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Comparison of thermal process evaluation methods for conduction heating foods in cylindrical containers Smith, Trudi 1981

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COMPARISON OF THERMAL PROCESS EVALUATION METHODS < CONDUCTION HEATING FOODS IN CYLINDRICAL CONTAIN by TRUDI SMITH .Sc.(Agr.) Honours, U n i v e r s i t y of B r i t i s h C o l u m b i a , 197 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Food S c i e n c e ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1981 © T r u d i S m i t h , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of f^oocX The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Da DE-6 (2/79) i i ABSTRACT F i v e f o r m u l a methods and t h r e e n u m e r i c a l g e n e r a l methods f o r d e t e r m i n i n g t h e r m a l p r o c e s s l e t h a l i t y were compared w i t h a r e f e r e n c e method t o demonstrate t h e i r a p p l i c a b i l i t y t o c o n d u c t i o n h e a t i n g f oods i n c y l i n d r i c a l c o n t a i n e r s . H y p o t h e t i c a l c e n t e r p o i n t t e m p e r a t u r e h i s t o r y c u r v e s f o r cans w i t h h e i g h t t o d i a m e t e r (H/D) r a t i o s of 0.1 t o 3.0 were g e n e r a t e d f o r a range of p r o c e s s i n g c o n d i t i o n s u s i n g computer s i m u l a t i o n . A f i n i t e - d i f f e r e n c e model based on T e i x e i r a e t a l . (1969b) was used as the b a s i s of the s i m u l a t i o n program. The d e l i v e r e d l e t h a l i t y t o an or g a n i s m c h a r a c t e r i z e d by Z=10C° was e v a l u a t e d u s i n g each of the methods and compared t o the l e t h a l i t y c a l c u l a t e d u s i n g the r e f e r e n c e method. For each of the t e s t methods, s i m u l a t i o n d a t a were p r o v i d e d f o r one minute i n t e r v a l s , but f o r the r e f e r e n c e method, d a t a were p r o v i d e d f o r i n t e r v a l s of 0.05 min. Each of the s e l e c t e d f o r m u l a methods was adapted t o a l l o w c a l c u l a t i o n s t o be done by computer w i t h o u t o p e r a t o r i n t e r v e n t i o n . The most s i g n i f i c a n t p a r t of t h i s a d a p t a t i o n was the development of a system t h a t e n a b l e d the computer t o s e l e c t the l i n e a r p o r t i o n of the heat p e n e t r a t i o n c u r v e t o f a c i l i t a t e c a l c u l a t i o n of the parameters f and j which a r e r e q u i r e d by a l l of t h e f o r m u l a methods. A method f o r h a n d l i n g l a r g e t a b l e s was a l s o d e v e l o p e d f o r use w i t h some of t h e f o r m u l a methods. For the g e n e r a l methods, the d e v i a t i o n s from the r e f e r e n c e method were g r e a t e s t when t h e h e a t i n g r a t e index ( f h ) and H/D were s m a l l and the u n a c c o m p l i s h e d temperature d i f f e r e n c e (g) was l a r g e . Whereas the t h e v a l u e of f h was the most s i g n i f i c a n t f a c t o r a f f e c t i n g the a c c u r a c y of c a l c u l a t i o n s done u s i n g the g e n e r a l method, i t d i d not g r e a t l y a f f e c t t h e performance of t h e f o r m u l a methods. The f a c t o r s t h a t most s i g n i f i c a n t l y i n f l u e n c e d d e v i a t i o n s between the f o r m u l a methods and the r e f e r e n c e method were H/D and g. The l a r g e s t d e v i a t i o n s i n a l l ca s e s o c c u r r e d when g was l a r g e and H/D was c l o s e t o u n i t y . These e r r o r s were m o s t l y on t h e " s a f e " s i d e , but the energy use i m p l i c a t i o n s c o u l d be s i g n i f i c a n t , e s p e c i a l l y f o r h i g h r e t o r t t e mperature p r o c e s s e s . i v TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i INTRODUCTION 1 LITERATURE REVIEW 3 A. F i n i t e - d i f f e r e n c e S i m u l a t i o n 3 B. G e n e r a l Methods 6 C. Formula Methods 7 1 . Background 7 2. Computer A p p l i c a t i o n s 10 3. U n c e r t a i n F a c t o r s i n Thermal P r o c e s s C a l c u l a t i o n s 10 4. P r e v i o u s Comparisons and Reviews 13 EXPERIMENTAL 15 A. G e n e r a t i o n of Heat P e n e t r a t i o n Data 15 1. I n t r o d u c t i o n 15 2. Theory 16 3. Time and Space Increment Study 17 4. Heat T r a n s f e r C o n s i d e r a t i o n s 22 5. P r o c e s s i n g C o n d i t i o n s Used 23 6. S i m u l a t i o n Program 24 B. P r o c e s s C a l c u l a t i o n Methods 27 1 . G e n e r a l Methods 27 a. Average t e m p e r a t u r e method 27 b. P a t a s h n i k ' s method 27 V c. C u b i c p o l y n o m i a l method 28 2. Formula Methods 29 a. B a l l ' s t a b l e method 29 b. B a l l ' s e q u a t i o n method 30 c. Stumbo's method 30 d. S t e e l e and Board's method 30 e. Hayakawa's method 31 C. A d a p t a t i o n f o r Computer S o l u t i o n 32 1 . T a b l e Access 32 2. D e t e r m i n a t i o n of f and j V a l u e s 32 3. R e f e r e n c e Method and C a l c u l a t i o n of D e v i a t i o n s 34 RESULTS AND DISCUSSION 35 A. G e n e r a l Methods 35 B. Formula Methods 39 1. I n i t i a l S t u d i e s 39 2. E f f e c t s of Can Shape and g V a l u e 43 3. Comparison of Methods 52 4. C a l c u l a t i o n E r r o r s i n Terms of P r o c e s s i n g Time 54 5. P o s i t i o n of C o l d Spot i n C o n t a i n e r 56 6. C o n v e c t i o n H e a t i n g P r o d u c t s 57 C. G e n e r a l C o n s i d e r a t i o n s and F u t u r e R e s e a r c h Needs 57 CONCLUSIONS 60 REFERENCES 61 L I S T OF TABLES T a b l e I E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a m e t h o d s f o r v a r i o u s c a n s i z e s a n d v a l u e s o f g T a b l e II E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a methods f o r v a r i o u s t h e r m a l d i f f u s i v i t i e s and f n  T a b l e I I I E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a methods f o r v a r i o u s v a l u e s o f g a n d two r e t o r t t e m p e r a t u r e s v i i ' LIST OF FIGURES F i g u r e 1. Space increment s t u d y 20 F i g u r e 2. Time inc r e m e n t s t u d y 21 F i g u r e 3. Flow c h a r t f o r s i m u l a t i o n program 26 F i g u r e 4. E f f e c t of h e a t i n g r a t e on e v a l u a t i o n e r r o r s u s i n g average t e m p e r a t u r e method 36 F i g u r e 5. E f f e c t of h e i g h t t o d i a m e t e r r a t i o on e v a l u a t i o n e r r o r s u s i n g average te m p e r a t u r e method 37 F i g u r e 6. E f f e c t of g on e v a l u a t i o n e r r o r s u s i n g average temperature method 38 F i g u r e 7. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g B a l l ' s t a b l e s 45 F i g u r e 8. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g B a l l ' s e q u a t i o n 46 F i g u r e 9. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g S t e e l e and Board's method 47 F i g u r e 10. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g Hayakawa's method 48 F i g u r e 11. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g Stumbo's method 49 F i g u r e 12. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s as r e l a t e d t o the c o o l i n g l a g f a c t o r 51 F i g u r e 13. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g f i v e f o r m u l a methods 53 F i g u r e 14. C a l c u l a t e d l e t h a l e f f e c t r e l a t i v e t o p r o c e s s i n g time 55 v i i i ACKNOWLEDGEMENTS The a u t h o r wishes t o e x p r e s s her s i n c e r e a p p r e c i a t i o n t o Dr. M a r v i n A. Tung f o r h i s i n t e r e s t , a d v i c e and encouragement throughout t h e c o u r s e of t h i s r e s e a r c h p r o j e c t . She i s t h a n k f u l t o the members of her committee: D r s . W.D. P o w r i e and J . V a n d e r s t o e p of the Department of Food S c i e n c e , and Dr. K.V. Lo, Department of B i o - r e s o u r c e E n g i n e e r i n g , f o r t h e i r i n t e r e s t i n and r e v i e w of t h i s t h e s i s . She i s a l s o e x t r e m e l y g r a t e f u l t o her husband, R i c h a r d S m i t h , f o r h i s encouragement, u n d e r s t a n d i n g and s u p p o r t , w i t h o u t which t h i s s tudy would not have been p o s s i b l e . 1 INTRODUCTION The p r e s e r v a t i o n of food t o ensure an adequate food s u p p l y t h r o u g h o u t the year has always been i m p o r t a n t . The e a r l i e s t p r e s e r v a t i o n methods were d r y i n g , s a l t i n g and c u r i n g , used m o s t l y t o p r e s e r v e meats and f i s h . F e r m e n t a t i o n p r o c e s s e s have a l s o been employed f o r c e n t u r i e s t o p r e s e r v e d a i r y p r o d u c t s , as yoghurt and cheese, and some v e g e t a b l e s , as p i c k l e d p r o d u c t s . The a r t of f o o d p r e s e r v a t i o n by means of heat s t e r i l i z a t i o n was not de v e l o p e d u n t i l the l a t e 18th c e n t u r y , and can thus be c o n s i d e r e d a r e l a t i v e l y new t e c h n i q u e . At t h a t t i m e , p r e s e r v a t i o n by c a n n i n g t r u l y was an a r t , s i n c e l i t t l e was u n d e r s t o o d about the p r i n c i p l e s which governed the s u c c e s s or f a i l u r e of p r o d u c t i o n of t h e r m a l l y s t a b i l i z e d f o o d s t u f f s t h a t were s a f e t o eat a f t e r c o n s i d e r a b l e p e r i o d s of s t o r a g e . S i n c e t h a t t i m e , p r e s e r v a t i o n by t h e r m a l p r o c e s s i n g has d e v e l o p e d t o a s t a g e of c o n s i d e r a b l e s o p h i s t i c a t i o n , and i s no l o n g e r an a r t , but a s c i e n c e . The m i c r o b i o l o g i c a l , k i n e t i c and e n g i n e e r i n g p r i n c i p l e s t h a t govern the e f f e c t i v e n e s s of the p r o c e s s a r e w e l l u n d e r s t o o d and r e s e a r c h i n the f i e l d i s now d i r e c t e d towards p r o c e s s i n g r e f i n e m e n t s i n terms of p r o d u c t q u a l i t y and p r o d u c t i o n e f f i c i e n c y . One of t h e i m p o r t a n t f a c e t s of t h i s r e s e a r c h i s the e v a l u a t i o n of t h e r m a l s t e r i l i z a t i o n p r o c e s s e s i n terms of t h e i r l e t h a l e f f e c t on t h e b a c t e r i a l s p o r e s which must be i n a c t i v a t e d t o render the p r o c e s s e d 2 food s h e l f s t a b l e and s a f e t o e a t . S i n c e the p u b l i c a t i o n of t h e f i r s t g e n e r a l method f o r t h e r m a l p r o c e s s e v a l u a t i o n by B i g e l o w and coworkers (1920), much has been w r i t t e n about the d e t e r m i n a t i o n of s a f e p r o c e s s e s f o r t h e r m a l l y s t e r i l i z e d f o o d s . S e v e r a l r e v i e w s and comparisons of p r o c e s s e v a l u a t i o n methods have appeared (Hayakawa, 1977,1978; Merson et a l . , 1978; Stumbo and L o n g l e y , 1966), a l t h o u g h none o f f e r s e x t e n s i v e n u m e r i c a l e v a l u a t i o n s . In the l a s t decade, more a t t e n t i o n has been g i v e n t o the e f f e c t s of p r o c e s s i n g on n u t r i t i o n a l and s e n s o r y q u a l i t y . The d e s i r e t o improve q u a l i t y and the need t o c o n t r o l p r o c e s s i n g c o s t s have r e s u l t e d i n a move t o reduce unnecessary o v e r p r o c e s s i n g . P a c k a g i n g changes, i n c o n t a i n e r t y p e s , s i z e s and shapes, and p r o c e s s c o n d i t i o n changes, such as e l e v a t e d r e t o r t t e m p e r a t u r e s and h i g h e r f i l l i n g t e m p e r a t u r e s , a r e becoming i m p o r t a n t . The i n t r o d u c t i o n of computer t e c h n o l o g y f o r r e t o r t c o n t r o l i s making p o s s i b l e more p r e c i s e c o n t r o l of p r o c e s s i n g c o n d i t i o n s . Because of t h e s e developments, t h e r e i s an i n c r e a s i n g i n t e r e s t i n the a c c u r a c y and r e l i a b i l i t y of the c u r r e n t l y a v a i l a b l e t h e r m a l p r o c e s s e v a l u a t i o n methods. Thus, the o b j e c t i v e of t h i s s t u d y was t o p r e p a r e an e v a l u a t i o n of the s e l e c t e d e v a l u a t i o n methods u s i n g a wide range of c o n d i t i o n s t h a t would i n c l u d e t r a d i t i o n a l and n o v e l t h e r m a l p r o c e s s i n g a p p l i c a t i o n s . 