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Thermo-physical properties of apples and prediction of freezing times Ramaswamy, Hosahalli Subrayasastri 1979

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THERMO-PHYSICAL PROPERTIES OF APPLES AND PREDICTION OF FREEZING TIMES by HOSAHALLI SUBRAYASASTRY^RAMASWAMY B.Sc., Bangalore U n i v e r s i t y , I n d i a , 1970 M.Sc. (Food Technology), FAO-IFTTC The U n i v e r s i t y o f Mysore, I n d i a , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Food Science We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December 1979 © H o s a h a l l i Subrayasastry |T<amaswamy In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . ( H . S . R A M A S W A M Y ) ^ , F o o d S c i e n c e D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 D a t e J a n u a r y 3 , 1 9 8 0 ABSTRACT In two v a r i e t i e s of apples, Golden Delicious and Granny Smith, the temperature dependence of d i f f e r e n t thermo-physical properties has been investigated. Detailed regress-ion equations are given to cover the variations of thermal conductivity, apparent s p e c i f i c heat and thermal d i f f u s i v i t y of apples with temperature both above and below the freezing point. Tissue density has been studied at four d i f f e r e n t temperatures. The thermo-physical properties determined i n t h i s study have been employed to predict the freezing times of apples under various conditions of freezing using d i f f e r e n t models reported in the l i t e r a t u r e . The freezing conditions included for both variety of apples were: f i v e freezing systems v i z . , freezing i n a i r at -21 to -25°C and at -28 to -30°C, freezing by immersion i n ethylene glycol at -18 to -20°C and -20 to -24°C and by immersion in l i q u i d nitrogen at -197°C; three container si z e s , v i z . , cans of size 300x407, 307x409 and 401x411; two i n i t i a l product temperatures, 16-25°C and 1-7°C; and two target temperatures, -10° and -18°C. Two types of pred i c t i o n methods were used, the ana-l y t i c a l methods of : Plank (194]) , Nagaoka et a l . (1955), I. I. R. (1972), Mellor (1976), Cleland and Earle . (1979b), and the numerical methods with constant as well as varying thermal properties. The predicted values of freezing times by the d i f f e r e n t models were compared with experimental values and the r e l a -t i v e merits of each model discussed. Based on an analysis of the pred i c t i o n errors, a modification of Plank's equation to give the l e a s t e r r o r was suggested as f o l l o w s : t = [0.3022 C 1 ( T i - T £) + L + 2.428 C 2 CT f - T ) ] f T P_\\ -[M i f c C T f V H k2 The mean o v e r a l l p r e d i c t i o n e r r o r of the suggested model was 6.64% which was l e s s than 5% beyond the e x p e r i -mental e r r o r o f 2.38%. i i TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES v i LIST OF FIGURES v i i i LIST OF SYMBOLS x i LIST OF APPENDICES x i v ACKNOWLEDGEMENT x v INTRODUCTION 1 LITERATURE REVIEW 5 D e f i n i t i o n of F r e e z i n g Time 5 P r e d i c t i o n Models 6 Plank's Model 7 Nagaoka M o d i f i c a t i o n 9 Levy M o d i f i c a t i o n 9 I.I.R. M o d i f i c a t i o n 10 M e l l o r M o d i f i c a t i o n 10 C l e l a n d and E a r l e M o d i f i c a t i o n 10 Gutschmidt Model 12 Mott's Procedure 12 i i i Page F o u r i e r Models 13 Newman's Equation 14 Tao's Charts 16 Numerical Methods 16 Surface Heat T r a n s f e r C o e f f i c i e n t 21 Thermo-Physical P r o p e r t i e s of Apples 22 EXPERIMENTAL M a t e r i a l 24 Methods 24 Thermal C o n d u c t i v i t y 24 Apparent S p e c i f i c Heat 26 Densit y 27 Thermal D i f f u s i v i t y 28 Moi s t u r e 28 Latent Heat 28 Texture 29 T o t a l S o l i d s and A c i d i t y 29 F r e e z i n g C o n d i t i o n s 29 Surface Heat T r a n s f e r C o e f f i c i e n t 31 F r e e z i n g Time P r e d i c t i o n s 32 RESULTS AND DISCUSSION 3 3 Thermo-Physical P r o p e r t i e s 33 i v Thermal C o n d u c t i v i t y Apparent S p e c i f i c Heat Density Thermal D i f f u s i v i t y Surface Heat T r a n s f e r C o e f f i c i e n t P r e d i c t i o n of F r e e z i n g Times Experimental F r e e z i n g Times The Zone of Maximum Ice C r y s t a l Formation. P r e d i c t i n g F r e e z i n g Times: A n a l y t i c a l Methods. Plank's Model I.I.R. M o d i f i c a t i o n Nagaoka M o d i f i c a t i o n M e l l o r M o d i f i c a t i o n General Considerations A Suggested M o d i f i c a t i o n P r e d i c t i n g F r e e z i n g Times: Numerical Methods . F i n i t e D i f f e r e n c e Scheme w i t h V a r i a b l e P r o p e r t i e s F i n i t e D i f f e r e n c e Scheme with Constant P r o p e r t i e s Constant Thermal Property Scheme V a r i a b l e Thermal Property Scheme CONCLUSIONS REFERENCES APPENDIX LIST OF TABLES Page Table 1 Physico-chemical p r o p e r t i e s of apples o du r i n g storage at 1-2 C 34 Table 2 Experimental c o n d i t i o n s and f r e e z i n g c h a r a c t e r i s t i c s of Golden D e l i c i o u s apples 53 Table 3 Experimental c o n d i t i o n s and f r e e z i n g char-a c t e r i s t i c s of Granny Smith apples 54 Table 4 Thermo-physical data f o r f r e e z i n g time computations 60 Table 5 Mean e r r o r s i n p r e d i c t i n g f r e e z i n g times of apples by d i f f e r e n t models under d i f f e r e n t c o n d i t i o n s of f r e e z i n g 62 Table 6 Comparison between the p r e d i c t e d f r e e z i n g times and p r e d i c t i o n e r r o r s of numerical methods and m o d i f i e d Plank's equation under d i f f e r e n t c o n d i t i o n s of f r e e z i n g of Golden D e l i c i o u s apples 78 v i Page T a b l e 7 C o m p a r i s o n b e t w e e n t h e p r e d i c t e d f r e e z i n g t i m e s and p r e d i c t i o n e r r o r s o f n u m e r i c a l m ethods and m o d i f i e d P l a n k ' s e q u a t i o n u n d e r d i f f e r e n t c o n d i t i o n s o f f r e e z i n g o f G r a n n y S m i t h a p p l e s 7 9 v i i LIST OF FIGURES Page Figu r e 1 Thermal c o n d u c t i v i t y of Golden D e l i c i o u s apples at v a r i o u s temperatures 36 F i g u r e 2 Thermal c o n d u c t i v i t y of Granny Smith apples at v a r i o u s temperatures 37 F i g u r e 3 Warming thermogram o f f r o z e n Golden D e l i c i o u s apple and of empty pan (blank) i n a Dupont DSC 42 F i g u r e 4 Mean apparent s p e c i f i c heat of apples at v a r i o u s temperatures 43 F i g u r e 5 F r e e z i n g curves f o r Golden D e l i c i o u s and Granny Smith apples under d i f f e r e n t c o n d i t i o n s i n a tinplate can of s i z e 300 x 407, with a product i n i t i a l temperature of 20- 23°C 55 F i g u r e 6 F r e e z i n g curves f o r Golden D e l i c i o u s and Granny Smith apples under d i f f e r e n t c o n d i t i o n s i n a tinplate can of s i z e 300 x 4 07, w i t h a product i n i t i a l temperature of 2-7°C 56 v i i i Page F i g u r e 7 V a r i a t i o n s i n the time taken to cross the zone of maximum i c e c r y s t a l forma-o t i o n (-1 to -5 C) with f r e e z i n g time to o reach -10 C, at two mean i n i t i a l temp-e r a t u r e s , 3.0 and 20.3°C 58 Fig u r e 8 Frequency histograms f o r p r e d i c t i o n e r r o r percentage i n Golden D e l i c i o u s apples u s i n g d i f f e r e n t models; Plank (1941), t p ; I.I.R. (1972), t j j Nagaoka et a l . (1955), t^jp", M e l l o r (1976), t ^ ; author's modi-f i c a t i o n , t g , under v a r i o u s c o n d i t i o n s : C l ( T i = 16 - 25°C, T c = -10°C), 02(1^ = 1-7°C, T c = -10°C), C3(,Ti = 16-25°C, T c = -18°C) and C4 (T^ = 1-7°C, T c = -18°C) ... 67 F i g u r e 9 Frequency histograms f o r p r e d i c t i o n e r r o r percentage i n Granny Smith apples u s i n g d i f f e r e n t models; Plank (1941), t p ; I.I.R. (1972), t p Nagaoka et a l . (1955), t N p ; M e l l o r (1976), t M ; author's m o d i f i c a t i o n , t g , under v a r i o u s c o n d i t i o n s : C1(T^ = 16-25°C, T c = -10°C), C2(T\ = 1-7°C, T c = -10°C), C3(T i = 16-25°C; T = -18°C) and C4(T. = 1-7°C, T c = -18°C) 68 i x Page F i g u r e 10 Frequency h i s t o g r a m s f o r p r e d i c t i o n e r r o r p e r c e n t a g e i n a p p l e s o f two v a r i e t i e s u s i n g d i f f e r e n t models; P l a n k (1941), t p ; I.I.R. (1972), t j ; Nagaoka e t a l . (1955), t N p ; M e l l o r (1976), t ^ ; a u t h o r ' s m o d i f i c a t i o n , t g , under v a r i o u s c o n d i t i o n s : C l (T\ = 16-25°C, T = -10°C), C2(T. = 1-7°C, T = ' c 1 c -10°C), C3CT± = 16-25°C, T c = -18°C) and C 4 ( T i = 1-7°C, T c = -18°C) 69 F i g u r e 11 R e l a t i o n s h i p between the e x p e r i m e n t a l f r e e z i n g time and the time p r e d i c t e d by u s i n g the suggested m o d i f i c a t i o n of P l a n k ' s e q u a t i o n 73 x LIST OF SYMBOLS A C r o s s s e c t i o n a l a r e a p e r p e n d i c u l a r t o heat f l o w , a c o n s t a n t i n eq. 19 B D i m e n s i o n l e s s number = hd/(2]<2) B i B i o t Number = k 2 / ( h d ) C S p e c i f i c heat c a p a c i t y , w i t h s u b s c r i p t s 1 and 2 f o r u n f r o z e n and f r o z e n s t a t e s , J/kgK° C In n u m e r i c a l methods, v o l u m e t r i c s p e c i f i c heat c a p a c i t y , J/m3K° D Co n s t a n t i n eq. 4 d Diameter o f a c y l i n d e r or t h i c k n e s s o f a s l a b , m dx V e l o c i t y o f the f r o z e n f r o n t , m/s AH Heat c o n t e n t or e n t h a l p y change , J/kg AT Temperature d i f f e r e n c e , C° At Time increment, s Ar Radius increment , m E-p C a l i b r a t i o n c o n s t a n t f o r DSC , (mg/rnin)- J/kgC ° e r f E r r o r f u n c t i o n a t Fo F o u r i e r ' s Number = — y d z G C o n s t a n t i n eq. 4 G D i m e n s i o n l e s s number determined by eq. 9 _ 2 h S u r f a c e heat t r a n s f e r c o e f f i c i e n t , W/m K° x i Time step superscript in f i n i t e d i f f e r e n c e methods K o s s o v i t c h Number = L / C 2 ( T f - T ) Thermal c o n d u c t i v i t y , with subscripts 1 and 2 r e s -p e c t i v e l y f o r unfrozen and f r o z e n s t a t e s , W/mK° Latent heat of f u s i o n of i c e , J/kg Length of a c y l i n d e r , m N a t u r a l l o g a r i t h m Mass (kg), moisture ( I ) or a number determined by R/Ar i n f i n i t e d i f f e r e n c e methods Space step subscript in f i n i t e d i f f e r e n c e methods Constant i n Plank's equation Plank's Number = C 1 ( T i - T f)/AH Quantity of heat, J Rate of heat flow, W Radius of a c y l i n d e r , m Dimensionless Number = A(^) Stefan's Number = C 2 ( T £ - T^)/AH Ambient temperature adjuce t to the can,°C Target temperature,°C F r e e z i n g point,°C I n i t i a l temperature,°C Temperature at the surface,°C Temperature at the center,°C Dimensionless number determined by equation 14. F r e e z i n g time, s Experimental f r e e z i n g time, h x i i t £ E f f e c t i v e f r e e z i n g time, s * f ( 10) F r e e z i n g time to reach -10°C, s t j F r e e z i n g time ( I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n , 1972), h tp F r e e z i n g time (Plank, 1941), h t^p F r e e z i n g time (Nagaoka et a l . 1955), h t N (-. F r e e z i n g time (Nagaoka et a l . 1955) y h tj^ F r e e z i n g time ( M e l l o r , 1976), h tg F r e e z i n g time (Suggested m o d i f i c a t i o n of Plank's e q u a t i o n ) , h V Volume, x Thickness of the i c e f r o n t , m a Thermal d i f f u s i v i t y with subscripts 1 and 2 r e s p e c t i v e l y f o r unfrozen and f r o z e n s t a t e s , m2/s $ Dimensionless number = Y^l(hd) = B i o t Number Y Dimensionless number = ^2^f " ^ a) ^ X Constant i n equations 13, 13a X^ F u n c t i o n of h and k 2 F u n c t i o n o f h and k 2 p Density with subscripts 1 and 2 r e s p e c t i v e l y f o r unfrozen and f r o z e n s t a t e s , kg/m x i i i Appendix 1 LIST OF APPENDICES Comparison between the experimental and p r e d i c t e d f r e e z i n g times: Golden D e l i c i o u s Page 95 Appendix 2 Comparison between the experimental and p r e d i c t e d f r e e z i n g times: Granny Smith 99 Appendix 3 A computer program f o r p r e d i c t i n g f r e e z -ing times of foods with v a r i a b l e thermal p r o p e r t i e s i n c y l i n d r i c a l c o n t a i n e r s . .. 103 Appendix 4 Schematic diagram of a c y l i n d e r f o r f i n i t e d i f f e r e n c e scheme 105 x i v ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. M.A. Tung, Research Supervisor, for his assistance and counsel during t h i s study. The advice of Professors W.D. Powrie, L.M. Staley, J. Vanderstoep and E.L. Watson i s sincerely appreciated. This research was financed in part by Canadian Commonwealth Fellowship and Scholarship Administration, Ottawa. 0 xv 1. INTRODUCTION Food preservation by freezing is becoming increas-ingly important as more foods i n larger quantities are being preserved, day after day, by freezing. Freezing i s a pro-cess of bringing down the temperature of the product below i t s freezing point to a temperature at which i t i s subse-quently stored. It has been generally recognized that the location and size of the ice crystals formed i s associated with the rate of freezing. In general, slow freezing results in the formation of large ice c r y s t a l s located i n extra-c e l l u l a r spaces while rapid freezing r e s u l t s i n small ice crystals i n both intra-and e x t r a - c e l l u l a r spaces (Fennema, 1966). Formation of ice c r y s t a l s , p a r t i c u l a r l y larger ones, damages the c e l l structure and thus the food material upon thawing w i l l have a poorer texture. This aspect has been reviewed by Fennema and Powrie (1964), Van Arsdel et a l . (1969), Tressler et a l . (1968) , and Fennema et a l . (1973). Food engineers dealing with freezing or defrosting of foods are often faced with the need to predict temperature history curves as well as freezing and thawing times. Therefore, the manner i n which the phase change takes place, the freezing times, rates of freezing and tempera-ture history of the material being frozen and subsequently stored are important i n the design and optimization of 2. processing equipment , and in exercizing control over the quality of the product' during freezing and subsequent storage. It has been a common pr a c t i c e , however, to r e l y on emperical experiences for these predictions. Over the years there have been numerous e f f o r t s to solve this problem. The complexity of the problem i s greatly enhanced by the dependence of the freezing times on d i f f e r e n t thermo-physical properties which often change during the freezing process. V a r i e t a l differences, a g r i c u l t u r a l prac-t i c e s , seasonal v a r i a t i o n s , growth locations, and other factors also influence these properties. This complexity has led to many assumptions and approximations. Hence, the accuracy of any prediction model depends on how close the experimental conditions match the assumptions. The model proposed by Plank (1941) i s one of the early and most widely used prediction models. This model, however, does not take into account the i n i t i a l super-heat or the l a t e r subcooling since i t assumes the material to be at the freezing point. Based on experimental results many modifications have been reported to include these factors (Nagaoka et a l . , 1955; Levy, 1958; Tanaka and Nishimoto, 1959, 1960, 1964; Cowell, 1967; Mott, 1964; International Institute of Refrigeration, 1972; Cleland and Earle, 1976, 1977, 1979a, 1979b). Another approach was to obtain solutions for the Fourier equation of heat conduction (Charm and Slavin, 1962). However, Cowell (1967) reported that t h i s method overestimates the freezing time when the Biot number (Bi) is less than 1. Tao (1967) developed charts for estimating the freezing times by numerically solving the Fourier equation. Two types of numerical f i n i t e difference schemes have been used. The f i r s t kind takes into account the i n i t i a l superheat as well as the convective boundary condition, but assumes that a l l the latent heat is released at a unique freezing temperature (Charm et a l . , 1972). However, foods do not exhibit sharp freezing points. The second way of approaching the problem i s to take into account the v a r i a t i o n of thermal conductivity and apparent s p e c i f i c heat with temperature thereby completely avoiding the phase change front (Comini et a l . , 1974). F i n i t e d i f f e r -ence schemes of th i s type are therefore expected to be more accurate. One other aspect of t h i s prediction time problem i s the lack of a standardized method for deter-mining the surface heat transfer c o e f f i c i e n t . Methods based on dependence of the surface f i l m conductance and the thermal d i f f u s i v i t y on the slope of the heating curve (Charm et a l . , 1972), cooling curve of a block of high thermal conductivity (Cowell and Namor, 1974; Earle, 1971), and f i n i t e difference method using a block of low thermal con-d u c t i v i t y (Cleland and Earle, 1976) have been suggested. The aim of the present research was to study the temperature dependence of the thermal properties of apples and examine the available prediction models for the