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Multiphase moisture transport in porous media under intensive microwave heating

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m u l t ip h a s e m o is t u r e t r a n s p o r t i n p o r o u s
MEDIA UNDER INTENSIVE MICROWAVE HEATING
A D issertation
resented to the Faculty o f the G raduate School
Cornell University
artial Fulfillment of the Requirements for the D egree of
Doctor of Philosophy
by
H aitao N i
Jan u ary 1997
Ih perm ission of the copyright owner Further reom H „w
reproduction prohibited without perm ission.
© H aitao N i 1997
ALL RIGHTS RESERVED
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BIOGRAPHICAL SKETCH
The au th o r w as b o m in Hefei, P. R. C hina, on June 5,1962. H e w as adm itted to the
Chinese U n iversity of Science and Technology in Septem ber o f 1979 and obtained
his B achelor's a n d M aster's degrees in T herm al Engineering, in July of 1984 and
in O ctober of 1987, respectively. A fterw ards, he h as served as a n assistan t profes­
sor in th at d ep artm en t. In Septem ber o f 1992, he w as adm itted to M echanical En­
gineering D ep artm en t at the Johns H opk in s U niversity as a Ph.D . stu d en t. Later
he transferred to th e D epartm ent of A gricultural an d Biological E ngineering at
Cornell U niversity in July of 1993.
iii
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To m y parents, wife, son an d daughter for th eir su p p o rt and patience
iv
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ACKNOWLEDGEMENTS
I w o u ld like to th an k m y m ajor ad visor P rofessor A shim K. D atta for bringing m e
into th is porou s m edia area, in w hich I had in terests fifteen years ago w hen I w as
in college. H is guidance, friend sh ip and especially, financial assistance through
th is stu d y w ill be alw ays ap preciated and rem em bered. I sh ould ad m it th at it is
m y great h o n o r to have Professor K enneth E. Torrance in m y com m ittee. H e is
very know ledgable an d 1 learn t a lot from him . H is patience m ade m e unforget­
table an d really sets m e a good exam ple in th e future. M y sincere g ratitu d e also
goes to Professor W illiam L. O lbricht for h is gu idin g, helping an d encouraging.
I w o uld also like to th an k th e C am pbell S oup C om pany for financially sup­
p o rtin g m e p art of the tim e d u rin g m y study. M y special th an ks go to the Cor­
nell T heory C enter for using th e ir sup ercom p uter an d to Professor C arlo M ontem agno for access to h is SGI w orkstation. W ithout th eir su p p o rt com putation
could n o t be finished a t th is tim e. Finally I also w an t to take th is o p p o rtu n ity to
th an k all m y labm ates: Steven Lobo, X iaolan Shi, M ontip C hanchong, and H ua
Z hang for th e ir h elp an d friend ship , and m y friend Dr. C hu-H ui C hen for his
valuable discussion a t th e in itial stage of code developm ent.
v
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Table of Contents
1 INTRODUCTION AND OBJECTIVES
1
2 LITERATURE REVIEW
4
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Sim ple M odels o f M oisture T ransport in
M icrow ave H e a tin g ................................................................................
M ore G eneralized M odels for M ultiphase T ransport in Porous m edia
A pplications of th e M ultiphase T ransport M odels to C onventional
H e a tin g ......................................................................................................
2.3.1 V ariations in the F orm ulations and S olution Techniques . .
2.3.2 Tem perature, M oisture, an d Pressure Profiles in C onventional
H e a tin g ..........................................................................................
A pplications of th e M ultiphase T ransport M odels to M icrow ave H eat­
in g ................................................................................................................
2.4.1 Tem perature, M oisture, an d Pressure Profiles in M icrow ave
H e a tin g ..........................................................................................
M ultiphase Porous M edia M odel in C om m ercial
CFD C o d e ...................................................................................................
M easurem ent in Food an d O th er M aterials d u rin g M icrow ave H eat­
in g ................................................................................................................
2.6.1 M oisture Loss in M icrow ave H e a tin g ...................................
2.6.2 M oisture Loss in M icrow ave-assisted C onvective H eating
Specific O b je c tiv e s...................................................................................
4
6
9
9
11
13
16
19
20
20
22
25
3 MATHEMATICAL FORMULATION OF MICROWAVE HEATING OF
POROUS MEDIA
27
3.1
3.2
3.3
A ssu m p tio n s.............................................................................................
E quilibrium State V ariables an d V apor P ressure .............................
3.2.1 Porosity, S aturation, an d C o n c e n tra tio n s .............................
3.2.2 Vapor P re s s u re .............................................................................
R ate L a w s ...................................................................................................
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27
31
31
35
36
3.4
3.5
3.6
3.7
3.8
G overning E quations fo r M ass an d Energy C o n s e rv a tio n
3.4.1 Vapor M ass C onservation E q u a tio n ......................................
3.4.2 W ater M ass C onservation E q u a tio n ......................................
3.4.3 A ir M ass C onservation E q u a tio n .............................................
3.4.4 Energy C onservation E q u a tio n ................................................
Initial C o n d itio n s.....................................................................................
B oundary C o n d itio n s ...........................................................................
3.6.1 Closed B oundary ......................................................................
3.6.2 O pen B o u n d a r y .........................................................................
Volum etric Energy A bsorption in M icrowave H e a tin g ..................
Sum m ary of G overning E quations, Initial C ondition an d Boundary
C o n d itio n s ................................................................................................
4 NUMERICAL SOLUTION AND INPUT PARAMETERS
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
G overning E quations an d B oundary C onditions in 1-D slab . . . .
D iscretization of th e G overning Equations .....................................
D iscretization of th e B oundary C o n d itio n s .....................................
M atrix A ssem bly an d S olution A lg o rith m .........................................
Convergence an d M esh R e fin e m e n t..................................................
Check for Energy a n d M ass C onservation of the N um erical M odel
Possible Existence of Sonic C ondition in
M icrow ave H e a tin g ................................................................................
4.7.1 Estim ation o f th e M axim um Mach N u m b e r..........................
4.7.2 N um erical C alculation of M ach N um ber an d
Reynolds N u m b e r ......................................................................
In p u t Param eters for th e M odel ........................................................
4.8.1 Equivalent P orosity an d Initial W ater S aturation of Porous
M a te ria ls ......................................................................................
4.8.2 Perm eability o f L iquid W ater and G a s ...................................
4.8.3 C apillary D iffusivity o f Liquid W a te r ...................................
4.8.4 Effective G as D iffusivity .........................................................
4.8.5 Sorption R e la tio n s h ip ...............................................................
4.8.6 Effective T herm al C onductivity and H eat C apacity . . . .
4.8.7 Thickness o f S a m p le ..................................................................
4.8.8 M icrow ave P en etratio n D e p t h ................................................
C onsideration of 2D G e o m e try ...........................................................
5 RESULTS AND DISCUSSION
5.1
40
41
42
43
44
46
47
47
47
52
54
56
56
58
60
62
65
67
68
68
73
75
76
77
84
91
93
94
95
96
96
99
Tem perature, P ressure, an d M oisture Profiles
in Convective H e a tin g ............................................................................
vii
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99
5.2
5.3
5.4
5.1.1 Low M oisture F o o d s ....................................................................
5.1.2 H igh-m oisture F o o d s ................................................................
5.1.3 C om parison of M odel w ith O ther N um erical S tudies . . .
T em perature, Pressure, a n d M oisture Profiles
in M icrow ave H e a tin g .............................................................................
5.2.1 L ow -m oisture F o o d s....................................................................
5 .2.2 H igh-m oisture F o o d s ................................................................
5.2.3 Foods Starting from a Spatially Varying M oisture C ontent
E xperim ental D ata on M icrow ave H e a tin g .......................................
S ensitivity A n a ly s is ................................................................................
5.4.1 C ontribution o f C onvection to Energy T ra n s p o rt.................
5.4.2 Effect o f Varying th e Thickness of S la b ....................................
5.4.3 Effect o f V ariation in the L iquid Intrinsic Perm eability of Very
Wet M a terials................................................................................
5.4.4 Effect o f V ariation in th e G as intrinsic P erm eability of Very
D ry M aterials .............................................................................
5.4.5 V ariations in Effective G as D iffu siv ity ....................................
5.4.6 Effect o f H eating R ate (M icrowave P ow er L e v e l).................
5.4.7 C om bined Effect o f Surface M ass and H eat T ransfer
C o efficien ts...................................................................................
5.4.8 C onsideration o f a 2D G e o m e try ..............................................
100
103
104
105
106
109
112
114
117
118
121
121
124
125
129
131
133
6
CONCLUSIONS
139
7
FUTURE WORK
141
A
ESTIMATION OF HEAT AND MASS TRANSFER COEFFICIENTS
142
B INPUT PARAMETERS FOR CONVECTIVE DRYING (Nasrallah et al.,
1988)
145
B.l
B.2
B.3
M aterial P roperties of C lay B ric k .........................................................
Initial value a n d bo un dary p a r a m e te r s ............................................
R e la tio n sh ip s............................................................................................
146
147
148
C GOVERNING EQUATIONS, INITIAL AND BOUNDARY CONDITIONS
IN 2D CYLINDER
150
C .l
C.2
G overning E q u a tio n s ............................................................................
In itial C ondition and B oundary C o n d itio n s......................................
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151
153
List of Tables
4.1
4.2
4.3
4.4
4.5
4.6
Param eters u sed in m esh convergence c h e c k ....................................
N um ber of n o d es an d space in c re m e n t..............................................
W ater loss conservation c h e c k .............................................................
Energy conservation c h e c k ....................................................................
Effect of perm eability on Ma and R e ....................................................
Porosity <t>an d saturation S w in the raw state of several com m on
foods. D ata o n m oisture content M an d density p from R ahm an
(1 9 9 5 ).........................................................................................................
Intrinsic perm eability d ata in porous m edia m o d e l .......................
Effective m oisture diffusivity in low m oisture ra n g e .......................
65
66
68
69
75
B oundary an d initial conditions in convective h eating .....................
B oundary an d in itial conditions in m icrow ave heating of low an d
high m oisture food (only properties different from Table 5.1 are
specified in th e table)...............................................................................
101
A .l M ass and h eat tran sfer c o e ffic ie n ts ....................................................
144
B .l
B.2
146
147
4.7
4.8
5.1
5.2
M aterial p ro p erties....................................................................................
Initial value an d boundary param eters.................................................
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78
79
89
108
List of Figures
3.1
3.2
3.3
3.4
4.1
42.
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
Schem atic of an u n satu rated poro us m ed ia.......................................
R epresentative elem entary volum e of poro us m edia.......................
Schem atic d iag ram of th e h eatin g process w ith the b o u nd ary con­
d itio n s..........................................................................................................
L iquid expulsion profile. D ata sh o w n here at 5s intervals...............
28
32
41
51
N odes d iag ram ...........................................................................................
58
M atrix stru ctu re..........................................................................................
63
Flow diag ram ............................................................................................
64
C onvergence check in 6 m inutes fo r different g rid sizes.................
67
H eating tim e taken to reach p ressu re 2.0 atm vs. different thick­
ness of m aterials........................................................................................
71
Pore velocity of v ap o r vs. different thickness of m aterials for d if­
72
ferent releasing tim e tr.............................................................................
Spatial d istrib u tio n of Ma a t differen t h eating tim e............................
73
Spatial d istrib u tio n o f Re a t d ifferent heatin g tim e............................
74
E quivalent porosity vs. ap p aren t d en sity a t different m oisture con­
ten t................................................................................................................
76
W ater satu ratio n vs. ap p aren t d en sity a t different m oisture content. 77
Total perm eability, calculated from in trin sic perm eability and rel­
83
ativ e perm eability, p lo tted ag ain st sa tu ratio n .....................................
Typical v ariatio n of capillary force as a function of liq uid satu ra­
tio n in a p o ro us m edia (Bear, 1972).......................................................
86
88
D iffusivity vs. liq u id saturatio n ............................................................
C apillary d iffusivity vs. m oisture co n ten t (d.b.)...............................
90
M oisture profiles after 1 h o u r of convective heating for different Dw. 91
Effective gas diffusivity vs. w ater satu ratio n .....................................
93
V apor pressure vs. m oisture content a t different tem peratures, as
show n by Eq. 4 .3 7 ....................................................................................
95
D ielectric p ro p erties an d p en etratio n d e p th vs. m oisture content. 97
x
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
T em perature, w ate r saturation, an d pressure profile, an d m oisture
loss in convective h eatin g of low m oisture potato. In p u t d ata from
Table 5.1...................................................................................................... 102
T em perature, w ate r saturation, an d pressure profile, an d m oisture
loss in convective heatin g of very w et potato. In p u t d ata from Ta­
ble 5.1........................................................................................................... 104
C om parison o f tem perature w ith N asrallah 's m odel........................ 105
C om parison of w ater saturation w ith N asrallah 's m odel................ 106
T em perature, w ater saturation, an d pressure profile, an d m oisture
loss in m icrow ave h eating of low m oisture potato. In pu t d ata from
Table 5.2...................................................................................................... 107
T em perature, w ater saturation, pressure, an d the loss of m oisture
in rapid m icrow ave heating of a high m oisture food. In p u t data
from Table 5.2. Increased m oisture loss after 3 m inutes is d u e to a
"pum ping effect" w hereby liquid w ater leaves the boundary w ith­
o u t being ev ap o rated ............................................................................... 110
T em perature, w ater saturation, pressure, an d the loss of m oisture
in m icrow ave heatin g of varying m oisture p o tato ............................. 113
C om parison of experim ental d ata and m odel prediction of m ois­
tu re loss in heatin g high m oisture food at 6 m inutes........................ 116
E xperim ental m oisture content history profiles for different initial
m oisture content in different oven in p u t po w er level....................... 117
C om parison of general profiles in low m oisture food w ith o u t con­
vection in energy equ ation (left) and w ith convection (right). . . . 119
C om parison o f general profiles in high m oisture food w ith o u t con­
vection in energy equ ation (left) and w ith convection (right) . . . 120
Effect of thickness on tem perature and w ater saturation ..............122
Effect of liq u id in trin sic perm eability of very w et m aterials k ^ . . 123
Effect of gas in trinsic perm eability of very d ry m aterials k ^ . . . . 126
Sensitivity of tem perature, pressure, satu ratio n and rate of m ois­
tu re loss to th e effective gas diffusivity for a low m oisture m aterial. 128
Effect of h eatin g rate on tem perature, pressure, saturation, an d to­
tal m oisture fo r a low m oisture food.................................................... 130
Effect of h eatin g rate on tem perature, pressure, saturation, and to­
tal m oisture for a high m oisture food................................................... 135
Effect of com bining effect of m ass and h eat tran sfer coefficients on
m axim um an d surface values................................................................ 136
Effect of hmv on m axim um and surface values................................ 137
Effect of h o n m axim um an d surface values.................................... 137
xi
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5.21 T em perature, w ater satu ratio n , an d pressure profile in m icrow ave
h eatin g of 2D cylindrical low m oisture foods a t 1 and 3 m inutes.
B.l
C .l
138
V apor pressure, v ap o r diffusivity and capillary diffusivity vs w a­
te r satu ratio n in clay brick. ..................................................................
149
Schem atic diagram o f 2D m odel...........................................................
151
xii
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List of Sym bols
a
Q-m
aw
b
C
°p
c
Cl —c10
D av
Deff,g
D eff
Dw
F
F0
h
h
b-mv
i
— ^18
k
k
kun
kgi
kgjkwr
J
J
M
M
Ma
sound velocity, ( m /s )
apparent d iffu siv ity coefficient due to capillary force, (k g /m s)
w ater activity
coefficients in fin ite difference equations
m olar density o f g a s m ixture
specific heat, (J / k g K)
m ass concentration, (kg / m3 total volum e)
coefficients in fin ite difference equations
binary diffusivity o f a ir and vapor, (m 2 / s)
effective gas d iffu siv ity in m oist m aterials, (m 2 / s)
effective d iffusivity in m oist m aterials, (m 2/ s)
capillary diffusivity, (m 2/s )
m icrow ave flux, (W / m 2)
m icrow ave flux o n th e surface, (W / m 2)
enthalpy, (J/k g )
h eat transfer coefficient on open boundary, (W /m 2 K)
vapor transfer coefficient on open boundary, (m /s )
volum etric ev ap o ratio n term , (k g /m 3 s)
coefficients in d ifferen tial equations
therm al conductivity, (W / m K)
intrinsic perm eability, (m 2)
liquid intrinsic p erm eab ility at very w et stage, (m 2)
gas intrinsic p erm eab ility at very d ry stage, (m 2)
gas relative perm eability, dim ensionless
w ater relative perm eability, dim ensionless
m ass flow on d ie bo undary, (k g /m 2)
m ass diffusive flu x , (k g /m 2 s)
m olecular w eight, (k g /k m o l)
m oisture content (d .b.) unless specified in context
M ach num ber
Xlll
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m
rh
n
n
P
p
pc
pa
Pm
q
q
Q
Qo
R
Rv
Ra
Re
V
v
S
T
th
tr
uav
x
Ax
m ass in representative elem en tary volum e
convective flux, (kg / m 2 s)
total flux, (k g /m 2 s)
ou tw ard vector norm al to th e surface
total pressure, (Pa)
partied pressure, (Pa)
capillary force, (Pa)
satu rated pressure of p u re w ater, (Pa)
gas pressure, (Pa)
volum etric flux, (m3 / m 2 s)
heat flow on the boundary, (W / m 2)
volum etric heat source term , (W / m 3)
m icrow ave surface flux, (W / m 2)
universal gas constant (J / km ol K)
v ap o r gas constant (J / kg K)
air gas constant (J / kg K)
R eynolds num ber, (uav\ / k / p )
volum e, m3
m ass flux, (k g /m 2 s)
saturatio n o r liquid sa tu ratio n
tem perature, (K)
heating tim e, (s)
releasing tim e of the v ap o r (chapter 4), (s)
average velocity of vapor, (m / s)
m olar fraction
m esh size of space
At
e
e*
e"
A
Ao
p
ps
Pd
tim e increm ent
volum e fraction
dielectric constant
dielectric loss
laten t of vapor, (J/k g )
m icrow ave w avelength in free space
intrinsic density, (kg/ m 3)
tru e density of solids, (k g /m 3)
density of solids in poro us m edia (k g /m 3 total volum e)
xiv
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(P9p)e//
Peff
<t>
a
T
P
6
effective h eat capacity o f th e m oist m aterials, (J/m 3)
a p p a re n t density of th e food, (k g /m 3)
p o ro sity
su rface tension, (N /m )
to rtu o sity
d y n am ic viscosity, (Pa s)
p en e tratio n d ep th of m icrow ave pow er, (m )
coefficient of Soret effect d u e to capillary force, (k g /m s
S u bscrip ts
a
V
9
w
am b
wev
wp
s
eff
1
0
air
v ap o r
gas
w ater
am bient
liq u id ev ap o ratio n on the surface
liq u id "p u m p in g " on th e surface
so lid m atrix
effective
n o d e on th e open surface
atm osp here
xv
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Chapter 1
INTRODUCTION AND
OBJECTIVES
M icrow aves pro v id e a kind of d e a n , fast, an d space-save h eatin g m eth od w hich
is challenging th e convective h eating m ethod in m any aspects from hom e reheat­
in g to industrial processing o f foods. A bout 87% of A m erican h ou seh o ld s have at
least one m icrow ave oven (Food Technology, 1994) and the m icrow avable frozen
foods has becom e an im po rtan t category of fast food in grocery store. D espite the
facts, how ever, th e hom e m icrow ave oven h as becom e a sim ple reh eatin g device
rath er than a m ethod of cooking. Even for reheating of frozen foods, the heat­
ing quality is often n o t quite satisfactory. In th e food in d u stry only tem pering of
frozen m eat an d d ry in g of p asta have been successful (Food Processing, 1993).
Therefore, increase o f the m icrow avable food m arket has presented a great chal­
lenge to to d ay 's food researchers.
1
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2
The m ajor problem associated w ith m icrow ave heating is non-uniform heat­
ing w hich resu lts in significant m oisture loss an d m akes the foods d ry an d tough
(N i an d D atta, 1994). W hen French M ed p o tato is m icrow ave heated it becom es
lim p an d soggy (Business w eek, 1988). W hen frozen food is h eated in th e m i­
crow aves, th e edges lose significant m o istu re and becom e dry (Ni an d D atta, 1995).
W hen the raw po tato w ith skin is m icrow ave heated, it can often blow up.
The m ajor reason is th a t m icrow ave heatin g produces strong in tern al vapor­
ization w hich pushes the liquid w ater a n d v apo r to th e edge. M eantim e, the sur­
face is cooler d u e to heat loss to the su rro u n d in g and as a result, reduces the m ois­
ture rem oval to th e surrou ndin g air. T herefore the surface becom es soggy. O n the
other h an d th e liqu id w ater can be d irectly "pum ped" o u t of the surface w ith o u t
having any phase change if the food is very w et and internal pressure is signif­
icantly high. T his accounts for the h ig h m oisture loss in a short p eriod of tim e
for heating of very w et foods. In the case o f heating of raw potato, th e very high
intern al pressure can cause it to blow u p .
Since th e pressure d riv en flow can n o t be lum ped into sim ple diffusional m odel
because of a com pletely different m echanism , a m ultiphase porous m edia m odel
is need ed to explicitly characterize the pressu re driv en flow. Since W hitaker (1977)
system atically described th e m echanistic porous m edia m odel, it h as been used
in m aterials such as sandstone (D avis a n d cow orkers, 1986), brick (Turner and
cow orkers, 1991) an d concrete (Perre a n d cow orkers, 1996). H ow ever, no w ork
has been d on e in food area w hich explicitly included the pressure d riv en flow.
Therefore, th e prim ary objectives o f th is w ork are described as follow ing:
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3
1. D evelop a m u ltip h ase porous m edia m odel to p redict m oisture tran sp o rt in
m icrow ave h eatin g of foods an d thereby u n d erstan d the fu ndam ental rea­
sons behind w et surface form ation an d high m oisture loss d u rin g m icrow ave
heating.
2. C haracterize th e effect of p rop erties and geom etry of the sam ple and oven
conditions o n th e m oisture tran sp o rt and p ressu re developm ent.
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Chapter 2
LITERATURE REVIEW
2.1 Simple M odels of Moisture Transport in
Microwave Heating
M oisture tran sp o rt in conventional heating o r conventional drying stu d ies often
use an effective diffusivity m odel, as given by
= V (Def f V M )
(2.1)
w here M is th e total m oisture content (liquid an d vapor) an d £>e/ / is an effec­
tive diffusivity th at includes the effects of all possible m echanism s of tran sp o rt
of m oisture in b o th liq u id an d vapor form . In alm ost all studies, the value o f ef­
fective diffusivity Deff is actually obtained from fitting experim ental d ata to the
solutions of th e sam e equation (Eq. 2.1).
Follow ing th is trend, m icrow ave heating, m icrow ave drying, and com bined
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5
m icrow ave an d conventional drying h ave also been m odeled using an effective
diffusivity value D ef f . For exam ple, S hivhare (1991) stu d ied m icrow ave d ryin g
of com u sin g an effective diffusivity m odel. In m icrow ave heating, how ever, the
internal ab so rptio n o f energy and in tern al ev aporation are often quite significant.
Thus, several au th o rs h av e u sed variations of th e follow ing equation for energy,
(pCp)ef
dT
f =
v ( kef / VT) + I + Q
(22)
together w ith Eq. 2.1 for m oisture tran sp o rt. In Eq. 2.2, / is the rate of internal
evaporation, an d Q is th e rate of volum etric h eatin g due to m icrow ave absorption.
The sim plicity o f th is form ulation is d eriv ed from assum ptions m ade in Eq. 2.2
to account for the in tern al evaporation / . R oques e t al. (1992) d id n o t include the
evaporation term / in the stu dy of com bined m icrow ave an d conventional h eat­
ing of a deform able p o ro u s m edia. In th e w ork of Tong et al. (1988), the evapora­
tion term / w as ap proxim ated as
t
\
=
dM
n
~dT
It is h ard to justify th is assum ption from physical considerations. Internal evapo­
ration / is an in d e p en d en t term , as described in the next section. W hereas, Eq. 2.3
assum es th a t all of th e m oisture change a t any location contributes to laten t h eat at
that location, w hich is physically unrealistic. For the m oisture transp ort, Tong et
al. (1988) u sed Eq. 2.1 fo r an effective diffusivity form ulation, w ith the diffusivity
param eter calculated from experim ental d ata, as noted earlier. For th eir physi­
cal situation, the effective diffusivity q u ite likely included pressure d riv en vapo r
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6
flow. In a sim ilar way, Z h o u e t al. (1994) u sed Eq. 2.3 to include ev ap o ratio n and
the sam e com m ents reg ard in g the v alid ity of such assum ption w o u ld apply.
A different b u t still sim ple assum ption regarding evaporation term w as m ade
by N i e t al. (1994). They considered th e follow ing governing equations
Q when T < 100°C
<PCr ) 'U % = V(fce //V7’) +
0
(2.4)
when T > 100°C
They also considered m oisture tran sp o rt to have no diffusional lim itation. In­
stead, m oisture at a location w ould change d u e to evaporation only, as given by
dM
0
when T < 100°C
(2.5)
—Q fA when T > 100°C
2.2 More Generalized M odels for M ultiphase Trans­
port in Porous media
Excellent review s of h eat an d m oisture tran sfer in porous m edia in th e context
of foods is contained in B ruin and L uyben (1980) and in Fortes an d O kos (1980).
