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1/5

F r e e z i n g t i m e p r e d i c ti o n s f o r r e g u la r sh a p e d f o o d s a s i m p l if ie d

g r a p h i c a l m e t h o d

V . O . S a l v a d o r i , R . O . R e y n o s o , A . d e M i c h e l i s a n d R . H . M a s c h e r o n i

C e n t r o d e I n v e s ti g a c i d n y D e s s a r r o l l o e n C r i o t e c n o l o g l a d e A l im e n t o s ( C I D C A ) , F a c u l t a d d e

C i e n c i a s E x a c t a s 4 7 y 1 1 6 ( 1 9 0 0 ) , L a P l a t a , A r g e n t i n a

R e c e i v e d 2 1 N o v e m b e r 1 9 8 6 : r ev i se d 6 M a r c h 1 9 8 7

A graphical meth od is propo sed for the es t imation o f f reezing t imes of foods with a high w ater content . I t has

been developed f rom the predict ions of a numer ical model which solves the heat balance for a food

und ergo ing freezing. The met ho d is valid for foods with different shapes (f iat plate, cylind er and sphere) and

covers a wide range o f working condit ions for industr ia l f reezers . I t a lso enables the predict ion to be mad e for

any given end tem perature in the therm al centre of the food. Freezing t imes predicted by this m etho d have

been com pared w ith published exper imental data , giving an average er ror in the predicted values o f only 4 9/0.

Keyworfls: reezing; r eezing im es; foo d products)

P r b v i s i o n s d u t e m p s d e c o n g b l a t i o n p o u r l e s a l i m e n t s d e f o r m e

r g u li r e m b t h o d e g r a p h i q u e s i m p l i f ir e

On propose une mdth ode graphique p our I 'estimation des temps de conodlation des alime nts d forte teneur en eau.

Cette mdthode est obtenue d partir d 'un moddle num~rique donnant le bilan thermique du produit en coms de

congdlation. O n peut appliquer ce tte mdthode d des alimen ts de diffdrentes fo rm es (p laqu e plane, cylindre et

sphbre). Elle se rap porte ~ diverses conditions du fon ctionn eme nt des congblateurs du shiels et pe rme t de prbvoir

une temperature finale donnde au centre thermique. En outre, on a compard les temps calculds avec les temps

exp~ rime ntaux publies , e t on a trouv~ une difference mo yenne de 4 seulemen t.

(Mots clds: congdlation; emp s de con gdlation; aliments)

N o m e n c l a t u r e

B i B i o t n u m b e r ( = h L / k o )

C C o n s t a n t i n t h e d e f i n i ti o n o f X

C po H e a t c a p a c i t y o f u n f r o z e n f o o d ( J k g - ~ ° C - 1)

e P e r c e n t a g e e r r o r o f p r e d i c t e d t f (9 /o )

F o F o u r i e r n u m b e r ( = tCtoL- 2)

h H e a t t r a n s f e r c o e f f ic i e n t ( W m - 2 ° C - ~ )

ko T h e r m a l c o n d u c t i v i t y o f u n f r o z e n f o o d

W m - 1 o f - 1 )

L F o o d d i m e n s i o n ( h a lf t h i ck n e s s o f s l ab ,

r a d i u s o f s p h e r e o r c y l i n d e r ) ( m )

m , n C o n s t a n t s i n t h e d e f i n i t i o n o f X

t T im e ( s)

t f F r ee z in g t im e ( s)

T T e m p e r a t u r e ( °C )

T c T e m p e r a t u r e o f t h e t h e r m a l c e n t r e ( °C )

Tcr I n i t i a l f r e e z i n g t e m p e r a t u r e ( ° C )

T i I n i t i a l f o o d t e m p e r a t u r e ( ° C )

T~ C o o l i n g m e d i u m t e m p e r a t u r e ( °C )

X C h a r a c t e r i s t i c v a r i a b l e ( - )

~to T h e r m a l d i ff u si v it y o f u n f r o z e n f o o d

m 2 s - l )

P o D e n s i t y o f u n f r o z e n f o o d ( k g m - 3 )

D u r i n g t h e l a s t f e w y e a r s t h e r e h a s b e e n a n o t a b l e i n c r e a s e

i n p u b l i s h e d l i t e r a t u r e r e f e r r i n g t o f r e e z i n g t i m e

p r e d i c t i o n s f o r f o o d s . T h e f o l l o w i n g t w o d i f f e r e n t

a p p r o a c h e s h a v e b e e n u s e d :

1 . n u m e r i c a l m e t h o d s : t h e se a r e b a s e d o n t h e s o l u t i o n o f

t h e d i f f e re n t ia l th e r m a l b a l a n c e f o r t h e f o o d t o b e f r o z e n .

