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Modeling heat and mass transfer in immersion frying. I, model development
Journrrl of‘ fi>od Eriginwnng 29 ( IYYb) 2 I I-226 Copyright 0 IVY6 Elscvicr Science Limited Printed in Great Britain. All rights reserved 0260-X774/96/$ IS.00 + O.tlO 0260-8774(95)00072-O
ELSFVIER
Modeling Heat and Mass ‘Ikansfer in Immersion Frying. I, Model Development B. E. Farkas,” R. P. Singh” & T. R. Rurnsey” “Department “Department
of Food Science, North Carolina State IJniversity, Raleigh, NC 276957624. USA of Biological and Agricultural Engineering, University of California, Davis, CA 956 16, USA
(Received 27 September lYY4:accepted I3 July 1995)
ABSTRACT This paper presents the development of the mathematical equations used for modeling immersion frying of an infinite slab. Immersion frying was viewed as a moving boundary problem similar to that found in freezing and freeze-drying. The infinite slab was divided into two regions, the crust and core, and macroscopic balances were used to develop the governing partial differential equations for heat and mass transfer in each region. Flux relations were proposed for the heat and mass transfer and incorporated into the partial differential equations. The final set of equations consisted of four non-linear partial differential equations and appropriate boundary conditions and initial conditions. Copyright 0 1996 Elsevier Science Limited. NOTATION
K L Nix P 91
Area (m’) Constant determined from steam table data Concentration of /j = I:/~/)/~ (kg /I/m:) Specific heat of species i (J/kg”(Z) Gravity in x-direction (m/s’) Enthalpy of species i (J/kg) Latent heat of vaporization (Jimol) Effective thermal conductivity of region i (W/m”(Z) Permeability (m’) Half-thickness of infinite slab (m) Flux of species i in x-direction (kg/m: s) Pressure (N/m’) Heat flux in x-direction (W/m’) 211
212
B. E. Farkas, R. I? S&h,
R
t
T (Ti)’ u= dW) dt viX
V, &,
Ideal gas constant (&314 J/m01 K) Time (s) Temperature (“C) Intrinsic phase average temperature
7: R. Rumsq
of species i (“C)
Velocity at which integace moves (m/s)
Velocity Specific Specific Position
of the species i (m/s) molar volume of liquid (m”/mol) molar volume of vapor (m”/mol) of crust/core interface (m)
Greek letters Ci Pi
d&b
Volume fraction of species i (m:/m:) Density of species i (kg of i/m:) Viscosity of the y species (Ns/m2)
Subscripts
Initial condition $I, Y, 4, CJ Liquid water, water vapor, oil, and solid, respectively
bP
Boiling point
INTRODUCTION Immersion frying, or deep-fat frying, may be defmed as the process of cooking foods by immersing them in an edible oil or fat which is at a temperature above the boiling point of water, typically 150-200°C. These conditions lead to high rates of heat transfer, rapid cooking, browning, and texture and flavor development. Due to the high cooking rates and desirable product characteristics, frying has become the mainstay of the snack food industry and one of the principal methods of cooking in fast food and large-scale catering industries (Varela et al., 1988). Frying can be considered a complex form of the Stefan class of problem. The generahzed Stefan heat transfer problem is characterized by the presence of a moving interface which divides two regions of distinct physical and thermal properties (Stefan, 1891; Carslaw and Jaeger, 1959). An example of the Stefan problem is the melting or freezing of water. The frying process is more complex than freezing due to the coupled heat and mass transfer, in two distinct regions separated by a moving boundary and temperatures well above the boiling point of water. During frying, various chemical reactions and physical changes occur. Chemical reactions in the form of surface pyrolization, gelation of starch, denaturation of protein, and flavor development take place. Physical changes are manifested as a decrease in moisture content, increase in temperature and oil content, development and growth of a crust layer, and possibly shrinkage or swelling of the product as a whole. It must be remembered that while frying is used to produce a desirable food product it is also used to lower the microbial count of the food rendering it safe to consume. Fried foods range from relatively ‘homogenous’ materials such as French fried potatoes to complex composites such as starch-based batter coated meats.
Heat and muss transfer in .fiyirzg
213
Past research on heat and mass transfer in frying has been limited to the formulation of empirical and semi-empirical equations. Mittelman et al. (1984) proposed a semi-empirical relationship for heat and mass transfer during frying. It was found that for a model food (sponge) the moisture content was proportional to the square root of frying time and the difference between the oil temperature and boiling temperature of water. Potatoes were fried and exhibited falling rate and constant rate stages typically found in drying. Using an assumption that all energy which entered a body was used for vaporization of water, an ordinary differential equation, similar to Plank’s equation for freezing (Singh and Heldman, 1993), was proposed to model moisture loss during frying. While the model was able to fit the experimental moisture data it did not address the transient temperature and moisture profiles in the material body or crust development. Keller and Escher (1989) proposed a mathematical model for the frying of potato sticks similar to that developed by Mittelman et al. (1984) with the addition of a term for the sensible heat required to heat the dry crust region from the boiling point of water to the oil temperature. Other researchers (Gamble et al., 1987; Rice and Gamble, 1989) proposed empirical equations relating the square root of time to moisture and oil content of the fried food and relating moisture to oil content. In developing an empirical relationship for the thermal conductivity of potato between 50 and lOO”C, Califano and Calvelo (1991) showed that frying oil temperature had little or no effect on heating rate at the center of a potato cylinder during frying. This was due to the presence of a moving crust/core interface at which vaporization of water took place. Kozempel el al. (1991) studied deep fat frying and modeled moisture loss using the first term of the solution to the diffusion equation, Fick’s second law (Bird et ui., 1960). Frying experiments were conducted at oil temperatures of 185, 195 and 201°C using French fries with a nominal minimum dimension of 1 cm. It was proposed that diffusivity was a function of oil temperature and could be related to oil temperature using an Arrhenius type equation. Using experimental data for oil temperature and product moisture content, an effective diffusivity, as a function of oil temperature, was found. Oil uptake was correlated with raw potato texture, fried potato moisture content and oil temperature. Past research in mathematical modeling of frying has been limited to empirical and semi-empirical relationships for heat and mass transfer. The transient temperature and moisture profiles inside the food during frying have not hecn studied and there has been little research on the factors which influence rate of crust development. The objective of this research was to formulate a mathematical model which would predict moisture profiles, temperature profiles, and crust thickness for the immersion frying of a one-dimensional model food system. The model would be based on fundamental principles of heat and mass transfer and utilize predetermined physical and thermal properties and process parameters. MODEL The ftying
DEVELOPMENT
process
We began by defining immersion frying as the process of fully submersing a food material in hot oil for a specified period of time. The length of this time is reflected
B. E. Farkas, R. If Singh, 7: R. Runmy
214
in the desired characteristics of the end product. This is in contrast to contact, convection, or long-wave infrared frying described by Dagerskog and Sorenfors (1978) Skjoldebrand (1979) and Hallstrom et al. (1988). Preliminary experiments (Farkas, 1994; Farkas et al., 1996) were conducted to visually study immersion frying and become familiar with the process. An infinite slab, of thickness 25 cm, was fried in 180°C canola oil for 960 s; slab composition is described in Farkas et al. (1996). Temperature and moisture data (Figs 1 and 2, respectively) were collected to assist in determining the type of driving forces for heat and mass transfer present during frying. The temperature at any location inside the material is limited to the boiling point of the liquid present (Fig. 1). When all liquid has been evaporated from the region, the temperature of the region may exceed the boiling point and approach the oil temperature. This increase, from the initial temperature to the boiling point (approximately lOO’C), is seen in Fig. 1. It is seen that at a distance of O-05 cm below the surface it requires approximately 60 s for the temperature to reach the boiling point, plateau for a short period of time, and then continue to rise once all liquid has been evaporated from the area. Using visual observations and analysis of the temperature and moisture data the frying process was described in terms of temporal stages and spatial regions of interest.