O ne of the w ell know n theories for porous m edia transport is the w ork of Luikov
(1975). H e derived the m acroscopic h eat an d m ass transfer governing equations
based on the phenom enological theory o f non-equilibrium therm odynam ics. By
choosing tem perature, m oisture content an d gas pressure as prim ary variables,
his final equations w ere
?L
at
=
K u V 2T + K 12V 2M + K l3V 2P
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7
—
at
dP
—
at
= K 2lV 2T + K22V2M + K ^ V 2P
= A31V2T + K32V2M + K ^ P
(2.6)
This m odel in d u d e s convective flow of gas as w ell as capillary flow of liquid.
T hree sim ple parallel final equations are very favorable to the analytical solution
in som e sim ple cases. H ow ever, the m odel has sev eral disadvantages. M ost im ­
portantly, physical in terp retatio n of the param eters is n o t dear. This is because
all flux expressions are based on phenom enological relationships. A nother im ­
p o rtan t draw back is th e use of a constant phase conversion factor that is a ra­
tio of w ater tran sp o rt in th e vapor phase to w ater tran sp o rt in the liquid phase.
A lthough phase conversion factor provides so lutio n w ith som e sim plidty, its as­
sum ed value really m akes the solution sem i-em pirical. Further, the gas diffusion
w as n o t described in diffusion tran sp o rt and liquid b u lk flow w as n o t described in
convective transp ort. Recent w orks th at are described in m ore details below u sed
th e theory to calculate th e relative effects of variou s param eters on the solution.
It is fair to say th a t th is theory is n o t being used w idely today.
Instead of a L uikov's phenom enological m odel, W hitaker (1977) used a m ech­
anistic m odel an d developed a set of heat and m ass tran sfer equations for porous
m edia. H e started from conservation equations fo r h e a t and m ass for each phase
(solid, liquid, gas p lus v apor) an d later volum e av erag ed the different phases. A l­
th o u g h the final equations look like nothing b u t sim ple continuity an d flux equa­
tions, the rigorous stu d y for the transition from th e in div id ual phase at the "m i­
croscopic" level to representative average volum e a t th e "m acroscopic" level p ro -
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8
v id es d ie fundam ental an d convincing basis. Thus the relatio n sh ip betw een the
po rou s m edium and its fictitious eq u iv alen t continuum is clearly show n. T he m a­
jo r assum ptions in his deriv ation s w ere local therm al equilibrium , v alid D arcy's
law, Fickian diffusion an d filtrational flow in gas transport, capillary flow in liq­
u id tran sp o rt (assum ing negligible effect o f gas pressure g rad ie n t on liq u id m ove­
m ent), rigid structure and absence of b o u n d w ater. The final equations w ere:
QT
(PCp)e//_^* ■+*(PwCpwVw + PgCpgVg)VT = V(Auc//V T ’) + Q — XI
A
— («,*) + V ( p vv,) = V(psDe//.3V ( ^ ) ) + /
+ V ( p„v,) = V ( p , D . , u V & )
d
-q j (^wPw)
V (p wvw) =
I
(2.7)
The big advantage of this m echanistic m odel is th at the physics of th e m odel is
b etter understood, the assum ptions are v ery clear and the p aram eters are w ell de­
fined. The above tw o sets o f general equ ation s (Eq. 2.6 o r 2.7) have set th e starting
p o in t for all literature stu d ies of m odeling of drying of p o ro u s m edia.
Several researchers have developed h e a t and m ass tran sfer equations for porous
m edia startin g from conservation equ atio ns and m echanistic flux m odels (Wei et
al., 1985; Stanish et al., 1986; N asrallah e t al, 1988; Die et al., 1989; C hen et al., 1989).
T heir equations look sim ilar to W h itak er's an d w ill be referred to as m echanistic
m odels hereafter. A lthough they do n o t alw ays discuss th e phase averaging ex­
plicitly, m uch of the steps a n d assu m p tio n in W hitaker's m o d el is im plied in these
form ulations. For this study, w e w ill b e u sin g th is m echanistic approach.
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9
2.3
Applications of the M ultiphase Transport M odels
to Conventional Heating
2.3.1 Variations in the Form ulations and Solution Techniques
V ariations in th e form ulation of the m u ltip h ase flow problem can be grouped along
the lines of w h eth er the researchers follow ed phenom enological (Luikov) or m ech­
anistic (W hitaker) ty p e form ulation. A lthough m ost of the researchers solved th e
equations n u m erically som e have dev eloped analytical solutions for restricted
cases. In the o rig in al w ork of Luikov (1975), Robbins and O zisik (1988) decou­
pled L uikov's eq u atio n s to get analytic solution and fu rth er studied the effect of
various dim ensionless param eters on d ry in g in constant gas pressure. Liu an d
C heng (1990) solved Luikov7s equations and d id param etric stud y of heat an d
m ass transfer in d ry in g of capillary-porous m edia. Irudayara and W u (1994,1996)
solved 1-D and 2-D Luikov7s equations b y finite elem ent m ethod. They used cen­
tral difference in space and C rank-N icholson schem e th at w as unconditionally
stable, to avoid num erical oscillations.
As m entioned earlier, recent stu d ies o f m ultiphase flow in porous m edia used
m ostly a m echanistic form ulation th a t is W hitaker type (Eq. 2.7). Wei e t al. (1985)
solved such eq u atio n s in 1-D cylindrical coordinates for conventional heating of
sandstone. They u se d the finite difference m ethod (FDM), w ith central differenc­
ing of space. A n experim ent-based h isto ry curve of w et surface percentage w as
used to account fo r total vapor surface flux in the m ass transfer b ou nd ary condi-
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10
tion. N asraU ah a n d Perre (1988) also u se d FDM to solve sim ilar se t o f equations in
1-D slab coordinates d u rin g conventional h eating of concrete. T hey used a nonuniform m esh w ith centered space difference an d fully im plicit tim e differencing
w ith u p w in d in g for th e convective term . Perre an d M oyne (1991) extended this
w ork to solve th e governing equations in 2-D rectangular dom ain. They consid­
ered b ound w ate r for conventional heatin g of softw ood w hich is hygroscopic and
anisotropic. B oukadida an d N asrallah (1995) follow ed sim ilar form ulation and
applied to a 2D m odel for th e p ro p erties of clay brick. Die an d T urner (1989) di­
vided the w hole region into w et an d d ry zones w ith m oving b o u n d ary and ap­
plied separate eq uatio ns in each zone. They u sed FDM to solve 1-D slab in con­
ventional h eatin g of brick. In th eir FDM, tim e w as considered fully im plicit, u pw inding w as u sed for th e convective term in th e energy eq uation , an d central dif­
ferencing in o th er term s. T hey found C rank-N icholson schem e to be unstable.
C hen and Pei (1989) also d iv id ed th e w hole region in to w et a n d sorp tio n zones
w here different sets of governing eq uation s w ere applied. They u sed a finite ele­
m ent m ethod w ith m oving boundary, w ith m ore elem ents in areas adjacent to the
m oving boundary. B ound w ater diffusion in hygroscopic m aterials such as w ool
and com kernels w as included. The m odel developed by P erre an d his cow orkers
w as com pared w ith th e m odel developed by T urner an d h is cow orkers in Turner
an d Perre (1995). The final form of equations w ere slightly different in the tw o
m odels alth o u g h th e u n d erly in g physics is exactly the sam e. T hey com pared th eir
2-D codes for conventional d ry in g of w ood and concluded reasonable qualitative
agreem ent b etw een them .
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11
Stanish et al. (1986) used a finite-space / continuous tim e approach w here equa­
tions w ere only sp atially discretized an d solved in term s of ordinary differential
equations in tim e. T hey claim th a t th is m ethod com bines th e conceptual sim plic­
ity of the finite difference m ethod w ith th e num erical pow er of a specialized and
robust solver for stiff equations. T hey u sed h y b rid upw inding for the convective
term . To m aintain discretization accuracy th ey used an adaptive m esh schem e
to concentrate the m esh points aro u n d regions w ith rapidly changing properties.
They also included bo u n d w ater into th e equ atio ns in conventional d ryin g of w ood.
M elaaen (1996) stu d ied the drying an d pyrolysis of w ood th at is intensive sur­
face heating. The coupled equations w ere discretized in space and in teg rated in
tim e like in the w o rk of Stanish e t al. (1986). A bdel-R ahm an et al. (1996) solved
the m echanistic equations for concrete w all exposed to fire. They used a FDM
w ith variable grid spacing, an im plicit tim e differencing and center differencing
for space.
2.3.2
Temperature, M oisture, and Pressure Profiles in Conven­
tional H eating
Temperature profiles
A s com pared to h eatin g w itho ut an y ev ap o ratio n or tran sp o rt (i.e., in a sim ply
conducting solid), tem perature profiles in the presence of evaporation ten d to be
m ore uniform after th e initial tim es. D uring th e in itial tim es, the profiles are closer
to those expected in a solid w ith o u t evaporation, i.e., stronger gradients tow ard
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12
th e surface. T he profiles becom e m ore uniform quicker in th e case of heating w ith
m oisture m igration. The tem perature a t th e surface approaches th e w et bulb tem ­
p eratu re. In th e later stages, the surface becom es d rier and th e tem perature starts
to increase. A ll o f these trends can be seen in the w orks of Wei e t al. (1985), N asrallah a n d P erre (1988), C hen and Pei (1989), Ilic an d Turner (1989). In the case of
very h ig h su rro u n d in g tem perature, the surface tem perature can keep increasing
w ith out approaching the w et bulb tem perature, as seen in A bdel-R ahm an e t al.
(1996).
M o istu re p ro files
A fter th e initial tim es, m oisture profiles are uniform un til the satu ratio n becom es
very low. In th e w ork of Wei et al. (1985), th is can be seen for m o st of the m oisture
profiles. For th e range of tim es they have provided the data, th e effective capil­
lary d iffusivity probably does n o t dro p dram atically. Therefore, a relatively hig h
diffusivity stay s for m uch of the d rying tim e in this study, resu ltin g in m ore u n i­
form m o istu re profiles. This is also seen in th e w ork o f N asrallah an d Perre (1988),
Die and T urner (1989), T urner et al. (1991). Eventually, the m o istu re level d ro p s
to values th a t lead to very sm all diffusivity and the m oisture level show s sh arp
d ro ps n e a r the surface, as in the w ork of N asrallah and Perre (1988), Chen and Pei
(1989), Ilic an d T urner (1989), an d T urner e t al. (1991).
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13
P ressure p ro files
Pressure profiles in th e literature show in terestin g trends. D uring m uch of the
heating, u n til a d ry zone develops at th e surface, the pressures a re negative, de­
creasing aw ay from th e surface. This is seen in both Wei et al. (1985) and N as­
rallah and Perre (1988). As the d ry zone m oves into the interior, significant posi­
tive pressures develop w ith higher values in th e interior, as seen in N asrallah and
Perre (1988), Die an d T urner (1989). The sam e tren d is seen in A bdel-R ahm an an d
A hm ed (1996), except th eir interior pressu res are m uch higher (> 6 atm ) due to
very high surface tem peratures.
2.4 Applications of the M ultiphase Transport Models
to Microwave Heating
M odeling of m icrow ave heating of porous m ed ia has been achieved in generally
tw o w ays. O ne is to d iv ide the entire h eatin g in to periods and to sim plify the gov­
erning equations in each period for solving. T he other is to sim ultaneously solve
these fully coupled equations for the en tire h eatin g period. The form er m ay n o t
need m uch com puter capability b u t require m any assum ption a n d sim plification
w hich restrict its m ore general use. The la tte r takes advantages o f the pow erful
capability of com puter to provide m ore accu rate and m ore general prediction.
H atcher e t al. (1975) follow ed the L uikov7s equations of capillary-porous m e­
dia u nd er tw o basic assum ptions: sm all R eynolds num ber for v a lid D arcy's law
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14
an d local therm al equilibrium , an d d iv id ed the drying history in to four periods
(initial adjustm ent, liqu id m ovem ent, constant rate, and falling rate). Each pe­
riod w as sim plified based on the L yon's experim ent and solved. T he prediction
agreed w ell w ith th e experim ent. H ow ever, the validity of the co n stan t pow er ab­
sorption assum p tio n is quite restricted. It m ay be good approxim ation in the low
m oisture m aterials. G enerally the thickness and the dielectric co n stan t and loss
w ill influence th e rate of exponential decay so th at it w ill change the total pres­
sure profile a n d rate of the m oisture loss. In addition, the solution d id not predict
liquid m ovem ent w hich did exist in L yon's experim ent. Perkin (1980) proposed a
sim plified m odel sim ilar to H atch er's to stud y the MW drying of non-hygroscopic
porous m aterials w ith pore size larger th an 10 °A. The drying h isto ry w as divided
into three characteristic regions: initial heating period, pressure generation pe­
riod and m oisture rem oval period. Each period w as sim plified a n d solved to get
eith er tem perature o r pressure history o r m oisture loss. Pressure b u ilt up soon
after the norm al boiling tem perature w as reached (sam e as H atcher, 1975). Sev­
eral param eters w ere grouped to characterize each period. The m odel considered
the E-field th ro u g h th e sam ple as constant and allow ed pow er ab so rp tio n to vary
on the m oisture and tem perature b u t no t location. N o liquid m ovem ent w as pre­
dicted either. C hen an d Schm idt (1990) considered the m icrow ave assisted drying
based on th e ir conventional heating m odel before. They ignored th e air flow in­
side the m edia to get th e ratio of v ap o r pressure gradient to gas p ressu re gradient
an d further converted gas pressure g rad ien t in D arcy's law in to v ap or pressure
gradient to sim plify v ap o r and w ater flux term s. In the energy eq u atio n they only
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15
considered th e energy tran sp o rte d b y v a p o r m ovem ent an d co n trib u tio n of heat
capacity of so lid m atrix so th a t m ass eq uatio n is uncoupled w ith en erg y equa­
tion. They sep arated the tw o regio n a n d solved by th e integral m eth od . T he MW
source term w as assum ed as d ifferen t constants in each region. Som e sim plifica­
tion procedure in th is w ork is good. B ut th e assum ption of ev ap o ratio n fro nt un ­
d er MW h eatin g is n o t very a p p ro p ria te because phase change can h ap p e n w ithin
a significant depth.
R esearchers w ho developed th e generalized m ultiphase tra n sp o rt equations
for porous m edia, only n eed ed to in clu d e a m icrow ave source term to m odel m i­
crow ave heating. Wei et al. (1985) assu m ed m icrow ave exponential decay of the
m icrow ave source term an d so lved th e equations in case of d ry in g of sandstone
a t 60 W m icrow ave pow er. T hey u sed a backw ard-difference im plicit m ethod to
solve th e equations, b u t also o b tain ed sim ilar results w ith a six p o in t orthogonal
collocation m ethod th at took con siderab ly less com puter tim e. T heir prediction
agreed w ell w ith Lyon an d H atch er (1972) experim ent an d th e p ressu re driven
flow of liq u id w as reported d u e to in tern al vaporization. C ontreras (1987) solved
sim ilar se t of m echanistic eq u atio n s, b u t h e considered the effect o f sw elling and
shrinkage in m aterials such as silicone rubber an d m eat. The sw elling pressure
w as m odeled. T urner an d Jolly (1991) u se d m icrow ave pow er d en sity calculated
by plane w ave assum ption a n d solved th e sam e equations as before in case of d ry ­
ing of b rick u n d er low m icrow ave pow er. P ressure d riv en flow o f liq u id w as also
reported. T urner and R udolph (1992) solved for th e com bination o f conventional
an d MW d ry in g of glass beads. T urner a n d Ferguson (1995) solved 2-D com bined
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16
m icrow ave and conventional d rying of softw ood.
Perre an d M oyne (1991) u sed the sim ple m icrow ave source term to solve 2-D
m icrow ave d ry in g of lig h t concrete. In th is p ap e r th e m odeling difficulties asso­
ciated w ith satu rated region w ere discussed an d som e m ethods w ere suggested.
The m icrow ave pow er w as intense. C onstant et al. (1992) solved the conventional
drying of light concrete w ith m icrow ave assisted a t a later stage w hich enhanced
pressure driven flow of vapor. C onstant et al. (1996) introduced the liq u id expul­
sion d u rin g 65 W m icrow ave drying of lig h t concrete. The criterion for w hich the
liquid w as expulsed w as p ro vid ed . The m oisture loss are very significant d u ring
the liquid expulsion period.
2.4.1
Temperature, M oisture, and Pressure Profiles in M icrowave
H eating
T em perature profiles
To have an idea of expected tem perature profiles in m icrow ave heating, the sim ­
plest situation is to consider exponential h eatin g of a solid w ithout any evapora­
tion o r tran sp o rt of m oisture. This w as do ne by D olande and D atta (1993) for slab
an d three characteristic tem p eratu re profiles w ere identified. As a first approxi­
m ation, therm al diffusion can b e ignored com pared to h eat generation d u rin g the
initial heating period and tem peratu re profiles show ed rapid drops from surface
to the interior, qualitatively sim ilar to conventional heating. Interm ediate heating
period w as characterized b y a tem perature p eak th at developed near th e surface
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17
an d propagated in w ard as h eatin g continued. A fter sufficiently long heating, the
profiles developed a concave d o w n shape, characteristic o f steady state w ith in­
ternal heat generation. In th e stu d y of Wei et al. (1985), th is type 3 tem perature
profile appeared, d u e to slow an d relatively uniform rate of heating (large pene­
tratio n depth). Sim ilar ty p e 3 tem perature profile can be seen in Perre an d M oyne
(1991) and C onstant e t al. (1996).
M oisture p rofiles
A t low pow er, m oisture profiles in m icrow ave heating are generally uniform . A t
low pow er, although th ere is som e filtrational flow an d condensation near the
surface due to the low er surface tem perature in m icrow ave heating, the am ount
reaching the surface is sm all an d increase in m oisture level tow ard th e surface is
sm all. This can be seen in the w orks of Wei et al. (1985), Turner e t al. (1991), an d
C onstant et al. (1996). In som e interm ediate pow er level, the surface m oisture is
m uch higher than the insid e m oisture b u t there is no liquid expulsion yet. The
m oisture decreases g rad u ally from the surface to the inside. This can be seen in
Perre and M oyne (1991).
M oisture m igration in very h ig h rate of heating has n o t been studied, except in
th e w ork o f C onstant e t al. (1996). W hen th e tem perature reached high enough,
significantly higher p ressu res are developed, causing strong filtrational flow b u t
the surface tem p erature stays low, as in m icrow ave heating. The com bination of
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18
these tw o causes a liquid p lu g to form a t the end o f sam ple an d significant expul­
sion occurs. As th e heating proceeds fo r longer tim es, absorbed p ow er decreases
due to reduced m oisture level an d in terio r pressures drop, decreasing the filtra­
tional flow. The v ap or rem oval capacity from th e surface stays approxim ately the
sam e, causing the m oisture levels at th e surface to d ro p eventually low er than the
values inside.
P ressu re p rofiles
In m icrow ave heating, pressures increase aw ay from the surface, w ith the m axi­
m um v alu e at the center. The m axim um value increases w ith tim e d u rin g the ini­
tial p erio d s of heating. Eventually, w hen the m oisture level drops, th e peak value
of p ressu re drops. This is seen in the w ork of Wei e t al. (1985) and in bo th low an d
high ra te s of heating of C onstant et al. (19%). A s expected, the m axim um pres­
sure reached for high rate o f heating w as m uch h ig h er (0.4 atm gauge for C onstant
et al., 1996) com pared to th a t for low er rate of heatin g (0.014 atm gauge for Con­
stan t e t al., 19%; 0.073 atm gauge in th e w ork of P erre and M oyne, 1991; and 0.04
atm g au g e in the w ork of Wei et al., 1985). All of these pressures are higher than
the m axim um pressure developed in convective heating (0.005 atm in the w ork
of N asrallah and Perre, 1988 and 0.013 atm in the w ork of Die and Turner, 1989)
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19
2.5 M ultiphase Porous Media M odel in Commercial
CFD Code
So far, the m ajor com m ercial CFD softw are such as FLUENT (Fluent Inc., Lebanon,
NH), FIDAP (Fluid D ynam ics International, Inc., E vanston, IL), CFDS-FLOW3D
(AEA-CFDS, Inc, P ittsburgh/ PA), NISA / 3D-FLUID (E ngineering M echanics Re­
search C orporation, Troy, M I), ADINA-F (ADINA R & D, Inc., W atertow n, MA)
and CFD-TW OPHASE (CFD Research C orporation, H untsville, AL) only have
the ability to solve th e flow th ro u g h satu rated porous m edia. G iven external pres­
sure and tem p eratu re b o u n d ary conditions, th e tem peratu re an d velocity fields
can be solved. FLUENT claim s to have the ability to solve th e m om entum equa­
tions in the tw o p h ase regions such as liquid an d gas soon b u t still assum e th at
these regions are im m iscible an d there is no evaporation.
FLOW-3D (FLOW SCIENCE, Inc., Los A lam os, NM ) h as som e ability to deal
w ith flow thro ug h u n sa tu ra ted porous m edia. H ow ever, th ere is no convective
flow of gas an d liq u id because of th e assum ption of th e constant to tal pressure.
There are several w ell developed num erical codes related to h eat and m ass
transport in u n satu rated , fractured porous m edia u n d er th e contract of Office of
N uclear R egulatory R esearch (W ashington, DC). The m athem atical description of
these w ork are com prehensive an d the code m ight have th e ability to handle the
heat and m ass tran sfer in u n satu rated porous m edia w ith internal heat source.
However, such calculation u sin g these codes have no t been reported .
Some basic features associated w ith m icrow ave heatin g of porous m aterials
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20
are th a t th e prim ary variab les for d riv in g force is w ate r satu ration , tem perature
an d to ta l pressure. T hree m ass conservation equations are for liqu id w ater, v ap o r
an d air. T here is no need to solve m om entum equ atio ns instead they are given
by D arcy's law. There is an in tern al evaporation term . A ll of these need to be
accounted for in future d ev elo pm en t o f num erical codes.
2.6
Measurement in Food and Other Materials during
Microwave Heating
2.6.1 M oisture Loss in M icrowave H eating
S hivhare e t al. (1990) m easu red th e m oisture loss curve in m icrow ave drying curve
of com a t very low p o w er (0.25 - 0.75 W / g) and different in itial m oisture content
(0.32 - 0.50 d.b.) w ith a ir tem p eratu re 30°C and velocity 0.5 m /s . A q u artz tu b e
contained grain w ith th e a ir flow ing through. M icorw ave cam e from one side.
G enerally th e rate of m oistu re loss gradually decreases w ith the heating tim e be­
cause th e v apo r pressure becom es low er as the m oistu re content decreases. For
the in itial m oisture content 0.50 (d.b.) the m oisture co ntent decreases to 028 (d.b.)
after 2 h o u rs at pow er level 0.5 W / g. The other com bination changed the rate of
m oisture loss accordingly. It w as fou nd th at m icrow ave d ry in g can reduce th e
equilib rium m oisture content (EMC) from 0.08 in convective d ryin g to 0.01- 0.32
(d.b.) dep en d in g u p o n th e p o w er level because it can raise the tem perature so
th at th e v ap o r p ressure is higher.
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21
G unasekaran (1990) stu d ied th e effect of m icrow ave p o w er cycling on th e ef­
ficiency of m oisture rem ove. C om sam ple of 25 g w ith m oisture content 0.21 0.29 (w.b.) w ere p u t in a Pyrex glass d ish form ing a single layer of kernels. M i­
crow ave in p u t pow er are set to 250 W an d 500 W w ith on an d off cycling. T he
d ry in g curves show s th at th e drying rate w ith in initial several seconds keeps in ­
creasing because the tem perature rises so th a t the v ap o r pressure increases a n d
the rate of m oisture loss increases. A fter the initial p erio d th e d rying rate becom es
constant for a w hile because the tem perature reaches steady state and m oisture
content in th is range does no t affect vapor pressure of com . Finally the d ry in g
rate sta rts to decline because fu rth er decrease o f m oisture content reduces the v a­
p o r p ressure of com . It w as found th at dow n to the sam e final m oisture content
0.15 (w.b.) d ryin g w as m ore rap id in the continuous m ode, how ever, operating a t
10 seconds o f pow er-on an d 75 seconds of pow er-off cycling resu lted in the low est
total pow er-on tim e and saves the energy. The fact is th a t rem oval of total am ou nt
of m oisture can be done eith er b y increasing th e rate of m oisture loss w hich re­
quires m ore energy to get higher tem perature and v ap o r pressure, o r by h avin g
the longer tim e.
Beke (1992) m easured th e drying curves of com w ith initial m oisture content
0.5 (d.b.) for several pow er density. It w as found th a t d ry in g rate decrease lin ­
early w ith th e decreasing m oisture content for several testin g pow er levels, w hich
m eans th a t th e drying rate decreases linearly w ith the d ry in g tim e. These resu lts
im ply th a t th e vapor density decreases linearly w ith th e d ry in g tim e, w hich m ig h t
be tru e because the tem perature increases w hile the v ap o r pressure decreases so
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22
th a t th e v ap o r d en sity decreases w ith die tim e. F urther m easurem ent show ed th a t
th e d ry in g rate a t m icrow ave pow er level 0.1 (kW h /k g com ) can be four tim es
h ig h er th a n th a t of 0.0017 ( g /g so lid /m in ) in convective drying w ith 120°C.
