- Email: [email protected]
Heat transfer during immersion frying of frozen foods
0
PII:
SO260-8774(97)00087-3
Journal of Food Engineering 34 ( lYY7) 2Y3--313 lYY7 Elscvier Science Limited. All rights reserved Printed in Great Brrtain 0260-8774107 $19.00 +O.l10
ELSEVIER
Heat Bansfer
during Immersion Frying of Frozen Foods J. Vijayan & R. P. Singh*
University
of California,
(Received
Department of Biological and Agricultural Davis, CA 95616, USA 9 August 1997; accepted
13 October
Engineering,
1997)
ABSTRACT A model was developed to predict heat transfer during frying of foods that are initially in a frozen state. The model involves two moving boundaries, one between j?o.zen and unfrozen regions, and the other between the dry crust and the moist core region. Heat transfer within the crust region was predicted by Fourier’s law, while heat transfer within the core region was formulated with the enthalpy method. Food composition and thermophysical properties were used to predict heat transfer within the food. The predicted results were experimentally validated. 0 1998 Elsevier Science Limited. All rights reserved
NOMENCLATURE
9 .\J+
t
At
Tt,
’
Area (m’) Apparent specific heat at node (J m ~’ K) Surface heat transfer coefficient (W rn-’ “C) Enthalpy (J m-“) Thermal conductivity at the interface (W rn~ ’ “C) Gross thermal conductivity of the crust (W m - ’ “C) Thermal conductivity of node m (W rn- “C) Half thickness of the product (m) Moisture content (decimal) Heat flux (W m ~ “) Position of the crust/core interface from the product surface (m) Position of the interface from the surface at the present time (m) Position of the interface after time At (m) Time (s) Time step (s) Temperature at the interface (“C)
*To whom correspondence
should be addressed 293
294
J. Vijayan and R. l? Singh
Temperature at the node next to the interface on the core side (“C) Temperature of the frying oil (“C) Temperature of the crust surface (“C) Distance measured along x-axis (m) Thickness of the crust (m) Distance between the interface and node m at timej (m) Density of water (kg rn-“) Latent heat of vaporization of water (J kg-‘)
INTRODUCTION Fried foods constitute a large percentage of the snack food industry. In 1993, snack food sales in the USA were valued at ca $60 billion (Stinson and Tomassetti, 1995). In the same year, 3.48 billion pounds of salty snacks were consumed by Americans, including potato chips, corn and tortilla chips, extruded and fabricated snacks, and multigrain chips (Shukla, 1994). Immersion frying or deep-fat frying is an important and growing food processing operation. This process results in cooking of the plant or animal tissues, extensive inactivation of enzymes and microorganisms, and partial or complete removal of moisture (Keller and Escher, 1989). Both fresh and processed foods are fried to achieve a desired end product. Frying of frozen food is common in both industrial and domestic sectors. Between 1970 and 1974, the per capita consumption of fresh potatoes (retail weight) declined 19%, while consumption of frozen potatoes more than doubled. This practice makes it possible to store food in a frozen state, which maintains a high level of quality for a longer duration, making the products available throughout the year. Information on the engineering aspects of frying is meager, although in the recent past, research on frying has been getting more attention (Singh, 1995). Typically a fried food will have an inner moist core and a dry outer crust such as a French fried potato strip. While the inner core is cooked by the heat conduction, the outer crust becomes crisp due to dehydration (Robertson, 1967). Frying of frozen food materials involves two moving boundaries, viz. the crust-core interface and the frozen-unfrozen interface. Problems involving moving boundaries are also referred as the classical Stefan problems. The solution of these types of problems are intrinsically difficult, because the interfaces between solid-liqui&vapor phases move as the latent heat is absorbed or released at the interface (Ozisik, 1993). This makes the position of the interface an unknown quantity, which also becomes a part of the solution. Carslaw and Jaeger (1959) have shown that these kinds of problems are non-linear in nature and analytical solutions are difficult to obtain for all the required conditions. When exact solutions are not available, approximate, semianalytic, and numerical methods can be used to solve the phase-change problems (Ozisik, 1993). I n many real life problems, explicit tracking of the interface is a difficult method to use. Enthalpy method bypasses explicit tracking of the interface and the jump condition (Stefan condition) is not forced into the solution, but it is obeyed automatically by its natural boundary conditions (Alexiades and Solomon, 1993). The objective of this study was to formulate a predictive model using fundamental physical laws governing heat transfer in immersion frying when the product undergoing frying is initially in frozen state.