3 LITERATURE REVIEW A. F i n i t e - d i f f e r e n c e S i m u l a t i o n The f i n i t e - d i f f e r e n c e s i m u l a t i o n model i n t r o d u c e d by T e i x e i r a e t a l . (1969b) has been adapted and used by s e v e r a l r e s e a r c h e r s t o study v a r i o u s a s p e c t s of t h e r m a l p r o c e s s i n g of f o o d s . The o r i g i n a l paper o u t l i n e d the t h e o r y b e h i n d t h e n u m e r i c a l m o d e l l i n g t e c h n i q u e and i n d i c a t e d t h a t t e m p e r a t u r e h i s t o r i e s o b t a i n e d u s i n g the s i m u l a t i o n program agreed w e l l w i t h c o r r e s p o n d i n g t e m p e r a t u r e s o b t a i n e d from H e i s l e r c h a r t s . The s i m u l a t i o n program was used as a means of c a l c u l a t i n g i n t e g r a t e d l e t h a l i t y and the r e s u l t s compared f a v o u r a b l y w i t h o t h e r methods a v a i l a b l e f o r thes e c a l c u l a t i o n s . The s i m u l a t i o n program was then used i n a stud y of the e f f e c t s o f v a r i o u s p r o c e s s i n g t e m p e r a t u r e s on n u t r i e n t r e t e n t i o n . O p t i m a l p r o c e s s i n g t e m p e r a t u r e s were shown t o be a f u n c t i o n of the d e g r a d a t i o n k i n e t i c s of the n u t r i e n t i n q u e s t i o n , but were g e n e r a l l y found t o be i n the range of c o n v e n t i o n a l p r o c e s s i n g t e m p e r a t u r e s . T h i s same program was used t o determine spore s u r v i v a l d i s t r i b u t i o n s i n a v a r i e t y of can s i z e s a f t e r t h e r m a l p r o c e s s i n g . The r e s u l t s i n d i c a t e d t h a t the l o c a t i o n w i t h i n a v c o n t a i n e r where s p o i l a g e would most l i k e l y f i r s t occur was not the c e n t e r of the c a n , but some o t h e r l o c a t i o n d e t e r m i n e d by the can geometry and p r o c e s s i n g c o n d i t i o n s ( T e i x e i r a e t a l . , 1969a). However, f o r most can shapes, the g r e a t e s t spore c o n c e n t r a t i o n per u n i t volume would be near 4 the c e n t e r p o i n t ; t h e e x c e p t i o n b e i n g a s h o r t can, l i k e a tuna c a n , f o r which the g r e a t e s t spore c o n c e n t r a t i o n may not f a l l on the c e n t e r p l a n e . These e f f e c t s a r e a r e s u l t of the t e m p e r a t u r e g r a d i e n t s t h a t e x i s t w i t h i n a can d u r i n g t h e h e a t i n g and c o o l i n g phases of a t h e r m a l p r o c e s s . T e i x e i r a e t a l . (1975) used the s i m u l a t i o n program t o i n v e s t i g a t e the f e a s i b i l t y of v a r y i n g the r e t o r t t e m p e r a t u r e d u r i n g the c o u r s e of a t h e r m a l p r o c e s s as a means t o improve t h i a m i n e r e t e n t i o n i n p r o c e s s e d f o o d s . They r e p o r t e d t h a t l i t t l e c o u l d be g a i n e d from t h i s a p p r o a c h , but t h a t c o n t a i n e r geometry was the l i m i t i n g f a c t o r , w i t h cans h a v i n g h e i g h t t o d i a m e t e r r a t i o s c l o s e t o u n i t y r e s u l t i n g i n the p o o r e s t t h i a m i n e r e t e n t i o n . The b e s t r e t o r t c o n t r o l programs f o r t h i a m i n e r e t e n t i o n i n pea puree had i n c r e a s i n g t e m p e r a t u r e p r o f i l e s d u r i n g the e a r l y p a r t of t h e p r o c e s s , w i t h a t e m p e r a t u r e peak a p p r o x i m a t e l y midway th r o u g h the p r o c e s s . From t h i s p o i n t , r e t o r t t e m p e r a t u r e was a l l o w e d t o d r o p s l o w l y u n t i l f i n a l c o o l i n g was t o b e g i n . I n t h i s way, heat t r a n s f e r i n t o the package was t r a d e d o f f a g a i n s t o v e r p r o c e s s i n g of the o u t e r l a y e r , t o o p t i m i z e the p r o c e s s i n terms of t h i a m i n e r e t e n t i o n . The s i m u l a t i o n concept was extended t o m o d e l l i n g of t h e r m a l p r o c e s s e s f o r r e c t a n g u l a r c o n t a i n e r s (Manson e t a l . , 1970) and t o pear shaped c o n t a i n e r s (Manson et a l . , 1974). In both c a s e s , the s i m u l a t i o n model was used t o e v a l u a t e t h e r m a l p r o c e s s e s i n terms of spo r e s u r v i v a l and n u t r i e n t r e t e n t i o n . Whereas b o t h of t h e s e models c o n s i d e r e d 5 t h r e e d i m e n s i o n a l heat t r a n s f e r , O h l s s o n (1980b) s i m p l i f i e d the s i m u l a t i o n model f o r r e c t a n g u l a r c o n t a i n e r s t o c o n s i d e r o n l y one d i m e n s i o n a l heat f l o w t h r o u g h t h e s m a l l e s t d i m e n s i o n . T h i s s i m p l i f i c a t i o n i s a c c e p t a b l e f o r v e r y t h i n packages, which behave i n a manner s i m i l a r t o an i n f i n i t e s l a b , but cannot be used i f one d i m e n s i o n i s not s i g n i f i c a n t l y l e s s than the o t h e r two. O h l s s o n r e p o r t e d t h a t the t e m p e r a t u r e s p r e d i c t e d by the model agreed w e l l w i t h heat p e n e t r a t i o n t e s t s conducted w i t h f l a t c a n s , r e t o r t pouches and s e m i - r i g i d t r a y s . The s i m u l a t i o n approach was used t o e v a l u a t e v a r i o u s r e t o r t p r o c e s s e s i n terms of t h e i r e f f e c t on q u a l i t y f a c t o r d e g r a d a t i o n , based on k i n e t i c f a c t o r s o u t l i n e d by O h l s s o n (1980a). The study showed t h a t package t h i c k n e s s was the s i n g l e most i m p o r t a n t f a c t o r i n o p t i m i z a t i o n s i n c e , a t h i g h p r o c e s s i n g t e m p e r a t u r e s , o v e r c o o k i n g a t the s u r f a c e became a problem i f the package was not v e r y t h i n . W i t h packages up t o 15mm i n t h i c k n e s s , p r o c e s s i n g t e m p e r a t u r e s of up t o 140°C d i d not r e s u l t i n s i g n i f i c a n t s u r f a c e damage. However, as package t h i c k n e s s was i n c r e a s e d , o p t i m a l p r o c e s s i n g c o n d i t i o n s tended towards c o n v e n t i o n a l p r o c e s s i n g t e m p e r a t u r e s of 120 t o 125°C. The f i n i t e - d i f f e r e n c e model has a l s o been adopted f o r t h e development of t a b l e s f o r p r o c e s s e v a l u a t i o n and e s t i m a t i o n . Jen e t a l . (1971) f i r s t used t h i s method t o d e v e l o p t a b l e s f o r c a l c u l a t i o n of mass average l e t h a l i t y and n u t r i e n t d e s t r u c t i o n . These t a b l e s were then r e f i n e d by 6 P u r o h i t and Stumbo (Stumbo, 1973) and p r e s e n t e d f o r c a l c u l a t i n g s i n g l e p o i n t l e t h a l i t y a c c o r d i n g t o Stumbo's f o r m u l a method. B. G e n e r a l Methods The g e n e r a l method has l o n g been c o n s i d e r e d the s t a n d a r d method f o r t h e r m a l p r o c e s s l e t h a l i t y d e t e r m i n a t i o n . T h i s method combines i n f o r m a t i o n about t h e r m a l d e s t r u c t i o n of b a c t e r i a l s p o r e s w i t h the measured c e n t e r t e m p e r a t u r e h i s t o r y of a c o n t a i n e r d u r i n g the h e a t i n g and c o o l i n g c y c l e s of a p r o c e s s . S i n c e i t s i n t r o d u c t i o n ( B i g e l o w e t a l . , 1920) many improvements and s i m p l i f i c a t i o n s have been i n t r o d u c e d , i n c l u d i n g the concept of an h y p o t h e t i c a l t h e r m a l d e a t h time c u r v e ( B a l l , 1928) and the use of l e t h a l r a t e paper ( S c h u l t z and O l s o n , 1940). The g e n e r a l method l e t h a l i t y d e t e r m i n a t i o n i s based on the t e m p e r a t u r e h i s t o r y of the c o l d e s t p o i n t w i t h i n a c o n t a i n e r and i t s performance s h o u l d not be a f f e c t e d by a t t r i b u t e s of the heat p e n e t r a t i o n c u r v e such as s l o p e and c u r v a t u r e . T h i s i s t r u e i f a c o n t i n u o u s r e c o r d of the t h e r m a l h i s t o r y i s used, b u t , i n most p r a c t i c a l s i t u a t i o n s , t e m p e r a t u r e d a t a a r e o b t a i n e d a t f i n i t e time i n t e r v a l s . Even so, the g r a p h i c a l forms of t h e g e n e r a l method e s t i m a t e the c o n t i n u o u s temperature f u n c t i o n and can be e x p e c t e d t o g i v e v e r y a c c u r a t e l e t h a l i t y v a l u e s . However, g r a p h i c a l methods a r e v e r y time consuming and t h e i r p r e c i s i o n i s degraded from the optimum by p h y s i c a l l i m i t a t i o n s on p l o t t i n g the d a t a and measuring a r e a s , as w e l l as the 7 i n e v i t a b i l i t y of human e r r o r . These f a c t o r s c o n t r i b u t e t o the p o p u l a r i t y of n u m e r i c a l g e n e r a l methods which a r e not s u b j e c t t o " o p e r a t o r " d i f f e r e n c e s . P a t a s h n i k (1953) i n t r o d u c e d the concept of c a l c u l a t i n g p r o c e s s l e t h a l i t y u s i n g t r a p e z o i d a l i n t e g r a t i o n of the l e t h a l i t y p l o t as l e t h a l r a t e (L) v s . t i m e . T h i s method i s much f a s t e r than the o r i g i n a l o r improved forms of the g e n e r a l method s i n c e g r a p h i n g of t h e d a t a i s not r e q u i r e d . Some degree of a c c u r a c y i s l o s t u s i n g the t r a p e z o i d a l i n t e g r a t i o n , depending on t h e time i n t e r v a l between te m p e r a t u r e measurements ( P a t a s h n i k , 1953). P a t a s h n i k 1 s s t u d y suggested 4 t o 5 min i n t e r v a l s t o be adequate, r e s u l t i n g i n e r r o r s t h a t were w i t h i n the i n h e r e n t v a r i a b i l i t y from can t o ca n . T h i s was a v e r y a p p r o p r i a t e c o n c l u s i o n i n 1953, but a t p r e s e n t , t e c h n o l o g i c a l advances make p o s s i b l e more a c c u r a t e a n a l y s i s , w i t h l i t t l e a d d i t i o n a l e f f o r t . C u r r e n t l y , time i n t e r v a l s of one minute a r e commonly used, due t o improvements i n d a t a l o g g i n g equipment. C. Formula Methods 1 . Background A l t h o u g h g e n e r a l methods f o r p r o c e s s e v a l u a t i o n a r e s u f f i c i e n t f o r d e t e r m i n i n g the l e t h a l e f f e c t of a g i v e n p r o c e s s , they a r e not v e r y u s e f u l f o r d e t e r m i n i n g the p r o c e s s i n g time r e q u i r e d t o d e l i v e r a g i v e n t a r g e t 8 l e t h a l i t y . I f a g e n e r a l method were t o be used f o r t h i s p u r p o s e , e x p e r i m e n t a l runs must be completed over a range of c o n d i t i o n s and e v a l u a t e d i n terms of d e l i v e r e d l e t h a l i t y . A r e q u i r e d p r o c e s s time c o u l d be d e r i v e d from t h e s e r e s u l t s u s i n g i n t e r p o l a t i o n . A l t e r n a t i v e l y , a g r a p h i c a l " t r i a l and e r r o r " t e c h n i q u e c o u l d be a p p l i e d i n which the e x p e r i m e n t a l heat p e n e t r a t i o n c u r v e i s extended or s h o r t e n e d t o g i v e an a p p r o x i m a t i o n of the r e q u i r e d p r o c e s s time f o r a s e l e c t e d t a r g e t l e t h a l i t y . T h i s approach f o r p r o c e s s e v a l u a t i o n and d e t e r m i n a t i o n i s t e d i o u s and time consuming. The f i r s t f o r m u l a method was i n t r o d u c e d by B a l l (1923) " i n o r d e r t o reduce the t i m e n e c e s s a r y t o o b t a i n r e s u l t s , as w e l l as t o o b t a i n a b a s i s f o r c o o r d i n a t i n g the v a r i o u s f a c t o r s which e n t e r i n t o t h e c a l c u l a t i o n " . These f a c t o r s i n c l u d e h e a t i n g b e h a v i o r , i n i t i a l t e m p e r a t u r e , f i n a l t e m p e r a t u r e and r e t o r t t e m p e r a t u r e . The b a s i s of the method was the almost l i n e a r r e l a t i o n s h i p t h a t was found t o e x i s t between