIII \\\\ Ilillllllllllh OVHH ‘I ‘\ " . \ . ‘\‘ Illl|\ II. II Ill-ilo'ok, ONNH - I I.II. Hui.“ I coma n P p n n p p - .m.m mucous EIII/fix 'xmrsuao monaoua 36 6.1.4 Thermal Conductivity Figure 6.4 illustrates the effect of depressed initial freezing temperature resulting from reduced initial water fraction on thermal conductivity of product at 25°C. As the initial freezing temperature is depressed by concentration, the thermal conductivity of the unfrozen product decreases. At higher initial freezing points, the thermal conductivity of the unfrozen product is most influenced by the thermal conductivity of water. With depression of freezing temperature, the products solutes exert greater and greater influence on thermal conductivity. Since product solids have lower thermal conductivity than water, the overall reduction in thermal conductivity of unfrozen product with decrease in initial water content should be expected. The influence of reducing the initial water content and depressing the initial freezing temperature on the variation of thermal conductivity with temperature is pre— sented in Figure 6.5. The change in thermal conductivity is quite dramatic immediately below the initial freezing temperatures and gradual over the whole range of freezing temperatures. For an initial freezing temperature of- -1.86°C (84.12% water), the thermal conductivity predicted by KOpelman's model increased from 0.588 W/m C to 2.3 W/m C for a 6°C change in temperature. This represents an almost fourfold increase. When the initial freezing temperature 37 0.2 P «I Thermal Conductivity W/m C I -14 -10 -6 -2 0 INITIAL FREEZING TEMPERATURE C Figure 6.4. Thermal conductivity as a function of initial freezing temperature (at 25°C) for grape juice. 38 .m0fism macnm Ho mousucummfimu mcwummuw Hafiuwcw pcoummmao How musumummawu mo coauocsu m on muw>wuosocoo daemon» oouowooum .m.m musmflm o .mmoe4mumzme N cl le NNI 0m! 1 a q q n q 4 4 Tm H 3 vm.mi .33).! l mm.®l llil l . .3- 2 m. u II.I 3.7 - m o .2. N r m 1 D m I .z. m I . I I Q, L ”OH I“ 0,0,0, 8",- llllll mm I m / all I at,” a !llliillI( .Lv.N 39 is depressed to -9.2 (50% water), the preducted thermal conductivity increased from 0.4 W/m C to 1.47 W/m C over a 6°C change in temperature, representing about three and a half-fold increase. Overall, the predicted thermal conductivity increased 314%, 353%, 355%, 351%, and 335% for initial freezing temperatures of -1.86°C, -2.5°C, -4.1°C, -6.3°C, and -9.2°C as the product is frozen from 25°C to -30°C. The rapid increase of thermal conductivity during freezing is due to the magnitude of thermal conductivity for ice as compared to water. At depressed freezing temperatures, there was a progressive increase in thermal conductivity, with the final values at -40°C being the highest. 6.1.5 Apparent Specific Heat The predicted relationship between apparent spe- cific heat and temperature is presented in Figure 6.6. It is evident that a large difference exists between the maximum values for the normal freezing point of -1.86°C (144.8 kJ/kg C) and the depressed freezing point of -9.24°C (22.2 kJ/kg C). Since apparent specific heat is a combination of sensible and latent heats, the value is always a maximum at the initial freezing temperature. The difference in the values of the apparent specific heat is the amount ice crystal formed within the freezing product. At the normal freezing temperature of -1.86°C, most of the 40 168 ' r t r l T 144( -1.86 . U U) i h 120 ., 53 U H 5 U 72" - In] Ga U) E 48 q 53 OI 33 24 . Figure 6.6. TEMPERATURE, C Predicted apparent specific heat versus temperature for different initial freezing temperatures of grape juice. 41 energy absorbed by the product is utilized for phase change from water to ice, rather than decreasing the temperature of product. A very high value of apparent specific heat at the initial freezing temperature is the result. However, as the initial freezing temperature is decreased, smaller amounts of heat are absorbed as latent heat thus resulting in lower apparent specific heat values. In all predictions, apparent specific heat decreased with decrease in temperature, although it was most dramatic for initial freezing point of -l.86 (from 144.8 to 2.8 kJ/kg C) and least dramatic for initial freezing temperature of -9.25°C (from 22.2 to 4.4 J/Kg). In conclusion, it can be said that more of the thermal energy removed results in reduction of the product tem- perature as the initial freezing temperature of the product is depressed. 