T h e y a l s o a l l o w f o r v a r i a t i o n s i n t h e t h e r m o p h y s i c a l

p r o p e r t i e s o f t h e f o o d w i t h t e m p e r a t u r e . I n g e n e r a l t h e y

r e q u i r e c o m p l e x c a l c u l a t io n s a n d a p o w e r f u l c o m p u t e r .

W i t h t h e s e m e t h o d s f r e e z i n g t i m e , t f, as w e l l a s

t e m p e r a t u r e p r o f il e s a s fu n c t i o n s o f t im e c a n b e p r e d i c t e d .

I t i s a l s o p o s s i b l e t o s t u d y t h e i n f l u e n c e o f e a c h i n d i v i d u a l

p r o c e s s p a r a m e t e r o n f r e e z in g ti m e r - 9 ; a n d

2 . a p p r o x i m a t e m e t h o d s : t h e s e a r e d e v e l o p e d t o

c a l c u l a t e f r e e z i n g t im e s i n a r e l a t i v e l y s i m p l e w a y , w h i l s t

a c c o u n t i n g f o r t h e i n f l u en c e o f w o r k i n g c o n d i t i o n s ( in i ti a l

a n d f i n a l f o o d t e m p e r a t u r e , h e a t t r a n s f e r c o e f f i c i e n t s ,

r e f r i g e r a n t t e m p e r a t u r e , c o m p o s i t i o n , s i z e a n d s h a p e o f

t h e f o o d ) . I n g e n e r a l t h e s e m e t h o d s a r e e x p r e s s e d a s a

c o m b i n a t i o n o f g r a p h s a n d m a t h e m a t i c a l f o r m u l a e o r a s a

ser ie s o f f o r m ula e r eq u i r ed to ach ieve g oo d pr ec i s ion 1°- 22 .

I t h a s b e e n a d e q u a t e l y p r o v e d a n d a c c e p t e d 17 t h a t ,

e v e n u s i n g t h e m o s t p r e c i s e a n d d e t a i le d m o d e l l i n g , th e

a c c u r a c y o f p r e d i c t i o n i s li m i t e d d u e t o t h e l a c k o f p r e c i s e

e x p e r i m e n t a l d a t a o r p r e d i c t i o n e q u a t i o n s f o r t h e

t h e r m o p h y s i c a l p r o p e r t i e s o v e r t h e r e q u i r e d t e m p e r a t u r e

r a n g e . T h u s , s t a n d a r d d e v i a t i o n s f r o m 2 . 6 t o 5 . 6 ~

b e t w e e n c a l c u l a te d a n d e x p e r i m e n t a l f r e e zi n g ti m e s a r e

0140-7007/87/060357-05503.00

©

1987 Butterw orth Co

Publishers)

Ltd and IIR Re v I n t F ro id 1 9 8 7 Vo l 1 0 No v e m b re 5 7

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F reez ing t ime p red ic t i ons V O Sa lvador i e t a l .

o b t a i n e d , d e p e n d i n g o n t h e f o o d s h a p e a n d c a l c u l a t i o n

m e t h o d u s e d .

A s w i l l b e s h o w n i n t h e p r e s e n t w o r k , a g o o d

a p p r o x i m a t e m o d e l c a n g i v e r e s u lt s o f si m i la r a c c u r a c y

10

o

D

¢ J

-10

-15

-20

25 t ~

~

:l

~.+~,~_ ~_--:

:+7. ~ ,

~ ~ ,

~ ¢ ~ ,

i i

1

\

\ \

\ \ \

I 3 I /4 1 i

F o

2 01 5 . . . . . . .. ~ . . . = ' = ' ' = , =

o S ~ c

3 N ':: .-~......... ' ,

\ - ; . - . . . • . . . .. . , . . , =

~ . , . . . . , ~

\ '~... ~,

,

- - 2 C I Z Io [ ~ 0 [ I 0 I ~ 0 - -

X (d imens ionless)

F i g u r e 1 Temperat ures during freezing at the centre of a slab of meat

cooled with heat flow perpendicular to fibres. (a) Independent variable:

dimensionless time, F o. (b) Independent variable: characteristic

parameter, X. Freezing conditions:- - -,

B i

= 1, Ti = 7, Tf = - 25; . . . . . ,

B i = 2 ,

T~=7, 7~=-45;

. . . . . , B i = 5 ,

Ti=7, 7~=35; -- +- -+ ,

Bi=13, 7]=25, Tf= -40 ; , Bi=13, 7]=15, Tf =- 40;

-- o-- o-- , Bi=13, Ti=2 , 7~= -40 ; . . .. . , Bi=35, ~= 7, 7~= -35

Figure 1

Evolu t ion de la tem p atur e au cours de la conqdlation d 'une

plaque de v iande refroidie avec f lu x therm ique perpendiculaire au x f ibres .