75-
0
0 0
’
A
0 A
50-0.0 3
o
A
25 $8 , ,
A A , ,
,
0
A
a
$ CI
,
, iz
,
,
,
5
, g
,
, 5
,
, 3
,
, g
Time, s Fig. 1. Experimentally determined temperature profiles at four positions in model food system during frying. Oil temperature 180°C; q 0.05 cm below surface; o 0.42 cm below surface; o O&5 cm below surface; al.27 cm below surface.
Heat and mass transfer
215
in ,fqing
Temporal stages of frying Immersion frying was broken down into four stages: (1) initial-heating, (2) surface boiling, (3) falling rate, and (4) bubble end point. The extent to which each of these stages exists depends upon the initial conditions, process parameters, and product size, shape, and composition. The following is a brief summary of these four stages, a complete discussion may be found in Farkas (1994). The first stage of frying is characterized by the initial immersion of the raw material into the hot oil and the absence of water vaporization. Initial-heating is the period of time during which the surface of the material heats from the initial temperature to the boiling point of the surface water. Heat is transferred from the oil to the food material via convection at the surface and conduction through the uncooked solid. Surface convective heat transfer is by free convection with no boiling present. This phase is short (on the order of 10 s) and a negligible amount of water is lost from the food. The second stage, termed surface boiling, is characterized by the sudden loss of free moisture at the surface, increase in surface heat transfer coefficient (Hallstrom, 1979), and the inception of crust formation. The onset of surface boiling changes the surface conditions from free convection to boiling conditions. This results in an increase in the convective heat transfer coefficient and, therefore, heat transfer to
q
Fig. 2.
Average
dry basis
moisture
content vs time during Oil temperature: 180°C.
frying
of a model food system.
B. E. Farkas, R. R Singh, T R. Rumsey
216
the food which leads to a further increase in surface turbulence. This cycle of increased heat transfer and vapor generation results in the ‘explosion’ of bubbles seen early in the frying processes. The third stage of frying is the longest and represents the period of time when the bulk of the moisture is lost and the core region temperatures approach the boiling point of the water present. The falling rate stage in frying parallels that found in drying and is characterized by the continued thickening of the crust layer, decrease in heat transfer due to the low thermal conductivity of the crust, and steady decrease in the rate of vapor mass transfer from the sample. As with drying, the falling rate period may actually be composed of several stages each determined by a reduction in the rate of moisture loss. Bubble end point is the final stage of frying. Bubble end point is a term used in the frying industry to describe the apparent cessation of moisture loss from the food during frying (Miller, 1993). The bubble end point may be caused by several factors ranging from the complete removal of all liquid water in the sample, such as with potato chips, to a reduction in heat transfer to the crust/core interface. Spatial
regions
of interest
In addition to increased heat transfer and initiation of mass transfer, a crust layer begins to develop during the second temporal stage. At this point it was proposed that the crust layer represents one of two regions within a fried food, the second being the core region. The crust region was defined in this study by two criteria: (1) temperature of the region is higher than the boiling point of the liquid present in the food material, and (2) the concentration of liquid water is negligible (Farkas, 1994). The crust region increases in thickness during frying and the interface between the crust and core regions may be viewed as a moving boundary. It was also proposed that moisture loss during frying is solely by transport of water, as vapor, from the interface to the surface and into the surrounding oil. Each region is in a dynamic state during frying, the crust becoming thicker and the core decreasing in thickness. Within each region simultaneous heat and mass transfer occur leading to thermal and moisture gradients. The regions are defined by a change in physical and thermal properties, or a change in the mass or energy flux of the system (such as that found at the centerline of the material). From a sensory standpoint, the crust provides most of the texture, color, and flavor for which fried foods are known. Also, it has been shown (Keller et al., 1986; Lamberg et al., 1990) that all oil absorbed during frying resides in the crust region. The core region is found in foods with a shape and size other than chip-like, which may be considered as pure crust. The core provides the bulk of the mass, moisture, and nutritional content found in fried foods and contributes to the texture of the food during consumption. Mathematical
model development
The following model is based on a one-dimensional, infinite slab analysis with the material being initially of homogenous composition and isothermal in nature. The development of the frying model draws heavily on the models of similar problems of freezing (Carslaw and Jaeger, 1959) and freeze-drying (Holland and Liapis, 1983).
.!I7
Heat atd muss tratzsfkr in f@ttg dX/dt 4
Region 1 Crust
Region II COK
.
NP
x=X(t)&
x=0-----------)
f3= liquid phaw y = vapor phase
Fig. 3.
Schematic
Ny
-
)
4
x=L
I$= oil phase
of an infinite
slab undergoing
frying.
The development of the mathematical model refers to the schematic of the process presented in Fig. 3. The following assumptions were made: 1. The core region is composed primarily of liquid water, solid material, and ;I negligible amount of gas. 2. The crust region is composed primarily of water vapor. oil, solid material, and a negligible amount of liquid water. 3. The advection of energy due to oil flux into the material is negligihlc. That is. energy flux into the material due to oil uptake during frying is much less than the energy flux due to convection and conduction into the material during frying. 4. The mass fraction of oil in the fried material is negligible and has a negligihlc effect on other mass and energy fluxes. The use of a thick infinite slab (3.5 cm) yields a mass fraction of oil much less than that of the total sample, that is: m,,,l < m5aInpic. Note that this would not be a valid assumption for a chip-like sample. 5. All energy and mass fluxes are orthogonal to the surface of the material. 6. Physical and thermal properties may change only along streamlines parallel to the x-axis. 7. Heat required for chemical changes such as starch gelatinization and prcjtein denaturation is small compared to that required for vaporization of water. 8. The process is symmetrical about the center-line at x = 0. In Fig. 3, terms N,, refer to the mass fluxes where i = p, 9, ;’ for liquid water, oil and water vapor respectively. the solid is referred to as o with N,,=O. Terms y’ refer
B. E. Farkas, R. I? Singh, 7: R. Rums?
218
to total heat transfer (by conduction and convection) in the i region where i = I, II. There are two distinct regions in the schematic: Region I, the crust X(t)_
Heat transfer:
core region
(0
Two types of heat transfer are considered in the core region, conduction and advection due to the movement of liquid water. While the energy flow due to advection is small, the term will be considered at this point. An energy balance on a segment in the core region gives
Letting ahi = C,$Ti yields:
a+
aT,i
- ax + N,d,p -+
ax
qp,&r
aT,j
at+““““““”
Assuming local thermal equilibrium, < T,> 1994) and Fourier’s law (qX = - kaT/ax) yields:
aT, at
(i = (To)", (Whitaker,
1977a; Farkas,
In the above expression kL\, is used, rather than k”, to remind the reader that due to the complex nature of the material an effective thermal conductivity is used.