p r o d u c t t e m p e r a t u r e and time when p l o t t e d on s e m i l o g a r i t h m i c c o o r d i n a t e s as the te m p e r a t u r e d i f f e r e n c e between the p r o d u c t and i t s environment ( B i g e l o w e t a l . , 1920). The e q u a t i o n f o r t h i s l i n e was combined w i t h the r e l a t i o n s h i p d e s c r i b i n g f i r s t o r d e r t h e r m a l d e a t h k i n e t i c s f o r b a c t e r i a l s p o r e s , and i n t e g r a t e d over t h e d u r a t i o n of the l i n e a r p o r t i o n s of the p r o c e s s c u r v e . A f t e r s t u d y of many heat p e n e t r a t i o n c u r v e s , B a l l d e v e l o p e d an e m p i r i c a l f o r m u l a t o d e s c r i b e the n o n - l i n e a r p o r t i o n of the c o o l i n g 9 c u r v e , which was found t o be w e l l e s t i m a t e d by an h y p e r b o l i c r e l a t i o n s h i p . F o r f u r t h e r d e t a i l s on t h i s method, the re v i e w of i t s t h e o r e t i c a l b a s i s by Merson e t a l . (1978) i s recommended. A l l of the s e d e t a i l s , however, a r e t r a n s p a r e n t t o anyone e m p l o y i n g B a l l ' s f o r m u l a method, s i n c e t h e r e s u l t s of the i n t e g r a t i o n s a r e p r o v i d e d i n t a b u l a r form. However, some of the u n d e r l y i n g a s s u m p t i o n s t h a t were i m p o r t a n t i n the development of the method a r e o f t e n not v a l i d , and the importance of meeting t h e s e assumptions i s not w e l l u n d e r s t o o d . One of the most s i g n i f i c a n t , and most o f t e n i n v a l i d , a s s u mptions i s t h a t the c h a r a c t e r of the c o o l i n g c u r v e does not v a r y . Board e t a l . (1960) d i s c u s s e d f a c t o r s , such as r e t o r t p r e s s u r e , which a f f e c t c o o l i n g b e h a v i o r of c o n d u c t i o n h e a t i n g f o o d s , and noted t h a t B a l l ' s method f o r p r o c e s s l e t h a l i t y c a l c u l a t i o n u s u a l l y u n d e r e s t i m a t e d p r o c e s s l e t h a l i t y , but t o a v a r i a b l e e x t e n t , depending on s e v e r a l f a c t o r s i n c l u d i n g the n a t u r e of the p r o d u c t and the p r o c e s s i n g c o n d i t i o n s . S i n c e the i n t r o d u c t i o n of B a l l ' s f o r m u l a method t h e r e have been many m o d i f i c a t i o n s s u g g ested ( B a l l and O l s o n , 1957; F l a m b e r t and D e l t o u r , 1972b; G i l l e s p y , 1951; G r i f f i n e t a l . , 1969,1971; Hayakawa, 1970; Herndon e t a l . , 1968; H i c k s , 1958; P f l u g , 1968; S t e e l e and Board, 1979; Stumbo and L o n g l e y , 1966; Stumbo, 1973). These m o d i f i c a t i o n s of the f o r m u l a method d e a l i n v a r i o u s ways w i t h t h e c o o l i n g 10 p o r t i o n of a t h e r m a l p r o c e s s , u s i n g t h e o r e t i c a l and e m p i r i c a l r e l a t i o n s h i p s between the c u r v e d and s t r a i g h t -l i n e p o r t i o n s of t h e t e m p e r a t u r e h i s t o r y c u r v e t o t r y t o more a c c u r a t e l y e s t i m a t e the l e t h a l e f f e c t of t h e c o o l i n g p o r t i o n of t h e r m a l p r o c e s s e s . 2. Computer A p p l i c a t i o n s One of t h e aims of the development of f o r m u l a methods was t o s i m p l i f y p r o c e s s c a l c u l a t i o n s . In r e c e n t y e a r s computer t e c h n o l o g y has become more a c c e s s i b l e and i n t e r e s t i n the use of the computer f o r p r o c e s s e v a l u a t i o n i s h i g h . Tung and G a r l a n d (1978) suggested a s e r i e s of computer o r i e n t e d p r o c e d u r e s t o f a c i l i t a t e r a p i d h a n d l i n g of l a r g e volumes of heat p e n e t r a t i o n d a t a and s t a t i s t i c a l t r e a t m e n t of the r e s u l t s of the c a l c u l a t i o n s . A number of o t h e r approaches f o r computer m a n i p u l a t i o n of t h e r m a l p r o c e s s c a l c u l a t i o n s have appeared, u s i n g e i t h e r a d a p t a t i o n s of f o r m u l a methods (Hayakawa, l977;Manson and Z a h r a d n i k , 1967; Timbers and Hayakawa, 1967) or f i n i t e - d i f f e r e n c e s i m u l a t i o n t e c h n i q u e s (Manson e t a l . , 1970; Manson e t a l . , 1974; T e i x e i r a e t a l . , 1969a, 1969b). 3. U n c e r t a i n F a c t o r s i n Thermal P r o c e s s C a l c u l a t i o n s S i n c e t h i s work d e a l s w i t h the " a c c u r a c y " of t h e r m a l p r o c e s s c a l c u l a t i o n methods, i t i s of some v a l u e t o put i n t o p e r s p e c t i v e v a r i o u s f a c t o r s which t e n d t o i n t r o d u c e u n c e r t a i n t y i n t o t h e r m a l p r o c e s s d e t e r m i n a t i o n s . T h i s i s a s u b j e c t t h a t has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n over the y e a r s , s i n c e i t i s o n l y on the b a s i s of a sound 11 u n d e r s t a n d i n g of t h e r o l e of these f a c t o r s t h a t s a f e t h e r m a l p r o c e s s e s can be d e v e l o p e d . U n c e r t a i n t i e s a r e i n t r o d u c e d i n t o t h e r m a l p r o c e s s d e t e r m i n a t i o n i n a number of d i f f e r e n t ways. F i r s t l y , t h e r e i s v a r i a b i l i t y from c o n t a i n e r t o c o n t a i n e r due t o the b i o l o g i c a l n a t u r e of the m a t e r i a l s which a r e b e i n g d e a l t w i t h (Herndon, 1971). T h i s i s one of t h e most i m p o r t a n t f a c t o r s s i n c e the v a r i a b i l i t y cannot be s i g n i f i c a n t l y r educed by improvement i n m a n u f a c t u r i n g p r a c t i c e s . V a r i a b i l i t y caused by poor f o r m u l a t i o n o r f i l l c o n t r o l can be g r e a t l y r e d u c e d , but t h e non-homogeneity of most t h e r m a l l y p r o c e s s e d p r o d u c t s w i l l s t i l l be an i m p o r t a n t f a c t o r (Evans and Board, 1954). S e c o n d l y , p r o c e s s c o n t r o l problems can a f f e c t the magnitude of s a f e t y f a c t o r s r e q u i r e d . Where c o n t r o l of p r o c e s s i n g t e m p e r a t u r e o r p r e s s u r e , c o o l i n g water t e m p e r a t u r e or p r o c e s s time i s not good, g r e a t e r s a f e t y f a c t o r s w i l l be r e q u i r e d . T h i r d l y , t h e a p p l i c a b i l i t y of b a c t e r i o l o g i c a l knowledge i s somewhat u n c e r t a i n , s i n c e v a r i o u s f a c t o r s such as the t h e r m a l h i s t o r y and s u s p e n s i o n media of b a c t e r i a l s p o r e s a r e known t o a f f e c t the t h e r m a l r e s i s t a n c e of the organisms i n q u e s t i o n ( H i c k s , 1961). H i c k s (1952) p r o v i d e d a b r i e f t a b l e of approximate e f f e c t s of some of t h e s e v a r i a b l e s on the d e l i v e r e d l e t h a l i t y of a t h e r m a l p r o c e s s . In terms of spore s u r v i v a l , minor v a r i a t i o n i n p r o c e s s c o n t r o l was shown t o be of l e s s 12 s i g n i f i c a n c e than e r r o r s i n the e s t i m a t e s of t h e r m a l death p a r a m e t e r s . Ross e t a l . (1979) a l s o p r e s e n t e d some s t u d i e s on the e f f e c t s of v a r i a b i l i t y i n measured q u a n t i t i e s . Lund (1978) p r e s e n t e d a s t u d y d e a l i n g w i t h the e f f e c t s of v a r i o u s f a c t o r s on c a l c u l a t e d p r o c e s s l e t h a l i t y u s i n g a Monte C a r l o p r o c e d u r e ; however, no c o n c l u s i o n was made as t o the a p p r o p r i a t e use of t h i s i n f o r m a t i o n i n d e t e r m i n i n g s a f e t y f a c t o r s . Tung and G a r l a n d (1978) suggest t h a t a " s t a t i s t i c a l approach t o d e a l w i t h t h i s v a r i a b l i t y i s more a p p r o p r i a t e than the c o n v e n t i o n a l approach of c o n s i d e r i n g o n l y t h e s l o w e s t - h e a t i n g c o n t a i n e r i n a heat p e n e t r a t i o n t e s t . T h i s approach r e s u l t s i n h e a t i n g regimes h a v i n g a g r e a t e r degree of c o n f i d e n c e of s a f e t y than those c a l c u l a t e d u s i n g the c o n v e n t i o n a l a p p r o a c h , w i t h o u t r e q u i r i n g more numerous heat p e n e t r a t i o n t e s t s . R e g a r d l e s s of how one chooses t o d e a l w i t h v a r i a b i l t y i n p h y s i c a l f a c t o r s which a f f e c t the c a l c u l a t i o n of p r o c e s s l e t h a l i t y , i t i s not r e a s o n a b l e t o i n t r o d u c e e r r o r u n n e c e s s a r i l y t h r o u g h i n a p p r o p r i a t e c a l c u l a t i o n methods. Hence, i t i s i m p o r t a n t t o d e t e r m i n e the a c c u r a c y and r e l i a b i l i t y of t h e s e methods i n o r d e r t o have c o n f i d e n c e t h a t the c a l c u l a t e d "answers" a r e as a c c u r a t e as they can be, g i v e n the i n f o r m a t i o n t h a t i s on hand. The d e t e r m i n a t i o n and a p p l i c a t i o n of s a f e t y f a c t o r s can then be d e a l t w i t h as a s e p a r a t e i s s u e . 13 4. P r e v i o u s Comparisons and Reviews A l t h o u g h Hayakawa (1978) produced a comprehensive r e v i e w of the development of a v a i l a b l e p r o c e s s c a l c u l a t i o n methods, d e a l i n g w i t h some of t h e i r a s s u m p t i o n s and l i m i t a t i o n s , no n u m e r i c a l d a t a were p r o v i d e d by which t o compare the methods. Merson et a l . (1978) b r i e f l y d i s c u s s e d v a r i o u s c a l c u l a t i o n methods, recommending Hayakawa's method (1970) as the most v e r s a t i l e ; however, n u m e r i c a l comparisons were not p r o v i d e d . Stumbo and L o n g l e y (1966) p r o v i d e d a summary of comparisons of a few methods which i n d i c a t e d t h a t l e t h a l i t y v a l u e s c a l c u l a t e d u s i n g B a l l ' s o r i g i n a l method (1923) were 5-15% lower than t h o s e c a l c u l a t e d u s i n g t h e i r new t a b l e s ; however, the range of c o n d i t i o n s i n v e s t i g a t e d was not c l e a r . Board e t a l . (1960) p r o v i d e d a comparison of the g e n e r a l method ( B i g e l o w e t a l . 1920), B a l l ' s method, and G i l l e s p y ' s method (1951) i n d i c a t i n g t h a t i n most ca s e s B a l l ' s method u n d e r e s t i m a t e d p r o c e s s l e t h a l i t y and G i l l e s p y ' s method o f t e n o v e r e s t i m a t e d p r o c e s s l e t h a l i t y . In view of the importance of t h e r m a l p r o c e s s i n g i n the f o o d i n d u s t r y , i t i s s u r p r i s i n g how l i t t l e a t t e n t i o n has been g i v e n t o the a c c u r a c y and r e l i a b i l i t y of methods used f o r d e t e r m i n i n g r e q u i r e d t h e r m a l p r o c e s s e s . Perhaps the r e a s o n i n g has been t h a t t h e r e have been so few problems w i t h commercial c a n n i n g s a f e t y i n r e c e n t y e a r s t h a t t h i s i n v e s t i g a t i o n would be u n n e c e s s a r y . However, the r e s e a r c h a c t i v i t y i n terms of development of new and m o d i f i e d 14 c a l c u l a t i o n methods would t e n d t o d i s p u t e t h i s c o n c l u s i o n . In any c a s e , l i t t l e n u m e r i c a l i n f o r m a t i o n i s a v a i l a b l e by which t o judge the m e r i t s of a v a i l a b l e t h e r m a l p r o c e s s c a l c u l a t i o n methods. I t was the g o a l of t h i s p r o j e c t t o s t a r t t o f i l l t h i s i n f o r m a t i o n gap. Of the many f o r m u l a methods p r e v i o u s l y c i t e d , f i v e were chosen, i n c l u d i n g b o t h the most w i d e l y used methods and the most r e c e n t l y d e v e l o p e d , as r e p r e s e n t a t i v e s of the v a r i o u s s t a g e s i n the e v o l u t i o n of p r o c e s s c a l c u l a t i o n methods. Computer s i m u l a t i o n was chosen as the means t o ge n e r a t e t e m p e r a t u r e h i s t o r i e s t o be e v a l u a t e d u s i n g t h e s e methods, s i n c e s i m u l a t i o n would not i n t r o d u c e e x t r a n e o u s e r r o r s i n t o the study and was w e l l a c c e p t e d i n the f i e l d . T h i s approach a l s o a l l o w e d t h e i n v e s t i g a t i o n of a wider range of p r o c e s s i n g c o n d i t i o n s than would o t h e r w i s e have been p o s s i b l e . 