UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Thermal processing of low-acid foods with sub-zero initial temperatures Sandberg, Gary 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A6_7 S36_5.pdf [ 6.44MB ]
Metadata
JSON: 831-1.0098677.json
JSON-LD: 831-1.0098677-ld.json
RDF/XML (Pretty): 831-1.0098677-rdf.xml
RDF/JSON: 831-1.0098677-rdf.json
Turtle: 831-1.0098677-turtle.txt
N-Triples: 831-1.0098677-rdf-ntriples.txt
Original Record: 831-1.0098677-source.json
Full Text
831-1.0098677-fulltext.txt
Citation
831-1.0098677.ris

Full Text

T H E R M A L P R O C E S S I N G O F L O W - A C I D F O O D S W I T H S U B - Z E R O INITIAL T E M P E R A T U R E S By Gary Sandberg B. Sc. (Microbiology) The University of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES FOOD SCIENCE We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1991 (c) Gary M. M. Sandberg, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Heat penetration t e s t s were c a r r i e d out f o r a flaked ham product and a 40% bentonite dispersion with i n i t i a l temperatures of 0°C, -5°C, -10°C, -15°C, and -20°C. The cans were processed at 124.4°C f o r 75 minutes i n a steam r e t o r t . Data was analyzed by the general method, B a l l ' s formula method and Stumbo's method. Thermal d i f f u s i v i t y and d i f f e r e n t i a l scanning calorimetry measurements were also made of the model food and the bentonite dispersion. From t h i s data, a f i n i t e d i f f e r e n c e model was generated to y i e l d an adequate thermal process time and temperature f o r meat products e x h i b i t i n g sub-freezing i n i t i a l temperatures. Substantial agreement was found between experimental cold spot temperatures within cans and c o l d spot temperatures calculated with the model. i i T A B L E O F C O N T E N T S Abstract i i List of Tables v i List of Figures v i i 1 I N T R O D U C T I O N 2 L I T E R A T U R E R E V I E W 5 2.1 Thermal Processes f o r Frozen Food Products 5 2.2 Meat Emulsions 7 2.2.1 Properties of Meat Emulsions 7 2.2.2 Structure and C h a r a c t e r i s t i c s of Meat Proteins 10 2.2.3 Formation of Meat Emulsions 15 2.3 Factors A f f e c t i n g Meat Binding Capacity 17 2.3.1 Protein Extraction 18 2.3.2 pH 19 2.3.3 S a l t s 20 2.4 Bentonite Models For Food Systems 24 2.5 P r e d i c t i o n of Food Product Thawing Rates 30 2.6 Plank's Equation ..32 2.7 Improvements on Plank's Equation 36 i i i 2.8 Heat Transfer 39 2.8.1 Conductive Heat Transfer 42 2.9 Thermal Properties of Food Products 45 2.9.1 S p e c i f i c Heat and Enthalpy 47 2.9.2 Thermal Conductivity 51 2.9.3 Thermal D i f f u s i v i t y 56 2.9.4 Surface Heat Transfer C o e f f i c i e n t s 63 2.10 Heat Transfer and F i n i t e Difference Models 68 2.10.1 Heat Transfer 68 2.10.2 F i n i t e Difference Analysis: Two Dimensions..72 3 T H E R M A L P R O P E R T I E S 3.1 Materials and Methods 86 3.1.1 D i f f e r e n t i a l Scanning Calorimetry 86 3.1.2 Thermal D i f f u s i v i t y 89 3.2 Results and Discussion 92 3.2.1 D i f f e r e n t i a l Scanning Calorimetry 92 3.2.2 Thermal D i f f u s i v i t y 103 4 D E V E L O P M E N T O F T H E FINITE D I F F E R E N C E M O D E L i 0 7 i v 5 E M P I R I C A L C O N F I R M A T I O N O F T H E M O D E L 5.1 Materials and Methods 116 5.2 Results and Discussion 117 6 C O N C L U S I O N S A p p e n d i x A 131 A p p e n d i x B 138 References i 4 3 V List of Tables 2.1 Thermal processes f o r canned tuna 6 3.1 Composition of chunk, emulsion and blend portions of flaked ham as well as the bentonite dispersion 95 3.2 S p e c i f i c heat and apparent s p e c i f i c heat f o r the chunk, emulsion and blend portions of the flaked ham product; and the bentonite dispersion f o r above free z i n g and the phase t r a n s i t i o n r e s p e c t i v e l y . 96 3.3 Thermal d i f f u s i v i t y of flaked ham and bentonite at various temperatures 104 5.1 Mean time to reach 0°C from i n i t i a l product temperatures of -20°C, -15°C, -10°C, -5°C, and 0°C 118 5.2 Mean values f o r i n i t i a l temperature, f h , and j f o r bentonite and ham at -20°C, -15°C, -10°C, -5°C and 0°C 120 5.3 Calculated process time comparison f o r flaked ham and bentonite 123 v i List of Figures 2.1 Structure of Muscle Fibre 12 2.2 Montmorillonite 28 2.3 I l l u s t r a t i o n of One-Dimensional Freezing 34 2.4 Conventions f o r Conductive Heat Flow 43 2.5 D i f f e r e n t i a l Scanning Calorimetry Curves 49 2.6 Hot Plate Method Schematic... 53 2.7 Hot Wire Method Schematic 54 2.8 Dickerson Apparatus 61 2.9 Nomenclature Used f o r Two-Dimensional Numerical Analysis of Heat Conduction 74 2.10 Nomenclature f o r Nodal Equation with Convective Boundary Condition ..81 3.1 DSC Thermogram Representative of Thawing 88 3.2 Thermal D i f f u s i v i t y Apparatus 91 3.3 DSC Thermograph f o r Flaked Ham — Chunk Portion 93 3.4 DSC Thermograph f o r Flaked Ham — Emulsion Portion...98 3.5 DSC Thermograph f o r Blended Flaked Ham 100 3.6 DSC Thermograph f o r Bentonite Dispersion../. 101 4.1 Nomenclature Used i n the Numerical Analysis of Heat Conduction i n the F i n i t e Cylinder.... 109 5.1 I n i t i a l Temperature Versus Process Time (Pt) f o r Flaked Ham and Bentonite 121 5.2 Heat Penetration Data Comparison — Actual and F i n i t e Difference 124 v i i Chapter 1 INTRODUCTION "No one s h a l l thermally process any canned food product having an i n i t i a l temperature below the f r e e z i n g point of water." This statement, although not e x p l i c i t y stated i n any l i t e r a t u r e or regulation, has come to be accepted as a tenet i n the commercial food processing industry. The reasons f o r t h i s acceptance are unclear; however there are several possible explanations. F i r s t , B a l l and Olson (1957) recommended the usage of thermocouples f o r heat penetration evaluations that were s u i t a b l e f o r the temperature range of 4.4 to 162.8°C. Second, the p o s s i b i l i t y of i c e being present i n the product and the required l a t e n t heat of r e c r y s t a l i z a t i o n required to change the i c e to water i s also a f a c t o r . The heat that goes into carrying out t h i s phase change may r e s u l t i n spore-forming organisms escaping the f u l l s t e r i l i z a t i o n treatment required to bring about a "commercially-safe process." Some preliminary work has been c a r r i e d out i n the laboratories of American National Can Company (Fairbrother, 1989) during the e a r l y 1960's, involving canned luncheon meat with an i n i t i a l temperature of -28.9°C. The r e s u l t s of the experiments suggested that while the f h and were d i f f e r e n t , 1 I N T R O D U C T I O N 2 the product didn't e x h i b i t a process time (Pt) any longer than that e x h i b i t i n g an i n i t i a l temperature above freezing (1°C). They observed a substantial increase i n the value, and a decrease i n f h . They also suggested that a reasonable time delay between f i l l i n g and processing of the product d i d not r e s u l t i n a frozen product, j u s t one with a low i n i t i a l temperature. Why then i s there a need to explore the processing of food products having i n i t i a l temperatures below freezing? Due to r i s i n g material costs i n the meat processing industry, the comminuting of meat and other ingredients to create meat emulsions r e s u l t i n g i n structures analogous to i n t a c t muscle pr o t e i n has become common. In t h i s process, protein e x t r a c t a b i l t y has been shown to be of prime importance (Schmidt, 1987). Maximum extraction of myosin protein occurs i n the range of -5°C to +2°C (Bard, 1965) . At the lower range of these temperatures the p o s s i b i l i t y of the presence of i c e i n the canned product must be considered. Furthermore, the thawing of meat materials r e s u l t s i n the loss of a c e r t a i n percentage of soluble p r o t e i n through d r i p -loss ( M i l l e r et a l , 1980) . The use of frozen ingredients could r e s u l t i n s i g n i f i c a n t cost-savings. In addition, p r i o r thawing of ingredients increases the p o t e n t i a l f o r microbial d e t e r i o r a t i o n of highly perishable ingredients before thermal processing. INTRODUCTION 3 Thus, a need e x i s t s to ensure that a product having an i n i t i a l temperature below fre e z i n g can be s a f e l y processed, and that a model can be structured that can be applied to a l l such products. This was accomplished by pursuing the following objectives; 1. Measurement of the heat flow into the two systems (a flaked ham product and a 36.2% bentonite dispersion), aimed at determining at what temperature the i c e c r y s t a l s underwent the phase change to l i q u i d water. 2. Measurement of the increase i n thermal process time required to accommodate t h i s phase change as the i n i t i a l temperature progressively decreased from 0°C to -20°C i n 5°C increments. 3. Assessment of the s u i t a b i l i t y of bentonite as a model, not only f o r food systems i n general, but f o r frozen foods as well . 4. Determination of the o v e r a l l behaviour of the temperature p r o f i l e of the flaked ham and the bentonite systems with respect to i n i t i a l temperatures i n the frozen range. 5. The f e a s i b i l i t y of an e f f e c t i v e thermal d i f f u s i v i t y when crossing the temperature range of the phase change. 6. To t e s t the effectiveness of the f i n i t e d i f f e r e n c e model i n simulating the thermal processing of food INTRODUCTION products with i n i t i a l temperatures i n the frozen range. Chapter 2 LITERATURE REVIEW In d e s cribing how a model has been applied to the s t e r i l i z a t i o n of meat products that e x h i b i t an i n i t i a l temperature below the freezing point of water, the following review f i r s t considers the system that e x i s t s i n a meat emulsion. The second area to consider i s that of the thermal properties involved, followed by the mathematical methods commonly used to model heat t r a n s f e r . 2.1 Thermal Processes f o r Frozen Products: A survey of government regulations and various processes l i s t e d i n l i t e r a t u r e indicates no scheduled thermal process f o r r e t o r t i n g food products e x h i b i t i n g an i n i t i a l temperature below the fr e e z i n g point of water. An exception comes from Lopez (1987) , who l i s t s processes f o r canning tuna with an i n i t i a l temperature of -1.1°C (30°F) (Table 2.1). However, under the sections concerning processing parameters, i t was mentioned that the f i s h was thawed f i r s t to -2.2°C (28°F) f o r ev i s c e r a t i o n , then precooked to a f i n a l backbone temperature of 71.1°C (160°F) . The f i s h was then cooled overnight before canning. Thus, even though a process f o r an i n i t i a l 5 LITERATURE REVIEW Table 2.1 Thermal processes for canned tuna (Lopez, 1987) TUNA, WAHOO, BONITO OR YELLOWTAIL, All Styles, In Oil (a) Minimum Initial TcmpCTimc Minutes at Retort Temperature Can Size CF) 230T 235T 240T 24J*P 230T 2 1 1 X 1 0 9 30 93 71 57 48 41 30 93 69 S3 46 40 70 92 68 34 43 39 3 0 7 X 1 1 3 30 121 93 80 69 62 30 119 93 78 67 60 70 117 91 73 63 38 307X200.23 30 132 106 89 78 70 30 129 103 87 76 68 70 127 101 83 73 65 307 x 207 30 146 120 103 92 83 30 144 117 101 89 80 70 141 113 98 86 78 4 0 1 X 2 0 3 30 136 129 112 100 91 30 133 127 109 97 88 70 ISO 124- 106 94 «* 4 0 1 X 2 1 1 30 177 149 130 117 107 30 174 . 146 127 114 104 70 170 142. 123 110 100 6 0 3 X 4 0 8 30 323 283 237 233 218 All Styles 30 317 278 249 228 211 Except 70 3°* 269 241 219 203 Gnted or Hiked 6 0 3 x 4 0 8 30 388 348 318 296 278 Grander SO 380 340 310 288 270 Ftaked 70 370 330 301 279 261 ( a) For processes in other sterilization systems or in other containers, consult processing gqwittfrjnF with a cunipftrnt thermal processing authority. (b) Initial temperature designates the average temperature of the coldest can in the retort at the time the steam is turned on for the process. LITERATURE REVIEW 7 temperature below 0°C i s l i s t e d , i n p r a c t i c e , the f i s h i s not placed i n the r e t o r t while frozen. 2.2 Meat Emulsions: 2.2.1 Properties of Meat Emulsions: The use of formed and sectioned meat products i s becoming incr e a s i n g l y popular i n today's food processes due to a number of i n t r i n s i c advantages i n t h i s type of product. These advantages include more accurate portion c o n t r o l , simulation of high q u a l i t y meat cuts, improved tenderness, improved e f f i c i e n c y , decreased processing time and increased uniformity of colour, texture, and f a t d i s t r i b u t i o n (Siegel et a l , 1978a). An emulsion i s an intimate mixture of two immiscible l i q u i d s . Since such i s not the case i n meat systems, the d e f i n i t i o n i s not r e a l l y applicable to the concept of a "meat emulsion." A better d e f i n i t i o n of such products would be a complex s o l with s o l i d meat p a r t i c l e s and f a t droplets dispersed i n an aqueous continuous phase. The l i q u i d component i s an aqueous s o l u t i o n of s a l t s and proteins and, at the same time, i t i s a medium i n which insoluble proteins and p a r t i c l e s of muscle f i b r e s and connective t i s s u e s are dispersed. The f a t comprises the dispersed phase of the LITERATURE REVIEW 8 emulsion. The continuous phase has also been r e f e r r e d to as the matrix (Schmidt, 1987; Rust, 1987). When f a t and water are dispersed as f i n e p a r t i c l e s i n the matrix, a multiple phase system i s formed, g i v i n g r i s e to the meat emulsion. Emulsions are made by grinding or chopping meat to a f i n e homogenate with water and the addition of sodium c h l o r i d e . The continuous phase i s not a simple l i q u i d . The symptoms of i n s t a b i l i t y i n the emulsion are not r e s t r i c t e d to phenomena such as f l o c c u l a t i o n and coalescence of dispersed p a r t i c l e s and the p r e c i p i t a t i o n of solutes from the aqueous phase, but also to possible d e s t a b i l i z a t i o n of the matrix i t s e l f , which leads to exudation of water. The s t a b i l i t y of a meat emulsion has been ascribed to two properties determined by the proteins: f i r s t , the water holding capacity (WHC) of the meat, which was mainly responsible f o r the s t a b i l i z a t i o n of the matrix; and second, the f a t holding capacity of the meat (FHC) , which was responsible f o r preventing coalescence of f a t p a r t i c l e s (Schmidt, 1987). The chemical properties of muscle t i s s u e , as well as those of meat homogenates, are determined by the macromolecular meat proteins and filaments b u i l t from them. Together they form a hy d r o p h i l i c system with the properties of a l i m i t e d swelling g e l (Schut, 1976; Siegel et a l , 1979a). Protein p a r t i c l e s are dispersed to such an extent that the properties of the system LITERATURE REVIEW 9 follow mainly from surface forces (Schut, 1976). A l t e r a t i o n of the i n t e r a c t i o n between adjacent protein molecules and between ions and proteins and other small polar molecules, such as ions, caused changes i n the amount of water that could be held within a p r o t e i n network (Hamm, 1971). Thus, by binding water, meat proteins contribute d i r e c t l y to the s t a b i l i t y of the continuous phase (matrix). However, from a thermodynamic standpoint, the two-phase system (matrix and fat) can only be stable i n bulk form with a minimal i n t e r f a c e (Schut, 1976) . I f one decreased the s i z e of f a t droplets into f i n e p a r t i c l e s , there was, of course, a corresponding increase i n the o v e r a l l surface area. This brought about the creation of a metastable emulsion which tended to separate into two phases (Schut, 1976). Some meat proteins, due to s t r u c t u r a l and s p e c i a l properties, however, are excellent emulsifying agents which are capable of s t a b i l i z i n g the emulsion system (Siegel and Schmidt, 1979b). Quantity, and e s p e c i a l l y q u a l i t y , of the proteins were of dec i s i v e importance i n y i e l d i n g a stable emulsion. The condition of the protein r e l a t e s to i t s s o l u b i l i t y , rate of hydration and swelling, and i t s emulsifying a b i l i t y . These facto r s depend on the species, sex, and age of the animal, and are influenced by the preslaughter and postmortem treatment of the animal and meat r e s p e c t i v e l y as well as pH, the ions present, i o n i c strength, etc. (Hamm, 1971; Siegel and Schmidt, LITERATURE REVIEW 10 1979b). The s t a b i l i t y of the matrix was the main fac t o r i n determining emulsion s t a b i l i t y . This i s determined to a great extent by the rate of hydration of the meat proteins, which i s required f o r the formation of a coherent and viscous matrix necessary to immobilize f a t p a r t i c l e s (Siegel and Schmidt, 1979b). In order to more f u l l y understand the properties of a meat emulsion, one must look at the structure of meat t i s s u e s as we l l as the properties of meat proteins. 2 . 2 . 2 Structure and C h a r a c t e r i s t i c s of Meat Proteins: On the average, lean meat i s composed of 75% water, 20% pro t e i n , 3% f a t , 2% nonprotein soluble substances such as glycogen, glucose, glucose-6-phosphate and l a c t i c a c i d as well as minerals (phosphorus, calcium, magnesium, zinc, i r o n , potassium, sodium and trace metals) (Schmidt, 1987). Of course meat ar i s e s from muscle t i s s u e and from s k e l e t a l muscle i n p a r t i c u l a r . S k e l e t a l muscle, that i s muscle t i s s u e involved i n the support and motion of the animal, i s composed of s t r i a t e d f i b r e s . This represents the e s s e n t i a l u n i t of muscle t i s s u e . Fibre c e l l s are multinucleated, long c e l l s which may a t t a i n a length of 1 to LITERATURE REVIEW 11 40 mm (the lonaissimus d o r s i f o r example) , with a diameter of 10 to 100 urn (Cassens, 1987). The f i b r e c e l l s are surrounded by a double-membrane sheath known as the sarcolemma. Each f i b r e i s composed of m y o f i b r i l s , which are surrounded by a f l u i d c a l l e d the sarcoplasm. The proteins present i n the sarcoplasm are soluble i n s a l t solutions of low i o n i c strength (Schut, 1976). Each of the m y o f i b r i l s are from 1 to 2 um i n diameter and are of a h a l f c r y s t a l l i n e , h a l f c o n t r a c t i l e protein g e l (Schut, 1976) . The m y o f i b r i l s occupy 55% of the volume of the muscle f i b r e (Rust, 1987). The proteins of the m y o f i b r i l are p r i n c i p a l l y responsible f o r the i n t e g r i t y of the meat emulsion (Siegel and Schmidt, 1979a). The fu n c t i o n a l u n i t of each m y o f i b r i l i s d i s t r i b u t e d between two z - l i n e s . The m y o f i b r i l l a r proteins are the s t r u c t u r a l proteins and are soluble i n concentrated s a l t solutions. They are thus known as the s a l t - s o l u b l e proteins. Muscle i s composed of bundles of f i b r e s , which consist of bundles of m y o f i b r i l s (Figure 2.1). Under the electron microscope, these m y o f i b r i l s can be seen to be composed of p a r a l l e l filaments. The t h i n filaments (about 50 A diameter) are a c t i n and the t h i c k filaments (about 100 A diameter) are myosin (Cassens, 1987; Bandman, 1987). Therefore, muscle i s composed of bundles of f i b r e s , the f i b r e s c o n s i s t of bundles of m y o f i b r i l s , and the m y o f i b r i l s c o n s i s t of numerous myosin myofibrils mitochondria myosin filaments actin filaments Z-llnej sarcoplasmic reticulum LITERATURE REVIEW 13 and a c t i n filaments ordered i n s p e c i a l arrangements (Huxley, 1953; 1957; Huxley and Hanson, 1957) , as can be seen i n Figure 2.1. The e n t i r e muscle i s surrounded by a sheath of connective t i s s u e known as the epimysium. Septa of connective t i s s u e penetrate from the inner surface of the epimysium in t o the muscle t i s s u e , thus d i v i d i n g the muscle into bundles of f i b r e s . This type of septum i s known as the perimysium, and contains nerves and the larger blood vessels. The connective t i s s u e enters further into the muscle t i s s u e to envelop each f i b r e and i s then known as the endomysium. Connective t i s s u e consists of formed e l a s t i c elements of e l a s t i n and s t r a i g h t inextensible and nonbranching f i b r e s of collagen and r e t i c u l i n . The properties of connective t i s s u e depend on the movements the muscles have to perform. For example, i n muscles performing gross movements, the amount of connective t i s s u e i s lower. The r e l a t i v e proportion of connective t i s s u e also v a r i es from muscle to muscle and to some extent determines the toughness of meat. The collagen p r o t o f i b r i l s c onsist of a t r i p l e h e l i x , which upon heating convert to g e l a t i n . The r e t i c u l i n and e l a s t i n remain stable. Connective t i s s u e e x h i b i t s l i m i t e d swelling and i s extremely r e s i s t a n t to a c i d and a l k a l i . The molecules have a small number of charged side chains. Although connective t i s s u e i s u s u a l l y regarded as an undesirable waste product, some work LITERATURE REVIEW 14 (Ladwig et a l , 1989) has been done on the use of phosphate to improve emulsion s t a b i l i t y i n high-collagen f r a n k f u r t e r s . Returning to the discussion of the s a l t - s o l u b l e proteins that comprise the m y o f i b r i l l a r unit, myosin accounts f o r 35% of muscle protein. I t i s a c o n t r a c i l e p r o t e i n with a molecular weight of 510,000 and assumes the conformation of an - h e l i x (Schut, 1976). The protein contains large amounts of a s p a r t i c a c i d and glutamic a c i d and su b s t a n t i a l amounts of h i s t i d i n e , l y s i n e and arginine. At p h y s i o l o g i c a l pH, the molecule has a negative charge. I t has an i s o e l e c t r i c point of 5.4. Myosin also has ATPase a c t i v i t y and a strong a f f i n i t y f o r di v a l e n t cations (Schut, 1976). A c t i n comprises 15% of the t o t a l muscle pro t e i n . I t i s also a c o n t r a c t i l e protein, e x h i b i t i n g a globular structure when i s o l a t e d (Schut, 1976). An estimate of molecular weight of r a b b i t s k e l e t a l muscle a c t i n was 42,000 (Elzinga et a l , 1973) . The molecule also has weak ATPase a c t i v i t y (Schut, 1976) . Myosin and F-actin together form a complex c a l l e d actomyosin. This i s the s t r u c t u r a l compound performing the contraction and r e l a x a t i o n of muscle i n the l i v i n g animal (Pearson, 1987). ATP has an influence on the condition of actomyosin. Phosphates have a r e l a x i n g and p l a s t i c i z i n g function as they bring about the separation of a c t i n and LITERATURE REVIEW 15 myosin. Thus, t h i s d i s s o c i a t i o n of a c t i n and myosin brings about a decrease i n v i s c o s i t y . Of the many proteins present i n the m y o f i b r i l l a r bundle, there are two a d d i t i o n a l molecules of i n t e r e s t . One i s tropomyosin, molecular weight 56,000, also known as tropomyosin B. I t i s a complex c o n t r a c t i l e protein with an amino a c i d sequence s i m i l a r to that of myosin (Bailey, 1948). I t does not combine with myosin although i t has been suggested that a c t i n filaments are attached to the z - l i n e by a meshwork of tropomyosin. The other molecule i s troponin, molecular weight 80,000. This protein i s believed to bind tropomyosin to a c t i n h e l i c e s . I t i s of importance i n muscle contraction, e s p e c i a l l y as a system f o r regulating calcium ions i n the l i v i n g muscle (Schut, 1976; Cassens, 1987). 