Mathematical Physics : A mathematical model of frying processes

Mathematical Physics:
A mathematical model of frying processes
A. Mancini
Univ. di Firenze, Dipartimento di Matematica “U.Dini”
Firenze - Italy
alberto.mancini@math.uni?.it
M. Primicerio
Univ. di Firenze, Dipartimento di Matematica “U.Dini”
Firenze - Italy
mario.primicerio@math.uni?.it
Yiqing Yang
Hangzhou Dianzi Univ.,Department of Mathematics
Hangzhou 310018, China
Abstract
We present a mathematical model for the process of frying a rather
thick sample of an indeformable porous material saturated with water
(e.g. a potato slice).
The model is based on thermodynamical arguments and results in a
initial-boundary value problem for a system of equations satis?ed by the
temperature and vapour content, with a free boundary separating the
region saturated with water and the vapour region.
We provide some results of numerical simulations.
In questo lavoro viene presentato un modello per il processo di frit-
tura per immersione. Si considera il processo applicato ad un campione
il cui spessore sia su?cientemente grande da rendere trascurabili le de-
formazioni dovute alla cottura come accade, per esempio, nelle comuni
patatine fritte.
Lo sviluppo del modello `e basato su considerazioni termodinamiche ed
ha la forma di un sistema di equazioni di?erenziali alle derivate parziali
con frontiera libera, quest’ultima rappresentata dal fronte di desaturazione
dovuta alla vaporizzazione dell’acqua contenuta nel campione (modellato
come un mezzo poroso non deformabile). Le incognite del sistema rap-
persentano la temperatura all’interno del campione ed il vapore contenuto.
Keywords and Phrases: mathematical model, frying, moving boundary
1

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1
Introduction
In the last decade, several papers have been devoted to the mathematical model
of frying, starting from [2] [3] that abandoned the purely phenomenological ap-
proach of previous attempts and analyzed and discussed the di?erent phenomena
of mass and energy transfer that are involved in the process.
Referring to one-dimensional geometry with planar symmetry, a slab x ?
(?L, L) of porous material saturated with water and having a given temperature
is considered to be put in contact, on the faces x = ±L, with an oil bath kept at
temperature T? above the boiling point of water. This is assumed to happen
starting from time t = 0 and frying is described as a coupling between heat
transfer (conduction and convection) and vapour migration in an undeformable
porous medium (see [1] and [8]).
As a matter of fact, this idealized situation exhibits most of the basic in-
gredients that seem to be relevant e.g. in the process of frying relatively thick
samples of potatoes. Moreover it is suitable for rather easy numerical simulation
and hence to possible experimental validation. Of course, once this preliminary
check is obtained, modi?cation induced in the organic material by the thermal
history (see [8] [9] and e.g. [7] for an introduction to the in?uence of the process
on the result) as well as more realistic geometry will be taken into account.
But, even in the idealized situation described above, the problem is far from
being trivial. In a recent paper [5] a complete analysis of the phenomenon,
based on a correct application of the basic balance laws, has been performed.
There the mathematical model is formulated in terms of a non-standard free
boundary problem for a system of parabolic equations. Indeed, the region x ?
(?L, L), t > 0 (of course, using symmetry the analysis is con?ned to (0, L)× +)
is the union of four sub-region separated by free boundary: the water-saturated
part, the region of coexistence of vapour and water, the pure vapour region and
the crust.
Here, we present a simpli?ed model in which the region of coexistence is
assumed to have a negligible thickness, while incorporating in the model the
correct Rankine-Hugoniot type conditions. Moreover, in the model that will be
presented and discussed, crust formation is neglected although a simple modi-
?cation of the boundary condition on x = L is suggested in order to take this
e?ect into account.
Thus, only one free boundary and two regions are considered: region 1,
where the porous medium is completely saturated by water, and region 2 in
which the pores are ?lled by water vapour in thermodynamical equilibrium.
A crucial point is to assume that the porous medium is non-deformable
and thus frying processes of thin layers (e.g. tortilla chips, see [9]) will need a
substantially di?erent model.
After evaluating the time scales, it will be possible to reach a partial dis-
entangling of the problem for the unknown temperature from the problem for
the unknown pressure, suggesting how to investigate the well-posedness of the
problem.
Finally, numerical simulations are shown for some speci?c cases.
2
The governing equations and the boundary
conditions
According to the discussion of the previous section, we write the balance equa-
tions in the two regions.
In region 1 (water-saturated porous medium), we have that pressure is con-
stant, since ?p = 0 at x = 0 and the medium is assumed non-deformable (and
?x
water compressibility is neglected). Thus, the only equation we have to consider
is the energy balance
2