Tong an d L und (1993) m easured th e tem perature and m oisture loss d u rin g
m icrow ave baking of b read . It w as found th at for bread w ith thickness 2.2 cm
an d pen etratio n d ep th 8 cm the center tem perature is no different w ith th e sur­
face tem p eratu re at larg er pow er level E = 42.5 V /m d ue to larger penetration
d ep th an d very short h eatin g tim e, b u t is m uch higher at sm aller pow er level E =
28.3 V /m d u e to surface evaporation an d convective cooling. The m oisture con­
te n t w as found from in itially 0.71(d.b.) dow n to 0.46 (d.b.) after 50 seconds cor­
respo nd ing to larger p o w er level w ith initial heating rate 8.53°C /s.
2.6.2
M oisture Loss in M icrowave-assisted C onvective H eating
B hartia e t al. (1973) m easu red the d rying curves in m icrow ave-assisted convec­
tive d ry in g of m aterials w ith different hygroscopidty such as sand, potato starch
an d silica gel. It w as fo u n d th a t the m icrow ave-assisted convective drying rate is
alm ost th e sam e near th e in itial m oisture content (w.b.) of each sam ple, i.e., 0.13
(sand), 0.23 (starch p o tato an d silica gel), w hich indicates th at m icrow ave can be
ad d ed in even later stag e in o rd er to im prove the energy efficiency. The reason
m ight be th a t the low rate of m oisture loss does n o t cause significant decrease in
the surface m oisture an d th e v apo r pressure at early stage. Therefore it is m ore
efficient to tu rn on the m icrow ave a t an even later stage. The d ry in g curves also
show ed th a t the im pact o f m icrow ave energy on increasing energy efficiency and
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23
low ering th e drying tim e is m ore prom inent for increasing hygroscopidty. T his
w ould b e th e case for m ost biological m aterials.
Jolly (1986) m easured th e tem perature an d d rying cu rv e of polyurethane foam
in m icrow ave-assisted convective drying. The foam slab w as p u t horizontally
w ith h o t air at 60°C flow ing o v er the top. The m icrow ave cam e from the top w ith
on-off according to tem perature controller w hich tu rn s off the pow er if the tem ­
p eratu re is over 63.5°C. The initial m oisture content w as 1.5 (d.b.). It w as found
that in o rd er for the final m oisture dow n to 0 2 , the d ry in g tim e is m uch reduced
from 8 h o u rs in convective d ry in g to 1.25 hours in m icrow ave-assisted convective
drying. T he reason is th a t the internal pressure p ressu re g rad ien t drives out the
vapo r m uch intensively th an th e vapor diffusion at th e d ry stage.
O lson e t al. (1983) stu d ied m icrow ave (915 M H z) assisted im pinging hot air
d ry ing o f sw eetgum veneer in a continuous m ode. In a d d itio n to drying tim e re­
duction w ith m icrow ave ad d ed , the drying curves sh o w th a t the rate of m oisture
loss approaches the constant from initial m oisture co n ten t 1.1 do w n to 0.15 (d.b.)
as the m axim um pow er level is used in th e study. T he reason m ight be at th is
pow er level equal am ount o f v ap or can be produced a t th e later state w hile th e
earlier d ry in g rate is controlled b y capillarity flow an d surface evaporation. A lso
the o ptim um m oisture uniform ity based o n the sta n d ard d ev iation of final m ois­
tu re of all the sam ple is obtained in a com paratively sm all pow er level. Pure h o t
a ir im pinging or higher m icrow ave pow er level d ecreased th e uniform ity. H ow ­
ever, it is realized th at uniform ity in this w ork is m ore related to oven uniform ity
instead of m oisture tran sp o rt inside the each sam ple.
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24
R oques e t al. (1992) m easu red the m oisture loss and tem perature in rigid porous
an d deform able non -po ro us m aterials d u rin g m icrow ave-assisted (1 W /m 3) con­
vective (38°C) drying. It w as fo u n d th at a high b u t short p erio d of liquid "pum p­
ing" w as observed afte r in itial heatin g period in rigid poro us m aterial (cellular
concrete). H ow ever, th e deform able porous m aterial (silica-alum ina) only show ed
a m uch sim pler g rad u ally increasing and decreasing rate, w hich m ight be due
to hygroscopidty of th e silica-alum ina resulting in low er v ap o r pressure. Gener­
ally th ere is no constant w et b u lb tem perature in m icrow ave heating and surface
tem p eratu re increases u n til a steady state and tend to be flat. Even in deform able
non-porous gel an o v erp ressu re period can be observed w hich increases the rate
of m oisture loss bu t th e u n d erly in g m echanism has not y et been w ell understood.
The tem perature can increase dram atically in later dry stage w hich cause the dam ­
age of therm osensitive m aterials.
P rabhanjan et al. (1995) m easured the drying curves of m icrow ave-assisted
convective drying of carrots. T he carrots w ith m oisture content 6.7 (d.b.) w ere
cut in to 12 x 12 x 12 m m cubes an d p u t in single layer w ith total w eight 60 g. The
m icrow ave pow er levels are 120 and 240 W, respectively an d th e air tem perature
is 45 an d 60° C, respectively. It w as found th at rate of m oisture loss decreases lin­
early w ith decreasing m o istu re content. N o constant rate w as observed for both
convective and m icrow ave-assisted convective heating at sm allest recording tim e
5 m inutes, w hich m ig ht b e d u e to the th in layer arrangem ent providing rap id dry­
ing conditions so th a t th e co n stan t rate period is shorter th an 5 m inutes. M oisture
content decreases from 6.7 (d.b.) to 5.3 for 240 W, 5.8 for 120 W an d 6.0 for con-
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25
vective d ry in g in 45°C after 5 m in u tes. A dding m icrow aves can reduce th e con­
vective d ry in g tim e by 25 - 90%.
2.7 Specific Objectives
A lthough m oisture m igration in m icrow ave h eatin g affects th e quality (often ad ­
versely) of n early all applications o f h eatin g of foods in a m icrow ave oven, there
has been n o stu d y describing th e m oisture tran sfer as a m ultip hase porous m edia
tran sp o rt process. Effect of th e in ten sive m icrow ave heating o n the m ultiphase
tran sp o rt process in a hygroscopic poro us m edia w as non-existent for any h eat­
ing ap p licatio n w hen this stu d y w as u ndertak en. Also, there h a s been alm ost no
attem pts a t including the effect o f stru ctu re change d u rin g h eating.
The p resen t w ork w ill fill th ese g ap s in the literatu re by considering the food
as a hygroscopic porous m edia an d th e tran sp o rt of liquid w ater, vapor, an d air
in this p o ro u s m edium d u e to ra p id in tern al evaporation from spatially v ary ing
intensive m icrow ave heating. T he m ass tran sp o rt m echanism s w ill include capil­
lary an d pressu re driven tran sp o rt of liquid, p ressu re driven an d diffusive tran s­
port of v a p o r an d air. The en erg y tran sp o rt w ill include diffusion, convection,
m icrow ave absorption, an d ev ap o ratio n . The specific objectives of this stu d y are
the follow ing:
1. D evelop the com plete set of governing equations for m ultiph ase tran sp o rt
in a hygroscopic porous m ed ia w ith in ternal heating an d internal evapora­
tion.
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26
2. D evelop th e app rop riate b o u n d ary conditions for ra p id heating.
3. Solve th e governing equation a n d boundary conditions u sin g a finite differ­
ence m ethod.
4. Select ap p ro p riate in p u t param eters for th e m odel.
5. V alidate th e num erical m odel com paring w ith com putational results from
sim ilar stu d ies in th e literature, an d w ith experim ental resu lts on total m ois­
tu re loss.
6. C om pute th e variables of im portance to m icrow ave processing of m oist m a­
terials — spatial distributions of tem perature, m oisture, and pressure, and
the to tal m oisture loss.
7. S tudy th e sensitivity of th e variables just m entioned to th e in p u t param eters
perm eability, gas diffusivity, thickness, initial w ater satu ratio n , and surface
heat an d m ass transfer coefficients.
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Chapter 3
MATHEMATICAL FORMULATION
OF MICROWAVE HEATING OF
POROUS MEDIA
The governing eq u atio n s an d the initial a n d boundary conditions for th e m u lti­
phase porous m edia m odel w ith m icrow ave heating are developed in th is chapter.
The form ulation of th e source term for m icrow ave heating is also described here.
3.1 Assumptions
A n schem atic of th e m icroscopic picture o f unsaturated porous m edia is show n
in Fig. 3.1. Four ph ases are considered here— solid m atrix, liquid w ater, vapor,
an d air. The m ajor assum ptio ns used in arriv ing at the governing equ atio n an d
27
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28
Water transport
• Pressure driven
•Capillarity (sat, temp) driven
Equilibrium
•water-vapor
mmammm
w m sm m
o * ° 9 Lo o
o
p
^
Energy transport
(Local Thermal Eq.)
•Diffusion
•Evaporation
•Microwave absorption
Airtransport
•Pressure driven
•Diffusion in vapor
Vapor transport
•Pressure driven
•Diffusion in air
Figure 3.1: Schem atic of an u n satu rated porous m edia.
boundary conditions are the follow ing:
1. The three p hases of solid, liquid and gas are continuous. As a m atter of fact,
the liquid can becom e disconnected in d ry in g of capillary-porous m aterials
w hen the liq u id satu ratio n level is less th an a certain threshold v alu e (Con­
treras, 1987). H ow ever, the prim ary interest o f this present w ork is th e m ois­
ture tran sp o rt d u rin g intensive b u t short tim e m icrow ave heating.
Therefore m ost of the m aterial w ill not be d rie d to the m oisture level w hen
the liquid becom es disconnected.
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29
2. Local th erm al equilibrium is v alid , w hich m eans th at the tem p eratu re in the
three p h ases are equal (W hitaker, 1977). This assum ption has been w idely
u sed in convective heating m o del (Wei et al., 1986; N asrallah et al., 1988; Die
et al., 1989) as w ell as in m icrow ave drying (Wei et.al., 1986; Perre e t al., 1991;
T urner e t al., 1991; C onstant e t al., 1996).
3. Isotherm s are used to describe th e v apor pressure as a function of tem per­
atu re as w ell as m oisture co n ten t (Rao and Rizvi, 1995; R ahm an, 1995). In
ad d itio n to C lausius-C lapeyron relationship in free w ater an d K elvin equa­
tion in capillary porous m aterials, isotherm s include the hy gro scop idty of
the m aterial w hich describes th e stro n g physicochem ical force betw een the
solid m atrix an d liquid (C ontreras, 1987), an d lum ps all of th e above factors
together. A lthough there are sev eral generalized isotherm m odels available,
coefficients in these m odels m u st be determ ined from experim ent for each
m aterial.
4. B inary m ixture of vapor an d a ir behaves like an ideal gas (W hitaker, 1977;
Wei e t al., 1986; N asrallah et al., 1988; Die et al., 1989).
5. D arcy's law is valid in describing th e convective flow of b o th gas an d liquid
in u n sa tu ra ted porous m edia. Perm eabilities of gas and liq uid can be ex­
pressed in term s of intrinsic a n d relative perm eabilities (Bear, 1972; Schdeggler, 1974; Wei e t al., 1986; N asrallah et al., 1988; Die et al., 1989).
6. L iquid m ovem ent results from convective flow due to the g rad ien t in total
gas p ressu re an d capillary flow d u e to the grad ient of capillary force th a t is a
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30
stro ng function of m oisture co n ten t a n d a w eak function of th e tem p eratu re
(Z hou et al., 1994).
7. G as m ovem ent results from convective flow d u e to to tal gas p ressu re g ra­
d ien t and m olecular (K nudsen) d iffusio n d u e to the concentration g rad ien t
(Wei, et al., 1986; N asrallah et al., 1988; Die et al., 1989).
8. C ontribution of the convection to energy tran sp o rt is ignored initially to sim ­
plify the solution procedure. Later, convective term s w ill be ad d e d to the
energy equation to check th is assum ption.
9. H eat conduction in the porou s m edia is described in term s of effective ther­
m al conductivity and is p ro p o rtio n al to th e m ass content of each phase
(W hitaker, 1977; Wei e t al., 1986; Ilic e t al., 1989).
10. E nthalpy of each phase is p ro p o rtio n al to th e tem p erature a n d th e m ass con­
ten t (W hitaker, 1977). The to tal en th alp y is ad ditio n o f th a t of all th e phases.
The latent h eat is included in th e enthalpy.
11. M ass transfer coefficient th a t quantifies the convective m oisture loss a t the
surface is independent of th e surface condition (N asrallah e t al., 1988; Die e t
al., 1989)
12. Isotropicity is assum ed for sim plicity, w hich is n o t alw ays tru e for biom ate­
rials such as food.
13. A lthough the m aterial is hygroscopic, shrinkage d u e to m o istu re loss is con­
sid ered insignificant In the p resen t w ork, tw o different po rosity are chosen
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31
to rep resen t hig h m oisture a n d low m oisture m aterials. For each m aterial,
the to tal d u ratio n of heating is relatively short an d therefore fu rth er shrink­
age of these m aterials is considered not significant.
14. G ravity is ignored. The sam ple size is sm all so that the effect of difference
in heig ht is insignificant.
3.2
Equilibrium State Variables and Vapor Pressure
This section w ill discuss the porosity, saturation, concentration an d vapor pres­
sure relationship.
3.2.1 Porosity, Saturation, and Concentrations
In rigid non-hygroscopic porous m edia, the porosity is defined as volum e fraction
0 of w ater p lu s gas, and is constant, as seen in Fig. 3.2a. H ow ever, food m aterials
are typically hygroscopic in w hich som e w ater is tightly bound to the solid m a­
trix. R ahm an (1995) listed several different porosity definitions (apparent poros­
ity, open pore porosity, closed po re porosity, bulk porosity and to tal porosity) in
the context o f foods. But all of th ese porosity definitions did n o t separate liquid
w ater from the solid m atrix, and is only defined in term s of th e gas porosity as
seen in Fig. 3.2b. The gas porosity is im portant to the gas phase tran sp o rt through
the porous m edia. Some recent stu d ies of such porosity change w ith drying are
those of M arousis et al. (1990) for starch m aterials an d Farkas et al. (1991) for air
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32
V apor + A ir
V apor + A ir
W ater
W ater +
Solid M atrix
Solid M atrix
non-hygroscopic
hygroscopic
F igure 3.2: R epresentative elem entary volum e of p o rou s m edia.
and freeze-dried chicken m eat.
In m u ltip h ase porous m edia m odel for hygroscopic m aterials, one can calcu­
late th e local m ass concentration in a sim ple w ay by defining app arent porosity
and w ater saturation analogous to those in non-hygroscopic porous m edia. Said
differently, th e vapor and air concentration can be determ ined from gas volum e
fraction, an d liquid w ater concentration from liq uid volum e fraction. As m en­
tioned u n d er assum ptions, shrinkage is ignored in this m odel because either the
heating d u ratio n is short w here th e average m oisture concentration does no t go
dow n to very low levels, o r m aterials are p ie-d ried w here fu rth e r shrinkage is
no t significant. U nder the conditions o f rig id stru ctu re and th e ap parent porosity
ju st m entioned, the volum e fractions in th e p o ro us m edia can be related as (see
Fig. 3.2a)
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33
AV' = AV; + AV W+ AVg
(3.1)
Porosity can b e d efin ed as
A 14 , + AV^
0=
KP—
.
.
(3-2)
Saturation of liq u id an d gas can then be defined as
Sw =
Sg
AVW
AVW+ AVg
AVa
AVW+ AVa
AVW
<t>AV
0AV
(3.3)
(3.4)
so that,
Sw + Sg = 1
(3.5)
For the gas m ix tu re of air and vapor, D alton's law states that the to ta l pressure is
equal to p artial p ressu re of vapor p lu s air.
P = Pv+Pa
(3.6)
If an ideal gas m ix tu re is assum ed, state equation for each com ponent is given by:
pvAVg = rrivRvT
(3.7)
paAVg = m aR aT
(3.8)
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34
T he m ass density of vapor, a ir a n d their m ixture are given by
Pv =
Pa =
Pg =
TTty
py
A Vg RuT
ma
pa
AVg R aT
7Ylv "|“ TTla
Av
- Pv+ P*
(3.9)
(3.10)
(3.11)
T he m olar density of g as m ixture is equal to,
C = £
(3.12)
T he m ass concentration of vapor, air and m ixture are given by
mv
PvSgfi
A V ~ R vT
ma
Pa$g&
ca —
A V ~ RaT
mv + ma
Cg =
= Cv
AV
(3.13)
CV --
(3.14)
(3.15)
Similarly, the liquid m ass concentration is given by
Cw
^0
PwObu;
t'i c.\
(3.16)
A fter the above d efin ition s are introduced, th e follow ing tw o sim ple m ass equa­
tio ns (m oisture content and ap p aren t density) can relate the ap p aren t porosity
an d w ater saturation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
M
=
Peff =
(3.17)
Pe//
(1 <t>)Ps "I" 0 S wpw
(3.18)
w here M is w ater con ten t (w.b.), peJf is th e ap p a re n t density of the food, pw an d pa
are the tru e density of w ater an d solids, respectively, <t>is th e equivalent porosity,
S w is the liquid w ater satu ratio n . The m ass of th e air an d vapor are ignored. The
w ater content and th e ap p aren t density d a ta are u su ally available in th e literature.
Solving Eq. 3.17 an d Eq. 3.18 for po rosity d>an d satu ratio n Sw,
$ =
5
=
i _£zffl±— M l
Ps
MpeffP*______
PwiPs — Peff { I — A / ) )
(3.19)
(3.20)
U sing Eq. 3.19 an d Eq. 3.20, th e equ iv alent porosity an d initial w ater satu ratio n
of biom aterials can be calculated.
3.2.2
Vapor Pressure
Vapor p ressure is very im p o rtan t in d eterm in in g d ie total gas p ressu re inside as
w ell as rate of m oisture loss from the surface. G enerally, v apo r p ressu re can be
characterized in term s of three m oisture con ten t ran g es . W hen th e m aterials is
very w et, the v apor pressu re obeys C lausius-C lapeyron equation, a n d is th e func­
tion of tem peratu re only, regardless of hygroscopic o r non-hygroscopic porous
m aterials. A s the m aterial dries, surface ten sio n o n th e liq u id by th e sm aller pores
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
starts to ho ld th e liq u id m ore tigh tly an d thereby reduce the v ap o r pressure. T his
can be described by th e K elvin equation. As the m aterials gets even drier, sev­
eral forces such as London-V an d e r W aals force, double layers force, short rang e
forces an d H ydrogen b o n d in g becom e im portant (C ontreras, 1987) w hich reduces
the v apo r pressure even m ore com pared w ith only th e effect of capillary pores. It
is in this range th at isotherm relationships are indispensible.
Practically, in food area, m easurem ent can be done continuously from very
w et stage to very low state. The w hole curve is referred to as isotherm relation­
ship (Rao and Rizvi, 1995) w hich is a function of tem perature, m oisture con ten t
an d the type of m aterial, i.e.,
Pv = Ps(T) * aw(M )
(3.21)
w here ps is th e v ap o r p ressure of free w ater, aw is w ater activity of the specific
m aterial, M is m oisture content.
3.3 Rate Laws
D uring the m icrow ave heatin g of the porous m edia, there w ill be diffusive tran s­
p o rt of vapor and air, capillarity driven tran sp o rt of liquid, an d total p ressu re
driv en flow of liquid, vapor, an d air. For the diffusive tran sp o rt of air and v ap o r
in th eir binary m ixture th ro u g h th e fraction of pores occupied by th e m ixture, th e
diffusive fluxes can be w ritten in term s of m ole fractions as (Bird e t al., 1960)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
Q2
— MaA/ vDe/ f <aV xa
(3.22)
C2
j v = — — A/a M vDef f,gV i r
(3.23)
_
3a =
w here Dej
j ,y
is effective gas d iffusivity in the porous m edia w hich takes into con­
sideration th e gas fraction of th e to tal cross-sectional area an d the tortuosity of the
gas pathw ays. The reason for choosing th e gradient of the m olar fraction as the
driving force instead of the g rad ien t of the m ass fraction is th at the m olar fraction
is directly related to the p artial pressu re of air and vapor, and the total pressure,
w hich m akes the subsequent d eriv atio n s m uch sim pler. T he reason for not choos­
ing m ass concentration o r m olar concentration g rad ien t as the driving forces is
that the m aterial has internal g rad ien t of total pressure an d tem perature.
The convective fluxes of th e gas m ixture are described using the D arcy's law
(Bear, 1972). A ccording to D arcy's law, volum etric flux of gas m ixture (air+vapor)
based on total cross-sectional area is given by
q, = - ^ V P
(3.24)
The relative perm eability takes in to consideration of th e volum etric fraction of th e
fluid in th e total void space. T he m ass flux of air an d vapo r based on the total
cross-sectional area are w ritten as
kk
rna = —pa— —V P
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.25)
C om bining the diffusive an d convective fluxes, the total flux (j + m) for a ir an d
vapo r can be w ritten as
kk
na = - P a — V P Pg
kk
n v = - Pv— £ - V P
Pg
C— MaM vDeff.gV x a
Pg
C~
MaM vDeff,gV x v
Pg
(3.27)
(3.28)
U sing the equilibrium relatio ns (Eqs. 3.10 - 3.12), the flux equation for a ir can
be w ritten as
- ( £ ) '
«. = - J ^ V P
R aT Pg
' ^ U . U . D. , u V ( & )
RvT
-r
S ubstituting pa = P - pv an d bringing in S w,
P
pv k kgf
RaT Pg
P \2
P
VP
1
Pv kkgr ,
•V P
RaT
Pg
R “Mg
T)effg _________
_
_
_
RT [{P —Pv) Ma + PvMv]
P Pv kkgr V P
RaT
/
V
_
_
pv \
p)
Pg
MaMvDef
f%
g
f n - " R T [ ( P - p v) M a + PvMv\
(PVpv - p „ V P )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.29)
39
-P pv kkgr
R aT pg V
MaMvD e ff,g
R T [ ( P - p v) M a + p vM vr \ {
_
p k
, v s ™+ p w
V T -
^
p )
(3.30)
Sim ilarly for vapor,
n„ -
- f ' ’ kk^ V P
R vT P g
- { w
f
RaT
Pv kkgr
R vT Pg
P
'
VP
RaT
Pr
& )
R„T
' R„T
fcfcgr
VP
Pt-P Pg
P 2MaM vD eff,g
-V
)
R T [(P —p„) M a + pvM ;]
u] \ p )
Pv kkgr,
RvT
Pg
-V P
A/unA/rn
dD e f,f,, g
(PV pr —pvV P )
R T [(P - pv) Af0 + Pt,M„]
Pv
PvP
-
kkgj. ,
-V P
Pg
(pH vs»+
P W
V T -
* vp)
(3.31)
L iquid tran sp o rt is described by D arcy's law. T he volum etric flux of liquid
based on to tal cross-sectioned area is given by
qw = - ^ L v Pttf
Pw
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(3.32)
40
The m ass flux of liquid b ased o n th e total cross-sectional area is w ritten as
_
pw
kkyjy __
V Pu)
fbv
= -P v ^ V (P -P ')
Pw
=
- p w— V P - a mV S w - 6 TV T
Pw
(3.33)
w here the capillary pressu re pc is a function of S w an d T, therefore, the coefficients
am and ST/ are equal to
Om = ~Pw
kk-urr dpc
TT^~
P-w C/jtif
6t = - P „ — ^
Pw o T
(3.34)
(3.35)
3.4 Governing Equations for Mass and Energy Con­
servation
A schem atic diagram of m o d el (ID slab) w as show n in Fig. 3.3. The m icrow ave
com es in from left side (open b o u nd ary ) w here energy an d m oistu re are converted
aw ay to the surrou nd ing . T he rig h t side (insulation b oundary) is insulated w here
energy an d m ass flux are zero. O n th e open boundary, diffusional flux as w ell as
convective flux (liquid "p u m p in g ") can occur.
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41
\
\
A / \ ____ \
AA
W ater
expulsion
MW
Sym m etry
(Im perm eable
surface)
V apor
diffusion
Energy,
m oisture
convected aw ay
/\A
x (cm)
0
Figure 3.3: Schem atic diagram o f th e heating process w ith the boundary condi­
tions.
3.4.1
Vapor Mass C onservation Equation
at
+ V . (S .) = /
(336)
From Eq. 3.13,
dcv
dt
9 ,PvSg((>.