6.1.6 Enthalpy The predicted relationship between enthalpy and temperature for different initial freezing temperatures of product is presented in Figure 6.7. In general, enthalpy for product increases moderately with increasing tempera- ture at lower temperatures,but increases dramatically at temperatures near the initial freezing temperature. Above the initial freezing temperatures, the change in enthalpy with increasing temperatures is moderate. Although the 42 400 300 200 PRODUCT ENTHALPY, kJ/kg 100 TEMPERATURE, C Figure 6.7. Predicted enthalpy as a function of tempera- ture for different initial freezing temperatures of grape juice. 43 predicted enthalpy magnitudes at different initial freez- ing temperatures follow the same general pattern, there are differences in enthalpy magnitudes at any given temperature. Except at the reference temperature of -40°C, enthalpy for product with depressed initial freezing temperature is always higher at temperatures below freezing, while at temperatures above freezing the opposite relationship is evident. It is interesting to note in Figure 6.7 the enthalpy magnitudes at the different initial freezing temperatures. As the initial freezing temperature is depressed, predicted product enthalpy at the initial freezing temperature decreases. By depressing the initial freezing temperature from -1.86 to -9.24°C, product enthalpy at the freezing temperature decreases from 308 kJ/kg C to 208 kJ/kg C. In addition, the enthalpy changes between initial freezing temperature and the reference temperature (-40°C) decreases as initial freezing temperature was decreased. 6.2 Influence of Freezing Point Depression on Product During Freezing 6.2.1 Temperature Histopy Figures 6.8, 6.9, and 6.10 illustrate the tempera- ture history at the center for different initial freezing temperatures and different thickness of product using a convective heat transfer coefficient of 40 W/m C during the freezing process. PRODUCT TEMPERATURE, C 44 2cm 40 :3‘ II II -30 0.5 1.0 TIME, HRS. Figure 6.8. Predicted center temperature history for grape juice product thickness of 2 cm and h of 40 W/mZC. .U~E\3 ov u n can 50 q no mmmchecu ocean ommum “Ow Snowman ousucuooEou Houcoo concaomum .m.m ousmfim .mmz .mzHe GMI d «I» 45 a d a w D 'HHOLVHEdWHI HHINHD IDDOOHd @N.ml mm.mi so a u a m~.«) meqNI UNE\3 0v u S om.Hi .0. we 46 .UNE\3 ow no a can So o «0 mmocxoenu ocean mocha How Houcoo um auoume: ousucuoosoa .oH.m ousmwm .mm: .MZHB owl m / m I I Tu m d 3 1 w. T... . n.. .0083 m fil 0~I\3°¢I1 lflu 47 At temperatures above the normal initial freezing temperature of the product,there are no significant differ- ences in the patterns of decreasing temperature. For the same product thickness and value of the convective heat transfer coefficient, the temperature at the center at any given time for the product with different initial freezing temperature do not differ by more than 2°C. During the cooling phase, rate of heat transfer is about the same for products with different initial freezing temperatures, although there may be a slight decrease as the initial freezing temperature is decreased. This observation during the prefreezing stage is probably due to changes in thermal properties. For example, when the initial freezing tempera- ture is depressed from -1.86°C to -9.2°C, the thermal con- ductivity above freezing decreases from 0.588.W/m C to 0.40 W/m C and specific heat changes from 3.758 kJ/kg C to 2.905 kJ/kg C. Although heat transfer by conduction is reduced with a depressed freezing temperature, the energy required for a unit weight of product to increase 1°C becomes smaller. The result of the counteracting effect of these changes is a nearly equal temperature at any given time above freezing when product thickness (weight) and convective heat transfer coefficients are not changed. Figures 6.11 and 6.12 show surface and center temperature histories for products with normal initial 48 I l h = 100W/m2C y b = 6 cm 0 a _ ~ Tf 1.86 c m d D: D 5 ) m cl) 3 B E" U .1 D 8 m 94 —3_D TIME, HRS. Figure 6.11. Predicted surface and center temperatures for initial freezing temperature of -1.86 C h a 100 W/m2C and thickness = 6 cm for grape juice. 49 * Cantu h = 100W/m2K -- lettuce = 6 cm 0 G . Tf = -9.24 c E E - '23 m B -. B U a D O a: .. m 1 2 3 TIME, HRS. Figure 6.12. Predicted surface and center temperature for initial freezing temperature of -9.24C h s 100 W/m2C and thickness - 6 cm for grape juice. 