Heat transfer during immersion frying of frozen foods
29s
THEORY Several changes occur in a food material during frying, the important one being the development of a crust on the surface of the food. In this section, heat transfer models for crust and core were developed separately and were combined with the model for crust growth to fully represent the immersion frying process. Model development
In a typical immersion frying process, at time = 0, a frozen food product is introduced into the frying oil. As the frying process continues, the temperature of the product increases and a crust layer is formed along the surface of the product due to dehydration. The thickness of the crust layer increases with frying time since the crust-core interface (evaporation front) moves towards the center of the product. The temperature within the frozen region rises during frying and the unfrozenfrozen interface (melting front) and the evaporation front move towards the center of the product. The melting front will be at ca 0°C and the evaporation front will be at the boiling point of water present in the food. In order to simplify a complex system, the following assumptions were made on the basis of observations: (1) one-dimensional heat transfer and energy fluxes are normal to the surface of the material; (2) thermophysical properties change only along streamlines parallel to the energy fluxes; (3) heating medium (oil) is at constant temperature; (4) heat transfer from the oil to the product surface is by convection; (5) heat required for chemical changes during frying, viz. starch gelation, and protein denaturation is negligible compared to that required for the vaporization of water (Farkas, 1994); (6) latent heat of evaporation is a constant; (7) the crust-core interface is sharp and of zero thickness; (8) at the crust-core interface, evaporation takes place at a constant temperature; (9) surface tension and curvature effects at the interface are insignificant; (10) at time = 0, the food material is homogenous and isothermal. The system that was considered for modeling one-dimensional heat transfer during frying is shown in Fig. 1. ABCD is the cylindrical food sample used for frying such that the axis of the cylinder coincides with the x-axis of a right handed Cartesian coordinate system xyz. Circular faces AD and BC are perpendicular to the x-axis. The circumferential surface of the sample was well insulated so that heat transfer takes place only along the x-axis. When this food sample is immersed in the frying oil, the circular surfaces AD and BC are subjected to the frying oil temperature. This results in symmetric distribution of temperature within the food material with point 0 as the center of symmetry. A system with two distinct regions, crust and core separated by an interface is shown in Fig. 2. This occurs after a short duration of frying. The crust is defined by the region 0 ix
J. Vijayan and R. I? Singh
296
Fig. 1. Schematic of the food system used for frying.
center kaT -0
&-v-
ax-
s(t + At) I
oil
TOil
unfrozen
/ frozen region
core 9
t
I x= 0
x = s(t)
-A%Fig. 2. Schematic of the system used for modeling.
Heat transfer during immersion ,fryingof frozen foods
297
center at a rate of dsldt. The core is defined by the region s(t)
(1) where, k,. is the gross thermal conductivity of the crust (W m lo C), A is the area (m’), h is the surface heat transfer coefficient (W mP2 “C), 7’<,ilis the temperature of the frying oil (“C) and T, is the temperature of the crust surface (“C). Because the crust layer gets filled with hot oil during frying, its porosity was neglected and a gross thermal conductivity value was used in the model. The temperature at the crust surface was determined using eqn (1) by approximating the temperature gradient within the crust by the relationship,
(2)
298
J. Vijuyun and R. P Singh
where, Ax,, is the thickness interface (“C).