0 Ptr
9S|[
RvT dt
(fypv dSfi
~RvT dt
)
(1 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dpv dSu
(3.37)
Sw)
dSu, dt
42
S ubstituting Eq. 3.31 into Eq. 3.36, th e follow ing equation is obtained
eft
+K - ^ +
eft
=
eft
V(ATIVV S„) + V(JC*VT) + V ( K 3vV P ) + t
(3.38)
T-
_
____ MgM»Deff,g______ p &Pv
lv
R T [ ( P - p v) M a + p vM v] d S w
_ _ _ _ _ _ A/q M vDeff,g_____________p dpv
2v ~ R T [ ( P - p v) M a + p vM v] d T
MaMVDeJf^g
r-____ Pv kkgjSv ~ RvT pg ~ R T [ ( P - p v) M a + p vM v]Pv
t'
Pv , 0 (1 —sw) dpv
t\.± v =
—<P n -rr, +
RvT
RvT
d S IV
0 ( 1 —Sw) d
K*
Rv
&T
m
K ev =
3.4.2
(3.39)
0
Water M ass C onservation Equation
^ = + V . ( n „) = - /
(3.40)
From Eq. 3.16 an d Eq. 3.33,
dc
yy
<t>Pw-£r = V(pw—
CFt
V P ) + V (atnV 5ti;) + V ( 6 t V T ) -
I
(3.41)
W ater + v ap o r m ass conservation eq u a tio n
The ev ap o ratio n term / can be elim inated by ad d in g Eq. 3.38 to Eq. 3.41. So that,
the final form of vapor plus water eq u atio n becom es
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43
QO
«Vp
ut
ut
r\ q
+ As— = V ( K \V S W) + V (K 2V T ) + V ( K 3V P )
eft
(3.42)
w here
A'i
3.4.3
=
Kiv + am
A'2 =
&2v + 6t
a
a 3„
3
=
kkyrr
+ Pw~ '
Pw
A*4 =
A'.,, + 0Pw
A5 =
A or
a
0
6
(3.43)
Air M ass Conservation Equation
^ + V . ( n o) =
0
(3.44)
U sing Eq. 3.14,
dca _
dt
_
d U P - pv) (1 - 5tt,) 0 l
d t\
R aT
J
<f> / P — pv
I — Sw dpv \ d S w
Ra\ T
T dSw) dt
I— - £ ^ + ( i - s t—
Ra \ T 2
T2
y
w) d T \ T / j d t
<t>(l-Sw) d P
RaT
dt
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.45)
44
the Eq. 3.44 becom es
(Jt
+ K n ^ - + K XJ ^ ~ = V (K 7V S „) + V(A'8 V r ) + V(AT9 V P )
ut
C/t
(3.46)
w here
____ _______ M g M y D e f f,g________p &Pv
7 ”
jv8 ~
r* “
r,
10
- / 2 T [ ( P - p v ) M a + p „ M v] d S w
_________ A /aA /| >Deff,g________ —C?Pt>
~ P r [ ( P - p v ) M a + p wA /v] d T
P Pv kkgr
A /a A /v£)e/ / tg
1 / " ^
( f f [ ( P - p .) M ,+ |n ,,tf .]
<t> ( P - p v t \ - S w dpv \
Pa \ T
T
d S w)
*" - - £ { ^ - # + (1- S»>l=(7)}
K 12
3.4.4
0 (1
“Z " *
(3.47)
tigl
Energy Conservation Equation
C?
ut
—(c„/iv + ca/ia + ctl,/i11, + ca/is) + V -(nt,/iv + n arta + ntt,/iw) = V(fce/ / V r ) + Q (3.48)
U sing
^-(Cvhv) + V • (n,,/^) =
at
=
Q
TvT(^tty^w) + V *
ctt
=
§ f ( c M + V • (n .h .) =
/ir ^ r +
+ ^ V n , + Hr • Vft*
at
at
dhv
r*
. f
Cy
"h TlvVftv “b hvI
at
Qftryy
”b V
at
ca ^
+ naV 6 „
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.49)
45
Eq. 3.48 can be sim plified as
+ Cpafia + C^n*,) • V T = V ( k e ffV T ) - \ I + Q
(^cp)e/ /
(3.50)
w here
(pCp)eyy
SgPg<f)Cpg + SWPW0CpXV + (l
kef f — Pg(f)kg ~f" PW(pku, *F (1
4>) PS^pS
(pi) ka
(3.51)
(3.52)
Substituting Eq. 3.41 to replace th e evaporation term / , Eq. 3.50 becom es
^ l6~dt~ + ^ l7~dt "**
~
® ~ (Cpv™v
cpu'^w) V T + V(A'i3V5u,)
+V(hT 14V r) + V (A '15V P )
A 13
=
(im A
A 14
=
k ej f —
A " io
=
(3.53)
kkyff
Pn , A
Pw
A 16
=
AT 1 7
=
A T is
=
— \<t>pw
(P °p \ff
0
To sum m arize, th e final equ atio ns w ith variables S w, T an d P are as follow s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
k
J^£- + k M - +
at
at
A 'i o ^ + A 'n ^
eft
K l^ W
eft
eft
+ A -,2^
eft
+ K n% + K k%
=
V (A ',V S„) + V ( K 2V T ) + V(A'3 V P )
=
V( tf 7 V S„) + V ( K aV T ) + V (K aV P )
=
Q - (c ^ b . + CpaRa + c*.b„) V T + V (A 13V S J
+ V (A 'UV T) + V (A '15V P)
(355)
w here K i - K ls are given in Eq. 3.43, Eq. 3.47 and Eq. 3.54.
3.5
Initial Conditions
H eating is assum ed to sta rt from an uniform initial tem perature T, an d th e pres­
sure th rou gh ou t the sam ple is equal to th e am bient pressure P. S aturation S w is
also uniform initially for all cases except th e special case described in section 5.2.3
w here th e initial m oisture can vary in space and th e surface is m ore d ry th an the
center. T hus, in general, th e initial conditions are as follow s
sw =
f{ x ,y )
T
= Tt
P
=
Pamb
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(3.56)
47
3.6 Boundary Conditions
Two sets o f b o u n d ary conditions are needed for th e sides of th e problem . O ne of
th e sides w ill be assu m ed a d o se d bo un dary w here no m ass an d h eat exchange
takes place. The o th e r boundary is open, i.e, energy, liquid, vapor, an d air ex­
change w ith the su rro u n d in g can occur at this surface.
3.6.1
C losed Boundary
nv =
- K lvV S w - K 2vV T — K ivV P = 0
nw =
-O nVSu, - &rVT - pw— V P = 0
Pw
- K 7V S w - K sV T - K 9V P = 0
na =
0 =
nvhv + nwhw + naha — kef j V T
(3.57)
B oundary condition Eq. 3.57 can be rew ritten as the follow ing, consistent w ith
Eq. 3.55,
3.6.2
—A'i V 5 —A 2 V T —A'3V P
=
nw + nv = 0
- A W 5 - A sV T - AT9V P
=
na = 0
—k e ffV T
=
0
(3.58)
O pen Boundary
In the open boundary, pressure driven flow or "pum ping" condition m ay occur
w here liqu id is p u sh ed across th e boundary w ith o u t any change of phase (this
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
can occur d u rin g intensive m icrow ave h eating o f high m oisture m aterials). N o
"pum ping" condition sim ply m eans n o liq uid m oves across the b o u n d ary w ith­
out phase change. Each o f these b o u n d ary conditions is described below
O pen b o u n d ary w ith o u t liq u id "p u m p in g "
In this case, m ass transfer o n the surface is assum ed to be in equilibrium w ith the
surrounding w hich m eans th at all m ass flux reaching the boundary from inside
is converted away. R egardless of volum etric ev aporation present inside, surface
evaporation occurs sim ultaneously w hich m eans th a t there is liquid flux crossing
the boundary and vaporizing instantly. The surface evaporation only affects the
boundary m ass and heat flux and is determ in ed based on th e surface area covered
by the liquid.
Tlv " f W® =
=
P
nvhv ■(■nwhw
naha
j j VT
Tlv "I" Tlwev —0 ( 5 g
+
Sw )
|
^mv
Pam b
n vhv 4" Tiyjhv 4“ Tiaha 4~ h, (T 'Pamb)
(3.59)
Eq. 3.59 can be rew ritten as the follow ing, consistent w ith Eq. 3.55,
- K xV S - K 2V T - K zV P
=
H-wev 4" Tlv —
P
=
Pam b
- k ef fV T
=
h ( T —Tamb) + nwA
^ F^T
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^mv
(3.60)
49
There are tw o w ays to replace riy, in th e energy b o u n d ary condition. O ne is to
substitute
= <f>Sw
— Pvo) fhnv- The other is to su b stitu te w ater flux (Eq. 3.33).
The second o n e is u se d here to be consistent w ith o u r final differential equations.
- K l3V S - K l4V T - A'15V P = h (T - Tamb)
(3.61)
O pen b o u n d a ry w ith liq u id "p um ping"
W hen th e in tern al p ressu re gradient pushes th e liq u id to th e surface at a h ig h
enough rate such th a t th e boundary evaporation cannot keep u p w ith it, th e su r­
face w ill b e fully satu rated . A t this point the w ater becom es free and there is no
capillary force w hich can hold the liquid w ithin th e m aterials. Even a little over­
pressure in sid e com pared to the boundary can cause th e liquid to flow out. T his
part of liq u id loss is called as "pum ping" and is tru ly due to the convective flow
and can n o t b e ev ap o rated so that no latent h eat is released. The necessary con­
dition for th e "p u m p in g " is (C onstant et al., 1996)
Sw =
1
(at surface)
Vpu, • n > 0
(at surface)
(3.62)
n is the o u tw ard vector norm al to the surface. T he boun dary conditions are d e­
scribed as
nv -I- riuj = nv +
-I-
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50
P
T i y -H nwhw ~f*n aha
=
(f){Sg + S w) ^ FLyT
—
Pam b
hjTlV
kej f V 1 — Tivhv •!■n wev^v "t" TiWphw "h n aha
+ h (T - Tamb)
(3.63)
w here n w = nwev + n ^ . E q .3 .63 can be rew ritten as the follow ing, co n sisten t w ith
Eq. 3.55,
-ATiVS —AaVT — A3 V P = nw -F n v = <t>(Sg + Sw) ^
P
=
Pamb
-k e ffV T
=
n^evA + h ( T — Tamb)
~
Pvi)^j ^lmv
nwp
(3.64)
It is understood th at th e th e liquid is p u m p ed o u t d u e to th e total p ressu re gradi­
ent. Therefore,
««, = -P v —
VP
(3.65)
Substituting Eq. 3.65 in to boundary co nd ition (Eq. 3.64) an d rearran g in g the
energy boundary condition, the final form of b o u n dary conditions w ith liquid
"pum ping" are w ritten as
-A ', V S - A'2V T - tf3„V P =
0 (S 9 + S„)
P
~
Pam b
K l3V S - K u V T
=
h { T - T {'„*)
- ft*,) Am„
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.66)
51
j:
03
500
bO
2 400
J 300
o>
3 200
a
1 100
<o
OS
1
4
2
3
H eating tim e (m in)
5
6
Figure 3.4: L iquid expulsion profile. D ata show n here at 5s intervals.
It is im p o rtan t to m aintain th e continuity of b oundary conditions from n o n ­
expulsion Eq. 3.61 to expulsion Eq. 3.66. The u n d erly in g physics is th at th e surface
liq u id satu ratio n first increases till unity w ith o u t expulsion. A fter th a t there is no
im m ediate expulsion u n til th e pressure g rad ien t is b u ilt up to start the expulsion.
It is ju st like a sw itch w hich controls the tw o regions w ith different m echanism s.
T herefore there is no discontinuity for the tw o ty p es of boundary condition b u t
th e slope of th e liquid expulsion vs. tim e is in d eed discontinuous d u e to th e dif­
ferent m echanism , as show n in Fig. 3.4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
3.7 Volumetric Energy Absorption in Microwave Heat
ing
A bsorption of m icrow ave energy (heat source term in th e energy equation) in the
m aterial can vary strongly d ep en d in g on several factors such as the dielectric p ro p ­
erties of food, the volum e an d the shap e of the food m aterial, the volum e o f the
m icrow ave cavity (oven), an d th e location of the food in sid e th e m icrow ave oven.
To m ake m atters w orse, as th e food is heated, the tem p eratu re field and the re­
sulting m oisture field pro duces a field o f dielectric p ro p erties th at is constantly
varying. This in tu rn changes the energy absorption th ro u g h o u t the m aterial. A
com prehensive analysis of th is process w ill require coupled solution of the elec­
trom agnetics an d the p o ro u s m edia tran sp o rt m odel. T he solution to 3D electro­
m agnetic m odeling of a cavity w ith continuously v ary in g dielectric properties h as
not been obtained, and is a m ajor research project b y itself.
For th is study, th e electrom agnetics is sim plified an d an exponential dro p of
the energy, as u sed by m ost of th e p revious studies (Wei e t al., 1985; D olande e t
al., 1993; Z eng e t al., 1994) involving m icrow ave h eat transfer, w as used. The m i­
crow ave flux at a position x is given by
(3.67)
w here Fq is the m icrow ave flux at the surface and 6 is th e penetration d ep th of en­
ergy th a t is related to th e dielectric constant and th e dielectric loss by th e relation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
6 = ^ ( 2 e '((l + (e"/e ' ) 2)a 5 ~ l ) ) - 0 '5
(3.68)
H ere A0 is the w avelength of m icrow aves in free space, e7 is dielectric constant and
e" is th e dielectric loss. The d ielectric constant and the dielectric loss w ill vary
as tem p erature and m oisture changes spatially, m aking th e penetration d ep th a
function of these tw o variables. T he m icrow ave flux term in From Eq. 3.67 can be
m odified (N i and D atta, 1994) fo r a varying penetration d ep th as
(3.69)
The volum etric heat source term in th e governing energy equation (Eq. 3.48) can
be d ed uced from the flux eq u atio n given by Eq. 3.69 as
(3.70)
The sim plified equation given b y Eq. 3.70 retains the essence of spatial variation
in m icrow ave heating for great m any situations. Use of th is sim ple b u t realistic
approxim ation of the m icrow ave source term w ill allow th is stud y to better focus
on the m ass and heat transfer aspects of the process.
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54
3.8 Summary of Governing Equations, Initial Condi­
tion and Boundary Conditions
The final eq u atio n s developed in the previous sections are sum m arized h ere for
clarity. The final th ree unknow ns are tem perature T , total pressure P , an d w ater
saturation Sw. T he governing equations in th e se three unknow ns are given by
+ Kh% + K * lit
KJ ^
ut
+K
*
ut
osw^ „ ar
16~ 0 T
^~dt
+K ”
ut
= V(A'>V S»> + V (A W r) + V(A'3 V P )
(3.71)
V(A'7V S „) + V (K aV T ) + V (i-,V P )
(3.72)
=
dP _ A
^
"dt™ —
\ xtt
■(“ ^pfl^a "I" CwTlw) v i
+V(AT13V St<,) + V(AT14V T) + V(ATl5 V P )
(3.73)
The boundary conditions for the closed side are
—A'i V S —K 2V T —A 3 V P
=
n,,, + nv = 0
- K j V S - A'8V T - A gV P
=
na = 0
—kef f V T
= 0
(3.74)
For the b o u n d ary conditions on th e open side, first the follow ing conditions are
checked to d ecid e if there w ill be "pum ping" a t the boundary.
Sw =
1
(a t surface)
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55
Vpw • n
> 0
(at surface)
(3.75)
If the conditions for "pum ping" are satisfied, th e follow ing b o u n d ary conditions
are used
- K tV S - K iV T
-
K 3vV P
= <t>(Sg + S„)
P
=
Pam b
K l3V S - K l4V T
=
h { T - T amb)
- Pu0)
(3.76)
O therw ise, the follow ing bo un dary conditions are used.
- K \ V S - K 2V T - A 3 V P
=
nwev + n v = <t>(Sg + Sw)
P
=
Pamb
- K l3V S - K UV T - K l5V P
=
h { T - T amb)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
~ Pvo) Knv
(3.77)
Chapter 4
NUMERICAL SOLUTION AND
INPUT PARAMETERS
In th is chapter, the finite difference equations for th e interio r and the boundary
are dev elo ped. The m atrix assem bly an d th e solution algorithm are described
next. C onvergence of th e solution and th e energy an d th e m ass conservation w ere
checked. Possibility of th e existence of sonic speed d u e to interned pressure devel­
opm ent is investigated. Finally, the in p u t d ata for food m aterials are described.
4.1
Governing Equations and Boundary Conditions in
l-D slab
A sum m ary of th e governing equations an d b o u n dary conditions are noted here
for th e special case of ID slab. From Eq. 3.73, ID g overning equations can be sim 56
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57
plified as
as
ar
ap
a ,„ds. a ^ a r, a,^ap,
*4at + AsaF+ KeaT = to(K*to) + aF(*2aF:1+di{K* 1 (41)
ft- 9 S + ft- 9 T + ft' 9 P
K l0 d t + K n ~m + Kl2~ft
^ as .. ar „ ap
Ari6^
+ A ,7 a r + K l8 a r
- 9 I K dS)+ ° ( K 9S\+ d t K dS\
~
+ t o (K * t e ) + & (A » t o )
=
a,., as,
a,„ as,
a
^
^
^
a
a
^
a.„ as,
u n
(42)
+ a ^ a ^
+<j
(4.3)
w here convective term in energy eq u atio n w as dropped because it is m uch sm aller
com pared w ith the laten t h eat (N asrallah e t al., 1988). The b o u n d ary conditions
are given as
x =
0
(w ithout "pum ping"):
dS
~dx
rr dS
3dx
x =
0
8T
9x
r, dT
4~dx
dP
9x
_
f pv
\
~ H K X ~ pM) h m
P = Pamb
dP
,
5~dx =
~ Tomb)
(4.4)
(w ith "pum ping"):
K
^
OX
+ K
^
+ K
dP
ox— r A3w“H~
ox
p
^
13——
ox
&r
I- K u -£ -
ox
**)
~
0 ( £
=
Pamb
=
^(T’ —TamD)
t
x = L:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.5)
58
(4.6)
4.2 Discretization of the Governing Equations
Finite difference m ethod w as u sed to discretize the governing equations w ith cen­
tral difference in space and the C rank-N icolson (C-N) schem e. Since C-N schem e
is unconditionally stable for a linear problem , it is hoped th a t it can also provide
better stab ility for a non-linear problem . A lthough both uniform an d non-uniform
m esh are u sed , an exam ple of the discretized equations are given here for the uni­
form m esh, as show n in Figure 4.1.
1
2
<l ■ o
i- 1
i
i+ 1
O I O I o
N -l
N
o
L
0
Figure 4.1: N odes diagram .
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59
■as = IC,+U2 s?+l - s?
K m
c\
"
^
nc
ax1
(4-7)
t^ri+1 on+1
( jy~ri+l
5xj
, jv-n+1 \ ot*+1 . fc'-n+l cn + l
2(A x )2
,
a?+i/2^ i - (*?+i/2 + *T -in )S? + ^ r - i/2^ - i
2(A x ) 2
(4 g)
An exam ple of the discretized form of Eq. 4.1 is show n in Eq. 4.9.
cio =
c \S i-i + czTi-\ + czPi-x + C4S 1
+C5 Ti + c^Pi + c7Si+i + csTt+ 1 + cgPi+i
where
r^n+l
=
61
c4 =
2
r'n + 1
3,—1/2
2(A x )2
3
* c rur r+—rs* c u H.--------« r m
2(A x )2
At
* k u + * & ; „ . i < r l/2
2(A x ) 2
At
=
* C U
+ * C i /2
2(A x ) 2
06
~
A 2,-l/2
2(A x ) 2
=
°5
7
r,Ti+l
‘.-1/2
2(A x )2
r/~n+ 1
ft+1/2
2(A x )2
8
, ^ n+1/2
At
~
t^n+ 1
%+l/2
2(A x )2
^
~
jv-«+l
^i+1/2
2 (A x ) 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.9)
60
■n-t-l/2
i+l/2
■n+l/2
n+ 1/2
2(A x )2
2(A x )2
2(A x ) 2
2(A x )2
w here A'i ± l / 2 = 0.5(A', + A'l±1). T he rest of th e equations in Eq. 4.1 - 4.3 w ere
discretized in a sim ilar way.
4.3 Discretization of the Boundary Conditions
The discretized equations for th e b o u n d ary are given as
O pen boundary, i
—
1 (w ithout "pumping"):
s r 1- s r 1^
2
Ax
—
I-**/* 4
2
Ax
Ar4AxST+ l - S ?
2
At
AT6 A x P T + l - ^ T
2
At
13l-H/2 *^2
2
t
Ax
i+l/2
2
Ax
A5A x7T +‘ - 7T
2
At
L? + l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rrn
?*'
61
P?+l - K +‘
s? - S f
2
Ax
2
Ax
T j ~ 7?
JOs.h.,,, / ? - / ?
2
Ax
2
Ax
K l6A x 5T+1 - S?
K l7A x Tp+l T?
2
At
2
At
K is A x PP +1 - P [
(4.11)
2
At
-
Open boundary, i
1 ( , p.
\~RvT
—
1 (w ith "pumping"):
, V
~
_
s ? +1 - s r 1 ,
7 T ‘ - 7 T +‘
) v -----------2----------A x ----- + ~ 2 -----------S J ----* & ! ,/, * r ' - p ? "
2
Ax
T ? -T ?
,
2
Ax
A 4 A x S ? +1 - S?
2
At
A'6A x Pp +1 - Pp
2
At
pn+I
•*1
_
2
s?-sr
Ax
K %Vi+i/2 p n - p n
2
Ax
A'5 A x 7 7 +1 - 77
2
At
p
—*ami
ATo
_ cn+1 , AT.+
1
_I T^R+l
L/TW —'T
13l5"1+
l/2 en
°2+1 —
l/2-*2T**+l
I
h(T?
T’am\* ) _
---------------- + —14l+
----------— ------
s? - s?
7? - 7?
2
Ax
2
Ax
A 16A x S ? +1 - S?ATl7Ax 77 +1 - 77
2
At
2
At
K lsA x Pp +1 - Pp
2
A*
(4.12)
For the convective h eat an d m ass tran sfer term on the open boundary, the values
a t the p rev io u s tim e step (n) is used for sim plicity. A lthough th ere is a tim e lag,
the erro r is sm all because th e rate of b o u n d ary value changes in th is case is very
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62
sm all.
C losed boundary, i =
0
—
A'!*+1
on+1
lN - i / 2 ^ N ~
2
Ni
pn+l
° N - l
.
2
~ S fr -1
2
K jA x S y ^ -S S
*
—
*tv—e/ 2 J Af ~
2
2
pn+l
, * v » t v - t / 3 ^ iV
Ax
* ? » - ■ /, ^
2
n
o
T>n+1
t ^i + 1
f.n + 1 •* /V______•*
1 .
C//
AX
A*
A s A j P % + '- P R
r .n
jw + l
1 N -l .
At
A7?+ *
At
pn+l
9 n - i / 3 * /V
2
~
pn+l
* N- 1
Ax
PZ - P S -1
Ax
2
'T>n
Ax
- 7y_,
K u & rT F '-T R
C//
A'16A x S ^ + 1 - 5 ^
2
2
pn+l
^ |V - l
Ax
2
Ax
- S fr-i
2
Ax
A TioAxSy1 - S %
2
At
~
~
^ 3 JV-1/2 P"n ~
2
rrn + l
pn+l
3 j v - i / 3 * iV
2
At
fC2+ *
^ A T -l
A ? +1
,
Ax
K s A x T p -' -T %
At
pn+l
’-pn+l
~ * N - 1
^ N - l / 2 T n ~ ^ iV -1 ,
2
A t
A.-r
2
'T’TH-l
2 S - i/ 2 1 N
Ax
■ ^ ljV -l/2
o
AT?’*’1
2
Ax
K n A x P% +[ - P R
2
At
'T ’n
W —1
At
AT17A x T%+ i - 7 %
2
At
A '18A x P ^ + 1 - P ^
2
At
1
4.4 Matrix Assem bly and Solution Algorithm
A ll equations are assem bled in one m atrix so th at all variables (S?+1, TJt+l, P*+l)
are solved sim ultaneously. T he m atrix w as form ed based on the o rd er of node
nu m b er {i = 1, N). For ev ery n o d e n u m b er there are three equations— the com ­
bin ed w ater an d vapor eq u atio n , th e a ir equation an d the energy equation. The
final m atrix is 3N x 3N , a s sh o w n in Fig. 4 2 .
The difference equ atio ns w ere coded in FORTRAN and the m atrix w as solved
by th e LAPACK library. T he nam e LAPACK is an acronym for L inear A lgebra
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63
”
w ater + v ap o r equation (S, T, P)
i = l ---- ►
—
bl -
air eq u atio n (S, T, P)
b2
energy eq u atio n (S, T, P)
b3
w ater + v ap o r equation (S, T, P)
b3N -l
air eq u atio n (S, T, P)
b 3N-2
energy eq u atio n (S, T, P)
b3N-3
__
3N x 3N
mm
3N x 1
F igure 4.2: M atrix structure.
PACKage. LAPACK is a library of FORTRAN 77 subroutines for solving linear
equations such as in this problem . It has been tested and optim ized through Basic
L inear A lgorim Subroutine (BLAS) in m any sup ercom pu ter an d w orkstations. In
th is w ork w e used the LAPACK installed on th e SP2 supercom puter in the C ornell
T heory C enter as well as th e H P735 and th e SG I w orkstations in th e d ep a rtm en t
The solution algorithm w as show n in Fig. 4.3. There are five subroutines (GRID,
INITIAL, PARAMETER, COEF, LAPACK). S ubroutine GRID is for creating u n i­
form o r non-uniform m esh. S ubroutine INITIAL is for setting th e initial values
of 5 , T an d P. Subroutine PARAMETER is fo r setting constant properties an d
p aram eters related to th e boundary. S ubroutine COEF is for calculating all the
m atrix coefficients. Subroutine LAPACK is for solving th e m atrix.
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64
call GRID
call INITIAL
call PARAMETER
DO 100 n=l, NP
time(n+l )=time(n)+dt
NO
IF(time(n+l )<TEND)
S11, T0, Pn| ^
call COEF
call LAPACK
IF(abs (S1+1-S1)<stol)
IF(abs (TI+1-T1)<ttol)
I£(abs (PI+1-Pl)<ptol)
NO
NO
YES
YES
100
CONTINUE
^ dt=dt/2 ^ ""
One time step finished
Figure 4 3 : Flow diagram .