50 freezing temperature of -l.86°C and a depressed initial freezing temperature of -9.24°C. At temperatures immediately below the initial freezing temperature, the product with depressed freezing temperature had smaller temperature differences than the product with normal initial freezing temperature. This relationship is caused by the differences in the amount of energy utilized for change of phase. At normal initial freezing temperature, the surface temperature keeps decreasing, but the temperature at the center remained relatively constant due to the energy being utilized for crystallization of ice. As the initial freezing tempera- ture is depressed, less energy is required for phase change and thermal energy removal results in center tem-: perature reduction. At temperatures below the initial freezing temperature, the difference between center and Surface temperatures again becomes larger for depressed freezing point predictions. This is caused by the high thermal conductivity of ice compared to properties of water and product solutes. The predicted temperatures at the product surface are quite different for the different initial freezing temperatures (Figure 6.13). For a product thickness of 6 cm and a convective heat transfer coefficient of 100 W/mC, the predicted surface temperature for the product having initial freezing temperature of -1.86°C is -12.8°C 51 SURFACE TEMPERATURE, C Figure 6.13. I fl .1 l 1 2 3 TIME, HRS. Predicted surface temperature for grape2 juice thickness of 6 cm and h = 100 W/m C. 52 after 1/2 hour and -18.48°C after one hour. For the same conditions, but with product having an initial freezing temperature of -9.24, the predicted surface temperatures are -22.07°C and -26.17°C after 1/2 and one hour, respec- tively. The predicted surface temperature is always lower for the depressed initial freezing temperatures than for product with normal initial freezing temperature. Further- more, the greater the initial freezing temperature depres- sion, the lower the surface temperature at any given time for same product size and convective heat transfer coeffi- cient. Figures 6.8, 6.9, and 6.10 illustrated that the temperature profile at temperatures above freezing did not differ significantly before commencement of freezing. As freezing begins, different temperature distribution patterns emerge. At normal freezing temperature, the center temperature remains virtually constant for an appre- ciable length of time as indicated by a plateau in Fig- ure 6.8. All the energy is being used for phase change rather than reduction of temperature. As initial freezing temperature is depressed, the time period during which the center temperature remains constant decreases until it is negligible when the freezing temperature is depressed to -9.4°C. This is due to reduced energy required for phase change and a larger prOportion of the energy is utilized 53 for reducing the center temperature of product. As the initial freezing temperature is depressed, the amount of water available for freezing decreases and less energy is required for latent heat. 6.2.2 Freezing Time Freezing time has been defined in Section 4.9 as the time required for the center temperature of product to reach the storage temperature of -l8°C. Table 6.1 shows the influence of freezing temperature depression on freez- ing time of product. I From Table 6.1, 6.2, and.6.3, it is obvious the freezing time is reduced as the initial freezing temperature is depressed. As the initial freezing temperature is depressed, the enthalpy or heat content of product is decreased (Section 6.1.6). Secondly, the peak apparent spe- cific heat of product also decreases drastically with the depression of initial freezing temperature (Section 6.1.5 and Figure 6.6). Removing the same amount of thermal energy from the product with normal freezing temperature and one with depressed freezing temperature results in the latter attaining a lower temperature more rapidly. This trans- lates into a shorter freezing time for product with depressed initial freezing temperatures. The magnitude of the reduction in freezing time increases as the convec- tive heat transfer coefficient decreases. One obvious 54 m.om m.mm m.cN vm.ml h.mm m.mh m.NN mm.ml c.5H o.mm m.a~ mH.a) m.o ~.ma m.a~ ma.~- a.o o.ooa m.a~ Gm.su ooa m.mm m.vw o.om vm.ml o.mm c.Nh m.mm mm.ol v.mH 0.0m m.hm mH.vl N m.m «.Hm m.~v mv.ml o o.ooa m.mv om.Hl om m.mm «.mm m.~v «N.ml H.mm m.oh m.hv mm.ml N.om m.mh m.mm mH.vl o.m o.Hm o.Hm mv.NI o OOH 0.5m om.Hl ov m OER. w U mcwumoum mafia mcwnooum mousse: musumuomfioa Ume\z A80. 9 cw mmcmcu w HcEHoz no a mafia mceuwmuh mcwnooum HmwuHcH n mmmcxowre 80 m mo mmocxoflcs ocean macho now needs mcwuomum omuowooumll.a.m mqm