Mass transfer: core region (0
Mass transfer in the core region is limited to the movement of liquid water, N,&. Assuming pp, Pi, c, are constant (Whitaker, 1981) a mass balance on a small segment in this region yields:
~,lO
Heat unrl mass transfer in ftying
(,4)
Several possible mechanisms for liquid transport were considered, the most well known being diffusion and capillary action. The possibility of two or more driving forces operating in series or parallel is likely, but due to the limited knowledge of the system only one theory will be tested. It is proposed that mass transfer in the core region during frying occurs by diffusion due to a concentration gradient. Diffusion was chosen due to the availability of diffusivity data for water/starch systems. At this time there exists very little data concerning capillary action in food systems and should be addressed in future work. Allowing ~/,p/~= cIj and N,], = -D,,,(&,,/?Ix), the transport expression with constant diffusivity may then be stated as (Bird et al., 1960):
a+
-=D
-
a*c,,
(5)
‘jmax2
at
The use of a constant diffusivity limits the model, it should be a function of temperature, moisture content, and physical structure/position, but at present this data is not available and this modification is left for future work. Heat transfer:
crust region
(X(t)
The crust region, Region I, is a thin layer (less than 2 mm for French fried potatoes) residing between the core and frying oil. Initially non-existent, it increases in thickness during frying and plays several key roles. In constructing a mathematical model of the process the crust represents a moving (thickening) layer which has thermal and physical properties similar to an insulating material. The low thermal conductivity and high porosity tend to slow heat transfer and therefore the rate at which water may be vaporized and the product cooked. It is assumed that the crust is composed only of solid (cr) and vapor (;t). An energy balance around a small segment of the crust yields the following equation: 3T.. aT, ‘:../);CP;,-’ + ,Q,Cprrat iit
= --
Assuming thermal equilibrium, (7’,)“= (r;,)” and Fourier’s law, eqn [6] simplifies to: (‘:;,&,;.+
Mass transfer:
crust region
/,J&,)~
= k&r $
aq, 8X
3T;. +N&,;.-
(Whitaker,
+ N&,,I
3X
1977~ ; Farkas,
(6) 19Y4),
(7)
(x(t) -CX-CL)
Mass transfer in the crust region is limited to the flux of water vapor from the moving crust/core interface to the surface and movement of oil into the crust from the surroundings. Using assumptions four and five the mass transfer of oil into the
220
crust region was neglected. region yields:
B. E. Farkas, R. P Singh, 7: R. Rumsey
Writing a mass balance on the water around
ap-,
al;;
pi, F+c..
)
the crust
aN..,
= L
’ at
ax
The first term on the left hand side may be dropped, Ck,./at=O, by assuming a constant porosity in the crust region, &$3t = 0, and the sum of the volume fractions is unity, e,+c;. = 1. It may also be assumed that the vapor in the crust region is incompressible, +,/at = 0, (Whitaker, 1981) therefore eqn [8] reduces to
The use of pressure gradient as a driving force for mass transfer has been studied by Evans et al. (1962) in the diffusion of gases in porous media. A diffusion equation, similar to that used for the case of uniform pressure, was found with the diffusion coefficient a function of position due to the presence of the pressure gradient. Gibson et al. (1978, 1979) and Cross et al. (1979) examined the drying of spherical iron ore pellets and a porous semi-infinite slab and found that the pressure gradients formed during drying could be determined through the use of Darcy’s Law. It is hypothesized at this point that the water vapor moves by a pressure gradient from the crust/core interface to the surface and that the process may be described by Darcy’s Law:
Multiplying both sides of eqn [lo] by p;: and assuming p,gX to be negligible gives the required expression for N,; an expression for the mass flux in the crust region: /jYKi. ap..
N;., = - -
p;,
Thus the equation for pressure distribution
)
ax
(11)
in the crust may be written as:
a
kap:.= -[ax pi,ax 1
0
The equation may then be reduced to a function of temperature and distance by relating vapor pressure and density with temperature through the use of the ideal gas law and Clausius-Clapeon equations and assuming viscosity constant over the temperature range in the crust region (Perry et al., 1984):
jyI 1 a
ap, Y,. ax
=0
(12b)
Having shown that the crust region mass transfer equation may be stated in terms of temperature alone it is now put forth that the equation may be dropped through
I!2 I
Hrut und muss trattsfcr. in .f-irzg
the use of thermal equilibrium. If the vapor pressure and density are a known function of temperature and the temperature is known through the crust region heat transfer equation, then the vapor flux may be determined from temperature alone. In summary, the heat and mass transfer equations which constitute the model are 131, [S], [7], and [12b] with unknowns T”, c,{, T’, and P:.. Note that the superscripts I and II are used here to designate temperature in the crust region and core region, respectively, although they were not used in the partial differential equations. Initial
and boundary
conditions
Four second order partial differential equations have been derived which describe the heat and mass transfer in the crust and core regions during frying. In order to solve these four equations, three initial conditions and eight boundary conditions will be needed. Core region
Equation [3] that models heat transfer in the core region and requires two boundary conditions. the first is a svmmetrv condition at the centerline of the slab (Birkhoff. 1960) and ‘is written as: ’ _ BCl:
E=t)
at x=O,t>O
(13)
The second boundary condition is a prescribed temperature at x = x(t). The crust/ core interface is a moving surface at which water from the core region is vaporized at the temperature T,,,, consequently the second boundary condition takes the form: BC2:
T = T,,,,
at x = X(t),t>O
( 14)
The mass transfer boundary conditions (for eqn (5)) are similar to those for the heat transfer equations and are stated as follows: BC3:
ac,, z=O
atx=O,t>O
BC4:
c,, = 0
at x = X(t), t >O
I
IS)
( 10)
The fourth boundary condition arises from the assumption that the liquid concentration in the crust region is negligible, or essentially zero, and the liquid concentration on the surface of the interface at x =x(t) is also negligible. or essentially zero. An alternate mass transfer boundary condition was considered using equality of flux across the interface, x =x(t), and was formulated as eqn (201. This formulation would allow a liquid concentration at x =x(t) greater than zero and is similar to the equality of heat flux boundary condition given by eqn 1191. Crust region
Equation boundary
(71 that models heat transfer in the crust region uses two, more complex, conditions each relying on the principal of continuity of flux (Whitaker,
B. E. Farkas, R. P Singh, 7: R. Rumsey
222
x=X(t+At)
x=0
x=X(t)
x=L
‘7
4”
I
-----A
*
I I
I
I
’ +
~*___--
I
I
~______~NY”~
I
I
I
NPxI -------l
I
I
I
I 4
I
I
I
Fig. 4.
Expanded
I
view of the crust/core
interface.