2.2.3 Formation of Meat Emulsions: In order to make an emulsion, a mechanical treatment i s applied to most processed meats. This mechanical treatment i s a p r e r e q u i s i t e f o r e f f e c t i v e binding and the most common methods are mixing, massaging, tumbling and mechanical tende r i z a t i o n . Measurements of the amount of nucl e i c a c i d present i n the cook-out of such processed meats have revealed that the mixing causes c e l l d i s r u p t i o n and breakage, with the LITERATURE REVIEW 16 subsequent release of c e l l constituents including, presumably, the m y o f i b r i l l a r proteins (Siegel and Schmidt, 1979a; Theno et a l . , 1978). In addition, most of the work that has been done with massaging and tumbling systems showed increasing protein e x t r a c t i o n with increasing time or s e v e r i t y of treatment. This was a t t r i b u t e d to greater c e l l d i s r u p t i o n and was supported by evidence from both l i g h t and scanning electron microscopy (Siegel and Schmidt, 1979a; 1979b). The muscle f i b r e s and m y o f i b r i l s , which are normally t i g h t l y packed, separate a f t e r massaging (Siegel et a l , 1978a). Once t h i s structure has been opened, the s o l u b i l i z e d proteins of the exudate are worked in t o the loose f i b r e s t ructure. This allows f o r a more cohesive bond to form between the protein matrix and the meat surface. This i s very s i m i l a r t o the concept envisioned f o r other emulsion products. I t has been suggested (Siegel and Schmidt, 1979a) that the s o l u b i l i z e d proteins i n t e r a c t with ions and form a f i n e p r o t e i n g e l during heating. While mixing increases the binding strength i n poultry and beef r o l l s during shorter mixing cycles, increasing the mixing time beyond a c e r t a i n l i m i t (approximately 30 minutes) gives r i s e to no increase i n cohesiveness and may be detrimental. The exception i s with the massaging of hams which may proceed fo r up to 12 hours (Siegel et a l , 1978a; 1978b). Prolonged chopping may bring about water LITERATURE REVIEW 17 exudation and f a t separation. The temperature of the s l u r r y i s not c r i t i c a l ( i . e . -4° to +26°C) as long as i t i s l e s s than 3°C when the f a t i s introduced (Schmidt, 1987). The meat should be c a r e f u l l y comminuted i n the presence of s a l t f o r a short time before i c e or water i s added. This r e s u l t s i n greater swelling and s o l u b i l i z a t i o n of the m y o f b r i l l a r proteins than i s obtained without s a l t . The more di s r u p t i o n of the meat structure by the uptake of water, the more e f f e c t i v e mechanical d i s i n t e g r a t i o n w i l l be. During mincing, muscle f i b r e s are separated and membranes disrupted. During f i n e comminution, l o n g i t u d i n a l and transverse c u t t i n g of f i b r e s and f i b r i l s occurs. The sarcolemma i s disrupted and m y o f i b r i l s and filaments are freed. Thus, there i s an increased i n t e r a c t i o n of the components of the m y o f i b r i l l a r system with the soluble ions present. The q u a l i t y of comminuted meats i s h i g h l y dependent on processing conditions such as time and temperature as well as on equipment design. Both speed and temperature of chopping may a f f e c t f a t and water release a f t e r heat treatment (Rust, 1987) . 2.3 Factors A f f e c t i n g Meat Binding Capacity: Binding i s defined as the f o r c e / c r o s s - s e c t i o n a l area required to p u l l apart bound pieces of meat. This includes a LITERATURE REVIEW 18 measure of both the cohesive force exerted between the binding matrix and the meat pieces and the strength of the matrix i t s e l f . The mechanism behind t h i s i s complex and not f u l l y understood, yet the following factors have been elucidated: p r o t e i n extraction, pH, presence and concentration of added s a l t s , mechanical treatment, temperature, and conditions of meat storage (Trout and Schmidt, 1984; S i e g e l et a l , 1978b). These w i l l be discussed i n turn. 2.3.1 Protein Extraction: According to Schmidt (1987), p r o t e i n extraction i s influenced by a number of f a c t o r s . F i r s t , a temperature i n the range of -5°C to +2°C gives maximum e x t r a c t a b i l i t y Research by G i l l e t et a l . (1977) found the optimum temperature f o r e x t r a c t i o n to be 7.2°C with a marked decrease i n e x t r a c t a b i l i t y at 0°C, which had been considered to be the optimum temperature. These differences were thought to have been the r e s u l t of v a r i a t i o n s i n the conditions of extraction (Schmidt, 1987). Second, increasing the extraction time re s u l t e d i n increased protein extraction up to a maximum of approximately 15 hours. Third, p r e - r i g o r meat had more extractable protein that post-rigor meat. Fourthly, a sodium ch l o r i d e concentration of 10% gave the highest l e v e l of p r o t e i n extraction (Bard, 1965). LITERATURE REVIEW 19 2.3.2 pH: When pH was increased from 5.5 to 6.5 i n small increments, there was an increase i n the amount of s a l t - s o l u b l e protein extracted (Schmidt, 1987). At the i s o e l e c t r i c point ( p i ) , muscle has a minimum water holding capacity (WHC) and swelling, as a r e s u l t of maximization of e l e c t r o s t a t i c i n t e r a c t i o n s between molecules (Hamm, 1971). At p i , net prot e i n charge i s at a minimum, there i s a maximum of intermolecular s a l t bridges between oppositely charged groups of p r o t e i n and a minimum of e l e c t r o s t a t i c repulsions (Hamm, 1971) . The addition of an aci d or base brings about a cleavage of the s a l t cross-linkages such that e l e c t r o s t a t i c repulsion i s increased. Therefore, WHC and swelling increase because the loosened p r o t e i n network can take up more water. Screening of the p o s i t i v e charges on the proteins by s a l t ions causes weakening of the e l e c t r o s t a t i c i n t e r a c t i o n s and therefore a loosening of the molecular structure and thus an increased uptake of water: |—COO"—+NH3—j + Na + CI" > |—COO +Na Cl+NI^—j In the a c i d i c range of p i , adding NaCl causes water loss because the binding of the anions decreases the e l e c t r o s t a t i c repulsion between p o s i t i v e l y charged groups: LITERATURE REVIEW 20 j—NH 3 + + N H 3 ~ | + 2C1" > |~NH3+-C1 Cl^NrL,— j Cations decrease the WHC of my o f i b r i l s i n the basic range of the p i . The reason f o r t h i s i s that there i s decreased e l e c t r o s t a t i c repulsion between negatively charged groups due to the screening e f f e c t s (Hamm, 1971). The amount of hydration water that i s present i s influenced l i t t l e by the structure and e l e c t r i c a l charges of the muscle proteins or by added s a l t s (Hamm, 1971). S o l u b i l i z e d meat proteins bind to the insoluble components i n the pro t e i n matrix to form a coherent stable combination. Increasing the amount of extracted myosin between meat surfaces r e s u l t e d i n a l i n e a r increase i n binding strength i n a model binding system (Siegel and Schmidt, 1979b). The use of polyphosphates also generally increases the extraction of the m y o f i b r i l l a r proteins. 2.3.3 S a l t s : The subject of added s a l t s has been b r i e f l y touched on i n the above discussion of pH. Due to taste and t o x i c o l o g i c a l considerations, only NaCl and sodium s a l t s of polyphosphoric a c i d are widely used. The main r o l e of these s a l t s i s to contribute i o n i c strength to the system, with the a d d i t i o n a l r o l e of a l t e r i n g the pH. For example, a l k a l i n e polyphosphates LITERATURE REVIEW 21 w i l l increase the pH (usually by 0.1 to 0.4 — depending on the type and concentration), while NaCl and other neutral s a l t s w i l l bring about a decrease i n pH (by 0.1 to 0.2). In the case of NaCl, etc., i t i s due to the displacement of H + by Na + on the meat protein surface with the l i b e r a t e d H + producing the drop i n pH (Schut, 1976). The mechanisms by which s a l t s increase the binding a b i l i t y of meat proteins are: 1) increase the amount of protein extracted; 2) a l t e r the i o n i c and pH environment such that the re s u l t a n t heat-set prot ein matrix forms a coherent three-dimensional structure (Schmidt, 1987). In the course of t h i s review, much mention has been made of heating the emulsion and heat-set proteins. Binding i s a h e a t - i n i t i a t e d reaction since no binding occurs i n the raw state (32°C) (Siegel and Schmidt, 1979a) . I t has been suggested that heating caused previously dissolved proteins to rearrange themselves so that they could i n t e r a c t with i n s o l u b l e proteins on the meat surface and thereby form the cohesive network structure required. Deatherage and Hamm (1960) found that the process began at 45°C, and involved non-covalent molecular i n t e r a c t i o n s . Heating to 70°C brought about no observable formation of intermolecular disulphide bonds. Therefore, they postulated that the s t a b i l i z i n g bonds were formed a f t e r heat denaturation. LITERATURE REVIEW 22 The degree to which heat-induced binding occurs i s a c h a r a c t e r i s t i c of the species of animal. For example, rabbit myosin bound at 40 to 60°C; f i s h m y o f i b r i l s at 30 to 80°C; and beef m y o f i b r i l s at 50 to 90°C. The optimum i o n i c strength also was c h a r a c t e r i s t i c of animal species (Schmidt, 1987). Therefore, heat-induced binding i s an i n t e r a c t i o n between the temperature of heating and the presence and concentration of d i f f e r e n t s a l t s . While the exact i n t e r a c t i o n has not been c l e a r l y elucidated, the temperature of maximum binding of the meat proteins i s dependent on the presence of s p e c i f i c s a l t s , i o n i c strength and pH. As can been seen from the above discussion, meat components perform many functions i n processed meats; i n t a c t muscle chunks, f i b r e s and my o f i b r i l s contribute d i r e c t l y to texture. Texture i s also affected by the i n c l u s i o n of connective t i s s u e and adipose t i s s u e . The i n c l u s i o n of a i r , water droplets and melted f a t within the product a f f e c t the texture of the cooked material, while the i o n i c environment, mechanical treatment, temperature, pH and r i g o r a f f e c t the release of the meat components from the t i s s u e structure. These components act to form the heat-set matrix and also to coat other structures. Species, composition, anatomical l o c a t i o n and m i c r o b i o l o g i c a l q u a l i t y of muscle a f f e c t the functioning of the meat ingredients. Frozen storage, reduced pH due to b a c t e r i a l growth, f a t LITERATURE REVIEW 23 oxidation and high temperature can lead to a decrease i n f u n c t i o n a l i t y of the meat ingredients. For example, Siegel and Schmidt (1979) , found that at temperatures below 45°C, the a b i l i t y of myosin f r a c t i o n s to bind meat pieces was absent however, at temperatures above 45°C; increasing temperature had a l i n e a r e f f e c t on binding, to a maximum temp, of 80°C. In addition, Kronman and Winterbottom (1960) demonstrated that sarcoplasmic proteins are l e s s soluble a f t e r f r e e z i n g of muscle t i s s u e and that there i s moreover a loss of s p e c i f i c components s e p a r a t e d by e l e c t r o p h o r e s i s and u l t r a c e n t r i f u g a t i o n . Awad et a l . (1968) found that t o t a l extractable protein, sarcoplasmic p r o t e i n , and actomyosin e x t r a c t a b i l i t y decreased with frozen storage. They also found a s l i g h t increase i n extractable l i p i d s from beef during eight weeks of frozen storage at -4°C. These degradative changes appear to be evidenced by changes i n the hydration of meat (Deatherage and Hamm, 1960) and i t capacity to hold water (Wierbicki et a l , 1957a; 1957b). Much has yet to be explained about how meat emulsions form and maintain t h e i r structure. The use of many of the pieces of processing equipment, the bowl chopper f o r example, s t i l l remains more of an a r t than a science. LITERATURE REVIEW 24 2.4 Bentonite Models f o r Food Systems: Bentonite-water dispersions have been used i n simulations of a v a r i e t y of heat-transfer applic a t i o n s i n additi o n to app l i c a t i o n s i n o i l - d r i l l i n g and wine c l a r i f i c a t i o n . The use of bentonite-water dispersions as models f o r heat-transfer studies of food systems has been due p r i m a r i l y to the low cost and chemical s t a b i l i t y of these types of dispersions i n r e l a t i o n to food systems (Unklesbay, 1982). Jackson and Olson (1953) were among the f i r s t to use a bentonite-water dispersion to simulate canned foods, although e a r l i e r studies into the rh e o l o g i c a l c h a r a c t e r i s t i c s of bentonite had been c a r r i e d out (Freundlich, 1937; Hauser and Reed, 1937) . Jackson and Olson used 1, 3.25, and 5% dispersions (on a weight basis) to simulate canned foods. The 3.25% dispersion exhibited a broken heating curve s i m i l a r to that found i n some food products upon heating. Dispersions were used to con t r o l v a r i a t i o n s inherent within cans and batches of food products. The researchers concluded that the change from convection to conduction during broken-curve heating was due to the s o l - g e l transformation of the bentonite-water dispersion. Yamano et a l . (1975) used 25% and 40% bentonite-water dispersions i n a r e t o r t study of f l e x i b l e food pouches and found that a 40% bentonite-water dispersion gave s i m i l a r LITERATURE REVIEW 25 thermal d i f f u s i v i t y values to those published f o r meat and f i s h . Due to the stable physiochemical properties and the a b i l i t y to accommodate a wide range of water contents, the researchers recommended the use of these dispersions. Darsch et a l . (1979) used modified bentonite-water dispersions to simulate soups and beverages when t e s t i n g a foodservice t r a y system. The f i r s t system consisted of 5% bentonite, 47.5% g l y c e r i n and 47.5% water, while the second system consisted of 1% bentonite, 49.5% g l y c e r i n and 49.5% water. They noted that the dispersions had heat retention c h a r a c t e r i s t i c s s i m i l a r to many foods. As i n the case of the study by Jackson and Olson (1953) , the use of homogeneous composition was seen as a major advantage of bentonite model foods. Unklesbay et a l . (1980) used bentonite-water dispersions as models f o r steaks during i n f r a r e d heat processing and found that there were no s i g n i f i c a n t d i f ferences i n energy use. A study by Unklesbay et a l . (1981) used an 18.2% bentonite (representing the t o t a l s o l i d s of sausage meat) and 81.8% water (representing the moisture and f a t content of sausage meat) demonstrated s i m i l a r r e s u l t s . The purpose was to model sausage p a t t i e s i n a convective heat processing environment. The r e s u l t was that f o r three d i f f e r e n t oven s i z e s , no s i g n i f i c a n t d i f ferences were found between actual sausage p a t t i e s and the bentonite models. LITERATURE REVIEW 26 In 1983, Niekamp et a l . used a bentonite dispersion to model a heterogeneous food (quiche). They found no s i g n i f i c a n t d i f f e r e n c e i n energy consumption on a per pie basis between the model and an actual quiche during convective heat processing. Peterson and Adams (1983) performed studies on r e t o r t pouches containing a 10% bentonite-water dispersion and experimentally determined values f o r moisture content, density, and thermal d i f f u s i v i t y of the dispersion. S p e c i f i c heat and thermal conductivity values were calc u l a t e d . Niekamp et a l . (1984) used bentonite-water dispersions prepared i n 5% increments from 5 to 50% bentonite to study the thermal properties. They found that as the bentonite concentration increased, the thermal conductivity and density increased and there was a decrease i n heat capacity. The o v e r a l l purpose of that study was to determine selected thermal properties (moisture content, density, heat capacity and thermal conductivity) i n order to f a c i l i t a t e mathematical modelling of foods. Researchers concluded that bentonite-water dispersions were e f f e c t i v e as models of food systems f o r food service energy research, and that the bentonite component of these models simulated the s o l i d s content of the foods. The water component simulated moisture and f a t contents. Bentonite i s a term that now commonly r e f e r s to any clay LITERATURE REVIEW 27 p r i m a r i l y composed of a smectite c l a y material (Grim and Guven, 1978). Although the chief constituent of bentonite i s montmorillonite, i t i s min e r a l o g i c a l l y a heterogeneous substance (Hauser and Reed, 1937). Montmorillonite i s a three-layer structure c o n s i s t i n g of a t e t r a h e d r a l sheet of S i 4 + , 02- and OIT ions between two octahedral sheets of A l 3 + , 02", and OH" ions. Figure 2.2 shows a proposed structure f o r montmorillonite. However, the po s t u l a t i o n of an i d e a l formula i s impossible due to a large amount of isomorphous s u b s t i t u t i o n that occurs among these sheets. Krauskopf (1967) acknowledged the continual presence of absorbed water with the following proposed r e a l i s t i c formula: (Al /Mg /Fe 3 +) 4(Si,Al) 8o 2 0(OH) 4 Hp. Kauskopf (1967) described montmorillonite clays as expanding l a t t i c e clays because water and other polar l i q u i d s can enter between the groups of S i - A l - S i planes and force them apart, thus causing large amounts of swelling. Bentonite c l a y has been found to absorb almost f i v e times i t s weight i n water. In the case of isomorphous replacements, aluminum i s supposedly capable of replacing s i l i c o n , i n which case an exchangeable cation would go along with the aluminum to n e u t r a l i z e what would otherwise be a net negative charge on the l a t t i c e . In addition, i t i s p o s s i b l e that the free oxygen bonds at the edges of the p a r t i c l e s can hold metals and hydrogen, and i t i s further possible that some of the oxygen LITERATURE REVIEW Figure 2.2: Montmorillonite (Freundlich, 1937). LITERATURE REVIEW 29 atoms li n k e d to s i l i c o n can react with water to form hydroxy1 groups, i n which hydrogen i s replaceable (Hauser and Reed, 1937) . In terms of i t s r h e o l o g i c a l properties, bentonite suspensions are t h i x o t r o p i c (Marshall, 1937; Freundlich, 1937). D i l u t e bentonite gels are capable of an isothermal, r e v e r s i b l e s o l - g e l transformation, being l i q u i f i e d on shaking and s e t t i n g spontaneously, showing no change of volume. Hauser and Reed (1937) demonstrated that bentonite dispersions of 1% also rheopectic e x h i b i t rheopectic behaviour ( i . e . i t turn to a g e l r a p i d l y i f sheared mechanically, but take a r e l a t i v e l y long time to s o l i d i f y spontaneously). This may be due to a coagulation of the p a r t i c l e s as soon as they have been made to approach each other by the movement of the l i q u i d (Freundlich, 1937). A poss i b l e explanation f o r these types of behaviour was advanced by Hauser and Reed (1937) as follows, the suspended p a r t i c l e become loosely locked into place to form a structure that w i l l r e s i s t shear. The p a r t i c l e s are not necessarily d i s t r i b u t e d uniformly throughout the t o t a l volume, but are grouped together i n primary c l u s t e r s , which i n turn coalesce to form a network throughout the whole volume interwoven with patches and channels of free d i s p e r s i n g medium. I t i s pos s i b l e that rheopexy manifests i t s e l f i n the presence of any gentle motion which tends to help the primary c l u s t e r s to LITERATURE REVIEW 30 aggregate i n t o the secondary network responsible f o r the r i g i d i t y of the structure. Thus, v i o l e n t mechanical action w i l l break down not only the secondary structure, but also the primary c l u s t e r s of p a r t i c l e s . 2.5 P r e d i c t i o n of Food Product Thawing Rates: In thawing of a food product, the most important consideration i s the thawing rat e . Further consideration must be given to the f a c t that, given equal temperature d i f f e r e n t i a l s , t i s s u e s , gels and a l l other aqueous materials which transmit heat energy p r i m a r i l y by conduction w i l l thaw more slowly than they w i l l freeze. Thus, the shape of the time-temperature curve f o r thawing e x h i b i t s l i t t l e s i m i l a r i t y to a time-temperature curve f o r f r e e z i n g (Fennema, 1975). Examination of thawing curves also reveals that the a d d i t i o n a l time required f o r thawing i s spent j u s t below the melting temperature. This subfreezing temperature zone i s undesirable f o r foods, since r e c r y s t a l l i z a t i o n and microorganism growth are both more l i k e l y i n t h i s temperature range than at any other subfreezing temperature. In addition, many chemical reactions proceed at s u r p r i s i n g l y r a p i d rates i n t h i s temperature zone (Fennema, 1975). Fennema and Powrie (1964) explained these differences, i n the nature of rates of free z i n g and thawing on the basis of LITERATURE REVIEW 31 several properties of water and i c e : 1) Latent heat of c r y s t a l l i z a t i o n of water (large). 