of the crust (m) and 7’,, is the temperature
at the
Initial crust thickness Until the product surface comes to the boiling point of water, the thickness of the crust will be zero (AXE,= 0). Once the surface comes to the boiling point of water, the crust begins to form. The Stefan condition was used for calculating the increase in crust thickness at any given frying time. For using this method, an initial value for crust thickness is necessary. This initial crust thickness value was obtained by introducing a function used by Farkas (1994). This function [eqn (3)J was derived by conducting an order of magnitude analysis. The function calculates an initial value of crust thickness just greater than zero. The initial guess of the crust thickness was at least one order of magnitude smaller than the distance traveled by the crust-core interface in one time step At used in the numerical solution. k,,hAt
h[Ax]*+k,.[Ax] = -
(Toil
-
Tb)
PA
Crust thickness The Stefan condition was used to locate the position of crust-core interface at any given time. Let & be the distance moved by the interface during the time interval At (Fig. 3). The quantity (&vpmAAs) is the heat required to evaporate the water
sj
,j+l
x=L
Fig. 3. Stefan condition
at the interface
to calculate crust-core
position.
Heat transfer during immersion frying of frozen foods
299
present in the area, between sj and sl+‘, perpendicular to the x direction. (qc,.AAt) is the amount of heat entering into the shaded element per unit area and (qOAAt) is the amount of heat escaping into the core region per unit area during the time At. Assuming there is no heat source at the interface, the heat balance for the crust increment As was written as: q,,AAt - qoAAt = &pmAAs
(4)
where m is the moisture content (decimal), p is the density of water (kg m-‘) and EL,is the latent heat of vaporization of water (J kg-‘). Dividing both sides of eqn (4) by (AAt) and letting At approach zero gives:
Solving the above equation by Euler’s method gave the following equation, which was used for calculating the position of the crust-core interface at any given time:
where si is the position of the interface and sj+ is the position of the interface were:
(Toil- Tb) qdr= kCT s
from the surface at the present time (m), after time At (m). The values of heat flux
(Tb- Tk) Ax;
(7)
where, s is the distance between the interface and the surface node, T,,, is the temperature at the node next to the interface on the core side (“C), k,, is the thermal conductivity at the interface (W m-’ “C), k,,, is the thermal conductivity of node m (W m-’ “C) and AX/ distance between the interface and node m at time j (m). Heat transfer in the core region Heat transfer in the core region is accompanied by a phase change process that involves thawing of a food material. Density and thermal conductivity of the product become temperature dependent during the phase change. These temperature dependencies are nonlinear in nature and exhibit discontinuities at the phase change temperatures (Mannapperuma, 1988). Strong discontinuity cause unstable solutions during the implementation of numerical techniques. In this research, the enthalpy method was used for the development of the model. In the enthalpy method, enthalpy is introduced into the problem as a dependent variable along with the temperature. Enthalpy becomes the primary dependent variable and temperature becomes the secondary dependent variable. The advantage
300
J. Vijayan and R. I? Singh
of using enthalpy method is that the continuously moving boundary need not be tracked over a discrete numerical grid (Voller, 1985). This method is relatively stable even when the phase change occurs over a range of temperatures (Voller and Cross, 1981). Enthalpy has a monotonic relationship with temperature from sub-freezing temperatures to ca 100°C. Once water starts to boil, the enthalpy of the system of interest will vary because of the vaporization process and in turn vapor leaves the system. Therefore, this part of analysis is restricted to the boiling point of water present in the food. The immersion frying process of a food material, when it is initially in frozen state, was formulated by an explicit enthalpy method by defining the equation that governs the process, initial and boundary conditions. In addition, a data file consisting of the thermophysical properties of the food material was used for interpolating between temperature and thermal conductivity at any given time for a given node. The governing equation was obtained by performing a heat balance over a small element in the core region. The equation in one dimension was of the following form (Mannapperuma, 1988):