1977b). Equating the convective heat flux from the oil to the material conductive heat flux from the material surface inwards yields: BC5:
k&z=
h(T,-TJ.=.)
and the
at x= L,t>O
(17)
Crustlcore integace A sixth boundary condition is used at the crust/core interface, the derivation refers to the expanded view of the interface given in Fig. 4 and follows that given by Holland and Liapis (1983). Define the following quantities as: = area of interface, m2 A = U = velocity at which interface moves, m/s dX(t)ldt = distance which interface moves from time t to t+At, m UAt uAAt(E~p~h~+cllpl~~,~)= energy in volume UAAt at tj, J/m3 UAAt(&~pg~+E,p,sr,.)= energy in volume UAAt at G+Atj, J/m” = energy into volume UAAt during At, J q’AAt+N,&AAt = energy out of volume UAAt during At, J qllAAt+NjxH>dAt Constructing
the balance equation yields:
AAt(q’ +N,j,h,j - q” -. N;,,H,.) = UAAt(a,p,h’,+a,.p,.H;,
-eVPnh’rO - appph,,)
(18)
Simplifying and combining like groups gives: BC6:
aT
- k& ax
aT
+ k% -+
ax
N;,,( h,{- H;,)
= U[&,p,(hf,-h~‘)+&,,p~(Hy-h,l)]
at x = X(t),t>O (19)
A mass balance around the control volume UdAt yields the following interfacial mass balance:
The crust region mass transfer boundary BC7:
conditions
are given as:
P;. = P ;,,, at x = L.t >O
(21) The final mass transfer boundary condition relates the pressure of the vapor, f;., at the crust/core interface to the temperature through the Clausius-Clapeon cyuation:
Equation
ilP;.
AI;r;
?T
T( ti;, - i/,,)
[22] is then solved (Felder and Rousseau, BC8:
P,. = exp (++Bj
(22) 1986) to yield:
at x = X(t),t>o
(23)
Initial conditions Initially, the material consists of only core region and two initial conditions are required for solution of the problem before a crust begins to form. In addition, BC6 is an ordinary differential equation therefore requiring a third initial condition. The assumed uniform initial conditions are: ICI:
T = T,,
for all x,t = 0
(2,4)
IC2:
c = c,,
for all x, t = 0
(25)
IC3:
X(t) = 0
at t = 0
(26)
DISCUSSION The process of immersion frying was found to be similar to that of high temperature air drying (Lebedev, 1960a, h) and freeze-drying (Holland and Liapis, 1984). A comparison of the transport equations developed by Holland and Liapis (1984) and Farkas (1994) was conducted by Farkas et al. (1995). The outer region of material in each mathematical model was considered a porous matrix through which Darcy flow of water vapor occurred. Heat conduction in this region was a combination of conduction through the solid matrix and advection by transport of water vapor. The inner layer of material in each process was quite different. The core region in freeze-drying consists of frozen water, hence no mass transport occurs, while in frying the core region contains liquid water and diffusion may be used to describe the transport of water to the interface. Core region heat transfer in each process utilizes conduction but the equations for transport during frying contain an additional term for advection of energy. Flux of liquid water was modeled using diffusion theory with constant diffusivity, D,,,. Past researchers (Rice and Gamble, 1989; Kozempel et al., 1991) have shown Califano and Calvelo (1991) that diffusivity, D,{,, is a function of oil temperature. have shown that oil temperature has no effect on core heating rate, therefore there
224
B. E. Farkas, R. P Singh, 7: R. Rumsey
should be little effect of oil temperature on moisture diffusivity and a more complex process is occurring. Analysis of the mathematical model for immersion frying reveals that liquid water is ‘fed’ to the interface by two modes, each possibly being dominant at different stages of frying. The first mode is a result of the movement of the interface into the core region of the material. The progression of the interface into the core region yields an apparent flux of water to the interface at a velocity NF”‘=c,jdX/dt The second mode of liquid transport is due to the movement of liqiid water from the core region to the interface by diffusion NF.= -D,,,&$ax. The magnitude and period during which each flux dominates and their imphcations is discussed in a companion paper (Farkas et al, 1996).
CONCLUSIONS The process of immersion frying was studied experimentally, both visually and by collecting temperature and moisture data. These observations were used to gain insight into immersion frying. The frying process was broken down into four temporal periods: initial heating, surface boiling, falling rate, and bubble end point. These periods were identified by the type of heat transfer at the surface and rate of mass transfer or vapor bubble formation at the surface. The fried material was divided into two distinct regions, the crust and core. These were identified by the distinct boundaries of the crust/oil interface, crust/core interface, and the centerline of the material. These temporal periods and regions and their implications will be discussed further in the companion paper (Farkas et al., 1996) with respect to the solution of the mathematical model developed herein.
REFERENCES Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (1960). Transport Phenomena. John Wiley and Sons, Inc., New York, N.Y. Birkhoff, G. (1960). Hydrodynamics, A Study in Logic, Fact, and Similitude. Princeton University Press, Princeton, NJ. Califano, A. N. & Calvelo, A. (1991). Thermal conductivity of potato between 50 and 100°C: A research note. J. Food Sci., 56 (2), 586. Carslaw, H. S. and Jaeger, J. C. (1959). Conduction of Heat in Solids, 2nd edn. Oxford University Press, New York, NY. Ceaglske, N. H. and Hougen, 0. A. (1937). Drying granular solids. Indust. Engng Chem. 805-13.
Cross, M., Gibson, R. D. & Young, R. W. (1979). Pressure generation during the drying of a porous half-space. Int. J. Heat Mass Transfer, 22, 47-50. Dagerskog, M. & Sorenfors, P. (1978). A comparison between four different methods of frying meat patties. I. Heat transfer, yielding and crust formation. Food Sci. Technol., 11 (6), 306.
Evans, III, R. B., Watson, G. M. & Mason, E. A. (1962). Gaseous diffusion in porous media. II. Effect of pressure gradients. J. Chem. Phys., 36 (7), 1894-902. Farkas, B. E. (1994). Modeling immersion frying as a moving boundary problem. Ph.D. Dissertation, University of California, Davis. Farkas, B. E., Singh, R. P. & Rumsey, T. (1995). Mathematical modeling of immersion frying: A novel use of drying theory. Proceedings of the 9th International Drying Symposium, Gold Coast, Australia, August l-4, 1994.
Hvut and muss trun.$er in ft?irtx
22.5
Farkas, B. E., Singh, R. P. & Rumscy, T. (10%). Modeling heat and mass transfer in immersion frying. II. Model solution and verification. 1. Food Engrg, 29, 227-148. Fclder. R. M. & Rousseau, R. W. (1986). Elemetttuty Prittci~ks of Chemical Procesws. 2nd edn. Wiley, New York, NY. Gamble, M. H.. Rice. P. & Sclman, J. D. (1987). Relationship hetwccn oil uptake and moisture loss during frying of potato slices from C.V. record U.K. tubers. In/. J. f+orl Sci. T~chnol.. 22, 2.33. (jibson, R. D., Cross, M. & Young, R. W. (197X). Pressure gcncration during the drying ot iron ore pellets. Ptw. lttl. Cot$ .4@. Nrrmcricul Mudelitzg. Pcntech Press, pp. X39-40. (iibson, R. D., Cross, M. & Young, R. W. (lY7Y). Pressure gradients gencratcd during Ihc drying of porous shapes. ItIt. J. Heat Muss Trutwfer, 22, X27-30. Hallstriim, B. (1 Y79). Heat and mass transfer in industrial cooking, chap. 57. In: I;ootl f-‘row.\.\ Engineering. Vol. I. Linko. P.. Miilkki. Y., Olkku, J. and Larinkari, J. (cds) Applied Scicncc Publishers Ltd. tIallstriim, B., Skjiildehrand. C. & Triigsrdh. c‘. IYXX. Ifear Trutt.$& uttd hxl Prrh~t.s. Elsevier Applied Science, London. tiolland. C. D. & Liapis. A. I. ( 1983). C’omprmr Methorl~ jix Sohittg Dytunric~ Sqxtratht Prohkms. McGraw-Hill Book Co.. New York. NY. Keller, Ch.. Escher, F. 8 Solms, J. A. (lY8h). Method for localizing fat distribution in dc.;pfat fried potato products. Lehetwx Wiss. U. Techttol., 19 (4), 346-8. Kcllcr, Ch. 8.~ Escher. F. (198’)). Heat attd Maxs Transfer Durittg Deep Ett Ftyittg of f’otato Prodtrcts. Int. Congress on Engineering and Food 5. Cologne, Germany. May 25June I. lY8Y. Kozempcl, M. F.. Tomasula. P. M. Jr & Craig, J. C. (IYYI). Correlation of moisture and oil concentration in French fries. Lehensm. Wiss. U. Techtrol.. 24 (S), 445-X. Lamhcrg. I., Hallstrom, B. Kr Olsson, H. (IWO). Fat uptake in a potato drying/frying process. Lihettsm. Wiss. U. Echtwl. , 23 (4). I.cbedcv, P. D. (1960). Heat and mass transfer during the drying of moist materials. ltlr ./. Heat Mass Trunsfer, 1. 2Y4-301.
