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HEAT TRANSFER DURING FREEZING OF FOODS AND PREDI CTI ON OF FREEZI NG THJES

A th e s i s pre s e n t e d in part i al ful f i lm en t of t h e re qui r em e n t s f o r the d e gre e of Do c t or o f Ph i l o s ophy in Bi o t e c hn o l ogy at M a s s e y Un ive r s i ty.

AN DREW CHARLE S CLELAND

1 9 7 7

HEAT TRANSFER DUR I NG FREE ZI N G OF FO ODS AND PREDI CTI O N OF FREEZING TD!ES

ABSTRA CT

A study o f m e thod s for pr e d i c t i n g the fre e z i n g t im e o f f ood s wa s m ad e . F our shap e s - fin i t e s l ab s , cy l i n d er s , s ph e r e s and r e c tan gul ar br i cks we r e c on s i d ere d .

For e ac h shap� e xperim e n t al m e asurement s o f the fr e e z ­ i n g t i m e were m ad e over a wi d e range o f c ond i t i on s u s in g Karl sruhe t e s t sub s t an c e , a d e f i n e d an alogu e mat e ri al . Exp e r im e nt s vri th slabs o f m in c e d l e an b e e f and m a sh e d p o t a t o we re al s o c ondu c t e d .

Prac t i c al food fre e z ing pro bl em s , wh ere t h e m a t e r i al i s i n i t i al ly sup erh e a t ed ab ove i t s fr e e zing p o i n t an d t h e t h i rd k i n d o f bound ary c ond i t i on

(

c onve c t ive c o o l in g

)

^{i s}

a p pl i ed , have n o t be e n s o lved an al y t i c ally be c au s e o f t h e n on - l in ear b ound ary c on d i t i o n s . Fo o d mat e r i al s whe n fre e z ­ i ng r e l e as e l at e n t h e at over a r ange o f temp e rature whi ch fur t h e r c om p l i cate s any at tempt at s o luti o n.

The a c c uracy o f t h e var i ou s s o lut i on s t o the fr e e z i n g p r o b l em pro p o s e d in t h e l i t e ratur e was evaluat e d b y c om ­ p ar i s o n o f the various c al c ulat e d fr e e z ing t im e s wi th t h e e x p erim e nt al ly d e t e rm i n e d value s ove r a t o tal o f

1 87

fre e z ine experiment s.

Fo r th o s e s o lut i on s r e quiring nume ri c al e valuat i on , t h e b e s t wa s found t o b e a thre e -l e vel fin i t e d i fferen c e s c h e m e whi c h g en erally gave a p re d i c t i on o f the fr e e z in g t im e t o wi thin

29%

o f the e xpe r i me n t al value s wi th

9 5%

c on fi d e n c e . With the r e gular g e ometri c shap e s inv e s t i ga t e d f i n i t e e l em e n t s have n o advan t ag e over fin i t e d i fferen c e s a n d w ere not c on s i d e r e d .

For the exi s t in g e xa c t an d approxim ate an alyt i c al s olut i on s , i t i s shown that th e s e d o not give ac cura t e

p r e d i c t i o n o f th e fre e z ing t im e for any of t h e four shape s ,

mai n ly b e c aus e all but t wo of th e s e solut i o n s d o not t ake ac c oun t of in i t i al sup erh e at in the mat e ri al to be fro z en, and t h e s e t wo s olution s are for i n i t i al sup e rheat in a s e m i -i n fi n i te s lab, n o t a fin i t e slab .

For the exi sting empiri c ally m o d i f i e d s olu t i on s and em p i r i c al relat i on sh i p s , i t i s shown that t h e y do not give ac curate pre d i c t i o n of fre e z i n g t im e ; at b e s t the 9 5%

c on f i d e n c e li mi t s of the p e r c e n t age d i ffe r e n c e b etwe en th e c alcula t e d and experim en t al fr e e z i n g tim e s ar e O% t o + 2 0%

f o r s l ab s , cylinde rs ·and sphere s .