2) Thermal conductivity (heat energy i s transmitted four times fa s t e r by ice than by water). 3) Thermal d i f f u s i v i t y (the rate of change i n temperature which ice undergoes i s approximately nine times fa s t e r than f o r water). In addition, four factors i d e n t i f i e d by Fennema and Powrie (1964) which influence freezing, and conversely, thawing rate were: a) the temperature d i f f e r e n t i a l between the product and the heating medium; b) the modes of heat transfer to, from, and within the product; c) the s i z e , type and shape of the package containing the product; and d) the si z e , shape and thermal properties of the product. Considerable information i s avail a b l e to a s s i s t i n describing the rates of heat tr a n s f e r i n various packages and products, however major l i m i t a t i o n s a r i s e i n describing the transient heat transfer with thermal properties as a function of temperature. This i s the case during the thawing of food products, since the apparent s p e c i f i c heat and thermal conductivity are both s i g n i f i c a n t functions of temperature i n the thawing zone or below the i n i t i a l freezing point of the product. Simplifying assumptions which do not account f o r the thermal d i f f u s i v i t y being a function of temperature have been u t i l i z e d i n many studies i n an e f f o r t to obtain a solut i o n to a complex heat LITERATURE REVIEW 32 conduction problem. This has the disadvantage that wherever any freezing time pred i c t i o n method i s used, some imprecision i s i n e v i t a b l e (Cleland and Earle, 1984a). The most straightforward expression a v a i l a b l e f o r computing the freezing time was derived by Plank (1913). By performing the expression i n reverse, an estimation of the thawing time may also be r e a d i l y achieved. 2.6 Plank's Equation: Plank's equation can be used f o r various geometries. As an example, the case of one-dimensional freezing of a product slab w i l l be considered. There are three basic equations involved i n the derivation which account f o r the heat transfer i n various phases of the product during freezing. The f i r s t expression concerns the basic heat-conduction equation f o r the frozen product region which has a variable thickness of x, where T F i s the i n i t i a l q = A(T. - T F )k/x (2.1) freezing point of the product and represents the temperature which e x i s t s i n a l l unfrozen regions of the product, k i s the thermal conductivity of the frozen material, A i s the area of the material being frozen, T, i s the temperature at the LITERATURE REVIEW 33 material surface and q i s the heat flow. Figure 2.3 i l l u s t r a t e s the nomenclature used i n t h i s example. The second expression describes the heat transfer from the product surface to the surrounding medium and i s expressed as q = - T. ) (2.2) where hc i s a convective heat transfer c o e f f i c i e n t at the product surface. Combining Equations 2.1 and 2.2 y i e l d s one expression that accounts f o r heat tr a n s f e r i n series and eliminates the need for knowledge of the surface temperature. A (T„ - T. ) q = (2.3) l / h c + x/k The t h i r d equation describes the rate at which heat i s generated at the freezing front and i s represented as follows dx q = A L n — (2.4) / dt where (dx/dt) represents the v e l o c i t y of the freezing front, p i s the density of the material, and L i s the latent heat of fusion. The next step i s to equate Equations 2.3 and 2.4 and by integrating between the appropriate l i m i t s , to derive Figure 2.3 I l l u s t r a t i o n of one-dimensional freezing of a product section used to derive Plank's equation LITERATURE REVIEW 35 an expression for the freezing time. 2hc a + 8k (2.5) The t o t a l thickness of the slab i s represented by "a". Introduction of the constants "P" and "R" gives the most general form of Plank's equation where P and R are constants which w i l l vary depending on the geometry of the material being frozen. In the case of an i n f i n i t e slab, the constants for P and R are 1/2 and 1/8 respectively (Equation 2.4). For a sphere, P = 1/8 and R = 1/24, and f o r an i n f i n i t e cylinder, P = 1/4 and R = 1/16. The "a" dimension, which i s the thickness of the i n f i n i t e slab, becomes the diameter of a cylinder and a sphere. However, there are major l i m i t a t i o n s i n the use of Plank's equation f o r predicting freezing, and thus thawing, times for food products. The assumption of some latent heat value i s required and the gradual removal of latent heat over a range of temperatures during the freezing process i s not considered. In the case of thawing, the gradual addition of latent heat i s (2.6) LITERATURE REVIEW 36 not considered. A second c r i t i c i s m i s that the procedure only u t i l i z e s the i n i t i a l freezing point of the product i n the equation and neglects the time required to remove sensible heat above the i n i t i a l freezing point. A t h i r d serious l i m i t a t i o n i s that a constant thermal conductivity i s assumed for the frozen region. This i s a serious l i m i t a t i o n from the standpoint that the freezing zone has a constantly changing temperature and phase during the freezing process. Therefore, thermal conductivity becomes a variable, dependent on temperature. The f i n a l factor i s that Plank's equation assumes the product i s i n the l i q u i d phase at the s t a r t of the freezing process. The accuracy of Plank's equation for food products decreases as the percentage of water i n the product decreases (Heldman and Singh, 1981). 2.7 Improvements on Plank's Equation: There have been a number of e f f o r t s on the part of researchers to overcome these l i m i t a t i o n s . Ede (1949) compared Plank's equation to predictions he obtained using a graphical method and f e l t that Plank's equation gave reasonable accuracy f o r predicting freezing times and that the compactness of the equation compensated f o r the s l i g h t inaccuracies. Nagaoka et a l . (1955) incorporated factors to account f o r sensible heat above and below the i n i t i a l freezing L I T E R A T U R E REVIEW 37 point, but the r e s u l t i n g equation s t i l l assumed that a l l of the latent heat was removed at a constant temperature (TF) . In addition, the desired f i n a l temperature of the product (T) could be established, and the value of the latent heat of fusion (L) f o r the water content of the product could be adjusted through the application of the Nagaoka equation. Cleland and Earle (1979a, 1979b) s i g n i f i c a n t l y modified Plank's equation. They wrote Plank's equation i n a dimensionless form where the rel a t i o n s h i p s among the variables were more evident and with the introduction of a new dimensionless number (N = Plank's Number), the influence of sensible heat above the i n i t i a l freezing point, could be incorporated. They were thus able to e s t a b l i s h empirical expressions which yielded accuracies of + 5.2% to ± 3.8% depending on the equations f o r slab or c y l i n d r i c a l geometries respectively. Most p r a c t i c a l freezing s i t u a t i o n s could be covered, but a moisture content of approximately 77% should be recognized. Neumann (1959) used an approach involving one-dimensional heat transfer i n a s e m i - i n f i n i t e body. As i n Plank's equations, three equations were involved i n developing the f i n a l equation. This represented a s l i g h t improvement over Plank's equation since there was not only a more accurate des c r i p t i o n of the freezing process i n foods, but also there was an allowance for d i f f e r e n t thermal c o n d u c t i v i t i e s within LITERATURE REVIEW 38 the frozen and unfrozen portions of the product. The applications, however, are limi t e d by the s e m i - i n f i n i t e body geometry. In addition, while there was an assumption that latent heat of fusion was removed at a constant temperature (TF) , there was no provision f o r d i r e c t incorporation of a surface heat transfer c o e f f i c i e n t into the freezing time computation. This solution to the Neumann problem has been used to compute food freezing times, but i n general c a l c u l a t i o n s are highly complex and involve t r i a l and error evaluation of various constants. Thus, t h i s approach i s u n l i k e l y to be used when other less complex procedures are ava i l a b l e . The f i r s t numerical solutions were derived by Tao (1967) and Joshi and Tao (1974) f o r the i n f i n i t e slab, i n f i n i t e c ylinder and sphere. Charts were then developed for computation of freezing times. The d i f f i c u l t y with the above described methods for the pred i c t i o n of freezing times and thawing i s that while they have reasonable accuracy under i d e a l conditions, a l l have l i m i t a t i o n s . In order to account for a l l the unique features of the food thawing process, the appropriate mathematical expressions must be solved numerically using computer simulation. To a r r i v e at the appropriate numerical mathematical expression, one must f i r s t determine the thermal properties of the food system under study. LITERATURE REVIEW 39 2.8 Heat Transfer: Heat transfer i s one of the most important u n i t operations i n the food industry i n that nearly every process requires eit h e r heat input or removal (Watson and Harper, 1988). This i s done i n order to a l t e r the physical, chemical or storage aspects of the product. Refrigerated or frozen storage are examples of removal of heat used to increase storage l i f e . Conversely, pasteurization and r e t o r t i n g are examples of the input of heat i n order to achieve preservation. In addition, the input or removal of heat can be used to develop colour and flavour or to a l t e r the physical and t e x t u r a l c h a r a c t e r i s t i c s of a food material. In a l l of these applications, physical laws govern the tra n s f e r of heat. Thus, an understanding of these physical p r i n c i p l e s enables one to predict heating phenomena and to determine optimum operation conditions. In understanding the physical laws of heat transfer, one must r e c a l l that heat i s defined as energy that i s transferred as a r e s u l t of a temperature difference. That i s to say, heat tra n s f e r i s a dynamic process wherein there i s a transfer of heat from a hotter body to a colder body. The rate of transfer depends on the temperature difference between the two bodies and t h i s temperature difference i s the " d r i v i n g force" of heat transfer. From a thermodynamic point of view, heat i s LITERATURE REVIEW 40 concerned with relationships between t h i s energy quantity and the equilibrium or steady-state properties of systems. Thus, nothing i s said about the mechanism of the heat transfer process or the rate at which the transfer occurs. Physical processes, of course, do not occur instantaneously. In a study of equilibrium properties, i t i s assumed that there i s no l i m i t on the length of time avai l a b l e for changes to take place. Many engineering problems require nothing more than the application of equilibrium thermodynamic rel a t i o n s h i p s i n order to a r r i v e at a solution. However, i n dealing with thermal energy, one i s faced with the problems of how and at what rate heat gets from one point to another. Heat transfer i s an energy-transfer phenomenon i n which the thermal energy i n a substance i s exhibited through the random motion of molecules, atoms and subatomic p a r t i c l e s . Thus, an increase i n the rate of heat transfer, causes a product's molecules or atoms to move more rap i d l y , r e s u l t i n g i n an increase i n the p a r t i c l e ' s k i n e t i c energy. Heat i s transferred when a f a s t moving p a r t i c l e c o l l i d e s with a slow moving p a r t i c l e , i n the case of gasous or l i q u i d systems. This causes the fa s t e r moving p a r t i c l e to lose k i n e t i c energy which i s gained by the slower moving p a r t i c l e . In a s o l i d material, the transfer of k i n e t i c energy i s through d i r e c t contact of adjacent molecules. Such tra n s f e r of k i n e t i c energy i n materials, whether gaseous, l i q u i d , or s o l i d , occurs LITERATURE REVIEW 41 by three broad mechanisms; these are conduction, convection and r a d i a t i o n . Heat transfer by conduction occurs through d i r e c t contact of p a r t i c l e to p a r t i c l e or by random c o l l i s i o n s , with no bulk movement of material. In the case of heat transfer by convection, the p a r t i c l e s of a f l u i d are free to move about, r e s u l t i n g i n a mixing of warmer and colder portions of the same material. Radiative heat transfer occurs by means of an electromagnetic mechanism through space that may or may not be occupied by matter (for example, through a i r or through a vacuum). Conversion to heat or any other form of energy does not occur u n t i l the ra d i a t i o n c o l l i d e s with matter, at which point i t may be transmitted, absorbed or r e f l e c t e d . Only absorbed energy can appear as heat. As alluded to e a r l i e r , when heat energy i s either added to or removed from a substance, there i s a change i n either the temperature, the physical state or both. A r i s e i n temperature which r e s u l t s from heat input i s known as "sensible heat," since the heat can be "sensed." Heat that i s added that does not contribute to a temperature increase i s known as "latent heat." Latent heat i s heat that brings about a change of state of a substance without a temperature change. The latent heat associated with a change of state of a substance from a s o l i d to a l i q u i d i s c a l l e d "heat of fusion." I f the change i s from a l i q u i d to a gas, i t i s c a l l e d "heat of LITERATURE REVIEW 42 vapourization." The reverse processes ( l i q u i d to s o l i d or gas to l i q u i d ) are c a l l e d "heat of s o l i d i f i c a t i o n (or heat of c r y s t a l l i z a t i o n ) " and "heat of condensation" respectively (Lund, 1975). Since t h i s study concerns the modeling of a conductive heat-transfer system, further discussion w i l l focus on t h i s system. 2.8.1 Conductive Heat Transfer: As has already been noted, the d r i v i n g force for the flow of heat i s temperature. Thus, the larger the temperature difference, or d r i v i n g force, the greater the rate of heat flow w i l l be. The rate of heat flow i s also proportional to the area perpendicular to the d i r e c t i o n of flow. F i n a l l y , as the length of the path f o r a given temperature difference increases, the rate of heat flow decreases. Figure 2.4 i l l u s t r a t e s these conditions i n which heat i s flowing through a d i f f e r e n t i a l section of a material. The distance dx separates the two planes of area A, and over t h i s distance dx, the temperature changes by an amount dT. The rate of heat flow i s thus expressed by Fourier's law (which applies to conductive heat transfer i n s o l i d s , l i q u i d s or gases): LITERATURE REVIEW 43 Figure 2.4 Conventions f o r conductive heat flow (Watson and Harper, 1988). LITERATURE REVIEW 44 dT q = - k A — (2.7) dx Where q i s expressed i n SI units of watts (W), A i n m2, x i n m, and T i n °C or °K. The factor k i s a p r o p o r t i o n a l i t y constant c a l l e d thermal conductivity, with units of watts m/m2 °K. The negative sign i s a convention used i n order to give q a p o s i t i v e value. The flow of heat i n the d i r e c t i o n of the arrow i n Figure 2.4 indicates that the temperature must decrease i n t h i s d i r e c t i o n and therefore dT i s a negative quantity. Thermal conductivity i s a property of a substance and i s rel a t e d to the transfer of k i n e t i c energy, which may be i n a t r a n s l a t i o n a l , v i b r a t i o n a l , or r o t a t i o n a l fashion. Thermal conductivity i s p a r a l l e l i n concept to e l e c t r i c a l conductivity i n metals. Gases have the lowest thermal co n d u c t i v i t i e s and metals have higher thermal conductivities (in f a c t , they may d i f f e r by a factor of 104 or more) . Furthermore, as with most properties, thermal conductivity depends on the temperature of the material. Equation 2.7 applies to heat conduction i n the x d i r e c t i o n at a point within a volume of some substance. A complete mathematical description for three dimensions would require s i m i l a r equations for the other two d i r e c t i o n s (y and LITERATURE REVIEW 45 z) and integration over the ent i r e volume. Since most processing applications are concerned with heat transfer i n only one d i r e c t i o n and are of p r i m a r i l y a steady-state heat conduction, one need not be concerned with a general mathematical solution and q i s constant, respectively. The rate of heat flow must be constant at every point i n a steady-state heat conduction system. Keeping these r e s t r i c t i o n s i n mind, Equation (2.7) can be rearranged as follows: (q/A) dx = - k dT (2.8) If A (area) i s constant over the length of the path of heat flow and k (thermal conductivity) can be considered to be constant, Equation (2.8) integrates to give (q/A) x = k (t t - t 2) (2.9) This i s the usual form of Fourier's law f o r u n i d i r e c t i o n a l steady-state conduction over a path of constant cross-sec t i o n a l area. 2.9 Thermal Properties of Food Products: As can be seen from the preceding discussion, the thermal properties of foods are of great importance i n p r e d i c t i o n of LITERATURE REVIEW 46 heat transfer rates i n foods. Early analyses, such as Plank's equation, required constant uniform values of thermal properties. These analyses, as exemplified by the numerous attempts to overcome inherent l i m i t a t i o n s , were t y p i c a l l y numerical techniques such as f i n i t e - d i f f e r e n c e and f i n i t e -element methods are much more sophisticated and can account for nonuniformity of thermal properties (Chen, 1985a; Chen, 1985b; Cleland and Earle, 1984b; de Cindo et a l . , 1985; Succar and Hayakawa, 1984) . Thus, allowance can be made for thermal properties that not only change with time, but also with temperature and location as the food material i s heated or cooled. This has resulted i n a greatly increased demand for accurate data on thermal properties. I t i s now necessary to know how thermal properties change during a process. A function can be written that describes the v a r i a b i l i t y of thermal d i f f u s i v i t y (a) with temperature during the conversion of i c e to water, oversimplified and inaccurate. The use of present-day k(T) <X(T) (2.10) where k i s the thermal conductivity, T i s the temperature (°K), p i s the density, and C^ i s the apparent s p e c i f i c heat. The s p e c i f i c heat used i n Equation 2.10 i s an apparent LITERATURE REVIEW 47 property since i t incorporates latent heat removed during the freezing process. This approach has lead to the d e f i n i t i o n of t h i s property as apparent s p e c i f i c heat (C^) (Heldman and Singh, 1981; Loncin and Merson, 1981). 2.9.1 S p e c i f i c Heat and Enthalpy: S p e c i f i c heat indicates how much heat i s required to change the temperature of a material. The heat required to heat a material of mass M from an i n i t i a l temperature Tl to a f i n a l temperature T2 i s equal to the product of the mass, the s p e c i f i c heat and (T2 - Tx) (Sweat, 1986) . S p e c i f i c heat i s independent of mass density and knowledge of the s p e c i f i c heat of each component i n a mixture i s usually s u f f i c i e n t to predict the s p e c i f i c heat of the mixture. The symbol Cp i s used as a designation for s p e c i f i c heat at constant pressure, which applies f o r almost a l l food heating or cooling applications. Only with gases i s i t necessary to d i s t i n g u i s h between Cp and Cv, the s p e c i f i c heat at constant volume (Sweat, 1986). Although s p e c i f i c heat has often been measured by some sort of a calorimeter, usually a simple "thermos" vacuum bot t l e , the d i f f e r e n t i a l scanning calorimeter (DSC) i s the instrument of choice. DSC i s a thermoanalytical technique f o r monitoring LITERATURE REVIEW 48 changes i n chemical or physical properties of materials as a function of temperature, by detecting heat changes associated with such processes. The measuring p r i n c i p l e i s to compare the rate of heat flow to the sample and to that of an i n e r t material, which i s heated and cooled at the same rate (Koga, and Yoshizumi, 1977). Resulting changes i n the sample that are associated with absorption or evolution of heat cause a change i n the d i f f e r e n t i a l heat flow which i s then recorded as a peak (Koga and Yoshizumi, 1977; Parducci and Duckworth, 1972). The area under the peak i s d i r e c t l y proportional to the enthalpic change and i t s d i r e c t i o n indicates whether the thermal event i s endothermic or exothermic (Figure 2.5) (Ladbrooke and Chapman, 1969). Absolute values f o r s p e c i f i c heat are determined v i a the measurement of the energy required to e s t a b l i s h a zero temperature difference between the sample substance and the reference material. The s p e c i f i c heat of the sample can be calculated from t h i s . Enthalpy i s the heat content or energy l e v e l of a material. Because i t i s very d i f f i c u l t to define the absolute value of enthalpy, a zero value i s usually a r b i t r a r i l y defined at -40°C, 0°C, or some other convenient temperature. Enthalpy has been used more for quantifying energy i n steam than i n foods. I t i s also very convenient for frozen samples (Parducci and Duckworth, 1972) because i t i s d i f f i c u l t LITERATURE REVIEW 49 U • H 20 40 60 80 Temperature (°C) Figure 2.5 Indication of exothermic and endothermic events as shown on DSC ( B i l i a d e r i s , 1983). LITERATURE REVIEW 50 to separate latent and sensible heats i n frozen foods, which often contain some unfrozen water even at very low temperatures. In fact, Fennema (1973) indicated that at -20°C, only 80% of the water i n beef i s present as i c e . The amount of energy required to heat a material from temperature Tt to temperature T2 i s simply M(H2 - HJ , where M i s the mass of material, and H2 and are enthalpies of the material at temperatures T2 and T 1 # respectively. When t h i s approach i s used, i t i s d i f f i c u l t to determine enthalpy for foods because enthalpy depends upon the amount of unfrozen water i n addition to the composition of the food. Very l i t t l e d i r e c t measurement of enthalpy has been done. Most of the basic data has been provided by Riedel (1969) . Heldman and Gorby (1975) developed a p r e d i c t i o n technique involving estimation of the amount of unfrozen water i n a food sample. For p r a c t i c a l purposes, d i f f e r e n t i a l scanning calorimetry can be used to measure the enthalpy of food products by scanning from about -60°C where nearly a l l water i n the product i s frozen (Fennema, 1973) , to 1°C (where a l l the water i s l i q u i d ) . The problem with t h i s approach i s that i t does not provide for a separate, independent measurement of percent water unfrozen; t h i s can only be i n f e r r e d from the enthalpy data. B i l i a d e r i s (1983) indicated that when water contents are s u f f i c i e n t l y low so that only unfreezable water i s present, there i s no DSC peak. Thus, by using samples of LITERATURE REVIEW 51 d i f f e r i n g water contents, i t i s possible to determine the point at which a l l the remaining water i s unfreezable. For foods that are cooled at a very slow rate, there may be an overestimation of the enthalpy as calculated from the area under the peak observed on a DSC curve, as the t r a n s i t i o n i s spread over a larger area than f o r a more rapid cooling rate. However, Koga and Yoshizumi (1979) found that while the temperature of the exotherm i s ascribable to the freezing of the mixture depended on the cooling rate of the calorimeter, the heat of freezing d i d not. I t becomes even more d i f f i c u l t i n p r a c t i c e as the temperature a c t u a l l y fluctuates i n a l l commercial frozen storage f a c i l i t i e s . This leads to changes i n i c e c r y s t a l structure, mass d i f f u s i o n , and presumably even the percentage of unfrozen water. 