---
where, H is the enthalpy (J rn-‘). Since enthalpy is used as a dependent variable along with temperature, the problem becomes equivalent to nonlinear heat conduction without phase change (Griffith and Nassersharif, 1990). Incorporation of enthalpy into the equation allows the use of the same governing equation for both phases, frozen and unfrozen, by implicitly incorporating the phase change boundary into the equation. Both enthalpy and thermal conductivity exhibits temperature dependencies. For using enthalpy as the primary dependent variable, temperature and thermal conductivity are expressed as functions of enthalpy as T = T(H) and k =k{H}. To solve eqn (8) two boundary conditions and one initial condition were required. The first boundary condition was written for the crust-core interface. B.C. 1
T = Tb at x = s(t), t > 0
(9)
The latent heat of vaporization for water is obtained from the sensible heat of oil present in the crust that is in contact with the crust-core interface. There may be a cooling effect or slight drop in the temperature at the interface due to the evaporation of water from the interface. It was assumed that the existence of vapor at the interface serves as a source term and maintains the interface temperature at the boiling point of water. Due to the symmetric nature of the problem (Fig. 2) heat flux at the center is zero. This condition was used as the second boundary condition. B.C.2
kz=Oatx=L,t>O
(10)
301
Heat transfer during immersion ftying of frozen foods
At the beginning of the process, product was assumed to be at uniform temperature, Tir1,hence, I.C. 1
T=Ti”
at t=O, O
(11)
To solve the above problem numerically, heat balance equations were written for each of the nodes. A schematic of the nodal representation followed in this problem is shown in Fig. 4. The product was divided into (n - 1) segments of Ax thickness. At time t = 0, enthalpy method was applied over the entire product. The heat balance equation for the segment of thickness Ax at node i between time intervals j and j+ 1 (Mannapperuma, 1988) was written as:
where A is the area Equation (12) was dimensional nature Therefore, eqn (12)
(m2), At is the time step (s) and Ax is the spatial increment (m). used to calculate enthalpy at all the interior nodes. The oneof the uroblem allowed the area terms to be canceled out. becomes’:
At HI+’ = H,‘+ (c’&(T:_, (Ax)’
-T;)-kkj+$T/-T,‘,,)}
( 1.3)
center
surface -7
i-l
i i_-
1 2
Fig. 4. Nodal representation
i+-
used for enthalpy
i+l 1 2
formulation
P-l
P P-i
at time = 0.
J. Vgayan and R. P Singh
302
Thermal conductivity, k, at half nodes is the average of the values at the two nodes on either side, obtained from an interpolation table created from a property prediction program as described later in this paper. Calculation of enthalpy at core node nearest to the interface Fig. 5(a) shows the nodal representation used in the model at time = 0, when there is no crust region. As the interface starts to move towards the center of the food product during frying, the position of the interface crosses the first core node from 1 to l,, [Fig. 5(b)]. This changes the nodal distance between the interface and the next node to AXE from its original value of AX. The enthalpy at the interface is always a constant because the interface temperature is assumed to be a constant, Tb. During every time step, the interface moves towards the center and takes a new position [Fig. 5(b)]. Depending upon the position of the interface, the nodal distance AX,, changes. This change must be incorporated into the equation, which calculates enthalpy of the core node next to the interface. Depending upon the position of the interface between the nodes m-l and m, the equation that calculates enthalpy at the node m also changes. Figure 6(a) shows the position of the interface between nodes m -1 and m in a generalized manner. m-i and m+h represent the half nodes. In Fig. 6(a), the element in which heat balance is carried
(a)
oil
center *, 2
1
-4
*s
Ax*
3
-----
P-l
L-
P
center
I -I I
Oil 1
Fig. 5. (a) Nodal
’ 1,
2
representation representation
3
--
-
-
-
P-l
P
used in the computer program at time = 0. (b) Nodal when the interface crosses the first node.
303
Heat transfer during immersion fying of frozen foods
out lies between half nodes m-i and m+$ . When the interface was between m - 1 and half node m-i, eqn (14) was used to calculate enthalpy at node m.