All s olu t i ons for fr e e z i n g of r e c t an gular bri ck s wi th t h e third k i nd of bound ary c on d i t i on use the ge o m e tri c fac t or s d e ri v e d by Pl ank . Th e s e fac t ors are shown to be s ubje c t to e rror, the e rror i n c r e a s ing as the rat i o s of the t wo l arger d im en s i ons to the sm alle s t i n c r e a s e .

A group o f formulae i s propos e d wh i ch ar e s imple to u s e and g i ve ac curat e pr e d i c t i on of fre ez i n g t im e . They m o d i fy the g e om e tric fac t o r s i n Pl ank ' s e q�ation , t ak i n g i n i t i al sup e rheat i n t o ac c oun t , and i n th e c a s e o f r e c t ­ an gular bri c k s c orre c t t h e e rr o r s inhe rent i n the s e ge o ­ m e t r i c fac t ors . Thi s group o f s im ple form ulae are shown t o p r e d i c t the fr e e z ing t im e wi th 9 5% c onfi d e n c e t o wi thin 5%

o f t h e expe r i mental val u e s for s labs, to wi t hin

7%

^{fo r}

c yli n d e r s and spher e s , and to wi thin 1 0% for r e c t an gul ar b r i ck s .

The pr e d i c t i on ac c urac y of the s imple formulae and the t hree le vel fin ite d i ffe r e n c e s ch e m e ar e s i m i lar but the s im ple formulae can b e c alc ulat e d qui ckly wi thout the us e of a c om pu t e r whi ch i s a b i g ad vantage . I n ad d i t i on t o s im pli c i ty and ac curacy t h e s i m p l e formulae are als o v e r s ­ a t i l e , and by use o f s u i t ab le approximat i on s c an handle s om e prac t i c al problem s i n whi ch c on d i t i on s chan g e wi th t im e .

ACKNOWLEDGEMENTS

I

wi sh t o acknowl edge the fol l owi ng:-

- Prof e s sor R . L . Earl e for hi s sup e rvi sion an d as s i s t an c e .

- Dr . S . H . Ri c h e r t and Dr . R . H . Vi l l e t for th e i r sup er­

vi s i on .

- Hoe c h s t N e w Ze aland· Lim i t e d for suppl yi n g m e thy l c e l lu­

l os e s ampl e s .

- Dr . I.F. Boa.g for valuab l e as s i s t an c e in the s t at i s t i c al an alys i s of d at a .

- M r . J.T. Al g e r and Mr . D . W . Coul i ng for t h e i r a s s i s t an c e i n bui l d in g an d main t aining e qu i pm ent .

- Ros em ary for moral support and for proof r e ad i ng .

1

2 2 . 1 2 . 1

^•

1 2 . 1

^•

2 2 . 1 . 3 2 . 2 2 . 2 . 1 2 . 2 . 2 2 . 2 . 2 . 1 2 . 2 . 2 . 2

2 . 2 . 2 . 3 2 . 2 . 2 . 4 2 . 2 . 3 2 . 2 . 4 2 . 2 . 5 2o 2 . 6 2 . 3

2 . 4 3

COHT ENT S ABS TRACT�

A CKN OWLEDG El'-IENT S ^• CONT EN T S .

L I S T OF FIGURES.

LI ST OF TABLES.

I N TRODUCT I ON.

LITER ATURE REVI EW.

Probl em De fini t i on .

De fin i t i on of fr e e z ing t i m e . Boun d ary c on d i t i on s .

I n i t i al c on d i ti on s .

Solution s U s in g t h e A s sumpt i on of a Un i que Fr e e z i n g T e mp e ratur e .

Exac t s ol u t i on s for s l ab s.

Approxim at e s olution s for s la b s . I n t e gral profi l e an d

vari at i on al t e chn i qu e s . Solu t i on s for all oy s ol i d i fi c ation.

Oth e r an al y t i c al approa c h e s .

2 4 5 1 0

1 4 1 6

1 8 1 8 1 8 1 9 20

2 1 2 1 2 1 2 1

2 2 2 3

Solution s for aqu e ous sy s t em s .