15 EXPERIMENTAL A. G e n e r a t i o n of Heat P e n e t r a t i o n Data 1. I n t r o d u c t i o n Thermal h i s t o r y c u r v e s f o r c o n d u c t i o n h e a t i n g foods i n c y l i n d r i c a l cans of v a r i o u s shapes and s i z e s were g e n e r a t e d u s i n g a FORTRAN language s i m u l a t i o n program adapted from T e i x e i r a e t a l . (1969b). S i m u l a t i o n t e c h n i q u e s a l l o w e d the i n v e s t i g a t i o n of a g r e a t many p r o c e s s i n g s i t u a t i o n s i n a s h o r t t i m e , and a t a r e a s o n a b l e c o s t . P i l o t p l a n t s t u d i e s would be v e r y time consuming and would be l i m i t e d i n scope by the a v a i l a b i l i t y of s u p p l i e s ( c o m m e r c i a l l y a v a i l a b l e can s i z e s ) and the p h y s i c a l l i m i t a t i o n s of a v a i l a b l e equipment ( r e t o r t s i n p a r t i c u l a r ) . P h y s i c a l e x p e r i m e n t a t i o n i s a l s o h i n d e r e d by measurement a c c u r a c y l i m i t s of a t be s t ± 0.5C° and by o t h e r e x p e r i m e n t a l e r r o r s . S t r i c t p r o c e s s c o n t r o l of p i l o t p l a n t equipment i s a l s o d i f f i c u l t t o a c h i e v e . S i m u l a t i o n , however, a l l o w s v e r y p r e c i s e " p r o c e s s c o n t r o l " and t h e r e f o r e the r e s u l t s a r e not c l o u d e d by u n c o n t r o l l a b l e f a c t o r s l e a d i n g t o e x p e r i m e n t a l e r r o r s . The r e s u l t s of a s i m u l a t i o n e x p e r i m e n t , however, a r e o n l y as good as the model used t o d e s c r i b e the system i n q u e s t i o n . Care must be ta k e n i n d e f i n i n g the system and how i t i n t e r a c t s w i t h i t s e n v i r o n m e n t , t o a v o i d u n n e c e s s a r y c o m p l i c a t i o n of the model or unwarranted s i m p l i f i c a t i o n . The f i n i t e - d i f f e r e n c e model used i n t h i s s t u d y has a sound b a s i s i n e n g i n e e r i n g p r i n c i p l e s of heat t r a n s f e r and has 16 been shown t o be an a p p r o p r i a t e system f o r s i m u l a t i o n of t h e r m a l p r o c e s s i n g of canned f o o d s . 2. Theory The f i n i t e - d i f f e r e n c e model was based on the F o u r i e r s i m p l i f i c a t i o n of t h e d i f f e r e n t i a l heat c o n d u c t i o n e q u a t i o n f o r a f i n i t e c y l i n d e r : d 2T 1 oT d 2T 1 dT o r 2 r or d y 2 o o t where T i s t e m p e r a t u r e , r i s r a d i a l d i s t a n c e from the c e n t r a l a x i s , y i s v e r t i c a l d i s t a n c e from the c e n t e r p l a n e , a i s t h e r m a l d i f f u s i v i t y and t i s t i m e . A n u m e r i c a l s o l u t i o n u s i n g t h e f i n i t e - d i f f e r e n c e form of t h i s e q u a t i o n was the b a s i s of the s i m u l a t i o n program. The f o l l o w i n g d e m o n s t r a t i o n (adapted from K r e i t h , 1964) of the f i n i t e - d i f f e r e n c e method f o r one d i m e n s i o n a l heat f l o w i s h e l p f u l i n u n d e r s t a n d i n g the more c o m p l i c a t e d two d i m e n s i o n a l model f o r heat t r a n s f e r i n a f i n i t e c y l i n d e r . The d e r i v a t i o n r e s u l t s i n an e q u a t i o n f o r c a l c u l a t i n g the t e m p e r a t u r e change i n volume element n over t h e time i n t e r v a l At from t=t t o t=t+1. E q u a t i o n f o r unsteady s t a t e heat c o n d u c t i o n i n one o 2T (2) o x 2 A X 2 T — (3) A x 2 d i m e n s i o n , x: 1 oT a ot For f i n i t e At and Ax: 1 A tT a At 1 7 The s u b s c r i p t s t and x i n d i c a t e whether t h e change of t e m p e r a t u r e w i t h t i m e , t , or l o c a t i o n , x, i s b e i n g r e f e r r e d t o . R e w r i t e l e f t - h a n d s i d e a s : 1 A T 1 T t + 1 - T1" 1 ^ t 1 = _ _n _n a At o At (4) R e w r i t e r i g h t - h a n d s i d e a s : t t t t ^ Tn+1 ' T n _ T n ~ T n - 1 Ax Ax Ax (5) A x 2 Ax Ax Recombining g i v e s : I T n + 1 - T n = T n + 1 ~ 2 T n + T n - 1 a At A x 2 (6) The f i n i t e - d i f f e r e n c e t e c h n i q u e was a p p l i e d t o the d i f f e r e n t i a l heat t r a n s f e r e q u a t i o n s f o r t h e f i n i t e c y l i n d e r t o d e v e l o p the f i n i t e - d i f f e r e n c e heat t r a n s f e r model used i n t h i s s t u d y . The FORTRAN language s i m u l a t i o n program d e v e l o p e d f o r t h i s s t u d y was adapted from T e i x e i r a e t a l . (1969b), a l t h o u g h t h e d e v e l o p e d program i s more v e r s a t i l e i n i t s a p p l i c a t i o n s . I t can s i m u l a t e p r o c e s s e s w i t h any t i m e -t e m p e r a t u r e program and i s e a s i l y m o d i f i e d f o r s p e c i f i c p u r p o s e s . 3. Time and Space Increment Study To a p p l y the f i n i t e - d i f f e r e n c e a p p r o a c h t o a c y l i n d r i c a l c o n t a i n e r , the c o n t a i n e r was d i v i d e d i n t o 18 l a y e r s of e q u a l t h i c k n e s s v e r t i c a l l y , and r i n g s of e q u a l t h i c k n e s s h o r i z o n t a l l y . The s i z e of t h e s e d i v i s i o n s has a b e a r i n g on the s t a b i l i t y of the model and the a c c u r a c y w i t h which i t s i m u l a t e s a heat t r a n s f e r p r o c e s s . A s t u d y was un d e r t a k e n t o d e t e r m i n e the a p p r o p r i a t e s i z e s of time and space i n c r e m e n t s t o be used. C a r s l a w and Jaeger (1959) i n d i c a t e d t h a t a s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y of the f i n i t e d i f f e r e n c e s o l u t i o n i s : aAt 1 M = < - (7) A x 2 2 where M i s c a l l e d the modulus and Ax i s the g r i d s p a c i n g i n e i t h e r r a d i a l or a x i a l d i r e c t i o n . The a c c u r a c y of the s o l u t i o n i n c r e a s e s as Ax and At d e c r e a s e , but s m a l l e r v a l u e s of Ax and At mean i n c r e a s e d c o m p u t a t i o n time and c o s t . I t i s t h e r e f o r e d e s i r a b l e t o s e l e c t Ax and At t o be as l a r g e as i s p e r m i s s i b l e f o r the degree of a c c u r a c y r e q u i r e d . Most of the t e s t runs f o r the increment study used can di m e n s i o n s of 4 cm r a d i u s and 8 cm h e i g h t and a t h e r m a l d i f f u s i v i t y of 0.100 cm 2/min. R e t o r t t e m p e r a t u r e was 121°C, i n i t i a l t e m p e r a t u r e 71°C, and water a t 21°C c o o l e d the cans t o 66°C a f t e r a 40 min p r o c e s s t i m e . Other p r o c e s s t i m e s and can s i z e s t e s t e d c o n f i r m e d t h a t d i f f e r e n c e s f o r o t h e r c o n d i t i o n s were not s i g n i f i c a n t . T e i x e i r a e t a l . (1969b) r e p o r t e d t h a t time i n c r e m e n t s of 0.125 min and a volume element m a t r i x w i t h 10 r a d i a l and 10 v e r t i c a l d i v i s i o n s were used f o r t h e i r work. A l t h o u g h 19 f i g u r e s were p r o v i d e d t o v e r i f y t h e i r c h o i c e , no d e t a i l s of the s t u d y were p r o v i d e d . The r e s u l t s of the increment study done he r e were l e s s o p t i m i s t i c than T e i x e i r a 1 s . They showed, f o r the main p r o c e s s used i n the t e s t s , t h a t c a l c u l a t e d i n t e g r a t e d l e t h a l i t y d i d not s t a b i l i z e u n t i l a 15x15 m a t r i x was used ( F i g u r e 1) whereas T e i x e i r a ' s r e s u l t s i n d i c a t e d a l e v e l i n g o f f f o r a 10x10 m a t r i x . T h i s apparent disagreement may be due t o d i f f e r e n c e s i n the t e s t c o n d i t i o n s between the two s t u d i e s . S i n c e the c o s t of the s i m u l a t i o n i n c r e a s e s as the p r o d u c t of t h e number of h o r i z o n t a l and v e r t i c a l i n c r e m e n t s , a 15x15 m a t r i x would c o s t 2.25 tim e s as much t o use as a 10x10 m a t r i x . F i g u r e 2 shows t h e r e s u l t s of the time increment study f o r a 10x10 m a t r i x . Whereas T e i x e i r a ' s f i g u r e s i n d i c a t e d l e v e l i n g o f f a t 8 i n c r e m e n t s per min, t h i s s t u d y i n d i c a t e d t h a t l e v e l i n g o f f d i d not occur a t fewer than 20 increments per min. Fo r t h i s work, o n l y the temperature a t t h e c e n t e r of the c y l i n d e r was of major i n t e r e s t . The c e n t e r p o i n t t e m p e r a t u r e was not a f f e c t e d by the g r i d s p a c i n g s as s t r o n g l y as the i n t e g r a t e d l e t h a l i t y v a l u e s were. At the end of a 40 min p r o c e s s , the p r e d i c t e d c e n t e r t e m p e r a t u r e s were 108.21°C and 108.23°C f o r a 10x10 and a 25x25 element m a t r i x , r e s p e c t i v e l y . I n o r d e r t o o b t a i n r e l i a b l e c e n t e r t e m p e r a t u r e p r e d i c t i o n s , a g r i d s p a c i n g of about 0.5 cm and time i n t e r v a l s of 0.05 min s h o u l d be s u f f i c i e n t . In t h i s s t u d y , time i n c r e m e n t s of 0.05 min and g r i d s p a c i n g s of 0.3 t o 0.4 cm were used, c o r r e s p o n d i n g t o M v a l u e s of 0.02 t o I n t e g r a t e d l e t h a l i t y (z«10C°). min OZ 8 12 Time i n c r e m e n t s 16 per 20 24 2 8 m i n F i g u r e 2. T i m e i n c r e m e n t s t u d y f o r 10x10 m a t r i x . 22 0.07 f o r the c o n d i t i o n s t e s t e d , w e l l below the r e q u i r e m e n t s f o r s t a b i l i t y of the s o l u t i o n . The number of v e r t i c a l and r a d i a l elements used ranged from 6 t o 25 and was d e t e r m i n e d f o r each s i m u l a t i o n by the c o n t a i n e r shape and s i z e . 4. Heat T r a n s f e r C o n s i d e r a t i o n s The r a t e of tem p e r a t u r e change a t the c e n t e r of a can d u r i n g a t h e r m a l p r o c e s s i s dependent upon the s u r f a c e heat t r a n s f e r c o e f f i c i e n t , the t h e r m a l d i f f u s i v i t y of the p r o d u c t and the tem p e r a t u r e g r a d i e n t s which b r i n g about heat t r a n s f e r . The te m p e r a t u r e g r a d i e n t s change over the c o u r s e of t h e p r o c e s s , b e i n g g r e a t e s t when the tem p e r a t u r e d i f f e r e n c e between the p r o d u c t and i t s environment i s l a r g e , and d e c r e a s i n g as the p r o d u c t t e m p e r a t u r e approaches t h e environment t e m p e r a t u r e . A l t h o u g h t h e r e i s reason t o b e l i e v e t h a t the t h e r m a l d i f f u s i v i t y of a p r o d u c t may change w i t h t e m p e r a t u r e (Evans, 1958), d a t a a r e l i m i t e d and the t h e r m a l d i f f u s i v i t y was t h e r e f o r e c o n s i d e r e d t o be c o n s t a n t t h r o u g h o u t the p r o c e s s f o r the purposes of t h i s work. T h i s assumption r e s u l t s i n the s i m p l e s t t ype of h e a t i n g c u r v e , which was d e s i r e d i n t h i s s t u d y t o f a c i l i t a t e comparison of the l e t h a l i t y c a l c u l a t i o n methods. In a l l of the a p p l i c a t i o n s of the f i n i t e - d i f f e r e n c e model p r e v i o u s l y c i t e d , the s u r f a c e heat t r a n s f e r c o e f f i c i e n t was c o n s i d e r e d t o be i n f i n i t e . T h i s s i m p l i f i c a t i o n i s c e r t a i n l y a p p r o p r i a t e f o r the h e a t i n g phase, when s a t u r a t e d steam i s used as the h e a t i n g medium. However, c o o l i n g i s u s u a l l y a c c o m p l i s h e d by f i l l i n g the 23 r e t o r t w i t h c o l d w a t e r , and s u r f a c e heat t r a n s f e r i n t h i s s i t u a t i o n i s not as good. However, s i n c e f o o d p r o d u c t s have r e l a t i v e l y low t h e r m a l c o n d u c t i v i t y , i n most c a s e s heat t r a n s f e r w i t h i n the p r o d u c t i s the l i m i t i n g f a c t o r . The B i o t number i s a d i m e n s i o n l e s s r a t i o of the i n t e r n a l r e s i s t a n c e t o heat t r a n s f e r t o the r e s i s t a n c e t o heat t r a n s f e r a t the s u r f a c e . For B i o t numbers of 40 or more, s u r f a c e heat t r a n s f e r can be c o n s i d e r e d t o be i n f i n i t e , and even f o r B i o t numbers as s m a l l as 10 l i t t l e e r r o r i s i n t r o d u c e d by t h i s a s s u m p t i o n (Heldman, 1975). Assuming an i n f i n i t e heat t r a n s f e r c o e f f i c i e n t d u r i n g the c o o l i n g c y c l e may t e n d t o s l i g h t l y o v e r e s t i m a t e the r a t e of heat t r a n s f e r i n some c a s e s . T h i s would r e s u l t i n more r a p i d c o o l i n g , i n d i c a t i n g a p r o c e s s t o have s l i g h t l y l e s s l e t h a l e f f e c t than i t would have i f c o o l i n g were s l i g h t l y s l o w e r . 