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65
A fter so lving the m atrix, the difference in values from th e tw o iteration levels I
an d l + l are com pared w ith the tolerance o f each v ariab les (sto/=1.0e-5, ttol=l.Oe5 an d ptol=1.0). If the tolerance is not m et, S l+l, T l+l a n d Pl+l are u sed to calculate
the new coefficient m atrix an d the above procedure is rep eated u n til the tolerance
is satisfied. If the iteration level I is larger th an N I T E R w ith o u t convergence, th e
program autom atically sto p s and prints o u t "solution is div erg ent". If tim e v ari­
able tim e(n) is larger th a n TEND, the program sto p s a n d the calculation is fin­
ished norm ally.
4.5
Convergence and Mesh Refinement
M esh refinem ent is used to check for convergence o f th e num erical results. T he
in p u t d ata fo r potato w ere used for these calculations, as show n in Table 4.1.
Table 4.1: P aram eters used in m esh convergence check
C apillary diffusivity
V apor pressure
Perm eability
P otato d ata
Potato data
5 x 10- 14m 2
(see Section 4.8)
(see Section 4.8)
Thickness
1
cm
End tim e
6
m in
The CPU tim e requirem ent as a function o f th e grid size is show n in Table 4.2- A s
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66
an exam ple, w h en th e n o d e num ber doubles from 21 to 41, CPU tim e increases
about 4 tim es because th e m atrix size increases by a factor 4. H ow ever, as the node
num ber d ou b le from 41 to 81, CPU tim e increases m ore than 4 tim es because the
tim e step m u st be red u ced as the grid size decreases so th at the total CPU time is
even longer.
Table 4.2: N um ber of n odes and space increm ent
N u m ber of nodes
21
41
61
81
101
A r (m m )
0.5
0.25
0.167
0.125
0 .1
CPU (s)
6.87
28.5
83.0
194.1
322.1
Effect of changing n o d e num bers on S, T , P, and m oisture content are show n in
Fig. 4.4. It ap p ears to be sufficient to use 21 nodes in the calculation.
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67
2 0 .4 4 -
N ode number=21,41,61,81,101
N ode number=21,41,61,81,101
038x (cm)
1.0081.0061.0041. 00 2 -
Node number=21,41,61,81,101
T------------1----------- 1------------1----------- 1
0.2
0.4
0.6
x (cm)
0.8
1.0
Figure 4.4: C onvergence check in
6
m in u tes for different g rid sizes.
4.6 Check for Energy and Mass Conservation of the
Numerical M odel
The num erical m odel w as checked for satisfyin g the conservation eq u atio n s for
energy an d m ass. In Table 4 3 , the v apo r lo ss from the surface com puted from
hmviPv —Po)&t is found to be very close to th e decrease in the liq u id w ate r inside,
as com puted from £ ( 0 pwA Sw). Table 4.4 sh ow s th at heat loss from th e surface,
as calculated from h A T is very close to th e sensible heat and laten t h eat insid e, as
calculated from Am + £ A (eipicpiT).
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68
Table 4.3: W ater loss conservation check
4.7
<f>pwA S w
tim e
hmvipv ~ P o)A t
Relative error
(m in)
(k g /m 2)
(k g /m 2)
15
0.45713708
0.45713741
6 .0
30
0.77028244
0.77028790
7.1 x 1 0 -‘
45
1.07525362
1.07526646
1 .2
60
1.37612997
1.37614887
1.4 x 10" 3
75
1.65833085
1.65835494
1.45 x 10~ 3
90
1.84268157
1.84272995
2 .6
(%)
x
x
x
1 0 '5
1 0 '3
10 "3
Possible Existence of Sonic Condition in
Microwave Heating
As the inside pressu re rises d u e to internal evaporation and gases are expelled
at increasing velocity, it is interesting an d necessary to address w h eth er a sonic
condition exists in th e porou s m edia u n d er intensive m icrow ave heating.
4.7.1 Estim ation o f the Maximum Mach Number
The first approach is n o t related to the num erical m odel, b u t it is described here
because of context.
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69
Table 4.4: Energy conservation check
tim e
kA T
Am + £ A{eiPiCpiT)
relative error
(m in)
(W /m 2)
(W /m 2)
(%)
15
83.79
83.83
4.8 x 10~ 2
30
83.68
83.72
4.8 x 10- 2
45
83.20
83.23
3.6 x 10~ 2
60
82.17
82.21
4.9 x 10~ 2
75
78.73
78.77
4.9 x 10" 2
90
60.85
60.87
3.3 x 10~ 2
We look at 1-D slab an d neglect all the sensible heat, therefore all the absorbed
energy is used for converting the liq u id into vapor, w hich predicts the m axim um
vapor generation. A furth er assum p tion is that n o vapor can be released u n til th e
vapor pressure reaches th e specific lim it pm after certain heating tim e t h. W hen
vapor releases, it is assum ed to have average velocity uav in a very sh o rt tim e tr,
w hich predicts th e m axim um v apo r velocity as w ell as m axim um M ach num ber.
We consider the effect of th e w ater saturation level on the vapor p ressu re in
term s of th e state eq uation b u t any sm all decrease in w ater saturation is ignored,
w hich is th e case in th is estim ation since the ratio o f liquid w ater to v ap o r is
1000
tim es. M icrow ave flux obeys L am bert's law. H ow ever, instead of using F = F0e~»
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70
w here 6 is a function o f w ater satu ratio n level, fo r sim plicity, w e u se th e constant
penetration d ep th 6 b u t m odify surface flux to b e <j>SwF0, w hich is q ualitatively
correct.
Ff = 4>SwF0e ~ f
(4.14)
For the region from th e surface (x = 0) to an y location x, the to tal p o w er ab­
sorption is given by
A F = <t>SwFo(e~* - e~f ) = <t>SwF0( 1 - e~f )
(4.15)
The rate of v ap o r generated (k g /m 2 /s ) after neglecting the sensible heat is
equal to A F div ided by latent heat of w ater A.
AT _ W
l- e - f )
A
(4 16)
A
A fter certain heatin g tim e £/, the stead y state v ap o r pressure is d eterm in ed by
state equation.
771
PmVg = — R T = pAx<t>( 1 - Sm)
(4.17)
w here M v is m olecular w eight of vapor; m is m ass o f vapor; Vg = Ax<t>{\ — S w).
Equating Eq. 4.16 w ith Eq. 4.17, w e have
m
pmx<t>(l - SW)M V
A ------------ R f ---------
AFfx
Su,Fo( l - e 7 )
= -----------A--------- th
0
Therefore,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<4-18)
71
i
i
i
0.0
0.5
i------------1------------1------------ 1
1.0
1.5
2.0
2.5
D epth of the region x (cm)
3.0
Figure 4.5: H eating tim e taken to reach pressure 2.0 atm vs. different thickness of
m aterials.
*
Pm^i 1 SW)A'IVA
th = g
----- 1 x
5 WF0(1 —e »)RT
(4.19)
The Eq 4.19 show s th at the h ea tin g tim e taken for the vap or to reach the specified
p ressu re for different w ater sa tu ra tio n level, as plotted in Fig. 4.5.
If w e denote uav as average velocity of vapor, tr as release tim e for th e vapor,
umPvtr<p(l - Sw) = — ^
lx - e ~T)th
(4.20)
S u bstitu te Eq 4.19 into Eq 4.20,
«- - ^
FT
R Tpvtr
(4.2D
Eq. 4.21 show s th e m axim um v a p o r velocity respective to different thickness and
different releasing tim e u n d er certain pressure lim it, as show n in Fig. 4.6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
Pm=2*° atm
F0=2.7W /cm ;
5 = 1.5 cm
T = 9 0 °C
0.0
0.5
1.0
1.5
2.0
2.5
D epth of the region x (cm)
3.0
Figure 4.6: Pore velocity o f vapor vs. different thickness of m aterials for different
releasing tim e tT.
The M ach num ber (A/a) is calculated.
m ax\M a\ —
a
~ 0.03
(4.22)
w here a is sound velocity. T hus far th e estim ation show ed th e sonic speed w ill
not occur u n d er th e above conditions d u rin g m icrow ave heating. H ow ever, there
m ight be som e cases w h ere th e velocity can be higher. For exam ple, for the real
m aterials w ith v ary ing p o re size distrib utio n, if th e sam e am ount of v ap o r p ush
o u t from the least size o f pore, it can produce m uch hig her velocity. In addition,
for som e m aterials th a t are m echanically stronger, th e in tern al p ressu re build-up
can cause the m aterials to blow u p, as a result, a very hig her v ap o r velocity can
be generated in a very sh o rt tim e. H ow ever, it is im possible to reach sonic speed
for o u r current heatin g situ ation .
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73
5 min
1510-
5-
0.0
0.4
0.8
0.6
1.0
x (a n )
Figure 4.7: S patial d istrib u tio n of Ma at different heating tim e.
4.7.2
Numerical Calculation of Mach Number and
Reynolds Number
A ccording to the com pressible flow dynam ics, the com pression w ork of the gas
can n o t be ignored and m u st be ad d ed into energy conservation equation once
M ach num ber larger th an 0.3. The com pression w ork is described by
DP
d
T =
dP
a T + (r* ' V ) P
(423)
w here P is total pressure an d vg is gas velocity. The M ach num ber w ith com pres­
sio n pressure w ork is sho w n in Fig. 4.7. The m axim um value u p to 5 m inutes of
m icrow ave intensive heatin g is a b o u t 2 0 x
1 0 -6.
The calculation also show ed th at
th ere is no significant difference betw een M ach num bers w ith com pression w ork
an d w ithout.
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0.0
0.2
0.4
0.6
x (cm)
0.8
1.0
F igure 4.8: Spatial d istrib u tio n o f Re at different heating tim e.
T he R eynolds num ber w ith com pression pressure w o rk is show n in Fig. 4.8.
The m axim um Re (< 5 m inutes) d u rin g intensive m icrow ave heating is 60 x 10“6.
A gain, n o difference w as found b etw een Reynolds num bers w ith com pression
w ork a n d w ithout.
In ad d itio n , it w as also found th a t M a and Re are no t sensitive to th e surface
m ass tran sfer coefficient and initial w a te r saturation. H ow ever, Re is q u ite sensi­
tive to perm eability show n in Table 4.5. Therefore the num erical calculation also
show ed th a t sonic speed w ill not o ccu r d u rin g m icrow ave heatin g and R e is m uch
less th a n 1 so th at the D arcy's law is v alid .
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75
Table 4.5: Effect o f perm eability on Ma an d Re
k ( x l 0 - 14m 2)
M a (X l 0 -5)
Re (x lO -5 )
0.25
1.26
1.14
2.5
2 .1 2
6 .1 0
25
2 .1 0
19.5
4.8 Input Parameters for the Model
In this section, the physical p ro p erties of food, especially of potato, required in th e
m odel are discussed. Porosity an d satu ratio n of the porous m edia are related to
the w ater content. T here are th ree ty p es of transport in th e porous m edia m odel
— m olecular diffusion, capillarity, an d pressure driven. For m olecular diffusion,
effective g as diffusivities of v ap o r in air an d vice versa are discussed. C apillary
diffusivity is discussed from theoretical considerations as w ell as available exper­
im ental d ata. For pressure d riven flow, perm eabilities of liquid w ater and gas are
discussed.
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76
4.8.1
Equivalent Porosity and Initial Water Saturation of Porous
Materials
The eq u iv alen t porosity <f>an d w ater satu ratio n S, as th ey relate to the app arent
density u sin g Eq. 3.19 an d Eq. 3.20, are p lo tted in Fig. 4.9 an d Fig. 4.10, respec­
tively. V egetables and fruits are generally m ade of carbohydrates and w ater. The
density of the carbohydrates ps is assum ed to be 1419 k g /m 3 a t T = 25°C (Choi et
al., 1986).
0
an d Sw of som e com m on fru its an d vegetables a t th eir typical m ois­
ture co nten ts and ap parent densities are calculated and show n in Table 4.6. Ac­
cording to th e table, apple is m ore porous th an banana, p o tato and carrot, w hich
conform s to w hat is expected.
1.0
.IT
Cfi 0 9-+■
2
a. 0 . 8 ^ : - — .......... /
+*
*
*
0 .7 M (w.bj~
<s
J
>
33
O'
w
0. 6 -
0.5
100% - - 70%
90% — 60%
80% - • 50%
—r~
600
...... .......
I
T
“ I---------- 1---------- 1
800
700
900 1000 1100
A pparent density pap (k g /m )
1200
Figure 4.9: E quivalent porosity vs. ap p aren t density at different m oisture content.
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77
1----------1---------- 1----------1--------- 1---------- 1----------1
600
700
800
900 1000 1100
A pparent density pap (k g /m )
1200
Figure 4.10: W ater satu ratio n vs. ap p aren t density a t d ifferent m oisture content.
4.8.2
Perm eability o f Liquid Water and Gas
Perm eability describes tran sp o rt d u e to th e g rad ien t in to tal pressure an d is per­
haps th e m ost im p o rtan t p aram eter in m icrow ave heatin g. Sm aller th e perm eabil­
ity less perm eable the m aterials an d higher th e in tern al pressure w ill be, w hich
can even cause th e m aterial to blow up . Larger th e p erm eab ility faster the m ois­
ture tran sp o rt an d therefore low er th e internal pressure.
Perm eability can be m easured directly. H ow ever, fo r th e deform able and hy­
groscopic m aterials such as m ost foods, m easurem ent is difficult since perm eabil­
ities are quite low an d the m aterials deform . W ater perm eability is n o t available
except for th e stu d y on m eat by C ontreras (1987). W ork o n gas perm eability is also
rare, w ith the exception o f th e stu d y on bread (Tong an d L und, 1993). Perm eabil­
ity d ata used in th ese an d o th er p o rou s m edia stu d ies are listed in Table 4.7. Thus,
reasonable approxim ations need to be m ade for th is study.
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78
Table 4.6: Porosity <pan d saturation Sw in the raw state of several com m on foods.
D ata on m oisture content M and density p from Rahm an (1995)
M aterials
P orosity
W ater satu ratio n
G as saturation
M (w.b.)
p( k g /m 3 )
<t>
Sw
Potato
0.83
1040
0.875
0.986
0 .0 1 2
C arrot
0.9
1040
0.927
1 .0
0
A pple
0.87
843
0.922
0.794
0.19
Banana
0.76
980
0.834
0.893
0.089
Intrinsic and relative permeability
To arrive at realistic perm eability values fo r food m aterials, it is n o ted th a t the per­
m eability o f a m aterial to a fluid, such as kw, is related (Bear, 1972) to th e intrinsic
perm eability k ^ of the m aterial and th e relative perm eability o f th e fluid to that
m aterial,
by
kyj — kwikwr
(4.24)
The intrinsic perm eability kw represents th e perm eability of a liq u id o r gas at fully
saturated state, an d corresponds to its m axim um value. It d ep en d s o n th e internal
structure of th e m aterial, particularly th e porosity and the po re size distribution.
A ttem pts to form ulate the intrinsic perm eability in term s of p o ro sity h as lim ited
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79
Table 4.7: In trinsic perm eability data in porou s m edia m odel
M aterials
G as intrinsic
L iquid intrinsic
perm eability
perm eability
kg x
ku. x
1 0 ~ 14m 2
References
1 0 ~l4 m 2
sandstone
1 .8
1 .8
Wei et al. (1986)
clay brick
2.5
2.5
N asrallah et al. (1988)
Die and T urner (1989)
light concrete
softw ood
0 .0 1
0 .0 1
N asrallah at al. (1988)
w ood
0 .1
0.05
Stanish e t al. (1986)
flour dough
2 .0
G oedeken et al. (1993)
(0 =0 .1 )
2300.0 (0=0.6)
m eat
15-46
C ontreras (1987)
success (Bear, 1972) fo r rigid porous m aterials and h as no t been attem pted for bio­
m aterials used as fo od . The relative perm eability such as
for a m ultiphase
flow is a function o f h o w m uch fluid is in the pore volum e. N o stud y exists for
relative perm eability values for food m aterials.
For hygroscopic m aterials such as food, there are additional m ajor com plica­
tions. Such m aterials shrink d u rin g drying o r h eatin g stage w hich changes the
pore structures (and therefore the intrinsic and th e relative perm eabilities). As an
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80
exam ple, consider raw po tato w ith m oisture content 0.83 (w.b.). T he liq u id per­
m eability at th is stage w o u ld b e alm ost th e sam e as th e intrinsic liq u id perm e­
ability because the liq uid satu ratio n is alm ost unity. H ow ever, a t th is h ig h m ois­
ture content, it h as only tin y pores an d th e liq u id intrinsic perm eability is very
sm all. As the p o tato dries, big m acropores are developed as w ell as th e solid m a­
trix shrinks. The liq u id intrinsic perm eability
should becom e larg er d u e to th e
larger pores as th e m aterial dries. H ow ever, the liq u id perm eability o f rig id nonhygroscopic m aterials decrease w ith decreasing liqu id content. For exam ple, for
a very dry potato, th e liq u id perm eability kw is even low er th an th a t of the w et
stage. This is because of very low liq u id relative perm eability k ^ , as explained
later. There m ight be a sm all region betw een very w et and very d ry stag es w here
kw is the highest. H ow ever, for th e sim plicity of m odel and also d u e to th e u n ­
know n n ature of variation o f kw as a function o f M , it is assum ed th a t kw is con­
stant and equal to th e k ^ a t th e raw state. This assum ption of fc*. can be fairly
reasonable approxim ation a t both very w et an d very d ry stages. T here m igh t be
som e underestim ation in betw een.
For the gas intrinsic perm eability kgi, there is less com plexity. Sim ilar to kw, kg
increases w ith decreasing liq u id content. H ow ever, th e gas content also increases
w ith decreasing liq u id content so th at the gas relative perm eability kgr also in­
creases. W hen th e p otato is very w et, kg is low an d also there is v ery little gas so
th at the total gas perm eability kg is close to zero. W hen the potato is v ery dry, kg
is m axim um an d gas content is also m axim um so th a t the total gas perm eability
kg is also m axim um . Therefore, kg consistently increases w ith decreasing liquid
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81
content till the m axim um value w here th ere is no liquid left. For th e sim plicity
o f the m odel and also d u e to th e u n k n ow n variation of
is assum ed th a t
k gi
is constant an d is equal to
kgi
as a function of M , it
at the very dry stage. This can
be fairly reasonable approxim ation a t b o th very w et and very dry stages. There
m igh t be som e overestim ation in betw een.
To sum m arize the inform ation on in trin sic perm eability data for th e hygro­
scopic food m aterials, tw o different in trin sic perm eability
w hich
k wi
kw
and
represents the liquid perm eability a t very w et stage and
k gi
are used in
kgi
represents
th e gas perm eability at very dry stage. In choosing the approxim ate k ^ of potato,
it is no ted th at it sho u ld be less th a n th e only experim ental data on m eat of C ontr­
eras (1987). T hus a sm aller value o f 5 x
1 0 “ l4 m 2
is used for the potato. In choosing
kgi of potato, there is no very good reference so th at a larger value of
10
x
1 0 “ l4m 2
th an kyn is used. Since these values are still som ew hat arbitrary, sensitivity of the
m odel to these param eters w ill be an im p o rtan t p art of the analysis.
For the relative perm eabilities, m any expressions exist that represen t th e ex­
perim en tal d ata for different situations. N one of these studies are in th e context
o f food or hygroscopic m aterials. For exam ple, for the gas relative perm eability
kgr, H arm athy (1969) presented th e follow ing em pirical relationship based on ex­
perim en tal d ata w ith clay brick
kgr
= exp(—5.5S2)
(4.25)
L ater on, sim pler analytical expression th a t represents experim ental d ata reason­
ably w ell w as suggested by Jones (1946,1949) an d is used w idely in th e porous
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82
m edia m odels (N asrallah e t £11., 1988; Hie e t al., 1989). In th is m odel, kgr increases
linearly w ith decreasing liquid content, as given by
kgr =
1 - 1.15*
kgr = 0
S w < 1/1.1
S w > 1 /1 .1
(4.26)
For the liquid relative perm eability k ^ , th e m ost w idely used expression (Scheidegger, 1972; N asrallah et al., 1988; Ilic et al., 1989) is as follow ing,
k
W
kwr
_ f Sw- $ r \ 3
- S ir )
S >S
~V 1
=0
Sw < Sir
(4.27)
w here S ir is irreducible liquid satu ratio n o r percolation threshold an d is assum ed
to be 0.09 (N asrallah et al., 1988; Ilic e t al., 1989). D ecrease in k ^ is very rapid
w ith decreasing liq u id content and approaches zero as th e percolation threshold
is reached since th e liquid w ater becom es h ydraulically disconnected.
The total perm eability calculated from Eq. 4.24 w ith th e choice of intrin sic per­
m eability as discussed above an d relative perm eability from Eq. 4.27, is show n in
Fig. 4.11.
Permeability in other areas
A lthough there is few perm eability d ata in Table 4.7, th e term permeability occurs
w idely in food-related literature w hich is n o t D arcy-based perm eability b u t m em ­
brane type o r packaging film based perm eability. Such perm eability d ata could
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83
86-
42-
0.0
0.4
0.6
Liquid saturation Sw
0.8
1.0
Figure 4.11: T otal perm eability, calculated from intrinsic p erm eab ility and relative
perm eability, p lotted against saturatio n.
not help in th e current study. For exam ple, in th e biological area, m em brane per­
m eability Lp is defined by
Jv = LpA P
(4.28)
w here Jv is th e flux, A P is the p ressu re drop, an d Lp is given by
.*1
L p = —^
(4.29)
w here k is perm eability thro ug h th e m em brane, /i is viscosity, an d A x is the thick­
ness of the m em brane. O bviously, L p lum ps th e D arcy based perm eability w ith
fluid p ro p erties and thickness of th e m em brane.
In food packaging area, the perm eability p-m thro ug h film s is defined by
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84
Ja = Pm&cA
(4.30)
w here JA is th e flux and A cA is th e concentration difference of th e gas A . This
perm eability is really based on the diffusive flow of the gas th ro u g h th e packaging
film and is eq u al to
*■ -
(431)
w here DAb is th e diffusivity of th e gas, A x is the thickness of th e film .
4.8.3
Capillary D iffusivity o f Liquid Water
Importance o f capillary force and diffusivity
C apillary d iffu siv ity of liquids is v ery im p o rtan t in convective d ry in g as w ell as
in m icrow ave heating. In convective heating, capillary force is th e driving force
for the liq u id to m ove from w et reg ion to d ry region. In m icrow ave heating, cap­
illary force is also the only driving force for th e liquid before the pressure gradi­
ent is d ev elo p ed initially o r after th e pressu re gradient is m uch reduced at a d ry
stage. D u rin g th e period of pressu re g rad ien t developm ent, the capillary force
can help th e m aterials to m aintain u n sa tu ra ted state n ear the surface to som e de­
gree so th a t th e governing equations described in this m odel are still valid. The
reason is th a t th e intensive p ressure d riv en flow from th e insid e can keep push­
ing the liq u id to the surface in a w e t m aterials such as raw p o tato w hile the sur­
face m oisture rem oval is lim ited b y th e convective tran sp o rt o f vapor. Therefore
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85
th e w ater satu ratio n level o n d ie surface w ill be u n ity w hich is defined as a fully
saturated state. O nce th e porous m edia is fully saturated, there is only one m ass
conservation equation, w hich is com pletely different from three m ass equations
in u n satu rated state. Even m ore com plicated is th e fact that th is fully satu rated re­
gion is n o t fixed in space. Furtherm ore, th is fully saturated region can no t rem ain
all the tim e an d w ill change back to u n satu rated state once th e pressure driven
flow decreases. Therefore, it is alm ost im possible to keep m odifying th e govern­
ing equations to accom m odate this change in physics.
O ne w ay to d eal w ith this is by greatly increasing the capillary diffusivity near
the fully satu ratio n region to increase th e capillary flow against th e p ressure driven
flow and to prev en t th e m aterial from becom ing fully saturated (Perre et al., 1991).
This idea can be justified from th e Fig. 4.12 in w hich the capillary force vs. w ater
saturatio n in soil is show n (Bear, 1972). N ear S w =
1,
dpc/d S w is alm ost infin­
ity. A ccording to Eq. 3.34, Dw can also go to infinity because of lim ited value of
kw. The u n d erly in g physics is th a t as Sw approaches 1, m ore w ater becom es free
w hile the resistance of th e solid m atrix to the flow of free w ater is alm ost zero.
Therefore, D w can be very large and, as a result, th e liquid concentration gradient
w ould be very sm all.
L everet fu n ctio n in p o ro u s m ed ia m odel
In order to describe th e capillary p ressure in a porous m edia m odel as show n in
Fig. 4.12, L everet function o r J-function (Leverett, 1941) has been w idely used. Ex­
am ples of u se in different m aterials include soils an d brick (Scheidegger, 1974;
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1
0
Water saturation
Figure 4.12: Typical variation o f capillary force as a function of liq u id satu ratio n
in a p o rou s m edia (Bear, 1972).
N ashallah e t al., 1988; Ilic et al., 1989), in w ood (Spolek et al., 1981) an d in other
general m aterials (H arm athy, 1969).