2 3

Sol u t i on s for other shap e s .

23

Empi r i c al r e l a t i on ships.

24

Us e of analogu e s.

Num er i c a l s ol ut i on s.

Solution s U s ing Changing Apparent Spe c i fi c H e at Cap ac i ty and Vary i n g Th erm al Con du c t i vi ty.

Summ ary .

PRELHTINARY CON S I DERATIONS

2 5 2 6

26

28

30

4 4 . 1 4 . 2 4 . 3 4 . 3 . 1 4 . 3 . 2 4 . 3 . 3 4 . 3 . 4

4 . 3 . 5 4 . 4

4 . 4 . 1 4 . 4 . 2 4 . 4 . 3 4 . 4 . 4 4 . 4 . 5 4 . 5

4 . 5 . 1 4 . 5 . 2 4 . 5 . 3 4 . 5 . 4 4 . 5 . 5

5 5 . 1 5 . 2 5 . 3 5 . 4

COLLECTI ON OF EXPERH;EI\TAL DATA .

3 2

I n troduc t i on .

3 2

Choice of Freez ing Mat er i al.

3 2

One-Dimen s i on al Heat Tran sfer i n Slabs.

3 4

The equi pmen t.

3 4

Di m en s i on al m easur ement and

con t rol .

3 5

Tem perature measuremen t and

control .

38

Measurem ent and c on t rol o f the

sur fac e heat transfer c oeff i c i ent .

39

Analysi s o f heat tran s fer i n s lab s .

4 1

Rad i al Heat Trans fer i n Cyl inders and Spheres.

The equi pmen t .

Di men s i on al measurement an d control .

Tem perature measuremen t and con t rol .

Measurement an d c on t rol of the

4 5 4 5

5 5 5 5

sur fac e heat tran s fer c oef f i c i en t .

5 5

Anal y s i s of heat t ran s fer i n

rad i al geometry .

58

Three-Dim en s i onal Heat Tran s fer i n

Rec tangular Bri c ks.

6 5

The equ i pm en t.

6 5

Di m en s i onal neasuremen t and

con t rol .

6 6

Tem perature m easuremen t and

con trol .

67

Measurem en t and con t rol of the

sur face heat t rans fer coef f i c i en t .

6 7

Analy s i s of heat tran s fer i n rec t an gular bri c ks .

EXPERIMEN TAL DES I GN AND RESULTS.

I n trodu c t i on . Slabs.

Cylinder s and Spheres.

Rec tangul ar Bri c k s .

6 8

7 1

7 1

7 1

7 3

74

6

6 . 1 6

^•

1

^•

1 6 . 1

^•

2 6 . 1 . 3

6 . 2 6 . 2 . 1 6 . 2 . 2 6 . 2 . 3

6 . 3 6 .3 . 1 6 .3 . 2 6 .3 .3

6 . 4 7

7 . 1 7 . 2 7 . 2 . 1 7 . 2 . 2 7 . 2 . 3

7 . 2 . 4 7 . 3 7 . 3 . 1 7 . 3 . 2

PREDI C TI ON OF FREEZI NG TD·iE BY NUNER I CAL JVIETHODS

S l ab s .

Se l e c t i on o f n um eri c al m ethod . Se l e c t i o n o f f i n i t e d i f f e r en c e appro ach .

Compar i s on o f f i n i t e d i f f e r e n c e s ch em e s .

Cylinders and Sph e r e s .

Se l e c t i o n o f n um erical m eth o d . S�l e ct i o n o f f i n i t e d i ff er e n c e appr o ach .

C o mpari s o n o f f i n i t e d i f f e r en c e s ch em e s .

R e c tangu l ar Bri ck s .

Se l e c t i o n o f n umeri cal m e th o d . S e l e ct i o n o f f i n i t e d i ff e r e nc e m e tho d .

Us e o f the fin i t e diff e r e n c e s c h em e .

Summary .

PREDI CT I ON OF FREE Z I NG TH·IE BY S I JV1PLE FORNULAE .