5. P r o c e s s i n g C o n d i t i o n s Used For most of the p r o c e s s e s s i m u l a t e d , the r e t o r t t e m p e r a t u r e was 120 °C, but some t e s t s a t 140 °C were performed t o t e s t whether the r e l a t i v e e r r o r s i n p r o c e s s l e t h a l i t y c a l c u l a t i o n were a f f e c t e d by r e t o r t t e m p e r a t u r e . C o o l i n g water t e m p e r a t u r e was 100 C° below r e t o r t t e m p e r a t u r e , t o be c o n s i s t e n t w i t h t h e assumptions of B a l l ' s and Stumbo's f h / U t o g t a b l e s . I n s t a n t comeup t o r e t o r t t e m p e r a t u r e and i n s t a n t environment t e m p e r a t u r e c o l l a p s e t o c o o l i n g water t e m p e r a t u r e were assumed, as would be t h e case f o r a c o n t i n u o u s r a t h e r than b a t c h p r o c e s s i n g s i t u a t i o n . Thermal p r o p e r t i e s were h e l d c o n s t a n t 24 w i t h i n each s i m u l a t i o n , and the i n i t i a l temperature d i s t r i b u t i o n was u n i f o r m . R e s i s t a n c e t o heat t r a n s f e r a t the s u r f a c e was c o n s i d e r e d t o be n e g l i g i b l e , and the e f f e c t of headspace on heat t r a n s f e r was not c o n s i d e r e d . Thermal d i f f u s i v i t y v a l u e s r a n g i n g from 0.075 t o 0.125 cm 2/min were used, c o v e r i n g the range of t h e r m a l p r o p e r t i e s e n c o u n t e r e d c o m m e r c i a l l y f o r c o n d u c t i o n h e a t i n g foods (Rha, 1975). V a r i o u s i n i t i a l t e m p e r a t u r e d i f f e r e n c e s ( r e t o r t t e m p e r a t u r e - p r o d u c t c e n t e r t e m p e r a t u r e ) from 15 t o 95 C° and g v a l u e s ( t e m p e r a t u r e d i f f e r e n c e a t steam o f f ) from 0.05 t o 15 C° were i n v e s t i g a t e d . H e i g h t t o d i a m e t e r (H/D) r a t i o s from 0.1 t o 3 were s t u d i e d . For most of the study can d i a m e t e r s of 8 cm were used, except f o r cans w i t h v e r y s m a l l H/D r a t i o s (0.1 and 0.25) f o r which l a r g e r d i a m e t e r s were used, t o a l l o w a minimum h e i g h t of 2 cm t o be used. 6. S i m u l a t i o n Program The s i m u l a t i o n program c o n s i s t e d of a d r i v e r program and a number of s u b r o u t i n e s t h a t performed i t e r a t i v e c a l c u l a t i o n s t o f o l l o w t h e tem p e r a t u r e changes, c a l c u l a t e p r o c e s s l e t h a l i t y , and c o n t r o l the program's o u t p u t . Input t o the program i n c l u d e d c o n t a i n e r d i m e n s i o n s , the number of space i n c r e m e n t s i n each d i r e c t i o n , the time increment t o be used and the t h e r m a l d i f f u s i v i t y of the p r o d u c t . R e t o r t t e m p e r a t u r e and o t h e r p r o c e s s s p e c i f i c a t i o n s were a l s o p r o v i d e d . Any u n i t s c o u l d be used f o r the s e v a l u e s , as l o n g as the u n i t s f o r a l l q u a n t i t i e s were c o n s i s t e n t w i t h each o t h e r . Program o u t p u t s i n c l u d e d c e n t e r p o i n t t e m p e r a t u r e s 25 l o g g e d a t one minute i n t e r v a l s , and the e q u i v a l e n t l e t h a l i t y of the p r o c e s s a t the can c e n t e r f o r a z v a l u e of 10C°. F i g u r e 3 i s a f l o w c h a r t t h a t o u t l i n e s t h e f l o w of c o n t r o l between t h e main program and the s u b r o u t i n e s . 26 F i g u r e 3. F l o w c h a r t f o r s i m u l a t i o n p r o g r a m . 27 B. P r o c e s s C a l c u l a t i o n Methods 1. G e n e r a l Methods Three " g e n e r a l " methods f o r p r o c e s s l e t h a l i t y d e t e r m i n a t i o n were examined t o det e r m i n e t h e i r a c c u r a c y r e l a t i v e t o the r e f e r e n c e method t h a t was c o n s i d e r e d t o be e q u i v a l e n t t o a c o n t i n u o u s i n t e g r a t i o n of the l e t h a l i t y h i s t o r y c u r v e . a. Average temperature method The average te m p e r a t u r e method s t u d i e d was i d e n t i c a l t o the r e f e r e n c e method ( d e s c r i b e d i n s e c t i o n C3) ex c e p t t h a t the i n t e r v a l between d a t a p o i n t s was i n c r e a s e d t o one minute. T h i s method was a v a r i a t i o n of P a t a s h n i k ' s method and an e x t r a p o l a t i o n of the t e c h n i q u e used f o r a c c u m u l a t i n g l e t h a l i t y f o r the s i m u l a t i o n model ( T e i x e i r a e t a l . , 1969b). b. P a t a s h n i k ' s method P a t a s h n i k ' s method employed t h e t e c h n i q u e o f t r a p e z o i d a l i n t e g r a t i o n ( P a t a s h n i k , 1953). I t was s i m i l a r t o the average t e m p e r a t u r e method e x c e p t t h a t s u c c e s s i v e d a t a p o i n t s were not averaged t o o b t a i n a temperature v a l u e , b u t , r a t h e r , the tem p e r a t u r e was c o n s i d e r e d t o remain c o n s t a n t a t the g i v e n d a t a v a l u e f o r one complete time i n t e r v a l . T h i s time i n t e r v a l c o u l d be c o n s i d e r e d t o be t h a t b e f o r e or a f t e r the i n s t a n t t h a t t h e measurement a p p l i e s t o or c o u l d be c o n s i d e r e d t o span midway i n t o each of these two i n t e r v a l s . As l o n g as the p r o c e s s s t a r t e d and ended a t s u b - l e t h a l t e m p e r a t u r e s , however, the r a t i o n a l i z a t i o n of 28 t h i s p o i n t was not s i g n i f i c a n t , c. C u b i c p o l y n o m i a l method The c u b i c p o l y n o m i a l method was a method t h a t was more s i m i l a r t o a g r a p h i c a l g e n e r a l method than most n u m e r i c a l methods s i n c e i t smoothed out the c u r v e r a t h e r than c o n s i d e r i n g i t as a s t e p f u n c t i o n . For each time i n t e r v a l , t o t i + l ' a c u b i c i n t e r p o l a t i o n p o l y n o m i a l based on the f o u r p o i n t s : ( t i - i ' L i - i ^ ' ( f c ± ' L l } ' ( t i + l ' L i + l * ' a n d ' ( t . . , _ , L . ), was i n t e g r a t e d . These p o i n t s were c a l c u l a t e d 1+2 1+z l e t h a l i t y v a l u e s c o r r e s p o n d i n g t o t e m p e r a t u r e s measured a t one minute time i n t e r v a l s . For t h i s s t u d y , the FORTRAN f u n c t i o n subprogram QINT4P (Madderom, 1978), a v a i l a b l e on t h e u n i v e r s i t y ' s computing system, was used; however, the method c o u l d be a d a p t e d f o r use w i t h any computing f a c i l i t y , or p o s s i b l y even a hand-held programmable c a l c u l a t o r . F i t t i n g of the c u b i c p o l y n o m i a l r e q u i r e d the s o l u t i o n of f o u r s i m u l t a n e o u s e q u a t i o n s i n f o u r unknowns u s i n g p r i c i p l e s of m a t r i x a l g e b r a . Once the p o l y n o m i a l was d e t e r m i n e d , the i n t e g r a t i o n was e a s i l y a c c o m p l i s h e d . A l l of t h e o p e r a t i o n s c o u l d be combined i n t o a s i n g l e program so t h a t the o p e r a t o r would o n l y e n t e r t i m e - t e m p e r a t u r e p a i r s , and i f t e m p e r a t u r e s were measured a t e q u a l time i n t e r v a l s , t h e system c o u l d be s i m p l i f i e d so t h a t o n l y temperature v a l u e s would have t o be e n t e r e d , w i t h the time base b e i n g g e n e r a t e d by the program. 29 2. Formula Methods F i v e f o r m u l a methods f o r c e n t e r p o i n t l e t h a l i t y d e t e r m i n a t i o n were examined. These were chosen t o be r e p r e s e n t a t i v e of c u r r e n t l y a p p l i e d methods as w e l l as the most r e c e n t developments i n p r o c e s s e v a l u a t i o n , a. B a l l ' s t a b l e method T h i s f o r m u l a method was a g r e a t m i l e s t o n e i n t h e r m a l p r o c e s s i n g and has been the i n d u s t r y s t a n d a r d s i n c e i t was f i r s t i n t r o d u c e d ( B a l l , 1923). B a l l d e v e l o p e d t a b l e s of p r o c e s s v a l u e (U) w i t h r e s p e c t t o h e a t i n g r a t e index ( f ^ ) and t e m p e r a t u r e d i f f e r e n c e between the p r o d u c t c o l d spot and the r e t o r t a t the end of the h e a t i n g c y c l e ( g ) . The s t r a i g h t - l i n e h e a t i n g and c o o l i n g p o r t i o n s of the t i m e -t e m p e r a t u r e h i s t o r y c u r v e s were e v a l u a t e d m a t h e m a t i c a l l y u s i n g e x p o n e n t i a l i n t e g r a l s . The c o o l i n g l a g f a c t o r ( J c c ) was assumed t o be 1.41 and the c u r v e was ap p r o x i m a t e d by an h y p e r b o l a . The s t a r t of s t r a i g h t - l i n e c o o l i n g was d e t e r m i n e d e m p i r i c a l l y t h r o u g h o b s e r v a t i o n of e x p e r i m e n t a l d a t a , and h e a t i n g and c o o l i n g r a t e s were assumed t o be e q u a l . The h e a t i n g l a g f a c t o r was used f o r t h e c a l c u l a t i o n of g but i t s l e t h a l e f f e c t was not a c c o u n t e d f o r . Merson e t a l . (1978) p r o v i d e d a good d e s c r i p t i o n of the p r i n c i p l e s of t h i s method. The American Can Company has s i n c e d e v e l o p e d more d e t a i l e d t a b l e s , i n t e r p o l a t i n g and e x t r a p o l a t i n g the t a b l e s t h a t were p u b l i s h e d by B a l l . These t a b l e s were used f o r the e v a l u a t i o n of t h i s method. 30 b. B a l l ' s e q u a t i o n method W h i l e t r y i n g t o d e v e l o p a method f o r i m p l e m e n t i n g B a l l ' s f o r m u l a method w i t h o u t u s i n g the t a b l e s , i t was d i s c o v e r e d t h a t t h e v a l u e s i n h i s t a b l e s d i d not agree w i t h the e q u a t i o n s t h a t were used t o d e v e l o p them (Smith and Tung, 1979). T h i s was c o n f i r m e d by the f i n d i n g s of S t e e l e e t a l . (1979). The second method i n v e s t i g a t e d , t h e r e f o r e , was B a l l ' s method u s i n g the e q u a t i o n s d e v e l o p e d f o r the p r o d u c t i o n of the t a b l e s ( B a l l and O l s o n , 1957). c. Stumbo's method Stumbo and L o n g l e y (1966) p u b l i s h e d t a b l e s f o r p r o c e s s e v a l u a t i o n t a k i n g i n t o a ccount the v a r i a b i l i t y of j c c v a l u e s . The v a l u e s i n t h e s e t a b l e s were o b t a i n e d t h r o u g h p l a n i m e t e r measurements of hand-drawn t e m p e r a t u r e h i s t o r i e s p l o t t e d on l e t h a l r a t e p a p e r , and subsequent i n t e r p o l a t i o n of g r a p h s . R e v i s e d t a b l e s (used i n t h i s e v a l u a t i o n ) were d e v e l o p e d through use of computer i n t e g r a t i o n of t h e r m a l h i s t o r i e s g e n e r a t e d from heat t r a n s f e r e q u a t i o n s u s i n g f i n i t e d i f f e r e n c e s i m u l a t i o n s (Stumbo, 1973). In a l l o t h e r p a r t i c u l a r s , the method was s i m i l a r t o B a l l ' s . d. S t e e l e and Board's method B a l l and Olso n (1957) d e v e l o p e d t a b l e s f o r e v a l u a t i o n of p r o c e s s e s e x h i b i t i n g broken h e a t i n g c u r v e s or unequal h e a t i n g and c o o l i n g r a t e s . These t a b l e s were based on the same c o n c e p t s as B a l l ' s o r i g i n a l t a b l e s e x c e p t t h a t the h e a t i n g and c o o l i n g p o r t i o n s of t h e p r o c e s s were kept s e p a r a t e . T h i s method was improved upon by G r i f f i n e t a l . 