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?T
?2T
(?c)1
? k
= 0,
(2.1)
?t
1 ?x2
where (?c)1 and k1 are respectively heat capacity and conductivity in region 1,
that are essentially given by
(?c)1 = (1 ? ?)?scs + ??wcw,
(2.2)
k1 = (1 ? ?)ks + ?kw.
(2.3)
In (2.2) and (2.3), we denoted by ? the porosity and ?, c and k represent
density, speci?c heat and conductivity respectively, while su?xes s and w refer
to solid matrix and water, respectively.
In region 2 (solid + vapour) we have to consider mass balance in addition to
energy conservation. Indeed, vapour migration is induced by pressure gradient,
so that Darcy’s law implies
??
K ?
?p
?
v ?
?
= 0,
(2.4)
?t
µ ?x
v ?x
where su?x v refers to vapour and K, µ denote intrinsic permeability of the
porous medium and dynamic viscosity of vapour.
Passing to energy balance, we should consider that, in principle, heat capac-
ity and conductivity depend on ?v, but it is reasonable to assume that
(?c)2 = (1 ? ?)?scs + ??vcv ? (1 ? ?)?scs,
(2.5)
k2 = (1 ? ?)ks + ?kv(?v) ? (1 ? ?)ks.
(2.6)
The most important di?erence with energy balance in region 1 is that now
a relevant role is played by convection. Indeed, we have
?T
?2T
K ?
?p
(?c)2
? k
? c
?
(T ? T
?t
2 ?x2
v µ ?x
v ?x
0)
= 0,
(2.7)
where T0 is the boiling point temperature at atmospheric pressure.
Let us pass to discuss the conditions on the external boundary and at the
initial time. For the latter, we have e.g.
T (x, 0) = ¯
T < T0,
0 < x < L,
(2.8)
where we may allow, in general, ¯
T to depend on x. Moreover,for symmetry
reasons,
?T (0,t) = 0, t > 0.
(2.9)
?x
On the boundary x = L, we write
?T
?k1
(L, t) = ?
?x
1(T (L, t) ? T?),
0 < t < t?,
(2.10)
and
?T
?k2
(L, t) = ?
?x
2(T (L, t) ? T?),
t > t?
(2.11)
where T? is the (prescribed) temperature of the oil bath, ?i are thermal ex-
change coe?cients (in principle, ?2 is dependent on the thermal history of the
surface and on the discharge of vapour) and
t? = sup {T (L, t) < T0}
t
3