accuracy of freezing time estimations in order to find an equation which would give the least prediction error. The variables included in the experiment are: variety, i n i t i a l temperature, freezing temperature, container size, the temperature to which the product i s ultimately cooled and the heat transfer c o e f f i c i e n t . LITERATURE REVIEW Freezing Time Definitions Lack of a consistent d e f i n i t i o n for freezing time i s one of the major problems in the published l i t e r a t u r e concerned with freezing of foodstuffs. This apparent lack of information i s caused by the fact that the temperature d i s t r i b u t i o n within the product during the freezing process varies s i g n i f i c a n t l y and therefore freezing time or rate has to be defined with respect to a given location and between two reference temperatures. The 'thermal center' or the location that cools most slowly i s commonly used as the reference point. Foods do not have a well defined freezing point unlike pure systems. Due to soluble components dispersed in the f l u i d s of foods, the latent heat released over a range of temperature (most of i t being released in the region -1 to -5°C, which is referred to as the zone of maximum ice c r y s t a l formation). Some authors use the duration of thi s phase change period as the freezing time while others (Brennan et a l . , 1976) define i t as the duration of the entire process including precooling, phase change and the subsequent cooling to the f i n a l temperature. The various methods that have been used by dif f e r e n t authors to express freezing rate have been reviewed 6. by Fennema and Powrie (1964). The 'thermal arrest time' (duration to cross the zone 0 to -5°C) has been shown to depend on the i n i t i a l product temperature (Long, 1955). Prediction Models The subject matter of freezing time prediction by d i f f e r e n t methods has been reviewed by Bakal and Hayakawa (1973), Brennan et a l . (1976), Charm (1978), Cleland and Earle (1976, 1977, 1979b), Heldman (1975), and Rebellato et a l . (1978). The prediction models are usually based on many assumptions. The body to be frozen is assumed to have a uniform i n i t i a l temperature and i s cooled by a constant temperature medium, thereby providing a uniform and constant surface heat transfer c o e f f i c i e n t between the cooling medium and the surface of the body. It i s also assumed in most models that the product w i l l have constant thermo-physical properties in the unfrozen and frozen states, and also possess a defined freezing point at which a l l the latent heat i s liberated. These assumptions enable the freezing process to be divided into three d i s t i n c t phases: the precooling period in which the temperature of the product i s lowered 7. from i t s i n i t i a l temperature (T\) to the f r e e z i n g p o i n t (T^.) , the phase change p e r i o d i n which the l a t e n t heat i s r e l e a s e d and a tempering p e r i o d i n which the temperature i s lowered from the f r e e z i n g p o i n t to the t a r g e t tempera-ture (T ). Another phenomenon observed d u r i n g the f r e e z i n g process i s the s u p e r c o o l i n g p e r i o d (Fennema and Powrie, 1964) i n which the temperature o f the product f a l l s w e l l below i t s f r e e z i n g p o i n t without the occurrence of f r e e z -i n g . F o l l o w i n g s u p e r c o o l i n g , the temperature i n c r e a s e s to the f r e e z i n g p o i n t and the normal f r e e z i n g process c o n t i n u e s . None of the f r e e z i n g time p r e d i c t i o n models takes t h i s s u p e r c o o l i n g p e r i o d i n t o c o n s i d e r a t i o n . Plank's model The model proposed by Plank (1941) i s one of the e a r l y and most widely used methods f o r f r e e z i n g time e s t i -mations. This model i s based on three b a s i c heat balance equations: Heat conduction: q = A ( T Q - T^) k 2 Heat c o n v e c t i o n : q = A n ( T a " T 0 ) Heat generated at the f r e e z i n g f r o n t * T dx q = ALp 2 ^ 8. where q i s the r a t e o f heat f l o w , x i s the t h i c k n e s s o f the i c e f r o n t , d x / d t , the v e l o c i t y o f the i c e f r o n t , A, a r e a o f c r o s s s e c t i o n p e r p e n d i c u l a r to the d i r e c t i o n o f heat f l o w , T a and T , the ambient and the s u r f a c e t e m p e r a t u r e s , T £, the f r e e z i n g p o i n t , L, the l a t e n t h e a t , p 2 and k 2 , the d e n s i t y and t h e r m a l c o n d u c t i v i t y o f t h e f r o z e n p r o d u c t and h, the s u r f a c e heat t r a n s f e r c o e f f i c i e n t . These t h r e e e q u a t i o n s are combined t o get the most g e n e r a l form o f P l a n k ' s e q u a t i o n ( P l a n k , 1941) t . — [— • sd] ( D (T f - T a) K k 2 where d i s the t h i c k n e s s o f a s l a b or d i a m e t e r o f a c y l i n d e r or s p h e r e , P and R are c o n s t a n t s depending on the p r o d u c t geometry. The v a l u e s o f P are 0.500, 0.250 and 0.167 f o r an i n f i n i t e s l a b , c y l i n d e r and sphere r e s p e c t i v e l y ; and the c o r r e s p o n d i n g v a l u e s o f R are 0.125, 0.0625 and 0.0417. A c h a r t f o r p r o v i d i n g P and R when a p p l i e d to a b r i c k or b l o c k geometry i s g i v e n by Ede (1949). P l a n k ' s model assumes t h a t the p r o d u c t i s i n i -t i a l l y a t i t s f r e e z i n g p o i n t and hence does not take i n t o account the p r e c o o l i n g or t empering p e r i o d . Hence the v a l u e p r e d i c t e d w i l l be g e n e r a l l y low when used under c o n d i t i o n s i n v o l v i n g p r e c o o l i n g or t e m p e r i n g p e r i o d s . 9. However, Ede (1949) and E a r l e and Fleming (1967) r e p o r t e d that t h i s formula g i v e s f a i r l y a ccurate e s t i m a t i o n of f r e e z -ing times. t i o n s have been r e p o r t e d to i n c l u d e the p r e c o o l i n g and tempering p e r i o d s i n Plank's equation. Nagaoka M o d i f i c a t i o n Nagaoka et a l . (1955) suggested the f o l l o w i n g m o d i f i c a -t i o n to Plank's equation: Based on experimental r e s u l t s many m o d i f i c a -t = [1 + 0.00445(T i-T £)] P 2 ' A H rPd + Rdjl (2) where AH = [ C 1 ( T i - T £ ) + L + C 2 ( T £ - T Q ) Levy M o d i f i c a t i o n Levy (1958) suggested that the L i n Plank's equation be r e p l a c e d by AH, the enthalpy change at the thermal center over the e n t i r e process to get the nominal f r e e z i n g time, t Then the e f f e c t i v e f r e e z i n g time i s c a l c u l a t e d as t e £ = t [1 + 0.0081 ( T i - T £ ) ] (3) 10. I.I.R. M o d i f i c a t i o n The I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n , 1972 recommended only r e p l a c i n g L i n Plank's equation by the enthalpy change over the e n t i r e process to get the f r e e z i n g time. M e l l o r m o d i f i c a t i o n M e l l o r (1976) suggested r e p l a c i n g L by a f a c t o r [j C 1 ( T i - T £) + L + j C2^-Tf " T a ^ i n P l a n k ' s equation to p r e d i c t the f r e e z i n g time. C l e l a n d and E a r l e M o d i f i c a t i o n s Plank's equation has been expressed i n terms of dimensionless numbers by Cowell (1967). F£ = D [ _ l + G ] (4) Ko B i where Fo i s the Fourier. Number = , B i , B i o t Number = and d 2 Ko is the K o s s o v i t c h Number = and D and G are C 2 ( T £ - T a) constants determined by the product geometry. 11. More r e c e n t l y , C l e l a n d and E a r l e (1976, 1979a, 1979b) e x p r e s s e d P l a n k ' s e q u a t i o n i n t h e d i m e n s i o n l e s s form F O = P < B I W + R ( s ^ } <5> C 2 ( T £ - T ) where Ste i s S t e f a n Number = . D e f i n i n g the AH s e n s i b l e heat d u r i n g the p r e c o o l i n g p e r i o d by a d i m e n s i o n l e s s C-, (T, - TV) number, P l a n k ' s Number, Pk = — L - — , the a u t h o r s AH suggested m o d i f i c a t i o n s o f the c o n s t a n t s P and R i n P l a n k ' s e q u a t i o n t o p r e d i c t the f r e e z i n g times more a c c u r a t e l y . S l a b : P = 0.5072 + 0.2018 Pk + Ste (0.3224 Pk + ° - ^ 2 5 + 0.0681) R = 0.1684 + Ste (0.2740 Pk - 0.0135) • C y l i n d e r : P = 0.3751 + 0.0999 Pk + Ste (0.4008 Pk + o ^ i o . 0 > 5 8 6 5 ) R = 0.0133 + Ste (0.0415 Pk + 0.3957) The f r e e z i n g times p r e d i c t e d , u s i n g t h e s e v a l u e s f o r P and R, were r e p o r t e d by the a u t h o r s t o be i n v e r y good agreement w i t h e x p e r i m e n t a l v a l u e s as w e l l as tho s e p r e d i c t e d 12. by f i n i t e d i f f e r e n c e n u m e r i c a l methods. C l e l a n d and E a r l e (1979a) a l s o found t h a t even a f t e r the above m o d i f i c a -t i o n s , the p r e d i c t e d v a l u e s f o r f r e e z i n g i n r e c t a n g u l a r b r i c k s were v e r y low. They r e p o r t e d , t h e r e f o r e , t h a t the assumptions made by P l a n k (1941) i n a r r i v i n g a t the geome-t r i c f a c t o r s a re u n j u s t i f i e d and hence the methods u s i n g these f a c t o r s (Nagaoka et a l . , 1955; M e l l o r , 1976) can c e r t a i n l y be expected to g i v e erronous r e s u l t s . The a u t h o r s have m o d i f i e d the e q u a t i o n s to f i n d P and R f u r t h e r , to c o v e r r e c t a n g u l a r b r i c k s . Gutschmidt E q u a t i o n For p r o d u c t s o f i r r e g u l a r shape Gutschmidt (1964) suggested the e q u a t i o n : AH V ( d + ^2, ( 6 ) (T f - T a ) k 2 A 2 h For o t h e r g e o m e t r i e s such as p a r a l l e l i p i p e d , r i g h t c i r -c u l a r cone and others,equations have been g i v e n by L o r e n t z e n and Rosvik (1960) and Tanaka and N i s h i m o t o (1959, 1960, 1964). M o t t ' s Procedure Mott (1964) d e v e l o p e d s e v e r a l t a b l e s , to get the t h e r m o - p h y s i c a l d a t a needed to use P l a n k ' s p r e d i c t i o n model, 13. by dimensional a n a l y s i s of experimental data which i n c l u d e d d i f f e r e n t product and package c h a r a c t e r i s t i c s and v a r i o u s c o n d i t i o n s of f r e e z i n g . In t h i s procedure, a f u n c t i o n a l r e l a t i o n s h i p among three dimensionless groups i s u t i l i z e d f o r f r e e z i n g time c a l c u l a t i o n . The d i f f e r e n t equations are given below. S - I L i l i - A ( d D ( 7 ) B = £ | (8) 2k 2 t h ( T f - T ) G = — - J (9) p Qd where S, B and G are the three dimensionless groups F o u r i e r Models A d i f f e r e n t approach i n s o l v i n g the p r e d i c t i o n time problem i s to o b t a i n s o l u t i o n s f o r Fourier's heat conduction equations under s u i t a b l e boundary c o n d i t i o n s . The v a l i d i t y of F o u r i e r ' s equation has been proven and t h i s equation has been widely used i n e n g i n e e r i n g s c i e n c e s . However, the s o l u t i o n s are r a t h e r complicated, and t h e r e f o r e , not many are a v a i l a b l e i n the p u b l i s h e d l i t e r a -t u r e . Carslaw and Jaeger (1959) and Muehlbauer and 14. Sunderland (1965) gave e x c e l l e n t reviews on the formulas f o r e s t i m a t i n g heat conduction i n a s o l i d when there i s a phase change i n the sample. Newman 's S o l u t i o n The Newman's s o l u t i o n p u b l i s h e d i n Carslaw and Jaeger (1959) u t i l i z e s a u n i d i m e n s i o n a l heat t r a n s f e r i n a s e m i - i n f i n i t e s l a b . /The assumptions which ho l d good f o r Plank's equation are made here a l s o . The p a r t i a l d i f f e r e n t i a l equations r e p r e s e n t i n g the temperature d i s -t r i b u t i o n i n the unfrozen and f r o z e n r e g i ons are repre-sented as: 8 2 T 1 x 9T 1 9x 7 " "o~ 9t~~ Unfrozen r e g i o n ( 1 0 ) 3 2 T 2 IT1 ~ a 2 3 t Frozen r e g i o n (11) The equation e x p r e s s i n g the heat f l u x between the f r o z e n and the unfrozen p o r t i o n which must be equal to the heat l i b e r a t e d at the f r e e z i n g f r o n t i s 9 T ? 8 T i 9x The Newman's s o l u t i o n u t i l i z e s s e v e r a l boundary c o n d i t i o n s such as 15. T x ( x , 0) = T. T^O, t) = T s ( t ) T 2 ( x , t) = T 1 ( x , t ) = T f 3T,(f, t) —±__£ = o Using these assumptions the equation expressing temperature as a f u n c t i o n of time and p o s i t i o n i n an i n f i n i t e s l a b i s g iven below. T t F T e r f 2 [ a 2 t ] Where T 2 = T £ , equation 13 reduces to x = 2 A ( a 2 t ) 1 / 2 (I3a) suggesting t h a t the l o c a t i o n of the f r e e z i n g f r o n t can be l i n e a r i z e d on a log l o g p l o t with a slope of 0.5 and A can be c a l c u l a t e d from the i n t e r c e p t (Bakal and Hayakawa, 1970) Charm and S l a v i n (1962) used equation 13 f o r c a l c u l a t i n g the f r e e z i n g time of cod f i l l e t s by u s i n g a m o d i f i e d 4 + h •2 h d ^1 t h i c k n e s s equal to (y + -==-). However, Cowell (1967) 16. r e p o r t e d that t h i s m o d i f i e d t h i c k n e s s r e s u l t s i n over-e s t i m a t i n g the f r e e z i n g time when the B i o t number i s l e s s than 1. T a o 1s Charts Tao (1967) developed c h a r t s f o r e s t i m a t i n g the f r e e z i n g times i n an i n f i n i t e s l a b , c y l i n d e r and a sphere by n u m e r i c a l l y s o l v i n g the F o u r i e r heat conduction equations using three dimensionless groups. t * = t k 2 ( T £ - T a ) / d 2 p 2 L (14) B = k 2/hd (15) Y = C 2 ( T £ - T a ) / L ' (16) The c h a r t s show a r e l a t i o n s h i p between t * and 3 at d i f f e r e n t values of Y. Numerical Methods The d i f f e r e n t methods d i s c u s s e d so far assume constant thermal p r o p e r t i e s . But the a c t u a l s i t u a t i o n r e q u i r e s s o l u t i o n of the equation of type 17. which i s a p a r t i a l d i f f e r e n t i a l e q u a t i o n of one dimen-s i o n a l heat c o n d u c t i o n i n a s l a b w i t h t h e r m a l d i f f u s i v i t y as a f u n c t i o n o f t e m p e r a t u r e . The more g e n e r a l way o f e x p r e s s i n g the r e l a t i o n s h i p i s as f o l l o w s f o r a s l a b , c y l i n d e r , and a sphere. 2 S l a b : C(T) | I = k(T) -8—J (18) fix C y l i n d e r (a-1) c m | T . |_ [ k [ T ) |I, t «k£rl |I (19) Sphere (a=2) The most common boundary c o n d i t i o n i s t h a t w i t h a s u r f a c e heat t r a n s f e r c o e f f i c i e n t , sometimes r e f e r r e d t o as Newton's law of c o o l i n g or as a boundary c o n d i t i o n o f the t h i r d k i n d t h a t uses the c o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t between the c o o l i n g medium and the s u r f a c e o f the p r o d u c t . T h i s i s r e p r e s e n t e d as h(T - T ) = [k(T) fl] (20) a s d x x = 0 f o r s l a b s h"(T a - T g) = [k(T) |I] r = d f Q r c y l i n d e r s (21) or s pheres 18. The s o l u t i o n of the above equations (17-21) are d i f f i c u l t without the use of numerical methods. Two types of numerical f i n i t e d i f f e r e n c e schemes have been used. The f i r s t type takes i n t o account the p r e c o o l i n g and tem-p e r i n g p e r i o d s as w e l l as the t h i r d k i nd of boundary con-d i t i o n but assumes a l l the l a t e n t heat to be r e l e a s e d at a unique f r e e z i n g temperature (Charm et a l . , 1972). However, foods do not e x h i b i t sharp f r e e z i n g p o i n t s , so the s o l u t i o n s of t h i s type depart s u b s t a n t i a l l y from the a c t u a l s i t u a t i o n . The second way that phase change i n f r e e z i n g foods can be accommodated i s to take i n t o account the v a r i a t i o n s o f thermal c o n d u c t i v i t y k ( T ) , and apparent s p e c i f i c heat c a p a c i t y C ( T ) . This completely avoids the need to d e f i n e a phase change f r o n t . The v e r s a t i l i t y of t h i s method has been demonstrated by Bonacina and Comini (1973), R e b e l l a t o et a l . (1978), C l e l a n d and E a r l e (1976, 1977, 1979a, 1979b). The f i n i t e element methods are more complex. For u n i d i m e n s i o n a l heat t r a n s f e r i t o f f e r s no d i s t i n c t advantage over the f i n i t e d i f f e r -ence schemes (C l e l a n d and E a r l e , 1979b). For a r a d i a l heat t r a n s f e r , a three t i m e - l e v e l f i n i t e d i f f e r e n c e scheme proposed by C l e l a n d and E a r l e (1979b) i s given below. 19. T i + 1 _ T i - 1 ri m m _ 1 r v i rr^- + ^ - T i + 1 , T i m 2 1 m+l/2 Um+l " m m+1 m m 2At 3(Ar) i - l _ T i - 1 . _ , i f T i + l _ T i + 1 !m+l m j Km-l/2 Um V l + T i - T i , + T i _ 1 - T1-"^) 1 + m m-l m m ' J k 1 m + 1 _ r p i + 1 + r j , i - T 1 •7 * o A m+1 m-l m+1 m-l 3mAr 2Ar + T 1 " ^ - T 1 " } ) (22) m+1 m-l' At the center (m = o ) , l i m i t I | I - ^ 1 (23) T i - 1 C 1 - 2 = — 7 [2k i + 1 / 7 ( T j + 1 - T j + 1 + T,1-2At 3 ( A r ) 2 m + 1 / 2 0 0 1 + T ^ 1 - T J " 1 ) ] (24) 20. i 9 T At the s u r f a c e (m = M) , h(T - T ) = k f , (-^r) , and a s w =MAr s e t t i n g k j J + 1 / 2 = hAr and T / J j + 1 = T &, T i + 1 _ T i - 1 r i M M _ 1 r, i r , T T i + 1 T i T ^ " - ^ M 2At 3 ( A r ) 2 M + 1 / 2 a M M " M k M - l / 2 ( T I M 1 " TM-1 + TiM " TM-1 + T x V 1 " T.VJ)] + h (3T - T . 