31 (1971) t h r o u g h the use of a r e l a t i o n s h i p between the c u r v e d and s t r a i g h t - l i n e p o r t i o n s of t h e c o o l i n g c u r v e . S t e e l e and Board (1979) adapted t h i s method f o r c a l c u l a t i o n u s i n g s t e r i l i z i n g r a t i o s , t o s i m p l i f y c a l c u l a t i o n s . T h i s method was e v a l u a t e d u s i n g t h e e q u a t i o n s d e v e l o p e d r a t h e r than the t a b l e s p r o v i d e d , e. Hayakawa's method Hayakawa (1970) d e v e l o p e d a method of l e t h a l i t y e v a l u a t i o n s i m i l a r t o those p r e v i o u s l y d e s c r i b e d except t h a t c i r c u l a r f u n c t i o n s were used t o e s t i m a t e t h e c u r v e d p o r t i o n s of the h e a t i n g and c o o l i n g c u r v e s . The p r o c e s s was d i v i d e d i n t o f o u r s e c t i o n s f o r e v a l u a t i o n : c u r v e d h e a t i n g , s t r a i g h t -l i n e h e a t i n g , c u r v e d c o o l i n g , and s t r a i g h t - l i n e c o o l i n g . The l e n g t h s of the c u r v e d p o r t i o n s were e s t i m a t e d u s i n g an e m p i r i c a l r e l a t i o n s h i p between f and j . D u r i n g t e s t i n g of the p r o c e d u r e s d e v e l o p e d f o r computer s o l u t i o n , e r r o r s i n the p u b l i s h e d t a b l e s were found. A c o r r e c t i o n t o the t a b l e s was found e l s e w h e r e (Downes and Hayakawa, 1977). T h i s method was e v a l u a t e d u s i n g t h e e q u a t i o n s d e v e l o p e d r a t h e r than the t a b l e s . 32 C. A d a p t a t i o n f o r Computer S o l u t i o n A d a p t a t i o n of the v a r i o u s f o r m u l a methods f o r s o l u t i o n s o l e l y by computer r e q u i r e d two major systems. ' One was a t a b l e a c c e s s i n g system and the o t h e r a system f o r d e t e r m i n i n g f and j v a l u e s from the t i m e - t e m p e r a t u r e i n p u t d a t a . 1. T a b l e A c c e s s T a b l e a c c e s s was a c c o m p l i s h e d by s e t t i n g up f i l e s o r g a n i z e d so t h a t t h e l i n e number c o r r e s p o n d e d t o the g-v a l u e f o r each p a r t of the t a b l e . The i n f o r m a t i o n on the f i l e - l i n e v a r i e d s l i g h t l y from one method t o a n o t h e r , but g e n e r a l l y i n c l u d e d the g v a l u e s and f^/U v a l u e s f o r the c u r r e n t and p r e v i o u s l i n e s of the t a b l e t o f a c i l i t a t e i n t e r p o l a t i o n where n e c e s s a r y , s i n c e i t i s d i f f i c u l t t o r e a d a f i l e backwards, i f the p r e v i o u s l i n e were needed. The f i l e l i n e s were read u s i n g an i n d e x e d r e a d and, i f n e c e s s a r y , the next l i n e of the t a b l e c o u l d be r e a d s e q u e n t i a l l y . For Stumbo's t a b l e s , the a c c e s s system took advantage of t h e l i n e a r r e l a t i o n s h i p between g and j c c f o r a g i v e n v a l u e of f^/U, c a l c u l a t i n g v a l u e s f o r i n t e r m e d i a t e j c c v a l u e s from the v a l u e s f o r j c c = 0 . 4 0 and j c c = 2 . 0 0 . In c a s e s where t a b l e s were t o be a c c e s s e d i n the c o u r s e of a l e t h a l i t y d e t e r m i n a t i o n , g v a l u e s were c o n v e r t e d from C° t o F° t o be c o m p a t i b l e w i t h a v a i l a b l e t a b l e s . 2. D e t e r m i n a t i o n of f and j V a l u e s D e t e r m i n a t i o n of f and j v a l u e s was a c c o m p l i s h e d u s i n g an i t e r a t i v e r e g r e s s i o n t e c h n i q u e t o l o c a t e the s t a r t of 33 the s t r a i g h t - l i n e p o r t i o n of the h e a t i n g and c o o l i n g c u r v e s ( l o g g v s . t and l o g m ( d i f f e r e n c e i n temp e r a t u r e between c o o l i n g water and can c e n t e r p o i n t d u r i n g c o o l ) v s . t , r e s p e c t i v e l y ) . L i n e a r r e g r e s s i o n was performed i t e r a t i v e l y , d e l e t i n g p o i n t s up t o t h e c r o s s o v e r of the f i t t e d s t r a i g h t l i n e and the d a t a c u r v e . In the f i r s t i t e r a t i o n , a l l of the d a t a f o r the c u r v e was used t o d e t e r m i n e the r e g r e s s i o n l i n e . The r e g r e s s i o n l i n e was then compared t o the d a t a c u r v e , and a l l of the d a t a p o i n t s up t o the c r o s s o v e r p o i n t were o m i t t e d f o r the next i t e r a t i o n . T h i s p r o c e d u r e was c o n t i n u e d u n t i l t he f i t t e d r e g r e s s i o n l i n e c r o s s e d the d a t a c u r v e c l o s e t o t h e f i r s t p o i n t i n c l u d e d i n t h e l a s t i t e r a t i o n (maximum r e l a t i v e d i f f e r e n c e 0.1 % ) . T h i s t e c h n i q u e r a p i d l y e l i m i n a t e d t h e n o n - l i n e a r p o r t i o n of t h e d a t a c u r v e , and s e l e c t e d t h e l i n e a r r e g i o n f o r t he c a l c u l a t i o n of the parameters f and j . The parameter f was then c a l c u l a t e d as the n e g a t i v e r e c i p r o c a l s l o p e of the r e g r e s s i o n l i n e , and j was c a l c u l a t e d as d e s c r i b e d by Stumbo (1973). The t e c h n i q u e was more d i s c r i m i n a t i n g than manual l i n e - f i t t i n g t e c h n i q u e s , o f t e n d i s c a r d i n g up t o the f i r s t 40 or more minutes of the d a t a as f a l l i n g i n the c u r v i l i n e a r p o r t i o n . The f v a l u e s o b t a i n e d t h e r e f o r e more c l o s e l y e s t i m a t e d t h e t r u e f v a l u e which i s t h e o r e t i c a l l y d e f i n e d as the asymptote of the h e a t i n g or c o o l i n g c u r v e . 34 3. R e f e r e n c e Method and C a l c u l a t i o n of D e v i a t i o n s The r e f e r e n c e method t o which the o t h e r s were compared was a n u m e r i c a l g e n e r a l method w i t h d a t a p o i n t s t a k e n e v e r y 0.05 min. The a r i t h m e t i c mean t e m p e r a t u r e over each time i n t e r v a l ( A t ) was d e t e r m i n e d and the l e t h a l r a t e (L) f o r t h i s t e m p e r a t u r e was c o n s i d e r e d t o a p p l y f o r the d u r a t i o n of the i n t e r v a l ( T e i x e i r a e t a l . , 1969b). A z v a l u e of 10 C° (18 F°) and a r e f e r e n c e t e m p e r a t u r e of 121.1 C° (250 F°) were used f o r l e t h a l r a t e c a l c u l a t i o n s . Thus, fT - 121.1 1° , L = 10 . (8) L e t h a l i t i e s were c a l c u l a t e d as LAt and summed t o determine t h e t o t a l p r o c e s s l e t h a l i t y ( F 0 ) . Because of the s m a l l time i n t e r v a l s (0.05 m i n ) , t h i s method was assumed t o e s t i m a t e the c o n t i n u o u s p r o c e s s c u r v e and hence the o r i g i n a l g e n e r a l method, and i t s l e t h a l i t y was used t o judge the a c c u r a c y of the a l t e r n a t e c a l c u l a t i o n methods. Each temperature h i s t o r y c u r v e was e v a l u a t e d f o r p r o c e s s l e t h a l i t y ( F 0 ) u s i n g the r e f e r e n c e method and each of the f o r m u l a ( t e s t ) methods. D e v i a t i o n s between F 0 v a l u e s of t h e r e f e r e n c e and t e s t methods were c a l c u l a t e d as p e r c e n t a g e s of the r e f e r e n c e F 0 u s i n g : F 0 " F 0 ( r e f ) ( t e s t ) D e v i a t i o n = x 100%. (9) F 0 ( r e f ) A p o s i t i v e p e r c e n t a g e d i f f e r e n c e would i n d i c a t e t h a t the t e s t method u n d e r e s t i m a t e d the a c t u a l p r o c e s s l e t h a l i t y . 35 RESULTS & DISCUSSION A. G e n e r a l Methods T h i s s t u d y showed t h a t the use of the average te m p e r a t u r e method r e s u l t e d i n s i g n i f i c a n t d e v i a t i o n s i n c a l c u l a t e d l e t h a l i t y compared t o the r e f e r e n c e method. The method's a c c u r a c y was s i g n i f i c a n t l y a f f e c t e d by both the s l o p e and c u r v a t u r e of the heat p e n e t r a t i o n c u r v e . When a time i n t e r v a l of 1 min was used, l a r g e r d e v i a t i o n s were noted f o r p r o d u c t s t h a t heat r a p i d l y ( s m a l l v a l u e s f o r the h e a t i n g r a t e i n d e x , f h ) than f o r s l o w e r h e a t i n g p r o d u c t s , when the can h e i g h t t o dia m e t e r r a t i o (H/D) and temp e r a t u r e d i f f e r e n c e between the r e t o r t and the c e n t e r p o i n t a t the end of the cook (g) were c o n s t a n t ( F i g u r e 4 ) . F o r an f h of 30 min, the p r o c e s s l e t h a l i t y u n d e r e s t i m a t i o n was t e n t i m e s l a r g e r than f o r an f h of 90 min (0.67% compared t o 0.063%). When £ and g were h e l d c o n s t a n t , and the h e i g h t t o di a m e t e r r a t i o v a r i e d , a l e s s d r a m a t i c but s t i l l a p p r e c i a b l e v a r i a t i o n i n a c c u r a c y was noted ( F i g u r e 5 ) . The g r e a t e s t e r r o r (0.99% f o r f = 30 min) was f o r s m a l l h e i g h t t o d i a m e t e r r a t i o s , and the l e a s t (0.66%) f o r a r a t i o c l o s e t o u n i t y . H o l d i n g f ^ and t h e h e i g h t t o d i a m e t e r r a t i o c o n s t a n t , the e f f e c t of g was a s s e s s e d . F i g u r e 6 shows t h a t the g r e a t e s t u n d e r e s t i m a t i o n of p r o c e s s l e t h a l i t y o c c u r r e d f o r l a r g e g, w i t h the e f f e c t more pronounced f o r s m a l l e r H/D r a t i o s . The h e i g h t t o d i a m e t e r r a t i o and v a l u e of g w i l l F i g u r e 5. E f f e c t o f h e i g h t t o d i a m e t e r r a t i o on e v a l u a t i o n e r r o r s u s i n g a v e r a g e t e m p e r a t u r e m e t h o d . (g=5 C ° ) U) D e v i a t i o n f r o m r e f e r e n c e l e tha l i t y . /, se 39 a f f e c t the c u r v a t u r e of the l e t h a l i t y p l o t , and t h u s , the a c c u r a c y w i t h which the average te m p e r a t u r e method e s t i m a t e s the l e t h a l v a l u e of the p r o c e s s . P a t a s h n i k ' s t r a p e z o i d a l i n t e g r a t i o n method was found t o be a more a p p r o p r i a t e method. E r r o r s were v e r y s m a l l ( u s u a l l y l e s s than 0.05%) when 1 min time i n t e r v a l s were used. However, when 5 min time i n t e r v a l s were used, as sugg e s t e d by P a t a s h n i k (1953), d e v i a t i o n s were as g r e a t as 4% o v e r e s t i m a t i o n of p r o c e s s l e t h a l i t y . However, t h i s d e v i a t i o n was s m a l l compared t o t h a t r e s u l t i n g from use of the average temperature method, which always u n d e r e s t i m a t e d p r o c e s s l e t h a l i t y , i n some c a s e s by as much as 27%. The c u b i c p o l y n o m i a l method was d e v e l o p e d i n an attempt t o improve the a c c u r a c y of l e t h a l i t y d e t e r m i n a t i o n s u s i n g the g e n e r a l method approach. However, e x p e r i m e n t a l r e s u l t s showed t h a t i t p r o v i d e d no improvement over P a t a s h n i k ' s method, even when 5 min d a t a i n t e r v a l s were used. B. Formula Methods 1 . I n i t i a l S t u d i e s I n i t i a l s t u d i e s were conducted t o de t e r m i n e which f a c t o r s had the g r e a t e s t e f f e c t on the a c c u r a c y of the v a r i o u s methods f o r p r o c e s s e v a l u a t i o n i n an attempt t o reduce the number of e x p e r i m e n t s r e q u i r e d t o compare t h e i r p e r f o r m a n c e . Even a f t e r t h i s r e d u c t i o n , over 200 t h e r m a l h i s t o r y c u r v e s were e v a l u a t e d . The i n i t i a l s t u d i e s showed 40 t h a t the temperature d i f f e r e n c e a t the end of the cook (g) and the h e i g h t t o d i a m e t e r r a t i o (H/D) were the most s i g n i f i c a n t f a c t o r s , r e s u l t i n g i n a wide v a r i a t i o n of e r r o r magnitude, w i t h c o n s i s t e n t e r r o r p a t t e r n s . Can s i z e , t h e r m a l d i f f u s i v i t y and f ^ d i d not have l a r g e e f f e c t s on e r r o r magnitudes compared t o the e f f e c