(a) Variable ind@endante: temps sans dimension,

Fo. (b)

Variable

indbpendante: par am re carac ter is t ique , X . Con di t ions de congblation:

, Bi=l, T+=7,

T y - ~ - 2 5 ;

. . . . . , Bi=2 ,

T ~ = 7 , T f = - 4 5 ;

. . . . . , Bi=5,

T i = 7 , T f = - 3 5 ,

-- + + , Bi=13,

T i = 2 5 ,

Ty=- 40; . . . . . . . ,Bi - 13 , T~=15, Ty=- 40; - -o o -- , Bi=13,

T~=2, Tf=

- 4 0 ;

..... , Bi=35, T~=7,

T y = - 3 5

w i t h m u c h l e s s c a l c u l a t i o n . I n t h i s P a p e r , a m e t h o d i s

p r o p o s e d t o e s t i m a t e f r e e z in g t i m e s f o r a n y p r a c t i c a l

c o n d i t i o n i n i n d u s t r i a l f r e e z i n g . I t i s b a s e d o n a

g e n e r a l i z e d g r a p h w h i c h g i v e s t e m p e r a t u r e v a r i a t i o n i n

t h e f o o d a s a f u n c t i o n o f o n l y o n e d i m e n s i o n l es s v a r i a b l e

X . T h e m e t h o d a l s o a c c o u n t s f o r p ro c e s s c o n d i t i o n s a n d

p h y s i ca l p r o p e r t i e s o f t h e u n f r o z e n p r o d u c t .

roposed method

I n p r e v i o u s s t u d i e s 4 ,8 .~ 0 n u m e r i c a l m o d e l s w e r e

d e v e l o p e d w h i c h a l l o w e d p r e d i c t i o n o f t e m p e r a t u r e

p r o f i l e s a t a n y t i m e , i n a d d i t i o n t o p r e d i c t i n g f r e e z i n g

t i m e s f o r d i f f e r e n t g e o m e t r i e s . T o e v a l u a t e t r i t is n e c e s s a r y

t o k n o w t h e v a r i a t i o n w i t h t i m e o f t h e t e m p e r a t u r e a t t h e

t h e r m a l c e n t r e o f t h e f o o d , T ~; t h i s c o i n c i d e s w i t h t h e

g e o m e t r i c a l c e n t r e f o r r e g u l a r s h a p e s.

T h e b a si s o f t h e p r o p o s e d m o d e l is to f i n d a n e w

i n d e p e n d e n t v a r i a b l e w h i c h m a y a d e q u a t e l y a c c o u n t f o r

t h e s i m u l t a n e o u s i n f lu e n c e o f ti m e , p r o c e s s p a r a m e t e r s ,

Tj, Tf, Bi, f o o d p r o p e r t i e s , S o, a n d d i m e n s i o n s , L , o n t h e

v a r i a t i o n i n t e m p e r a t u r e o f th e t h e r m a l c e n t r e , T~. T h u s ,

a l l pos s ib le work ing condi t ions (any T i , T~ , Bi , ~o o r L )

m u s t l e a d t o s i m i l a r c u r v e s o f T~ v e r s u s X . T h i s v a r i a b l e i s

d e f i n e d a s :

F o [ T r -

T c r ) / T c r ]

X - 1 /B i +

C ) [ T c r - T i ) /T c r ] n

( l )

w h e r e T c r i s t h e i n i t i a l f r e e z i n g t e m p e r a t u r e ( w h i c h f o r

m e a t w a s e s t i m a t e d a s T +r = - I ° C ) a n d C , m a n d n a r e

e m p i r i c a l c o n s t a n t s w h o s e v a l u e s a r e d e p e n d e n t o n t h e

f o o d a n d i t s g e o m e t r y . T h i s d e f i n i t io n o f X a r i se s f r o m t h e

e x i s t e n c e o f s e v e r a l r e g r e s s i o n s f o r t h e c a l c u l a t i o n o f

f r e e z i n g t i m e o f t h e f o l l o w i n g t y p e l ° 2 2 :

F o = A 1 / B i + C ) [ T c r - T i) /T c r] n[ T f -

T c r ) / T c r ] - m

(2)

w h i c h , w h e n r e a r r a n g e d , l e a d t o a n e q u a t i o n s i m i l a r t o

E q u a t i o n ( 1 ) .

T h e t h r e e c o n s t a n t s w e r e a d j u s t ed t o g i v e t h e l o w e r

s p r e a d i n t h e n u m e r i c a l v a l u e s o f T~ v e r s u s X (see F i g u r e

l b ) , i n t h e f i n a l f r e e z i n g z o n e . T h e r a n g e s o f p r o c e s s

p a r a m e t e r s , Bi, T~ a n d T r, c o v e r e d i n t h e o p t i m i z a t i o n o f

t h e m e t h o d a r e s h o w n i n

T able 1

f o r t h e g e o m e t r i e s

s t u d i e d . I n a l l c a s e s , t h e r m a l p r o p e r t i e s o f b e e f w e r e u s e d

f o r t h e f o o d .