Th ermal Dat a . S l abs .

S o lut i on s for th e first k i n d o f boun d ar y c on d i t i on .

Solut i o n s f o r th e third k i n d o f bound ary c o nd i t i on .

Empi r i c al m o d i fi c at i o n s and formul ae .

Pr e s ent d e ve l o pm ents . Cyl inder s and Sph e r e s .

Solut i o n s for th e fi rst k i n d o f b oun d ar y c on d i ti on .

Solut i o n s f o r th e thi rd k i n d o f b ound ar y c on d i t i on .

93 9 3 9 3 9 5

9 6 1 05 1 05 1 06

1 06 1 1 2 1 1 2 1 1 2

1 13 1 1 9

1 2 1

1 2 1

1 2 1

1 23

1 23

1 28

1 29

1 3 1

1 3 1

1 34

7 . 3 . 3

7 . 3 . 3 . 1 7 . 3 . 3 . 2 7 . 3 . 4 7 . 4 7 . 4 . 1 7 . 4 . 2 8

9

9 . 1 9 . 2 9 . 3 9 . 4

9 . 5 9 . 6 9 . 7 1 0

1

2

3 4 5

Empi r i c al modifi c ation s and

formul ae .

1 3 4

Cyl i n d e r s .

1 34

S ph e r e s .

1 3 5

Pr e s e n t d evelopm en t s .

1 36

Re c t angul ar Bri c k s .

1 3 7

Exi s t i n g formul ae .

1 3 7

Pr e s e n t d eve lopm ents .

1 4 1

COJV:PARI SON OF NUTv:ERI CAL AND SI MPLE f,lETHODS

FOR PREDI CTING FREEZING TD,lE S .

1 4 5

PREDI CTICN OF FOOD FREE ZI NG TifviES FOR

S I TUATIONS WI TH NON-CONSTANT COND I TI ONS.

1 6 1

I n troduc t i on .

Vary ing Am b i ent T em perature . Non-Uni form I n i t i al Tem peratur e . Ch an �ing Sur fac e Heat Trans fer Coe ffi c i en t .

I r r e gul ar G e ome t ry.

Non -H om og e n e ou s Food Mat erial . Summ ary .

CONCLUSIONS . NOT-!EI\CLATURE. REFERENCES.

APPENDICES .

G e n e ral De s c ri pti on of Fini t e Di fferen c e Program s .

One -Dim en s i on al Fini t e Diffe r en c e Program s .

T wo- and Thr e e - Di m en s i onal Fi n i t e Di ff eren c e Program s .

Rad i al Fi n i t e Di ffe r e n c e Program . Deri vati on of M e l l or ' s Formul a .

1 6 1 1 6 1 1 6 2

1 6 3 1 6 4 1 6 5 1 6 5 1 6 7 1 69 1 7 2

1 8 5 1 94

204

2 1 7

2 2 2

6 7

I nve s t i gati on of Changing Ambi ent T em perature .

I nve s t i gati on o f Non-Uni form I n i t i al Temperature .

2 2 5

2 3 1

LI ST OF FI GURES

4 . 1 Schemati c outline o f the experim ental plat e fre e z er .

4 . 2 Test s l ab s .

4 . 3 Typi c al t emperature/ tim e profi l e s for

therm o c oupl e s placed at , or near , the surfac e o f a fre ezi�g slab .

4 . 4 Schem ati c outline o f the experim ental l i quid immers i on fre e z er .

4 . 5 The s arri?l e o s c i � lator us ed in liquid imm ersi on fre e z ing experiments

4 . 6 Schem at i c outline o f th e sy stem used to o s ci l l ate cylinders in the liquid imm ersi on fre e z er

4 . 7 Arrangement o f the polystyrene foam c aps and therm o c oupl e l e ad s for cylinders .

4 . 8 Insert i on o f therm o c oupl e s in the sphere s .

4 . 9 Schemati c outline of the experimental air blast free z er .