t s r e s u l t i n g from v a r i a t i o n s i n g and c o n t a i n e r d i m e n s i o n s . Because of the n a t u r e of t h e s e s i m u l a t i o n e x p e r i m e n t s , s t a t i s t i c a l t r e a t m e n t of the d a t a was not a p p r o p r i a t e , t h e r e f o r e the d e c i s i o n s t o i n c l u d e or e x c l u d e f a c t o r s from f u r t h e r e x p e r i m e n t s were made a f t e r v i s u a l i n s p e c t i o n of the r e s u l t s of s i m u l a t i o n e x p e r i m e n t s f o r a sample of runs c o v e r i n g a wide range of c o n d i t i o n s . The d a t a i n Ta b l e I show t h a t whereas t h e p e r c e n t a g e e r r o r i n l e t h a l i t y c a l c u l a t i o n v a r i e d w i d e l y as g was v a r i e d , t h e e f f e c t of v a r y i n g the c o n t a i n e r s i z e was r e l a t i v e l y s m a l l . T a b l e I I i s a summary, f o r one can s i z e and v a l u e of g, of d a t a f o r e x p e r i m e n t s c o v e r i n g a range of f ^ v a l u e s and a range of t h e r m a l d i f f u s i v i t i e s . The e r r o r v a r i a b i l i t y i n d i c a t e d w i t h i n any one column of t h i s t a b l e i s s m a l l compared t o the v a r i a b i l i t y due t o v a r i a t i o n s i n g i n d i c a t e d i n the c o r r e s p o n d i n g column i n T a b l e I . The f a c t o r s of can s i z e , f , and t h e r m a l d i f f u s i v i t y were t h e r e f o r e c o n s i d e r e d t o be r e l a t i v e l y i n s i g n i f i c a n t and were not i n c l u d e d as v a r i a b l e s i n f u r t h e r e x p e r i m e n t s . V a r y i n g the i n i t i a l t e m perature d i f f e r e n c e r e s u l t e d i n s m a l l d i f f e r e n c e s , m o s t l y due t o the e f f e c t on the e v a l u a t i o n of f h and Jcht* A range of i n i t i a l 41 T a b l e I - E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a methods f o r v a r i o u s can s i z e s and v a l u e s of g-(H/D=1. 35) P e r c e n t e r r o r g r a d i u s B a l l ' s B a l l ' s S t e e l e Hayakawa Stumbo ( c ° ) (cm) T a b l e s E q u a t i o n & Board 1 5 3 59.4 67.8 56.6 46.8 17.1 15 4 60.3 68.6 57.3 •47.5 15.9 15 5 60.7 69.0 57.7 47.5 13.8 5 3 31.3 40.4 29.2 20.9 10.1 5 4 32. 1 41.2 30.0 21.8 10.7 5 5 32.4 41 .4 30.0 21 .4 9.2 1.5 3 18.0 22.7 15.1 9.5 4.9 1.5 4 18.4 23.0 15.6 10.2 5.9 1.5 5 18.5 23.1 15.7 10.2 5.5 0.5 3 12.1 14.5 9.4 5.7 2.1 0.5 4 12.3 14.7 9.6 5.8 2.1 0.5 5 12.3 14.7 9.7 6.0 2.2 0.15 3 9.1 9.8 6.3 3.8 1 .9 0.15 4 9.2 9.8 6.4 3.8 1 .6 0.15 5 9.2 9.7 6.4 3.8 1.5 42 T a b l e I I - E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a methods f o r v a r i o u s t h e r m a l d i f f u s i v i t i e s and f h . (H/D=1.0; g=5 C°) P e r c e n t e r r o r £ . c B a l l ' s B a l l ' s S t e e l e Hayakawa Stumbo n (min) (cmVmin) T a b l e s E q u a t i o n & Board 30 0.075 35.2 43.8 32.6 24.2 11.4 30 0. 100 35.6 44.2 32.9 24.4 1 1 .4 30 0.1 25 35.9 44.5 33.2 24.7 11.6 50 0.075 35.7 44.3 33.1 24.7 11.8 50 0.1 00 36.2 44.7 33.7 25.5 12.8 50 0.125 36.4 44.9 33.9 25.7 13.1 70 0.075 36. 1 44.7 33.3 24.7 11.0 70 0. 1 00 36.4 44.9 33.6 25. 1 11.1 70 0. 125 36.7 45.2 33.9 25.4 1 1 .7 90 0.075 36.4 44.9 33.7 25.2 11.7 90 0. 100 36.6 45.1 33.9 25.4 1 1.9 90 0. 125 36.8 45.2 34. 1 25.6 12.2 43 te m p e r a t u r e d i f f e r e n c e s from 20 t o 95 C° was i n c l u d e d i n the s t u d y ; but d a t a from i n i t i a l t e m p e r a t u r e d i f f e r e n c e s of 20 C° were l a t e r e x c l u d e d because e s t i m a t e s of f n were poor, e s p e c i a l l y when g was l a r g e , s i n c e s t r a i g h t - l i n e h e a t i n g b e h a v i o r was not w e l l e s t a b l i s h e d b e f o r e the end of the h e a t i n g c y c l e . The e f f e c t s were s m a l l compared t o the e f f e c t s of o t h e r f a c t o r s b e i n g s t u d i e d . P a r a l l e l p r o d u c t t h e r m a l h i s t o r i e s f o r r e t o r t t e m p e r a t u r e s of 120 and 140 °C were e v a l u a t e d and the c a l c u l a t e d p e r c e n t a g e e r r o r s were found t o be comparable ( T a b l e I I I ) . A l l subsequent t h e r m a l h i s t o r i e s were g e n e r a t e d u s i n g a r e t o r t t e m p e r a t u r e of 120 °C but the p e r c e n t a g e e r r o r s p r e s e n t e d can be c o n s i d e r e d t o a p p l y f o r h i g h e r r e t o r t t e m p e r a t u r e s as w e l l . T h i s i s of p a r t i c u l a r s i g n i f i c a n c e when c o n s i d e r i n g l a r g e g v a l u e s s i n c e t h e s e ar e of g r e a t e r importance f o r h i g h e r r e t o r t t e mperature p r o c e s s e s than f o r p r o c e s s e s a t 120 °C. 2. E f f e c t s of Can Shape and g V a l u e F i g u r e s 7 t h r o u g h 11 show t h e e r r o r s , r e l a t i v e t o the r e f e r e n c e method, t h a t r e s u l t e d from c a l c u l a t i o n of p r o c e s s l e t h a l i t y u s i n g each of the f i v e t e s t methods. Each p o i n t r e p r e s e n t s an average e r r o r f o r i n i t i a l t e m perature d i f f e r e n c e s of 35,65, and 95 C°. The d e v i a t i o n s a r e f u n c t i o n s of both can shape (H/D) and g. The shapes of the e r r o r c u r v e s a r e s i m i l a r f o r B a l l ' s t a b l e method, B a l l ' s e q u a t i o n method, S t e e l e and Board's method and Hayakawa's 44 T a b l e I l l - E r r o r s i n c a l c u l a t e d l e t h a l i t i e s u s i n g 5 f o r m u l a methods f o r v a r i o u s v a l u e s of g and two r e t o r t t e m p e r a t u r e s . (H/D=1.35) P e r c e n t e r r o r g r e t o r t B a l l ' s B a l l ' s S t e e l e Hayakawa Stumbo (C°) (°C) T a b l e s E q u a t i o n & Board 15 120 60.3 68.6 57.3 47.5 15.9 15 140 60.3 68.7 57.5 47.6 16. 1 5 120 32. 1 41.2 30.0 21 .8 10.7 5 140 32.1 41 .2 30.0 21.8 10.7 1 .5 120 18.4 23.0 15.6 10.2 5.9 1 .5 140 18.5 23.2 15.8 10.5 6.4 0.5 120 12.3 14.7 9.6 5.8 2. 1 0.5 140 12.5 14.9 9.8 6.1 2.5 0.15 120 9.2 9.8 6.4 3.8 1 .6 0.15 140 9.4 10.0 6.6 4.0 1.9 0.05 120 5.8 7.0 4.7 2.9 0.9 0.05 140 6. 1 7.4 4.8 2.9 0.8 H / D r a t i o F i g u r e 7. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i B a l l ' s t a b l e s . F i g u r e 8. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g B a l l ' s e q u a t i o n . I I I I I 1 I 0 0.5 1.0 1.5 2.0 2 5 3.0 H / D r a t i o F i g u r e 9. E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s u s i n g S t e e l e and B o a r d ' s m e t h o d . 8fr 6 f r 50 method, a l t h o u g h the magnitudes of the e r r o r s d i f f e r . The e r r o r c u r v e s f o r Stumbo's method have s i m i l a r c h a r a c t e r i s t i c s , but do not appear t o be s i m p l e f u n c t i o n s of H/D, e s p e c i a l l y f o r l a r g e v a l u e s of g. In a l m o s t a l l c a s e s t h e s e e r r o r s were u n d e r e s t i m a t i o n s of p r o c e s s l e t h a l i t y , t h a t i s , e r r o r s on the " s a f e " s i d e . These d e v i a t i o n s were s m a l l e s t f o r s m a l l H/D, i n c r e a s e d t o a maximum when H/D was near u n i t y , then d e c r e a s e d s l o w l y t o an i n t e r m e d i a t e v a l u e as H/D became l a r g e . I t might be e x p e c t e d t h a t the e r r o r s i n f l u e n c e d by can shape would be a f u n c t i o n of the c o o l i n g l a g f a c t o r , s i n c e a p l o t of j as a f u n c t i o n of H/D has a shape s i m i l a r t o the p l o t of e r r o r magnitude ( B a l l and O l s o n , 1957). F i g u r e 12 shows t h a t a l t h o u g h the e r r o r s tended t o be g r e a t e r f o r l a r g e r j c c v a l u e s , the t r e n d was not a smooth f u n c t i o n . T h e r e f o r e , some o t h e r f a c t o r or f a c t o r s , must a l s o be i n f l u e n c i n g the e r r o r magnitude, and t h e s e would be e x p e c t e d t o be r e l a t e d t o the H/D r a t i o , s i n c e the p l o t of e r r o r magnitude as a f u n c t i o n of H/D i s a smooth and c o n t i n u o u s c u r v e i n most c a s e s . The shape e f f e c t i s b e l i e v e d t o be a r e s u l t of d i f f e r e n c e s i n c o o l i n g c u r v e shape r e s u l t i n g from the t e m p e r a t u r e g r a d i e n t t h r o u g h the can a t the end of the h e a t i n g c y c l e . E r r o r s i n l e t h a l i t y d e t e r m i n a t i o n were a l s o found t o be a f u n c t i o n of g, i n c r e a s i n g as the v a l u e of g i n c r e a s e d . The e f f e c t of g v a l u e on l e t h a l i t y d e t e r m i n a t i o n e r r o r s may be a r e s u l t of the t e m p e r a t u r e g r a d i e n t from the s u r f a c e t o • Ball A S t u m b o A A A 1.2 L_ 1.4 16 18 ice F i g u r e 1 2 . E r r o r s i n p r o c e s s l e t h a l i t y d e t e r m i n a t i o n s a s r e l a t e d t o t h e c o o l i n g l a g f a c t o r . ( g =5 C ° ) 52 the c e n t e r of the c o n t a i n e r a t the end of the h e a t i n g c y c l e . D u r i n g the h e a t i n g c y c l e , the temperature near the s u r f a c e o f the can i s h i g h e r than t h a t near the c e n t e r . I f the h e a t i n g c y c l e i s c o n t i n u e d u n t i l g i s s m a l l , t h i s t e m p e r a t u r e g r a d i e n t becomes i n s i g n i f i c a n t . However, i f the p r o c e s s i s stopped when g i s l a r g e , t h e g r a d i e n t causes the tem p e r a t u r e a t the c e n t e r of the can t o c o n t i n u e t o r i s e f o r a p e r i o d of time a f t e r the s t a r t of the c o o l i n g c y c l e and the c e n t e r t e m p e r a t u r e of t h e can may not b e g i n t o drop f o r s e v e r a l m i n u t e s . T h i s e f f e c t i s not a c c o u n t e d f o r i n any of the f o r m u l a methods t e s t e d . 3. Comparison of Methods F i g u r e 13 shows a co m p a r i s o n of the e r r o r s a s s o c i a t e d w i t h t h e f i v e f o r m u l a methods t e s t e d . E r r o r s f o r o n l y one v a l u e of g a r e shown, but the t r e n d s were s i m i l a r f o r o t h e r v a l u e s of g, a l t h o u g h t h e e r r o r magnitudes d i f f e r ( F i g u r e s 7-11). The v a l u e of g=5C° used i n F i g u r e 13 i s i n the range of the l a r g e s t g v a l u e s t h a t would be e n c o u n t e r e d i n c o n v e n t i o n a l p r o c e s s i n g . C a l c u l a t i o n of p r o c e s s l e t h a l i t y u s i n g B a l l ' s t a b l e method r e s u l t e d i n r e l a t i v e l y l a r g e e r r o r s , i n d i c a t i n g t h a t the f o r m u l a method approach has been s i g n i f i c a n t l y improved by more r e c e n t m o d i f i c a t i o n s . S t e e l e and Board's method, h a v i n g e l i m i n a t e d the a s s u m p t i o n s of o n l y one v a l u e f o r j c c and e q u a l h e a t i n g and c o o l i n g r a t e s , performed s l i g h t l y b e t t e r than B a l l ' s t a b l e method under most c o n d i t i o n s . 54 Hayakawa's method, which e s t i m a t e d the c o o l i n g l a g u s i n g c i r c u l a r f u n c t i o n s , r e s u l t e d i n s m a l l e r e r r o r s than e i t h e r of t h e s e methods. The l a r g e s t e r r o r f o r a g v a l u e of 5 C° u s i n g Hayakawa's method was 2 5 % compared t o 36% u s i n g B a l l ' s t a b l e s and 33% u s i n g S t e e l e and Board's method. Use of B a l l ' s e q u a t i o n method r e s u l t e d i n the l a r g e s t e r r o r s (up t o 4 4 % f o r g=5 C°). Stumbo's method appears t o be the most a c c u r a t e f o r m u l a method of tho s e t e s t e d . However, when g was v e r y s m a l l , t h i s method o v e r e s t i m a t e d the p r o c e s s l e t h a l i t y s l i g h t l y , i n some c a s e s . Stumbo's method a l s o r e s u l t e d i n h i g h l y v a r i a b l e a c c u r a c y when g was l a r g e ( F i g u r e 1 1 ) . 