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Moreover for t > t?
p(L, t) = p0,
t > t?,
(2.12)
where p0 is the atmospheric pressure.
Finally, we have to write the conditions on the free boundary x = s(t), t > t?,
separating the two regions. To simplify notation we write T ± ? T (s(t) ± 0, t)
and similarly for the other quantities. Thus, we have
T ? = T0,
(2.13)
Pressure is continuous and given by the Clapeyron’s law
?
1
1
p+ = p? = p0 exp
?
,
(2.14)
R
T0
T +
where ? is the latent heat of vapourization and R is the gas constant. Note
that we had to write on the r.h.s. the temperature of vapour, since temperature
will be discontinuous across the free boundary, as a consequence of the fact
that in our scheme the thickness of the region occupied by saturated vapour is
neglected.
Imposing global mass and energy balance and di?erentiating as in the pro-
cedure currently used to obtain Rankine-Hugoniot jump conditions, we obtain
K
?(?w ? ?+
v )s (t) =
?+
µ v p+
x ,
t > t?
(2.15)
that will be approximated by
K
??ws (t) =
?+
µ v p+
x ,
t > t?
(2.16)
K
????ws (t)+k1T ?
x ?k2T +
x ?(?c)2(T +?T0)s (t)?cv (T +?T0)
?+
µ v p+
x = 0,
t > t?,
(2.17)
We will ?nally assume that, in region 2, ?v, p and T are related by a known
state equation. For simplicity, we take
p = RT ?v.
(2.18)
Of course, the ”initial” condition for s(t) is
s(t?) = L.
(2.19)
3
Rescaling
We set
x = ?L,
(3.1)
T = uT0,
(3.2)
p = vp0,
(3.3)
and we rescale time by a constant ? that will be chosen later
t = ??.
(3.4)
Then, we de?ne
?0 = po/(RT0),
(3.5)
4

---

and
? ? = t?/?,
(3.6)
so that ? ? = sup? {u(1, ?) < 1}. We also de?ne the rescaled free boundary
between the water-saturated region and the vapour region
1,
? ? ? ?,
z(? ) =
s(?? )/L,
? > ? ?
(3.7)
Then, after simple calculation, we get
?u
? ?2u
=
,
0 < ? < z(? ), ? > 0,
(3.8)
??
t1 ??2
?u
? ?2u
?
? ?
?v
1
=
+ ? 0cv
v
1 ?
, z(? ) < ? < 1, ? > ? ?,
(3.9)
??
t2 ??2
(?c)2 t3 ??
??
u
?v
v ?u
?
?
v ?v
?
=
u
, z(? ) < ? < 1, ? > ? ?,
(3.10)
??
u ??
t3 ?? u ??
where
L2(?c)
t
1
1 =
,
(3.11)
k1
L2(?c)
t
2
2 =
,
(3.12)
k2
L2? µ
t3 =
.
(3.13)
p0 K
Initial and ?xed boundary conditions read
u(?, 0) = ¯
u(?) = ¯
T (?)/T0, 0 < ? < 1,
(3.14)
?u (0,?) = 0, ? > 0,
(3.15)
??
?u
?
(1, ? ) = ?
??
1[u(1, ? ) ? u?],
0 < ? < ? ?,
(3.16)
?u
?
(1, ? ) = ?
??
2[u(1, ? ) ? u?],
? > ? ?
(3.17)
v(1, ? ) = 1,
? > ? ?.
(3.18)
where ?1 = ?1L/k1, ?2 = ?2L/k2, u? = T?/T0.
Next, we consider the interphase conditions. From (2.16), we have
dz
? ? v+
?v +
=
0
, ? > ? ?,
(3.19)
d?
t3 ?w u+
??
From (2.17), we obtain
dz
(?c)
?
(?c)
?
(?c)
dz
?
?
u+ ? 1
=
1T0
u??
2T0
u+?
2T0 (u+?1) ? 0cvT0 v+v+
,
d?
???
?
?
?
w
t1
???w t2
???w
d?
??w t3
u+
? > ? ?.
(3.20)
5