(4.32)
w here a is d ie surface tension, 0 is th e porosity, and kw is the liq u id in trin sic per­
m eability. The Leveret function is like a dim ensionless param eter an d represents
only th e w ate r satu ratio n effect w ith o u t including the effects of th e p o ro us struc­
tu re an d th e fluid properties. It is m ean t to be a generalized relation ship regard­
less of th e stru ctu re an d fluid. O nce th e J-function is obtained b y regression of
experim ental d ata such as in Fig. 4.12, it can be applied to other cases w ith differ­
ent p o ro u s stru ctu re an d fluid p ro p erties so th a t the capillary force an d capillary
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87
diffusivity can be calculated. In th e case w h ere Leveret function lacks o f gener­
ality or there is an exp erim en tal curve o f Fig. 4.12, capillary force curve can be
directly used to calculate th e capillary diffusivity (Wei e t al. 1985).
Capillary diffusivity vs. effective moisture diffusivity
Food is a highly hygroscopic m aterial. L everet function is n o t general enough to
be used for such m aterials. T here is also n o experim ent d ata for capillary force
available. H ow ever, larg e am o u n t of effective m oisture diffusivity d ata have been
reported in the literatu re. T hese w ere o btained m ostly by fitting diffusion m odels
to experim ental d ry in g curves (M arousis e t al., 1989; X iong et al., 1991; M arousis
et al., 1991; K arathanos et al., 1991; K aratas e t al., 1991; V agenas et al., 1991; Litch­
field et al., 1992; M oreira e t al., 1993; Pel e t al., 1993; Kim e t al., 1995).
The effective m o istu re diffusivity is related to th e totcil m oisture tran spo rt and
therefore it lum ps cap illary flow of liquid an d diffusional flow of v ap o r together.
In fact, it is very close to capillary diffusivity w hen the m aterial is very w et be­
cause the vapor diffusion is insignificant. H ow ever, it can b e q u ite different from
capillary diffusivity as th e m aterial dries because the v ap o r diffusion reaches the
m axim um w hile th e liq u id capillary flow becom es m uch sm aller. Therefore only
De/ f in w et range can b e related to Dw. C onceptually, curves o f capillary diffu­
sivity, v ap or diffusivity an d effective m o istu re diffusivity w o u ld look like th at
show n in Fig. 4.13.
In Fig. 4.13, for com parison, all the diffusivity including v ap o r diffusivity are
based on g rad ien t of to tal m oisture content o r liq u id content (Both are alm ost equal).
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88
m2 / s
_
a?
.
Effective gas diffusivity
0
1
Water saturation Sw
0
1
Water saturation Sw
Figure 4.13: D iffusivity vs. liquid saturation.
If the gradient of v ap o r content is chosen as the driving force, the v a p o r diffusiv­
ity can be as high as the ord er of 10- 6 m2/s show n in Fig. 4.16, w hich is d u e to the
vapor m ass content being as low as 1 / 1 0 0 0 o f liquid m ass content, an d a s a result,
low er driving force.
In food literatures, m ost of the data in effective m oisture diffusivity w ere given
in low m oisture range, as show n in Table 4.8. The capillary d iffusivity o f potato
(G oring, 1958) covering a large m oisture ran g e show s that Du, increases v ery rapidly
w ith th e m oisture content near the sa tu ratio n b u t the m axim um v alu e is about
5.0 x
1 0 “ 9 m2 /s.
In th eir study, Dw decreased to zero w ith decreasing m oisture
content. However, there is a m axim um v alu e occurring in betw een w hich w as
explained to be d u e to increasing effect of fine capillaries at this m axim um hygro­
scopic m oisture content. This data w as n o t u sed in o ur m odel because n o sim ilar
trend w as found in o th er w orks. Besides th is study, som e Def / d a ta o f potato,
green pepper, carrot an d onion w ere rep o rted for a large m oisture ran g e (Kira-
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89
Table 4.8: Effective m oisture diffusivity in lo w m oisture range
m aterial
m oisture content
effective diffusivity
M (d.b.)
Deff (m 2 /s )
source
pasta
0.05-0.25 (d.b.)
potato
0.67-1.22 (d.b.)
5.128 x lO - 13e lAi-M
R uan et al. (1991)
0.54-1.86 (d.b.)
025-2.3 xlO - 9
U m bach et al. (1992)
1 0 " 11
X iong et al. (1991)
- 1 0 " 10
starch
w ith glu ten
noudis e t al., 1994). P otato d ata from their study is show n in Fig. 4.14.
The experim ental d a ta in Fig. 4.14 show s Defj to be the o rd er of 10“ 9 m2/ s
in highly w et region. H ow ever, calculated m oisture profiles based on this data
show s th a t this diffusivity valu e is likely to be ra th e r low. For exam ple, if w e cal­
culate th e m oisture profiles in a slab u n d er convective heating w ith thickness of
lcm an d a su rro u n d in g tem perature of 177°C, as sh o w n in Fig. 4.15, th e m oisture
content d ro p s uniform ly for Dw = 1.0 x 10“ 7 w hich corresponds to the constant
drying rate. H ow ever, for a valu e of Dw =
1 .0
x
1 0 -9,
the m oisture content drops
very slow a t the center, corresponding to th e foiling rate of d ry ing . It is felt th at a
capillary diffusivity D w = 1.0 x 10~ 9 is unusually lo w for the h ig h m oisture range.
For exam ple, in the w o rk on clay by N asrallah an d Perre (1988), constant rate of
drying w as observed initially, corresponding to a diffusivity valu e greater th an
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90
CO
O R uan e t al. •
— T his w ork
K iranoudis e t al.
D w = 10.0*exp(-2.8+2.0 M ) /
M oisture content M (d.b.)
Figure 4.14: C apillary diffusivity vs. m oisture content (d.b.).
1.0 x 10~6.
Based on this discussion, it appears th at the liquid w ater diffusivity can be
characterized by three regio ns given by
1. C onstant rate region: D w > 1.0 x 10~ 7
2. Falling rate region: D w < x 10- 9
3. Transient region from constant to falling rate w ith diffusivity values in be­
tw een the above tw o.
T hese values w ere incorp orated in one equation to represent the en tire region from
low to high m oisture co n ten t
D w = 1.0 x 10- 8 exp(—2.8 + 2.0 * M )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.33)
91
Time=0
sV
C
ou
£
1 .0 e-8
2
(lhr)
1.0e-7 (lhr)
1.0e-9 (lhr)
0.0
0.4
0.6
0.8
1.0
Distance (x=0: open surface; x=lcm: symmetry line)
Figure 4.15: M oisture profiles after
1
h o u r of convective heating for different D w.
w here M is th e m oisture content (d.b.) an d Dw is th e capillary diffusivity in m 2 /s.
N ote th at th e effect of tem perature o n capillary diffusivity, i.e. Soret effect, is in­
significant in m icrow ave heating (Z hou e t al., 1994) an d is ignored in this w ork.
4.8.4
E ffective Gas D iffu sivity
The effective diffusivity of gas d eterm in es the rate of gas tran sp o rt d ue to the gas
concentration gradient. It is im p o rtan t especially in the later stage of convective
o r m icrow ave d ry in g because capillary flow o f liqu id and pressure driven flow of
gas an d liq u id are m uch sm aller.
In som e special cases as in Section 4.8.3, the effective gas diffusivity can be d e­
fined based o n the liquid concentration g rad ien t provided the m ass relationship
betw een liq u id an d gas is know n. T he p u rp o se is to describe th e com bining flux
of liquid a n d gas in a very sim ple w ay.
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92
M ore general form ula fo r th e effective diffusivity o f b in ary gas in porous m e­
dia is given (G eankoplis, 1978) as
(434)
T
w here eg is gas volum e fraction o f to tal volum e w hich is eq u al to <t>Sg. Tortuosity of
the gas p ath , r , d epends o n th e stru ctu re of each m aterial an d m u st be determ ined
from experim ent (K arlsson, 1985). Since r is very h ard to get, the expression of gas
effective diffusivity in soils (Baver an d G ardner, 1972) is u sed in th is m odel.
Deff,g = D va(Sg<t>)A/z
(4.35)
C om paring Eq. 4.35 w ith Eq. 4.34, {Sg4>)~l/3 is sim ilar to r , b u t it is not equal.
For the real food m aterials, r varies from 2 to
6
(G eankoplis, 1978). Binary diffu­
sivity of v ap o r in the air, D va, w hich also d epends on th e tem perature and total
pressure, is given by
D „ = 2 .3 x l t r 5^
‘ 81
(436)
w here 7o=256 K and P0= l atm . For sim plicity, constant v alu e Dva (2.6 x 10-5 m 2 / s)
is used in th e m odel (G eankoplis, 1978) a n d the effect of tem p eratu re and the total
pressure o n binary d iffusivity is considered in the sensitivity analysis. The effec­
tive gas diffusivity vs. w ater satu ratio n is show n in Fig. 4.16.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
20 - r
CO
N
S 1 5inO
X 10Q
0.0
0.4
0.6
0.8
W ater saturation Sw
1.0
Figure 4.16: Effective gas diffusivity vs. w ater saturation.
4.8.5
Sorption R elationship
T he vapor pressure in th e food dep en d s on tem p erature as w ell as th e m oisture
c o n te n t w hich is called sorption relationship. A s m entioned in Section 4.37, the
sorption relationship is one of the m ost im portant param eters in d ryin g of foods
because it can affect th e surface v ap o r condensation, internal pressure d riv en flow
an d th e rate of m oisture loss from d ie surface. U sually, low er the surface tem p era­
tu re, low er the surface v ap o r pressure. A s a result, v ap o r condensation can occur
a t the surface if there is m ore vapor com ing from th e inside to the surface cau sin g
th e surface m ore w et, w hich can be th e case in m icrow ave heating. O n th e o th e r
hand , drier the surface, low er the surface vapor p ressure. Therefore, th e ra te of
m oisture loss from th e surface is also lower, w hich is th e case of convective h ea t­
ing. M eantim e, if the in sid e tem perature is higher an d th e inside is m ore w et, th e
pressure w ill be higher too w hich can cause intensive pressure driven flow from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
inside to th e surface.
In ad d itio n to its im portance in tran sp o rt process, th e vapor p ressure is even
m ore im po rtant for m icrobial grow th in food storage in term s o f w ater activity
(Rizvi, 1995). B ecause o f this, m any em pirical equations w ere developed such as
the BET equation, H enderson correlation a n d GAB equation etc. (Rizvi, 1995).
However, th ese param etric equations are o n ly good in a certain range and also
require experim entation to determ ine the coefficients for each m aterial. Since the
potato is te sted as an exam ple in this w ork, th e sorption relationship of potato
(Ratti et al., 1989) is u se d an d is show n in Fig. 4.17.
In - 7 ^— = —0.0267A/-1656 + 0.0107e-1287A/A /1513 In pa[ T )
Ps(T)
(4.37)
w here pv is v ap o r p ressu re in kPa, pa is sa tu ra ted vapor pressure in kPa, and M is
m oisture co n ten t (d.b.).
4.8.6 E ffective Thermal C onductivity and Heat Capacity
Effective th erm al conductivity can be assum ed to be add itio n of therm al conduc­
tivities of all th e com ponents (W hitaker, 1977).
kef f = ka(l — <t>) + kw<t>Sw + kg<t>(l —Sw)
(4.38)
w here ka is th e th erm al conductivity of so lids a n d is about 0.21 W /K m for carbo­
hydrate (Choi e t al., 1986), kw is therm al conductivity of liquid w ate r and is about
0.64 W /K m , a n d kg is therm al conductivity o f gas and is about 0.026 W /K m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
1 .0 - i
0 .8
-
0 .6
-
0 .4 -
Ou
0.0
0.0
1.5
1.0
0.5
M oisture content M (d.b.)
2.0
Figure 4.17: Vapor pressure vs. m oisture content at different tem peratures, as
show n by Eq. 4.37
Effective heat capacity is equal to th e m ass average of the heat capacities of all
the com ponents
(P^p)e// = Ps^psi. 1
0)
Pw^pw^^w
PgCpg${ 1
S w)
(4.39)
w here pa is density of solids an d is assum ed to be 1419 k g /m 3 for carbo hy drate
(Choi e t al., 1986), and Cp* is specific h eat of solids an d is assum ed to be 1566 J / kg
K for carbohydrates (Choi e t al., 1986).
4.8.7
Thickness o f Sam ple
The thickness of sam ple in th is w ork is chosen to be 1cm. This w as first based
on practical utility, for exam ple, m icrow avable frozen dinners are d o se to th is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
thickness. T he non-uniform h eatin g can increase w ith th e thickness in m icrow ave
heatin g . R easonable thickness for th e solid foods v aries h o rn 1 cm to 2 cm. The
effect of thickness variation w ill be accounted for in th e sensitivity analysis.
4.8.8
M icrowave Penetration D epth
The m icrow ave p en etratio n d ep th is very im p o rtan t to the m oisture transport.
H ow ever, it is also very difficult to m easure th e dielectric properties in the very
low m oistu re ran ge because it has very low loss so th a t the sam ple should be very
thick to elim inate th e reflection from the o th er sid e (H P 85070A D ielectric Probe
M anual, H ew lett-Packard, E ast Syrause, NY). T here are som e d ata reported (Tulasid as e t al., 1995; H aynes et al., 1995), how ever, th ese d ata can no t be used in th is
w ork because o f su g ar effect o r lack of th e dielectric constant d ata. For this w ork,
th e general tren d in dielectric p roperties of leath er (M etaxas an d M eredith, 1988)
is u sed as a reasonable variation of dielectric p ro p erties of potato. By know ing the
dielectric p ro p erties of raw p otato, dielectric p ro p erties are form ulated as show n
in Fig. 3.68. T he pen etratio n d ep th is calculated from th is dielectric property d ata
u sin g Eq. 3.68.
4.9
Consideration of 2D Geometry
T he tem p eratu re, pressure, an d m oisture are affected w hen the m icrow aves are
in cid en t from tw o directions. The sim plest tw o-dim ensional m odel developed
here is an axisym m etric geom etry w ith a right circu lar cylinder. It is expected th a t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
10-.
80 -i
CO
£
8eo
S-6 -
Constant
60-
I.
2
Q.
0 40-
1
1
o
20 H
0
£4
Loss
1
2
3
4
5
Moisture content M (d.b.)
1
2
3
4
5
Moisture content M (d.b.)
Figure 4.18: D ielectric p ro p erties an d penetration d e p th vs. m o isture content.
the consideration of th e tw o-dim ensional effects w ill n o t resu lt in qualitatively
new inform ation. H ow ever, th e significantly increased h eatin g rates in a 2D sit­
u atio n brings the sim ulation closer to reality in term s of tem p eratu re an d pres­
sure rise an d m oisture loss. O f course, th e com putational tim e an d challenges are
likely to increase for th is 2D problem as com pared to th e ID problem .
The origin of cylindrical coordinate system is placed o n the b o tto m center (r=0,
z=0). A rad iu s of 0.5 cm an d a heig h t o f 2 cm is considered. It is assu m ed th a t little
o r no m icrow aves are com ing from the bottom . The m icrow aves com ing from the
to p an d the side of th e cylinder are o f sam e intensity a t th e surface. T here is no
m ass an d h eat exchange o n th e bottom w hile the to p an d the side are op en to the
air. T he detailed governing equations, boundary conditions an d th e discretized
equations can be seen in A ppendix B.
A non-uniform g rid system is u sed to concentrate m ore g rid n e a r the surface
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98
an d reduce th e com putation tim e. A grid system o f 11 in the radial d irectio n and
21 in the v ertical direction w as used. Fully im plicit m ethod is u sed th a t m ade
the code m uch sho rter th an u sin g the C rank-N icholson scheme. Inclusion of the
boundary conditions w as ted iou s since the g rid p o in ts a t the com er a n d th e bound­
aries had to b e treated individually.
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Chapter 5
RESULTS AND DISCUSSION
In this chapter, tem p eratu re, m oisture, an d pressure profiles as w ell as m oisture
loss are described first for convective and later for m icrow ave heating. Results
are presented for m aterials w ith tw o different initial states— sem i-dry a n d very
w et. Sensitivity an aly sis is th en perform ed to determ ine th e m ost significant pa­
ram eters.
5.1 Temperature, Pressure, and Moisture Profiles
in Convective Heating
Since conventional h eatin g h as been stu died in the p ast an d it is expected to be
sim pler than m icrow ave heating, conventional heating is first stu died as a pre­
lu de to b etter u n d ersta n d in g of the m icrow ave heating. C alculations for conven­
tional heating w ere perform ed for sem i-dry an d very w et m aterials for th e in pu t
99
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100
d ata show n in Table 5.1. T he eq uiv alent porosity in sem i-dry m aterials w as low ­
ered to account for shrinkage. T he h e a t an d m ass tran sfer coefficients are calcu­
lated for typical p aram eter values, as show n in A ppendix A. These transfer co­
efficients are close to th e values u sed in th e literatu re for sim ilar heating situ a­
tions. For exam ple, h=23 an d 15 u sed by Perre et al. (1988), h=20.92 used by W ei
et al. (1985), h=24 used b y C hen e t al. (1990), ^ ^ = 0 .0 2 used by Perre et al. (1988),
hmV=0.014 used by Perre (1988), an d fimv=0.025 used by Chen et al. (1990).
5.1.1
Low M oisture Foods
Tem perature profiles for low m o istu re potato are show n in Fig. 5.1a. The su r­
face tem perature increases very rap id ly b u t the tem perature inside rem ains a t a
low value of 70°C. Such profiles are characteristics of capillarity driven d ry in g
processes, as can b e seen, fo r exam ple, in concrete w all exposed to fire (A bdelR ahm an et al., 1996). T he low th erm al conductivity o f the d ry surface region, to ­
gether w ith evaporation, leads to th e significant d ro p in tem perature from the su r­
face to die interior.
The w ater satu ratio n profile fo r th e low m oisture p otato is show n in Fig. 5.1b.
T he initial saturation corresponds to a m oisture content of ab ou t 1.0 (dry basis).
The surface m oisture d ro p s very fast d u e to the quick rise in surface tem perature.
H ow ever, the inside m o isture d ro p s slow ly because of a m uch reduced capillary
diffusivity in these regions. In tw o h o u rs, the center m oisture content drops o n ly
slightly, w hich is th e characteristics o f diffusion controlled m oisture transport, as
can be seen in d ry ing o f brick an d w ood (N asrallah a n d Perre, 1988) an d in d ry in g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 5.1: B oundary an d initial conditions in convective heating.
sem i-dry
very w et
P orosity <p
0.75
0 .8 8
Initial satu ratio n Sw
0.5
0 .8
10
10
5
5
177
177
0
0
20
20
0 .0 1
0 .0 1
Intrinsic perm eability a t
very d ry kgi (x
1 0 _l 4 m2)
Intrinsic perm eability at
very w et
(x
1 0 _l 4 m2)
S urrounding tem perature
Ta (°C)
S u rro u n d in g v ap o r
d en sity pv (k g /m 3)
H eat tran sfer coefficient
h (w /m 2 K)
M ass tran sfer coefficient
kmv (® / s)
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102
2 0 3 - 20 min
6020 min
0.0
0.4
0.6
x (cm)
1.0
0.0
0.4
60
2 0. 8 -
1.003-
|
s 1.002-
a. 1.001 -
(d)
-6 5 g
0.6 -
1.0
o
- 6.o c*
-5 5 3
-5.0 »
o
-4 .5o
--o
! 03-
I
0.4
0.6
x (cm)
1.0
| 0.4-
20 min
5 1.000
0.0
0.8
0.6
x (cm)
0.0 -I
0.6
0.8
0.4
Heating time (hr)
-4.0*]
1.0
Figure 5.1: Tem perature, w ater satu ration , and pressure profile, and m oisture loss
in convective heating of low m oisture potato. In p u t d ata from Table 5.1.
of brick (Die an d Turner, 1989).
The pressure profiles for th e low m oisture p o tato are show n in Fig. 5.1c. Ini­
tially th e pressure peak is n ea r th e surface. This is d u e to th e evaporation be­
ing m ostly n ear the surface from a m uch h igher tem p erature, com bined w ith the
b o u nd ary pressure set at the atm ospheric pressure. L ater on, th e inside pressure
starts to equilibrate. The m axim um overpressure th at develops is about 450 Pa
in one hour. The sm all o verpressure probably m akes the pressu re driven flow in­
significant, also seen in the stu d ies m entioned in th e p arag rap h above.
T he tran sien t average m o isture content and th e rate of m oisture loss are show n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
in Fig. 5.1d. The rate of m oisture loss increases fo r the first 10 m inutes o r so w h en
th e tem p eratu re is increasing, th e surface m oisture is still high and therefore the
v ap o r p ressu re is com paratively high. A fterw ard, the rate decreases because of
a m uch lo w er surface v ap or pressure. E ventual decrease in the rates of m o istu re
loss can be seen in other stu dies, for exam ple in th at of Stanish et al. (1986) an d
in U m bach e t al. (1990).
5.1.2
H igh-m oisture Foods
H ig h m oisture foods o r very w et m aterials such as raw potato, on the o th er h an d ,
are expected to have different heating or drying characteristics. T em perature p ro­
files in su ch m aterials are show n in Fig. 5 .2 a. T he interior tem perature ten d s to be
co n stan t at its w et bulb tem perature, irrespective of boundary tem perature. This
is d u e to th e dynam ic equilibrium betw een the surface evaporation an d th e cap­
illary diffusion of liquid w ater. The w ater satu ration profiles of the sam e m aterial
are sh o w n in Fig. 5.2b. The m oisture levels d ro p quite uniform ly d u e to th e large
capillary diffusivities a t high initial m oisture contents. The non-uniform ity in the
m oistu re profile starts to d evelop once th e m oisture level near the surface red u ces
to a valu e th a t the capillary diffusivity decreases significantly. The p ressure p ro­
files for th is heating process are show n in Fig. 5.2c. The inside pressure is sligh tly
below atm ospheric pressure, w hich w as also reported by Wei et al. (1985).
T he average m oisture content and the rate of m oisture loss as a fu n ctio n of
tim e are show n in Fig. 5.2d. T he rate of m oisture loss increases initially (d u rin g
th e first 24 m inutes) w ith increase in surface tem perature. As the surface tem p er-
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104
5 0-
0.8- |
tr?
c
o
0 .6 f*
.
3
S 0 .4 41
**
40,60
20 min
40&. 3 0 Time=0
0.0
0.4
0.6
x (cm)
£
0.8
1.0
1.00000
Time=0
0.99999
0.999980.999970.99996-
07-
Time=0
-----20 m in
40^
T
60
I
(b)
i
0.4
0.6
x(cm)
■a 4 .0 g 3 .0 -
20 min
0.99995x cm)
i--------- 1--------- 1--------- r
0.2
0.4
0.6
0.8
Heating time (hr)
Figure 5.2: T em perature, w ater saturation, an d pressu re profile, and m oisture loss
in convective h eating of very w et potato. In p u t d ata from Table 5.1.
atu re becom es constant at the w et bulb an d th e v ap o r pressure is alm ost in depen ­
dent of m oisture in th is high m oisture range, ra te of m oisture loss becom es con­
stant. As the m oisture level reduces, conditions described in Fig. 5.1 are reached.
5.1.3
Com parison o f M odel w ith O ther Num erical Studies
In this section, the m odel w as com pared w ith th e num erical prediction of conven­
tional d ryin g of clay brick (N asrallah et al., 1988). A ll the in p u t param eters w ere
given in A ppendix B. The tem perature com parison w as show n in Fig. 5.3. The
trend is correct. H ow ever, the w et bulb tem p eratu re in o u r prediction w as ab ou t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
4°C low er th an theirs. T he w ate r saturation com parison w as show n in Fig. 5.4.
The tren d is also correct. H ow ever, th e w ater saturatio n n e a r th e surface p red icted
by o u r m odel is m ore g rad u al th an theirs, w hich m ight b e d u e to uniform g rid w e
used. G ravity force and convective energy transfer w ere ignored in o u r m odel­
ing. A ccording to N asrallah e t al. (1988), neglecting these term s does not lead to
significant error.
6°-i
---- This model (1)
---- Nasrallah et al. (1988) (2)
Time-0
5040-
75 min (2)
75 min (1)
3020-
\
^
30-60 min (2)
15-45 min (1)
100 - i------------- 1------------- r~
0.0
0.2
0.4
r ----------- 1------------- 1
0.6
0.8
1.0
x(cm)
Figure 5.3: C om parison of tem perature w ith N asrallah 's m odel.
5.2 Temperature, Pressure, and Moisture Profiles
in Microwave Heating
This section parallels th e p rev io u s section on convective heating. Two different
initial m oisture contents w ere considered, for w hich th e in p u t d ata are show n in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
101
» 0.8-
CA
e
o
2
3
Time-0
/
15 min
---- This model (1)
-~N asaH ahetaL(1988)(2)
30 m in 4 5lnin
go min 75nun
* 0.4- .............................
£*S
2 0.2- ^
--------.-------90 nun (1) 105 mm (1)
0.0- 1------------- 1------------- 1------------- 1-------------1------------- 1
0.0
0.2
0.4
0.6
0.8
1.0
x (cm )
Figure 5.4: C om parison of w ater satu ratio n w ith N asraU ah's m odel.
Table 5.2.
5.2.1
Low-m oisture Foods
T em perature profiles for m icrow ave h eatin g of low m oisture m aterials are show n
in Fig. 5.5a. T em perature increase is slow since the rate o f m icrow ave absorption
is lo w er a t this low m oisture content. T he surface tem p erature stays colder d u e
to cold su rro u n d in g air. These profiles are characteristic of fairly uniform vol­
u m etric heating, as show n b y D olande an d D atta (1993). The m oisture profiles
are sh o w n in Fig. 5.5b. In first six m inutes, the m oisture d ro p s sim ilar to convec­
tiv e h eatin g w here low er in side tem p eratu re causes low er rate of evaporation an d
low er pressu re generation an d therefore insignificant pressure driv en flow.