1 / 1 M M-1-'J v a M T i i - T M " 1 ) / 3 R ( 2 5 ) where / 9 = i ( k i + k * ) (26) ni+1/2 2 m m+1 These n u m e r i c a l methods r e q u i r e the use o f a computer. Charm (1978), C o r d e l l and Webb (1972), Bonacina and Comini (1973), Fleming (1971), C l e l a n d and E a r l e (1977, 1979a, 1979b) have used computer programs to s o l v e the above 21. equations, some u s i n g constant thermal p r o p e r t i e s (Charm, 1978) while others u s i n g v a r i a b l e thermal p r o p e r t i e s . R e b e l l a t o et a l . (1978). extended the f i n i t e element a n a l y s i s to i n c l u d e i r r e g u l a r shaped products a l s o . A l b i n et a l . (1976, 1979) used Goodman's i n t e g r a l t e c h n i -que (Goodman, 1964) to s o l v e the p a r t i a l d i f f e r e n t i a l equa-t i o n s (17-21) u s i n g f o u r dimensionless numbers. Surface Heat T r a n s f e r C o e f f i c i e n t Determinations One of the major problems i n v o l v e d i n p r e d i c t i o n models i s the l a c k of a standard method f o r determining the s u r f a c e heat t r a n s f e r c o e f f i c i e n t . A commonly used approach i s the s o - c a l l e d heat p e n e t r a t i o n method where the c o o l i n g curve of a b l o c k of metal of high thermal c o n d u c t i v i t y i s found under the c o n d i t i o n s that w i l l apply i n the f r e e z i n g system. This method has been used by Cowell and Namor (1974) i n p l a t e f r e e z i n g and E a r l e (1971) i n a i r b l a s t f r e e z i n g . Charm (1971) suggested the use of an equation employed by B a l l and Olson (1957) d e s c r i b i n g the e f f e c t of s u r f a c e conductance and thermal d i f f u s i v i t y on the slope of the h e a t i n g curve, 2"I05 = A 2 + y 2 (28) o^f 1 1 where f i s the n e g a t i v e r e c i p r o c a l of the slope of the 22. h e a t i n g curve o b t a i n e d - by u s i n g a Jackson P l o t , \ 1 = f i r s t r o o t of cot X(1/2) = ( k 2 / E ) X y^ = f i r s t r o o t of J Q ( y r ) = y ( k 2 / h ) J-^(yr), 1 r e p r e s e n t s the the depth of a s l a b or c y l i n d e r , r = r a d i u s of c y l i n d e r . Use of the f i n i t e d i f f e r e n c e methods u s i n g g e l samples of thermal p r o p e r t i e s c l o s e to those of food m a t e r i a l s i s recommended by C l e l a n d and E a r l e (1976) . T h i s method i n v o l v e s the d e t e r m i n a t i o n of s u r f a c e temperatures and then use of an e x p l i c i t e f i n i t e d i f f e r e n c e scheme with the t h i r d k i nd of boundary c o n d i t i o n s to get an estimate of h at each step. Thermo-physical P r o p e r t i e s of Apples Very l i t t l e work has been p u b l i s h e d on the eva-l u a t i o n of thermo-physical p r o p e r t i e s or p r e d i c t i o n of f r e e z i n g times f o r apples. One of the e a r l i e s t r e p o r t s on the thermal c o n d u c t i v i t y of apples comes from Gane (1936) who gave values f o r thermal c o n d u c t i v i t y , apparent d e n s i t y , mean s p e c i f i c heat and thermal d i f f u s i v i t y at 15.6°C f o r apples, apple j u i c e , apple j u i c e c oncentrate and apple sauce. In a survey of thermal c o n d u c t i v i t y of f r u i t s and v e g e t a b l e s , Sweat (1974) r e p o r t e d values f o r d e n s i t y and thermal c o n d u c t i v i t y at 28°C f o r green and red apples 23. ( p = 790 and 840 kg/nT and k = 0.422 and 0.513 W/mK° resp e c t i v e l y ) . He also gave a regression equation express-ing the relationship between thermal conductivity and moisture content ( I wet basis) at room temperature. k = 0.148 + 0.00493 (% Moisture) (29) Lozano et a l . (1979) expressed the relationship between the moisture content ( f r a c t i o n , X, dry basis) and thermal conductivity of Granny Smith apples i n the following equation k = [0.283 - 0.256 exp(-0.206X)] x 1.731 W/mK° (30) Values obtained by Sweat. (1974) were s i g n i f i c a n t l y higher than those reported by Lozano et a l . (1979). Riedel (1951) has given tables for the enthalpy of apple j u i c e , apple juice concentrate and apple sauce at d i f f e r e n t temperatures. Riedel (1949) expressed a general relationship between k and moisture content at temperatures above freezing point by the quadratic equation (with temperature in F°) k x = [307 + 0.645T - 0.00105T2] [0.46 + 0.054 (% Moisture)] x I O - 3 BTU/h ftF° (31) 24. EXPERIMENTAL Material Two v a r i e t i e s of apples, v i z . , Golden Delicious and Granny Smith, obtained from the l o c a l market, were used in the study. The apples were stored at 1-2°C u n t i l use. Methods Thermal Conductivity [k(T)] Thermal conductivity was measured by the transient method using a thermal conductivity probe 3.81 cm long and 0.0813 cm diameter as described by Sweat and Haugh (1972). Apples were mechanically peeled and sliced into octets. The probe was inserted into a s l i c e in the longitudinal d i r e c t i o n , the s l i c e with probe was placed i n a long cl o s e l y f i t t i n g r e tort pouch (12 ym polyester/9 ym A l fo i l / 5 0 ym polypro-pylene) and clamped at both the ends to secure the po s i t i o n . The s l i c e s were then cooled to di f f e r e n t temperatures in a constant temperature bath, and held at least for 1 h,for e q u i l i b r a t i o n . Each measurement was made as follows: when the temperature of the sample was steady, a current of 160 mA was applied to the probe. The res u l t i n g temperature r i s e 25. was recorded as the thermocouple m i l l i v o l t output using a Digitec data logger (United Systems Corp.) The millivolt record thus obtained was used to provide temperatures occurring at one second intervals up to 40 s of the heating time. The time-temperature data [in the form InGp-) vs T] were 1 subjected to a linear regression through a least squares procedure using the University of B r i t i s h Columbia computer (Amdahl 470V/6-II). To avoid points which do not f a l l in a straight l i n e , the f i r s t 10 s of heating time was not used i n the c a l c u l a t i o n of the slope. Further, the data with a co r r e l a t i o n c o e f f i c i e n t of less than 0.90 were rejected. The thermal conductivity was calculated using the relationship Q In (^) k = - — (31a) 4TT (T 2 - T 1) where Q i s the power consumed by the probe heater, T^ and T 2 are the temperatures of the probe thermocouple at time t^ and t 2 respectively. With the calculated slope, the equation reduces to the form k = Q(slope) /4TT . Experiments were conducted in 3 re p l i c a t e s at d i f f e r e n t temperatures between 25 and -25°C. 26. Apparent S p e c i f i c Heat [C(T)] D i f f e r e n t i a l thermal analysis (DTA) i s a technique for recording the difference in temperature, AT, between a test substance and an inert reference material as samples of the two are warmed or cooled, at a constant rate. If the test substance i s thermally active, then the curve obtained by p l o t t i n g AT, against temperature shows irregu-l a r i t i e s or peaks or v a l l e y s . These peaks indicate the occurrence and measure the extent of energy-involving reactions, t r a n s i t i o n s or phase changes within the test sample. When techniques to measure these changes d i r e c t l y in energy units are available, the measurement i s referred to as d i f f e r e n t i a l scanning calorimetry (DSC). The d i f f e r e n t aspects of DSC and DTA have been discussed in d e t a i l in the book of MacKenzie (1970). The Dupont 900 DSC was used to obtain the warming thermograms. This instrument was c a l i b r a t e d to obtain the c a l i b r a t i o n constant, E^ ,, using zinc (purity 99.9991). Samples of apple tissues [8 to 14 mg] were cooled i n the DSC c e l l to -100°C using l i q u i d nitrogen. The cooling curves were not recorded. The samples were held for s u f f i c i e n t time (-^  to 1 h) to achieve e q u i l i b r a -t i o n . The warming thermograms were recorded while heating at 20C°/min. 27. The s p e c i f i c heat data were obtained as follows: Two empty pans were f i r s t warmed at 20C°/min on the DSC from -100 to 80°C to obtain a blank thermogram. Then the sample was placed i n one of the pans and the sample and reference pans were again warmed from -100 to 80°C. From the r e s u l t i n g thermograms of sample and blank, the apparent s p e c i f i c heat at any temperature i s calculated by measur-ing the AT values for sample and blank at that temperature, and using the following equation: (AT n, , + ATC , ) T E T ( C ) T = S a m P l e C32) (Heating Rate) (Mass) Density ( p ) For determining density, apples were cut into cylinders of 1.9 cm diameter and 1.6 cm length and weighed (M) i n d i v i d u a l l y . Density was then calculated using the equation: p = $ = ™ _ (33) TTd 1 The temperatures used were 2 and 25°C. For determining density in the frozen state, the apples were cut into brick shaped s l i c e s of approximate dimensions 1.0 x 1.0 x 3.0 cm , frozen to -20 and -35°C in freezer rooms, where they were 28. l e f t packaged for 1 wk for e q u i l i b r a t i o n . Six of the above s l i c e s were placed in a measuring cylinder previously tared with a sinker and weighed (M). To this, 50 ml of water at 2-3°C were added and the volume recorded. From th i s volume, the volume of 50 ml water plus the sinker was subtracted to y i e l d the volume (V) of the s l i c e s and the density was determined using equation 33. Eight r e p l i c a t e s were used at each temperature. Thermal d i f f u s i v i t y [ a(T)] Thermal d i f f u s i v i t y ( a ) was calculated using the equation, Moisture (M) Moisture was estimated by drying a known weight of the sample in a vacuum oven at 70°C for 24 h. Twelve repl i c a t e s were used each time. Latent Heat (L) Latent heat of fusion (L) was calculated based on the moisture content {% wet basis) using the equation: C a ) T = ( J | ) T (34) L = % Moisture 100 x 334.9 x 10 3 J/kg (35) Texture Texture of whole apples was measured using a Magness-Taylor puncture probe attached to an Instron tester. Two readings were taken on each f r u i t and eight f r u i t s were sampled at each time. The load (kg) required to puncture the apple tissue (with a thin section of the skin removed at the point of puncture before the measurement) was taken as an index of the texture. Total Soluble Solids and A c i d i t y Total soluble solids and t i t r a t a b l e a c i d i t y were determined using the juice obtained from two s l i c e s taken from eight d i f f e r e n t f r u i t s by the methods suggested by Ruck (1969). Moisture, t o t a l soluble s o l i d s , a c i d i t y and texture values were determined four times during the period of study for each var i e t y as a measure of the quality of the apples. Freezing Conditions Freezing experiments were carried out under five freezing systems, three container sizes, two d i s t i n c t i n i t i a l and f i n a l temperatures for both v a r i e t i e s of apples. The 30. d i f f e r e n t c o n d i t i o n s are d e s c r i b e d below: 1. F r e e z i n g Systems: a) b) c) d) e) Freezer room at -21 to -25°C Freezer room at -28 to -30°C Immersion i n ethylene g l y c o l (609o) at -18 to -20°C Immersion i n ethylene g l y c o l C100°6) at -20 to -24°C Immersion i n l i q u i d n i t r o g e n at - 197°C 2. Contai n e r : T i n - a) 300 x 407 p l a t e cans of three b) 307 x 409 d i f f e r e n t s i z e s c) 401 x 411 o 3. I n i t i a l Temperatures ( T i ) a) 16-25 C b) 1-7°C 4. F i n a l Temperature (T ) aj -10°C -18°C For f r e e z i n g , apples were peeled and s l i c e d i n t o o c t e t s , dipped i n a potassium m e t a b i s u l f i t e s o l u t i o n c o n t a i n i n g approximately 200 ppm of s u l f u r d i o x i d e . The s l i c e s were o r d e r l y and t i g h t l y packed i n t o the cans so as to minimize the extent of empty spaces i n s i d e the can. 31. The can ends were i n s u l a t e d with 3 cm t h i c k cardboard planks. A needle type Eklund copper-constantan thermo-couple was i n s e r t e d i n t o the can i n such a way that the t i p of the thermocouple was at the geometric c e n t e r of the can, embedded i n s i d e one of the apple s l i c e s . Temperature at the c e n t e r of the can as w e l l as that of the f r e e z i n g medium were recorded as the thermocouple m i l l i v o l t output u s i n g a D i g i t e c data logger (United Systems Corp.) at 2 min. i n t e r v a l s . These data were used to ev a l u a t e the o experimental f r e e z i n g times to c o o l the m a t e r i a l to -10 C and -18°C and a l s o to determine the time taken to cross the zone of maximum i c e c r y s t a l f o r m a t i o n (-1 to -5°C). Surface Heat T r a n s f e r C o e f f i c i e n t Surface heat t r a n s f e r c o e f f i c i e n t s were determined u s i n g Plank's equation as w e l l as by the method of Charm (1972). For u s i n g Plank's equation, the apples were packed i n t o cans as i n the f r e e z i n g experiments, s t o r e d i n a room at 1 to 2°C u n t i l equilibration was reached, and then frozen i n the d i f f e r e n t f r e e z i n g systems. Using the f r e e z i n g time to reach -10°C Uf(-_i0))> t n e heat t r a n s f e r c o e f f i c i e n t (h) was c a l c u l a t e d by the equation 32. K = Pd (-10) ( T f (36) Rd' When using Charm's (1972) method the slope of the heating curve was c a l c u l a t e d both from the f r e e z i n g data and by a l l o w i n g the f r o z e n sample to warm up i n the f r e e z i n g system. F r e e z i n g Time P r e d i c t i o n s F r e e z i n g time p r e d i c t i o n s were made us i n g the models suggested by Plank (1941), Nagaoka et a l . (1955), I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n (1972), M e l l o r (1976), C l e l a n d and E a r l e (1979b), and a suggested m o d i f i c a t i o n of Plank's equation. Numerical f i n i t e d i f -ference methods f o r a r a d i a l heat t r a n s f e r with a two time l e v e l scheme with constant as w e l l as v a r y i n g thermal p r o p e r t i e s were a l s o used to p r e d i c t the f r e e z i n g times. The schemes were s i m i l a r to that proposed by C l e l a n d and E a r l e (1979b) us i n g a three time l e v e l scheme. 33. RESULTS AND DISCUSSION Thermo-Physical P r o p e r t i e s i The d i f f e r e n t p h y s i c o - c h e m i c a l p r o p e r t i e s , moisture, t o t a l s o l i d s and the Magness-Taylor t e x t u r e values f o r the two v a r i e t i e s of apples d i d not change s i g n i f i c a n t l y d u r i n g the p e r i o d of study (Table 1). Golden D e l i c i o u s was, however, higher i n moisture content and lower i n t o t a l s o l i d s , a c i d i t y as w e l l as t e x t u r e value as compared to Granny Smith. From the observed moisture and t e x t u r e values i t was presumed that the q u a l i t y of the apples remained e s s e n t i a l l y constant throughout the p e r i o d of study (approximately two months). Thermal C o n d u c t i v i t y The probe method of c o n d u c t i v i t y measurement was found to g i v e accurate and c o n s i s t e n t r e s u l t s . The thermal c o n d u c t i v i t y values f o r a 0.4% agar g e l i n e i g h t d i f f e r e n t runs were found to be 1.48, 1.51, 1.57, 1.53, 1.54, 1.41, 1.43 and 1.45 W/mK° with a mean value of 1.49 W/mK (and a standard d e v i a t i o n of 0.057) as compared to the expected value of 1.52 W/mK°. Thus, the experimental value was found to be w i t h i n 2% of the a c t u a l v a l u e . With food samples, the accuracy of the probe methods have been r e p o r t e d to ±5% at temperatures above f r e e z i n g and Table 1. Physio-chemical properties of apples during storage at 1-2°C. PROPERTY Golden Delicious Granny Smith J u l 31 Aug 2 Aug 14 Sep 29 Aug 7 Aug 9 Aug 16 Oct 5 Moisture (%) 86.21 87.81 87.89 87. 52 85.46 85.64 86.31 85.40 Total Soluble Solids m b 11.55 11.00 10.40 10. 20 12.10 12.10 11.75 12.65 Acidit y (mg malic/100 g ) b 190 170 150 96 506 481 453 480 Texture (Magness-Taylor puncture probe, kg) c 2.45 2.32 2.34 2. 