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Finally, (2.13) and (2.14) become respectively
u? = 1,
(3.21)
?
1
v(z(? ), ? ) = exp
1 ?
, ? > ? ?.
(3.22)
RT0
u+
At this point, it is natural to choose
???
? =
wL2 ,
(3.23)
k1T0
so that, setting
(?c)
? =
2T0 ,
(3.24)
???w
condition (3.20) becomes
dz
k
dz
?
t
u+ ? 1
= u? ? 2 u+ ? ?(u+ ? 1)
? ? 0cv 1 v+v+
, ? > ? ?.
(3.25)
d?
?
k
?
?
1
d?
(?c)1 t3
u+
4
A reasonable simpli?ed model
It is immediately checked that, while ? and the characteristic di?usion times t1
and t2 are of the same order of magnitude (using the data of [2] we have indeed
?
325, t1
846, t2
870), t3 = 0.38 and thus ?/t3 is of the order 103.
Therefore, it is reasonable to substitute (3.10) with the much simpler rela-
tionship
?
v ?v
= 0, z(? ) < ? < 1, ? > ? ?,
(4.1)
??
u ??
Note that (3.9), (3.19) and (3.25) contain the ratios ?/t3, t2/t3, t1/t3, but there
are also multiplying factors of the order ?0/?w ? 10?3.
Now, we use again symbols x and t instead of ? and ? and denoting by
capital letters A, B, C, ... the constants O(1) appearing in the equations written
in Section 3, we can state the following classical formulation of our problem:
Find a constant t?, a decreasing function z(t), z(t?) = 1, z(t) > 0 and two
functions u(x, t), v(x, t) possessing all the regularity we will need and such that:
?
?
? u
?
t = Auxx,
in D0 = (0, 1) × (0, t?),
u(x, 0) = ¯
u,
x ? (0, 1),
(4.2)
?
? u
?
x(0, t) = 0,
t ? (0, t?),
?ux(1, t) = ?1[u(1, t) ? u?],
t ? (0, t?),
where ¯
u ? (0, 1) and A = ?/t1.
0 < u(x, t) < 1,
in D0,
(4.3)
u(1, t?) = 1.
(4.4)
?
?
? u
?
t = Auxx,
in D1 = {(x, t) : 0 < x < z(t), t > t?},
u(x, t?+) = u(x, t??),
x ? (0, 1),
(4.5)
?
? u
?
x(0, t) = 0,
t > t?,
u(z(t), t) = 1,
t > t?.
6

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0 < u(x, t) < 1,
in D1,
(4.6)
ut = Buxx + C vvx 1 ? 1
in D
u
x
2 = {(x, t) : z(t) < x < 1, t > t?}
?ux(1, t) = ?2[u(1, t) ? u?]
t > t?,
(4.7)
B = ?/t2, C = ??0cv?/[(?c)2t3].
vvx
= 0, in D
u
2,
(4.8)
x
v(z(t), t) = exp G(1 ? 1/u+) , t > t?,
(4.9)
v(1, t) = 1, t > t?,
(4.10)
and
z (t) = Hv+v+
x /u+,
t > t?,
(4.11)
u+ ? 1
z (t) = u?
x ? M u+
x ? ?(u+ ? 1)z (t) ? N v+v+
x
, t > t?,
(4.12)
u+
1 < u(x, t) < u?, in D2,
(4.13)
v(x, t) > 1, in D2.
(4.14)
where G = ?/[RT0], H = ??0/[t3?w], M = k2/k1 and N = ??0cvt1/[(?c)1t3].
Since (4.2) is a standard heat conduction problem we can solve it. Then (4.3) is
a consequence of maximum principle and the existence of t? follows immediately
from (4.4) if u? > 1.
Let us write, from (4.8)
vvx = ?f(t), in D
u
2,
(4.15)
for an unknown positive function f (t) and consider the free boundary problem
for u(x, t) consisting in (4.5) and
ut = Buxx ? Cf(t)ux, in D2.
(4.16)
?ux(1, t) = ?2(u(1, t) ? u?), t > t?,
(4.17)
z (t) = ?Hf (t), t > t?,
(4.18)
z (t) = u?
x ? M u+
x ? ?(u+ ? 1)z (t) + N f (t)(u+ ? 1), t > t?,
(4.19)
with the condition on f (t)
(v2)x = ?2f(t)u(x, t), in D2,
(4.20)
i.e.
1 ? exp{2G(1 ? 1/u+)}
f (t) = ?
.
(4.21)
2 1 u(x, t)dx
z(t)
Thus, for any given f (t), we have transformed the original problem in a free
boundary problem (in two ”phases”) for a single unknown function u(x, t),
whose solution in turn gives f (t). The analysis of the well-posedness of this
non-standard problem will be performed in a forthcoming paper.
7