A s tem peratures in side reaches closer to 100°C, evap oration increases an d p res­
su re sta rts to build. E ven sm all am ounts of pressure cause enough m oisture to
reach th e surface, exceeding its m oisture rem oval capacity. This causes m oisture
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107
TemperatureT(°C)
100-1
602 min
0.0
(a)
Time=0
0.4
0.6
x (cm)
0.8
2
3
Sjj
5
5
035030
0.450.40-
Time=0
2 min
1.0
x (cm)
1.020 -I
J3 1.10-1
1.015-
2 1.05-
1.010 -
c 1 .0 0 -
-8
§ 0.95-
1.005-
-4
(d)
2 min Time=0
0.90
0.4
0.6
t (cm)
0.8
1.0
2
4
6
Heating time (min)
8
Rate of moisture loss (0.1g/s/m 2 )
Total pressure P(O.IMPa)
* 0.65-
Figure 5.5: T em perature, w ater satu ratio n , an d pressure profile, an d m oisture loss
in m icrow ave heating of low m oisture potato. In p u t data from Table 5.2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
108
Table 5.2: B oundary a n d initial conditions in m icrow ave heating of low an d h ig h
m o istu re food (only p ro p erties different from Table 5.1 are specified in th e table).
sem i-dry
very w et
20
20
30000
30000
S urrounding tem perature
Ta (°C)
Surface flux
Fq (W /m 2)
accum ulation near the surface. Thus, at around 7 m inutes, the first signs of m ois­
tu re accum ulation seen in Fig. 5.5b correspond to a n excess pressure of only ab o u t
2000 Pa o r about 0.02 atm . Therefore, even a sm all pressure differential can le ad to
a soggy surface. W ith tim e, the surface m oisture keep s increasing an d after ab o u t
eig h t m inutes the surface is m uch w et than initial. C ontinued heating w o u ld even­
tu ally cause the surface m oisture to drop since m oisture level inside w o u ld de­
crease an d lead to decreased m icrow ave absorption, reduced evaporation an d pres­
su re generation. These satu ratio n profiles are fundam entally different from p ro ­
files d u e to capillary diffusion m echanism w h ere d ie m oisture m oves from h ig h
m oisture to low m oisture region.
Total pressure profiles for this heating process are show n in Fig. 5.5c. T here is
n o t m uch pressure b u ild in g u p before six m inutes because the m axim um tem per-
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
109
ature in th e sam ple is still below 100°C. H ow ever, th e pressure increases there­
after w ith m axim um overpressure of 2 0 0 0 p a in ab o u t eight m inutes, w hich re­
sults in all th e m oisture m ovem ent described befo re. P ressure increases m onotonically inside, w ith the m axim um value being a t th e farth est location from surface.
The average m oistu re an d rate of m oisture lo ss d u rin g heating are show n in
Fig. 5.5d. T he rate of to ta l m oisture loss increases u p to 1.4 g / s / m 2 in 8 m in as the
surface gets hotter a n d w etter, increasing th e v a p o r pressure at the surface. As
m entioned before, for lo n g er heating, the rate o f m oistu re loss decreases ev entu­
ally.
5.2.2 H igh-m oisture Foods
As m entioned earlier, food m aterials such as ra w vegetables are considered here
to be of very high in itial m oisture content. If th e m icrow ave pow er is low, the
tem perature, m oisture, an d pressure profiles are sim ilar to th at of low m oisture
food. A t h ig h pow er, th e se profiles are expected to be m uch different because th e
evaporation bo un dary condition restricts th e m o istu re rem oval from the surface
and cannot explain th e larg er m oisture loss o b serv ed experim entally. Therefore, it
is necessary to introduce th e liquid pum ping co n d itio n (see problem form ulation
in chapter 3) on the su rface w hich allows liquid w a te r to leave the sam ple w ith o u t
change of phase. O ther param eters used for th is section are show n in Table 5.2.
Tem perature profiles fo r heating of high-m oisture foods are show n in Fig. 5.6a.
Tem perature quickly (in ab o u t three m inutes) reach es the boiling point. The m ax­
im um tem perature occu rs in about 3 m inutes a n d is ab o u t 107° C. The heating rate
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1 10
u sed in this situation is m u ch h ig h er th an th e heating rate of low m o istu re m ate­
rial. A fter about 3 m inutes, the tem perature sta rts to d ro p because th e m oisture
inside becom es low er, decreasing m icrow ave absorption an d therefore, evapora­
tion rate and pressure developm ent. Surface tem perature is alw ays lo w er than
inside, due to cooling by su rro u n d in g a ir as discussed earlier for lo w m oisture
foods.
„
1 0 -v
Time=0
100
2 min
60-
0.41 min
(a)
0.0
0.0
1.0
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0.8
0.4
0.6
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-r
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5
Figure 5.6: T em perature, w ater saturatio n, pressure, an d the loss o f m oistu re in
rap id m icrow ave heatin g of a high m oisture food. In p u t d ata from T able 5.2. In­
creased m oisture loss after 3 m inutes is d u e to a "pum ping effect" w h ereb y liquid
w ater leaves the b o u n d ary w ith ou t being evaporated.
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I ll
W ater satu ratio n profiles for th e h eatin g are show n in Fig. 5.6b. In th e very ini­
tial stages (first m in ute) of heating, m oistu re level dro ps som ew hat sim ilar to con­
vective heating of a hig h m oisture food (see Fig. 5.2). This is w h en h ig h pressure
is yet to b u ild u p . H ow ever, in a b o u t 2 m inutes, th e surface sta rts to accum ulate
m oisture u n til alm ost full. The p ressu re generated continues to p u sh inside w a­
te r to the surface u n til the surface can n o t hold any m ore (because th e capillary
force tends to be zero w hen th e m aterial is fully saturated). A fter th is tim e, liq­
uid is "pu m p ed" o u t directly w ith o u t h av in g phase change an d convective m ass
transfer lim itations in the air. The larg e dro p in internal m oisture betw een three
and four m inutes is d u e to th is "p u m p in g " phenom ena.
Total p ressure profiles for th e h eatin g are show n in Fig. 5.6c. Pressure inside
reaches quickly (in abo ut 3 m inutes) an overpressure of ab o u t 32000 Pa, w hich
is m uch higher th a n any oth er h eatin g situation discussed earlier. T he rap id de­
velopm ent of h ig h p ressure is th e fundam ental cause for the m o istu re profiles
discussed in the previous p arag rap h . Like other heating situations, th e pressure
eventually d rop s as m oisture is d ep leted .
The pressure developm ent also explains the decrease in m o isture content and
the rate of m oisture loss as show n in Fig. 5.6d. The rate of loss slow ly increases
in the first three m inutes w hen th ere is n o significant "pum p in g" effect and fol­
low ing th a t it increases rapid ly as "p u m p in g " becom es very significant. The rate
of m oisture loss eventually d ro p s b u t do es it slow ly since th e tem p eratu re and
m oisture at th e surface are still com paratively m uch higher.
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1 12
5.2.3
Foods Starting from a Spatially Varying M oisture Content
Perhaps th e g reatest use o f food m icrow ave heatin g is the reheating of foods al­
ready processed. Processes su ch as baking and frying often resu lt in a m oisture
profile in the m aterial a t th e en d o f th e process, w ith m ore m oisture inside and
less outside (leading to p referred crunchy texture). To stu d y th e heating of such
a m aterial, w e assum e an in itial m oisture content as show n in Fig. 5.7b.
T em perature profiles in Fig. 5.7a show th at w ith in the first six m inutes, the su r­
face tem perature is little low er an d th e inside tem peratures little high er than those
of Fig. 5.5a. This is d u e to th e difference in m oisture content th at lead to m ore
m icrow ave pow er ab sorp tion in sid e th an near th e surface. T his w ould be the de­
sired selective h eatin g in a d ry in g process.
W ater satu ratio n profiles are show n in Fig. 5.7b, starting from m oisture profile
w ith low er value (Sw = 0.3) a t th e surface and h ig h er inside (Sw = 0.55). The
initial (up to about six m inutes) m oisture equilibration occurs p rim arily d u e to the
capillary d riv en flow. The surface g ets d rier d u e to m oisture rem oval as enough
m oisture cannot reach th e surface from inside. A fter about five m inutes, pressure
developm ent (Fig. 5.7c) an d p ressu re d riven flow becom es significant, form ing a
saturation peak n ear th e surface. A s th e inside pressure increases w ith tim e (due
to increased evaporation a t h ig h er tem peratures), th e satu ratio n peak intensifies
and m oves tow ard th e surface. By ab o u t eight m inutes, the surface m oisture is
m ore th an its initial value. T hus, an in itial dry (and crunchy) surface has started
to becom e soggy. E ventually th e surface can be v ery w et (quite soggy).
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113
100- i
8-10.
uo
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H
&
2 0.5-
2
5
0.42 min
0.0
13 03
(a)
3 min
H
0.4
0.6
* (cm)
1.0
0.0
0.4
0.6
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20 «
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3 min
£ 1.000
oe
i
0.0
0.4
0.6
x (cm)
0.8
1.0
Heating time (min)
Figure 5.7: T em perature, w ater satu ratio n , pressure, and th e loss of m oisture in
m icrow ave heatin g of varying m oisture potato.
The overpressure developed here is m uch low er than the 32000 p a reached in
heating h ig h m oisture foods, as show n in Fig. 5.6c. Thus, th e "p u m p in g '' effect at
the bo un dary occurring for high m oisture foods is n o t presen t here. The internal
pressure d ro p s eventually as significant m oisture is lost from th e inside, leading
to less abso rptio n of m icrow ave energy. M oisture content an d th e rate of m oisture
loss (Fig. 5.7d) show th at the rate increases w ith th e tim e an d reaches m axim um
in about ten m inutes. It should d ro p eventually.
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114
5.3 Experimental Data on Microwave Heating
In th is section w e w ill present th e experim ental data o n m oisture loss an d pres­
sure profiles. The com parison o f th ese data w ith the m od el prediction is also pre­
sented.
T here are tw o m ajor problem s associated w ith the experim ental validation of
the m icrow ave heating m odel. O n e is the difficulty of accurately m easuring the
tem p eratu re, pressure and m o istu re loss data. The reason is that w hen the food
is h eated u p to the boiling p o in t th e internal pressure p u sh es out the fiberoptic
tem p eratu re probe. U nlike rig id m aterials, the food becom es soft u p o n heating,
therefore, it is very h ard to fix th e p ro b e in the sam ple. T he second problem is that
the real p o w er absorption stro n g ly depends on m any factors such as the oven in­
p u t pow er, geom etry, p rop erties a n d location o f sam ple. Therefore it is very dif­
ficult to m atch th e surface flux in th e m odel w ith the experim ental situation.
To circum vent th is problem it is suggested in this w o rk th at heating rate (rate
of tem p eratu re rise) is a better in d ic ato r for com parison. N o m atter how com plex
the real situ atio n is, the heating ra te is unique. M oreover, it is the initial rate of
tem p eratu re increase th at co rrespo nds to the m agnitude o f tru e pow er absorption
because th ere is not m uch laten t h e a t effect. A s a result, it should be enough to
m easure th e tem perature u p to 80°C and avoid the problem near 100°C w hen the
probe gets p u sh ed out. A fter estab lish in g this base, a series of com parison can be
m ade su ch as m oisture loss a n d p ressu re profiles.
T he sam p les in th e experim ent are cylinders of raw p o ta to w ith m oisture con­
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115
tent 83% (w.b.). The cylinder is ab o u t 1cm thick an d 2.54 cm in diam eter. T he ini­
tial tem p eratu re varies from 26-28° C. The bottom a n d th e sid e of th e sam ple are
w rapp ed w ith alum inum foil to p rev en t the m icrow ave p en etratio n an d m oisture
loss. T he to p o f th e sam ple is open. Therefore, th e sam ple approxim ates a ID sys­
tem . The reason for n ot choosing larg er diam eter is th e non-uniform d istribu tion
of m icrow ave surface flux on th e surface. The sam ple is placed ho rizon tally on
a sm all ceram ic plate. The larg er thick ceram ic p late in th e oven is taken o u t be­
cause it can also absorb som e m icrow aves.
The fiberoptic probe is in serted in to the center h alf w ay d ow n from th e top.
The probe ten d s to loose and com e a p a rt from th e sam ple w hen th e tem perature
is in th e range of 85-100°C. H ow ever, the tem perature tim e relationship is found
to be a fairly straig h t line u p to 80°C. Therefore th e slope of th e curve is defined
as the in itial rate of tem perature rise (or heating rate) w hich determ ines th e m i­
crow ave energy absorption level in th is sam ple. H eating is continued for 6 m in­
utes, alth o u g h the probe com es o u t w hen th e tem p eratu re reaches close to 100°C.
The sam ple w eight is m easured im m ediately an d th e m oisture loss percentage
(initial w eig h t basis) is calculated, as show n in Fig. 5.8. T he oven d oes n o t have
facilities fo r continuous w eight m easu rem en t The m odel predictions in Fig. 5.8
have th e sam e tren d w ith the experim ental data an d is ab o u t 12% higher, w hich
is reasonable, considering the difficulties in obtaining accurate d ata.
In ad d itio n to th e above experim ent, the m oisture content histories are m ea­
sured by a series of individual experim ents at different en d in g tim es keeping th e
sam ple, location an d oven in p u t p o w er are exactly th e sam e. Several sam ples
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116
SR 6 O -1
3
.5 5 0 a>
4-»
Fitting experim ent d ata
0.4
0.6
0.8
1.0
1J2
1.4
R ate o f Initial tem perature rise (°C / s)
Figure 5.8: C om parison of experim ental data and m odel prediction of m oisture
loss in h eatin g high m oisture food a t 6 m inutes.
w ith different initial m oisture level (by drying in the conventional oven) are heated
u n d er th ree different level o f m icrow ave in p u t pow er. The purpose of these ex­
perim ent is to show the qu alitativ e behavior of the m oisture loss profiles, as com­
p ared w ith the m odel prediction.
For th e raw potato, as show n in Fig. 5.9a, the initial m oisture decrease is slow
w hich corresponds to p ressu re b u ilt u p stage w hen th e convective m ass transfer is
th e only w ay to rem ove th e m oisture from the surface, an d then it becom es m uch
foster corresponding to intensiv e pressure driven flow an d subsequent "pum p­
ing" on th e surface. This h ap p en s sooner a t low er pow er. Finally it becom es slow
again w hich is d u e to the reduced "pum ping" effect a t th e low er m oisture.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
117
400W
n
620'
0
•d
TJ
1
400W
4
2
3
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5
6
0
1
3
4
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5
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400W
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1
2
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4
Heating time (min)
5
0. 0 -
0
1
2
3
4
5
Heating time (min)
6
F igure 5.9: Experim ental m oisture content history profiles for different initial
m o istu re content in different oven in p u t pow er level.
5.4
Sensitivity Analysis
T he properties such as perm eability an d diffusivity can vary trem endously for
food m aterials, and, as m entioned previously, little o r n o m easured d ata is avail­
able. A lso, w ith so m any relevant param eters, it is im p o rtan t to know w hich ones
are d ie m ost influential. S ensitivity analysis, w here sensitivity of one param eter
is stu d ied w hile keeping o th e r param eters constant, is o n e w ay to answ er som e
o f th ese questions. This ch ap ter rep orts th e sensitivity of tem perature, saturation,
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118
m oisture loss an d pressu re to m aterial p ara m e te rs— liquid an d gas perm eability,
capillary and gas diffusivity, the thickness o f the m aterial, an d oven param eters
— heatin g rate (pow er level) and the su rface heat and m ass tran sfer coefficients.
5.4.1
Contribution o f C onvection to Energy Transport
As stated earlier u n d er th e assum ptions, convection term has b een ignored in the
energy equation in o rd er to sim plify th e solution. In order to ju stify th is assum p­
tion, the convective term s are added to th e energy equation. T he solution w ith
the convective term are presented in Fig. 5.10 and Fig. 5.11.
For low m oisture food, since there is a large spatial tem perature variation, the
convection term in energy equation b rin g s m ore energy from th e in side to th e su r­
face so that the surface tem perature is a little higher. C onsequently, the internal
m oisture is p u sh ed tow ard the surface so th a t there is a m ore sh a rp interface in­
side. Surface w ater satu ratio n is also so m ew hat higher. The m esh size need to be
fu rth er reduced generally in order to b e tte r resolve the sh arp interface.
For high m oisture food, since th e tem p eratu re profile is q u ite uniform spa­
tially, so there is less effect of th e convection term . The tem p eratu re an d w ater sat­
u ratio n profiles rem ain alm ost the sam e. T hus, inclusion of the convection keeps
the solution qualitatively th e sam e w ith sm all difference in m agn itud e. T his w ould
justify dropping th e convective term in th e rest of this w ork.
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119
100100-1
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Figure 5.10: C om parison of general profiles in low m oisture food w ith o u t convec­
tion in en erg y equation (left) an d w ith convection (right).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
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Figure 5.11: C om parison of general profiles in h ig h m o isture food w ith o u t con­
vection in energy equation (left) an d w ith convection (right)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 21
5.42
Effect o f Varying the Thickness o f Slab
In o rd er to justify th e thickness u sed in th e m odel, tem perature an d w ater sat­
uratio n are calculated for th ree different thickness values, as show n in Fig. 5.12.
The change of thickness does n o t change any profile qualitatively. For th e thicker
m aterial, m ore grid p oints (sm aller A x) are required to obtain the sam e precision,
therefore, the m uch sm aller tim e step A t is also necessary for the convergence.
G enerally, if the CPU tim e tak en for 1 cm thickness is less th an 1 hr, th e CPU tim e
taken for 3 cm could be be m ore th a n 10 hr. Therefore m uch CPU tim es could be
saved u sin g the th in n er m aterials for the calculation.
5.43
Effect of Variation in the Liquid Intrinsic Perm eability o f
Very Wet M aterials
The liq u id intrinsic perm eability
is an im portant p aram eter to pressure driven
flow. A s m entioned in Section 4 8 2 , in this m odel th e gas intrinsic perm eability
is kept in dep en den t of the liq u id intrinsic perm eability to include th e effects of
stru ctu re change in w et vs. d ry m aterials. A possible exam ple of different liq­
u id intrinsic perm eability is th e raw potato and raw ap p le in their very w et stage.
Later, at a very d ry stage, they are likely to have sim ilar gas intrinsic perm eability.
To find o u t the effect o f differences in the liquid in trin sic perm eability o n th e heat
and m oisture tran sp o rt is the goal of th is section.
W hen kni is larger
= 20), the surface satu ratio n increases, as show n in
Fig. 5.13b. C alculation o f pressu re gradient show s th a t although the pressure gra-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
122
1 mm
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,3,4,5
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Time=0,1,2 min
1 0 0 -^
5 0.8
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Temperature T (°C)
Time=0,l,2 min
« 0.68 0.4£ 02
T"" T
1.0 1.5 2.0
x(cm)
x (cm)
Figure 5.12: Effect o f thickness on tem p eratu re and w ater saturation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2
£
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6
Heating time (min)
Heating time (min)
a. 1-02-
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Heating time (min)
100 n
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6 min
2 80 2 m in
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0.0
0.2
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Figure 5.13: Effect of liq u id intrinsic perm eability of very w et m aterials kun.
dient decreases w ith h ig h er k ^ , product of k ^ an d th e pressure g rad ien t increases,
increasing the flux o f liq u id tow ard the surface. Since the rem oval of liquid from
surface w ou ld n o t change appreciably, liquid w o u ld accum ulate n ear the surface,
increasing surface sa tu ra tio n greatly. There is n o appreciable convective drying
stage initially because very sm all overpressure can cause the liq u id flow. O n the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
o th e r hand, w hen
is sm aller, th e m aterial is less perm eable to th e liq u id . As a
result, th e pressure d riv en flow is less so th a t th e surface can stay drier, as show n
in Fig. 5.13b and Fig. 5.13e. It is m uch like convective drying.
A s kun increases, th e m axim um tem perature te n d s to be a little low er, as show n
in Fig. 5.13a and 5.13f. L ow er w ater saturation in sid e leads to decreased m icrow ave
absorption. That is also the reason th at the m axim um pressure is a little low er
as kni increases. The effects of
on m oisture loss is not significant, as show n
in Fig. 5.13d. The p ressu res developed inside is n o t high enough fo r th e blow ­
ing condition at the surface. Thus, m oisture rem oval is lim ited by the convective
m ass transfer at th e surface.
5.4.4
Effect o f Variation in the Gas intrinsic Perm eability o f Very
Dry M aterials
T he gas intrinsic perm eability kgi refers to intrinsic perm eability at a v ery d ry stage.
It is also an im portant param eter to pressure d riv en flow. Gas intrinsic perm eabil­
ity is m uch related to th e air porosity and different kgi can be obtained b y control­
ling the processing su ch as d rying rate, extrusion etc. To find o u t the effect o f dif­
ferent gas intrinsic perm eabilities on the h eat an d m oisture tran sp o rt is the goal
of th is section.
W hen
decreases (k^ = 1), th e m aterial is less perm eable to the g as and the
insid e total pressure is m uch h igh er ( Fig. 5.14c) because less air is d riv en o u t of the
m aterial. The m oisture level (saturation) is affected m ostly by th e liq u id co n ten t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
A s th e p ressu re increases rapidly, p ressure g rad ien t also increases. For a fixed w a­
te r perm eability, the w a te r flux can increase trem endously, as show n in Fig. 5.14b
an d Fig. 5.14e. The surface can g et fully satu rated (Fig. 5.14b) an d can experience
th e "p u m p in g ". M oisture loss increases m uch d u rin g th is "p u m p in g '' p erio d , as
sho w n in Fig. 5.14d. A t v ery h ig h values o f
on th e o ther hand, there is less
liq u id w ater m oving to w ard th e surface so th a t th e surface can get d ry like th at
d u rin g convective heating.
5.4.5
Variations in E ffective Gas D iffa sivity
This effective gas diffusivity, De/ j gr u sed in th e m odel is a function of the b in ary
diffusion of v apor in a ir (an d vice versa) th ro u g h the gas volum es of th e pores.
Such b in ary diffusivity is a function of tem peratu re as given by
(5.1)
w here T is in K. The effective diffusivity Def f g is also a function of the to rtu o sity of
th e p o rou s m edia. There is n o t m uch d ata on to rtuo sities in food system s. U sing
diffusivity in the range 2.6 — 4.6 x 10-5m2/s from Eq. 5.1 an d tortuosity rang e
o f 1-6 from G eankoplis (1978), w e get effective diffusivity varying from 0.43 —
4.6 x 10_5m2/s . Effect o f th is rang e v ariatio n in th e effective diffusivity is sh o w n
in Fig. 5.15b.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
Temperature (°C)
100-1
surface
.
maximum
>
100
80-
Maximum total pressure (0.1 MPa)
g oj&A
2 0. 6 -
40-
0.4-
2
4
6
Heating time (min)
•Hi
100
2
4
6
H eating time (min)
1.04-
1.03-
c 1.02-
1.0 2 -
8 10°-
8
(d)
10,100
1 0.98-
100
0.96-
1.00
2
4
6
Heating time (min)
10,100
2 min
5 min
70-
2
4
6
Heating time (min)
8
80-
50-
1.0-1
cn
e 0.8■oo
0.4
0.6
x (cm)
1.0
100
0.0
8
2 min
5 min
S
2 0.6
S
3IB 0.4
0.2
0.0
100
8
1.04-
90-i
Temperature (°C)
maximum
cn
6 0-
1.01 -
surface
1 0 -1
10,11
10,100
0.4
0.6
x (cm)
0.8
1.0
F igure 5.14: Effect of gas intrinsic perm eability o f very dry m aterials
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
T he tem p eratu re profiles are no t affected greatly d u e to the change in the m ois­
tu re diffusivity. The surface tem perature increases slightly w ith higher diffusiv­
ity, since m ore m icrow aves are absorbed d u e to increased saturation near the su r­
face. T he m axim um tem perature decreases w ith larger diffusivity since m ore m ois­
tu re from in sid e is lost an d therefore th e m icrow ave absorption inside is lower.
As th e diffusivity increases, vapor transports a t a faster rate from the inside to
the surface. H ow ever, since the capacity to rem ove m oisture from the surface
stays approxim ately the sam e, the excess vapor is condensed on the surface. This
increases surface w ater satu ratio n significantly. The higher surface tem perature
and surface satu ration increases m oisture loss slightly for higher diffusivity, as
show n in Fig. 5.15d. At low er diffusivity, surface becom es d rier since not enough
m oisture can reach the surface.
Initially, before pressure b uilds u p (Fig. 5.15c) d u e to higher internal tem per­
atures, th e surface does n o t get enough m oisture from inside and therefore the
surface w ater content decreases. A fter certain heatin g tim e it m ight increase be­
cause of m ore intensive pressure driven flow. The increase in diffusivity also "re­
leases" the pressures developed. Conversely, at low er diffusivity, significantly
h ig h er p ressu res can get developed, as is show n for the low est value of diffusivity.
H ow ever, for th e low est diffusivity, th e num erical solutio n diverges after som e
heatin g w h en th e pressure b u ild u p is very sharp. T he pressures drop eventually
as th e m oisture inside gets d ry and does not absorb m uch m icrow ave energy.