30 2.98 2.87 2.86 2.66 a average of nine re p l i c a t e s b average of duplicates from pooled pulp c average of 12 repl i c a t e s 35. +_]0% at temperatures below f r e e z i n g p o i n t (Sweat and Haugh, 1972). Vos (1955) r e p o r t e d that only when the expression 4cc t — 2 was g r e a t e r than 0.6 the e r r o r due to f i n i t e sample a s i z e was n o t i c e a b l e . In the present set of experiments - 7 2 a v a r i e d from 1.342 to 9.422 x 10 m / s ; t , the h e a t i n g time c o n s i d e r e d was 25 s; and a, the s h o r t e s t d i s t a n c e from the thermocouple to the boundary = 1.5 cm or .015 m. Hence the f a c t o r ranged from 0.0596 to 0.418. The r e f o r e , the boundary i n f l u e n c e was taken to be n e g l i g i b l e . The v a r i a t i o n s i n the thermal c o n d u c t i v i t y of Golden D e l i c i o u s and Granny Smith apples with temperature are shown i n F i g u r e s 1 and 2. A comparison of the two f i g u r e s i n d i c a t e s that the two v a r i e t i e s showed s i m i l a r conductivity v a r i a t i o n s with temperature. The thermal c o n d u c t i v i t y values of both the v a r i e t i e s showed a more c o n s i s t e n t v a r i a t i o n with temperature above f r e e z i n g p o i n t ( c o r r e -l a t i o n c o e f f i c i e n t of 0.735 and 0.815 r e s p e c t i v e l y f o r Golden D e l i c i o u s and Granny Smith r e s p e c t i v e l y ) . The l i n e a r r e g r e s s i o n equations f o r thermal c o n d u c t i v i t y at temperatures above f r e e z i n g were: Golden D e l i c i o u s : k(T) = 0.394 + 0.00212T (W/mK°), T > T £ (37) T E M P E R A T U R E [ °C ] F i g u r e 1. T h e r m a l c o n d u c t i v i t y o f G o l d e n D e l i c i o u s a p p l e s a t v a r i o u s t e m p e r a t u r e ' L I 38. Granny Smith: k(T) = 0.367 + 0.00250T (W/mK°), T > T £ (38) Below the f r e e z i n g p o i n t , thermal c o n d u c t i v i t y values showed a l a r g e r s c a t t e r i n g with temperature. L i n e a r r e g r e s s i o n a n a l y s i s of the data gave c o r r e l a t i o n c o e f f i c i e n t s of 0.394 and 0.650 f o r Golden D e l i c i o u s and Granny Smith r e s p e c t i v e l y , which were however s i g n i f i c a n t at 5% l e v e l . The l i n e a r r e g r e s s i o n equations under these c o n d i t i o n s were: Golden D e l i c i o u s : k(T) = 1.289 - 0.0095T (W/mK°), T < T £ (39) Granny Smith: k(T) = 1.066 - 0.0111T (W/mK°) , T <_ T £ (40) For o b t a i n i n g the mean values of thermal conduc-t i v i t y i n the unfrozen and f r o z e n r e g i o n , a l l the observed values at temperatures above and below the f r e e z i n g p o i n t (-1°C) r e s p e c t i v e l y were used. 39. The greater v a r i a b i l i t y observed in the values of thermal conductivities at temperatures below the freezing point are presumed to be related to the complexities of the freezing system, p a r t i c u l a r l y to the varying degree of ice crystallization at d i f f e r e n t temperatures. In Figures 1 and 2, the li n e s represent the regression equations 37 to 40. These data were u t i l i z e d in prediction of freezing time by f i n i t e difference numeri-cal methods. However, for use in other prediction models a mean value of 0.427 and 0.398 W/mK° respectively in the unfrozen state and 1.445 and 1.220 W/mK° respectively in the frozen state for Golden Delicious and Granny Smith were used. The observed values of thermal conductivity (mean values at temperatures above freezing point) were s l i g h t l y higher than those computed by equation 30, given by Lozano et a l . (1979). Using the equation for moisture contents (g ^ O/g dry matter) of 6.874 and 7.033 corres-ponding to Golden Delicious and Granny Smith, the thermal conductivities were found to be 0.383 and 0.368 W/mK°, as compared to observed values of 0.42 7 and 0.398 W/mK°. The values reported by Sweat (1974) were, however, s i g n i -f i c a n t l y higher (0.578 and 0.571 W/mK°, calculated from equation 29, and 0.513 W/mK° reported for red apples with 40. 84.91 moisture c o n t e n t ) . The equation of R i e d e l (1949) a l s o gave a h i g h e r value f o r apples (0.55 W/mK° at 20°C and 85% m o i s t u r e ) . There has been no p u b l i s h e d i n f o r m a t i o n on the thermal c o n d u c t i v i t y of apples at temperatures below the f r e e z i n g p o i n t . The values of thermal c o n d u c t i v i t i e s both below and above f r e e z i n g p o i n t f o r Golden D e l i c i o u s were more than those f o r Granny Smith. Since the slopes of the r e g r e s s i o n l i n e s f o r the two v a r i e t i e s were observed to be very c l o s e (equations 37 to 40) both at temperatures above and below f r e e z i n g , i t i s reasonable to assume that the d i f f e r -ences i n the thermal c o n d u c t i v i t y between the two v a r i e -t i e s at d i f f e r e n t temperatures a r i s e mainly because of the d i f f e r e n c e s i n moisture content. Golden D e l i c i o u s had a mean moisture content of 87.3% and Granny Smith, 85.8%. Based on a r e g r e s s i o n a n a l y s i s of the thermal c o n d u c t i v i t y data f o r the two v a r i e t i e s pooled together, an equation e x p r e s s i n g the dependence of thermal c o n d u c t i v i t y on moisture content and temperature can be w r i t t e n as f o l l o w s . k(T,M) = 0.667M (0.027 - 0.00038T) + 0.0213T - 1.13 W/mK°, T > T £ (41) k(T,M) = 0.667M (0.223 + 0.0016T) - 0.10T - 11.63 W/mK°, T < T £ (42) 41. Apparent S p e c i f i c Heat A t y p i c a l thermogram recorded u s i n g the Dupont DSC f o r a sample of Golden D e l i c i o u s apple and a blank are given i n F i g u r e 3 and the computed values of apparent s p e c i f i c heat f o r both the v a r i e t i e s as a f u n c t i o n of temp-er a t u r e are g i v e n i n F i g u r e 4. The values f o r temperatures above and below f r e e z i n g were obtained from d i f f e r e n t thermograms. Each p o i n t i n F i g u r e 4 r e p r e s e n t s the mean value of f o u r r e p l i c a t e s . The p a t t e r n of the apparent s p e c i f i c heat curves f o r both v a r i e t i e s were s i m i l a r . However, f o r Golden D e l i c i o u s the observed values were higher than those observed f o r Granny Smith as with thermal c o n d u c t i v i t y data. T h i s d i f f e r e n c e presumably a r i s e s from the d i f f e r e n c e s i n the moisture content of the two v a r i e t i e s . The r e g r e s s i o n equations are expressed, here, o on the b a s i s of four l e v e l s of temperature, below -25 C, -25 to -10°C, -10 to -1°C and above -1°C. The equations are: Golden D e l i c i o u s : C(T) = 3.36 + 0.0075T (kJ/kgC°), T > -1°C (43) C(T) = 2.18 - 1.484T (kJ/kgC°), -1 > T > -10°C (44) g o l d e n d e l i c i o u s 5 . 9 5 m g F i g u r e 3. Warming t h e r m o g r a m o f f r o z e n G o l d e n D e l i c i o u s a p p l e s and o f empty pan ( b l a n k ) i n a Dupont DSC. (AT s c a l e o f 2C°/unit r e f e r s t o o n l y t h e s m a l l p e a k a r e a a t t h e c e n t e r w h i l e 0.5C°/unit r e f e r s t o t h e r e s t o f t h e c u r v e ) . Figure 4. Mean apparent s p e c i f i c heat of apples at various temperatures. 44. C(T) = [24 .40 + 0 . 791T] x 10 3 (J/kgC°), -10 > T > -25°C (45) C(T) = [2.89+0.0138T] x 10 3 (J/kgC°), T < -25°C (46) Granny Smith: C(T) = [3.40+0.0049T] x 10 3 (J/kgC°), T > -1°C (47) C(T) = [2.65 = 1.421T] x 10 3 (J/kgC°) , -1 > T>-10°C (48) C(T) = [24.93+0.760T] x 10 3 (J/kgC°), -10 > T>-25°C (49) C(T) = [2.50 + 0.0118T] x 10 3(J/kgC°), T < -25°C (50) These equations expressing the apparent s p e c i f i c heat as a function of temperature are used in the estimation of freezing time by f i n i t e difference numerical methods. For the purpose of freezing time computations by other models, the mean value of apparent s p e c i f i c heat between 20 and 60°C was taken for temperatures above the freezing point and the mean value between -30 and -80°C was taken for temperatures below the freezing point. At temperatures between 0 and -25°C, the effect of the release of latent heat was re f l e c t e d in the apparent s p e c i f i c heat curve 45. and hence were not used for cal c u l a t i n g the mean values. The mean s p e c i f i c heats were 1.946 and 1.678 kJ/kgK° below the freezing point and 3.690 and 3.578 kJ/kgK° above the freezing point respectively, for Golden Delicious and Granny Smith apples. The apparent s p e c i f i c heat above freezing point was about 1.9 to 2.1 times more than that below the freezing point. There is not much information available in the published l i t e r a t u r e regarding the speci-f i c heat data for apples. Gane (1936) reported a value of 3.768 kJ/kgK° for apples without mentioning the mois-ture content. Ordinanz (1946) gave a range of s p e c i f i c heat for apples with moisture contents between 75-85% as 3.73 - 4.02 kJ/kgK°. It has been generally recognized that the speci-f i c heat in foods containing high moisture can be e s t i -mated using a simple relationship (at temperatures above freezing point) C = ~Q x 0.8 (4187) + 0. 2 (4187) J/kgK° (51) which gives a value of 3.76 and 3.71 kJ/kgK° for Golden Delicious and Granny Smith with 87.31 and 85.8% moisture content respectively. The observed values were s l i g h t l y lower than the values reported in l i t e r a t u r e . The 46. observed values of apparent s p e c i f i c heats of apples were consistent with the observations of Short and B a r t l e t t (1944), at temperatures below -7°C. Short and B a r t l e t t (1944), however, reported a constant value of 3.73 kJ/kgK° at temperatures above freezing point. Density For determination of density, the methods used were d i f f e r e n t for frozen and unfrozen samples. For the unfrozen material the method was straight forward. There was no d i f f i c u l t y i n cutting the apples into cylinders using a cork borer of known internal diameter and then to cut the cylinders to a known length using two spaced knives. The r e p r o d u c i b i l i t y of the results was good. This method was not suitable for frozen samples because of the d i f f i -c u lty in cutting the samples into proper size. The modi-f i c a t i o n suggested to accommodate the frozen samples is simple enough. The main disadvantage of the method was that the temperature of material did not remain constant over the period of measurement, since the volume displacement was measured using water at 1-2°C. The experi-ment needed only a few seconds and the temperature change estimated during that time was less than 5C°. Hence the temperature of measurement could be taken as +_3C° of the 47. specified temperature. This method was not recommended for use with unfrozen samples because of the porosity of the apple s l i c e s . In the frozen state, however, because of the ice formation, the structure would be harder and less porous. The mean values of density for Golden Delicious and Granny Smith respectively were 843 and 837 kg/m3 at 25°C, 847 and 820 kg/m3 at 2°C, 785 and 789 kg/m3 at -20°C and 791 and 787 kg/m3 at -35°C. The values for the two v a r i e t i e s were more comparable in the frozen state than i n the unfrozen state. The changes in the density below and above freez-ing point with respect to temperatures were not s i g n i -f i c a n t . However, the density in the frozen state was approximately 5.2 to 6.8% less than the density in the unfrozen state. For the purpose of computation of freez-ing times by d i f f e r e n t models a mean value of 845 and 788 3 3 kg/m for Golden Delicious and 829 and 786 kg/m for Granny Smith were taken for temperatures above and below freezing point respectively. Thermal D i f f u s i v i t y Thermal d i f f u s i v i t y of apples was calculated using the relationship given i n the equation 34. The mean values of thermal d i f f u s i v i t y above and below freezing 48. p o i n t r e s p e c t i v e l y were 1.371 x 10 ' and 9.430 x 10 ' m z/s - 7 - 7 f o r Golden D e l i c i o u s and 1. 342 x 10 and 9.257 x 10 m / s f o r Granny Smith. The mean value r e p o r t e d by Gane o - 7 (1936) f o r unfrozen apples (0 to 32°C ) was 1.265 x 10 2 m / s . Thermal d i f f u s i v i t y of f r o z e n apples was about 6.9 times that of unfrozen samples. Using equations (37-40, 43-50) which express thermal c o n d u c t i v i t y and s p e c i f i c heat as f u n c t i o n s of temperature, the v a r i a t i o n s of thermal d i f f u s i v i t y with temperature (assuming d e n s i t y of the m a t e r i a l to be constant i n unfrozen and f r o z e n s t a t e s ) are shown i n the f o l l o w i n g equations Golden D e l i c i o u s : a(T) (0.00278T+1.389) x 1 0 " 7 (m 2/s) T > T £ (52) a(T) (-0.109T + 5 .085) x 10" 7 (m 2/s) T < T f (53) Granny Smith: a(T) (0.00556T+1.309) x 10" 7(m 2/s) T > T £ (54) a(T) (-0.130T + 4.745) x 1Q~ ( m 2 / ) T < T £ (55) 49. These equations (52-55) assume the l a t e n t heat to be r e l e a s e d at the f r e e z i n g p o i n t . Hence the apparent s p e c i -f i c heat at temperatures between -30 to -80°C was c o n s i -dered to cover the whole f r o z e n r e g i o n . To get the more r e a l i s t i c approach o f l a t e n t heat r e l e a s e over a range of temperature, the apparent s p e c i f i c heat between -1 to -30°C which r e f l e c t s the l a t e n t heat e f f e c t had to be c o n s i d e r e d . The r e g r e s s i o n equations f o r thermal d i f f u s i v i t y depen-dence on temperature i n t h i s r e g i o n are given below: Golden D e l i c i o u s a(T) = (0 .437T + a (T) = (-0.187T Granny Smith a(T) = (0.339T + a(T) = (-0.123T 4.367) x 10" 7 (m 2/s) -1.215 x 10~ 7 (m 2/s) 3.656) x 10" 7 (m 2/s) - 0.603) x 10~ 7 (m 2/s) T £ > T > -10°C (56) -10 > T > -25°C (57) T f > T > -10°C (58) -10 > T > -25°C (59) Equations 52 and 54 hold good f o r T > T f and 53 and 55 f o r T < -25°C to complete the spectrum of temperature -80 to 60°C. 50. Surface Heat T r a n s f e r C o e f f i c i e n t Two methods were t r i e d f o r determining s u r f a c e heat t r a n s f e r c o e f f i c i e n t s a s s o c i a t e d with the f r e e z i n g systems. The method suggested by Charm (1972) i n v o l v e d warming up of a m a t e r i a l of known thermal p r o p e r t i e s i n the f r e e z i n g system, thus n e c e s s i t a t i n g c o o l i n g of the t e s t m a t e r i a l below the temperature of the f r e e z i n g system. Hence t h i s was not s u i t a b l e f o r use i n determining the h a s s o c i a t e d w i t h l i q u i d n i t r o g e n . A l t e r n a t i v e l y , the slope could be determined from the l a t t e r p a r t of the f r e e z i n g data. Experiments c a r r i e d out i n two systems ( f r e e z i n g i n o o a i r at -21 C and immersion i n ethylene g l y c o l at -18 C) showed that the r e c i p r o c a l s of the slopes c a l c u l a t e d u s i n g the warming and f r e e z i n g data were f a i r l y c l o s e . F u r t h e r , the mean value of s u r f a c e heat t r a n s f e r c o e f f i c i e n t s f o r the system of f r e e z i n g i n ethylene g l y c o l (100%) at -20 to -24°C c a l c u l a t e d by the method of Charm (1972) was 54.06 2 W/m K° and, by u s i n g Plank's formula (equation 35) was 2 55.59 W/m K°. Since these two methods gave s i m i l a r r e s u l t s , f o r other f r e e z i n g systems, Plank's method was used. Measurement of s u r f a c e temperature was not attempted because of the non-homogeneity of the apple pack to have a uniform c o n t a c t at the s u r f a c e . Hence, the method suggested by C l e l a n d and E a r l e (1976) was not employed. 51. The mean values of four to e i g h t r e p l i c a t e s of the s u r f a c e heat t r a n s f e r c o e f f i c i e n t s f o r the d i f f e r e n t f r e e z i n g systems were as f o l l o w s : Immersion i n ethylene g l y c o l (100%) at -20 to -24°C, 55.59 W / m 2 K ° ; immersion i n ethylene g l y c o l (60%) at -18 to -20°C, 59.68 W / m 2 K ° ; f r e e z i n g i n a i r at -21 to -25°C, 17.83 W / m 2 K ° ; f r e e z i n g i n a i r at -28 to -30°C, 13.85 W / m 2 K ° ; and immersion i n l i q u i d n i t r o g e n at -197°C, 68.42 W / m 2 K ° . The heat t r a n s f e r c o e f f i c i e n t i n 100% ethylene g l y c o l was r e l a t i v e l y s m a l l compared to that f o r 60% ethylene g l y c o l although the f r e e z i n g temperature was lower i n the l a t t e r . T h i s was probably due to higher v i s -c o s i t y a s s o c i a t e d with 100% ethylene g l y c o l at lower temperatures. A l s o the heat t r a n s f e r c o e f f i c i e n t asso-c i a t e d with the f r e e z e r room at -21 to -25°C was higher than t h a t provided in the f r e e z e r room at -28 to -30°C because of a s l i g h t l y h i g h e r a i r v e l o c i t y i n the former. P r e d i c t i o n of F r e e z i n g Times Experimental F r e e z i n g Times T h i r t y three f r e e z i n g experiments were conducted w i t h each v a r i e t y of apples. The experimental d e s i g n was set up to cover a wide range of conditions that are 52. commonly encountered i n food f r e e z i n g . The c o o l i n g medium temperature was v a r i e d from -18 to -30°C (and -197°C i n o l i q u i d n i t r o g e n ) , i n i t i a l temperature from 1 to 25 C, d i a -meter of the c o n t a i n e r 0.076 to 0.103m, s u r f a c e heat t r a n s -f e r c o e f f i c i e n t from 13.85 to 68.42 W / m 2 K ° . Tables 2 and 3 show the experimental c o n d i t i o n s f o r each run, the experimental f r e e z i n g times to reach -10 C and -18°C from the onset of c o o l i n g and a l s o the time taken to cross the zone of maximum i c e c r y s t a l formation (-1 o to -5 CJ m each run. Based on f o r t y f o u r r e p l i c a t e s o f d u p l i c a t e values under the d i f f e r e n t experimental c o n d i t i o n s shown i n Tables 2 and 3, the mean experimental e r r o r was e s t i -mated to be 2.38%. T h i s low experimental e r r o r r e f l e c t e d c o n s i d e r a b l e u n i f o r m i t y i n the method of packing the apples i n t o the can. The f r e e z i n g curves f o r the two v a r i e t i e s of apples i n 300 x 407 cans i n f i v e d i f f e r e n t f r e e z i n g sys-tems are given i n F i g u r e 5. The i n i t i a l temperature of o the apples v a r i e d from 20 to 23 C. F r e e z i n g curves f o r the two v a r i e t i e s under s i m i l a r c o n d i t i o n s w i t h an i n i t i a l o temperature of 3.5 to 7.0 C are given i n F i g u r e 6. 