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5
Numerical simulation
We show here some numerical simulations where the simpli?ed adimensional
model has been used. With reference to the situation sketched in Fig. 1 (where
the conditions on the ?xed boundaries are shown) we have solved
ut = Auxx
in D0 ? D1
(5.1)
ut = Buxx ? Cf(t)ux
in D2
(5.2)
where f (t) is given by (4.21) and z(t) (z(t?) = 1) is the free boundary on which
the following conditions are prescribed
u(z(t)?, t) = 1
(5.3)
z (t) = ?Hf (t)
(5.4)
z (t) 1 + ?(u+ ? 1)
= u?
x ? M u+
x + N f (t)(u+ ? 1).
(5.5)
The following values have been taken for the physical quantities (taken from
[10] [11]).
?s
1.6
g/cm3
?v
0.001
g/cm3
?w
1.0
g/cm3
cs
0.74
cal/(g K)
cv
0.48
cal/(g K)
cw
1.0
cal/(g K)
ks
0.00136
cal/(sec cm K)
kw
0.00127
cal/(sec cm K)
L
1.0
cm
?
0.3
K
1.0E ? 9
darcy
µ
0.0013
poise
R
0.11
cal/(g ? K)
?1
0.002
cal/(sec ? cm2 ? K)
?2
0.003
cal/(sec ? cm2 ? K)
T0
373.16
K
?
540.2
cal/g
p0
1013250.0
g/(cm ? sec2)
Consequently the constants appearing in (5.1)-(5.5) and (4.21) are the fol-
lowing:
A =
0.38,
B = 0.37
C =
0.08,
H =
0.49,
? = 1.9
M =
0.71,
N = 0.16
G = 13.16.
The constants appearing in the initial and boundary conditions are
u0 = 0.745
u? = 1.428
?1 = 1.50
?2 = 3.15
8

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x = s(t)
D2
?ux = ?2(u ? u?)
0
D
=
1
u x
t = t?
D0
?ux = ?1(u ? u?)
u(x, 0) = ¯
u
L
Figure 1: Sketch of the problem
In Figure 2 we show tha evolution of the rescaled free boundary as a function
of the rescaled time.
Figure 3 gives the rescaled temperature pro?le in the two zones, at di?erent
times.
In Figures 4 and 5 we display the same quantities, but we simulate the crust
formation as a relaxation in the heat exchange coe?cient ?2. In particular we
assumed
?2(0) = ?2
(5.6)
?
?2(t) = ? (u(1, t) ? 1)
(5.7)
+
Acknowledgments. We are indebted to Prof. A. Fasano for many interesting
discussions.
References
[1] A. Bouddour, L. Auriault, M. Mhamdi-Alaoui, Heat and mass transfer in
wet porous media in presence of evaporation-condensation, J.Heat Mass
Transfer 41 (1998) 2263-2277.
[2] B.E.Farkas, R.P.Singh and T.R.Rumsey, Modeling heat and mass transfer
in immersion frying. I,model development, J.Food Eng. 29 (1996) 211-226.
[3] B.E.Farkas, R.P.Singh and T.R.Rumsey, Modeling heat and mass transfer
in immersion frying. II,model solution and veri?cation, J.Food Eng. 29
(1996) 227-248.
9

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1.6
s(t)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Figure 2: Evolution of the free boundary
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0
0.2
0.4
0.6
0.8
1
Figure 3: Evolution of temperature pro?le
10

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