Lcbedev. P. D. (1960). Heat and mass transfer bctwcen moist solids and air. lnt. 1. flcut hfa.\s Trutzsfk-, 1, 302-S. Ixvine. 1~.(1990). Understanding frying operations, part I. (‘ctwtl Foods World, 35 (9). 273 Luikov, A. V. (1964). Heat and mass transfer in capillary-porous bodies. In: Ad~nttccs irt Hwt Truttsfer, Vol. 1. Thomas F. Irvine, Jr. and P. James Hartnett (eds), pp. 123-84. Miller, K. S. (1993). Personal Communication. Department of Biological and Agricultural Engineering. University of California, Davis. Mittclman, N., Mizrahi. S. & Bcrk, Z. (1984). Heat and mass transfer in frying, chap. 17. 111: Etzgitzeerittg und hod. McKcnna, B. M. (cd.). Elsevier Applied Science Publishers. Perry, R. H., Green, D. W. & Maloney, J. 0. ( 1984). PerTi; C’hcmic.al Etgittecrs Hunt/hook. 6th edn. McGraw-Hill Book Co., New York. Rice, P. Kr Gamble, M. H. (198Y). Modcling moisture loss during potato slice frying. Ittr. ./. Fk~l Sci. Techol., 24, I X3. Singh. R. P. & Heldman, D. R. (1093). ltttrodacfiott to Food Etrgittwrittg, 2nd cdn. Ac;.Ldemic Press, Inc., New York, NY. Skjcildcbrand. C. (1979). Frying and reheating in a forced convection oven. Ph.D. thesis. I.und Institute of Technology, L>und, Sweden. Siirenfors. P. VBrme-och masstransport vid stekning av fiirsproduktcr. Dissertation, Alnarp. 1078. Stcfan, J. (1XYI). Ann. Phys. Chcmie (Wicdemannschc Annalen), 42, 269-86. Varela, G., Bender, A. E. & Morton, I. D. (1988). Ftyvitzgqf’ Food. Pritzciples. Chuttgcs, IV~M, Approaches. Ellis Horwood Series in Food Science and Technology. VCH Publishers. Whitakcr, S. (1977a). Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying. In: Ad~wrtws itt ffeat 7httsfer. Vol. 13. James P. Hartnctt and Thomas F.
226
B. E. Furkas, R. I? Singh, 7: R. Rumsey
Irvine, Jr (eds). Academic Press, pp. 119-202. Whitaker, S. (19776). Fundamental Principles of Heat Trunsfer. Pergamon Press, New York, NY. Whitaker, S. (1981). ltztroductiotz to Fluid Mechanics. R. E. Krieger Publishing Co., Malabar, FL.
ELSFVIER
Modeling Heat and Mass ‘Ikansfer in Immersion Frying. I, Model Development B. E. Farkas,” R. P. Singh” & T. R. Rurnsey” “Department “Department
of Food Science, North Carolina State IJniversity, Raleigh, NC 276957624. USA of Biological and Agricultural Engineering, University of California, Davis, CA 956 16, USA
(Received 27 September lYY4:accepted I3 July 1995)
ABSTRACT This paper presents the development of the mathematical equations used for modeling immersion frying of an infinite slab. Immersion frying was viewed as a moving boundary problem similar to that found in freezing and freeze-drying. The infinite slab was divided into two regions, the crust and core, and macroscopic balances were used to develop the governing partial differential equations for heat and mass transfer in each region. Flux relations were proposed for the heat and mass transfer and incorporated into the partial differential equations. The final set of equations consisted of four non-linear partial differential equations and appropriate boundary conditions and initial conditions. Copyright 0 1996 Elsevier Science Limited. NOTATION
K L Nix P 91
Area (m’) Constant determined from steam table data Concentration of /j = I:/~/)/~ (kg /I/m:) Specific heat of species i (J/kg”(Z) Gravity in x-direction (m/s’) Enthalpy of species i (J/kg) Latent heat of vaporization (Jimol) Effective thermal conductivity of region i (W/m”(Z) Permeability (m’) Half-thickness of infinite slab (m) Flux of species i in x-direction (kg/m: s) Pressure (N/m’) Heat flux in x-direction (W/m’) 211
212
B. E. Farkas, R. I? S&h,
R
t
T (Ti)’ u= dW) dt viX
V, &,
Ideal gas constant (&314 J/m01 K) Time (s) Temperature (“C) Intrinsic phase average temperature
7: R. Rumsq
of species i (“C)
Velocity at which integace moves (m/s)
Velocity Specific Specific Position
of the species i (m/s) molar volume of liquid (m”/mol) molar volume of vapor (m”/mol) of crust/core interface (m)
Greek letters Ci Pi
d&b
Volume fraction of species i (m:/m:) Density of species i (kg of i/m:) Viscosity of the y species (Ns/m2)
Subscripts
Initial condition $I, Y, 4, CJ Liquid water, water vapor, oil, and solid, respectively
bP
Boiling point
INTRODUCTION Immersion frying, or deep-fat frying, may be defmed as the process of cooking foods by immersing them in an edible oil or fat which is at a temperature above the boiling point of water, typically 150-200°C. These conditions lead to high rates of heat transfer, rapid cooking, browning, and texture and flavor development. Due to the high cooking rates and desirable product characteristics, frying has become the mainstay of the snack food industry and one of the principal methods of cooking in fast food and large-scale catering industries (Varela et al., 1988). Frying can be considered a complex form of the Stefan class of problem. The generahzed Stefan heat transfer problem is characterized by the presence of a moving interface which divides two regions of distinct physical and thermal properties (Stefan, 1891; Carslaw and Jaeger, 1959). An example of the Stefan problem is the melting or freezing of water. The frying process is more complex than freezing due to the coupled heat and mass transfer, in two distinct regions separated by a moving boundary and temperatures well above the boiling point of water. During frying, various chemical reactions and physical changes occur. Chemical reactions in the form of surface pyrolization, gelation of starch, denaturation of protein, and flavor development take place. Physical changes are manifested as a decrease in moisture content, increase in temperature and oil content, development and growth of a crust layer, and possibly shrinkage or swelling of the product as a whole. It must be remembered that while frying is used to produce a desirable food product it is also used to lower the microbial count of the food rendering it safe to consume. Fried foods range from relatively ‘homogenous’ materials such as French fried potatoes to complex composites such as starch-based batter coated meats.