Table 1 Extreme ranges of processing conditions used in the definition of the metho d and values of empirical consta nts used in the calculation of X

Tableau 1

Dom aines ex trem es de condi t ions de t rai tem ent ut i li s~es dans la dbf init ion d e la re bode e t des valeurs des constantes empir iques ut il i sbes dans le

calcul de X

Geometry and type of food

Range of freezing paramete rs

Value of

constants in X formula

7] (°C) Tf (°C) B i m n C

Flat plate (lean beef, heat flow perpen dicula r to fibres) 2-25

Flat plate (lean beef, heat flow parallel to fibres) 2-20

Infinite cylinder (lean beef, heat flow perpend icular to fibres) 7 25

Sphere (minced meat) 2-25

- 2 5 - 4 5 1 - 5 0 1 . 0 4 0 . 0 9 0 . 1 8

- 2 5 - 4 5 1 - 6 0 1 . 0 3 0 . 1 0 0 . 1 6

- 2 ~ -

40 1-30

1 . 0 0 0 . 0 9 0 . 1 7

- 25- - 45 1-20 0.90 0.06 0.18

3 5 8 I nt J R e f ri g 1 9 8 7 V o l 1 0 N o v e m b e r

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Freezing time predictions: V O Salvadori

et al

e s u l t s a n d d i s c u s s i o n

F i g u r e l a

s h o w s p r e d i c t e d t i m e - t e m p e r a t u r e c u r v e s ,

c a l c u l a t e d w i t h t h e n u m e r i c a l m o d e l , f o r t h e t h e r m a l

c e n t r e o f a fl a t p la t e o f b e e f w h i c h i s f r o z e n w i t h h e a t f l o w

i n a d i r e c t i o n p e r p e n d i c u l a r t o th e f ib r e s . E a c h c u r v e

r e p r e s e n t s t h e o u t p u t o f o n e r u n o f t h e c o m p u t e r p r o g r a m

o f t h e m o d e l . T h e s e t h e r m a l h i s t o r i e s c o r r e s p o n d t o v e r y

d i f f e r e n t f r e e z i n g c o n d i t i o n s , c a l c u l a t e d u s i n g t h e m o s t

e x t r e m e v a l u e s o f Ti , T f a n d

B i

f o u n d i n i n d u s t r i a l f r e e z i n g

p r o c e s s e s .

F i g u r e l b

s h o w s t h e s a m e c u r v e s p l o t t e d i n

t e r m s o f t h e n e w v a r i a b l e , X . I n t h e t e m p e r a t u r e r a n g e i n

w h i c h t h e p r o d u c t c a n b e c o n s i d e r e d a s p a r t i a l l y o r

w h o l l y f r o z e n , t h e m a x i m u m s e p a r a t i o n b e t w e e n c u r v e s

( in t e r m s o f X ) i s o n l y 1 2 ~ .

I n

F i g u r e 2

t h e r e g i o n f r o m - 2 t o - 1 8 ° C is s h o w n f o r

t h e s a m ~ p r o d u c t a n d w o r k i n g c o n d i t i o n s a s i n

F i g u r e 1 .

T h e c u r v e s e l e c t e d f o r t h e c a lc u l a t i o n s i s s h o w n ; t h i s w a s

u s e d t o a v e r a g e t h e v a r i a t i o n r a n g e o f X f o r e a c h

t e m p e r a t u r e . I t s h o u l d b e n o t e d t h a t t h i s m e t h o d a l l o w s

o n e t o e s t i m a t e t t f o r a n y f i n a l T o , w h e n m o s t o f t h e

a p p r o x i m a t e m e t h o d s a r e o n l y v a l i d fo r a s e t f i n a l

t e m p e r a t u r e , g e n e r a l l y - 1 0 o r - 1 8 ° C .

I n

T a b l e 2

e x p e r i m e n t a l a n d t h e o r e t i c a l ( n u m e r i c a l )

v a l u e s o f tr a r e c o m p a r e d t o t h e e v a l u a t i o n m a d e w i t h t h e

p r o p o s e d m e t h o d f o r d i f f e r e n t f o o d s w i t h fl a t p l a t e

g e o m e t r y . T h e a v e r a g e p e r c e n t a g e e r r o r f o r a l l t h e

p u b l i s h e d T y l o s e 11 r e s u l t s i s 6 . 9 ~ o , w h i l s t f o r t h o s e

p u b l i s h e d i n R e f e r e n c e 1 9 t h e e r r o r i s 2 . 9 ~ o. A s c a n b e

o b s e r v e d p r e d i c t i o n i s v e r y g o o d , e v e n f o r n o n - b e e f

p r o d u c t s . T h e s a m e c u r v e i s v a l i d f o r s u b s t a n c e s w i t h h i g h

w a t e r c o n t e n t a n d s t r u c t u r e s w h i c h a r e n o t v e r y re g u l a r i n

t h e d i r e c t i o n o f h e a t f l u x ( f or e x a m p l e , m i n c e d m e a t s ,