53. Table 2. Experimental data for freezing of Golden Delicious apples Code Freezing Conditions Can Size * 1 C O 'a (°C) Freezing C h a r a c t e r i s t i c s Time to reach Time to reach Time to cross Mode _ 1 0 o C _ 1 8 . c _ 5 < > c (h) y (h) ( h ) GDI 300x407 21. 0 -20 I GD2 300x407 21. 0 -20 I GD3 300x407 21. 0 -27 A GD4 300x407 21. 0 -27 A GD5 300x407 21. 0 -18 I GD6 300x407 21. 0 -18 I GD7 300x407 23. 0 -21 A GD8 300x407 23. 0 -21 A GD9 300x407 20. 0 -197 L GD10 300x407 20. 0 -197 L GDII 300x407 4 . 5 -24 I GD12 300x407 5. 5 -24 I GD13 300x407 4. 0 -29 A GD14 300x407 4. .0 -29 A GD15 300x407 3. , 5 -20 I GD16 300x407 3. .5 -20 I GD17 300x407 5. .0 -22 A GD18 300x407 7. .0 -22 A GD19 300x407 6. ,0 -197 L GD20 307x409 18. .0 -23 I GD21 307x409 17, .0 -23 I GD22 307x409 18. .0 -22 A GD23 307x409 18 .0 -22 A GD24 307x409 18 .5 -197 L GD25 307x409 2 .0 -23 I GD26 307x409 2 .0 -24 A GD27 . 307x409 2 .0 -197 L GD28 401x411 19 .0 -20 I GD29 401x411 16 .0 -23 A GD30 401x411 19 .0 -197 L GD31 401x411 2 .0 -22 I GD32 401x411 1 .0 -25 A GD33 401x411 2 .0 -197 L 03 06 13 13 47 27 25 47 0.242 0.23 77 70 47 53 13 23 20 33 0.193 73 83 37 30 0.325 2.33 5.47 0.267 3.86 6.36 0.360 ,60 .97 2.33 2.33 5.57 5.63 .93 .93 ,08 .25 0.250 0.23 95 87 90 97 50 53 20 97 0.205 3.00 3.07 6.03 5.90 0.328 60 10 0.268 27 13 0.310 0.408 3.97 6.06 0.340 0.37 0.40 67 57 00 13 83 05 0.013 0.012 0.90 0.74 3.20 3.20 1.27 ,30 .53 .57 0.062 23 33 ,13 ,20 0.033 70 97 0.072 2.00 2.90 0.032 ,23 .10 0.095 I = Immersion i n ethylene g l y c o l A = A i r L = L i q u i d Nitrogen Immersion 5 4 . T a b l e 3. E x p e r i m e n t a l d a t a f o r f r e e z i n g o f G r a n n y S m i t h a p p l e s . Freezing Conditions Code Can Size T i T a Mode CC) CC) GS1 300x407 24. 5 -22.0 I GS2 300x407 24. 5 -22.0 I GS3 •300x407 24. 5 -28.0 A GS4 300x407 24. 5 -28.0 A GS5 300x407 23. 0 -18.0 I GS6 300x407 23. 0 -18.0 I GS7 300x407 23. 5 -21.0 A GS8 300x407 23. . 5 -21.0 A GS9 300x407 23. ,5 -197 L GS10 300x407 23. .5 -197 L GS11 300x407 2. .0 -21.0 I GS12 300x407 2. .0 -21.0 I GS13 300x407 1, .5 -30.0 A GS14 300x407 1. .5 -30.0 A GS15 300x407 1. .5 -20.0 I GS16 300x407 1 . 5 -20.0 I GS17 300x407 3 .0 -22.0 A GS18 300x407 3 .0 -22.0 A GS19 300x407 4 .0 -197 L GS20 307x409 16 .0 -22.0 I GS21 307x409 18 .0 -22.0 I GS22 307x409 18 .5 -22.0 A GS23 307x409 18 . 5 -22.0 A GS24 307x409 18 .5 -197 L GS25 307x409 2 .0 -23.0 I GS26 307x409 2 .0 -24.0 A GS27 307x409 4 .5 -197 L GS28 401x411 19 .0 -20.0 I GS29 401x411 21 .0 -23.0 A GS30 401x411 19 .0 -197 L GS31 401x411 2 .0 -22.0 I GS32 401x411 3 .0 . -25.0 A GS33 401x411 2 .0 -197 L Freezing C h a r a c t e r i s t i c s Time to reach Time to reach Time to cross -10°C (h) -18°C (h) -1° to -5°C (h) 2.20 2.43 0.43 2.10 2.33 0.33 4.70 5.20 2.60 4.67 5.17 2.60 2.30 2.93 x 1.07 2.37 2.97 1.10 5.30 6.43 2.25 4.80 5.93 1.93 0.215 0.222 0.017 0.260 0.272 0.025 2.17 2.47 1.50 2.17 2.50 1.47 3.50 3.77 2.63 4.03 4.43 2.90 2.30 2.73 1.33 2.17 2.60 1.50 4.00 4.83 2.80 4.47 5.40 2.93 0.198 0.208 0.063 2.90 3.26 1.53 2.90 3.23 1.50 5.53 6.33 2.97 5.70 6.37 3.43 0.310 0.322 0.029 2.33 2.53 1.67 5.44 6.00 3.73 0.270 0.302 0.075 4.00 4.33 2.03 6.56 7.70 3.33 0.380 0.419 0.035 3.73 4.20 2.56 6.17 7.10 3.83 0.330 0.362 0.100 I = Immerson in ethylene g l y c o l A = A i r L = L i q u i d Nitrogen Immersion 5 5 . Figure 5. Freezing curves for Golden Delicious and Granny Smith apples under d i f f e r e n t condi-tions in a tinplate can of size 300x407 , with a product i n i t i a l temperature of 20-23°C. (LN, Immersion i n l i q u i d nitrogen at -197°C, II, 12, immersion in ethylene glycol at -20 and -18°C, Al and A2, freezing in a i r at -27 and -23°C respectively.) 56. 1 0 • LN o I 1 • I 2 o A l v A 2 granny smith 1 0 - 1 0 - 2 0 golden delicious 0 1 2 3 4 T I M E [ H O U R S J Figure 6. F r e e z i n g curves f o r Golden D e l i c i o u s and Granny Smith apples under d i f f e r e n t c o n d i t i o n s i n a t i n -plate can of s i z e 300x407 wi t h a product i n i t i a l temperature o f 2-7°C . (LN, l i q u i d n i t r o g e n f r e e z i n g at -197°C, I I , 12, immersion i n ethylene g l y c o l at -21 and -20°C, A l and A2, f r e e z i n g i n a i r at -30 and -22°C r e s p e c t i v e l y ) . 57. The Zone of Maximum Ice Crystal Formation A comparison of Figures 5 and 6 and Tables 2 and 3 indicates that the time required to cross the zone of temperature of the freezing medium, heat transfer c o e f f i -cient as well as the i n i t i a l temperature of apples. For higher i n i t i a l temperature, the time taken to cross the zone was shorter provided the freezing system remained the same This time has been termed the 'thermal arrest time' because the temperature i n th i s zone changes more slowly with time or is arrested until the latent heat i s released. This finding is in agreement with the findings of Long (1955) on the freezing of f i s h . In order to provide a better com-parison, the 'thermal arrest time' has been plotted against o the freezing time to reach -10 C (a function of freezing temperature as well as the heat transfer c o e f f i c i e n t ) at two mean i n i t i a l temperatures, 20.3°C and 3.0°C in Figure 7, for the data of two v a r i e t i e s pooled. The regression equations showed very high li n e a r correlations (0.97 at 20°C and 0.99 at 3°C). The equations were: maximum ice c r y s t a l formation (-1 to -5°C) depended on the At 3 C: t (-1 to -5) 3600 £(-10) - 0.12 (h) (60) At 20°C: t 0.57 (-1 to -5) 3600 f(-10) - 0.28 (h) (61) 58. m i i to to O U 0 0 . Tj= 20.3 °C • T; = 3.0 °C 2 3 4 5 F R E E Z I N G T I M E [h] F i g u r e 7. V a r i a t i o n s i n the time taken to cross the zone of maximum i c e c r y s t a l formation (-1 to -5 C) with f r e e z i n g time to reach -10°C, oat two mean i n i t i a l temperatures, 3.0 and 20.3 C. 59. P r e d i c t e d Freezing'Times: A n a l y t i c a l Methods F r e e z i n g times were c a l c u l a t e d u s i n g the d i f f e r e n t methods proposed i n the l i t e r a t u r e and the r e s u l t s com-pared with the experimental f r e e z i n g time to c a l c u l a t e the percentage d i f f e r e n c e between the two which i s termed as 'the p r e d i c t i o n e r r o r ' of the p a r t i c u l a r model. The methods i n v e s t i g a t e d were d i v i d e d i n t o two groups, those r e q u i r i n g numerical e v a l u a t i o n by a computer and those r e q u i r i n g o n l y a simple c a l c u l a t i o n . For a l l p r e d i c t i o n models i n v o l v i n g constant thermo-p h y s i c a l p r o p e r t i e s , the data obtained from p a r t A o f t h i s i n v e s t i g a t i o n were made use o f . These values have been summarized i n Table 4. Most of the p r e d i c t i o n models (other than the numerical a n a l y s i s methods) are based on Plank's equation. The p r e d i c t e d f r e e z i n g times by models proposed by Plank (1941), t p >Nagaoka et a l . (1955), 't^p, t N C , I.I.R. (1972), t j , and M e l l o r (1976) , t ^ , along with the p r e d i c t i o n e r r o r s are summarized i n Appendix 1 and 2 f o r the v a r i e t i e s Golden D e l i c i o u s and Granny Smith r e s p e c t i v e l y . When the pooled data from a l l the experimental c o n d i t i o n s f o r the two v a r i e t i e s of apples were analyzed, the mean p r e d i c t i o n e r r o r s ( t a k i n g only the ab s o l u t e value 60. Table 4. Thermo-physical data for freezing time computations. Parameter Goden Delicious Granny Smith Unfrozen Frozen Unfrozen Frozen Thermal conductivity (W/mk°) a S p e c i f i c heat x 10 CJ/kgK°) b Density (kg/m 3)xl0 2 Thermal d i f f u s i v i t y (m2/s x IO" 7) d Moisture (%je Latent heat (J/kg x 10 3) f g Freezing point (°C) Surface f i l m conduc-tance (W/m 2K°) n I n i t i a l temperature (°C) Final temperature (°C) Ambient temperature (°C) Can Size (Diameter) (m) 0. 427 3.690 8.450 1. 371 1.445 1.946 7.880 9.430 81.3 292 .4 -1.0 12.7-68.4 1-25 -10, -18 -18 to -30 0.076-0.103 0.398 3.578 8.290 1.342 1.220 1.678 7.860 9.257 85.8 287 .4 -1.0 12.7-68.4 1-25 -10, -18 -18 to -30 0.076-0.103 a average of b average of c average of d calculated e average of f calculated g taken from h average of three re p l i c a t e s at d i f f e r e n t temperatures four r e p l i c a t e s at d i f f e r e n t temperatures eight r e p l i c a t e s at 4 temperatures from a. b and c 36 re p l i c a t e s from latent heat freezing curves 4-8 re p l i c a t e s moisture 100 x 334.9 kJ/kg 61. of the e r r o r s ) i n using the d i f f e r e n t models were: tp, 18.2%; t j , 9.0%; t N p , 17.0%; t N C , 12.9% and t M , 13.9%. t^p i s the f r e e z i n g time i n hours by Nagaoka's model c a l -c u l a t e d u s i n g the Br i t i s h units and t ^ , the same us i n g the M e t r i c u n i t s . The above-mentioned v a l u e s , however, do not represent the true performance of the d i f f e r e n t p r e d i c t i o n models i n t h e i r c a p a c i t y to p r e d i c t the f r e e z i n g times under the d i f f e r e n t c o n d i t i o n s of f r e e z i n g . In Table 5, the ob s e r v a t i o n s have been grouped i n t o four general f r e e z i n g c o n d i t i o n s f o r each v a r i e t y based on the i n i t i a l and f i n a l temperatures as f o l l o w s C l = : F r e e z i n g from T. = I = 16 to 25°C to T = c = -10°C C2 = = F r e e z i n g from T. = l = 1 to 7°C to T c . = : -10OC C3 = = F r e e z i n g from T. * l = 16 to 25°C to T = c o = -18 C C4 = = F r e e z i n g from T. = l = 1 to 7°C to T c 8 = -18°C The mean percentage p r e d i c t i o n e r r o r s under these four conditions obtained by us i n g the d i f f e r e n t p r e d i c t i o n models on the two v a r i e t i e s o f apples are giv e n i n Table 5. Under these d i f f e r e n t c o n d i t i o n s s i g n i f i c a n t d i f f e r e n c e s i n the p r e d i c -t i o n e r r o r s were observed between the c o n d i t i o n s C l and C2 Table 5. Mean errors in predicting freezing times of apples by di f f e r e n t models under dif f e r e n t conditions of freezing. Freezing Conditions Golden Delicious Granny Smith  T. T tr, t T t_, t t r t n t t.. t t c i c P I M NF S P i M NF s (° C) (° C^ (%) (%) (%) (I) (%) ( l ) w C%) (%) m Cl 16-25 -10 C2 1-7 -10 C3 16-25 -18 C4 1-7 -18 Cl 16-25 -10 C2 1-7 -10 C3 16-25 -18 C4 1-7 -18 17.7 12.3 11.2 28.4 9.2 14.1 15.6 13.2 36.1 7.3 12.4 7.2 14.0 7.9 6.4 9.7 5.5 13.0 5.9 6.6 26.7 6.1 12.5 18.2 6.9 24.6 7.2 12.6 23.4 4.8 20.8 8.3 17.7 7.5 5.3 19.4 9.7 17.3 8.5 6.6 Two v a r i e t i e s mixed *P *I ^ tNF *S 112 (U Q J m (IJ 15.9 14.0 12.2 32.3 8.3 11.1 6.4 13.5 6.9 6.5 25.6 6.7 12.6 20.8 5.8 20.1 9.0 17.5 8.0 6.0 tp Freezing time model, Plank (1941) t T Freezing time model, International Institute of Refrigeration (1972) tg Suggested model t^p Freezing time model, Nagaoka et a l . (1955) t ^ Freezing time model, Mellor (1976) 63. i n most of the above-mentioned models and between C3 and C4, CI and C3, and C2 and C4 i n some models (Table 5). This c l e a r l y i n d i c a t e s the d i f f e r e n c e s i n the c a p a b i l i t y of the d i f f e r e n t models i n accounting f o r the p r e c o o l i n g and tempering p e r i o d s . Plank's Model Plank's equation assumes that the m a t e r i a l i s i n i t i a l l y at the f r e e z i n g p o i n t and estimates the time r e q u i r e d to complete the f r e e z i n g (to reach a temperature of -10°C). Hence, t h i s method had the l e a s t e r r o r under c o n d i t i o n C2 (T. = 1-7°C and T = -10°C) which was about ^ 1 c o 11.1%, and maximum e r r o r under c o n d i t i o n C3 (T^ = 16-25 C and T c = -18°C) which was about 24.6 to 26.7% (Table 5). I.I.R. M o d i f i c a t i o n The p r e d i c t i o n e r r o r i n the model proposed by the I n t e r n a t i o n a l I n s t i t u t e of R e g r i g e r a t i o n (1972) was found to be the l e a s t among the four models. This model does take i n t o account the p r e c o o l i n g as w e l l as the tempering p e r i o d by r e p l a c i n g the l a t e n t heat f a c t o r i n Plank's equation by the enthalpy from T^ to T c . Even so, the p r e d i c t i o n e r r o r s were q u i t e h i g h . Under the c o n d i t i o n CI which had a g r e a t e r i n f l u e n c e due to the p r e c o o l i n g period (T i = 16-25°C, T = -10°C), the . mean p r e d i c t i o n error 64. ranged from 12.3 to 15.6%. S i g n i f i c a n t d i f f e r e n c e s i n the p r e d i c t i o n e r r o r s were observed only when C l was compared with C2, C3 and C4 (Table 5). Nagaoka M o d i f i c a t i o n s The Nagaoka et a l . (1955) m o d i f i c a t i o n s r e s u l t e d g e n e r a l l y i n a c o n s i d e r a b l e o v e r e s t i m a t i o n of the f r e e z -ing time except under c o n d i t i o n s where the i n i t i a l temp-er a t u r e was low. T h i s model under low T., however, was no d i f f e r e n t from the one proposed by the International I n s t i -t u t e of R e f r i g e r a t i o n (1972) which had a l r e a d y been shown to give low p r e d i c t i o n e r r o r s under the c o n d i t i o n C2. Another p o i n t worth n o t i c i n g i n the model of Nagaoka et a l . (1955) i s that an a d d i t i o n a l dimensional p r o p e r t y i n the form of [1 + 0.00445 (^ - T f ) ] has been used as a f a c t o r to be m u l t i p l i e d by t j to get the modi-f i e d f r e e z i n g time. Because of the dimensional p r o p e r t y (temperature i n t h i s case) , the f a c t o r assumes d i f f e r e n t values depending on the u n i t i n which the temperature i s measured. I f (T^ - T £) i s equal to 25" C°, the f a c t o r would be 1.111 and f o r a corresponding temperature d i f f e r e n c e i n F°, the f a c t o r would be 1 . 200. Hence t X I T- which was obtained ' NF by u sing F° i n the above f a c t o r was always g r e a t e r than t^p where temperature was taken i n C°. Model t^p r e s u l t e d 65. i n mean p r e d i c t i o n e r r o r s of 20.8 to 32.31 when T\ was 16-25°C (Table 5) . M e l l o r M o d i f i c a t i o n Mellor's (1976) m o d i f i c a t i o n showed a c o n s i s t e n t l y higher p r e d i c t i o n e r r o r under a l l four c o n d i t i o n s f o r both the v a r i e t i e s . T h i s model s u f f e r s from two drawbacks. F i r s t l y , i t does not have any f a c t o r to account f o r d i f f e r e n c e s i n the tempering p e r i o d . Hence the f r e e z -o o ing time estimated to -10 C would be same as that f o r -18 C. Secondly, because of the f a c t o r (T,- - T ) C2 i n the AH c a l c u l a t i o n , under c o n d i t i o n s when the f r e e z i n g temperatures were very low (e.g. l i q u i d n i t r o g e n , -197°C), the model r e s u l t e d i n c o n s i d e r a b l e o v e r e s t i m a t i o n of the f r e e z i n g time (as high as 501). The p r e d i c t i o n mean e r r o r i n t h i s model was 11.2 to 14.01 under CI to C3 and 17.3 to 17.7% under the c o n d i t i o n C4 (Table 5). General C o n s i d e r a t i o n s In g e n e r a l , between the two v a r i e t i e s there were no s i g n i f i c a n t d i f f e r e n c e s i n the mean p r e d i c t i o n e r r o r s by using the d i f f e r e n t models under the four d i f f e r e n t c o n d i t i o n s . The mean e r r o r s f o r the two v a r i e t i e s pooled are a l s o i n c l u d e d i n Table 5. 66. On the basis of the r e s u l t s given i n Table 5, i t would not be p o s s i b l e to determine whether the models over-estimate the f r e e z i n g time or underestimate i t because the t a b l e was based on the absolute values of the pred i c -t i o n e r r o r s g i v i n g only t h e i r magnitudes. Figures 8, 9 and 10 represent the frequency histograms of the pr e d i c -t i o n e r r o r s , under the four d i f f e r e n t c o n d i t i o n s , asso-c i a t e d with the d i f f e r e n t p r e d i c t i o n models f o r Golden D e l i c i o u s , Granny Smith and the two pooled together r e s p e c t i v e l y . In these diagrams the ordinate s c a l e f o r frequency i s an a r b i t r a r y one such that the area under each histogram i s the same. These f i g u r e s could be used to obtain the information on the q u a l i t y of the p r e d i c t i o n e r r o r of the d i f f e r e n t models under the d i f f e r e n t f r e e z -ing c o n d i t i o n s as to whether i t overestimated or underestimated the f r e e z i n g time. Plank's equation thus r e s u l t e d i n gross underestimation while Nagaoka et a l . (1955) model under T^ = 16-25°C l a r g e l y overestimated the f r e e z i n g time, and under c o n d i t i o n C4 r e s u l t e d in considerable underestimation. Mellor's(1976) model showed a l a r g e r s c a t t e r under a l l the co n d i t i o n s w i t h a peak frequency at 0% e r r o r under condi-t i o n s where T = -10°C. With T - -18°C the model l a r g e l y c C b J underestimated the f r e e z i n g time f o r obvious reasons. 