Heat and muss transfer in .fiyirzg
213
Past research on heat and mass transfer in frying has been limited to the formulation of empirical and semi-empirical equations. Mittelman et al. (1984) proposed a semi-empirical relationship for heat and mass transfer during frying. It was found that for a model food (sponge) the moisture content was proportional to the square root of frying time and the difference between the oil temperature and boiling temperature of water. Potatoes were fried and exhibited falling rate and constant rate stages typically found in drying. Using an assumption that all energy which entered a body was used for vaporization of water, an ordinary differential equation, similar to Plank’s equation for freezing (Singh and Heldman, 1993), was proposed to model moisture loss during frying. While the model was able to fit the experimental moisture data it did not address the transient temperature and moisture profiles in the material body or crust development. Keller and Escher (1989) proposed a mathematical model for the frying of potato sticks similar to that developed by Mittelman et al. (1984) with the addition of a term for the sensible heat required to heat the dry crust region from the boiling point of water to the oil temperature. Other researchers (Gamble et al., 1987; Rice and Gamble, 1989) proposed empirical equations relating the square root of time to moisture and oil content of the fried food and relating moisture to oil content. In developing an empirical relationship for the thermal conductivity of potato between 50 and lOO”C, Califano and Calvelo (1991) showed that frying oil temperature had little or no effect on heating rate at the center of a potato cylinder during frying. This was due to the presence of a moving crust/core interface at which vaporization of water took place. Kozempel el al. (1991) studied deep fat frying and modeled moisture loss using the first term of the solution to the diffusion equation, Fick’s second law (Bird et ui., 1960). Frying experiments were conducted at oil temperatures of 185, 195 and 201°C using French fries with a nominal minimum dimension of 1 cm. It was proposed that diffusivity was a function of oil temperature and could be related to oil temperature using an Arrhenius type equation. Using experimental data for oil temperature and product moisture content, an effective diffusivity, as a function of oil temperature, was found. Oil uptake was correlated with raw potato texture, fried potato moisture content and oil temperature. Past research in mathematical modeling of frying has been limited to empirical and semi-empirical relationships for heat and mass transfer. The transient temperature and moisture profiles inside the food during frying have not hecn studied and there has been little research on the factors which influence rate of crust development. The objective of this research was to formulate a mathematical model which would predict moisture profiles, temperature profiles, and crust thickness for the immersion frying of a one-dimensional model food system. The model would be based on fundamental principles of heat and mass transfer and utilize predetermined physical and thermal properties and process parameters. MODEL The ftying
DEVELOPMENT
process
We began by defining immersion frying as the process of fully submersing a food material in hot oil for a specified period of time. The length of this time is reflected
B. E. Farkas, R. If Singh, 7: R. Runmy
214
in the desired characteristics of the end product. This is in contrast to contact, convection, or long-wave infrared frying described by Dagerskog and Sorenfors (1978) Skjoldebrand (1979) and Hallstrom et al. (1988). Preliminary experiments (Farkas, 1994; Farkas et al., 1996) were conducted to visually study immersion frying and become familiar with the process. An infinite slab, of thickness 25 cm, was fried in 180°C canola oil for 960 s; slab composition is described in Farkas et al. (1996). Temperature and moisture data (Figs 1 and 2, respectively) were collected to assist in determining the type of driving forces for heat and mass transfer present during frying. The temperature at any location inside the material is limited to the boiling point of the liquid present (Fig. 1). When all liquid has been evaporated from the region, the temperature of the region may exceed the boiling point and approach the oil temperature. This increase, from the initial temperature to the boiling point (approximately lOO’C), is seen in Fig. 1. It is seen that at a distance of O-05 cm below the surface it requires approximately 60 s for the temperature to reach the boiling point, plateau for a short period of time, and then continue to rise once all liquid has been evaporated from the area. Using visual observations and analysis of the temperature and moisture data the frying process was described in terms of temporal stages and spatial regions of interest.
75-
0
0 0
’
A
0 A
50-0.0 3
o
A
25 $8 , ,
A A , ,
,
0
A
a
$ CI
,
, iz
,
,
,
5
, g
,
, 5
,
, 3
,
, g
Time, s Fig. 1. Experimentally determined temperature profiles at four positions in model food system during frying. Oil temperature 180°C; q 0.05 cm below surface; o 0.42 cm below surface; o O&5 cm below surface; al.27 cm below surface.
Heat and mass transfer
215
in ,fqing
Temporal stages of frying Immersion frying was broken down into four stages: (1) initial-heating, (2) surface boiling, (3) falling rate, and (4) bubble end point. The extent to which each of these stages exists depends upon the initial conditions, process parameters, and product size, shape, and composition. The following is a brief summary of these four stages, a complete discussion may be found in Farkas (1994). The first stage of frying is characterized by the initial immersion of the raw material into the hot oil and the absence of water vaporization. Initial-heating is the period of time during which the surface of the material heats from the initial temperature to the boiling point of the surface water. Heat is transferred from the oil to the food material via convection at the surface and conduction through the uncooked solid. Surface convective heat transfer is by free convection with no boiling present. This phase is short (on the order of 10 s) and a negligible amount of water is lost from the food. The second stage, termed surface boiling, is characterized by the sudden loss of free moisture at the surface, increase in surface heat transfer coefficient (Hallstrom, 1979), and the inception of crust formation. The onset of surface boiling changes the surface conditions from free convection to boiling conditions. This results in an increase in the convective heat transfer coefficient and, therefore, heat transfer to
q
Fig. 2.
Average
dry basis
moisture
content vs time during Oil temperature: 180°C.
frying
of a model food system.
B. E. Farkas, R. R Singh, T R. Rumsey
216
the food which leads to a further increase in surface turbulence. This cycle of increased heat transfer and vapor generation results in the ‘explosion’ of bubbles seen early in the frying processes. The third stage of frying is the longest and represents the period of time when the bulk of the moisture is lost and the core region temperatures approach the boiling point of the water present. The falling rate stage in frying parallels that found in drying and is characterized by the continued thickening of the crust layer, decrease in heat transfer due to the low thermal conductivity of the crust, and steady decrease in the rate of vapor mass transfer from the sample. As with drying, the falling rate period may actually be composed of several stages each determined by a reduction in the rate of moisture loss. Bubble end point is the final stage of frying. Bubble end point is a term used in the frying industry to describe the apparent cessation of moisture loss from the food during frying (Miller, 1993). The bubble end point may be caused by several factors ranging from the complete removal of all liquid water in the sample, such as with potato chips, to a reduction in heat transfer to the crust/core interface. Spatial
regions
of interest
In addition to increased heat transfer and initiation of mass transfer, a crust layer begins to develop during the second temporal stage. At this point it was proposed that the crust layer represents one of two regions within a fried food, the second being the core region. The crust region was defined in this study by two criteria: (1) temperature of the region is higher than the boiling point of the liquid present in the food material, and (2) the concentration of liquid water is negligible (Farkas, 1994). The crust region increases in thickness during frying and the interface between the crust and core regions may be viewed as a moving boundary. It was also proposed that moisture loss during frying is solely by transport of water, as vapor, from the interface to the surface and into the surrounding oil. Each region is in a dynamic state during frying, the crust becoming thicker and the core decreasing in thickness. Within each region simultaneous heat and mass transfer occur leading to thermal and moisture gradients. The regions are defined by a change in physical and thermal properties, or a change in the mass or energy flux of the system (such as that found at the centerline of the material). From a sensory standpoint, the crust provides most of the texture, color, and flavor for which fried foods are known. Also, it has been shown (Keller et al., 1986; Lamberg et al., 1990) that all oil absorbed during frying resides in the crust region. The core region is found in foods with a shape and size other than chip-like, which may be considered as pure crust. The core provides the bulk of the mass, moisture, and nutritional content found in fried foods and contributes to the texture of the food during consumption. Mathematical
model development
The following model is based on a one-dimensional, infinite slab analysis with the material being initially of homogenous composition and isothermal in nature. The development of the frying model draws heavily on the models of similar problems of freezing (Carslaw and Jaeger, 1959) and freeze-drying (Holland and Liapis, 1983).