Table 2 C o m p a r i s o n o f p u b l i s h e d f re e z i n g t i m e s w i t h t h o s e p r e d i c t e d b y t h i s w o r k f o r a f la t p la t e w i t h h e a t f lo w p e r p e n d i c u l a r t o t h e f ib r e s

T a b l e a u 2

Com paraison des temps de congblation publibs avec ceux prbvus d' aprbs cette ude pour une plaque plane avec flu x thermique perpendiculaire

aux fibres

L h ~ 7~ Tc tf,exp

t f p r e d

E r r o r

M a t e r i a l ( m ) ( W m - 2 ° C - ' )

B i

( °C ) ( °C ) ( °C ) (h ) ( h ) (%)

L e a n b e e f 1 6 0 .0 3 1 6 8.5 1 0 .5 3 1 8 .3 -3 8 . 7 -1 8 1 .3 0 1 .2 9 -0 .8 6

k o = 0 . 4 8 W m - 1 ° C - 1 0 . 0 3 1 46 .9 9 . 1 8 1 8 . 0 - 3 3 . 3 - 1 8 1 . 52 1 . 59 4 . 5 0

~ o = 1 .3 11 x 1 0 - 7 m 2 s - t 0 . 03 8 8 .1 5 . 51 1 7. 8 - 3 9 . 9 - 1 8 1 . 57 1 . 6 4 4 . 2 2

0 .03 72 .2 4 .51 22 .0 - 27 .3 - 18 2 .47 2 .78 12 .63

0 .047 120.5 11 .80 8 .0 - 36 .5 - 18 2 .83 3 .03 6 .97

M u t t o n 1 ° 0 .0 3 7 9 .5 4 .8 2 1 0 .6 - 3 7 .0 - 1 8 1 .9 0 1 .9 2 1 .2 8

k o = 0 . 4 9 5 W m - 1 ° C - 1 0 . 03 9 9 . 5 8 6 . 0 4 9 . 6 - 4 0 . 1 5 - 1 8 1 . 47 1 . 56 6 . 1 3

c t o = 1 . 24 x 1 0 - 7 m 2 s - I 0 . 0 3 1 2 0 .5 3 7 . 3 0 1 0. 3 - 3 3 . 1 5 - 1 8 1 . 92 1 .7 7 - 7 . 9 8

0 .0 3 1 0 2 .8 9 6 .2 4 9 .4 - 3 9 .9 - 1 8 1 .65 1 .5 4 -6 .4 3

M i n c e d m e a t 11 0 . 0 2 4 25 2 1 . 6 1 . 19 3 .0 - 2 8 . 4 - 1 0 3 . 3 0 3 . 4 2 3 . 8 0

k o = 0 . 4 4 W m - 1 ° C - 1 0 . 03 6 2 2 0 . 0 1 8 . 00 2 8 . 0 - 2 5 . 0 - 1 0 2 . 3 2 2 . 3 8 2 . 6 5

~ o = 1 .3 0 2 x 1 0 - ~ m z s - 1 0 .0 3 6 a 51 .9 4 .25 3 .0 - 23 .9 - 10 3 .84 3 .69 - 3 .94

0 .0 2 4 2 5 9 0 .0 4 .9 6 3 0 .0 - 2 5 .4 - 1 0 1 .6 8 1.7 3 3 .0 5

0 .0125 30 .6 0 .87 28 .6 - 25 .7 - 10 1 .54 1 .58 2 .31

0 .0125 a 16 .7 0 .47 16 .5 - 28 .8 - 10 2 .34 2 .31 - 1 .42

M a s h e d p o t a t o 1 1' 1° 0 . 0 2 4 2 5 9 0 . 0 4 . 1 2 1 8 .3 - 2 6 . 7 - 1 0 1 . 58 1 . 57 - 0 . 5 4

k o = 0 .5 3 W m - 1 oC - 1 0 .0 2 4 2 5 2 1 .6 0 .9 9 1 5 .0 - 2 5 .0 - 1 0 4 .4 8 4 .6 7 4 .2 6

~ = 1 . 44 8 x 1 0 - 7 m 2 s - 1 0 . 0 2 2 1 0 6 .0 4 . 4 1 6 .8 - 2 5 . 5 - 1 8 1 . 47 1 . 4 4 - 1 . 7 3