2.9.2 Thermal Conductivity: Thermal conductivity i s a measure of the a b i l i t y of a material to conduct heat. In foods, the thermal conductivity depends mostly on composition, but also on any factor that a f f e c t s the path of heat flow through the material, such as percent void spaces, shape, si z e and arrangement of void spaces, homogeneity, orientation of f i b e r s i n frozen meats, etc. Determination of thermal conductivity can be performed using either of two ways. The most common i s by steady-state LITERATURE REVIEW 52 methods. An independent heat flow i s created between a heat source and a heat sink, with the material to be measured placed i n between. The guarded hot-plate method (ASTM, 1955) i s a standard method fo r many non-food materials (Figure 2.6) (Ohlsson, 1983). The advantages of the steady-state method l i e i n the s i m p l i c i t y of processing the r e s u l t s mathematically, the ease of control of experimental conditions, and the high p r e c i s i o n of the r e s u l t s . The disadvantages are that a long time i s required f o r e q u i l i b r a t i o n of the temperature f i e l d , there can be moisture migration during the long measuring times and there i s a need to prevent heat losses to the environment during the long required measuring times. F i n a l l y , since only k i s determined, the r e s u l t i s a mean value f o r the temperature i n t e r v a l used i n the experiments. Transient methods are based on subjecting the sample to a well-defined heat flow and monitoring the temperature at one or more points within the sample and/or at i t s surface. The hot wire method of Vos (1955) , which employs a heater wire and thermocouple wire adjacent to the heater wire, i s the method of choice (Figure 2.7). The hot-wire method operates on the p r i n c i p l e that, as energy i s supplied at a constant rate to the heater wire, the energy must be dissipated i n the sample near the wire. The temperature of the sample near the wire w i l l depend on how well the sample conducts heat away from the LITERATURE REVIEW 53 Figure 2.6 Schematic of the guarded hot-plate method f o r determining thermal conductivity (Ohlsspn, 1983). LITERATURE REVIEW 54 D.C. power supply set a t 7 v o l t s A / W Heater wire 10.2 Ohms/foot 0 to 30 Ohms 0 to 3 Ohms D.C. Ammeter Range 0 to 1000 milliamperes Heater leads Thermocouple leads Thermocouple Figure 2.7. Schematic of the hot wire method f o r determining thermal conductivity (Sweat, 19.86). LITERATURE REVIEW 55 wire. Vos (1955) demonstrated that, a f t e r a b r i e f i n i t i a l t r ansient period, thermal conductivity of the sample may be evaluated from the data obtained while heating the wire. The p l o t of the natural logarithm of time versus temperature i s l i n e a r , having a slope equal to Q/4 k. Thermal conductivity can thus be written as: l n [ ( t 2 - t 0 ) / ( t t - t 0)] k = Q • (2.11) 4 T f(T 2 - TJ where k i s the thermal conductivity (W/m C) , Q i s the power generated by the probe heater (W/m) , t x and t 2 are the time since the probe heater was energized ( s e c ) , t 0 i s a time correction factor ( s e c ) , Tj i s the temperature of the probe thermocouple at time t t (°C) and T2 i s the temperature of the probe thermocouple at time t 2 (°C). The accuracy of the method i s approximately + 2%, which i s adequate f o r most food applications. An alternate method of determining thermal conductivity i n frozen foods involving p r e d i c t i o n from appropriate mathematical models was proposed by Heldmann and Gorby (1975) . This method required knowledge of the frozen water f r a c t i o n as a function of temperature. LITERATURE REVIEW 56 2.9.3 Thermal D i f f u s i v i t y : Thermal d i f f u s i v i t y i s a very important thermal property of materials. I t determines the heat f l u x i n nonsteady state processes according to the general heat tr a n s f e r equation. I t i s a physical parameter which i s defined as the r a t i o of thermal conductivity to the product of the heat capacity (Cp) and density. I t may be calculated, as indicated i n Equation (2.10), or measured. Thermal d i f f u s i v i t y i s required to predict the temperature hi s t o r y curves of foods during various heating or cooling treatments. Calculation requires determination of heat capacity and thermal conductivity by methods s i m i l a r to those already mentioned. The second approach i s to measure thermal d i f f u s i v i t y d i r e c t l y . Singh (1982) i d e n t i f i e d four methods i n common use for determining thermal d i f f u s i v i t y . The f i r s t method, that of least-squares estimation, required the measurement of the temperature h i s t o r y at the center of an object of well-defined shape, such as an i n f i n i t e cylinder, an i n f i n i t e slab, or a sphere. An appropriate a n a l y t i c a l s o l u t i o n of the p a r t i a l d i f f e r e n t i a l equation was then programmed into a computer to predict temperature at various times f o r some a r b i t r a r i l y selected value of thermal d i f f u s i v i t y . I t became obvious that the chosen value of thermal d i f f u s i v i t y , , the predicted temperature and the experimental temperature may not coincide. LITERATURE REVIEW 57 However, using i t e r a t i v e techniques, the value of thermal d i f f u s i v i t y was changed u n t i l the differences between the predicted and experimental temperatures were n e g l i g i b l e . Thus, an estimate of the thermal d i f f u s i v i t y that gave the minimum sum of squared deviations was obtained. The second method used heat penetration curves. B a l l and Olson (1957) plotted experimental heat penetration curves on semilog graph paper and determined the following r e l a t i o n : f h = 2.303/CX[ (2.405)2/R2 + TT 2 /J, 2 ] (2.12) where « i s the thermal d i f f u s i v i t y , R i s the radius (m) and i i s the length of a f i n i t e cylinder. Equation (2.12) allows the determination of thermal d i f f u s i v i t y i f the heating rate parameter f h i s obtained from a heat-penetration study. The t h i r d method involved the time-temperature charts that Schneider (1963) derived from a study of the a n a l y t i c a l solutions of the p a r t i a l d i f f e r e n t i a l equations that describe heat transfer. These time-temperature charts contain dimensionless numbers such as the Fourier number, Biot number and temperature r a t i o . I f the temperature r a t i o i s determined experimentally f o r a p a r t i c u l a r l o c a t i o n within the object under study at a known time, the time-temperature charts may then be used to estimate the Fourier number, from which the LITERATURE REVIEW 58 thermal d i f f u s i v i t y can then be determined. The fourth method used series solutions of a n a l y t i c a l equations (Carslaw and Jaeger, 1959; Uno and Hayakawa, 1980). The series solutions converged r a p i d l y a f t e r a c e r t a i n time had elapsed. Therefore, a f t e r s u f f i c i e n t time had passed, the experimentally determined temperature at a known location was used i n the f i r s t term of the series to evaluate thermal d i f f u s i v i t y . Methods one, three and four require accurate measurement of the temperature of the i n i t i a l product, the surrounding medium, and of a known location within the product. Method two has the advantage that the slope from a large number of data points i s used rather than a single-point determination. These techniques are based on gathering time-temperature re l a t i o n s h i p s during the heating (unsteady or steady state) of food materials i n known configurations (slab, cylinder or sphere) and estimating an average thermal d i f f u s i v i t y value over a range of temperatures. A number of important applications of the above methods have been described i n the l i t e r a t u r e . Hayakawa (1972) discussed computer-aided methods to determine the apparent thermal d i f f u s i v i t y of foods using heat conduction expressed i n c y l i n d r i c a l coordinates. Hayakawa and Bakal (1973) used a sample of i n f i n i t e slab geometry to predict thermal d i f f u s i v i t y . They cautioned that thermocouple p o s i t i o n must be LITERATURE REVIEW 59 p r e c i s e l y determined i n order to accurately estimate thermal d i f f u s i v i t y values using the ent i r e temperature h i s t o r y curves for the sample material during heating. The advantage of t h i s computational procedure was that the sample of food could be exposed to any time-variable heating or cooling temperature during the experiment and surface temperature need not be monitored. Further refinement of t h i s type of modelling was done by Hayakawa and Bakal (1973) and Bhowmik and Hayakawa (1979). Lenz and Lund (1977) developed a method c a l l e d the " l e t h a l i t y - F o u r i e r number method" i n order to study transient heat tr a n s f e r and l e t h a l i t y i n food products during s t e r i l i z a t i o n processes. This method has been used i n estimating confidence i n t e r v a l s of l e t h a l i t y or process time of a thermal process, including the influence of v a r i a b i l i t y i n thermal d i f f u s i v i t y among samples. Lenz and Lund (1977) assumed that the thermal d i f f u s i v i t y remained constant throughout the thermal h i s t o r y of the product. Several empirical models useful f o r p r e d i c t i n g the thermal d i f f u s i v i t y of foods have appeared i n the l i t e r a t u r e . Most of these models are s p e c i f i c to the products studied. The model proposed by Riedel (1969), however, allows p r e d i c t i o n of thermal d i f f u s i v i t y as a function of water content (wet b a s i s ) . This expression allows f o r the encompassing of a wider range of food products, LITERATURE REVIEW 60 fX = 0.088 X lO"6 + (a w - 0.088 X 10-*)W (2.13) where W i s the water content, % by weight (wet basis) . Dickerson and Read (1975) used the above model and found good agreement with experimental values of thermal d i f f u s i v i t y for a v a r i e t y of meats. Martens (1980) investigated the influence of water, f a t , protein, carbohydrate, and temperature on thermal d i f f u s i v i t y . Using s t a t i s t i c a l analysis, he found that temperature and water content were the major factors a f f e c t i n g thermal d i f f u s i v i t y . V a r i a t i o n within the s o l i d f r a c t i o n of f a t , protein and carbohydrate had a small influence on thermal d i f f u s i v i t y . Martens (1980) performed multiple regression analysis on 246 published values of thermal d i f f u s i v i t y f o r a var i e t y of food products and obtained the following regression equation: (X = [0.057363 W + 0.000288 (T + 273)] X 10"6 (2.14) Dickerson (1965) developed an experimental apparatus to measure the thermal d i f f u s i v i t y of foods (Figure 2.8). The apparatus consisted of a constant-temperature water bath and a thermal d i f f u s i v i t y tube that was f i l l e d with the food sample. Thermocouples located at the center and the surface of the tube provide a complete temperature h i s t o r y when the LITERATURE REVIEW 61 Dickerson (1965) Figure 2.8. Experimental apparatus of Dickerson. LITERATURE REVIEW 62 tube, containing the food sample was immersed i n the water bath, which was heated at a constant rate. Dickerson developed the following equation to compute the thermal d i f f u s i v i t y from h i s experimental apparatus: B R2 CX = (2.15) 4 (T R - T0) Determination of thermal d i f f u s i v i t y was possible i f the constant rate of temperature increase (B) and the constant temperature difference between the outside surface of the tube and the center of the tube (T R- T0) were known based on the radius of the cylinder (R). The constant temperature values were determined a f t e r the e f f e c t of the i n i t i a l t ransient was eliminated ( i . e . a f t e r the dimensionless r a t i o o(t/R2 > 0.55). Although the mathematical derivation i s f o r an i n f i n i t e cylinder, i t s ap p l i c a t i o n to the d i f f u s i v i t y tube (9 inches long, 1 1/4 inch inside radius (R) , f i l l e d and f i t t e d with Teflon end caps) resulted i n an error of less than 2%. This apparatus allowed the determination of a thermal d i f f u s i v i t y value over a range of temperatures. Thus, i f the temperature range of the ultimate process was known, a s i m i l a r range could be used for d i f f u s i v i t y measurement. In processes involving heating and holding however, Dickerson's method cannot be used over the entire heat/hold period. The method LITERATURE REVIEW 63 of Pflug et a l . (1965) may be used for the holding period. The data useful i n c a l c u l a t i n g thermal d i f f u s i v i t y was obtained i n less than 2 hours from the s t a r t of a t e s t . The method required that the surrounding medium be heated at a constant rate, although no heat flow measurements were required. Dickerson recommended the use of t h i s apparatus for measuring thermal d i f f u s i v i t y of foods with an accuracy of about 5%. I t i s well known that the most dramatic influence of temperature on thermal d i f f u s i v i t y occurs at temperatures below the freezing point. Heldman and Singh (1981) reviewed the importance of variable thermal d i f f u s i v i t y i n freezing processes. 2.9.4 Surface Heat Transfer C o e f f i c i e n t s : The surface heat transfer c o e f f i c i e n t i s not a c t u a l l y a property of a food material or any other material. I t i s used to quantify the rate of heat convection to or away from the surface of an object. I t i s needed, however, f o r quantifying heat tr a n s f e r i n most food heating or cooling applications. The surface heat transfer c o e f f i c i e n t i s defined by Newton's Law of Cooling as the p r o p o r t i o n a l i t y constant that r e l a t e s the heat f l u x to or from a surface, to the temperature difference between the surface and the f l u i d stream moving LITERATURE REVIEW 64 past the surface. I t depends mostly on the f l u i d v e l o c i t y , but also on the f l u i d properties, surface texture and shape, and even on the temperature difference i t s e l f . Steam i s present as a vapour when i t i s used i n thermal processing. Heat may be released by condensation onto the surfaces that are at a lower temperature than the steam. While the exact mechanism of the condensation event i s not well known, two mechanisms have been described. In the f i r s t mechanism, the condensing l i q u i d does not wet the s o l i d surface, but tends to form microscopic droplets that are removed continuously from the surface. This i s known as "dropwise condensation." New droplets are i n i t i a t e d at numerous nucleation s i t e s over the surface, leading to the bulk of the surface's area being covered by droplets of very small s i z e (less than 0.1 mm i n diameter). A f r a c t i o n of the water molecules present as steam may enter the small droplets upon c o l l i s i o n with the surface and lose enthalpy with the change i n phase from gas to l i q u i d ; t h i s t r a n s i t i o n i s promoted by surface temperatures that are lower than the steam temperature adjacent to the surface. In a transient-heating s i t u a t i o n , as the surface temperature r i s e s , the rate of water molecules leaving the droplets increases u n t i l an equilibrium i s achieved when no temperature gradient e x i s t s . As long as the surface i s colder than the surrounding steam, there i s a very rapid transfer of heat when condensation continues by the LITERATURE REVIEW 65 dropwise mechanism. The second mechanism i s more common i n thermal processing, and i s thought to p r e v a i l i n r e t o r t processing. In t h i s mechanism, a layer of condensed l i q u i d builds up at the surface to form a continuous f i l m . This i s c a l l e d "filmwise condensation." The f i l m thickness i s much greater than the droplet thickness described e a r l i e r , and heat transfer rates can be one or two orders of magnitude lower than f o r dropwise condensation (Tung et a l . , 1990). Measurement of the surface heat transfer c o e f f i c i e n t i s quite d i f f e r e n t from the measurement of other thermal properties because i t i s not a material property, as mentioned above. I t i s a measure of convective heat transfer between a surface and the convecting medium. Newton's law of cooling describes convective transfer q = h A (t, - t) (2.16) where h i s a pro p o r t i o n a l i t y constant representing thermal conductivity of the f l u i d f i l m across the temperature change (t, - t) and A i s the area. This equation can be rearranged to show that the surface heat transfer c o e f f i c i e n t i s equal to the heat f l u x through a surface, divided by the difference between the surface temperature (t 8) and the convecting medium temperature (t) . In practice, the heat f l u x and surface temperature are very d i f f i c u l t to measure without LITERATURE REVIEW 66 a f f e c t i n g heat transfer. Heat convection i s often accompanied by mass transfer f o r many foods having high water contents; t h i s further complicates measurement of the surface heat tra n s f e r c o e f f i c i e n t . Measurement of the surface heat transfer c o e f f i c i e n t was treated rather thoroughly by Arce and Sweat (1980) who presented a comprehensive compilation of surface heat transfer c o e f f i c i e n t data for foods. Through the use of surface heat transfer c o e f f i c i e n t equations f o r the Nusslet number (Nu) as a function of Reynolds number (Re) and the Prandtl (Pr) or Grashof (Gr) number, one may obtain an estimate of the surface heat transfer c o e f f i c i e n t f o r a broad range of f l u i d v e l o c i t i e s and product dimensions Nu = f (Re, Pr) (2.17) with f representing the heating or cooling rate constant (in seconds) . Thus, for heat transfer from a f l a t p late to a f l u i d during laminar glow (Heldman and Singh, 1981), the equation becomes Nu = 0.664 (Re1'2 Pr 1 / 3) (2.18) The decision to use the Prandtl number or the Grashoff number i s dependent upon how the f l u i d i s being treated. For example, i f the f l u i d being heated i s at a very low v i s c o s i t y during turbulent flow, the Prandtl number i s applicable. On the other hand, during free or natural convection, when f l u i d LITERATURE REVIEW -a 67 flow i s not induced by external forces and the motion within the f l u i d i s brought about by the influence of temperature on f l u i d density and the development of a buoyant force, then the Grashoff number i s applicable. Thus, the Nusselt number for the average convection heat-transfer c o e f f i c i e n t (Nu) becomes Nu = K (Gr Pr) a (2.18) with K and a being constants that vary with the geometry and orie n t a t i o n of the surface. The Nusselt number i s the product of surface heat tra n s f e r c o e f f i c i e n t and a c h a r a c t e r i s t i c length of the object, divided by the thermal conductivity of the f l u i d . The c h a r a c t e r i s t i c length i s t y p i c a l l y the diameter of the object. M NU = (2.19) with h c equal to the convective heat transfer c o e f f i c i e n t , j] i s length (m) and kf i s the thermal conductivity of f l u i d . The Reynolds number i s the product of the f l u i d v e l o c i t y , mass density of the object, and c h a r a c t e r i s t i c length of the object, divided by the f l u i d v i s c o s i t y . The Prandtl number i s made up e n t i r e l y of f l u i d properties and forms an e s s e n t i a l part of a l l general convective heat transfer c o r r e l a t i o n s . LITERATURE REVIEW 68 Pr = (2.20) kf with Cp as the s p e c i f i c heat (kJ/kg °K) , and u equal to the v i s c o s i t y (Pa sec). The Grashof number involves the c o e f f i c i e n t of volumetric expansion and the temperature difference that i s the source of the convection and i s determined from the following equation Gr = -I (2.21) f with equal to the density (in kg/m3) , g i s the acceleration due to gravity (m/sec.2) , j$ i s the c o e f f i c i e n t of expansion for the f l u i d being heated (1/°C) , X i s the c h a r a c t e r i s t i c length (m) , T i s the temperature gradient between the surface and the f l u i d (°C) and JU i s the v i s c o s i t y (Pa s e c ) . 