H,‘,:’ = H,‘,,+ s
k;,
CT:, - Tl;,) _ k,i
Ax,,
node
CT:,, - Tl,,+, 1
‘I’+’
(14)
Ax
where, H,,,is the enthalpy at the node next to the interface (J m -‘), AX,, is the distance between the interface and the node next to the interface (m), and k,, is the average of the thermal conductivities at the interface and at the node m ~’ “C). The thermal conductivity at node m+$ is the average of the thermal (Wm conductivities at nodes m and m+l. When the interface crosses the half node m-i [Fig. 6(b)], eqn (15) was used to calculate enthalpy at node m:
(a)
interface
m-l
m--
m
1
m+-
2
1
m+l
2
interface
(h)
+
j++&++ m-l
Fig. 6. (a) Position
of the interface
m--
m
1 2
m+I
1
m+l
2
before the half node. (b) Position the half node,
of the interface
after
304
J. Vijayan and R. l? Singh
0 81.5 mm
1
- Thermocouple
-
,-
Nylon rod
0 0.51 mm
Fig. 7. Sample holder (mold) used for experimental
validation.
.105
Heat transfer during immersion ,frying qf,frozen foods
TABLE 1 Property Values and Other Parameters
used in the Model
Property
Source(s)
Density of water (p), 1000 kg m ’ Thermal conductivity of the crust (restructured potato), 0.219 W m ~’ “C Heat transfer coefficient, h, 500 W m 2 “C Initial moisture content (decimal wet basis) O-73 Oil temperature, T<>il170, 180, 190°C Boiling point of water, T,,, 102°C Composition of the food used for generating the property file. Water (73.00/o), carbohydrate (21.8%), protein (3.0%), fat (c).5%), fiber (0.2%) and ash (15%)
Singh and Heldman Estimated value
Porosity of the food material,
H,‘,:’ = H,‘,,+
Farkas ( 1994) Actual measurement Actual measurement Estimated value Estimated value
0.0
Estimated
At k!
AX i A41+ 2
(Ti, - T:,,,
A-r,,
( 1993)
_k,
value
_ CT!,-T:,,,,)
“‘+’
(1.s)
AX
i
Once the interface crosses node m, node m+l in Fig. 6(b) becomes to the interface. The steps were then repeated as explained above.
the node next
Calculation of enthalpy at the center node At the center, the heat balance was carried out in the half element, of thickness Ax/Z, to determine the enthalpy at nodep. Due to the symmetry boundary condition, heat flux at the center node is equal to zero. In that case, the equation for enthalpy at the center node becomes:
H
k;,
+
CT:,
I - T;,)
AX
l<](7)
Stability criterion The stability criterion [eqn (18)] used for the apparent adapted from Mannapperuma and Singh (1988).
specific heat formulation
was
C/,1,/+1 = ST;_ ,+( 1 - 2S)T;+ST,!+, Cl
(17)
.I. VTayan and R. I? Singh
306
g
50
0
-25
. ..................................... ......................................
c
t ........................___............ ................... ...........
_______.......__.___.................. / .._.._.__.__.___..........
1 0
I 250
I
500
1000
750
Time (s) Fig. 8. Predicted
and experimentally
determined temperature frozen food material.
profiles at the center
of the
TABLE 2
Slope and Intercept Oil temperature (“C) 170 180 190
Values Obtained
from the Regression
Analysis
?
Slope
Intercept
0986 0.982 0,983
0.987 1.065 1.189
- 2.162 - 1.935 -0492
307
Heat transfer during immersion frying of frozen foods
where Ci is the apparent
specific heat at node i (J mP3 K), and s=
kibt (18)
(Ax)‘Cl’
The stability of the problem is preserved if the right-hand side of the eqn (17) is always positive (Mannapperuma, 1988). Hence the stability criteria is expressed by the following function.
S
(19)
L
At the center node p, the denominator in eqn (18) becomes [O~.~(AX)~C/]because only a half element is available for heat balance. The value of AX at the crust-core
/
............................
.................................
l-
2
3
4
Model crust thickness, mm Fig. 9. Linear regression analysis between the predicted and experimental crust thickness (Farkas, 1994) of the unfrozen food material. Initial temperature = 24”C, frying oil temperature = 18O”C, food slab thickness = 25.4 mm.