0 .0 2 3 6 9 .0 2 .9 9 4 .0 - 3 0 .6 - 1 8 1 .5 0 1 .4 6 - 2 .5 7

0 . 0 36 9 0 . 0 6 . 11 2 8 . 0 - 2 4 . 3 - 1 0 3 . 1 2 3 . 2 2 3 . 2 7

0 .036 a 51 .9 3 .53 11 .8 - 24 .9 - 10 3 .72 3 .93 5 .64

0 .0125 30 .6 0 .72 28 .4 - 25 .9 - 10 1 .76 1 .66 - 5 .85

0 .0125 13 .6 0 .32 4 .7 - 25 .9 - 10 3 .02 3 .01 - 0 .28

T y l o s e t l ( s e l ec t ed v a l u e s ) 0 .0 3 6 5 1 .9 3 .0 6 1 0 .0 - 4 0 .0 - 1 0 2 .2 2 2 .2 5 1 .4 9

k o = 0 . 6 1 W m - I ° C - x 0 .0 3 6 4 1 0 .0 2 4 . 20 3 . 0 - 2 2 . 0 - 1 0 1 .9 2 1 .7 0 - 1 1 . 2 7

= 1 .6 4 4 x 1 0 - 7 m 2 s - ~ 0 .0 3 6 5 1 .9 3 .0 6 3 0 .0 - 2 0 .0 - 1 0 4 .8 0 5 .2 0 8 .4 4

0 .0 3 6 2 1 .6 1 .2 7 1 0 .0 - 4 0 .0 - 1 0 4 .0 2 4 .2 8 6 .5 8

0 . 0 2 4 2 5 a 4 1 0 . 0 1 6 . 30 1 1 . 0 - 2 1 . 0 - 1 0 1 . 00 0 . 9 8 - 2 . 1 0

0 .0 2 4 2 5 3 0 .6 1 .2 2 2 0 .0 - 3 0 .0 - 1 0 2 .7 4 2 .9 0 5 .7 6

0 .0 2 4 2 5 3 0 .6 1 .2 2 2 0 .0 - 3 0 .0 - 1 0 2 .6 6 2 .9 0 9 .0 2

0 .0 1 2 5 a 4 3 0 .0 8 .81 3 .0 - 1 9 .0 - 1 0 0 .3 4 0 .3 2 - 5 .9 6

0 . 0 1 2 5 a 5 1 . 9 1 . 0 6 3 0 . 0 - 4 0 . 0 - 1 0 0 . 6 4 0 . 6 6 2 . 8 9

0 . 0 1 2 5 2 1 . 6 0 . 4 4 3 0 . 0 - 4 0 . 0 - 1 0 1 . 42 1 . 44 1 . 2 4

0 . 0 1 2 5 1 6. 7 0 . 3 4 2 8 . 7 - 2 6 . 0 - 1 0 2 . 6 8 2 . 8 9 7 . 9 9

C a r p 1 9 0 . 0 2 4 6 8 .5 3 . 4 3 3 . 9 - 2 8 . 7 - 1 8 1 .6 5 1 . 68 2 . 1 8

k o = 0 . 4 8 W m - 1 o C -1

c t o = 1 . 3 4 1 x 1 0 - ~ m 2 s - 1

T y l o s e 1 9 ( s e le c t e d v a l u e s ) 0 . 0 1 45 1 0 8 .0 2 . 8 5 3 . 5 - 2 9 . 3 - 1 8 0 . 5 5 0 . 61 1 0 . 5 0

k o = 0 . 5 5 W m - 1 ° C - 1 0 . 0 0 6 1 0 8 .0 1 . 18 1 6. 8 - 3 0 . 9 - 1 8 0 . 2 2 0 . 2 2 0 . 0 0

= 1 .4 8 2 x 1 0 - 7 m 2 s - 1 0 .0 0 5 5 1 0 6 .0 1 .0 6 4 .4 - 2 5 .7 - 1 8 0 .2 1 0 .2 2 3 .2 0

0 .013 5 68 .0 1 .67 17 .4 - 25 .2 - 18 1 .00 1 .03 3 .25

N u m e r i c a l m e t h o d 0 . 03 6 1 4 .2 5 1 . 0 7 . 0 - 2 5 . 0 - 1 8 1 2 .6 3 1 2 .1 7 - 3 . 6 6

E x t r e m e v a l u e s o f p r o c e s s c o n d i t i o n s 0 . 0 3 6 1 4 .2 5 1 . 0 7 . 0 - 4 5 . 0 - 1 8 6 . 2 9 6 . 4 8 2 . 9 9

k o = 0 . 5 1 3 W i n - 1 ° C - 1 0 . 03 6 1 8 5. 9 6 1 3 .0 5 2 . 0 - 4 0 . 0 - 1 8 1 .4 7 1 .4 6 - 0 . 5 3

= 1 .2 6 8 x 1 0 - 7 m 2 s - 1 0 .0 3 6 1 8 5 .9 6 1 3 .0 5 2 5 .0 - 4 0 .0 - 1 8 1 .7 6 1 .7 8 0 .9 0