.!I7
Heat atd muss tratzsfkr in f@ttg dX/dt 4
Region 1 Crust
Region II COK
.
NP
x=X(t)&
x=0-----------)
f3= liquid phaw y = vapor phase
Fig. 3.
Schematic
Ny
-
)
4
x=L
I$= oil phase
of an infinite
slab undergoing
frying.
The development of the mathematical model refers to the schematic of the process presented in Fig. 3. The following assumptions were made: 1. The core region is composed primarily of liquid water, solid material, and ;I negligible amount of gas. 2. The crust region is composed primarily of water vapor. oil, solid material, and a negligible amount of liquid water. 3. The advection of energy due to oil flux into the material is negligihlc. That is. energy flux into the material due to oil uptake during frying is much less than the energy flux due to convection and conduction into the material during frying. 4. The mass fraction of oil in the fried material is negligible and has a negligihlc effect on other mass and energy fluxes. The use of a thick infinite slab (3.5 cm) yields a mass fraction of oil much less than that of the total sample, that is: m,,,l < m5aInpic. Note that this would not be a valid assumption for a chip-like sample. 5. All energy and mass fluxes are orthogonal to the surface of the material. 6. Physical and thermal properties may change only along streamlines parallel to the x-axis. 7. Heat required for chemical changes such as starch gelatinization and prcjtein denaturation is small compared to that required for vaporization of water. 8. The process is symmetrical about the center-line at x = 0. In Fig. 3, terms N,, refer to the mass fluxes where i = p, 9, ;’ for liquid water, oil and water vapor respectively. the solid is referred to as o with N,,=O. Terms y’ refer
B. E. Farkas, R. I? Singh, 7: R. Rums?
218
to total heat transfer (by conduction and convection) in the i region where i = I, II. There are two distinct regions in the schematic: Region I, the crust X(t)_
Heat transfer:
core region
(0
Two types of heat transfer are considered in the core region, conduction and advection due to the movement of liquid water. While the energy flow due to advection is small, the term will be considered at this point. An energy balance on a segment in the core region gives
Letting ahi = C,$Ti yields:
a+
aT,i
- ax + N,d,p -+
ax
qp,&r
aT,j
at+““““““”
Assuming local thermal equilibrium, < T,> 1994) and Fourier’s law (qX = - kaT/ax) yields:
aT, at
(i = (To)", (Whitaker,
1977a; Farkas,
In the above expression kL\, is used, rather than k”, to remind the reader that due to the complex nature of the material an effective thermal conductivity is used.
Mass transfer: core region (0
Mass transfer in the core region is limited to the movement of liquid water, N,&. Assuming pp, Pi, c, are constant (Whitaker, 1981) a mass balance on a small segment in this region yields:
~,lO
Heat unrl mass transfer in ftying
(,4)
Several possible mechanisms for liquid transport were considered, the most well known being diffusion and capillary action. The possibility of two or more driving forces operating in series or parallel is likely, but due to the limited knowledge of the system only one theory will be tested. It is proposed that mass transfer in the core region during frying occurs by diffusion due to a concentration gradient. Diffusion was chosen due to the availability of diffusivity data for water/starch systems. At this time there exists very little data concerning capillary action in food systems and should be addressed in future work. Allowing ~/,p/~= cIj and N,], = -D,,,(&,,/?Ix), the transport expression with constant diffusivity may then be stated as (Bird et al., 1960):
a+
-=D
-
a*c,,
(5)
‘jmax2
at
The use of a constant diffusivity limits the model, it should be a function of temperature, moisture content, and physical structure/position, but at present this data is not available and this modification is left for future work. Heat transfer:
crust region
(X(t)
The crust region, Region I, is a thin layer (less than 2 mm for French fried potatoes) residing between the core and frying oil. Initially non-existent, it increases in thickness during frying and plays several key roles. In constructing a mathematical model of the process the crust represents a moving (thickening) layer which has thermal and physical properties similar to an insulating material. The low thermal conductivity and high porosity tend to slow heat transfer and therefore the rate at which water may be vaporized and the product cooked. It is assumed that the crust is composed only of solid (cr) and vapor (;t). An energy balance around a small segment of the crust yields the following equation: 3T.. aT, ‘:../);CP;,-’ + ,Q,Cprrat iit
= --
Assuming thermal equilibrium, (7’,)“= (r;,)” and Fourier’s law, eqn [6] simplifies to: (‘:;,&,;.+
Mass transfer:
crust region
/,J&,)~
= k&r $
aq, 8X
3T;. +N&,;.-
(Whitaker,
+ N&,,I
3X
1977~ ; Farkas,
(6) 19Y4),
(7)
(x(t) -CX-CL)
Mass transfer in the crust region is limited to the flux of water vapor from the moving crust/core interface to the surface and movement of oil into the crust from the surroundings. Using assumptions four and five the mass transfer of oil into the
220
crust region was neglected. region yields:
B. E. Farkas, R. P Singh, 7: R. Rumsey
Writing a mass balance on the water around
ap-,
al;;
pi, F+c..
)
the crust
aN..,
= L
’ at
ax
The first term on the left hand side may be dropped, Ck,./at=O, by assuming a constant porosity in the crust region, &$3t = 0, and the sum of the volume fractions is unity, e,+c;. = 1. It may also be assumed that the vapor in the crust region is incompressible, +,/at = 0, (Whitaker, 1981) therefore eqn [8] reduces to
The use of pressure gradient as a driving force for mass transfer has been studied by Evans et al. (1962) in the diffusion of gases in porous media. A diffusion equation, similar to that used for the case of uniform pressure, was found with the diffusion coefficient a function of position due to the presence of the pressure gradient. Gibson et al. (1978, 1979) and Cross et al. (1979) examined the drying of spherical iron ore pellets and a porous semi-infinite slab and found that the pressure gradients formed during drying could be determined through the use of Darcy’s Law. It is hypothesized at this point that the water vapor moves by a pressure gradient from the crust/core interface to the surface and that the process may be described by Darcy’s Law:
Multiplying both sides of eqn [lo] by p;: and assuming p,gX to be negligible gives the required expression for N,; an expression for the mass flux in the crust region: /jYKi. ap..