2.10 Heat Transfer and F i n i t e Difference Modelling: 2.10.1 Heat Transfer: Steady-state heat transfer e x i s t s when the temperature at any point does not change with time, i . e . the system i s i n equilibrium. However, there are many important s i t u a t i o n s i n which conditions of steady-state do not p r e v a i l . The problems that a r i s e from these sit u a t i o n s f a l l into the category of LITERATURE REVIEW 69 transient-heat transfer, or unsteady-state heat transfer. For example, regular day to night changes i n a i r temperature create many problems of transient-heat t r a n s f e r i n agri c u l t u r e . E s s e n t i a l l y , any time a processing operation i s started, there w i l l be a period of transience u n t i l a steady-state condition i s reached — when equilibrium i s attained. A most important food application i s i n the thermal processing of foods i n i n d i v i d u a l containers, that i s to say, i n the r e t o r t s t e r i l i z a t i o n of packaged foods. In the basic approach to t h i s problem, an object of i n i t i a l l y uniform temperature i s suddenly placed i n an environment of a d i f f e r e n t temperature. One i s then required to determine the temperature of the object as a function of time. One important point to note i s that the mathematical so l u t i o n to the problem proceeds i n the same fashion, regardless of whether the process i s one of heating or cooling. The solution f o r the cooling process i s the reverse of the solut i o n for the heating process. Thus, a s l i g h t rearrangement of the integrated form of Fourier's law (Equation 2.16) w i l l give (t t - t 2) (t, - t 2) q = = (2.22) Ax/kA R where the thermal resistance R equals ^ x/kA. This provides an LITERATURE REVIEW 70 extremely powerful experimental t o o l . In heat flow there exists a convection resistance between the surface of the object and the surrounding medium and an i n t e r n a l resistance within the object. Thus, there w i l l be a temperature difference between the medium and the surface of the object and there w i l l also be a temperature difference within the object which w i l l vary continuously from the surface to the centre. In the case of l i q u i d food i n a can, i f there i s s u f f i c i e n t a g i t a t i o n to provide good mixing, the temperature throughout the i n t e r i o r w i l l be uniform, except f o r a t h i n f i l m adjacent to the surface. In t h i s case, the external resistance i s a series consisting of the outside convection resistance, the resistance of the container wall, and the inside f i l m resistance. To determine the influence of each of these resistances, the Biot number (or modulus), designated as B i , has been developed. Thus, a very low value of the Biot modulus means that internal-conduction resistance i s n e g l i g i b l e i n comparison to surface-convection resistance. This i n turn implies that the temperature w i l l be nearly uniform throughout (Loncin and Merson, 1981; Heldman and Singh, 1981). In considering a f l u i d with no mixing, or a s o l i d , such as the flaked ham and bentonite preparations the assumption of uniform i n i t i a l temperature applies. However, as heating i s LITERATURE REVIEW 71 i n i t i a t e d and progresses, temperatures near the surface w i l l change more r a p i d l y that those toward the centre. Thus, while the product may s t a r t o f f at a uniform i n i t i a l temperature, since there are no convective mixing forces occurring, heating must be by conduction. Since the thermal conductivity of food materials i s much lower than that of metals and the heat flow i s from the outside to the centre, the r e s u l t i s that the outer edges of the product are exposed to higher temperatures f o r a much longer time than the center of the product. The general s o l u t i o n to t h i s problem must therefore give temperature as a function of both time and p o s i t i o n . Thus, the a n a l y s i s of the t r a n s i e n t heating or cooling processes which take place i n the interim period before equilibrium i s estab l i s h e d must be modified to take the change i n i n t e r n a l energy of the body int o account, and the boundary conditions must be adjusted to match the p h y s i c a l s i t u a t i o n which i s apparent i n the unsteady-state heat-transfer problem. The objective of any heat-transfer analysis i s to p r e d i c t heat flow or the temperature which r e s u l t s from a c e r t a i n heat flow (Holman, 1976a). In d e r i v a t i o n of the unsteady-state heat-transfer model used i n t h i s study, the general case of two-dimensional steady-state heat flow was considered. Therefore i f a steady st a t e e x i s t s and one assumes constant thermal conductivity, LITERATURE REVIEW 72 then the Laplace equation applies d*T + 0 (2.23) dx2 dy2 and the temperature i n a two-dimensional body w i l l r i s e as a function of the two independent space coordinates x and y. The solu t i o n to Equation 2.18 may be obtained by a n a l y t i c a l , numerical or graphical techniques. In considering the choice of solution technique, one has to remember that even though an immense number of a n a l y t i c a l solutions for conduction and convection heat-transfer problems have accumulated i n the l i t e r a t u r e (Datta and T e i x e i r a , 1987; Naveh et a l . , 1984; Pham, 1987; Lenz and Lund, 1977; Tan and Ling, 1988; Richard et a l . , 1991), there are many p r a c t i c a l s i t u a t i o n s where the geometry or boundary conditions are such that an a n a l y t i c a l solution either has not been obtained or involves such a complex series solution that numerical evaluation becomes extremely d i f f i c u l t . For these situations the techniques of f i n i t e - d i f f e r e n c e analysis y i e l d the most f r u i t f u l r e s u l t s . 2.10.2 F i n i t e Difference Analysis — Two Dimensions: Consider a two-dimensional body which i s divided into equal increments i n both the x and y d i r e c t i o n s (Figure 2.9). LITERATURE REVIEW 73 The nodal points are designated as shown, with x locations i n d i c a t i n g the x increment and y locations i n d i c a t i n g the y increment. Using Equation 2 . 1 8 as a governing condition, one can now e s t a b l i s h the temperature at any of these nodal points. F i n i t e differences are used to approximate d i f f e r e n t i a l increments i n the temperature and space coordinates. The smaller the f i n i t e increments that are chosen, the more c l o s e l y the true temperature d i s t r i b u t i o n w i l l be approximated. The Fourier equations can then be used to calc u l a t e the heat flow i n the x and y di r e c t i o n s . For the x d i r e c t i o n , the appropriate equation appears i n Equation 2 . 7 . In the y d i r e c t i o n , the equation becomes The p a r t i a l derivatives of the temperature gradients for Figure 2 . 9 may be written as follows (for the centre node 2 with x = 2+1/2 to 2 - 1 / 2 and y = 2+2A/2 to 2 - 2 B / 2 as x and y straddle the node 2 ) . dT = - k Ay — dy ( 2 . 2 4 ) ( 2 . 2 5 ) LITERATURE REVIEW 74 Figure 2.9 Nomenclature used i n two-dimensional numerical analysis of heat conduction. LITERATURE REVIEW 75 dx /2-1/2 *" T2 - T3 (2.26) x dT \ T2A - T2 (2.27) dy /2+2A/2 ^ dT \ T2 - T2B I ~ (2.28) dy /2-2B/2 ^ y Combining these equations gives, for the centre node, i n the x dimension: d*T \ dT \ dT \ ~ " (2-29) dx2/2 dx /2+1/2 dx /2-1/2 Substituting the approximations of the p a r t i a l derivatives (Equations 2.25 and 2.26) into and sim p l i f y i n g the r e s u l t i n g equation y i e l d s T l + T3 - 2(T2) (2.30) (4x) S i m i l a r l y , i n the y dimension: LITERATURE REVIEW 76 d'T T2A + T2B - 2(T2) (2.31) dy 2 2 (Ay) 2 Thus, f o r Equation 2.23 the f i n i t e - d i f f e r e n c e approximation becomes: T l + T3 - 2(T2) T2A + T2B - 2(T2) + 0 (2.32) (Ax) 2 (Ay) 2 I t can be seen also that i f x and y are equal, then the f i n i t e - d i f f e r e n c e approximation s i m p l i f i e s further to become: Since the case of constant thermal conductivity i s being considered, the heat flows may a l l be expressed i n terms of temperature d i f f e r e n t i a l s . The net heat flow into any node i s zero at steady-state conditions, as expressed by Equation 2.33. In order to u t i l i z e the numerical method, Equation 2.33 must be written f o r each node within the material and the resultant system of equations must be solved for the temperature at the various nodes. The s i z e of the subdivision chosen may r e s u l t i n a large T l + T3 + T2A + T2B - 4(T2) = 0 (2.34) LITERATURE REVIEW 77 number of nodes and equations that may begin to become unwieldy. There are two solutions. One i s the use of a tedious numerical solution encompassing every node, while the second i s to optimize the number of nodes (Holman, 1976b). One may f i n d that there i s a point at which increasing the number of nodes w i l l not add appreciably to the accuracy of the sol u t i o n . Related to t h i s i s a technique c a l l e d " g r i d -stretching" (Kublbeck et a l . , 1980; Vinokur, 1983) which attempts to solve the problem of unwieldy c a l c u l a t i o n s . The technique of grid-stretching i s to generate f i n e grids near the boundary walls where large step increases i n temperature are experienced. Larger grids are generated i n the core region where variatio n s of temperature and v e l o c i t y of heat transfer are small. The non-uniform g r i d i s generated through the use of a simple algebraic method. Further discussion w i l l not be pursued here and the interested reader i s directed to the references l i s t e d (Kublbeck et a l . , 1980; Vinokur, 1983). As was previously mentioned, steady-state heat-transfer i s not applicable i n the case of retorted food products that heat by conduction. Thus the case of unsteady-state heat transfer must now be considered. In order to deal with t h i s s i t u a t i o n the transient numerical method i s adopted. This approach i s handled i n a s i m i l a r fashion to the discussion j u s t completed f o r steady-state heat transfer, with LITERATURE REVIEW 78 the exception that the boundary conditions vary with time. Returning to the concept of the two-dimensional g r i d shown i n Figure 2.9, the nomenclature depicted w i l l again denote the x p o s i t i o n and the y p o s i t i o n . Within the s o l i d body, the d i f f e r e n t i a l equation which governs heat flow, assuming constant properties (such as constant environmental temperature, uniform i n i t i a l temperature) for an unsteady state, i s : where k i s thermal conductivity, p i s density, Cp i s heat capacity, and X. i s the time increment. S i m i l a r l y to the equations f o r steady-state heat transfer, the p a r t i a l d erivatives may again be approximated as follows: + (2.34) T l + T3 - 2(T2) (2.35) T2A + T2B - 2(T2) (2.36) This leaves only the need to approximate the time increment i n Equation 2.34. The approximation i s LITERATURE REVIEW 79 dT T2 t + 1 - T2l ~ (2.37) df ~ A t In t h i s notation, the superscripts designate the time increment. Thus, i f t represents the present time, then t+i represents one time increment into the future. Combining the above r e l a t i o n s gives the f i n i t e d ifference equation equivalent to Equation 2.34. T l 1 + T3l - 2(T2)' T2A1 + T2B1 - 2(T2)' +  (Ax) ; (Ay); T2l T2l C< A t (2.38) One can see that i f the temperatures of the various nodes are known at any p a r t i c u l a r time, the temperatures a f t e r a time increment may also be calculated by wr i t i n g an equation such as Equation 2.38 for each node and obtaining the values of T2 at t+1. The procedure may be repeated to obtain the d i s t r i b u t i o n a f t e r any desired number of time increments. Allowing f o r a choice of the s p a c i a l increments to be equal, the r e s u l t i n g equation for T2 at t+1 becomes: T 2t + 1 = (Tl 1 + T3l + T2A1 + T2B') + (4x)2 4 OCAt (AX) 2J T2l (2.39) LITERATURE REVIEW 80 I f Ax = Ay/ then the r e l a t i o n f o r T2 at t+1 becomes T2 t+i • _ oc/vc (Ax) 2 h A x 2 T + 2(T3) 1 + T2A' + T2B' k (Ax) 2 hAx — - 2 4 ocAr k T2l (2.41) With t h i s analysis i n place, an extension into three dimensions can be made quite e a s i l y . The container used i n t h i s study was a c y l i n d r i c a l can. In order to generate a numerical model f o r t h i s geometric shape, two general heat t r a n s f e r equations were combined. The f i r s t equation i s f o r one-dimensional (x) heat t r a n s f e r i n an i n f i n i t e slab under unsteady-state heat t r a n s f e r conditions and can be expressed as d ^ dx2 1 dT (X d t (2.42) A s i m i l a r expression f o r heat conduction i n the r a d i a l d i r e c t i o n (r) i n an i n f i n i t e c y l i n d e r i s dr 2 1 + — dT dr 1 dT (2.43) LITERATURE REVIEW 81 e—* Figure 2.10 Nomenclature f o r nodal equation with convective boundary condition. LITERATURE REVIEW 82 With t h i s analysis i n place, an extension into three dimensions can be made quite e a s i l y . The container used i n t h i s study was a c y l i n d r i c a l can. In order to generate a numerical model f o r t h i s geometric shape, two general heat transfer equations were combined. The f i r s t equation i s for one-dimensional (x) heat transfer i n an i n f i n i t e slab under unsteady-state heat transfer conditions and can be expressed as d*T 1 dT - — . = (2.42) dx2 (x d t A s i m i l a r expression for heat conduction i n the r a d i a l d i r e c t i o n (r) i n an i n f i n i t e cylinder i s d2? 1 dT 1 dT + — = (2.43) dr 2 r dr ot Combining these two equations y i e l d s the heat-transfer equation for a f i n i t e cylinder d ^ d ^ 1 dT 1 dT + + — = — : (2.44) dx 2 dr 2 r dr Oc d t The e x p l i c i t f i n i t e difference solution f o r Equation 2.44 w i l l be presented i n Chapter 4, Development of the F i n i t e  Difference Model. LITERATURE REVIEW 83 The power of t h i s technique i n dealing with problems of heat transfer should be obvious to the reader. With computers, cal c u l a t i o n s that once were laborious can now be performed with r e l a t i v e ease. These techniques may also be applied to v i r t u a l l y any s i t u a t i o n with just a l i t t l e patience and care i n s e l e c t i o n of boundary conditions and equations for the shape of the container so that the appropriate s e l e c t i o n of nodes may be undertaken. Equations e x i s t to account f o r a great many geometric shapes and boundaries. Chapter 3 THERMAL PROPERTIES The p r e d i c t i o n of heat transfer rates i n foods has gained increased importance (Ramaswamy and Tung^ 1986). This has resulted from a need to know the e f f e c t s of composition on thermal properties of the food and packaging a l i k e . In fact, knowing the composition, temperature, density and porosity of the product i s s u f f i c i e n t , i n theory, f o r p r e d i c t i o n of the thermal properties of a food material (Ohlsson, 1983). Early heat tr a n s f e r analyses for the heating or cooling of food products required that the thermal properties have constant uniform values. However, one can now account f o r non-uniform thermal properties through much more sophisticated techniques such as f i n i t e difference and finite-element methods (Sweat, 1986; Sastry et a l . , 1985). Thus, one can model changes i n these properties with time, temperature and location i n a food material as i t i s heated or cooled. Coupled with these improved a b i l i t i e s comes an increased demand for more accurate data not only on the thermal properties, but also on how they change during a process. In order to develop a model that accurately simulates the 84 THERMAL PROPERTIES 85 thermal process required to achieve "commercial s t e r i l i t y " i n a canned meat product exh i b i t i n g a sub-zero i n i t i a l temperature, c e r t a i n s p e c i f i c thermal properties must be determined, such as thermal d i f f u s i v i t y , thermal conductivity, s p e c i f i c heat, enthalpy and density (Levy, 1979; Ohlsson, 1983). M i t t a l and B l a i s d e l l (1984) examined the heat and mass transfer properties of meat emulsions during thermal processing. Dickerson (1968) examined the mass and heat transfer c h a r a c t e r i s t i c s for frozen foods. This portion of the thesis describes the measurement of the heat transfer properties of the systems used i n the study. Two main properties were considered. F i r s t l y , changes i n heat flow of a bentonite dispersion and a "flaked" ham product (comminuted meat emulsion with i n t a c t t i s s u e chunks suspended i n the matrix) as they passed from the frozen to the thawed state were evaluated. The purpose of t h i s portion of the study was to ascertain at what temperature the phase change of bulk water from a frozen state to a l i q u i d state occurred. The second portion concerned the measurement of the thermal d i f f u s i v i t y of the bentonite dispersion and the flaked ham product. The achieve these objectives, determination of the heat tr a n s f e r properties of flaked ham and bentonite was required. This was necessary f o r the development of the f i n i t e difference model of canned, conduction heating foods processed with a THERMAL PROPERTIES 86 sub-freezing i n i t i a l temperature. 3.1 MATERIALS AND METHODS: 3.1.1 DIFFERENTIAL SCANNING CALORIMETRY: For the purposes of t h i s study, a bentonite dispersion of 40% bentonite and 60% d i s t i l l e d water was prepared. The moisture content of the bentonite was 9.6%, thus y i e l d i n g an e f f e c t i v e moisture content of 36.2% f o r the bentonite dispersion. The bentonite used was designated as "X-tra Gel" and obtained from The American C o l l o i d Company (Arlington Heights, IL). The flaked ham system consisted of a 50:50 blend of 95% lean ham (ground through a 19 mm plate) and a comminuted meat emulsion containing lean (85%) ham trim, water, lean (94%) ham shanks, s a l t , smoke flavour, sodium phosphate, sodium erythorbate, and sodium n i t r i t e . The f i n i s h e d product was then allowed to cure for 36 hours to develop colour and maximum formation of the emulsion matrix. D i f f e r e n t i a l Scanning Calorimetry (DSC) was performed using a Du Pont 1090 Thermal analyzer with a 910 D i f f e r e n t i a l Scanning Calorimeter module. The samples were sealed i n hermetic pans of anodized aluminum (I.D. 6.6 mm) and the system was cooled to s t a r t i n g temperature with a dry THERMAL PROPERTIES 87 ice/ethanol mixture. The temperature was raised at 2°C per minute from approximately -3 0°C to room temperature. Analyses were performed i n duplicate on the bentonite dispersion and on the chunk, emulsion and f i n a l blend of the flaked ham product. The heat of thawing was estimated by integrating the peak area between the thermogram and a base-line under the peak (Koga and Yoshizumi, 1977) . This was then expressed i n terms of joules per unit weight of a sample (J/g). Figure 3.1 shows a representative DSC thermogram representative of thawing. When the sample and reference materials were heated at a constant rate (2 °C/minute) , the DSC maintained tham i n an equal l e v e l or fi x e d d i f f e r e n t i a l condition throughout the analysis. As the reference material had no enthalpy change i n the temperature range of the measurement, the DSC curve obtained was a recording of heat flow (dH/dt) i n mW/sec as a function of temperature (Wendlant, 1973). Heat capacity i s defined as the energy (joules) that must be added to one kilogram of a substance i n order to r a i s e i t s temperature by 1°K. This y i e l d s the following Cp = Q / WAT (3.1) where Cp i s the heat capacity at constant pressure, Q i s the t o t a l heat transferred (joules), W i s the mass (kg), and AT i s the temperature difference i n °K (Dickerson, 1968) . THERMAL PROPERTIES 88 Figure 3.1 DSC thermogram representative of thawing. THERMAL PROPERTIES 89 Attempts to measure the s p e c i f i c heat of processes such as freezing and thawing use an apparent s p e c i f i c heat. The apparent s p e c i f i c heat incorporates the heat involved i n the change of state of the substance i n addition to the sensible heat and can change d r a s t i c a l l y with freezing and thawing over a wide range of temperatures (Chen, 1985a). In addition, i t i s dependent on the chemical composition of the food material (Staph and Woolrich, 1951). Providing the sample composition i s known, the following can be used to calculate s p e c i f i c heats above freezing Cp = 1.424 mc + 1.549 m,, + 1.675 m, + 0.837 m, + 4.187' m,,, (3.2) where m = the mass f r a c t i o n , and the subscripts c, p, f, a, and m are carbohydrate, protein, f a t , ash, and moisture respectively (Heldman and Singh, 1981). 3 . 1 . 2 THERMAL DIFFUSIVITY: Thermal d i f f u s i v i t y measurements were performed by the method of Dickerson (1965). This method involved the measuring of temperatures at the centre (Tc) and surface (TJ of a food material packed i n the d i f f u s i v i t y c e l l , which was then heated at a constant rate (B) . Knowing the difference i n THERMAL PROPERTIES 90 temperature, (T, - Tc) a f t e r an i n i t i a l temperature lag, the thermal d i f f u s i v i t y was calculated. The values for (T, - Tc) ranged from 37.2 °C during the transient portion to 3.78 °C during the constant portion of the curve. Thermal d i f f u s i v i t y was assumed to be r e l a t i v e l y constant over narrow temperature ranges and were computed used Equation 3.3 as per Tung et a l . (1984) using C< = B R2/4 (T. - Tc) (3.3) In t h i s study, the equipment layout was modified as shown i n Figure 3.2, using an Elevated Temperature Coliform Incubator water bath (GCA Precision S c i e n t i f i c , Chicago, IL), which features an i n t e g r a l submersible pump to provide the a g i t a t i o n . The water bath was f i t t e d with a Model 70 D5X thermostat control (United E l e c t r i c Controls (Canada) Ltd., Mississauga, ON) i n order to s a t i s f y the requirement of a constant dT/d . F i n a l l y , the water bath was f i l l e d with a mix of 50% of a commercial ethylene g l y c o l mixture (Prestone II anti-freeze, F i r s t Brands (Canada) Corp., Scarborough, ON) and 50% d i s t i l l e d water. The freezing point was -3 6°C. Thermocouples were connected as per Dickerson (1965) and the c e l l was immersed i n the l i q u i d . The apparatus was then put into a commercial freezer set at -18°C for 24 hours. Following the 24 hour period i n the freezer, the apparatus was brought out and the thermocouple leads were connected to THERMAL PROPERTIES Figure 3.2 Thermal d i f f u s i v i t y apparatus. THERMAL PROPERTIES 92 a Calwest Model 3 2 datalogger (Calwest Technologies, Canyon Country, CA) . Data was stored on a Toshiba T1000 portable computer (Toshiba Canada Ltd., Willowdale, ON). The computer software used to c o l l e c t , store and analyze the data was the CalSoft program available from Technical Inc. (Metairie, LA) . Further data analysis was conducted using Lotus 1-2-3 software (Lotus Development Corporation, Cambridge, MA) . Center and surface temperature measurements were gathered at one minute i n t e r v a l s as the food sample was heated i n the ethylene glycol/water bath at a constant rate. The thermal d i f f u s i v i t y was then calculated f o r each measurement using Equation (3.3). For values calculated i n the frozen range of the product and e s p e c i a l l y across the t r a n s i t i o n from frozen to thawed product, the transient was incorporated into the thermal d i f f u s i v i t y value. For t h i s portion of the measurements, the calculated value f o r thermal d i f f u s i v i t y yielded an e f f e c t i v e thermal d i f f u s i v i t y rather than the true thermal d i f f u s i v i t y . 