0 .0 3 6 4 9 0 .9 1 3 4 .4 5 7 .0 - 3 5 .0 - 1 8 1 .4 4 1 .5 4 .2 0

° P a r a m e t e r s o u t s i d e t h e r a n g e g i v e n i n

Table 1

Re v I n t F ro id 1 9 8 7 Vo l 1 0 No v e m b re 5 9

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Free zing t ime predictions. V. O. Sa lvad or i et al

T a b l e

3 Co mp arison of published freezing t imes with those predic ted by this work for a fla t pla te with heat f low para l le l to the f ibres

Ta ble a u 3

Comparaison des temps de congklation publibs avec ceux prbvus d aprbs cette btude pour une plaque plane avec flux thermique parallble aux

fibres

L h Ti Tff Tc tf,exp tf,pred E rr o r

M ater ia l (m) (W m - 2 °C - ~)

Bi

(°C) (°C) (°C) (h) (h) ( )

Le a n be e f19 0.024 106.0 4.99 17.4 - 25.2 - 18 1.81 1.65 - 8.99

k o = 0 . 5 1 W m 1 o C - 1 0 .0 2 35 6 9 .0 3 .1 8 4 .3 - 2 7 . 6 - 1 8 1 .7 5 1 . 6 7 - 4 . 8 4

~o = 1.39 7 x 10 7m2 s 1

Num e r ic a l m e thod 0 .036 15 .56 1 .00 7 .0 -2 5 - 18 11 .55 11 .19 -3 .1 2

Extrem e values of process con dit ion s 0.036 777.78 50.00 7.0 - 25 - 18 1.77 1.74 - 1.91

0.036 535.89 34.45 7.0 - 45 - 18 0 .94 0.98 3.89

0.036 535.89 34.45 2.0 - 40 - 18 1.02 1.00 - 1.72

0.036 77.78 5.00 20 .0 - 45 - 18 2.01 2.05 1.90

c o r r e s p o n d i n g t o p r o d u c t s w i t h d i ff e re n c es in w a t e r

c o n t e n t , h o m o g e n e i t y , s t r u c t u r e , e t c . , w e r e u s e d . T h e

m a x i m u m e r r o r i n t h e c a l c u l a t i o n o f tf w a s 1 3 , t h e

o v e r a l l m e a n e r r o r w a s 4 . 1 , a n d 9 0 o f t h e p o i n t s l a y

w i t h i n a n e r r o r o f - 8 . 0 t o + 8 .2 . T h e s e r e s u l t s i n d i c a t e

t h a t t h i s m e t h o d h a s a v e r y s a t i s fa c t o r y p r e c i s i o n .

T h e f o l l o w i n g is a p r a c t i c a l g u i d e t o t h e s t e p s r e q u i r e d

t o u s e th i s m e t h o d :

1 . t h e p r o d u c t i s d e f i n e d a n d i ts t h e r m a l p r o p e r t i e s i n

t h e u n f r o z e n s t a t e a r e o b t a i n e d f r o m t h e l i t e ra t u r e o r b y

m e a s u r e m e n t ;

2 . o n c e t h e g e o m e t r y i s g i v e n , th e d e s i r e d f i n a l

t e m p e r a t u r e i s s e le c t ed f o r t h e t h e r m a l c e n t r e ;

3 . w o r k i n g c o n d i t i o n s a r e d e f i n e d ( T~ , T f a n d

Bi);

4 . t h e X v a l u e is o b t a i n e d f r o m t h e c o r r e s p o n d i n g

c u r v e ; a n d

5 . tf i s c a l c u l a t e d , r e a r r a n g i n g E q u a t i o n ( 1) t o o b t a i n :

t f

X 1/Bi + C ) [ T c r - T i ) / T c r ] n [ T f - T c r) /T c r - m 0 ~o 1 L2

3 )

T h u s , i t c a n b e s e e n t h a t t h i s m e t h o d c o m b i n e s

s i m p l i c i t y w i t h a d e g r e e o f p r e c i s io n c o m p a r a b l e t o t h a t o f

m u c h m o r e i n v o l v e d a p p r o x i m a t e o r ev e n n u m e r i c a l

m e t h o d s . I n a d d i t i o n , o n l y o n e g r a p h is n e c e s s a ry f o r a

w i d e v a r i e t y o f f o o d s h a v i n g t h e s a m e g e o m e t r y o r fo r a n y

f i na l t e m p e r a t u r e .

F i n a l l y , i t s h o u l d b e n o t e d t h a t a l t h o u g h v a l u e s o f m , n

a n d C w e r e o b t a i n e d f o r t h e r a n g e s o f p r o c e s s p a r a m e t e r s

s p e c i f i e d i n

Table 1,

w o r k i n g o u t s i d e t h o s e l i m it s w il l n o t

l e a d t o h i g h e r e r r o r s i n t h e e s t i m a t i o n o f t f, a s s h o w n b y

s o m e o f t h e d a t a ( m a r k e d a ) i n

Table 2.