4 . 1 0 Typi c al fini t e di fferen c e re sults for fre e z ing of a sph ere .

4 . 1 1 Typi c al experimental r e sul ts for air blast free z ing o f a cylind er .

4 . 1 2 At tachment o f l i d s t o the plastic boxe s .

4 . 1 3 Typi c al polypropy l en e c o nt ainers used in the experim ental inve stigat i on into freezing o f rectangular bri ck s .

3 6 3 7

4 6

47

48

4 9

5 0 5 3

5 4

6 1

6 2 6 3

6 4

5 . 1

5 . 2

5 . 3

5 . 4

Typi c al t emperature curve s for fre e zing of a slab o f K arl sruhe t e s t sub s tanc e ( Run F 1 7 ) .

Typi c al t em perature curves for fre e zing of a slab o f K arl sruhe t e st sub s tance ( Run F 1 4 ) .

Typi c al t emperature curve s for fre e zing o f a cylinder o f Karl sruh e t e st sub s tan c e ( Run C3 ) .

Typi c al t emperature curve s for fre ezing o f a s phere o f K arl sruhe test subs tanc e ( Run 8 1 ) .

5 . 5 Typi c al temperature curves for fre ezing o f a rectangular bri ck o f K arl sruhe t e s t

sub s t an c e ( Run B7 2 ) . 5 . 6

5 . 7

Typi c al temperature curve s for fr e e z in g o f a rectangul ar bri ck o f Karlsruhe t e st

substan c e ( Run B5 5 ) .

Typi c al t emperature curves for freezing o f a rectangular bri ck o f Karl sruhe t e st

sub s t an c e ( Run B4 3 ) .

6 . 1 Therm al pro perty curves u s ed t o approximat e

8 6

8 7

88

89

90

9 1

9 2

fre e zing at a uniQue phas e change t emperatur e . 98

6 . 2 The fin i t e di fference gri d at the surfac e o f a fre e z ing slab .

6 . 3 Thermal conductivi ty dat a .

6 . 4 Apparent spe ci fi c heat c ap ac i ty data for K arlsruhe test substanc e .

6 . 5 Appare nt s p e c i fi c heat c apacity data for minced l ean b e e f .

6 . 6 Apparent spe cific heat c apacity data for mashed p otat o .

98 9 9

9 9

1 00

1 00

6 . 7 Frequen cy diagram o f the p erc entage di fferen c e s between experimental fre e zing tim e s for

slabs , and tim e s cal cul at e d by finite differen c e s .

6 . 8 Typi cal t emperature and therm al c onduc tivi ty profil e s through a fre e z in g s phere or cylind er shov1ing the effect of a l i n e ar approximation

1 04

o f the thermal c onductivi ty . 1 09

6 . 9 Frequency d i agr·am o f the p erc entage di fferen c e s betwe en experimental fre e z ing tim e s for

cyl ind ers , and tim e s cal culated by fin i t e differen c e s .

6 . 1 0 Frequenc y d iagram of the p erc entage di fferenc e s betwe en experim en tal fre e z ing tim e s for

spher e s , and tim e s calcul ated by fini t e di fferen c e s .

7 . 1 Fre quency di agram of the perc entage d i fferenc e s betw e en experimental fre e zing times for s lab s , and t i m e s c al cul ated by s o luti ons with the first kind o f b oundary c ondi ti on .

7 . 2 Fre quency di agram of the p ercentage di fferen c e s b e tween experimental fre e zing tim es for slab s , and tim e s cal culated by s ol uti ons with the third k ind of b oundary c ondi tion .

7 . 3 Schemat i c di agram showing the heat t o be rem oved in fre e zing of s l ab s .

7 . 4 Frequency d iagram o f the p er c entage differenc e s b etwe en experimental fre e zing t im e s for slab s , and t im e s c alcul ated by empiri c al m odifi c at i on s .and formulae .

1 1 0

1 1 0

1 2 5

1 2 5

1 26

1 27

7 . 5 Frequen cy di agram o f the perc entage d i fference s between experim en tal fre e z ing tim e s for

cylind ers , and tim e s calculated by vari ous m ethod s .