3.2 RESULTS AND DISCUSSION: 3.2.1 DIFFERENTIAL SCANNING CALORIMETRY: Analysis of the thawing process of the systems measured involved the determination of the thawing point (t h) and THERMAL P R O P E R T I E S 93 GO in CO t-« I CD •• CD V £ C J u en tn m a x c o o 3: i O) in in I" s u a } 3E X in + 00 m I X a o 0 -»J - i 0 -n a u. T3 0 a a. U • X •) u . C\J CM u Cf) . w a * z e x > 1 0 a L 0 *+ CM at a. a 6 N 0 at -+> o Q a .a c 0 a. 3 . .a a r*<" ..a to .ta ? .ca CM 0 L 3 a) L V Q. E • I CM i ra i cn tn xx. o. « I I CD 1 I -+-<MW) M O I d Y^H Figure 3 . 3 DSC thermograph f o r f l a k e d ham — chunk po r t i o n . THERMAL PROPERTIES 94 heat of thawing (AH) . Figure 3.3 shows a thermograph f o r the chunk portion of the flaked ham product with an endotherm peak at -0.3°C. The thawing point (t h) was -4.9°C. This endotherm was ascribed to the melting of the i c e that was formed from the bulk water present i n the t i s s u e s . Riedel (1957) indicated that for beef muscle, at -3 0°C, 89% of the water i n the tissues was i n the form of i c e . Reidel.further indicated that approximately 10% of the t o t a l water i n the tissues existed as bound water (Reidel, 1956; 1957). An analogous system was expected f o r ham meat. The heat of thawing, from the endotherm, was 187 J/g. The endothermic peaks were reproducible to within 0.7°C. Table 3.1 l i s t s the composition of the chunk, emulsion and blend portions of the flaked ham product as well as f o r bentonite (using A.O.A.C. methods). From t h i s data, an estimate of the s p e c i f i c heat of the bentonite and ham product was obtained using equation 3.2 (Table 3.2). However, during the phase change from the frozen state to the thawed state, heat energy was involved i n the change of state and could change d r a s t i c a l l y over a given temperature range (Levy and Lund, 1977) . This gave r i s e to the apparent s p e c i f i c heat which was the s p e c i f i c heat i n addition to the transient heat involved i n the phase change. Table 3.2 indicates the apparent heat capacity for the t r a n s i t i o n , and the calculated heat capacity above freezing for the chunk, emulsion and blend THERMAL PROPERTIES 95 Table 3.1 Composition of chunk, emulsion and blend portions of flaked ham as well as the bentonite dispersion. Percent Composition Material Fat Moisture Ash Protein Chunk 5.0 73.5 1.0 22 . 0 Emulsion 10.0 69.0 7.5 13.5 Blend 6.7 71.0 1.0 21.3 Bentonite 0.0 63.8 36.2 0.0 THERMAL PROPERTIES 96 Table 3.2 Calculated s p e c i f i c heat and apparent s p e c i f i c heat for the chunk, emulsion and blend portions of the flaked ham product; and the bentonite dispersion for above freezing and the phase t r a n s i t i o n respectively. Material Calculated S p e c i f i c Heat (kJ/Kg °K) Apparent S p e c i f i c Heat (kJ/Kg °K) Chunk 3.51 5.54 Emulsion 3.27 1.78 Blend 3.42 4.52 Bentonite 2.53 12.94 THERMAL PROPERTIES 97 portions of the flaked ham product, as well as f o r the bentonite-water dispersion. In a l l cases, except for the emulsion portion of the flaked ham, the apparent s p e c i f i c heat was greater than the calculated s p e c i f i c heat from Equation 3.2. This difference may r e l a t e to the precent unfrozen water i n the material. The equation of Heldman and Singh (1981) does not d i s t i n g u i s h between bound and free water. Thus, according to B i l i a d e r i s (1983), i f there i s a high percentage of unfrozen water, there w i l l be no DSC peak and thus no measureable H. In the emulsion portion of the flaked ham, due to not only the entrappment of water molecules i n the structure of the matrix, but also binding of water molecules to negative charges on the proteins (Schmidt, 1987), as well as interactions with other solutes that are present (phosphates, e t c . ) , t h i s may be the case. Mohsenin (1980) has also discussed the speculations of researchers on the va r i a t i o n s of bound and free water i n a sample i n terms of heat capacity. Figure 3.4 shows the DSC thermogram of the emulsion portion of the flaked ham product. The heating pattern was s i m i l a r to that observed for the chunk portion. The endotherm indicated a thawing point of -10.7°C, although i t would appear that some thawing began at approximately -3 0°C. The endothermic peak of the thermogram appeared at -5.1°C. Again, t h i s r e l a t e d to the i c e to water t r a n s i t i o n . The heat of THERMAL PROPERTIES 9S in CO CO .. V E U tn Q f (M i m ! CD I X a u O J cn 0 a 1 xm a: i a a ui cn 1 3 CD < V 0 a a u . 01 0 Q . - H a a. U n a ( J "X cn x fl tn X CM UJ . 01 a. a; 6 N a •*< a CM > • u in CM a -z V X > -l - l 21 0 a Il u • c a M CM E — 0 I. • or 0) 0 a t. r r a . -+-ca CM 4 a cn a a «i m c 0 Q . -KJ C .ta co in „ u . Q o CO v> « L 3 CM I 0 Q E 0 .ta H • Q 4.Q I . .ta CM I cn CM I CM"0 *°Id V»»H-Figure 3 . 4 DSC thermograph f o r f l a k e d ham — emulsion p o r t i o n . THERMAL PROPERTIES 99 thawing f o r t h i s thermogram was 62 J/g. The lowered t r a n s i t i o n temperature for the phase change was due to the presence of the various additives: s a l t , phosphate, n i t r i t e , and erythorbate, which served to depress the freezing point. S i m i l a r i l y , t h i s may have accounted for the f a c t that thawing appeared to have i n i t i a t e d at a temperature s u b s t a n t i a l l y lower than -10.7°C. In solution, sodium n i t r i t e , has a eutectic point of -18°C; sodium chloride has a eutectic point of -21.13°C; and the eutectic points for most of the phosphate s a l t s are i n the range of -13 to -23.5°C (Fennema, 1973) . Figure 3.5 shows the DSC thermograph f o r the blended product (50% chunk/50% emulsion). The endothermic peak occurred at -2.5°C, with the thawing point occurring at -8.2°C. The heat of thawing for the thermogram was 167 J/g. The DSC t r a n s i t i o n temperature of the blended product f e l l between the chunk and emulsion portion t r a n s i t i o n temperatures. Overall these r e s u l t s suggest that at a temperature of -2.5°C only l i q u i d water was present i n the flaked ham product. Figure 3.6 shows the DSC thermograph for the bentonite dispersion. The endothermic peak appeared at +2.1°C, the thawing point (t h) appeared at -0.7°C, and the heat of thawing was 194 J/g. The reason for the endothermic peak appearing above 0°C may l i e i n the heterogeneous nature of bentonite. Thawing began below 0°C, but the endothermic peak crossed the THERMAL PROPERTIES 100 « O i f* m m L ( L i o n E H -*» 0 0 •* 0 L vi oirr a. CM") MOX . 3 Figure 3.5 DSC thermograph for blended flaked ham. THERMAL P R O P E R T I E S 101 (M«U) M O X J ^ D 9 H Figure 3.6 DSC thermograph f o r bentonite d i s p e r s i o n . THERMAL PROPERTIES 102 normal freezing range. Replicate t e s t i n g exhibited the same pattern. The extremely hygroscopic nature of bentonite (which i s capable of adsorbing f i v e times i t s weight i n water (Krauskopf, 1967)) may play a r o l e . Thus, most of the water was present as bound water as opposed to the greater percentage of free bulk water present i n food systems such as muscle t i s s u e . Mohsenin (1980) discussed the speculations of researchers on the v a r i a t i o n s caused by bound water and free water i n a sample. Some researchers have maintained that bound water has a higher heat capacity than free water, while others take an opposing view. A second factor involved may be the heating rate of the DSC. A rate of 2°C/min. may be either to slow or to f a s t to provide an accurate representation of a rapid phase change. Thus the data was spaced over a wider range than a c t u a l l y existed i n the sample. However, Koga and Yoshizumi (1979) made mention of the f a c t that while exotherms ascribable to the freezing of a mixture of ethanol and water depended upon the cooling rate of the calorimeter, the heat of freezing did not. One can assume that the same argument applies to thawing as i t proceeds i n more or le s s the reverse fashion. According to Levy (1977), one d i f f i c u l t y that can a r i s e i n these studies i s due to the rapid change i n H when approaching the thawing point from the frozen state. The measurements taken i n t h i s region are rather d i f f i c u l t and the accuracy of experimental values could s u f f e r . THERMAL PROPERTIES 103 Niekamp et a l (1984) performed a comprehensive study on the thermal properties of bentonite-water dispersions i n environments above freezing. Further study, however, on the thermal behaviour of bentonite-water dispersions as they pass through a phase change between the thawed and frozen states appears to be warranted, i n order to determine t h e i r s u i t a b i l i t y as models for frozen food products. 3.2.2 THERMAL DIFFUSIVITY: The dependence of thermal d i f f u s i v i t y on temperature has been well documented i n the l i t e r a t u r e . The c a l c u l a t i o n used by Dickerson (1965) yielded an average thermal d i f f u s i v i t y across the environmental temperature range. Ramaswamy and Tung (1986) demonstrated the temperature dependence of thermal d i f f u s i v i t y and noted the f a c t that as temperature increased, so d i d thermal d i f f u s i v i t y values. Thus, f i n i t e difference techniques can achieve greater accuracy i f thermal d i f f u s i v i t y i s considered as a function of temperature rather than a constant value. I n i t i a l t e s t s performed with the apparatus described i n t h i s study yielded a re l a t i o n s h i p between thermal d i f f u s i v i t y and temperature. Table 3.3 indicates thermal d i f f u s i v i t y values for the flaked ham product and f o r the bentonite dispersion. THERMAL PROPERTIES 104 Table 3.3 Thermal d i f f u s i v i t y of flaked ham and bentonite at various temperatures. TEMPERATURE THERMAL DIFFUSIVITY (°C) ( x 107 m2/s) Flaked Ham Bentonite -20 11. 00 5.88 -15 6.14 4.33 -10 1.22 2.79 - 5 0.52 1.67 0 0.40 0.45 5 0.42 0.48 10 0.46 0.53 15 0.50 0.60 20 0.54 0.66 25 0.58 0.74 30 0.64 0.84 35 0.70 0. 95 40 0.76 1.08 45 0.83 1.22 50 0.91 1.39 55 0.97 1.57 60 1.10 1.71 65 1.24 2.00 70 1.43 2.20 75 1.68 2.45 80 1.99 2 . 60 85 2.51 3.10 90 3.00 4.07 95 3.61 4.15 100 4.10 4.65 105 4.50 5.03 110 5.20 5.57 115 5.84 5.98 120 6.15 6.50 125 6.74 6.92 LITERATURE REVIEW 105 Values for the thermal d i f f u s i v i t y of the flaked ham product and f o r the bentonite dispersion can be seen to change with increasing temperature. Ramaswamy and Tung (1985) also noted that thermal d i f f u s i v i t y varied with the moisture content and formulation of the food material. Niekamp et a l (1984) l i s t e d the density, thermal conductivity and heat capacity for a 50% bentonite dispersion as 1.50 g/ml, 0.788 W/cm °C, and 3151 J/Kg°C respectively at 66°C. Using these parameters (with conversion to appropriate u n i t s ) , and the re l a t i o n s h i p between thermal d i f f u s i v i t y and these parameters, a check of the thermal d i f f u s i v i t y of the bentonite system was made. The calculated thermal d i f f u s i v i t y , from the data of Niekamp et a l (1984), corresponding to the percent bentonite dispersion used i n t h i s study was 2.78 x 10"7 m2/s. This i s an average value and corresponds to an average value of 2.74 x 10"7 m2/s f o r the measurements obtained i n t h i s experiment across the temperature range presented i n Table 3.3. The average thermal d i f f u s i v i t y of the flaked ham fo r the temperature range of -20°C to +125°C was 2.39 x 10"7 m2/s. The l i t e r a t u r e values of 1.00 to 1.70 x 10~7 m2/s are common f o r most food materials i n that temperature range, however the incorporation of the transient across the phase change served to give a higher average thermal d i f f u s i v i t y than expected. The range of thermal d i f f u s i v i t y values determined indicates the dependence of thermal d i f f u s i v i t y on temperature THERMAL PROPERTIES 106 and had important implications i n the modelling of heat tr a n s f e r i n food materials using such techniques as f i n i t e d ifference. Chapter 4 DEVELOPMENT OF THE FINITE DIFFERENCE MODEL This chapter concerns the development of the e x p l i c i t f i n i t e difference solution to the relevant equation f o r a f i n i t e cylinder presented i n the discussion i n Chapter 2. This equation resulted from the combination of two general heat tr a n s f e r equations. The f i r s t equation was f o r one-dimensional heat transfer (in the x dimension) i n an i n f i n i t e slab under unsteady-state conditions and was expressed as d*T 1 dT dx2 o( d f The second expression was f o r the heat transfer i n the r a d i a l d i r e c t i o n (r) i n an i n f i n i t e c ylinder and was expressed as cVT 1 dT 1 dT dr 2 r dr cx d t Equations (4.1) and (4.2), i n combination, give the 107 DEVELOPMENT 108 equation for the cylinder d*T d*T 1 dT 1 dT dx2 dr 2 r dr Oi d% In pursuing the analysis of unsteady-state heat transfer i n a f i n i t e cylinder such as a can, one must again denote a nomenclature f o r the necessary increments i n the l i n e a r d i r e c t i o n (x) and the r a d i a l d i r e c t i o n (r) . Thus, the c y l i n d r i c a l container can be imagined as being divided into d i s c r e t e volumes that appear as layers of concentric rings having rectangular cross-sections. The nomenclature used i s displayed i n Figure 4.1. One can see from Equation 4.3 and Figure 4.1, that t h i s was e s s e n t i a l l y a problem i n two dimensions. In re-writing Equation 4.3, the assumptions of constant environmental temperature and uniform i n i t i a l temperature were made i n order to adopt the transient numerical method. The assumption of constant environmental temperature was made i n that whatever s l i g h t f l u ctuations i n r e t o r t temperature may occur, the subsequent e f f e c t on the heat transfer c o e f f i c i e n t of the saturated steam environment would be minimal (Ramaswamy et a l . , 1983). Equation 4.3 thus became (4.4) DEVELOPMENT 109 Figure 4.1 Nomenclature used i n the numerical analysis of heat conduction i n the f i n i t e c y l i n d e r DEVELOPMENT 110 where k i s the thermal conductivity, p i s the density, Cp i s the heat capacity, and X i s the time increment (usually i n seconds). Application was again made of the Fourier equation dT q = - k A (4.5) dx to c a l c u l a t e heat flow i n the x and r d i r e c t i o n s . Thus, for the x d i r e c t i o n , the equation became dT q x = - k A, (4.6) dx and f o r the r d i r e c t i o n , the equation, i n s i m i l a r fashion, became dT qr = - k A, (4.7) d r The p a r t i a l derivatives of the temperature gradients f o r Figure 4.1 may be written as follows (for the centre node 2 with x = 2+2A/2 to 2-2B/2 and r = 2+1/2 to 2-1/2 as x and r straddle the node 2) . DEVELOPMENT 111 dT \ T l - T2 dx /2+1/2 ~ Ax (4.8) tl.) dx /2-1/2 "° T2 - T3 >x (4.9) dT \ dr / T2A - T2 2+2A/2 Ar (4.10) dT \ dr /2-2B/: T2 - T2B (4.11) Combining the equations f o r the approximations of the p a r t i a l derivatives [Equations 4.8 and 4.9] i n the x dimension and si m p l i f y i n g the r e s u l t i n g expression y i e l d s d*T \ T l + T3 - 2(T2) J = (4.12) dx2/2 (Ax) 2 In a s i m i l a r fashion, the expression f o r the r dimension can be written as d^TX T2A + T2B - 2(T2) 1 = (4.13) dr 2/2 ( A r ) 2 DEVELOPMENT 112 The term dT/dr can be approximated with T2A - T2B (4.14) 2/Ar since the temperature difference was calculated between nodes 2A and 2B which were two nodes apart. This l e f t only the need to apprimate the time increment. This was done with the following expression dT \ dr/2 dT \ T2 t + 1 - T2l T<) z A t ( 4 - 1 5 > with the superscript t denoting the present time and the superscript t+1 representing one time increment into the future. Combining Equations 4.12), (4.13), (4.14) and (4.15) gives the f i n i t e difference equation equivalent to Equation (4.3). T2A + T2B - 2(T2) + l/T2A - T2B\ + T l + T3 - 2(T2) (Ar) 2 ! r V 2 r / (&x): 1 T2 t + 1 - T2l 0< (4.16) DEVELOPMENT 113 This equation corresponded to the node designated T2 as described i n Figure 4.1. Equation 4.16 y i e l d e d the equation f o r the node T2 (4.17) By choosing the appropriate r a d i a l and l i n e a r increments ( i n t h i s study, 0.00218 m and 0.0010715 m respectively) and a s u i t a b l e time increment (1 second), the unsteady-state heat t r a n s f e r could be simulated. The system modelled was a 0.0873 m by 0.027 m s t e e l can undergoing thermal processing i n a static-cooking, v e r t i c a l saturated steam r e t o r t . Some i m p l i c i t assumptions were made i n the usage of t h i s model as a simulation of the above-mentioned process. F i r s t l y , i t was assumed that the i n i t i a l product temperature was uniform throughout the container due to the long time allowed f o r the actual product to e q u i l i b r a t e p r i o r to the s t a r t of the ac t u a l heat penetration studies. The second assumption was that there was a perfe c t thermal contact at the surfaces of the container, i n an e f f o r t to s i m p l i f y the model. L a s t l y , due to the large temperature d i f f e r e n c e between the DEVELOPMENT 114 i n t e r n a l temperature of the container and the external temperature of the environment, i t was assumed that the convective boundary condition could be e s s e n t i a l l y ignored and that the f i r s t node of the f i n i t e d i f f e r e n c e g r i d would be located at the container surface. As was previously mentioned, any minor f l u c t u a t i o n s i n r e t o r t temperature would have minimal e f f e c t s on the heat t r a n s f e r c o e f f i c i e n t of the saturated steam environment being simulated. The e x p l i c i t computer program used i s presented i n Appendix A. Using the e f f e c t i v e thermal d i f f u s i v i t y values obtained i n Chapter 3, and the above mentioned g r i d dimensions f o r the r and x d i r e c t i o n s , a temperature h i s t o r y was obtained f o r the centre-point of the simulated container with i n i t i a l temperatures of -20°C, -15°C, -10°C, -5°C, and 0°C. The temperature h i s t o r y was analyzed with the CalSoft program and the a p p l i c able thermal process parameters were then derived v i a the B a l l ( B a l l and Olson, 1957), Stumbo (Stumbo, 1973) and general methods (Bigelow et a l , 1920). This was done f o r both the bentonite d i s p e r s i o n and the flaked ham. Results are discussed below. Chapter 5 EMPIRICAL CONFIRMATION OF THE MODEL In determining the required thermal process f o r food s t u f f s , the aim i s to maximize destruction of microorganisms and spores while minimizing the loss of nu t r i e n t s and other de s i r a b l e a t t r i b u t e s . These processes are governed by the temperature h i s t o r y of the product, which may be estimated from the temperature at the geometric centre, f o r conduction-heating products. From t h i s temperature h i s t o r y , l e t h a l i t y can be obtained by numerical or graphical i n t e g r a t i o n , a technique known as the general method (Bigelow et a l , 1920). Hayakawa (1978) divided formula methods f o r estimating l e t h a l i t y into two groups. The f i r s t group consisted of those methods that c a l c u l a t e d l e t h a l i t y at the thermal centre of the product. The second group consisted of methods that c a l c u l a t e d mass average l e t h a l i t y f o r whole containers. There i s another group of methods, by which l e t h a l i t y may be c a l c u l a t e d from f i r s t p r i n c i p l e s ( i . e . the thermal properties of the food material) using e i t h e r f i n i t e d i f f e r e n c e 115 CONFIRMATION 116 computations (Teixeira et a l , 1969) or numerical int e g r a t i o n (Hayakawa, 1977). V e r i f i c a t i o n of the f i n i t e d i f f e r e n c e model as presented i n Chapter 4, was required. This was done by performing heat penetration experiments on the two systems involved i n t h i s study and comparing the r e s u l t s to the time/temperature p r o f i l e estimated from the f i n i t e d i f f e r e n c e model. 5.1 MATERIALS AND METHODS: Thermocouples were located at the geometric centre of twelve 307 X 111 two-piece t i n - p l a t e cans per t r i a l . Cans were f i l l e d with a minimal headspace and hermetically sealed on an Angeles 61H steam-flow c l o s i n g machine (Angeles Corp, Anaheim, CA). The cans were then placed i n a freezer set to the desired i n i t i a l temperature (-20°C, -15°C, -10°C, -5°C, or 0°C) and allowed to e q u i l i b r a t e over a twenty-four hour period. Temperature f l u c t u a t i o n i n the f r e e z e r was +/- 2°C. Next, cans were placed i n t o a v e r t i c a l , saturated steam, non-a g i t a t i n g r e t o r t . Centrepoint thermocouples were attached to a Calwest model 32 datalogger (Calwest Technologies, Canyon Country, CA) , connected i n turn to a Toshiba T1000 computer equipped with the C a l s o f t software program f o r a c q u i s i t i o n and a n a l y i s of the heat penetration data (Technical Inc., Me t a i r i e , LA). CONFIRMATION 117 A t o t a l of 50 containers, f o r each i n i t i a l temperature, were processed at 124.4°C u n t i l a l e t h a l i t y (FJ of at l e a s t 6.0 minutes was achieved as calcula t e d by the Ca.lSof t program using B a l l ' s formula. At t h i s point, the co o l i n g phase of the thermal process was i n i t i a t e d . Temperature data was corrected f o r the e f f e c t of the thermocouple (Ecklund, 1956). The data was then p l o t t e d and analyzed v i a the general (Bigelow et a l . , 1920), B a l l ( B a l l and Olson, 1957) and Stumbo (Stumbo, 1973) methods and compared to the r e s u l t s of the f i n i t e d i f f e r e n c e computations as ca l c u l a t e d by the same methods. 