4 . C a l c u l a t i o n E r r o r s i n Terms of P r o c e s s i n g Time S i n c e a l l of the methods u n d e r e s t i m a t e d t h e l e t h a l i t y of t h e r m a l p r o c e s s e s f o r c o n d u c t i o n h e a t i n g f o o d s , t h e p r o c e s s t i m e s c a l c u l a t e d u s i n g t h e s e methods would be l o n g e r than r e q u i r e d t o a c h i e v e a s p e c i f i e d t a r g e t p r o c e s s l e t h a l i t y . F i g u r e 14 shows t h a t B a l l ' s method o v e r e s t i m a t e d r e q u i r e d p r o c e s s t i m e s by 6 t o 7 min f o r a p r o c e s s of one hour or more. A l t h o u g h t h i s f i g u r e shows d a t a f o r o n l y one can s i z e and i n i t i a l t e m p e r a t u r e , e r r o r s were s i m i l a r f o r o t h e r c o n d i t i o n s . Stumbo's method o v e r e s t i m a t e d r e q u i r e d p r o c e s s t i m e s by o n l y about 2 min, which i s a p p r o a c h i n g the a c c u r a c y of p r o c e s s c o n t r o l f o r manually o p e r a t e d r e t o r t s . A l t h o u g h t h e s e e r r o r s c o u l d be c o n s i d e r e d t o be e x t r a s a f e t y f a c t o r s , t h e y a re i n f l u e n c e d by can shape and Heat ing t i me, m i n F i g u r e 14. C a l c u l a t e d l e t h a l e f f e c t r e l a t i v e t o p r o c e s s i n g t i m e . (H/D=1.0, Ih=95 C ° ) 56 p r o c e s s i n g c o n d i t i o n s and t h e r e f o r e do not p r o v i d e a c o n s t a n t margin f o r e r r o r . S a f e t y margins a r e c e r t a i n l y n e c e s s a r y , but s h o u l d be w e l l d e f i n e d , t o h e l p a s s u r e p r o d u c t s a f e t y . S t a t i s t i c a l a n a l y s i s of v a r i a b i l i t y , as s u g g e s t e d by Lund (1978) may be a r e a s o n a b l e approach t o t h i s problem. 5. P o s i t i o n of C o l d Spot i n C o n t a i n e r For c o n d u c t i o n h e a t i n g p r o d u c t s , the c o o l i n g p o r t i o n of a t h e r m a l p r o c e s s can account f o r a s i g n i f i c a n t p a r t of the d e l i v e r e d l e t h a l i t y of a p r o c e s s (Board e t a l . , 1960). The c o o l i n g p o r t i o n i s most s i g n i f i c a n t f o r s h o r t e r p r o c e s s e s which r e s u l t i n l a r g e r g v a l u e s . For economic r e a s o n s , i t i s i m p o r t a n t t o t a k e the l e t h a l i t y of the c o o l i n g p o r t i o n i n t o account when d e t e r m i n i n g r e q u i r e d t h e r m a l p r o c e s s e s . However, i f t h i s approach i s t a k e n , i t may be n e c e s s a r y t o d e t e r m i n e the p r o c e s s f o r a " c r i t i c a l p o i n t " which i s not a t the c e n t e r of the c o n t a i n e r . The p o s i t i o n of the c r i t i c a l p o i n t i n the c o n t a i n e r , which r e c e i v e s the l e a s t l e t h a l e f f e c t from a p r o c e s s , depends on v a r i o u s f a c t o r s i n c l u d i n g the n a t u r e of the h e a t i n g and c o o l i n g program and the shape of the c o n t a i n e r ( F l a m b e r t and D e l t o u r , 1972a). A l t h o u g h t h i s p o i n t may be d i s p l a c e d c o n s i d e r a b l y from the c e n t e r of the c o n t a i n e r , the d i f f e r e n c e between th e l e t h a l i t y d e l i v e r e d a t t h i s p o i n t and a t the c e n t e r i s u s u a l l y s m a l l ( H i c k s , 1951). 57 6 . C o n v e c t i o n H e a t i n g P r o d u c t s I t must be noted t h a t the r e s u l t s r e p o r t e d here a p p l y t o foods t h a t heat and c o o l by c o n d u c t i o n o n l y . P r o d u c t s or p r o c e s s i n g methods f o r which n a t u r a l or f o r c e d c o n v e c t i o n a r e s i g n i f i c a n t f a c t o r s i n heat t r a n s f e r w i l l e x p e r i e n c e a s h o r t e n e d l a g b e f o r e c o o l i n g b e g i n s . Thus the f o r m u l a methods can be e x p e c t e d t o e s t i m a t e l e t h a l i t y more a c c u r a t e l y f o r c o n v e c t i o n h e a t i n g p r o d u c t s than f o r c o n d u c t i o n h e a t i n g p r o d u c t s . In o r d e r t o a c h i e v e s i m i l a r a c c u r a c y i n p r o c e s s e s t i m a t i o n f o r both c o n d u c t i o n and c o n v e c t i o n h e a t i n g p r o d u c t s , d i f f e r e n t e v a l u a t i o n methods may be r e q u i r e d . C. G e n e r a l C o n s i d e r a t i o n s and F u t u r e R e s e a r c h Needs Some of the most s i g n i f i c a n t f a c t o r s i n p r o c e s s c a l c u l a t i o n u s i n g f o r m u l a methods may be the d e f i n i t i o n of the parameter g and the c a l c u l a t i o n of f ^ , f , and j • The una c c o m p l i s h e d t e m p e r a t u r e d i f f e r e n c e , g, i s o f t e n d e f i n e d as the d i f f e r e n c e i n t e m p e r a t u r e between the r e t o r t and the h i g h e s t temperature reached a t the c e n t e r of the c o n t a i n e r i n q u e s t i o n . However, i t i s a l s o o f t e n d e s c r i b e d as the tem p e r a t u r e d i f f e r e n c e a t t h e time steam i s t u r n e d o f f . In some s i t u a t i o n s , n o t a b l y w i t h c o n v e c t i o n h e a t i n g p r o d u c t s , t h e s e two d e f i n i t i o n s may c o i n c i d e , but i n most c a s e s , the tem p e r a t u r e a t the c e n t e r of a c o n t a i n e r w i l l c o n t i n u e t o r i s e f o r some time a f t e r the steam i s t u r n e d o f f due t o the temp e r a t u r e g r a d i e n t from the s u r f a c e t o the c e n t e r of the 58 c o n t a i n e r . In b a t c h - t y p e p r o c e s s i n g s i t u a t i o n s , t h e r e i s a l s o a time l a g between when the steam i s t u r n e d o f f and the c o n t a i n e r i s immersed i n c o o l i n g w a t e r . The d e l a y r e s u l t s from the time r e q u i r e d t o blow down t h e r e t o r t and t o a l l o w the r e t o r t t o f i l l w i t h water. D u r i n g t h i s t i m e , t h e r e i s o f t e n s t i l l s i g n i f i c a n t heat t r a n s f e r i n t o the c o n t a i n e r from i t s s u r r o u n d i n g s . Board e t a l . ( i 9 6 0 ) have sug g e s t e d t h a t much of t h i s time can be c o n s i d e r e d e q u i v a l e n t t o a c o n t i n u a t i o n of the p r o c e s s a t r e t o r t t e m p e r a t u r e i n many c a s e s , w i t h o u t i n t r o d u c t i o n of s i g n i f i c a n t e r r o r . I n c a s e s where t h e r e i s a s i g n i f i c a n t l a g a t the s t a r t of c o o l i n g , e i t h e r due t o a temperature g r a d i e n t w i t h i n the c o n t a i n e r , or due t o a d e l a y i n the e s t a b l i s h m e n t of c o o l i n g c o n d i t i o n s , i t i s c l e a r t h a t the d e f i n i t i o n of g i s ambiguous. F u r t h e r m o r e , the d e f i n i t i o n s of the h e a t i n g r a t e i n d e x , f h , and t h e c o o l i n g r a t e i n d e x , f c , a r e not unambiguously d e f i n e d . T h e o r e t i c a l l l y , f i s d e f i n e d as the n e g a t i v e i n v e r s e s l o p e of the asymptote t o t h e h e a t i n g or c o o l i n g c u r v e . However, i n p r a c t i c e , f i s c a l c u l a t e d as the n e g a t i v e i n v e r s e s l o p e of a l i n e f i t t e d t o the " l i n e a r " p o r t i o n of the c u r v e . A l t h o u g h the d i f f e r e n c e may not be l a r g e when l o n g p r o c e s s e s and l o n g c o o l i n g p e r i o d s a r e c o n s i d e r e d , s i n c e the " l i n e a r " p o r t i o n of the c u r v e approaches more c l o s e l y the asymptote as the p r o c e s s goes on, i n most p r a c t i c a l s i t u a t i o n s h e a t i n g p e r i o d s a re s h o r t enough t o r e s u l t i n c o n s i d e r a b l e d i s c r e p a n c i e s . C o o l i n g 59 r e c o r d s a r e r a r e l y l o n g enough t o g i v e a r e a s o n a b l e e s t i m a t e of f c and any d i s c r e p a n c y i n the d e t e r m i n a t i o n of f c i s t r a n s l a t e d a l s o i n t o an e r r o r i n the c a l c u l a t i o n of D cc* The parameter, j C c r a s w e l l as b e i n g v e r y s e n s i t i v e t o e r r o r s i n o t h e r c a l c u l a t e d p a r a m e t e r s , i s i t s e l f not a v e r y a p p r o p r i a t e d e s c r i p t o r of the i n i t i a l p o r t i o n of the c o o l i n g c u r v e . A l t h o u g h i t does l o c a t e the p o s i t i o n of the " l i n e a r " p o r t i o n of t h e c o o l i n g c u r v e , i t does not a d e q u a t e l y d e s c r i b e t h e n a t u r e of the c u r v i l i n e a r p o r t i o n of t h e c o o l i n g c u r v e , s i n c e any number of unique c u r v e s c o u l d be drawn, a l l h a v i n g t h e same v a l u e s f o r g, f c and j c c but d i f f e r e n t l e t h a l e f f e c t s . The same i s t r u e f o r the h e a t i n g c u r v e , but i s not so i m p o r t a n t s i n c e l i t t l e l e t h a l e f f e c t i s d e l i v e r e d i n the i n i t i a l p a r t o f t h e p r o c e s s . However, the most i m p o r t a n t p a r t s of a p r o c e s s , p a r t i c u l a r l y f o r c o n d u c t i o n h e a t i n g f o o d s , a r e the end of t h e h e a t i n g c y c l e and the s t a r t of t h e c o o l i n g c y c l e , s i n c e i t i s d u r i n g t h i s p e r i o d t h a t t h e l e t h a l r a t e i s h i g h e s t . I t i s c l e a r from the above d i s c u s s i o n t h a t r e s e a r c h e f f o r t s h o u l d be d i r e c t e d towards the development of a p p r o p r i a t e p r o c e d u r e s f o r the d e t e r m i n a t i o n of t h e p a r a m e t e r s g, f h , f c , and j c c r and perhaps t o the i n t r o d u c t i o n of a n o t h e r parameter as a d e s c r i p t o r of the c u r v i l i n e a r p o r t i o n of the c o o l i n g c u r v e . U n t i l t h i s r e s e a r c h i s s a t i s f a c t o r i l y c o m p l e t e d , f u r t h e r comparison of t h e r m a l p r o c e s s c a l c u l a t i o n methods i s not w a r r a n t e d . 6 0 CONCLUSIONS The n u m e r i c a l g e n e r a l method u s i n g t r a p e z o i d a l i n t e g r a t i o n of l e t h a l h i s t o r y d a t a d e v e l o p e d by P a t a s h n i k (1953) was found t o agree w e l l w i t h the r e f e r e n c e method when one minute time i n t e r v a l s were used. However, use of l a r g e r time i n t e r v a l s r e s u l t e d i n g r e a t e r e r r o r , e s p e c i a l l y f o r p r o d u c t s which heat and c o o l r a p i d l y . The f i v e f o r m u l a methods f o r d e t e r m i n i n g p r o c e s s l e t h a l i t y showed d e v i a t i o n s from the r e f e r e n c e method f o r c o n d u c t i o n h e a t i n g foods i n c y l i n d r i c a l c o n t a i n e r s under a wide range of c o n d i t i o n s . D e v i a t i o n s were g r e a t e s t when g v a l u e s were l a r g e , as might be e n c o u n t e r e d i n h i g h r e t o r t t e m p e r a t u r e p r o c e s s i n g . E r r o r s a l s o v a r i e d a c c o r d i n g t o the shape of t h e can, w i t h the g r e a t e s t e r r o r s c o r r e s p o n d i n g t o H/D c l o s e t o u n i t y . Can s i z e , t h e r m a l d i f f u s i v i t y , h e a t i n g r a t e , i n i t i a l t e m perature d i f f e r e n c e and r e t o r t t e m p e r a t u r e d i d not g r e a t l y a f f e c t e r r o r magnitudes. Under a l l c o n d i t i o n s examined, Stumbo's method was found t o g i v e the be s t e s t i m a t e s of p r o c e s s l e t h a l i t y , but the method was more s e n s i t i v e t o s l i g h t v a r i a t i o n s i n f h and j c c than were the o t h e r methods. Use of t h e s e f o r m u l a methods t o c a l c u l a t e p r o c e s s t i m e s would r e s u l t i n s l i g h t o v e r e s t i m a t e s of r e q u i r e d p r o c e s s i n g t i m e f o r c o n d u c t i o n h e a t i n g p r o d u c t s . T h i s o v e r p r o c e s s i n g r e p r e s e n t s an e x t r a s a f e t y m a r g i n , but c o u l d a l s o be s i g n i f i c a n t i n terms of a d d i t i o n a l energy use and reduced p l a n t t h r o u g h p u t . 61 REFERENCES B a l l , C O . 1923. Thermal p r o c e s s time f o r canned f o o d . B u l l . 7-1 (37) N a t ' l . Res. C o u n c i l , Washington, DC. B a l l , C O . 1928. M a t h e m a t i c a l S o l u t i o n of Problems on Thermal P r o c e s s i n g of Canned Foods. U n i v . C a l i f . P u b l . P u b l i c H e a l t h 1, 230pp. B a l l , C O . , and O l s o n , F.C.W. 1957. " S t e r i l i z a t i o n i n Food Technology", M c G r a w - H i l l Book Company, I n c . New Y o r k , NY. 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