67. •p \ *NF f M *S U l -I* - + i ci Q Z o u o z M LU - 1 + -1 + - 1 + - 1 + 1 "I* -1 + - 1 + - 1 + TO m o c m Z n -< I L L 50 F i g u r e 8 50 0 50 0 50 0 PREDICTION ERROR [%] 50 50 Frequency h i s t o g r a m s f o r p r e d i c t i o n e r r o r p e r c e n t a g e i n Golden D e l i c i o u s a p p l e s u s i n g d i f f e r e n t models; P l a n k (1941), t P ; I.I.R. (1972), t : ; Nagaoka e t a l . (1955), t i j r ; M e l l o r (1976), t M ; a u t h o r ' s m o d i f i c a t i o n , t s , under v a r i o u s c o n d i t i o n s : C l ( T i = 16-25°C, T c = - 1 0 6 C ) , C2 ( 1 ^ = 1-7 WC, C 4 ( T i = 1 7°C, 10°C), C3 ( T i = 16-25°C, T c = -18°C) and T c = -18 WC) 50 0 50 0 50 0 PREDICTION ERROR [%] 50 Frequency histograms for prediction error percentage in Granny Smith apples using d i f f e r e n t models; Plank (1941), t P ; I..I.R. (1972), t j ; Nagaoka et a l . (1955), tflp; Mellor (1976), t^; author's modification, t s , under various conditions: CI (Ti = 16=25°C, T c = -10 6C), C2 C3(Ti = 16-25°C, T c = -18°C)and (Ti = 1-7WC, T c = C4(T i = l-7 bC, T( ) 69. NF -1+ -1+ 11 -I* -It is Q Z o u 1 I 1 o z UJ -1+ -I* -1+ -I* -I* 70 tn -o C tn z n *< 1 U "I* -I* -1 + -1 + 1 1 50 Figure 10 50 50 50 50 50 PREDICTION ERROR 1 % Frequency histograms for prediction error percentage in apples of two v a r i e t i e s using d i f f e r e n t models Plank (1941), t P ; I.I.R. (1972), t T ; Nagaoka et a l t N F ; MellorJ1976), tgi ^ " " ^ ^ | various conditions (Ti = 1-7°C, T C4(T i = 1-7°C, (1955) , under C2 = -10°C), C3(Ti = 16-25*C, T c = -18°C) and T = -18°C). c 70. A Suggested M o d i f i c a t i o n Most of the m o d i f i c a t i o n s of Plank's equation attempt at a l t e r i n g AH, the amount of heat to be r e l e a s e d from the m a t e r i a l i n order that i t i s cooled from i t s i n i t i a l temperature down to the f i n a l temperature. This i n c l u d e d the l a t e n t heat L, the temperature d i f f e r e n t i a l (T^ - T £ ) and s p e c i f i c heat before f r e e z i n g (C-^) f o r the p r e c o o l i n g p e r i o d , and the temperature d i f f e r e n t i a l ( T £ -T ) and the s p e c i f i c heat a f t e r f r e e z i n g (C? ) f o r the temp-c ^ e r i n g p e r i o d . Taking i n t o c o n s i d e r a t i o n the model proposed by I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n (1972) i n Table 5 and F i g u r e s 8 and 9, i t can be observed that c o n d i t i o n s CI and C3 r e s u l t e d i n o v e r e s t i m a t i o n of f r e e z -ing time (with mean e r r o r s of 14.0 and 6.7% r e s p e c t i v e l y ) and C2 and C4 r e s u l t e d i n u n d e r e s t i m a t i o n (mean e r r o r s of 6.4 and 9.0% respectively). The i n f l u e n c i n g f a c t o r i n CI was e s s e n t i a l l y the s e n s i b l e heat d u r i n g the precooling p e r i o d and i n C4, the s e n s i b l e heat i n the tempering period• These two r e p r e s e n t the extreme c o n d i t i o n s on e i t h e r s i d e o f the f r e e z i n g p o i n t , thus experience the maximum e r r o r s of o v e r e s t i m a t i o n of the p r e c o o l i n g p e r i o d and under-e s t i m a t i o n of the tempering p e r i o d . The other two i n t e r -mediate c o n d i t i o n s C2 and C3 which have these two i n f l u e n c e s a c t i n g together r e s u l t e d i n i n t e r m e d i a t e e r r o r s . 71. The c o r r e c t i o n f a c t o r should, t h e r e f o r e , be aimed at minimizing the i n f l u e n c e of the f a c t o r s (T^ - T £) and C 2(Tf _ T ). This c o u l d be achieved by s u i t a b l y r e d u c i n g the magnitude of the former and i n c r e a s i n g the magnitude of the l a t t e r by a m u l t i p l e r e g r e s s i o n a n a l y s i s of the experimental values of C 1 ( T i - T f) and C 2 ( T f - T ) on the expected value of t h e i r sum (AH - L ) , obtained by us i n g the a c t u a l experimental f r e e z i n g times i n the p r e d i c -t i o n model of I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n (1972) under each of 33 d i f f e r e n t runs f o r both the v a r i e t i e s . The r e g r e s s i o n equation gave a s i g n i f i c a n t l y h i g h c o r r e l a t i o n c o e f f i c i e n t (0.635) with 40.3 % o f the va r i a n c e being e x p l a i n e d by the two v a r i a b l e s . Based on the above r e g r e s s i o n equation the suggested m o d i f i c a t i o n to take i n t o account the d i f f e r e n t f r e e z i n g c o n d i t i o n s i s given below. t = [0.3022 C l i T i - T £ ) + L + 2.428 C 2 ( T f - T c ) ] • [f a h K2 The mean percentage e r r o r and the frequency h i s t o -trams under the d i f f e r e n t c o n d i t i o n s of f r e e z i n g f o r the suggested model are a l s o shown i n Table 5 and F i g u r e s 8-10. 72. The mean e r r o r s under the four c o n d i t i o n s C l to C4 were 7.3 to 9.2% ( C l ) , 6.4 to 6.6% (C2) , 4.8 to 6.9% (C3) and 5.3 to 6.6% (C4) which were more c o n s i s t e n t than those observed f o r any other model. The o v e r a l l mean e r r o r observed was 6.6 % • C o n s i d e r i n g that the mean experimental e r r o r was 2.38%, the mean p r e d i c t i o n e r r o r a s s o c i a t e d with the suggested model was l e s s than 5% beyond the experimental e r r o r . F i g u r e 11 shows a r e l a t i o n s h i p between the e x p e r i -mental f r e e z i n g times and the values p r e d i c t e d by the suggested m o d i f i c a t i o n ( t g ) . A l e a s t squares r e g r e s s i o n a n a l y s i s gave a c o e f f i c i e n t of d e t e r m i n a t i o n of 0.984 with the equation f o r the f i t t e d l i n e as f o l l o w s : t c = 0.986 t a + 0.0784 (h) (63) P r e d i c t i n g F r e e z i n g Times: Numerical Methods The numerical methods c o n s t i t u t e a second major group of methods used f o r p r e d i c t i n g f r e e z i n g times of foods. In the present study only one type of numerical 7 3 . EXPERIMENTAL FREEZING TIME[h] F i g u r e 11. R e l a t i o n s h i p b e t w e e n t h e e x p e r i m e n t a l f r e e z i n g t i m e a n d t h e t i m e p r e d i c t e d b y u s i n g t h e s u g g e s t e d m o d i f i c a t i o n o f P l a n k ' s e q u a t i o n . 74. method has been t r i e d , the f i n i t e d i f f e r e n c e method. The f i n i t e element methods are a l s o a p p l i c a b l e , but are more complex and o f f e r no d i s t i n c t advantage when used f o r u n i d i m e n s i o n a l heat t r a n s f e r s as i n the case of c y l i n d e r s (Myers, 1971 and A l b a s i n y , 1956). The f i n i t e d i f f e r e n c e method us i n g constant as w e l l as v a r i a b l e thermal p r o p e r t i e s has been used i n t h i s study. For c a l c u l a t i n g the f r e e z i n g times of s l a b s , c y l i n d e r s and spheres, i t has been r e p o r t e d that the three time l e v e l scheme of Lees (1966) was more accurate than the two time l e v e l scheme because of s m a l l e r trunca-t i o n e r r o r s ( C l e l a n d and E a r l e , 1979b). However, i n t h i s study only the two time l e v e l scheme has been employed. F i n i t e D i f f e r e n c e Scheme with V a r i a b l e Thermal P r o p e r t i e s The general r a d i a l heat conduction equation can be approximated by a two l e v e l f i n i t e d i f f e r e n c e scheme as f o l l o w s : T i + 1 _ I k 1 ( T 1 ^ 1 - T 1 ) - k 1 ( T 1 - T 1 ,) p i _m m _ m^  m+1 nr mv m m- 1J Cm At 7-^2 2m(Ar) 2 (64) 75. where m Ar = r i s the d i s t a n c e from the c e n t e r . This r e p r e s e n t s the i n t e r n a l nodes. At the cente r (m = 0), the equation takes the form: r i T o + 1 - T O k o ^ T i - T Q ) - k X - Tib At - ( A r ) 2 i i i because T^ +^ = T m_^ = T^ at m = 0 due to the symmetry. Th i s equation can be s i m p l i f i e d f u r t h e r to ( A r ) 2 At the s u r f a c e (m = M), the t h i r d k i n d o f boundary c o n d i t i o n i s taken i n t o account by u s i n g , r i T M + 1 - TM E ^ T a - TM^ CI + g)  LM At [Ar) V 1 ^ " T M - 1 ^ (Ar) 1 (66) where MAr = R i s the r a d i u s of the c y l i n d e r . One other assumption t h a t has been made here i s t h a t k* + i - k m 7 76. k* 1_ w h i c h e n a b l e s f u r t h e r s i m p l i c a t i o n o f t h e c o m p u t a t i o n p r o g r a m . By s e l e c t i n g A t and A r s u c h t h a t t h e e x p r e s s i o n = \ , e q u a t i o n s 64 t o 66 r e d u c e t o k ™ A t m M W 2 T 1 . , + T 1 , T* - T 1 c • i j ~v -i + 1 m+1 m-1 , m+1 m-1 f f i l ^ ( i n t e r n a l n o d e) T m = 2 4 i i t- b /- ) ( c e n t e r ) T J + 1 = l\ (68) r c ^ T 1 + 1 - T M * TM-1 h A r r T _ T i . r f i Q ^ ( s u r f a c e ) T M 2 T ~ C a W C J k M The c o m p u t e r p r o g r a m t h a t was u s e d f o r t h e ab o v e f i n i t e d i f f e r e n c e scheme has b e e n given i n Appendix 3 (see a l s o Appendix 4 f o r a schematic diagram o f the f i n i t e d i f f e r e n c e method). F i n i t e D i f f e r e n c e Scheme w i t h C o n s t a n t T h e r m a l P r o p e r t i e s The e q u a t i o n s u s e d i n t h i s m e t hod w e r e s i m i l a r t o t h e ones u s e d i n t h e scheme w i t h v a r y i n g t h e r m a l p r o p e r t i e s . C o n s t a n t v a l u e s o f t h e r m a l c o n d u c t i v i t y (k-^, k 2 ) and s p e c i f i c h e a t (C-j_., C 2 ) were p l u g g e d i n t o t h e 77. equations above and below the f r e e z i n g p o i n t . The major problem here was to accommodate the r e l e a s e of the l a t e n t heat. For t h i s purpose the l a t e n t heat was assumed to be r e l e a s e d i n a range of temperature (5C° taken a r b i t r a r i l y ) below the f r e e z i n g p o i n t . Then the l a t e n t heat was con-v e r t e d i n t o apparent s p e c i f i c heat. The s p e c i f i c heat and thermal c o n d u c t i v i t y d u r i n g the phase change p e r i o d were c a l c u l a t e d as C = L + 5 5 ° Q * C - l j / k g K ° (70) and k -| + k~ o k = 2- W/mK (71) Other than t h i s m o d i f i c a t i o n , the procedure was e x a c t l y the same as that d e s c r i b e d f o r the scheme with v a r y i n g thermal p r o p e r t i e s . The p r e d i c t e d values of f r e e z i n g times by the two numerical f i n i t e d i f f e r e n c e methods as w e l l as those by the suggested m o d i f i c a t i o n of Plank's equation along with the percentage p r e d i c t i o n e r r o r c a l c u l a t e d on the b a s i s of experimental values of f r e e z i n g time are given i n Table 6 f o r Golden D e l i c i o u s and Table 7 f o r Granny Smith. These t a b l e s do not c o n t a i n a l l the experimental c o n d i t i o n s 78. Table 6. Comparison between the p r e d i c t e d f r e e z i n g times and p r e d i c t i o n e r r o r s of numerical methods and modified Plank's equation under d i f f e r e n t c o n d i t i o n s of f r e e z i n g of Golden D e l i c i o u s apples. h tS E E E Code (W/m2K°) (h) (h) (%) (h) (%) Ch) m GDI 55.59 2.03 2.46 + 21.2 3.54 + 74.4 1.49 -26.7 2.33 2.72 + 16.7 3.82 + 63.9 3.26 + 39.9 GD4 13.85 5.13 4.92 -4.1 6.40 + 24.8 2.75 -46.4 5.57 5.43 -2.5 6.77 + 24.7 5.28 -5.2 GD7 17.83 5.25 5.23 -0.3 6.74 + 28.4 2.96 -43.6 6.08 5.77 -5.2 7.34 + 20.7 7.02 + 15.5 GDII 55.59 1.77 1.93 + 9.0 1.25 -29.4 1.00 -43.5 1.95 2.13 + 9.3 1.40 -28.1 1.89 -3.0 GD12 55.59 1.70 1.94 + 13.8 1.36 -20 .0 1.04 -38.8 1. 87 2 .14 + 14.3 1.51 -19.3 1.92 + 2.7 GD13 13.85 4.47 4.30 -3.8 4.76 +6.5 2.00 -55.3 4.90 4.80 -2.1 5.08 + 4.8 4.12 -15.9 GD16 59.68 2.23 2.24 + 0 . 3 1.47 -34.1 1.14 -48.9 2.53 2.48 -2.0 1.73 -31.6 2.81 + 11.1 GD17 17.83 4. 20 4.71 + 12.1 5.34 + 27.1 2.28 -45.7 5.20 5.22 + 0.3 5.85 + 12.5 5.69 + 9.4 GD18 17. 83 4.33 4. 74 + 9.4 5.70 + 31.6 2.36 -45.5 4.97 5.25 + 5.5 6.21 + 24.9 5.77 + 16.0 GD20 55.59 2.73 2.56 -6.4 3.21 + 17.6 1.55 -43.2 3.00 2.82 -6.0 3.41 + 13.7 2.74 -8.7 GD21 55.59 2.83 2.54 -10.1 3.07 + 8.5 1.55 -45.2 3.07 2.81 -8.5 3.27 +6.5 2.73 -11.0 GD22 17.83 5.37 5.77 + 7.4 7.42 + 38. 2 3.21 -40.2 6.03 6.37 • +5.6 8.01 + 32.8 7.14 + 18.4 GD25 55.59 2.33 2.42 + 4.0 1.09 -53.2 1.18 -49.4 2.60 2.69 + 3.4 1.29 -50.4 2.37 -8.8 GD26 17.83 5.47 5.00 -8.6 4.55 -21.0 2.29 -58.1 6.10 5.54 -9.2 5.00 -18.0 5.30 -13.1 GD28 55.59 3.86 3.82 -1.0 4.28 + 10.8 1.97 -49.0 4.27 4.23 -0.3 4.64 + 8.7 4.23 0.0 GD29 17.83 6.36 6.74 + 5.9 7.74 + 21.7 3.26 -54.3 7.13 7.44 + 4.3 8.28 + 16.1 6.91 -3.1 GD31 55.59 3.60 3.27 -9.0 1.31 -63.6 1.35 -62.5 3.97 3.62 -8.7 1.55 -61.0 2.85 -28.2 GD32 17.83 5.97 5.88 -1.5 3.83 -35.8 2.34 -60.8 6.06 6.53 + 7.8 4.27 -29.5 5.20 -14.2 Footnote: as i n Table 7. 79. Table 7. Comparison between the predicted freezing times and prediction errors of numerical methods and modified Plank's equation under different conditions of freezing of Granny Smith apples. * s E l nc E \v E C o d e (*) (h) m (W/jn K°) (h) (h) (I) (h) GS1 55.59 2.20 2.35 + 6.8 3.37 + 53.2 1.48 -32.7 2.43 2.56 + 5.5 3.54 +45.8 2.57 + 5.8 GS3 13.85 4.70 4.77 + 1.5 6.18 + 30.0 2.81 -40.2 5.20 5.20 0.0 6.49 + 24.8 4.96 -4.6 GS7 17.83 5.30 5.25 -1.0 6.64 + 25.3 3.05 -42.5 6.43 5.73 -11.0 7.16 + 11.4 6.80 + 5.8 GS11 55.59 2.17 2.30 + 5.8 1.05 -51.6 1.12 -48.4 2.47 2.51 +1.8 1.25 -49.4 2.44 -1.2 GS13 13.85 3.50 4.12 + 17.6 3.54 + 1.1 1.89 -46.0 3.77 4.51 + 19.6 3.80 + 0.7 3.73 -1.1 GS15 59.68 2.30 2.32 +0.7 1.05 -54.3 1.12 -51.3 2.73 2.55 -6.7 1.29 -52.7 2.68 -1.8 GS17 17.83 4.00 4.68 -17.0 4.75 + 15.8 2.25 -43.8 4.83 5.12 +6.0 5.19 + 7.5 5.42 + 12.2 GS20 55.59 2.90 2.79 -3.8 3.12 + 7.6 1.67 -42.4 3.26 3.05 -6.5 3.32 + 1.8 2.96 -9.2 GS21 55.59 2.90 2.81 -3.1 3.42 + 17.9 1.72 -40.7 3.23 3.07 -5.0 3.62 + 12.1 3.00 -7.1 GS22 17.83 5.53 5.80 + 5.0 7.45 + 34.7 3.34 -39.6 6.33 6.33 0.0 7.96 + 25.8 6.96 + 10 .0 GS25 55.59 2.33 2.54 + 8.8 1.05 -58.7 1.21 -48.1 2.53 2.79 + 10.3 1.23 -51.4 2.30 -7.9 GS26 17.83 5.44 5.02 -7.7 4.33 -13.7 2.38 -56.3 6.00 5.50 -8.3 4.73 -14.0 5.15 -14.2 GS28 55.59 4.00 4.04 +1.0 4.34 + 7.4 2.09 -47.8 4.33 4.40 + 1.7 4.66 + 7.6 4.20 -3.0 GS29 17.83 6.56 6.92 + 5.5 8.10 + 17.1 3.55 -48.7 7.70 7.54 -2.0 8.58 + 13.8 6.89 -10.5 GS31 55.59 3.73 3.45 -7.4 1.29 -62.6 1.43 -61.7 4.20 3.78 -10.1 1.61 -57.4 2.82 -32.9 GS32 17.83 6.17 5.97 -3.2 3.73 -37.5 2.54 -58.8 7.10 6.55 -7.8 4.01 -38.8 5.18 -27.0 First line under each code refers to freezing time to reach -10"C Second line under each code refers to freezing time to reach -18°C t e Experimental freezing time. tg Freezing time by the suggested modification of Plank's equation. E Prediction error. ''nc *r\v Freezing times by numerical finite difference scheme with constant and variable thermal properties respectively. 80. d i s c u s s e d e a r l i e r f o r the o t h e r a n a l y t i c a l methods based on P l a n k ' s e q u a t i o n f o r c e r t a i n reasons which w i l l be d i s c u s s e d l a t e r . C o n s t a n t Thermal P r o p e r t y Scheme The p r e d i c t i o n e r r o r v a r i e d from -63.6 t o +74.4% f o r Golden D e l i c i o u s and -62.6 t o +53.2% f o r Granny Smith by u s i n g the n u m e r i c a l method w i t h c o n s t a n t t h e r m a l p r o p e r t i e s , as compared t o -10.1% t o +21.2% and -17.0 t o + 19.6% r e s p e c t i v e l y by u s i n g the suggested modif i c a t i o n (Tables 6 & 7). Examination of the a n a l y s i s showed t h a t c o n d i t i o n s C l and C3 {Ti - 16-25°Cand T c = -10 and -18°C r e s p e c t i v e l y ) r e s u l t e d i n o v e r e s t i m a t i o n o f the f r e e z i n g t i m e , w h i l e the f r e e z i n g c o n d i t i o n s C2 and C4 ( T i = 1-7°C, and T c = -10 and -18°C) r e s u l t e d i n u n d e r e s t i m a t i n g the f r e e z i n g t i m e . The mean e r r o r s ( a b s o l u t e v a l u e s ) o f f r e e z i n g time e s t i m a t i o n by t h i s method f o r c o n d i t i o n s T. = 16-25°C (T = -10 o r 3 l ^ c -18°C) and T. = 1-7°C (T = -10 or -18°C) r e s p e c t i v e l y were 25.7% and 30.1% f o r Golden D e l i c i o u s and 21.0% and 35.5% f o r Granny Smith a p p l e s . I t has been g e n e r a l l y r e c o g n i z e d t h a t the f i n i t e d i f f e r e n c e schemes w i t h c o n s t a n t t h e r m a l p r o p e r t i e s based on the l a t e n t heat b e i n g r e l e a s e d a t a unique f r e e z i n g p o i n t o f t e n r e s u l t i n o v e r e s t i m a t i n g the f r e e z i n g t i m e s (Charm e t a l . , 1972). The r e s u l t s under 81. c o n d i t i o n s C l and C3 supported t h i s o b s e r v a t i o n i n s p i t e of the modification made to accommodate the r e l e a s e of the l a t e n t heat u n i f o r m l y over a temperature range of 5C° below the f r e e z i n g p o i n t . Another important phenomenon that has been f r e q u e n t l y encountered with numerical f i n i t e d i f f e r e n c e schemes i s commonly known as "jumping" of the l a t e n t heat peak. T h i s problem a r i s e s where the c a l c u l a t i o n procedure does not f o l l o w the s p e c i f i c heat curve and undercuts a p o r t i o n o f the peak. T h i s problem worsens as the B i o t Number i n c r e a s e s because the c o o l i n g r a t e a l s o i n c r e a s e s . F u r t h e r , t h i s can a l s o occur when the l a t e n t heat i s assumed to be r e l e a s e d i n a short range of temperature and the i n i t i a l temperature i s c l o s e to the f r e e z i n g p o i n t with a low ambient temperature. This was probably one of the reasons f o r the under p r e d i c t i o n of the f r e e z i n g times under c o n d i t i o n s C2 and C4. The problem became worse when used to p