5.2 RESULTS AND DISCUSSION: Table 5.1 shows the time taken f o r frozen flaked ham to reach a temperature above the fr e e z i n g point of water. As the i n i t i a l temperature of the product decreased there was a corresponding increase i n the time required to reach a temperature above freez i n g . This pattern also held true f o r the f i n i t e d i f f e r e n c e generated data (Table 5.1). This supported the hypothesis that the i n i t i a l assumptions made with respect to boundary conditions and determination of thermal d i f f u s i v i t i e s were v a l i d . In the range of -10°C to -15°C, the increase i n time to reach the thawing point was s l i g h t . This was explicable i n that within the temperature range of 0°C to -5°C there was a 118 CONFIRMATION 119 phase change from frozen water to l i q u i d water which represents the transformation of pure or nearly pure i c e c r y s t a l s to water. This i s consistent with the data generated from the DSC measurements which showed an endothermic peak i n the same range. In the range of -10°C to -15°C the system was r e l a t i v e l y stable, i n that a s u s t a n t i a l phase change d i d not occur. As previously mentioned however, below -15°C there was again an increase i n the thawing time, which may have r e l a t e d to the presence of phosphate s a l t s i n the ham, and to s a l t s of the various contaminants present i n bentonite. The e u t e c t i c points f o r phosphate s a l t s are i n the range of -13°C to -20°C (Fennema and Powrie, 1973). The net r e s u l t , as indicated i n Table 5.2 was that there was an increase i n the 11 j h " value (heating l a g factor) with decreasing i n i t i a l temperature of the product. The value f o r f h (which i s the slope of the heating curve) was v i r t u a l l y unaffected. This was expected as once the products have passed through the phase change from a frozen s t a t e to a thawed state, they should behave i n a s i m i l a r way. Figure 5.1 shows the increase i n process time (Pt) , to an F„ of 6.0 minutes, observed as a r e s u l t of decreasing i n i t i a l product temperature. The dominant f a c t o r i n t h i s delay was the l a t e n t heat of melting (Richard et a l , 1991) since the lag phase f o r samples with progressively lower i n i t i a l temperatures increased. An anomaly was evident i n the range of -15°C to -20°C f o r CONFIRMATION 120 Table 5.2: Mean values f o r i n i t i a l temperature, f h, and j h f o r bentonite and ham at -20°C, -15°C, -10°C, -5°C, and 0°C. I n i t i a l Temperature f h(min.) (°C) Ham 0 26. ,57 1. ,32 -5 25. .26 (2. .72) 0 0 1. ,93 (0. .61) -10 26. .52 (4. .84) 2. ,23 (0. .54) -15 23. .95 (2. .52) 2. ,30 (0. ,62) -20 24. .44 (0. .83) 2. .78 (0. .30) Bentonite 0 24. .20 0. .92 -5 25. .55 (2. .47) 1. .02 (0. .16) -10 25. .69 (3. .56) 1. .02 (0. .23) -15 29. .24 (4. .71) 0. .88 (0. .49) -20 29. .95 (4. .40) 0. .79 (0. .51) (a) Standard Deviation. CONFIRMATION 121 - 2 0 INITIAL TEMPERATURE (°C) + Flaked Ham (Actual) O Flaked Ham (Finite) ^ Bentonite (Actual) A Bentonite (Finite) Figure 5.1. I n i t i a l temperature versus process time _(Pt) f o r an F 0 = 6.0 minutes f o r flaked ham and bentonite, corresponding to data c a l c u l a t e d using the B a l l formula method ( B a l l and Olson, 1957). CONFIRMATION 122 the bentonite system. For t h i s temperature range, a decrease i n " j h " values was observed. Using the experimental values f o r j h and f h one could c a l c u l a t e the required process time to achieve a l e t h a l i t y (F0) of 6.0 with the B a l l and Stumbo formula methods and Bigelow's (1920) general method. The r e s u l t s of these c a l c u l a t i o n s are summarized i n Table 5.3. These r e s u l t s served as a check of the accuracy of the f i n i t e d i f f e r e n c e model as an estimate of the heating p r o f i l e of the fl a k e d ham and bentonite d i s p e r s i o n systems with i n i t i a l temperatures i n the fr e e z i n g range. Indeed, while the r e s u l t s of the f i n i t e d i f f e r e n c e c a l c u l a t i o n s v a r i e d by a few minutes i n some cases, they were, i n most cases, within one standard d e v i a t i o n of the empirical values. Graphical representation of the data i n comparison with actual heat penetration data supports t h i s (Figure 5.2). In most cases, the model underestimated the experimental process time. A pos s i b l e reason f o r t h i s may be that there was a head-space i n the a c t u a l container. Thus, there was an i n s u l a t i n g layer of a i r on one surface which would serve to impede heat flow across that surface. Secondly, the average thermal d i f f u s i v i t y f o r water i s 1.60 x 10 7 m2/sec. over the temperature range modelled i n t h i s study. Thus, the thermal d i f f u s i v i t y of water was lower than that of the ham. Therefore, the presence of an exudate from the product would CONFIRMATION 123 Table 5.3: Calculated process time comparison f o r flaked ham and bentonite, using the general method60, B a l l 0 0 and Stumbo(c) formula methods f o r act u a l heat penetration data and from the f i n i t e d i f f e r e n c e model. A l l c a l c u l a t i o n s are done to an F„ of 6.0 minutes. PROCESS TIME (min.l I.T. General B a l l Stumbo CO Method Method Method HAM 0 34.0 (3.3) ( d ) 41.0 (4.0) 39.8 (4.0) 31.0(e) 38.3 35.0 -5 35.0 (5.5) 43.4 (3.9) 44.3 (3.9) 34.5 42.5 39.2 -10 43.2 (6.6) 43.6 (3.9) 47.8 (3.9) 36.5 44.4 41.1 -15 37.0 (6.7) 43.9 (3.8) 47.8 (3.8) 32.1 45.2 41.9 -20 41.8 (7.2) 47.1 (3.8) 47.9 (3.8) 37.2 47.5 44.2 BENTONITE 0 25.1 (3.8) 33.6 (2.5) 33.1 (2.5) 22.0 34.5 31.3 -5 28.4 (4.9) 37.4 (3.0) 38.1 (3.0) 27.7 34.9 31.6 -10 30.1 (5.5) 37.3 (3.2) 39.9 (3.2) 28.2 35.2 32.0 -15 31.1 (5.6) 38.9 (4.0) 42.1 (4.0) 28.3 38.1 34.7 -20 29.5 (5.0) 41.5 (4.1) 40.5 (4.1) 30.4 38.5 36.3 (a) Bigelow, 1920. (b) B a l l and Olson, 1957. (c) Stumbo, 1973. (d) Standard deviations appear i n parentheses. (e) Calculated from f i n i t e d i f f e r e n c e model. CONFIRMATION 124 Figure 5.2 Heat: penetration data comparison — a c t u a l and f i n i t e d i f f e r e n c e (Bentonite: I.T. -20°C) . CONFIRMATION 125 acted as an i n s u l a t o r with respect to the tr a n s f e r of heat r e l a t i v e to the flaked ham product. Thus, t h i s would r e s u l t the model underestimating the experimental data. For the bentonite samples, where there was no apparent exudate, there was a clo s e r f i t of the model to the experimental data. Chapter 6 CONCLUSIONS A number of conclusions can be drawn from t h i s study. F i r s t l y , the phase change as the product crossed from the frozen state to the thawed state caused an i n i t i a l r e t a r d ation of heating (lengthened lag phase) of canned foods. However, once t h i s point was passed, the curve resembled the c h a r a c t e r i s t i c heating curve obtained f o r a canned food material that was i n i t i a l l y thawed. The second conclusion that arose from t h i s study was that the thawing time required to melt a product with an i n i t i a l temperature ranging down to -20°C added up to 8.1 minutes to the processing time f o r the ham product, and 7.4 minutes f o r the bentonite dispersion. This compared with 7.4 and 5.0 minutes r e s p e c t i v e l y , f o r the f i n i t e d i f f e r e n c e estimates of the process. As mentioned above, t h i s increase i n processing time was due to the l a t e n t heat of melting. The t h i r d conclusion concerned the usage of bentonite as a model f o r food systems. Niekamp et a l (1984) and numerous other researchers demonstrated the a p p l i c a b i l i t y of various percentage dispersions of bentonite i n water as models f o r food systems. The present study also showed that bentonite 126 CONCLUSIONS 127 could be used f o r the study of the thermal properties of food systems. There i s some question as to i t s s u i t a b i l i t y i n the modelling of frozen foods, and c e r t a i n l y to s p e c i f i c food products. The fourth conclusion a r i s e s from a comparison of the temperature p r o f i l e s of flaked ham and bentonite systems with respect to i n i t i a l temperatures i n the frozen range. The o v e r a l l temperature p r o f i l e generated by the f i n i t e d i f f e r e n c e model was s i m i l a r to the actual experimental data f o r the bentonite and flaked ham systems. However, c e r t a i n anomalies d i d a r i s e , apparently due to the di f f e r e n c e s i n structure of the two materials. The flaked ham product, being a meat emulsion, r e l i e d on a c r o s s - l i n k i n g of s a l t - s o l u b l e proteins to form a g e l structure, with water trapped i n the i n t e r s t i t i a l spaces. Heating r e s u l t e d i n the s t a b i l i z a t i o n of the g e l . However, following r e t o r t i n g , water was present i n the form of an exudate (the e f f e c t s of which have already been discussed), leaving the mean drained weight of the product at 85% of the i n i t i a l product weight. I t was surmised that t h i s was due to thermal destruction of the g e l . The bentonite dispersion, with i t s high water-binding capacity, d i d not behave i n t h i s way. In f a c t , the material, following r e t o r t i n g , exhibited the same appearance as i t d i d p r i o r to the commercial s t e r i l i z i n g treatment. Thus, while there were s i m i l a r i t i e s between the two systems, one cannot CONCLUSIONS 128 use a bentonite dispersion as a p e r f e c t model to mimic a s p e c i f i c food system. Differences with respect to water binding behaviour do point out some p o t e n t i a l problems, p a r t i c u l a r i l y with respect to the modelling of frozen foods. For example, the nature of the m e t a l l i c ion contaminants commonly found i n bentonite and i t s hi g h l y hygroscopic behaviour point to some p o t e n t i a l d i f f i c u l t i e s , p a r t i c u l a r l y i n the study of t a n s i t i o n temperatures. While there may be s u f f i c i e n t d i f f e r e n c e s i n the s p e c i f i c behaviour of water molecules i n the two systems to warrant further study on the v a l i d i t y of bentonite dispersions as models f o r frozen food systems, both the bentonite system and the f l a k e d ham system demonstrated s i m i l a r heating behaviour. The i n i t i a l s tate and temperature of the food material a f f e c t e d only the process duration and not the end r e s u l t , i . e . the f i n a l product temperature. Thus, there was a l a g i n the heating p r o f i l e as the product underwent thawing, which increased as the i n i t i a l temperature decreased. However, once t h i s i n i t i a l thawing time was completed, the remainder of the heating p r o f i l e remained r e l a t i v e l y s i m i l a r , no matter what the i n i t i a l temperature was. The f i f t h conclusion of t h i s study i n d i c a t e s that the use of thermal d i f f u s i v i t y values across the temperature range of i n t e r e s t y i e l d s a good approximation of the model to the acut a l data. This i n d i r e c t l y supports the suggestion by CONCLUSIONS 129 Ramaswamy and Tung (1986) that the use of an average thermal d i f f u s i v i t y value would r e s u l t i n a gorss over-estimation of the process time required to achieve commercial s t e r i l i t y . The f i n a l conclusion of t h i s study concerns the f i n i t e d i f f e r e n c e model. The model developed here presented a consistent p o r t r a y a l of the events occuring i n a canned food with a subfreezing i n i t i a l temperature undergoing a thermal process. In addition, the underlying assumptions with respect to boundary conditions, and use of an e f f e c t i v e thermal d i f f u s i v i t y incorporating the transient, were v a l i d f o r the purposes of t h i s study. The assumption of constant thermal contact may not have been as accurate as was o r i g i n a l l y surmised. The key to the correct use of t h i s model was accurate determination of thermal d i f f u s i v i t y . The only l i m i t a t i o n of the model appears to be a need to run the program on a high-speed computer. This allowed the program to proceed at a speed equivalent to r e a l time or f a s t e r . O v e r a l l , the main th r u s t of t h i s study was to demonstrate that food materials which s t a r t o f f i n i t i a l l y frozen can be thermally processed s a f e l y i n order to achieve commercial s t e r i l i t y . While the model demonstrates t h i s i n a mathematical fashion, further work with innoculated packs and thermal death times with spore suspensions would be needed i n order to confirm t h i s . However, the r e s u l t s have shown that only the l a g f a c t o r and duration of the process were affe c t e d CONCLUSIONS 130 and the effectiveness of the process was not impaired. Appendix A EXPLICIT FINITE DIFFERENCE MODEL (BENTONITE) 10 CLS 20 LOCATE 1,23:PRINT "FINITE DIFFERENCE METHOD CALCULATIONS" 30 LOCATE 2,34:PRINT "BENTONITE MODEL" 40 LOCATE 3,28:PRINT "TRANSIENT NUMERICAL METHOD" 50 LOCATE 9,1:PRINT "PLEASE ENTER THE FOLLOWING PARAMETERS:" 60 Z=.0010715 70 R=.00218 80 INPUT "ENTER A VALUE FOR INITIAL BODY TEMPERATURE (C):";IT 90 INPUT "ENTER A VALUE FOR N (NUMBER OF LOOPS): ";N 100 INPUT "ENTER A VALUE FOR T (T = TIME INCREMENT IN SEC.):";T 110 TT=0:T1=ET:T2=IT:T3=IT:T4=IT:T5=IT:T6=IT:T7=IT:T8=IT: T9=IT:T10=IT:T11=IT:T12=IT:T13=IT:T14=IT:T15=IT:T16= IT:T17=IT:T18=IT:T19=IT:T2 0=IT:T21=IT 120 T2A=IT:T2B=IT:T3A=IT:T3B=IT:T4A=IT:T4B=IT:T5A=IT:T5B= IT:T6A=IT:T6B=IT:T7A=IT:T7B=IT:T8A=IT:T8B=IT:T9A=IT: T9A=IT:T10A=IT:T10B=IT:T11A=IT:T11B=IT:T12A=IT:T12B= IT:T13A=IT:T13B=IT:T14A=IT:T14B=IT:T15A=IT:T15B=IT: T16A=IT:T16B=IT:T17A=IT:T17B=IT:T18A=IT: 130 T18B=IT:T19A=IT:T20A=IT:T20B=IT: 140 RI=.0436 150 B=0 170 FOR Q=l TO 60 180 Al=l/(2*RI*R) 190 A2=1/RA2 200 T l = ET 210 RI=RI-R 220 M=T2 230 GOSUB 280 240 GOTO 590 250 IF M<-20 THEN TD=5.88E-07 260 IF M>=-20 AND M<-15 THEN TD=4.33E-07 270 IF M>=-15 AND M<-10 THEN TD=2.79E-07 280 IF M>=-10 AND M<-5 THEN TD=1.67E-07 290 IF M>=-5 AND M<0 THEN TD=4.5E-08 300 IF M>=0 AND M<5 THEN TD=4.8E-08 131 APPENDIX A 132 310 IF M>=5 AND M<10 THEN TD=5.3E-08 320 IF M>= =10 AND M<15 THEN TD= 6E 1-08 330 IF M>= =15 AND M<20 THEN TD= 6. 63-08 340 IF M>= 20 AND M<25 THEN TD= 7. 4E-08 350 IF M>= =25 AND MOO THEN TD= 8. 4E-08 360 IF M>= =30 AND M<35 THEN TD= 9. 5E-08 370 IF M>= =35 AND M<40 THEN TD= =1. 08E-07 380 IF M>= =40 AND M<45 THEN TD= 1. 22E-07 390 IF M>= =45 AND M<50 THEN TD= 1. 39E-07 400 IF M>= =50 AND M<55 THEN TD= 1. 57E-07 410 IF M>= =55 AND M<60 THEN TD= •1. 71E-07 420 IF M>= =60 AND M<65 THEN TD= .0000002 430 IF M>= =65 AND M<70 THEN TD= 2. 2E-07 440 IF M>= =70 AND M<75 THEN TD= 2. 45E-07 450 IF M>= =75 AND M<80 THEN TD= 2. 6E-07 460 IF M>= =80 AND M<85 THEN TD= 3. 1E-07 470 IF M>= =85 AND MOO THEN TD= A. 07E-07 548 IF M>== =90 AND M<95 THEN TD= 4. 15E-07 490 IF M>= =95 AND M<100 THEN TD 1=4 .65E-07 500 IF M>=100 AND M<105 THEN TD=5.03E-07 510 IF M>=105 AND M<110 THEN TD=5.57E-07 520 IF M>=110 AND M<115 THEN TD=5.98E-07 53 0 IF M>=115 AND M<120 THEN TD=6.5E-07 540 IF m.=120 AND M<125 THEN TD=6.92E-07 550 RETURN 560 TD2=TD 570 N2=(T*TD2*(A2+A1))*(T2A) + (T*TD2*((1/R A2)-(1/2*RI*R)))* (T2B) + ((T*TD2)/(Z A2))*(T1+T3)+ (1—((T*TD2)*((2/R A2))+ (2/(Z-2)))))*T2 580 M=T3 590 GOSUB 280 600 TD3=TD 610 IF N2>ET THEN N2=ET 620 IF N2>ET THEN T2A=ET 630 IF N2>ET THEN T2B=ET 640 IF N2>ET THEN T2=ET 650 RI=TI-R 660 N3=(T*TD3*(A2+A1))*(T3A) + (T*TD3*((1/R A2)-(1/2*RI*R)))* (T3B) + ((T*TD3)/(Z~2))*(T2+T4)+ (1-((T*TD3)*((2/R~2))+ (2/(Z-2)))))*T3 670 M=T4 680 GOSUB 280 690 TD4=TD 700 IF N3>ET THEN N3=ET 710 IF N3>ET THEN T3A=ET 720 IF N3>ET THEN T3B=ET 730 IF N3>ET THEN T3=ET 740 RI=TI-R 750 N4=(T*TD4*(A2+A1))*(T4A) + (T*TD4*((1/R A2)-(1/2*RI*R)))* (T3B) + ((T*TD4)/(Z A2))*(T3+T5)+ (1-((T*TD4)*((2/R A2))+ APPENDIX A 133 (2/(Z*2)))))*T4 760 M=T5 770 GOSUB 280 780 TD5=TD 790 IF N4>ET THEN N4=ET 800 IF N4>ET THEN T4A=ET 810 IF N4>ET THEN T4B=ET 820 IF N4>ET THEN T4=ET 830 RI=TI-R 840 N5=(T*TD5*(A2+A1))*(T5A) + (T*TD5*((1/R A2)—(1/2*RI*R)))* (T5B) + ( (T*TD5) / (Z*2) ) * (T4+T6) + (1— ( (T*TD5) * ( (2/RA'2) ) + (2/(Z*2)))))*T5 850 M=T6 860 GOSUB 280 870 TD6=TD 880 IF N5>ET THEN N5=ET 890 IF N5>ET THEN T5A=ET 900 IF N5>ET THEN T5B=ET 910 IF N5>ET THEN T5=ET 920 RI=TI-R 930 N6=(T*TD6*(A2+A1))*(T6A) + (T*TD6*((1/R A2)-(1/2*RI*R)))* (T6B) + ((T*TD6)/(Z A2))*(T5+T7)+ (1-((T*TD6)*((2/R A2))+ (2/(Z-2)))))*T6 940 M=T7 950 GOSUB 280 960 TD7=TD 970 IF N6>ET THEN N6=ET 980 IF N6>ET THEN T6A=ET 990 IF N6>ET THEN T6B=ET 1000 IF N6>ET THEN T6=ET 1010 RI=TI-R 1020 N7=(T*TD7*(A2+A1))*(T7A) + (T*TD7*((1/R A2)-(1/2*RI*R)))*(T7B) + ((T*TD7)/(Z A2))*(T6+T8)+ (1-((T*TD7)*((2/R"2))+ (2/(Z"2)))))*T7 1030 M=T8 1040 GOSUB 280 1050 TD8=TD 1060 IF N7>ET THEN N7=ET 1070 IF N7>ET THEN T7A=ET 1080 IF N7>ET THEN T7B=ET 1090 IF N7>ET THEN T7=ET 1100 RI=TI-R 1110 N8=(T*TD8*(A2+A1))*(T8A) + (T*TD8*((1/R A2)-(1/2*RI*R)))*(T8B) + ((T*TD8)/(Z A2))*(T7+T9)+ (1-((T*TD8)*((2/R A2))+ (2/(Z^2)))))*T8 1120 M=T9 1130 GOSUB 280 1140 TD9=TD 1150 IF N8>ET THEN N8=ET 1160 IF N8>ET THEN T8A=ET APPENDIX A 134 1170 IF N8>ET THEN T8B=ET 1180 IF N8>ET THEN T8=ET 1190 RI=TI-R 1200 N9=(T*TD9*(A2+A1))*(T9A) + (T*TD9*((1/R A2)-(1/2*RI*R)))*(T9B) + ((T*TD9)/(Z A2))*(T8+T10)+ (1-((T*TD9)*((2/R A2))+ (2/(Z A2)))))*T9 1210 M=T10 1220 GOSUB 280 1230 TD10=TD 1240 IF N9>ET THEN N9=ET 1250 IF N9>ET THEN T9A=ET 1260 IF N9>ET THEN T9B=ET 1270 IF N9>ET THEN T9=ET 1280 RI=TI-R 1290 N10=(T*TD10*(A2+A1))*(T10A) + (T*TD10*((1/R A2)-(1/2*RI*R)))*(T10B) + ((T*TD10)/(Z A2))*(T9+T11)+ (1-((T*TD10)*((2/R"2))+ (2/(Z A2)))))*T10 13 00 M=T11 1310 GOSUB 280 1320 TD11=TD 1330 IF N10>ET THEN N10=ET 1340 IF N10>ET THEN T10A=ET 1350 IF N10>ET THEN T10B=ET 1360 IF N10>ET THEN T10=ET 1370 RI=TI-R 1380 N11=(T*TD11*(A2+A1))*(T11A) + (T*TD11*((1/R~2)-(1/2*RI*R)))*(T11B) + ((T*TD11)/(Z A2))*(T10+T12)+ (1-((T*TD11)*((2/R A2))+ (2/(Z A2)))))*T11 1390 M=T12 1400 GOSUB 280 1410 TD12=TD 1420 IF N11>ET THEN N11=ET 143 0 IF N11>ET THEN T11A=ET 1440 IF N11>ET THEN T11B=ET 1450 IF N11>ET THEN T11=ET 1460 RI=TI-R 1470 N12=(T*TD12*(A2+A1))*(T12A) + (T*TD12*((1/R"2)-(1/2*RI*R)))*(T12B) + ((T*TD12)/(Z A2))*(T11+T13)+ (1-((T*TD12)*((2/R~2))+ (2/(Z A2)))))*T12 1480 M=T13 1490 GOSUB 280 1500 TD13=TD 1510 IF N12>ET THEN N12=ET 1520 IF N12>ET THEN T12A=ET 1530 IF N12>ET THEN T12B=ET 1540 IF N12>ET THEN T12=ET 1550 RI=TI-R 1560 N13=(T*TD13*(A2+A1))*(T13A) + (T*TD13*((l/R'2)-(1/2*RI*R)))*(T13B) + ((T*TD13)/(Z A2))*(T12+T14)+ (1-((T*TD13)*((2/R A2))+ (2/(Z A2)))))*T13 APPENDIX A 135 1570 M=T14 1580 GOSUB 280 1590 TD14=TD 1600 IF N13>ET THEN N13=ET 1610 IF N13>ET THEN T13A=ET 1620 IF N13>ET THEN T13B=ET 1630 IF N13>ET THEN T13=ET 1640 RI=TI-R 1650 N14=(T*TD14*(A2+A1))*(T14A) + (T*TD14*((1/R"2)-(1/2*RI*R)))*(T14B) + ((T*TD14)/(Z A2))*(T13+T15)+ (1-((T*TD14)*((2/R A2))+ (2/(Z A2)))))*T14 1660 M=T15 1670 GOSUB 280 1680 TD15=TD 1690 IF N14>ET THEN N14=ET 1700 IF N14>ET THEN T14A=ET 1710 IF N14>ET THEN T14B=ET 1720 IF N14>ET THEN T14=ET 1730 RI=TI-R 1740 N15=(T*TD15*(A2+A1))*(T15A) + (T*TD15*((1/R A2)-(1/2*RI*R)))*(T15B) + ((T*TD15)/(Z A2))*(T14+T16)+ (1-((T*TD15)*((2/R~2))+ (2/(Z*2)))))*T15 1750 M=T16 1760 GOSUB 280 1770 TD16=TD 1780 IF N15>ET THEN N15=ET 1790 IF N15>ET THEN T15A=ET 1800 IF N15>ET THEN T15B=ET 1810 IF N15>ET THEN T15=ET 1820 RI=TI-R 1830 N16=(T*TD16*(A2+A1))*(T16A) + (T*TD16*((1/R"2)-(1/2*RI*R)))*(T16B) + ((T*TD16)/(Z"2))*(T15+T17)+ (1-((T*TD16)*((2/R*2))+ (2/(Z A2)))))*T16 1840 M=T17 1850 GOSUB 280 1860 TD17=TD 1870 IF N16>ET THEN N16=ET 1880 IF N16>ET THEN T16A=ET 1890 IF N16>ET THEN T16B=ET 1900 IF N16>ET THEN T16=ET 1910 RI=TI-R 1920 N17=(T*TD17*(A2+A1))*(T17A) + (T*TD17*((1/R A2)-(1/2*RI*R)))*(T17B) + ((T*TD17)/(Z A2))*(T16+T18)+ (1-((T*TD17)*((2/R*2))+ (2/(Z"2)))))*T17 1930 M=T18 1940 GOSUB 280 1950 TD18=TD 1960 IF N17>ET THEN N17=ET 1970 IF N17>ET THEN T17A=ET 1980 IF N17>ET THEN T17B=ET APPENDIX A 136 1990 IF N17>ET THEN T17=ET 2000 RI=TI-R 2010 N18=(T*TD18*(A2+A1))*(T18A) + (T*TD18*((1/R A2)-(1/2*RI*R)))*(T18B) + ((T*TD18)/(Z A2))*(T17+T19)+ (1-((T*TD18)*((2/R A2))+ (2/(Z A2)))))*T18 2020 M=T19 2030 GOSUB 280 2040 TD19=TD 2050 IF N18>ET THEN N18=ET 2060 IF N18>ET THEN T18A=ET 2070 IF N18>ET THEN T18B=ET 2080 IF N18>ET THEN T18=ET 2090 RI=TI-R 2100 N19=(T*TD19*(A2+A1))*(T19A) + (T*TD19*((1/R A2)-(1/2*RI*R)))*(T19B) + ((T*TD19)/(Z A2))*(T18+T20)+ (1-((T*TD19)*((2/R A2))+ (2/(Z A2)))))*T19 2110 M=T20 2120 GOSUB 280 2130 TD20=TD 2140 IF N19>ET THEN N19=ET 2150 IF N19>ET THEN T19A=ET 2160 IF N19>ET THEN T19B=ET 2170 IF N19>ET THEN T19=ET 2180 RI=TI-R 2190 N20=(T*TD20*(A2+A1))*(T20A) + (T*TD20*((1/R A2)-(1/2*RI*R)))*(T20B) + ((T*TD20)/(Z~2))*(T19+T21)+ (1-((T*TD20)*((2/R A2))+ (2/(Z~2)))))*T20 2200 M=T21 2210 GOSUB 280 2220 TD21=TD 2230 IF N20>ET THEN N20=ET 2240 IF N20>ET THEN T20A=ET 2250 IF N20>ET THEN T20B=ET 2260 IF N20>ET THEN T20=ET 2270 N21=((T*TD21)/(Z A2))*(T20)+ (1-((T*TD21)*(1/(Z A2))))*T21 2280 IF N21>ET THEN N21=ET 2290 IF N21>ET THEN T21=ET 2300 T1=N1:T2=N2:T3=N3:T4=N4:T5=N5:T6=N6:T7=N7:T8=N8:T9=N9: T10=N10:T11=N11:T12=N12:T13=N13:T14=N14:T15=N15: T16=N16:T17=N17:T18=N18:T19=N19:T2 0=N2 0:T21=N21: 2310 T2A=N2:T2B=N2:T3A=N3:T3B=N3:T4A=N4:T4B=N4:T5A=N5: T5B=N5:T6A=N6:T6B=N6:T7A=N7:T7B=N7:T8A=N8:T8B=N8: T9A=N9:T9B=N9:T10A=N10:T10B=N10:T11A=N11:T11B=N11: T12A=N12:T12B=N12:T13A=N13:T13B=N13:T14A=N14:T14B=N14: T15A=N15:T15B=N15:T16A=N16: 2320 T16B=N16:T17A=N17:T17B=N17:T18A=N18:T18B=N18:19A=N19: T19B=N19:T2 0A=N2 0:T2 0B=N2 0: 233 0 TT=T+TT 2340 RI=R+RI APPENDIX A 137 2350 NEXT Q 2360 B=l 2370 B=B+60 2380 PRINT "TIME = ";TT 2390 PRINT "T21 = ";T21 2400 IF T21=ET GOTO 2450 2410 NEXT 2420 END The d i f f e r e n c e between the f i n i t e d i f f e r e n c e model f o r the bentonite and that f o r the flaked ham was i n the d i f f e r i n g values of thermal d i f f u s i v i t y f o r the corresponding temperatures. The values displayed i n Table 3.3 were the ones used i n the model. Appendix B NOMENCLATURE A Area (m2) a w Water a c t i v i t y B Rate of temperature increase ( C°/sec.) B B Thermal process time as c a l c u l a t e d by B a l l ' s formula method ( B a l l and Olson, 1957) (min.) B i Biot number C p S p e c i f i c heat (constant pressure) (kJ/Kg °K) Cp, Apparent s p e c i f i c heat (kJ/Kg °K) Cv S p e c i f i c heat (constant volume) (kJ/Kg °K) F The t o t a l l e t h a l e f f e c t of heat applied at d i f f e r e n t temperatures expressed as minutes at some reference temperature. 