A ccording to the above analysis, change in effective diffusivity can change the
surface w etness significantly w hereas surface tem perature, internal pressure pro-
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
128
100u
I£
3
surface
maximum
803
0.43
40?0 -
4.6
0.6
0*4.6
60-
8.
t-
0.43
8w
0)
(a)
0.4D=4.6
5
“1---------1----------- 1-------- 1
2
4
6
Heating time (min)
2
8
4
Heating time (min)
6
8
o 1.020 0.43
3 1.015-
eSI
1.0 1 0 -
e
8
£3
2.0
UB
E 1.005
3
E
X
10
5
0.43
.2.0
1.00
(d)
0.98
5 0.%
2
4
Heating time (min)
6
8
2
4
Heating time (min)
0.43
2 min
6 min
100
055-|
^ 050- ZO
§ 0.452 0.40
3
035030
1.02
0.43
8
Z0
4.6
83
D=4.6, Z0. 0.43
S.
E
_____
£
0.0
6
0.4
0.6
x (cm)
1.0
0.0
0.4
0.6
x (cm)
0.8
1.0
Figure 5.15: Sensitivity o f tem perature, p ressu re, saturation an d rate of m oisture
loss to th e effective gas diffusivity for a low m oisture m aterial.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
file and m oisture loss are less affected. T herefore, the effect of v ap o r diffusion
need to be considered in th e governing eq u atio n for heating of low m oistu re m a­
terials. It is likely th a t this term can be ig n o red for the very w et m aterials w here
gas porosity is very sm all and therefore th e effect of diffusion is insignificant.
5.4.6
Effect o f H eating Rate (M icrowave Power Level)
H igher rate of h eating can be achieved by increasing the pow er level o f the mi­
crow ave oven. H eating rate is also a function o f the dielectric p rop erties and the
volum e of the m aterial. In the m odel, h eatin g rate can be changed by varying the
surface flux an d th e penetration d ep th . In th is study, only the surface flux w as
varied but, instead o f using the Fo values representing indirectly the v ario u s heat­
ing rates, heating rates them selves w ere u sed . The sensitivity to v ario u s heating
rates are show n in Fig. 5.16 and Fig. 5.17 for low an d high m oisture food, respec­
tively.
T em perature values for the low m o istu re food show n in Fig. 5.16a, show s that
the tem perature rise is closer to being lin ear for the highest heating rate. This is
characteristic of h ig h rate of m icrow ave h eatin g w here heat conduction does not
becom e significant, as discussed by D olande an d D atta (1993). Before pressure
builds up, surface m oisture an d the to tal m oistu re keeps dropping. T he pressure
build u p for the hig her heating rates is extrem ely rap id com pared to th e slow heat­
ing rate. This also causes a very sh arp increase in the saturation levels. These
sharp increases lead to divergence of th e num erical solution eventually. T he trends
in Fig. 5.16d seem to indicate a m uch g reater total m oisture loss a t h ig h er heating
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
rates.
100 H
> 0.70c 0.65 « 0.602 055-
0.27
U
o
H 80£
3<o 60h»
8L
E
{2
1.82
surface
maximum
4
6
Heating time (min)
2
£
0.45-
0-27°C/s
T
4
6
Heating time (min)
8
1.101
o.
£3 1.081.06Cl.
1.042
1.02E
3
E
0.27cC /s
2
'x
IQ
2
0.98-
-i--------- 1--------- T
2
4
6
Heating time (min)
0.96-
T--------- 1--------- T
2
4
6
Heating time (min)
Figure 5.16: Effect of heating rate on tem perature, pressure, satu ratio n , an d total
m oisture fo r a low m oisture food.
The h ea tin g rates for high m oisture foods are generally higher a n d therefore
a high er set o f heatin g rates are used in Fig. 5.17. A s show n in Fig. 5.17a, tem per­
atu res quickly reach 100°C an d slightly hig her a t these high rates. M uch higher
pressures are d ev elo p ed th a t can p u sh w ater q u ite effectively a n d w ate r satu ra­
tion quickly reaches its m axim um possible v alu e (Fig. 5.17c). This p ressu re b u ild u p
and th e com plete surface saturation lead s to significant "pu m pin g" effect at the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
boundary for all th e three h eating rates stu d ied . P ressures increase w ith heating
rate and and reach th eir m axim um value quicker. C onsequently th e steep reduc­
tion in the average m oisture content d u e to the "p u m p in g '' effect sta rts sooner at
higher heating rates an d leads to a g reater reduction in th e to tal m oisture. Even­
tu ally as m oisture in th e m aterial reduces significantly th e pressure b u ild u p is re­
duced, w ater satu ratio n d ro p an d the rate of reduction in th e total m o isture also
slow s dow n (Fig. 5.17d).
5.4.7 Com bined Effect o f Surface M ass and H eat Transfer
C oefficients
Surface heat an d m ass transfer coefficients can be m an ipu lated by oven design.
O ne sim ple w ay to change th e surface transfer coefficients is by changing air ve­
locity inside th e oven. In this section, h eat and m ass tran sfer coefficients corre­
sponding to th ree air velocities of 0.5,1.5, and 3 m /s th a t cover a feasible range
w as used. W hen velocity changes, b o th th e heat and th e m ass transfer coefficients
change sim ultaneously. Fig. 5.18 show s th e effects on tem perature, pressure, sat­
uration, and m oisture loss for th e values o f h eat transfer coefficients (8.7,15.1,21.4
W /m 2K) and m ass tran sfer coefficients (0.010,0.018, and 0.026 m /s ) correspond­
ing to the three velocities. H ow ever, to u n d erstan d w h eth er a p articu lar effect is
contributed m ore d u e to th e h eat tran sfer coefficient th an th e m ass tran sfer coef­
ficient, sim ulations w ere also ru n keeping one of the coefficients artificially fixed.
Thus, Fig. 5.19 an d Fig. 5.20 show the effects of change in ju st th e h eat or th e m ass
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
tran sfer coefficient respectively.
Surface tem p eratu re increases w ith decreases in velocity (transfer coefficients).
A t low er h eat tran sfer coefficient h ea t loss to th e colder surro un din gs is reduced,
w hich increase th e tem perature. A t lo w er m ass transfer coefficient less m oisture
is rem oved from the surface, increasing th e surface w etness and th e m icrow ave
absorption. Less m oisture rem oval from surface also m eans that less en erg y goes
into evaporation, so tem peratures w o u ld be higher. A lthough, b o th factors can
cause the increase in tem perature, com paring the tem perature increases in Fig. 5.20
an d Fig. 5.19, it is seen th at the m ass tran sfer coefficient has a greater effect on the
tem perature. T he m axim um tem p erature is not affected m uch since th e surface
effects do n o t p en etrate m uch into th e m aterial due to sm all therm al conductivity
an d capillary diffusivity.
T he increase in surface m oisture w ith decrease in transfer coefficients in Fig. 5.18
is m ore due to th e decrease in m ass tran sfer coefficient rather than th e h eat trans­
fer coefficient. The surface is w etter fo r decreased m ass transfer coefficient d u e to
decreased rem oval of surface m oisture, as show n in Fig. 5.19. The h ig h e r surface
tem perature d u e to low er heat tran sfer coefficient, on the other h a n d , increases
the vapor pressure of w ater, thereby increasing the m oisture rem oval from the
surface, as show n in Fig. 5.20. H ow ever, th e com bined effect of decrease in heat
an d m ass transfer coefficients is to increase the surface m oisture.
In the later stages of heating (at approxim ately 6 m inutes), as th e p ressu re bu ilds
up , the pressure driv en flow becom es significant and the surface m o istu re starts to
increase. A t these tim es and beyond, th e h eat transfer coefficient seem ed to have
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
less effect as in earlier tim es, on th e surface satu ratio n. M ass transfer coefficient
has m uch stro n g e r effect an d its decrease w o u ld cause the surface m oisture to in­
crease. A t a lo w enough m ass transfer coefficient, th e large am ount of m o istu re
brought in to th e surface b y th e pressure d riv e n flow causes w ater to accum ulate
on the surface b ey o n d in itial saturation. A t v ery large m ass transfer coefficients,
the surface can rem ain q u ite dry.
5.4.8
C onsideration o f a 2D G eom etry
This section w ill discuss how th e tem perature, pressure, an d m oisture are affected
w hen the m aterial is considered to be h eated from tw o perpendicular d irectio n s
by m icrow aves. The sim plest tw o-dim ensional m odel developed here is a n axisym m etric geom etry w ith a rig h t circular cylinder, w ith the bottom receiving n o
m icrow aves a n d having n o energy o r m o istu re transfer. The top an d th e sid es
are heated by th e sam e m icrow ave flux. F or m ore details o n the m odel, see sec­
tion 4.9 an d A ppendix C. T he tem perature, w ate r saturation and total p ressu re
profiles for h ea tin g a low m oisture m aterial a re show n in Fig. 5.21. D ue to th e m i­
crow aves com ing from tw o directions, th e tem p eratu re is m uch higher in sid e a s
com pared to th e ID m odel (Fig. 5.5). T em perature rise is ab o u t 90°C in 1 m in u te as
com pared w ith 6 m inutes for th e ID m odel. M ore intense heating is th e p rim ary
effect of h eatin g from tw o directions as o p p o sed to one. The top com er sh o w s
the low est tem p eratu re because it loses h e a t from both sides. The h ig hest tem ­
perature occurs o n the centerline close to th e to p surface in tw o m inutes. A fter 4
m inutes the te m p eratu re field inside tends to b e uniform d u e to therm al equilib ­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
rium , as can b e seen in ID profiles.
Initially, before p ressu re b u ild s u p , m oisture loss is by diffusion an d the top
com er loses th e m o st am ount of m oisture (low est surface saturation). In about
3 m inutes, significant p ressure bu ilds u p an d th e p ressure driven flow leads to
m uch higher su rface satu ration . The to p com er still stays the driest since m oisture
is rem oved m o st effectively from th e com ers. The pressure increase is also m ore
rapid th an in th e ID m odel. The m oisture loss is also m uch m ore rap id . Therefore
in 2D system , h ig h er heatin g rates cause all of th e changes to be faster.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
0 .6 9
0.69°C/s
surface
maximum
r-N ^ l. 1 2
« 0.7-
1 .4 6
0.69°C/s
I------- 1----- 1------1------ 1
1
2
3
4
5
Heating time (min)
I--------1---- 1-------1------ 1
1
2
3
4
Heating time (min)
5
0.69°C/s
0.69°C/s
1.46
I'
1
I
i
1.46
1
-----1
1------ 1------ 1
2
3
4
Heating time (min)
2
3
4
5
Heating time (min)
2 min
6 min
0.69°C/s
1.12,1-46
1.12,1.46
0.69°C/s
2 min
6 min
0.69=^
(0
x (cm)
x (cm)
Figure 5.17: Effect of heatin g rate o n tem perature, pressure, saturation, an d total
m oisture for a high m oisture food.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
136
0.7
surface
m aximum
m*
e0 0.6 H
•s
2 05
5
S
Im
£
1
0.4 H
03
■-B
a.
2
o,
V 1.0203
1.015a.
1.0102
1.005f
3
E 1.000X
2
2«t
o.
E
£
20
1.04
c 1.00
O 0.96% 0.92-
T--------- 1--------- T
2
4
6
H eating tim e (m in)
/
3 2
r-
i
1"
0.0
--*2
n ------------ 1----------- r
2
4
6
H eating tim e (m in)
4
6
H eating tim e (m in)
2
2 m in
8 min
H
a3
(b)
03 H
t ------------ 1------------ r
2
7 \
* « * * .• * » ,
02
8
6
4
H eating tim e (m in)
Time=0
2 m in
8 m in
/
0 .4 03-
(f)
'""I
' " l'~
0.4
0.6
x (cm)
T
I
0.8
1.0
x (an )
Figure 5.18: Effect of com bining effect o f m ass and heat transfer coefficients on
m axim um an d surface values.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0.1 MPa)
Maximum total pressure
0.015
100 -
Surface temperature
(°C)
137
x / .* ...•..**•
TOO 2.
3
80 3
S'
60 3
_v ...*
— \
h,= 0.03
nn
40
0.007
20
1---------1--------r
2
4
6
Heating time (min)
IfX£,
43
2 0.8a 0.6-
S
‘iiiiiii..:__
&
Ii 0.4-
{? 8
3 42
Oa
n cn
2
A
1. 020 -
1.0051.000
2
4
6
Heating time (min)
8
o3
4
6
Heating time (min)
-I
S
1c 1.00 8 0.96s
2
0.92'o<o
2
h ,- 0.03
f
v 'o.oo7 o.os ..........
0.015
■0.2 |D
“i--------- 1---------r
3 1.04-
1. 010 -
i
**»...
K,= 0.03
0.2-
1.025-f
1.015-
— surface
maximum
(b)
2
0.007
.0.015
(d)
2
4
Heating time (min)
6
8
Surface temperature (°C)
F igure 5.19: Effect o f hmV o n m axim um an d surface values.
100
surface
maximum
s
3 1 0.7e
3
40-
1w 0.6*
2
■S <B
3 I O'5'
c 8«
h~30
3
n cn
20
2
Maximum total pressure (0.1 MPa)
100 5
0X» c0 0.8-
surface
3, ll
maximum >
4
6
Heating time (min)
a>)
0.4-
0.4
h«T
h - r ^ ••••cr*....
"i--------- 1---------r
2
8
4
6
Heating time (min)
Si
3 1.04-
1 .0 2 0 -
2
!
1.015
1. 00 -
8
sa
1.005
2
4
6
Heating time (min)
8
(d)
0.96-
h=3
4
Heating time (min)
Figure 5.20: Effect of h o n m axim um an d surface values.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
8
b
,<cm)0-5
(Ml
(a)
Time-1 min
(b)
Time-3 min
Water saturation Syy
Water saturation S**
0.450
0.400
Top
comer Top
center
0350
Bottom ^
center
1.0
05' 0j)
Bottom
Z(cm)
(c) Time-1 min
Total pressure (O.IMPa)
Z(cm)
(d)
1.0
0.o
0.0 ricm)
Tim e-3 min
Total pressure (O.IMPa)
1.0075
1.0050
1.0025
1 .0
(e)
Z(cm)
Time-1 min
Top
comer
1.005
r
Bottom
2-0
center
0.0
ID
dcm ).
(f)
Z(on)
Time-3 min
Figure 5.21: T em perature, w ate r saturation, an d p ressu re profile in microwave
heating o f 2D cylindrical lo w m oisture foods a t 1 a n d 3 m inutes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 6
CONCLUSIONS
T he follow ing conclusions are m ade from die study:
1. A porous m edia m odel w ith internal heat generation an d w ith D arcy's flow
d u e to internal p ressu res w as developed to p redict th e m oisture tran sp o rt
in foods u n d er m icrow ave heating.
2. Pressures developed d u e to internal evaporation fundam entally changes th e
nature of m oisture m ig ration in a m icrow ave heatin g process, as com pared
to a conventional h eatin g process. As tem peratures inside approach the boil­
ing point of w ater, p ressu re developm ent becom es significant. This p u sh es
m oisture tow ard th e surface. Generally, it results in a m uch higher surface
m oisture level th an d u e to th e diffusive m ovem ent alone.
3. A colder food surface d u e to th e surrounding air a t room tem perature re­
duces the m oisture rem oval capacity of the surface an d aid s in the m oisture
build up near the surface.
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
140
4. In a very w et m aterial, an d u n d e r h ig h rates o f m icrow ave heating, large
en o u g h pressures are d ev elo p ed th a t lead to "p u m p in g ou t" of w ater a t the
surface. This leads to a m u ch h ig h er rate of m oisture loss u n d er such heating
conditions.
5. L arger liquid perm eability a t w et stag e increases th e in tern al m oisture trans­
p o rt th a t leads to w etter surface. L arger gas perm eability a t a very d ry stage
red u ces the in ternal m o isture tran sp o rt.
6. S m aller gas diffusivity can cause num erical divergence. Therefore, its value
n eed s to be carefully chosen.
7. H ig h er heating rate increases m oisture tran sp o rt from the inside to th e sur­
face.
8. Increasing air velocity aro u n d th e m aterials can reduce the surface m oisture
a s w ell as surface tem perature.
9. T he thickness of the m aterials d oes n o t change th e profiles qualitatively.
10. C onvective term in th e en erg y equ ation can change th e tem perature and
m o istu re profiles qualitatively for low m oisture foods b u t affects very little
th e h ig h m oisture foods.
11. In a 2D geom etry, heatin g ra te s are faster d u e to m icrow aves reaching from
b o th sides. T hus its effects are equivalent to those o f h igher heating rates in
th e ID heating. This lead s to faster changes in tem perature, m oisture and
pressu re. The com ers becom e d rier w hile the surface becom es m ore w et.
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Chapter 7
FUTURE WORK
1. Q uality im provem ent by com bining m icrow ave heating w ith h o t a ir or in­
frared heating, an d by using pressurized cham ber to decrease th e in tern al
m oisture tran sp o rt.
2. U se ad ap tiv e m esh to h an d le the shai p interface.
3. Incorporate source term from solving M axw ell's equations.
4. B etter hand le th e fully satu rated region.
5. M easure in p u t p aram eters such as perm eabilities and diffusivities.
6. E xperim entally m easure m oisture profiles.
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix A
ESTIMATION OF HEAT AND
MASS TRANSFER COEFFICIENTS
142
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143
It is assum ed th a t the a ir flow o ver th e sam ple is lam inar
Nu = ~
K
= 0 .6 6 4 (-)3 ite |
Of
(A.1)
so th at
h = 0.664 £ 0
5 q 0 333^(0 5_0 333)
(A.2)
By analogy to h eat transfer, the m ass tran sfer Num can be w ritten b y
Num =
hrtaiL
‘- 'A B
U
= 0.664( - ) i R e l
(A.3)
U AB
so that
T \ 0.67m0.5
hjnV = 0.664 ^0.5^5-0^333)
^A *4^
Take u=2.l x 10-7 m 2/ s a t 77°C (Incropera et al.,1990), Dva= 4.6 x 10~5 m2/ s a t
77°C (Eq. 4.36), a = 3.0 x 10-5 m 2/ s a t 77°C (Incropera et al.,1990), an d L=0.1m.
The m ass and h ea t tran sfer coefficients w ere calculated for air for d ifferent veloc­
ities, as show n in Table A .I.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table A.1: M ass an d h eat tran sfer coefficients
A ir velocity (m /s)
0.1
3.0
M ass tran sfer coefficient h m (m /s)
0.003
0.03
H eat tran sfer coefficient h (W / m 2 K)
32
25.7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix B
INPUT PARAMETERS FOR
CONVECTIVE DRYING (Nasrallah
et al., 1988)
145
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146
B .l Material Properties of Clay Brick
Table B .l: M aterial properties.
P aram eter
Symbol
Value
porosity
<t>
0.26
Intrinsic perm eability
K
2.5 x 10"14
m2
1
Vapor diffusivity in a ir
Dva
2.3 x 10-5
m 2/ s
2
D ensity of solid m atrix
Ps
2600
k g /m 3
1
Specific h eat of solid m atrix
Cs
879
J/k g K
1
Therm al conductivity o f m atrix
ka
1.442
W /m K
1
Coefficient in Eq. B.3
<?0
0.1212
N /m
2
Coefficient in Eq. B.3
0
1.67 x 10"4
N /m K
2
D ynam ic viscosity o f w ater
Pw
5.468 xlO "4
P aS
D ynam ic viscosity o f gas
Ps
1.8 xlO -4
P aS
L atent of v ap o r
A
2,257
k j/k g
Irreducible w ater satu ra tio n
s*
0.08
Thickness of slab
L
0.01
U nits
Source
1
1
m
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147
B.2 Initial value and boundary parameters
Table B.2: Initial value an d b o u n d ary param eters.
Sym bol
V alue
Units
Source
Initial T em perature
Ti
47
°C
1
Initial w ater satu ratio n
S%Vi
0.9
Initial total pressu re
Pi
101,000
Pa
1
Surrounding tem p eratu re
To
87
°C
1
S urrounding pressure
Po
101000
Pa
1
Surrounding v ap o r density
Pvo
0
k g /m 3
1
M ass transfer coefficient
K
0.014
m /s
1
H eat transfer coefficient
h
15.0
W / m 2/ K
1
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B.3 Relationships
Capillary force is given by Leveret's relation ship ,
Pc — Y
w here
J ( S W) =
0.364(1 —exp(—40(1 —Sw))) + 0.221(1 —Sw)
+0.005/(5* - 0.08)
er(T) = O o - 0 T
Vapor pressure is given by K elvin's eq u atio n , as show n in Fig. B.l a.
pv = pV3exp(—2 o M v/r p wR T )
w here
log(r)
= 2.16 x 10"2 + 43.85* - 253.55£ + 794.545;
-1333.75?, + 11115® - 352.55?, - 10
Relative permeabilities of w ater and gas are given by
kg =
1 - 1.15*
fc* =
(5* — 5*r)/(l —Sir)
Effective gas diffusivity is given by
d ./u
= £ W i - s „ ) 2(0 (i - s „ ))V3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Effective thermal conductivity is given by
(B.8)
kejf = (Af25# ! - Sw) + k%*<t>Sw + A f*( 1 - 0))4
The cap illary diffusivity (D w =
) is calculated based o n Eq. 3.34 and is
show n in Fig. B .lb along w ith effective gas diffusivity.
Clay
0.4
Clay
- capillary diffusivity
- vapor diffusivity
60°C
i
0.0
0.2
0.4
0.6
Water saturation Sw
0.8
1.0
0.0
0.4
0.6
0.8
Water saturation Sw
1.0
Figure B .l: V apor pressure, v ap o r d iffusivity an d capillary d iffusiv ity vs w ater
satu ratio n in clay brick.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix C
GOVERNING EQUATIONS,
INITIAL AND BOUNDARY
CONDITIONS IN 2D CYLINDER
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C.1 Governing Equations
T he 2D geom etry is show n in F igure C .l. We only consider a rectangular slice
w ith length R a n d H because o f axisym m etry. The m icrow ave com es from top,
bottom an d th e side.
MW
Z
H i
Top com er
MW
Bottom center
MW
Figure C.1: Schem atic diagram o f 2D m odel.
Based on Eq. 3.73, the governing equations in cylindric coordinate w ithout
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
convective term in energy equation are given as
dS
dT
dP
f t 4 - s r + n 5 T - + a 6 -57-
Ot
at
at
as
ar
ap
at
at
at
A U>"aT + n n — + n i 2 - r -
r dr
I d
+rar(
I d
^ rd r
dr
dz
dz
dT, a .
a7\
2rd r ) + d2(
dP. d
dP.
r dr
dz
dz
1a
as. a as,
r a r ( 7 r a r ) + a 2( W
1a a r . a ,,, ar.
+ rar
8 r a r ) + a 2( 8a2 )
i a ... ap a ap
+ ; a ? (A (,ra r ) + a ; (A9a r )
„ as „ dr .. ap
id,,,, as, a,., as,
A,6'¥ + AlTar + Al8ar = ?a;(/fl:,raF>+ aI(Ar,3 aT)
i a, „ ar, a (t, ar,
+ ; a ? ( /f w r a T 1 + a J ( " a T *
ia
ap.
#px a
+;a?(A15’"aF) + aI( A ‘5 ai-’+ 5
w here S is w ater satu ratio n.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.1)
(C.2)
(C.3)
153
C.2 Initial Condition and Boundary Conditions
io
II
II
(f = 0)
(t = 0)
II
Co
Initial conditions:
(t = 0)
(C.4)
Boundary conditions:
1. O pen b o u n d ary w ith o u t “pum ping"
Vaporflux
|r=« =
h0(pv - p 00) (r = R.O < z < H)
(—A'1^ - — Ko-zp — K 3-^ -) |.=w =
ft Q
r¥T*
n
( A i — -f A 2-^ - + A 3 — ) |;=o =
hu(pv — px ) (z = H . O < r < R )
— A'2 ^ 7 - A'3 ^
7)
(c =
0
0 .0
< r < /?)
(C.5)
Heat flux
( - t f i 3 § ^ - A',4| £ - A 'i s ^ ) |r=R =
r\ p
ryy»
qq
( - A - , 3 ^ - - A' u 3 7 - A'15 g 7 i U h =
r\ o
(A'i3 -^— h A i 4 —— Ic/c
c/c
h ( T - T x ) (r = R , 0 < z < R )
H T - T x ) ( z = H.O < r < R)
on
oz
) |r=o =
0
(c =
0 .0
< r < /?)
(C.6 )
Total pressure
P = Poe {r = R.O < z < H; z = H ,0 < r < R: z = 0.0 < r < R)
2. Insulated b o u n d ary
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(C.7)
154
Vaporflu x
|r= 0
=
0 (r = 0,0 < z < H )
(C.8 )
+ K , ^ ) |„ „
=
0 (r =
(C.9)
(A 'i— 4- A 2-^—+ A 3 ——)
<7T
£7T
or
A ir flu x
(K ~ +
0 ,0
<;<//)
Heaf /Zmx
(A 13— +
4- K is-^p ) Ij—o =
0 (r = 0,0 < z < H )
(C.10)
w here hu and hQare m ass tran sfer coefficients on the to p surface a n d side, respec­
tively. h is h eat tran sfer coefficient of to p surface an d side. It is assu m ed th at there
is no m ass an d energy exchange o n th e bottom . The b o u n dary conditions w ith
"pum ping" are n ot explicitly given an d can be referred in ch ap ter 3.
The m icrow ave surface is the sam e o n the to p surface and sid e. H ow ever, the
flux on th e bottom is assum ed to b e o ne ten th of the o ther flux.
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