r e d i c t f r e e z i n g times i n l i q u i d n i t r o -gen immersion systems (T = -197°C). These r e s u l t s were not i n c l u d e d because of gross underestimations (up to 100%). V a r i a b l e Thermal Property Scheme The p r e d i c t i o n e r r o r v a r i e d from -62.5 to +12.2% f o r Golden D e l i c i o u s and -61.7 to +39.9% f o r Granny Smith 8 2 . apples by using the numerical method with variable thermal properties. A break up of analysis showed, i n this case, however, that there i s no s i g n i f i c a n t difference in the freezing times between conditions Cl and C3, contrary to the model with constant thermal properties. However, condi-tions Cl and C3 (freezing time to reach -10°C) resulted in large underestimationsof freezing times (-26.7 to -62.5% for Golden Delicious and -32.7 to -61.7% for Granny Smith) while conditions C2 and C4 (freezing time to reach -18°C) gave f a i r l y accurate predictions (-10.5 to +5.8% for Granny Smith and -14.2 to 16.0% for Golden Delicious except in two cases where the values were -32.9% and +39.9%). The mean errors (absolute values) of freezing time estimations by this method were 47.6% and 11.7% for Golden Delicious and 46.8% and 8.6% for Granny Smith respectively under conditions of T = -10 C (T. = 16-25°C or 1-7°C) and T c = -18°C (T i = 16-25°C or 1-7°C). Thus the mean error for the two v a r i e t i e s i n estimating freezing time to reach -18°C was approximately 10% pre-dominantly towards underestimation. This error could have been caused probably by the p a r t i a l skipping of the latent heat peak or the incomplete release of latent heat at -18°C. However, the phenomenal underestimatimation of 83. o about 45% in freezing time computation to reach -10 C was obviously due to incomplete l i b e r a t i o n of latent heat o at -10 C. The apparent s p e c i f i c heat curve (Figure 4) c l e a r l y shows that the release of latent heat i s complete o o only at temperatures below -25 C. Up to -10 C probably one half of the latent heat was not accounted for, and hence the underestimation of about 45%. This problem does not arise in the f i n i t e difference scheme with constant thermal properties because of the assumption made that the entire latent heat was released within 5C° below the freezing o point (-1 C) or in the other prediction models where the t o t a l latent heat was considered. Further, for the conditions involving freezing o at an ambient temperature of -18 C (GD5, GD6, GS5 and GS6), the numerical methods were not used because the o freezing times to reach -18 C would be become a very large number. The method was also not useful for estima-ting of freezing times when the ambient temperature was o very low ( l i q u i d nitrogen freezing at -197 C). Under conditions similar to those i n which the numerical methods were used, the suggested modifications of Plank's model gave mean errors of 6.67% for Golden. Delicious and 6.2% for Granny Smith with a mean value of 6.46% for the two v a r i e t i e s pooled, which was much better than 10% for the numerical method with variable thermal properties and 23.4% for the numerical method with con-stant thermal properties. 85. CONCLUSIONS There has not been much published information available on the thermo-physical properties of apples. In this study, the temperature dependence of the thermo-physical properties of two v a r i e t i e s of apples, Golden Delicious and Granny Smith, have been investigated. Detailed regression equations by the method of least squares fitting to cover the variations of thermal conductivity, thermal d i f f u s i v i t y and apparent s p e c i f i c heats with temperature have been presented. Density varia-tions at four d i f f e r e n t temperatures are given. The thermo-physical properties determined i n this study have been used to predict freezing times of apples under d i f -ferent conditions of freezing by using the various predic-tion models available i n the l i t e r a t u r e . The predicted values of the freezing times by the d i f f e r e n t models have been compared with the experi-mental values of the freezing times and based on the predic-tion error analysis, an equation which gives the least error has been suggested. Plank's model (1941) has been the basis for the determination of freezing times of foods. However, this model assumes that the material i s i n i t i a l l y at i t s 86. f r e e z i n g p o i n t and hence does not take i n t o account the p r e c o o l i n g or tempering p e r i o d s . Hence, t h i s model r e s u l t s i n c o n s i d e r a b l e u n d e r e s t i m a t i o n of the f r e e z i n g time when used under the c o n d i t i o n s of i n i t i a l temperatures higher than the f r e e z i n g p o i n t and f i n a l temperatures below the zone o of maximum i c e c r y s t a l formation (-1 to -5 C). Over the yea r s , many m o d i f i c a t i o n s have been suggested to Plank's model to i n c l u d e the p r e c o o l i n g and tempering p e r i o d s . A l l these m o d i f i c a t i o n s are based on some e m p i r i c a l r e l a t i o n s h i p s used to accommodate the d e v i a t i o n s of the value p r e d i c t e d by Plank's model from the experimental v a l u e s . Most of these m o d i f i c a t i o n s (Nagaoka et a l . , 1955; I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n , 1972; M e l l o r , 1976; Levy, 1958) are aimed at s u b s t i t u t i n g the l a t e n t heat (L) i n Plank's equation by d i f f e r e n t p r o p o r t i o n s of the t o t a l amount of heat to be removed from the body i n order to b r i n g i t down from i t s i n i t i a l temperature to the f i n a l temperature. The present suggested m o d i f i c a t i o n a l s o belongs to t h i s category. The simple f a c t that the performance of the d i f f e r e n t models i s so d i f f e r e n t under the d i f f e r e n t con-d i t i o n s of f r e e z i n g i n d i c a t e s t hat no such simple empiri-c a l m o d i f i c a t i o n can be used to cover a l l the p r a c t i c a l s i t u a t i o n s . Each of the suggested m o d i f i c a t i o n s to Plank's equation must have been the optimum s o l u t i o n when 87. used under the c o n d i t i o n s tested by the r e s p e c t i v e authors. Obviously, the d i f f e r e n c e s i n the behavior of the d i f -f e r e n t models appear to be due to the f a c t t h at only the p r o p e r t i e s of the f r o z e n m a t e r i a l (p ^ and Y^) are used even to c o n s i d e r the s i t u a t i o n s where the product i s i n i -t i a l l y at temperatures much higher the f r e e z i n g p o i n t . A model which accounts f o r a l l the three d i s t i n c t phases: p r e c o o l i n g , phase change and tempering p e r i o d s , on the mer i t s of the thermal p r o p e r t i e s of these phases should be the i d e a l one. A f t e r e x t e n s i v e r e s e a r c h on a model system as w e l l as food m a t e r i a l s under v a r i o u s c o n d i t i o n s of f r e e z i n g , C l e l a n d and E a r l e (1977, 1979a, 1979b) suggested modifying the values of P and R on the b a s i s of three dimensionless q u a n t i t i e s , B i o t Number, Plank's Number and Stefan's Number and u s i n g the t o t a l q u a n t i t y of heat to be removed over the e n t i r e range of temperature i n s t e a d of L. Even i n t h i s model the s t r u c t u r e i s essen-t i a l l y based on the thermal p r o p e r t i e s o f the f r o z e n m a t e r i a l . Hence, i t cannot be an i d e a l one. This model when used under the c o n d i t i o n s mentioned i n t h i s i n v e s t i g a t i o n r e s u l t e d i n about 20-301 o v e r e s t i m a t i o n i n most cases and as high as 100-150% when used to p r e d i c t the f r e e z i n g times i n l i q u i d n i t r o g e n f r e e z i n g at 88. - 197°C. The Stefan's Number under the conditions of l i q u i d nitrogen freezing was very large (close to 1.0) and the accuracy of the model has not been v e r i f i e d by the authors (Cleland and Earle, 1979b) to cover this condition. The numerical methods have the v e r s a t i l i t y of accommodating any kind of boundary condition as well as variations i n the thermal properties of the product. By id e n t i f y i n g the nature of the problem these methods could be e a s i l y modified to y i e l d accurate freezing time e s t i -mations. The major disadvantages of these methods are, however, the absolute necessity of access to a computer, time required to program the computer (packaged programs are rarely available) and the need to have detailed information on the v a r i a t i o n of thermal properties with temperature without which the accuracy of the method suffers. 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The c a l c u l a t i o n of food f r e e z i n g time. Proc. 12th Int. Congr. R e f r i g . 2: 667. Cowell, N.D. and Namor, M.S.S. 1974. Heat t r a n s f e r c o e f f i -c i e n t s i n p l a t e f r e e z i n g - E f f e c t of packaging m a t e r i a l s , I.I.R. Meeting Commissions B l , C l , and C2, Bressanone, 17-20 Sept. E a r l e , R.L. 1971. Simple probe to determine c o o l i n g r a t e s i n a i r . Proc. XIII I n t . Congr. R e f r i g . 2: 373. E a r l e , R.L. and Fleming A.K. 1967. C o o l i n g and f r e e z i n g of lamb and mutton c a r c a s s e s . I. C o o l i n g and f r e e z i n g r a t e s i n l e g s . Food Technol. 21: 79. Ede, A .J. 1949. The c a l c u l a t i o n of f r e e z i n g and thawing of f o o d s t u f f s . Modern R e f r i g . 52: 52. Fennema, O.R. 1966. An o v e r a l l view of low temperature food p r e s e r v a t i o n . C r y o b i o l o g y 3: 197. Fennema, O.R. and Powrie, W.D. 1964. Low temperature food p r e s e r v a t i o n . Adv. Food Res. 13: 219. Fennema, O.R., Powrie, W.D. and Marth, E.H. 1973. "Low tempera-tu r e p r e s e r v a t i o n o f foods and l i v i n g matter'.' M a r c e l Dekker Inc. NY. Fleming, A.K. 1971. The numerical c a l c u l a t i o n of f r e e z i n g p r o c e s s e s . Proc. 13th Int. Congr. R e f r i g . 2:303. Gane, R. 1936. The thermal c o n d u c t i v i t y of the t i s s u e of f r u i t s . Ann. Rep. Food Invest. Board, Great B r i t a i n , p. 211. Goodman, T.R. 1964. A p p l i c a t i o n of i n t e g r a l methods to t r a n s i e n t n o n l i n e a r heat t r a n s f e r . Advan. Heat T r a n s f e r 4: 52. Gutschmidt., J . 1964 in MProbleme der Ernhrung durch G e f r i e r -kost" D i e t r i c h S t e i n k o p f , Darmstadt. C i t e d i n " C o o l i n g Technolog, i n the Food I n d u s t r y " . 1975 ed. Ciobanu, A., Lascu, G., Bercescu, V. and N i c u l e s c u , L. Abacus Press, Tunbridge Wells , Kent. 92. Heldman, D.R. 1975. "Food Process E n g i n e e r i n g " . A v i , Westport, Conn. I n t e r n a t i o n a l I n s t i t u t e of R e f r i g e r a t i o n . 1972. "Recom-mendations f o r the P r o c e s s i n g and Handling of Frozen Foods". 2nd ed. I n t . I n s t . R e f r i g . P a r i s . Lees, M. 1966. A l i n e a r three l e v e l d i f f e r e n c e s scheme f o r ' q u a s i - l i n e a r p a r a b o l i c equations.Maths. Comput. 20: 516. Levy, F.L. 1958. J . R e f r i g . 1: 35 c i t e d by Brennen et a l . 1976. "Food E n g i n e e r i n g Operations" 2nd ed. Chapter 14. A p p l i e d Science Publ. L t d . London. Long, R.A. 1955. Some thermo-dynamic p r o p e r t i e s of f i s h and t h e i r e f f e c t on the r a t e of f r e e z i n g . J . S c i . Food A g r i .. 6 : 621. Lorentzen, G. and Rosvik, S. 1960. I n v e s t i g a t i o n of heat t r a n s f e r and weight l o s s d u r i n g f r e e z i n g of meat. Progr. R e f r i g . S c i . Technol. Proc. I n t . Congr. R e f r i g . 10th.1959 V o l . 3, p. 120. Lozano, J.E., U r b i c a i n , M.J. and R o t s t e m , E. 1979 . Thermal c o n d u c t i v i t y of apples as a f u n c t i o n of mois-ture content. J . Food S c i . 44: 198. MacKenzie, R.C. (Ed.) 1970. " D i f f e r e n t i a l Thermal A n a l y s i s " . V o l . 1, Academic P r e s s , N.Y. M e l l o r , J.D. 1976. Personal communication, c i t e d by C l e l a n d , A.C. and E a r l e , R.L. J . Food S c i . 44: 958. Mott, L.F. 1964. P r e d i c t i o n of product f r e e z i n g time. 'Aust. R e f r i g . , A i r Cond. Heat. 18: 16. 93. Muehlbauer, J.C. and Sunderland, J.E. 1965. Heat conduc-ti o n with freezing or melting. Appl. Mech. Rev. 18: 951. •, G.E. 1971. "Analyt i c a l Methods in Conduction Heat Transfer", p. 339. McGraw H i l l , New York. Nagaoka, J . , Takaji, S. and Hohani, S. 1955. Experiments on the freezing of f i s h in an a i r blast freezer. Proc. 9th Int. Congr. Refrig. 95C: 42. Ordinanz, W.O. 1946. Spe c i f i c heat of foods in cooking. Food Ind. 18: 101. Plank, R. 1941. Bietrage zur berechnung und bewertung der gefriergeshwindigkeit von leben smittein. Z. ges Kalteind. 10, Beih Reihe (3): 1. Rebellato, L., Del Giudice, S. , and Comini, G. 1978 . F i n i t e element analysis of freezing processes in foodstuffs. J. Food S c i . 43: 239. Riedel, L. 1949. Measurement of thermal conductivity of sugar solutions, f r u i t juices and milk. Chemie-Ingenieur-Technik. 21, 340. Cited by R.W. Dickerson 1968 in "The Freezing Preservation of Foods", eds. Tressler, D.K., Van Arsdel, W.B. and Copley, M.J. IV ed. Avi, Wesport, Conn. Riedel, L. 1951. The r e f r i g e r a t i n g e f f e c t required to freeze f r u i t s and vegetables. Refrig. Eng. 59: 670. Ruck, J.A., 1969. Chemical methods for analysis of f r u i t and vegetable products. SP 20,Research Branch, Canada Dept. Agriculture. Short, B.E., B a r t l e t t , L.H. 1944. "The S p e c i f i c Heat of Food Stuffs", Univ. Texas, Bio. Eng. Res. Publ. No. 4432. Sweat, V.E. 1974. Experimental values of thermal conduc-t i v i t y of selected f r u i t s and vegetables. J. Food S c i . 39: 1080. 94. Sweat, V.E. and Haugh, C.G. 1972. A thermal c o n d u c t i v i t y probe f o r small food samples. Paper 72-376 presented to 1972 Ann. Meet. Amer. Soc. A g r i c . Eng., Hot Springs, Ark. Tanaka, K. and Nishimoto, J - i . 1959. Determination of time r e q u i r e d f o r f r e e z i n g of s h i p j a c k . J . Tokyo Univ. F i s h . 45: 205. Tanaka, K. and Nishimoto, J - i . 1960. Determination of the time r e q u i r e d f o r f r e e z i n g of whale meat. J . Tokyo Univ. F i s h . 47: 81. Tanaka, K. and Nishimoto, J - i . 1964. Determination of the time r e q u i r e d f o r c o n t a c t f r e e z i n g of whale meat. J . Tokyo Univ. F i s h . 50: 49. Tao, L.C. 1967. G e n e r a l i z e d numerical s o l u t i o n s of f r e e z -ing a s a t u r a t e d l i q u i d i n c y l i n d e r s and spheres. Am. I n s t . Chem. Eng. J . 13: 165. T r e s s l e r , D.K., Van A r s d e l , W.B., and Coopley, M.J. 1968. "The F r e e z i n g P r e s e r v a t i o n of Foods". V o l . 2. A v i , Westport, Conn. Van A r s d e l , W.B., Coopley, M.J. and Olson, R.L. 1969. " Q u a l i t y and S t a b i l i t y of Frozen Foods. Time-Temperature Tolerance and I t s S i g n i f i c a n c e " . Wiley ( I n t e r s c i e n c e ) , N.Y. Vos, B.H. 1955. Measurements of thermal c o n d u c t i v i t y by a non-steady s t a t e method. Appl. S c i e n t i f i c Res. A5: 425. A p p e n d i x 1 C o m p a r i s o n b e t w e e n t h e e x p e r i m e n t a l and p r e d i c t e d f r e e z i n g t i m e s . Code (./Hv, („? (?) (h! (» > »' (h?c ct) (hM „", GDI 55 .59 2 .03 2 .00 -1. 1 2 .68 + 32 .4 3. 16 + 55 .8 2 .95 + 45.3 2 .41 + 18. 9 55 .59 2. .33 2 .00 -13. 8 2 .79 + 19 .8 3. 28 + 40 .9 3 .06 + 31.6 2 .41 + 3. 6 GD2 55 .59 2 .06 2 .00 -2. 5 2 .68 + 30 .4 3. 16 + 53 .5 2 .95 + 43.2 2 .41 + 17. 1 55 .59 2 .33 2 .00 -13. 8 2 .79 + 19 .8 3. 28 + 41 .0 3 .06 + 31.6 2 .41 + 3. 6 GD3 13 .85 5 .13 4 .01 -21. 7 5 .36 + 4 .6 6. 31 + 23 .1 5 .89 + 14.9 4 .91 -4. 1 13 .85 5 .57 4 .01 -28. 0 5 .57 + 0 .1 6. 56 + 17 .8 6 .12 + 10.0 4 .91 -11. 7 GD4 13 .85 5 .13 4 .01 -21. 7 5 .36 + 4 .6 6. 31 + 23 .1 5 .89 + 14.9 4 .91 -4. 1 13 .85 5 .63 4 .01 -28. 7 5 .57 -1 .0 .6. 56 + 16 .6 6 .12 + 8.8 4 .91 -12. 7 GD5 59 .68 2 .47 2 .15 -12. 6 2 .88 + 16 .8 3. 39 + 37 .5 3 .17 + 28.3 2 .57 + 4. 4 59 .68 2 .93 2 .15 -26. 4 2 .99 + 2 .3 3. 52 + 20 .4 3 .29 + 12.4 2 .57 -12. 0 GD6 59 .68 2 .27 2 . 15 -4. 9 2 .88 + 27 .1 3. 39 + 49 .6 3 .17 + 39.6 2 .57 + 13. 6 59 .68 2 .93 2 .15 -26. 4 2 .99 + 2 .3 3. 52 + 20 .4 3 .29 + 12.4 2 .57 -12. 0 GD7 17 .83 5 .25 4 .23 -19. 2 5 .77 + 10 .0 6. 89 + 31 .3 6 .39 + 21.8 5 . 16 -1. 7 17 .83 6 .08 4 .23 -30 . 3 5 .99 -1 .4 7. 15 + 17 . 7 6 .64 + 9.2 5 . 16 -15. 1 GD8 17 .83 5 .47 4 .23 -22. 5 5 .77 + 5 .6 6. 89 + 26 .0 6 .39 + 16.9 5 .16 -5. 6 17 .83 6 .25 4 .23 -32. 2 5 .99 -4 .0 7. 15 + 14 .5 6 .64 + 6.2 5 .16 -17. 4 GD9 68 .42 0 .242 0 .172 -28. 6 0 .228 -5 .4 0. 267 + 10 .5 0 .250 + 3.4 0 .308 +27 .3 68 .42 0 .250 0 .172 -30 . 9 0 .237 -4 .9 0. 278 + 11 .2 0 .260 + 4.1 0 .308 +23 .3 GD10 68 .42 0 .230 0 . 172 -24: •9 0 .228 -0 .5 0. 267 + 16 .3 0 .250 + 8.8 0 .308 +34 .0 68 .42 0 . 230 0 .172 -24. 9 0 .237 + 3 .4 0. 278 + 20 .9 0 . 