138 APPENDIX B 139 F 0 F at the geometric centre of the container when T x = 121.1°C and z = 10 C° (min.) f The time f o r the l i n e a r s e c t i o n of a heating or cooling curve p l o t t e d on semi-log coordinates to traverse one log cycle (min.) f h Heating rate index, f associated with the heating curve (min.) g Acceleration due to gr a v i t y (cm/sec2) Gr Grashoff number h Surface heat t r a n s f e r c o e f f i c i e n t (W/m2 C°) H Enthalpy (J/g) ^H Enthalpy change ( i . e . Hj - HJ (J/g) h c Convective heat t r a n s f e r c o e f f i c i e n t (W/m2 C°) IT I n i t i a l product temperature (°C) APPENDIX B 140 j h Lag f a c t o r f o r heating = (RT - Tpih) / (RT - T^) k Thermal conductivity (W/cm C°) kf Thermal conductivity of a f l u i d (W/cm C°) M Mass (g) Nu Nusselt number q Heat flow (watts) Pr Prandtl number pt Process time, the time a f t e r venting required to achieve a desired F 0 (minutes). R Radius (m) Re Reynolds number RT Retort processing temperature (°C) T x Reference temperature (°C) NOMENCLATURE 141 t h Thawing point (°C) T F i n a l product temperature (°C) T F I n i t i a l f r e e z i n g point of product (°C) Tj,, I n i t i a l product temperature (°C) TpiK P s e u d o - i n i t i a l product temperature (°C) T s Temperature at material surface (°C) W Water content (% by weight, wet basis) z Number of degrees required f o r the thermal death time curve to traverse one log cycle (C°) rx Thermal d i f f u s i v i t y (m2/s) P C o e f f i c i e n t of expansion f o r a f l u i d being heated (1/°C) P Density (g/cm2) APPENDIX B 142 JUL V i s c o s i t y (dyne sec./cm2) C h a r a c t e r i s t i c length f o r Grashoff equation (cm) REFERENCES Arce, J.A., and Sweat, V.E. 1980. Survey of published heat t r a n s f e r c o e f f i c i e n t s encountered i n food r e f r i g e r a t i o n processes. ASHRAE Trans. 86. ASTM, 1955. Standard methods of t e s t f o r thermal conductivity of materials by means of the guarded hot p l a t e . ASTM Standards 3:1084. Awad, A., Powrie, W.D. and Fennema, O. 1968. Chemical d e t e r i o r a t i o n of frozen bovine muscle at -4°C J . Food S c i . 33:327. B a l l , CO. and Olson, F.C.W. 1957. S t e r i l i z a t i o n i n Food  Technology. McGraw H i l l Book Co. Inc. New York, NY. Bailey, F. 1948. Tropomyosin: a new asymmetric protein component of the muscle f i b r i l . Biochem J . 183:339. Bandman, E. 1987. Chemistry of Animal Tissues — Part I, Proteins. Chapter 3 i n The Science of Meat and Meat Products, 3rd e d i t i o n . J.F. P r i c e and B.S. Schweigert, eds. Food and N u t r i t i o n Press, Inc. Westport, CN, U.S.A. pp. 61-101. Bard, J.C. 1965. Some fa c t o r s i n f l u e n c i n g e x t r a c t a b i l i t y of s a l t - s o l u b l e proteins. In Proceedings of the Meat  Industry Research Conference. American Meat I n s t i t u t e Foundation. Arlington, V i r g i n i a . Bhowmik, S.R. and Hayakawa, K. 1979. A new method f o r determining the apparent thermal d i f f u s i v i t y of thermally conductive food. J . Food S c i . 44:469. Bigelow, W.D., Bohart, G.S., Richardson, A.C. and B a l l , CO. 1920. Heat penetration i n processing canned foods. B u l l e t i n No. 16-L. Res. Lab. Nat. Canners Assoc., Washington, D.C B i l i a d e r i s , C.G. 1983. D i f f e r e n t i a l scanning calorimetry i n food research — a review. Food Chemistry 10:239. Carslaw, H.S. and Jaeger, J . C 1959. Conduction of Heat  i n S o l i d s . Clarendon Press, Oxford, England. Cassens, R.G. 1987. Structure of muscle. Chapter 2 i n The Science of Meat and Meat Products. 3rd e d i t i o n . J.F. P r i c e and B.S. Schweigert (eds.). Food and N u t r i t i o n Press, Inc. Westport, CN, U.S.A. pp. 11-60. 143 REFERENCES 144 Chen, C.S. 1985a. Thermodynamic analysis of the freezing and thawing of foods: enthalpy and apparent s p e c i f i c heat. J . Food S c i . 50:1158. Chen, C.S. 1985b. Thermodynamic analysis of the freezing and thawing of foods: i c e content and m o l l i e r diagram. J . Food S c i . 50:1163. Cleland, A.C. and Earle, R.L. 1979a. A comparison of methods fo r p r e d i c t i n g the freezing times of c y l i n d r i c a l and sp h e r i c a l foodstuffs. J . Food S c i . 44:958. Cleland, A.C. and Earle, R.L. 1979b. P r e d i c t i o n of freezing times f o r foods i n rectangular packages. J . Food S c i . 44:964. Cleland, A.C. and Earle, R.L. 1984a. Assessment of freezing time p r e d i c t i o n methods. J . Food S c i . 49:1034. Cleland, A.C. and Earle, R.L. 1984b. Freezing time pre d i c t i o n s f o r d i f f e r e n t f i n a l product temperatures. J . Food S c i . 49:1230. Darsch, G.A., Shaw, C P . and Tuomy, J.M. 1979. Examination of patient t r a y food service equipment/an evaluation of the Sweetheart food service c a r t . T e c h i c a l Report Natick/TR-79/016. Food Engineering Laboratory, U.S. Army Natick Research and Development Command. Natick, MD, U.S.A. Datta, A.K. and T e i x e i r a , A.A. 1987. Mathematical modeling of natural convection heating i n canned l i q u i d foods. Transactions of the ASAE. 30(5):1542. Deatherage, F.E. and Hamm, R. 1960. Influence of free z i n g and thawing on hydration and changes of the muscle proteins. Food Res. 25:623. De Cindio, B., I o r i o , G., and Romano, V. 1985. Thermal ana l y s i s of the free z i n g of i c e cream b r i c k e t t e s by the f i n i t e element method. J . Food S c i . 50:1463. Dickerson, R.W., J r . 1965. An apparatus f o r the measurement of thermal d i f f u s i v i t y of foods. Food Technol. 19(5):198. REFERENCES 145 Dickerson, R.W., J r . , 1968. Thermal properties of foods. Chapter 2 i n The Freezing Preservation of Foods. 4th e d i t i o n . D.K. T r e s s l e r , W.B. Van Arsdel, M.J. Copley (eds.). AVi Publishing Co., Westport, CN, pp. 26-51. Dickerson, R.W., J r . , and Read, R.B., J r . 1975. Thermal d i f f u s i v i t y of meats. ASHRAE Trans. 81(1):356 Ecklund, O.F. 1956. Correction f a c t o r s f o r heat penetration thermocouples. Food Technol. 10(1):43. Ede, A.J. 1949. The c a l c u l a t i o n of the f r e e z i n g and thawing of foodstuffs. Mod. R e f r i g . 52:52. Elzinga, M., C o l l i n s , J.H., Kuehk, W.M. and Adelstein, R.S. 1973. Complete amino ac i d sequence of a c t i n of ra b b i t s k e l e t a l muscle. Proc. Nat. Acad. S c i . USA 70:2687. Fairbrother, R. 1989. American National Can Company. Personal Communication. Fennema, O.R. and Powrie, W.D. 1964. Low temperature food preservation. Advan. Food Res. 13:219. Fennema, O.R. 1973. S o l i d - L i q u i d E q u i l i b r i a . Chapter 3 i n Low Temperature Preservation of Foods and L i v i n g Matter. O.R. Fennema, W.D. Powrie, and E.H. Marth, eds. Marcel Dekker, Inc. New York, NY. pp. 107-108. Fennema, O.R. 1975. Freezing preservation. Chapter 6 i n P r i n c i p l e s of Food Science: Part I I Ph y s i c a l P r i n c i p l e s  of Food Preservation. M. Karel, O.R. Fennema, D.B. Lund, eds. Marcel Dekker, Inc. New York, NY, pp. 205-210. Freundlich, H. 1937. Some recent work on gels. J . Phys. Chem. 41:901. G i l l e t , T.A., Meiburg, D.E., Brown, C.L., and Simon, S. 1977. Parameters a f f e c t i n g meat p r o t e i n extraction and i n t e r p r e t a t i o n of model system data f o r meat emulsion formation. J . Food S c i . 42:1606. Grim, R.E. and Guven, N. 1978. Bentonites: Geology, Mineralogy. Properties and Uses. E l s e v i e r S c i e n t i f i c P ublishing Co. Amsterdam, Holland. REFERENCES 146 Hamm, R. 1971. Interactions Between Phosphates and Meat Proteins. Chapter 5 i n Symposium: Phosphates i n Food  Processing. J.M. deMan and P. Melnychn, eds. AVi Publishing. Westport, CN. pp. 65-82. Hauser, E.A. and Reed, C.E. 1937. Studies i n thixotropy. I I : the t h i x o t r o p i c behaviour and structure of bentonite. J . Phys. Chem. 41:911. Hayakawa, K. 1972. A new method f o r the computerized determination of the apparent thermal d i f f u s i v i t y of food. In Proc. of 1st P a c i f i c Chem. Engineering Congress, Part I I , p. 129. Soc. Chem. Eng., Japan. Hayakawa, K. 1977. Mathematical methods f o r estimating proper thermal processes and t h e i r computer implementation. Adv. Food Res. 23:75. Hayakawa, K. 1978. A c r i t i c a l review of mathematical procedures f o r determining proper heat s t e r i l i z a t i o n processes. Food Technol. 32(3):59. Hayakawa, K., and Bakal, A. 1973. Formulas f o r p r e d i c t i n g t r a n s i e n t temperatures i n food during f r e e z i n g or thawing. SICHE Symposium Series 69(132):4. Heldman, D.R. and Gorby, D.P. 1975. P r e d i c t i o n of thermal conductivity i n frozen foods. Trans. ASAE. 18:740. Heldman, D.R. and Singh, R.P. 1981. Food Process Engineering. 2nd e d i t i o n . AVi Publishing Co., Westport, Connecticut, U.S.A. pp. 176-197. Holman, J.P. 1976a. Steady-state conduction — two dimensions. Chapter 3 i n Heat Transfer. 4th e d i t i o n . McGraw H i l l , New York, NY. pp 57-79. Holman, J.P. 1976b. Unsteady-state conduction. Chapter 4 i n Heat Transfer. 4th e d i t i o n . McGraw H i l l , New York, NY. pp 95-135. Huxley, H.E. 1953. X-ray analysis and the problem of muscle. Proc. Roy. Soc. B141:59. Huxley, H.E. 1957. The double array of filaments i n cross-s t r i a t e d muscle. J . Biophys. Biochem. Cy t o l . 3:631. REFERENCES 147 Huxley, H.E. and Hanson, J . 1957. Preliminary observations on the structures of insect f l i g h t muscle. Proc. Stockholm Conf. Elec. Microscopy. F.S. Sjostrand and J . Rhodin (eds.). Jackson, J.M. and Olson, F.C.W. 1953. Thermal processing of canned foods i n t i n containers. Chapter 4 i n S t e r i l i z a t i o n of Canned Foods. T h e o r e t i c a l  Considerations i n the S t e r i l i z a t i o n of Canned Foods. American Can Co., Barrington, IL. p. 35. Joshi, C. and Tao, L.C. 1974. A numerical method of simulating the axisymmetrical f r e e z i n g of food systems. J . Food S c i . 39:623. Koga, K. and Yoshizumi, H. 1977. D i f f e r e n t i a l scanning calorimetry (DSC) studies on the structures of water-ethanol mixtures and aged whiskey. J . Food S c i . 42(5):1213. Koga, K. and Yoshizumi, H. 1979. D i f f e r e n t i a l scanning calorimetry (DSC) studies on the f r e e z i n g processes of water-ethanol mixtures and d i s t i l l e d s p i r i t s . J . Food S c i . 44:1386. Krauskopf, K.B. 1967. Introduction to Geochemistry. McGraw-Hill Book Co. New York, NY. Kronman, M.J. and Winterbottom, R.J. 1960. Post-mortem changes i n the water-soluble proteins of bovine s k e l e t a l muscle during aging and freez i n g . J . Agr. Food Chem. 8:671. Kublbeck, K., Merker, G.P., and Straub, J . 1980. Advanced numerical computation of two-dimensional time-dependent fre e convection i n c a v i t i e s . Inter. J . Heat Mass Trans. 23:203. Ladbrooke, B.D. and Chapman, D. 1969. Thermal analysis of l i p i d s , proteins and b i o l o g i c a l membranes: a review and summary of some recent studies. Chem. Phys. L i p i d s 3:304. Ladwig, K.M., Knipe, C.L., and Sebranek, J.G. 1989. E f f e c t s of sodium tripolyphosphate on the p h y s i c a l , chemical and t e x t u r a l properties of high-collagen f r a n k f u r t e r s . J . Food S c i . 54:505. REFERENCES 148 Lenz, M.K. and Lund, D.B. 1977. The l e t h a l i t y — f o u r i e r number method: experimental v e r i f i c a t i o n of a model f o r c a l c u l a t i n g temperature p r o f i l e s and l e t h a l i t y i n conduction-heating canned foods. J . Food S c i . 42:989. Levy, F.L. 1979. Enthalpy and s p e c i f i c heat of meat and f i s h i n the f r e e z i n g range. J . Fd Technol. 14, 549-560. Loncin, M. and Merson, L. 1981. Food Engineerincr. Academic Press, San Francisco, CA. pp. 282-297. Lopez, A. 1987. A Complete Course i n Canning and Related  Processes: B o o k l l l — Processing Procedures f o r Canned  Food Products. 12th e d i t i o n . Canning Trade Inc. Baltimore, MD. pp. 286-287. Lund, D. 1975. Heat Transfer i n Foods. Chapter 2 i n P r i n c i p l e s i n Food Science: Part II P h y s i c a l P r i n c i p l e s  of Food Preservation. O.R. Fennema, ed. Marcel Dekker, Inc. New York, NY. pp. 11-30. Marshall, C.E. 1937. The c o l l o i d a l properties of the clays as r e l a t e d to t h e i r c r y s t a l structure. The Journal of Phys. Chem. 41:935. Martens, T. 1980. Mathematical model of heat processing i n f l a t containers. Ph.D. t h e s i s . Katholeike Univ., Leuven, Belgium. M i l l e r , A.J., Ackerman, S.A., and Palumbo, S.A. 1980. E f f e c t s of frozen storage on f u n c t i o n a l i t y of meat f o r processing. J . Food S c i . 45:1466. M i t t a l , G.S. and B l a i s d e l l , J.L. 1984. Heat and mass t r a n s f e r properties of meat emulsion. Lebensm.-Wiss. u. Technol. 17:94. Mohehsin, N.M. 1980. Thermal Properties of Foods and  A g r i c u l t u r a l Materials. Gordon and Breach Science Publishers, New York, NY. Nagaoka, J . , Takagi, S., and Hotani, S. 1955. Experiments on the f r e e z i n g of f i s h i n an a i r - b l a s t freezer. Proc. 9th Intern. Congr. R e f r i g . (Paris) 2:4. REFERENCES 149 Naveh, D., Pflug, I.J., and Kopelman, I.J. 1984. S t e r i l i z a t i o n of food i n containers with an end f l a t against a r e t o r t bottom: numerical analysis and experimental measurements. J . Food S c i . 49:461. Niekamp, A.A., Unklesbay, N., Unklesbay, K. and E l l e r s i e c k , M. 1983. Bentonite-water dispersions — an e f f e c t i v e d i e t e t i c research t o o l . J . Amer. Dietet. Assoc. 82:516. Niekamp, A., Unklesbay, K., Unklesbay, N., and E l l e r s i e c k , M. 1984. Thermal properties of bentonite-water dispersions used f o r modeling foods. J . Food S c i . 49:28. Ohlsson, T. 1983. The Measurement of Thermal Properties. Chapter 17 i n Physical Properties of Foods. J o u i t t et a l . (eds.) pp 313-325. Olivares, M., Guzman, J.A., and Solar, I. 1986. Thermal d i f f u s i v i t y of nonhomogeneous food. J . Food Process. Pres. 10:57. Parducci, L.G. and Duckworth, R.B. 1972. D i f f e r e n t i a l thermal analysis of frozen food systems I I . Micro-scale studies on egg white,, cod and c e l e r y . J . Food Technol. 7:423. Pearson, A.M. 1987. Muscle Function and Postmortem Changes. Chapter 4 i n The Science of Meat and Meat  Products. 3rd e d i t i o n . J.F. P r i c e and B.S. Schweigert, eds. Food and N u t r i t i o n Press, Inc. Westport, CN, U.S.A. pp. 155-191. Perez, M.G.R. and Calvelo, A. 1984. Modeling the thermal conductivity of cooked meat. J . Food S c i . 49:152. Peterson, W.R. and Adams, J.P. 1983. Water v e l o c i t y e f f e c t on heat penetration parameters during i n s t i t u t i o n a l s i z e r e t o r t pouch processing. J . Food S c i . 48:457. Pflug, I . J . , B l a i s d e l l , J.L., and Kopelman, I . J . " 1965. Developing temperature-time curves f o r objects that can be approximated by a sphere, i n f i n i t e p l a t e or i n f i n i t e c y l i n d e r . ASHRAE Trans. 71:2.38. Pflug, I . J . 1975. Procedures f o r c a r r y i n g out a heat penetration t e s t and analysis of the r e s u l t i n g data. Department of Food Science and N u t r i t i o n , U n i v e r s i t y of Minnesota. Minneapolis, Minnesota. REFERENCES 150 Pham, Q.T. 1987. Ca l c u l a t i o n of thermal process l e t h a l i t y f o r conduction-heated canned foods. J . Food S c i . 52(4):967. Plank, R.Z. 1913. Ges. Kalte-Ind. 20:109. Powrie, W.D. 1973. C h a r a c t e r i s t i c s of Food Myosystems and Their Behaviour During Freeze-Preservation. Chapter 6 i n Low Temperature Preservation of Foods and L i v i n g  Matter. O.R. Fennema, W.D. Powrie, and E.H. Marth, eds. Marcel Dekker, Inc. New York, NY. pp 87-100. Quinn, J.R., Raymond, D.P. and Harwalkar, V.R. 1980. D i f f e r e n t i a l scanning calorimetry of meat proteins as affec t e d by processing treatment. J . Food S c i . 45:1146. Ramaswamy, H.S. and Tung, M.A. 1986. Modelling heat tr a n s f e r i n steam/air processing of t h i n p r o f i l e packages. Can. Inst. Food S c i . Technol. J . 19:215 Ramaswamy, H.S., Tung, M.A. and Stark, R. 1983. A method to measure surface heat t r a n s f e r from steam/air mixtures i n batch r e t o r t s . J . Food S c i . 48(3):900. Richard, P., Durance, T.D., and Sandberg, G.M.M. 1991. A computer simulation of thermal s t e r i l i z a t i o n of canned foods with sub-freezing i n i t i a l temperatures. Can. Inst. Food S c i . Technol. J . 24:95. Riedel, L. 1956. Calorimetric i n v e s t i g a t i o n of the freez i n g of f i s h meat. Kaltetechnik. 8:374. Riedel, L. 1957. Calorimetric i n v e s t i g a t i o n of the meat free z i n g process. Kaltetechnik. 9:38. Riedel, L. 1969. Measurements of thermal d i f f u s i v i t y i n foodstuffs r i c h i n water. Kaltetechnik-Klimatisierung 21(11):315. Rust, R.E. 1987. Sausage Products. Chapter 13 i n The Science of Meat and Meat Products. 3rd e d i t i o n . J.F. P r i c e and B.S. Schweigert, eds. Food and N u t r i t i o n Press, Inc. Westport, CN, U.S.A. pp. 457-486. Sastry, S.K., Beelman, R.B., and Speroni, J . J . 1985. A three-dimensional f i n i t e element model f o r thermally induced changes i n foods: a p p l i c a t i o n to degradation of a g a r i t i n e i n canned mushrooms. J . Food S c i . 50:1293. R E F E R E N C E S 151 Schmidt, G.R. 1987. Functional behavior of meat components i n processing. Chapter 11 i n The Science of  Meat and Meat Products. 3rd e d i t i o n . J.F. P r i c e and B.S. Schweigert, eds. Food and N u t r i t i o n Press, Inc. Westport, Connecticut, U.S.A. pp. 413-429. Schut, J . 1976. "Meat emulsions." Chapter 8 i n Food  Emulsions. S. Friberg, ed. Marcel Dekker Inc. New York, NY. pp. 385-458. Si e g e l , D.G., Theno, D. M., Schmidt, G.R. 1978a. Meat massaging: the e f f e c t s of s a l t , phosphate and massaging on the presence of s p e c i f i c s k e l e t a l muscle proteins i n the exudate of a sectioned and formed ham. J . Food S c i . 43:327. S i e g e l , D.G., Theno, D.M., Schmidt, G.R., and Norton, H.W. 1978b. Meat massaging: the e f f e c t s of s a l t , phosphate and massaging on cooking l o s s , binding strength and exudate composition i n sectioned and formed ham. J . Food S c i . 43:331. Si e g e l , D.G. and Schmidt, G.R. 1979a. Ionic, pH and temperature e f f e c t s on the binding q u a l i t y of myosin. J . Food S c i . 44:1686. Si e g e l , D.G. and Schmidt, G.R. 1979b. Crude myosin f r a c t i o n s as meat binders. J . Food S c i . 44:1129. Singh, R. P. 1982. Thermal d i f f u s i v i t y i n food processing. Food Technol. 36(2):87. Schneider, P.J. 1963. Temperature Response Charts. John Wiley and Sons, Inc. New York. Staph, H.E. and Woolrich, W.R. 1951. S p e c i f i c and l a t e n t heats of foods i n the freezing zone. R e f r i g . Eng. 59:1086. Stumbo, C.R. 1973. Thermobacteriolocrv i n Food Processing. 2nd ed. Academic Press, New York, NY. Succar, J . and Hayakawa, K. 1984. Parametric analysis f o r p r e d i c t i n g f r e e z i n g time of i n f i n i t e l y slab-shaped food. J . Food S c i . 49:468. Sweat, V.E. 1986. Thermal Properties of Foods. Chapter 2 i n Engineering Properties of Foods f ed. M.A. Rao and S.S.H. R i z v i . Marcel Dekker Inc, New York, NY. pp 49-87. REFERENCES 152 Tan, C.-S. and Ling, A.C. 1988. E f f e c t of non-uniform heat t r a n s f e r through can surfaces on process l e t h a i l i t y of conduction heating foods. Can. Inst. Food S c i . Technol. J . 21:378. Tao, L.C. 1967. Generalized numerical solutions of fre e z i n g a saturated l i q u i d i n cy l i n d e r s and spheres. Am Inst. Chem. Eng. J . 13:165. T e i x e i r a , A.A., Dixon, J.R., Zahradnik, J.W. and Zinmeister, G.E. 1969. Computer optimization of nutrient r e t e n t i o n i n the thermal processing of conduction heating foods. Food Technol. 23(6):848. Theno, D.M., Sie g e l , D.G., and Schmidt, G.R. 1978. Meat massaging: e f f e c t s of s a l t and phosphate on the microstructural composition of the muscle exudate. J . Food S c i . 43:483. Trout, G.R. and Schmidt, G.R. 1984. E f f e c t of phosphate type and concentration, s a l t l e v e l and method of preparation on binding i n restructured beef r o l l s . J . Food S c i . 49:687. Tung, M.A., Morello, G.F. and Ramaswamy, H.S. 1986. Food properties, heat t r a n s f e r conditions and s t e r i l z a t i o n considerations i n r e t o r t processes. Unpublished paper. Tung, M.A., B r i t t , J.A. and Ramaswamy, H.S. 1990. Food s t e r i l i z a t i o n i n steam/air r e t o r t s . Food Technol. 44(12):105. Unklesbay, N., Unklesbay, K, and Henderson, J . 1980. Simulation of energy used by foodservice i n f r a r e d heating equipment with bentonite models of menu items. J . Food Prot. 43:789. Unklesbay, N., Unklesbay, K., Buergler, D., and Stringer, W. 1981. Bentonite-water dispersions simulate foodservice energy consumption of sausage p a t t i e s . J . Food S c i . 46:1808. Unklesbay, N. 1982. Overview of foodservice energy research: heat processing. J . Food Prot. 45:984. Uno, J . and Hayakawa, K. 1980. A method f o r estimating thermal d i f f u s i v i t y of heat conduction food i n a c y l i n d r i c a l can. J . Food S c i . 45:692. REFERENCES 153 Vinokur, M. 1983. On one-dimensional s t r e t c h i n g functions f o r f i n i t e d i f f e r e n c e c a l c u l a t i o n s . J . Comp. Phys. 50:215. Vos, B.H. 1955. Measurements of thermal conductivity by a nonsteady-state method. Appl. S c i . Res., Haugue A5:425. Watson, E.L. and Harper, J.C. 1988. Elements of Food Engineering, 2nd e d i t i o n . AVi Publishing Co. Westport, CN. pp. 120 - 140. Wendlant, W.W. 1973. D i f f e r e n t i a l thermal analysis and d i f f e r e n t i a l scanning calorimetry. In Thermal Methods  of Analysis. Wiley-Interscience Ltd. New York, NY p. 193. Wierbicki, E., Kunkle, L.E., and Deatherage, F.E. 1957. Changes i n water-holding capacity and c a t i o n i c s h i f t s during heating and fr e e z i n g and thawing of meat as revealed by a simple c e n t r i f u g a l method f o r measuring shrinking. Food Technol. 11:69 Yamano, Y., E j i r i , K., Endo, T., and Senda, M. 1975. Heat processing i n t o f l e x i b l e packages with food-simulated materials. Jap. J . Food S c i . Technol. 22:199. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0098677/manifest

Comment

Related Items