N;., = - -
p;,
Thus the equation for pressure distribution
)
ax
(11)
in the crust may be written as:
a
kap:.= -[ax pi,ax 1
0
The equation may then be reduced to a function of temperature and distance by relating vapor pressure and density with temperature through the use of the ideal gas law and Clausius-Clapeon equations and assuming viscosity constant over the temperature range in the crust region (Perry et al., 1984):
jyI 1 a
ap, Y,. ax
=0
(12b)
Having shown that the crust region mass transfer equation may be stated in terms of temperature alone it is now put forth that the equation may be dropped through
I!2 I
Hrut und muss trattsfcr. in .f-irzg
the use of thermal equilibrium. If the vapor pressure and density are a known function of temperature and the temperature is known through the crust region heat transfer equation, then the vapor flux may be determined from temperature alone. In summary, the heat and mass transfer equations which constitute the model are 131, [S], [7], and [12b] with unknowns T”, c,{, T’, and P:.. Note that the superscripts I and II are used here to designate temperature in the crust region and core region, respectively, although they were not used in the partial differential equations. Initial
and boundary
conditions
Four second order partial differential equations have been derived which describe the heat and mass transfer in the crust and core regions during frying. In order to solve these four equations, three initial conditions and eight boundary conditions will be needed. Core region
Equation [3] that models heat transfer in the core region and requires two boundary conditions. the first is a svmmetrv condition at the centerline of the slab (Birkhoff. 1960) and ‘is written as: ’ _ BCl:
E=t)
at x=O,t>O
(13)
The second boundary condition is a prescribed temperature at x = x(t). The crust/ core interface is a moving surface at which water from the core region is vaporized at the temperature T,,,, consequently the second boundary condition takes the form: BC2:
T = T,,,,
at x = X(t),t>O
( 14)
The mass transfer boundary conditions (for eqn (5)) are similar to those for the heat transfer equations and are stated as follows: BC3:
ac,, z=O
atx=O,t>O
BC4:
c,, = 0
at x = X(t), t >O
I
IS)
( 10)
The fourth boundary condition arises from the assumption that the liquid concentration in the crust region is negligible, or essentially zero, and the liquid concentration on the surface of the interface at x =x(t) is also negligible. or essentially zero. An alternate mass transfer boundary condition was considered using equality of flux across the interface, x =x(t), and was formulated as eqn (201. This formulation would allow a liquid concentration at x =x(t) greater than zero and is similar to the equality of heat flux boundary condition given by eqn 1191. Crust region
Equation boundary
(71 that models heat transfer in the crust region uses two, more complex, conditions each relying on the principal of continuity of flux (Whitaker,
B. E. Farkas, R. P Singh, 7: R. Rumsey
222
x=X(t+At)
x=0
x=X(t)
x=L
‘7
4”
I
-----A
*
I I
I
I
’ +
~*___--
I
I
~______~NY”~
I
I
I
NPxI -------l
I
I
I
I 4
I
I
I
Fig. 4.
Expanded
I
view of the crust/core
interface.
1977b). Equating the convective heat flux from the oil to the material conductive heat flux from the material surface inwards yields: BC5:
k&z=
h(T,-TJ.=.)
and the
at x= L,t>O
(17)
Crustlcore integace A sixth boundary condition is used at the crust/core interface, the derivation refers to the expanded view of the interface given in Fig. 4 and follows that given by Holland and Liapis (1983). Define the following quantities as: = area of interface, m2 A = U = velocity at which interface moves, m/s dX(t)ldt = distance which interface moves from time t to t+At, m UAt uAAt(E~p~h~+cllpl~~,~)= energy in volume UAAt at tj, J/m3 UAAt(&~pg~+E,p,sr,.)= energy in volume UAAt at G+Atj, J/m” = energy into volume UAAt during At, J q’AAt+N,&AAt = energy out of volume UAAt during At, J qllAAt+NjxH>dAt Constructing
the balance equation yields:
AAt(q’ +N,j,h,j - q” -. N;,,H,.) = UAAt(a,p,h’,+a,.p,.H;,
-eVPnh’rO - appph,,)
(18)
Simplifying and combining like groups gives: BC6:
aT
- k& ax
aT
+ k% -+
ax
N;,,( h,{- H;,)
= U[&,p,(hf,-h~‘)+&,,p~(Hy-h,l)]
at x = X(t),t>O (19)
A mass balance around the control volume UdAt yields the following interfacial mass balance:
The crust region mass transfer boundary BC7:
conditions
are given as:
P;. = P ;,,, at x = L.t >O
(21) The final mass transfer boundary condition relates the pressure of the vapor, f;., at the crust/core interface to the temperature through the Clausius-Clapeon cyuation:
Equation
ilP;.
AI;r;
?T
T( ti;, - i/,,)
[22] is then solved (Felder and Rousseau, BC8:
P,. = exp (++Bj
(22) 1986) to yield:
at x = X(t),t>o
(23)
Initial conditions Initially, the material consists of only core region and two initial conditions are required for solution of the problem before a crust begins to form. In addition, BC6 is an ordinary differential equation therefore requiring a third initial condition. The assumed uniform initial conditions are: ICI:
T = T,,
for all x,t = 0
(2,4)
IC2:
c = c,,
for all x, t = 0
(25)
IC3:
X(t) = 0
at t = 0
(26)
DISCUSSION The process of immersion frying was found to be similar to that of high temperature air drying (Lebedev, 1960a, h) and freeze-drying (Holland and Liapis, 1984). A comparison of the transport equations developed by Holland and Liapis (1984) and Farkas (1994) was conducted by Farkas et al. (1995). The outer region of material in each mathematical model was considered a porous matrix through which Darcy flow of water vapor occurred. Heat conduction in this region was a combination of conduction through the solid matrix and advection by transport of water vapor. The inner layer of material in each process was quite different. The core region in freeze-drying consists of frozen water, hence no mass transport occurs, while in frying the core region contains liquid water and diffusion may be used to describe the transport of water to the interface. Core region heat transfer in each process utilizes conduction but the equations for transport during frying contain an additional term for advection of energy. Flux of liquid water was modeled using diffusion theory with constant diffusivity, D,,,. Past researchers (Rice and Gamble, 1989; Kozempel et al., 1991) have shown Califano and Calvelo (1991) that diffusivity, D,{,, is a function of oil temperature. have shown that oil temperature has no effect on core heating rate, therefore there
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should be little effect of oil temperature on moisture diffusivity and a more complex process is occurring. Analysis of the mathematical model for immersion frying reveals that liquid water is ‘fed’ to the interface by two modes, each possibly being dominant at different stages of frying. The first mode is a result of the movement of the interface into the core region of the material. The progression of the interface into the core region yields an apparent flux of water to the interface at a velocity NF”‘=c,jdX/dt The second mode of liquid transport is due to the movement of liqiid water from the core region to the interface by diffusion NF.= -D,,,&$ax. The magnitude and period during which each flux dominates and their imphcations is discussed in a companion paper (Farkas et al, 1996).
CONCLUSIONS The process of immersion frying was studied experimentally, both visually and by collecting temperature and moisture data. These observations were used to gain insight into immersion frying. The frying process was broken down into four temporal periods: initial heating, surface boiling, falling rate, and bubble end point. These periods were identified by the type of heat transfer at the surface and rate of mass transfer or vapor bubble formation at the surface. The fried material was divided into two distinct regions, the crust and core. These were identified by the distinct boundaries of the crust/oil interface, crust/core interface, and the centerline of the material. These temporal periods and regions and their implications will be discussed further in the companion paper (Farkas et al., 1996) with respect to the solution of the mathematical model developed herein.
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Hvut and muss trun.$er in ft?irtx
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