7 . 6 Frequency di agram o f the p erc entage d i fferen c e s b etwe en experim ental fre e z in g tim e s for

spher e s , and tim e s cal cul at ed by vari ous method s .

7 . 7 Plo t o f the ave�age predi c t i on error

(

wh en comparing calcul at ed fre e z ing tim e s t o

experim ental re sul t s for r e c tangular bri cks ) versus number o f e quival ent heat trans fer dimen s i on s .

8 . 1 Fre1uency di agram o f the p er c entage di fferen c e s betwe en experimental fre e z in g tim e s , and

cal cul ated fre e zing time s .

A 5 . 1 Schemat i c repre s entati on o f typi c al t emperature profi l e s within a free zing m at erial where

fre e z in g o ccur s at a uni qu e phas e change te�perature T f .

A6 . 1 Ambient t emperature profi l e s .

1 33

1 33

1 40

1 46

2 23

2 26

LI ST OF TABLES

5 . 1 Typi cal c on di t i on s in food fre e z ers .

5 . 2 De sign o f th e fact orial experiment for

inve stigat i on of the fre ezing t im e s of s l abs o f Karl sruhe test subs tance .

5 . 3 Experim en t al data for fre e z ing of slabs o f Karl sruhe t e st sub s t ance .

5 . 4 Experim en tal data for fre e z ing of min c ed lean beef ( M ) and mashed po tato ( P ) in slab s .

5 . 5 Exp erimental data for fre e z ing of cylind ers o f Karlsruhe test substan c e .

5 . 6 Exp erimental data for fre e zing of spheres of Karlsruh e t e s t substance .

5 . 7 Experimental d ata for fre e z ing of r e c t angul ar bricks o f Karl sruh e t e s t substance .

6 . 1 Compari s on o f resul t s from the thre e­

dimen si on al program t o a known analyt i c al soluti on for c o oling o f a cube .

6 . 2 R e sul t s from the fini te d i fference s imul at i on of the fre e zing o f rectangul ar bri ck s .

7 . 1 Thermal data for fre e z ing food materi al s .

7 .2 Compar i s on o f experimental freezing tim e s with t i m e s calculated b y Neumann ' s m e th o d .

7 . 3 Pre d i c t i on of fre e zing t im e s o f slabs o f min c e d lean b e e f and mashed potat o by e quati on s 7 . 4 and 7 . 5 .

7 6

7 7

7 8

80

8 1

8 2

8 3

1 1 6

1 1 7 1 2 2

1 24

1 3 2

7 . 4 M ean s and 9 5% confi d en c e lim i t s for th e applicabil i ty of e quat i on s 7 . 1 9 t o 7 . 27 t o the experimental dat a .

8 . 1 Experim ental and c al cul ated fre e zing tim e data for all shapes .

8 . 2 Means and standard d eviati ons o f the perc entag e di fferen c e s between experimen tal fre e zing

times , and �im e s cal c ulat e d from ( a ) the three-level fin�te di fferen c e s chem e , ( b ) e quations 7 . 1 9 to 7 . 27 .

8 . 3 C ompari s on of m etho d s for pred i c ti on o f food freezing t im e s .

A6 . 1 C ondi ti o n s used for the inve stigat ion o f the e ffect o f changing ambi en t t emperature on fre ezi.::1g tim e .

A6 . 2 Resul t s o f the fini t e d i fferen c e simulation o f fre e z ing of a s l ab sub j e c t t o a cycl ing ambi ent t emperature .

A6 . 3 Resul t s o f the fin i t e di fferen c e simul ation of fre e zing o f a slab sub j e c t to an exponential fall

L'1

runbi ent temperature .

A7 . 1 R e sul t s from the finite d i fferen ce simulat i on of fre e z ing of a slab o f K arlsruhe t e s t

1 44

1 5 1

1 58

1 5 9

2 28

2 2 9

230

substan c e wi th n on -uni form ini tial t em perature . 2 3 2