260 + 13M 0 .308 +34 .0 J o-= ~Z.r~, J JZ +J sz 4-» SZ o -o o u O L O Cn CT1 r~J L O l / l N H O O L O O O r T O st O ° ^ ° \ O (O « * L O oo o o r~- L O i — ( L O L O o o ^ cn T^^t oo o + 1 + 1 I i—I I H I H I H + H + 1 ' ^ I H I I I I I I r o r o L O L O cn O I oi cn o o o o r- r-~ rsi rsi cn cn o o oooo oooo H H i—i r-i I - H i—t i-Ht—I L O L O rsi rsi L O L O H H I - H I - H "sT>sr rsi rsi rsi rsi -si- *=f- rr^3- O O rg r g ^tcri o o rsi o L O o> rsi \o r o i—i i—i r o I ( o r o o oo oo r g *d-cn oo i n ai o r - H O I r s i L O rsi rsi O ' S - oo rsi + + i—I + I I I I I I I I i - H | I - H + + + + + + + + «=f o i — i o -si- ro r- r-- t-^r~- O i o c i o o tsi L O r- O H O O O cn O oi O r s i ^ r r s i H r o H ro r- cn o o o rsi rsi o> o H r s i H (N rj- T J - r s i r s i r s i r s i ^ L O O O r s i r o L O O l rs) L O O ro cn L O H O LOro r s i o i O cn «3- ro O ^ - t ~ - H r s i r s . . o o 3^- O O r - - ro u i < * oo o L O cn H + H H I I r o i + i + i H I H + + + H + + + + I + + + L O ^ 3 - O i oo L O L O L O L O r o r o r o r o o r g cn H O H m N C I o cn o ro L O r O L O rs] ro r s i r o o o o C T i r g r s i r s i H N H r s i H N «st^- ' S-'st rsi rsi r j rsi ^ L O L O O O r o r o r-'a- ro oo ^ - r - o o> H O O «t oi H oo rsi L O rs. H H r s i L O O H L O O O f s - H H O I M O 0 » r - ~ 0 0 H rs) H O L O + + H + I H I H + 1 I H + 1 + 1 + + + 1 + 1 1 I oo r-~ t\ L O oi r - oo r - oo r - L O L O L O L O O O cn oo cn o>o ro "3-oo O i oo cn H r o H r o H N H N W r . O o o H r s i r - - o o H H H H -st"^ r s i r s i r s i r s i •=3-'* ^ - O O r s i r s i LO r o o •'sT o o r s - H ^ i - cn L O O O en's- o o o o m oo oi oi O L O I H O - s f t^- L O CTirsi r o r o r o r s i o o o o L O N O i I H H H r s i H N i r s i H r s i i rsi I H H H r s i r s i I I l I l l i l l l l I I l l rsi rsi L O L O L O L O rsi rsi r s i r s i r s i r s i rsi rsi r o r o r o r o r-~. t-. O O o o o o r - - r - ~ r>-rs- cn cn cn cn o o o o H H H H H H H H r o r o r o r o H H H H -3- O O r s i r s i r o L O L O or-- r^-o M N r o o r o r o O O r o r - C T i O r o o r-. cn r- oo %t Oi L O O I H L O rsiLO r s i r s i ro cn H r s ) t— o H H H H •"st-«3- •st'st r s i r s i rslOsl rj-LO -s* rtf- O O rsi ro cn cn cn cn L O L O L O L O o o o o o o o o r o r o r o r o r s i r s i a> O i L O L O L O L O o o o o o o o o o o o o o o o o o o o o •si- >sf L O L O L O L O L O L O r o r o ro ro cn cn cn cn rs. r-- rs- r-. o o o o L O L O LOLO LOLO H H H H LOLO LOLO H H H H O O LOLO Q Q ro H Q Q L O H Q O H Q O cn H Q O rsi Q 97. r- r- H ^ L O to »° to r- N N ^ 0 t o r - ^ ^ • • • • • • H CO Lf) vO O rtH tO rs] tO N IO S ~2 ~1 H H + 1 + 1 I rH H rs! ^ 1 ^ + I ^ ^ I I * * I I I + + I + - r a i O i " 3 - r j- r-. a i d ^j-^i- L O L O rsi rsi oo oo oo oo *a- o O O o to to rsi rsi r-- r- t o i o r-~ r- L O L O *-T rsi rs] L O L O L O L O o o rsi rsi •**• o o to to o o o o tO O l LO rsi O l P - ~ 0 LOt~~ OOtO O l rsi O H rsi o i-^  to t o o -st L O o r - r - - L O O L O H H H O O O O > O H M O + 1 (S H f s ) H I I + 1 H H H I H + M H + 1 + + + + I I I + + + Oi H *r)> O H O rsi o i L O O I L O O I H T J - L O H ^ J - to t oo L O ^ o 01 o O l O O O i O O l rsi tO tO'O- C O O rsi rsi " * 3 - O O O l IO rsi to vo o o o o o cxi rsi L O O O ^ r-- t-- o o LO H « 3 - 0 H t O 0 0 H O O O i ^ i - O N r - rs) O O OOtO H • • • • » • • • • • • • • • • O I L O r s i r s l T L O H H H •«-)- o o H r - ~ t o o r- oo L O O + + t o r s i t o r s i i + + i H H H I r s i H N H H + + + + + I I I + + + + + oi H r-~ o t-- to O r s I H O i H O i H t O o t-^  r-~ o t o •ct r-~ o 0 0 H H t O H ts H t o H t o to to to^ i - O O H rsi rsi r - - O i O " * ^ -^a-to to r-» r- t-» r-- o o rsi rsi s t io o o oooo o o t o rsi oo to to co H O o o o o H L O H r-- r-- H L O oo ^ j - o o o o L o r s i o to oo too> o o H t o o o o I I H + H + H H I I H H H I + + H + I I + + I I I I I + L O O H rs) Oi to O H O H O H CO O H LOr~- tO H O i O O i L O t s r~-oo H H«rj- r s i r s i to r--. 0 1 r s i r s l H N H t o to to rsirs) o o O O O O r s i r s i O O r-^  r-- O O O L O O ^ LOr~- O r - - t - - H o o o O i t O o o o t o c o O l O l L O H H H O O i LOLO O l O l O O l O H CO O N H t o r s i r s i t o H N H H to to I H r s i r s i r s i r s i H r s i H r s ] r s i t o I I I I I I I I I I I I I I I I I I I H H H H tO IO O O « 5 j - ^ t H H O O r s i r s l H H t O t O r - t - - r-- t--H H t--r-- r - - r - r s i r s i H H to to r s i r s i H H L O L O r s i r s i rsirs) n- * 3 - " 3 - T O O r s i r s i o o to to L O L O O O L O co r- oo o oo to r~- r » to o o r s i r s i t o o r- o o o o o t o o o o o o t o o to O i to to t o o "=r H r s i r v i o o r s i t o n to 'tf-r s i t o L O O L O L O o o r s i r s l L O O o o to •rt o t - ~ o o O i O ) to to to to r s i r s i O i O i to to r s i r s i o> O i to to rsirs) L O L O oooo oooo 'a-'rr L O L O oooo T L O L O O O O O -rt rj-L O L O r~-r~ t-» r- oooo L O L O r--r-~ oooo L O L O r - r - oooo L O L O H H H H O O L O L O H H O O L O L O H H O O T3 O c_> rsi Q CJ rsi rsi Q CJ to rsi a CJ rsi Q CJ L O rsi Q CJ O rsi Q CJ rsi Q CJ 00 rsi Q CJ O l rsi Q CJ o to Q CJ Code (W/ni K° ) (hj Ii (h) E i l ) (h) E (I) NC (h) E (h) E GD31 GD32 GD33 55. 59 3 .60 2 .83 -21 .3 3. 11 55. 59 3 .97 2 .83 -28 .6 3. 25 17. 83 5 .97 5 .10 -14 .4 5. 54 17. 83 6 .06 5 .10 -15 .7 5. 81 68. 42 0 .310 0 .273 -11 .7 0. 300 68. 42 0 .340 0 .273 -19 . 5 0. 314 13.5 17.9 -7.0 -4.1 -3.0 -7.4 3.18 3.33 5.64 5.90 0.308 0.322 11.4 •15.9 -5.5 -2.5 -0.6 -5.1 3.15 3.30 5.59 5.86 12.3 16.8 -6.2 -3.2 0.304 -1.6 0.319 -6.1 3.08 -14.2 3.08 -22.3 5 . 58 5.58 -6.4 -7.9 0.457 +47.5 0.457 +34.6 1 2 3 4 5 6 F i r s t l i n e under each code refers to freezing time to reach -10°C Second lin e under each code refers to freezing time to reach -18°C t e represents experimental value of freezing time tp = value predicted by Plank's equation (Plank, 1941) t j = value predicted by a modified Plank's equation (International Institute of Refrigeration (1972)) t^p and t^jc, Nagaoka et a l . modification (1955)of Plank's equation t[vj, Mellor (1976) modification E i s the.prediction error based on experimental value. OO Appendix 2 Comparison between experimental and pr e d i c t e d f r e e z i n g times. Granny Smith GS1 55.59 55.59 2.20 2.43 1.92 1.92 GS2 55.59 55.59 2.10 2 .33 1.92 1.92 GS3 13.85 13.85 4.70 5.20 3.89 3.89 GS4 13.85 13.85 4.67 5.17 3.89 3.89 GS5 59.68 59.68 2 .30 2.93 2.28 2 . 28 GS6 59.68 59.68 2.37 2.97 2.28 2.28 GS7 17.83 17.83 5.30 6.43 4.30 4.30 GS8 17.83 17.83 4 .80 5.93 4.30 4.30 GS9 68.42 68.42 0 .215 0.222 0.184 0 .184 GS10 68.42 68.42 0 . 260 0 .272 0.184 0.184 -12.7 -21.0 2.63 2.72 + 19.7 + 11.9 3.17 3.28 + 44.2 + 34.9 -8.5 -17.6 2.63 2.72 + 25.3 + 16.7 3.71 3.28 + 51.1 + 40.7 -17.0 -25.0 5.34 5.52 + 13.8 + 6.2 6.44 6.65 + 37.1 + 28.0 -16.5 -24.6 5.34 5.52 + 14.4 + 6.8 6.44 6.65 + 37.9 + 28.7 -0.5 -21.9 3.09 3.19 + 34.5 + 9.2 3.69 3.81 + 60.5 + 30.2 -3.4 -22.9 3.09 3.19 + 30.5 + 7.7 3.69 3.81 + 55.7 + 28.5 -18.7 -33.1 5.84 6.04 + 10.3 -6.0 6.99 7.23 + 32 .0 + 12.5 -10.3 -27.4 5.84 6.04 + 21.8 + 1.9 6.99 7.23 + 45.8 + 22.0 -14.2 -17.0 0.250 0.258 + 16.4 + 16.6 0.300 0.309 + 39.3 + 39. 5 -29.1 -32.3 0.250 0 .258 -3.7 -4.9 0.300 0.309 + 15.2 + 13.8 2. 93 + 33. 3 2. 34 + 6.5 3. 03 + 24. 7 2. 34 -3.5 2. 93 + 24. 7 2. 34 + 11.6 3. 03 + 30. 0 2. 34 + 0.6 5. 95 + 26. 7 4. ,82 + 2.7 6. 15 + 18 . 3 4. ,82 -7.1 5. 95 + 27. 4 4. ,82 + 3.3 6. 15 + 18. 9 4. ,82 -6.7 3. ,42 + 48. 9 2. ,74 + 19.3 3. ,54 + 20. 9 2. .74 -6.3 3. ,42 + 44. ,5 2, .74 + 15.8 3. ,54 + 19. ,2 2, .74 -7.6 6. .48 + 22. .4 5 .21 -1.6 6, .70 + 4, .3 5 .21 -18.9 6 .84 + 35, .1 5 .21 + 8.6 6 .70 + 13 .1 5 .21 -12.0 0 .277 + 29 . 1 0 .317 +47.8 0 .286 + 29 .3 0 .317 +43.2 0 .277 + 6 .8 0 .317 +22.2 0 .286 + 5 .5 0 .317 +16.9 100. H O rH rH rHCTl O l O i IV t— O t— ( O 00 s t i n ^ ^ O O o r g o t o r f L O o 0 1 • « » 0 1 n m H o r - «-r o + r H + i—t r H + I I I r H + r H r H I I r H " T T I H 1 1 + 1 1 + I I U O L O r— r— r— r— o i 0 1 0 1 0 1 0 1 0 1 O I O I L O L O L O L O 0 1 0 1 t o t o H H H H O l O l C l O l H H H H "<3- " * N N r - r— N N r g r g t o t o t o t o N N r g r g r t -3- o o r g r g oo Oi r— H r— o r g «a- ^3-tO * * H H O O o r g r g r g O " - ! -r g u o r g r - t o o H \ D r g ^ - t o o L O O r o H o m O H + 1 + 1 H H I I I H + H H I + H + + H + + + 1 1 + I + O 00 t o r g t o r g oo <3- oo «a- t o "3- t o o c i o cn • H H O I O r g t o r g t o O i H cn H r g t o r g to o r — o r - - r g r g H t o r g r g r g r g t o t o r g r g r g r g **• <3- O O t o t o c i o cn o o o o t o L O i n i n « 3 - r g r— r— m o H H I O H t o m t o o * 3 - H o w H r o rs-cn o o "-ro M N o r — + 1 + 1 H H I I I H + 1 H + + H + + H + + + I + 1 + t o r g ui Tt L O T J - H O O H O O O O O O O O O O H N r— cn r g t o (M t n O H O H r g t o r g t o o o o o o o r g r g t o r j -r g r g r g r g r t - "=t r g r g r g r g r i - ^ - ^ j - o o t o t o ^ j - r g t o t o rt- t— t o L O m r o t o n o o H O O o o o o t o oo H r - H O O r g co r g r— t n i n N H t o r g H r g r o r g N i n + 1 + 1 H + I I I H + H H I + H + + + 1 + 1 1 + I t_n t o o oi o oi t o o i t o o i H H H H r g o N O O H o r -r g r g r g r g O i o O i o r g t o r g t o m r - - i n t— r g r g c n o r g r g r g r g t o r o " - f r g r g r g r g ^a-^r o o r g t o o t o o t o r - oo O I H o o r— t o L O H t o H on tr N H r— oo t— c n t o r n o i o o H m L O H N m oo O H c n o o I H I H + i I H H r g i r g + H I r g i i—i H r g i i i l l l l l i l l r f H H H H r g r g r g r g <3- 'd- c n c n c n o i o o o o -rj-^-o o o o o o o o o o o o o o o o H H t o t o r g r g r g r g t o t o t o t o r g r g r g r g -a- r j - o o r g r g oo oo r— r— r - o o r - r o r o o t o r— o o t o r— O O i o o o H r f H L O m r — O ^s- t o r— H O O O O « t r f H r g c n r g r g r g r g r g t o t o n-^s- r g r g r g r g r r L O o o r g t o t o r o t o t o cncn cncn m m m m o o o o o o o o m m m m o o o o o o o o o o o o o o o o o o o o «-r cn cn m m m m m m m m m m oo r o ro ro o i O i m m Ol o i m m oo co o o m m m m C3 00 t o H oo ID 00 u m H 00 ID o H 00 00 oo O l H OO o r g 00 u 101. o US r g so T L O O T r g  CC.'~*. o rsj 0 1 H  <X>, U] Ol LO t— 0 0 R " ^ 0 O J T T ro' Ol" O O T" 2 O O O * T O £ ° } I H + 1 + r - t ™ * ^ + . H N ™ I I + r H ^ ro to ro to rsi rg o o rsi rsi rsi rsi oooo r s i r s i cn o i o o o o o o o o t— r— r— r— r— r— r o r o T T r— r— r o r o Oi CTi oooo L O L O rsi rsi L O L O L O L O O O r s i r s i T T O O r o r o O O O O r-t rsi rg T L O r s o i o rsi co to to o t o H C O o d %t s t O L O t o r - l o i o r s i r s i O r - 1 O L O r o o cn ro r— H o ro H + rs) i—I H H + + + + H H I rH H H rs],—I rH + + + + + + I I I + + + + + cn o o o cn T cocn H L O H L O r H t o r - r — t— oo o r — o r g to f—I H to r s i r o o o o o o o t o t o T L O o o o rsi rsi r— cn t o o <t st t o t o o r — o r — o o r s i r s i T L O O O T T oooo o o rg r— r H O rsioo O N r o o to T r — L O cn to H O I r — L O O H H O O r — r — cncn r — r o o i T H O O O H o c n r— o r s j H t o n r s i H + + + + I H I I r g r g t o n H H + + + + + + l + + + + + + cn H L O o t— rg o o o L O O L O O to L O o o t o t o o r — r— L O r g t o T O T O rg L O rg L O t o t o L O O Oi H r g r g o r g c n r g T T to to r — r — r — r — o o r g r g T L O O O L O L O co cn o o t o n r o L O o o o t o o r — L O L O T O I L O I O L O o c n T O I T t o t o r g O H L O L O T O H O L O N O i T L O H H T + 1 H + H + I I + + H H I H + + H + + 1 + + I I I + to to T T L O 0 0 r g t o o o o o o o O i o t O T H H L O O r— rg O O T oo oi O H N < * rg T r g t o T L O O O O r g r g to L O L O O O t o t o r o r o O O O O O O r g r g T L O O O T T t — r — o o r g T o o N H c n r — o o cn T H O H L O cnoo rg T cn r— r g t o L O T o c n T H O O O O L O o r g N U I r g c n H r g H r g H r g r g r g I H H r g H r g H r g H r g rg rg I I I I I I I I i I I I I I I I I I I O O O O L O L O T ^ S - t o t o r o r o r g r g r o r o H H r g r g L O L O H H O I O I r o r o oooo oooo r g r g r g r g T T rg rsi t o t o r — r — r g r g r g r g T T T T o o r g r g T T o o t o t o L O L O o o o rg o rg o o i O t o r o r o o r — H N r o r o T O r — o o t o o o oo H cn rg L O I O r— to ro to I O L O T O r g t o o t o L O r— to T r g t o L O O L O O o o r g r g L O O O O T T o r — o o Oi Oi t o t o r o r o r g r g O i O i r o r o r g r g o i Oi r o r o r g r g L O L O 0 0 0 0 O O O O T T L O L O 0 0 0 0 T T L O L O 0 0 0 0 T T L O L O r — r — r — r — oooo L O L O r — r — oooo L O L O t — r — oooo L O L O H H H H O O L O L O H H O O L O L O H H O O O U rg 00 CJ rg rg CO CJ to r g 00 CJ T rg 00 CJ LO r g 00 CJ o rg 00 CJ t— rg 00 CJ oo r g 00 CJ Ol rg 00 CJ o to 00 CJ Code ( W / n T K ° ) (h) (hj E (%) (h) E tNF (h) E NC (h) E (h) E ( ? o ) GS31 GS32 GS33 55. 59 55 .59 17.83 17.83 68.42 68.42 3.73 3.03 4.20 3.03 6.17 5.23 7.10 5.23 -18.6 -27.7 -15.1 -26.3 0.330 0.295 -10.4 0.362 0.295 -18.3 3 .31 -11 .3 3 .39 -9 .0 3 .35 -10 .0 3. 27 -12. 1 3 .44 -17 .9 3 .53 -15 .9 3 .49 -16 .8 3. 27 -21. 9 5 .77 -6 .4 5 .96 -3 .3 5 .88 -4 .7 5. 73 -7. 0 6 .01 -15 .3 6 .21 -12 .5 6 .12 -13 .7 5. 73 -19. 2 0 . 322 -2 .2 0 .330 + 0 .2 0 .327 -0 .9 0. 470 +42 .6 0 .336 -7 .2 0 .344 -4 .9 0 .340 -5 .9 0. 470 +30 .0 1 F i r s t l i n e under each code refers to freezing time to reach -10°C 2 Second l i n e under each code refers to freezing time to reach -18°C 3 t e represents experimental value of freezing time 4 t p = value predicted by Plank's equation (Plank, 1941) 5 t j = value predicted by a modified Plank's equation (International Institute of Refrigeration (1972)) 6 tj^p and t j j c , Nagaoka et a l . modification (1955)of Plank's equation 7 tj^, Mellor (1976) modification E i s the prediction error based on experimental value. 8 i — 1 o Is) 103. Appendix 3 A computer program for predicting freezing times of foods with variable thermal properties in c y l i n d r i c a l containers. C THIS PROGRAM CALCULATES THE FREEZING TIME IN FOODS IN C CYLINDRICAL CONTAINERS BASED DN VARYING THERMAL PROPERTIES C C IMPLICIT REAL *3(A-H,O-Z> DIMENSION T l l O O l , X K ( 1 0 0 ) , C ( 1 0 0 ) 500 READ!5,1,END=1 6)RM,DR,RHOI,RH02,HTCtTI,TF,TA,TM,TC, ENDT i FORMAT ( F 5 . 4 , F 6 . 5 , 2 F 5 . l , F 5 . 2 , 5 F 5 . 1 , F 3 . 1 ) N=RM/DR+2 DO 10 1=1,N T(I»=TI 10 CONTINUE 990 WRITE (6,2000) T I M E , ( T i l ) , 1 = 1 , N ) 2000 FORMAT!• TI ME(HOURS)•,20X,•TEMPERATURE C',/15X, 2* T l T2 T3 T4 T5',/23X, 3» T6 T7 T8 T9 T i O T i l 4///2X,F8.4,5X,5F8.I,/23X,7F8.1) XK ( I ) = 3 . 6 * ( 0 . 3 67+0.03250*T(I)) C(I)=3.40+0.0049*T(I> D T H = ( R H 0 1 * C < 1 ) * ( D * * * 2 ) ) / ( X K ( l ) * 2 . 0 ) K=ENDT/DTH TIME=0.0 DO 19 J=l,K 11 DO 15 1=1,N X K ( I ) = 3 . 6 * l 0 . 3 6 7 + 0 . 0 0 2 5 0 * T l I ) ) C U ) = 3.40+0.0049*T(I| XN=XK{ 11/(HTC*DR) r IF IT( I ) - T F ) 20,20, 12 ! 12 IF ( I - l ) 25,25,30 30 IF (I-N) 204,40,40 C 25 T l 1 ) = 2 . * T ( 2 ) - T ( 1 ) M T A - T ( l ) )*<N-l)/(XN*(N-2) ) D T H = ( R H 0 1 * C ( I ) * ( 0 R * * 2 ) ) / ( X K ( I ) * 2 . 0 ) I F ( T d ) . L T . T F ) GOTO 26 GO TO 15 26 T(1)=TF GO TO 15 204 T ( I 1 = ( T < l + l ) + T ( I - l ) )/2. T X = ( T ( I - l ) - T ( I + l ) ) / ( 4 . 0 * ( N - I ) ) T ( I )=TI I)+T X TY=T(N-1) IF( T( D . L T . T F J GOTO 27 GOTO 15 27 T ( I ) = T F GO TO 15 40 T ( N ) = ( T ( N - l ) + T Y ) / 2 . 0 I F t T I N I . L T . T F ) GOTO 28 GOTO 15 28 T(N»=TF GO TO 15 104. 2 0 T Z = T M - T ( I ) I F ( T Z . G T . 0 . 0 ) GOTO 2 2 0 XK( I } = 3 . 6 * ( 1 . 0 6 6 - 0 . O i l 1 * T t I ) ) C ( I . = 2 . 6 5 - 1 . 4 2 1 * T l I ) XN=XK( 1) / ( HTC*DR ) I F ( I - l ) 1 2 5 , 1 2 5 , 1 3 0 1 3 0 I F U - N ) 3 0 4 , 1 4 0 , 1 4 0 12 5 T ( 1 ) = 2 . * T ( 2 ) - T ( 1 ) + ( T A - T ( 1 ) ) * ( N - 1 ) / ( X N * ( N - 2 ) > D T H = ( R H 0 2 * C ( I ) * ( D R * * 2 ) ) / < X K U ) * 2 . 0 ) GO TO 15 3 0 4 T U t = ( T ( I + l ) + T ( I - L ) } / 2 . 0 TX=( T( I - l » - T t I + l ) ) / ( 4 . 0 M N - I ) ) T U ) = T ( I ) + T X TY= T( N-1 ) GO TO 15 1 4 0 T(N> = ( T ( H - l ) + T Y ) / 2 . 0 GO TO 15 C 22 0 XK( I ) = 3 . 6 * ( 1 . 0 6 6 - 0 . 0 1 U * T ( I ) ) C ( I ) = 2 4 . 4 0 + 0 . 7 9 1 0 * T ( I ) X N = X K ( 1 ) / ( H T C * D R ) I F ( I - l ) 2 2 5 , 2 2 5 , 2 3 0 2 3 0 IF ( I - N J 4 0 4 , 2 4 0 , 2 4 0 2 2 5 T{ 1) = ? . * T { 2 ) - T { 1 ) • ( T A - T ( 1) ) * ( N - 1 ) / i X N M N - 2 ) ) D T H = ( R H 0 2 * C ( I ) * ( D R * * 2 ) ) / ( X K I I | * 2 . 0 ) GO TO 15 4 0 4 T U ) = ( T< 1*1 ) * T I 1-11 1 / 2 . 0 T X = ( T ( I - l . - T l 1 + 1 ) ) / ( 4 . 0 * ( N - I ) > T ( I ) = T ( I J + T X TY=T< N-1 ) GO TO 15 2 4 0 T ( N ) = ( T ( N - l ) + T Y ) / 2 . 0 15 CONTINUE C I F I T ( N ) - T C ) 5 0 0 , 8 5 , 8 5 8 5 T IME=TIME+DTH 9 9 9 WRITE ( 6 , 2 0 0 1 ) TI M E , t T ( I ) , I = 1 , N ) 2 0 0 1 FORMAT ( 2 X , F 8 . 4 , 5 X , 5 F 8 . 1 / 2 3 X , 7 F 8 . 1 ) 19 CONTINUE 16 STOP END C c c C R M = R A D I U S , DR=RADIUS I N C R E M E N T , R H 0 1 = D E N S I T Y DF UNFROZEN M A T E R I A L C R H 0 2 = = D E N S I T Y OF FROZEN M A T E R I A L , HTC=HEAT T R A N S F E R C O E F F I C I E N T , C T I = I N I T I A L T E M P E R A T U R E , T F= FREE Z I N G P O I N T , T A = A M B I E N T TEMPER A T J R E , C T M = I N T E R M E D I A T E T E M P E R A T U R E , TC=TARGET T E M P E R A T U R E , ENDT= T IME C L I M I T , D T H= TIM E I , X K U I AND CCD ARE TEMPERATURE F'JNCT I S DF C THERMAL C O N D U C T I V I T Y AND APPARENT S P E C I F I C HEAT Appendix 4 Schematic diagram of the se c t i o n s of a c y l i n d e r f o r a f i n i t e d i f f e r e n c e scheme. Center line r = mAr m Centre ; Surface ; I n t e r i o r ; m = 0 m = M M > m > 0 

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