R e f e r e n c e s

1 B o n a c in a C . C o m i n i G . O n the so lu t ion o f the non- l ine a r he a t

c onduc t ion e qua t ions by num e r ic a l m e thods

Int J Heat Mass

Transfer

(1973) 16 581-589

2 F le m ing, A . K . The Ne w Ze a la nd a pproa c h to m e a t f re e zing

Meat Freezing Why and How

MR I Sym pos ium No . 3 (Ed C . L .

Cutl ing ) (1974) 24.01-24.13

3 Joshi , C . , Ta o , L . C . A num e r ic a l m e thod of s im ula t ing the

axisymmetr ica l f reez ing of food systems

J Food Sci

(1974) 3 9

6 2 3 ~ 2 6

4 Mascheroni , R. H. Transferencia de ca lor con s imult /meo cam bio

de fase en la congelac ibn de productos cb. rneos

PhD Thesis

Unive r s ida d Na c iona l de L a P la ta , Arge nt ina (1977)

5 Hsieh, R. C. , Lerew, L. E~ Heldm an, D. R. Predic t ion of f reez ing

times for foods as inf luenced by p roduc t p ropert ies

J Food Proc

Eng

(1977) 1 183-197

6 S ch w a rtzb erg H . G . R o s en a u J . R . H e i g h t J . R .

The pre dic t ion

of freezing and thaw ing temp era tures

versus

t im e be ha viour

through the use of effec t ive heat capacity equation

IIR

Commissions C1 and C2

Ka r l s ruhe , F RG (1977) 311-318

7 Rebellato, L., De l Giudice, S., Cnmini, G. Fin ite elem ent analysis

of freezing processes in foodstuffs

J Food Sci

(1978) 43 239-243

8 Mascheroni , R. H. , Caivelo, A. Rela t ionsh ip between heat

transfer parameters and the character is t ic damage var iables for

the freezing of beef

Meat Science

(1980) 4 267-285

9 Purwadaria , H. K. , Heldm an, D. R. A f ini te e lement mo del for

pre dic t ion of f re e zing r a te s in food prod uc t s w i th a nom a lous

shapes

ASAE Trans

(1982) 25 827-832

10 Bazim, H. C. , Mascheroni , R. H. Hea t t ransfer with s imultaneo us

c ha nge of pha se in f r e ez ing bo ne d m ut ton

Latin American J Heat

Mass Transfer

(1984) 8 5 5-76

11 Cle land, A. C. , Earle , R. L. A com pariso n of analyt ica l and

num erica l method s for predic t ing freezing t imes of foods J Food

Sci

(1977) 42 1390-1395

12 Cle land, A. C. , Earle , R. L. Predic t ing freezing t imes of foods in

rec tangular packages

J Food Sci

(1979) 44 964-970

13 Cle land, A. C. , Earle , R. L. A com parison of meth ods for

predic t ing the freezing t imes of cyl indrica l and spherica l

foodstuffs

J Food Sci

(1979) 44 958-963

14 Cle land, A. C. , Earle , R. L. Freezin g t ime predic t ion for foods - a

s implif ied procedure

Int J Refrig

(1982) 5 134-140

15 Mascheroni, R . H., Calvelo, A. A simplified mo del for freezing

t im e c a lc u la t ions in foods

J Food Sci

(1982) 47 1201-1207

16 de Michelis , A. , Calvelo, A. Ma them atica l mod els for non -

symm etr ic f reezing of beef

J Food Sci

(1982) 47 1211-1217

17 Cieland A. C. Earle R. L . Cleland D . J . Th e effect of freezing

ra te on th e accuracy of numerica l f reez ing t ime predic t ions

Int J

Refrig

(1982) 5 294-301

18 de Michelis , A. , Calvelo, A. Freezin g t ime predic t ions for br ick

and cylindrical-shaped foo ds

J Food Sci

(1983) 48 909-913

19 Hung , Y. C. , Thompson, D. R. Freezing t ime predic t ion for s lab

shape foodstuffs by an improved analyt ica l method

J Food Sci

(1983) 48 555-560

20 Pha m , Q . T . Exte ns ion to P la nk ' s e qua t ion for pre dic ting

freezing times of foodstuffs of simple shapes

Int J Refrig

(1984) 7

377-383

21 de Michel i s A. Mascheroni R. H. C alvelo A. Es t im a c ion de

t i e m pos de c onge la c i6n de produ c tos c / i rne os La

Alimentacibn

Latinoamericana

(1985) 20(151) 52 ~ 6

22 Baz in, H. C. , Flores , E. S. , Mascheroni , R. H. Freezing t ime of

s labs of boneless o r mince d meats . Measurem ent and pred ic t ion

for different types of freezers

J Food Techn

subm i t t e d for

publ ic a t ion

Rev In t F ro id 198 7 Vo l 10 Nov em bre 36