J. Vzjayanand R. I? Singh
308
interface varies depending upon the interface position at any given time. The solution to the model equations was stable if the stability criterion was satisfied at the boundaries of the core region since maximum and minimum apparent specific heat occur at the core boundary. Using eqn (18), depending upon the value of AX, the value of At could be calculated to ensure stability. Food properties Thermophysical properties such as, enthalpy, density, thermal conductivity and specific heat, of a food product were required for the simulation of the immersion frying process by the enthalpy method. In this study, property data files for the frying simulation were created using Industrial-Scale Food Freezing - Simulation and ProcessTM (PROPSIMTM), a commercial simulation program developed by Singh and Mannapperuma (1994). This simulation program requires following inputs: composition of the food product (%); unfreezable water content (%); initial freezing point (OF); porosity (%); and upper and lower temperature limit (“F). Using the input values, the program generates an interpolation table where thermal con-
.. .._ __ ___.
._ __
_
/
_ .. .._.. .
I--
..
..I
L
500 Time (s)
Fig. 10. Predicted
crust thickness values. Initial temperature = -2O”C, ature = 18O”C, food slab thickness = 25.4 mm.
frying oil temper-
Heat transfer during immersion fying of frozen foods
ductivity and temperature are arranged (Mannapperuma and Singh, 1989).
in equal increments
of volumetric
309
enthalpy
Computer simulation To simulate the frying process, a computer program was written in Microsoft Quick Basic. The input items for the simulation model included: total thickness of the slab; initial temperature of the product; temperature of the frying oil; thermal conductivity of the crust; surface heat transfer coefficient; moisture content of the food; boiling point of water; time at which the program should stop; name of the property input file; how many times the results must be printed; and name of the output file.
MATERIALS
AND METHODS
A restructured dried potato mixture (Basic American Foods, Blackfoot, ID, US4), commercially used in making french fries, was used for experimental validation of
Location of the frozen/unfrozen Surface
interface; Center
Surface
Thickness (mm)
Fig. 11. Movement of unfrozen-frozen interface and crust-core interface towards the center of the product with frying time. Initial temperature = -2O”C, frying oil temperature = 180°C. food slab thickness = 25.4 mm.
J. Vijayan and R. P Singh
310
computer predicted results. The dry product, stored under refrigerated conditions, was rehydrated before use. The restructured dried potato granules were powdered using a blender and sieved thorough a mesh (No. 40) to obtain uniform particle size. Warm water (70-75°C) was then added to the powder (30 g of water per 70 g of potato starch powder) and mixed using a mixer. The mixture was then kneaded to obtain a uniform dough. The dough was wrapped in a polyethylene film (to avoid drying) and was allowed to stand for at least 8 h in an incubator at 5°C to allow complete rehydration. A slab configuration was used to study the one-dimensional heat transfer. For this purpose, a mold (Fig. 7) was fabricated from a 25.7 mm thick TeflonTM sheet to withstand higher temperatures encountered during frying. Thermocouples (Type T, 20 gauge) were inserted through a nylon rod (2.04 mm diameter) which was attached to the mold to place the thermocouples at the desired location. The rehydrated potato mixture was added to the cavity of the mold around the thermo-
‘-
0
~~
-1
500
II lo
Time (s) Fig. 12. Temperature profiles () obtained from the mathematical model at various locations from the surface in the unfrozen food material during frying. Initial temperature = -2O”C,
frying oil temperature = 18O”C, food slab thickness = 25.4 mm. Solid lines represent temperature profiles at various frying times.
Heat transfer during immersion frying of frozen foods
311
couples in small amounts and compacted to remove air pockets. After filling the cavity and leveling the surface using a straight edge, the mold was placed in an airblast freezer maintained at - 20°C for 3 h to insure complete freezing of the product. A commercial deep fat fryer (Capacity: 19 1; Hobart Corporation, Troy, OH, USA) was used for frying the frozen samples. Canola oil (Chefs pride, Wilsey Food, CA, USA) was used as the frying medium. A total of 21 frying experiments were conducted for the validation purpose, of which six experiments were conducted at an oil temperature of 170+ 15”C, eight experiments at 180+ 135°C and seven experiments at 190f 1.5”C. The experiments were started with fresh oil and the residual oil was retamed for all the remaining experiments at various temperatures. Temperature data were collected at the center of a given test sample during the frying process.
-
0
-25
1
0
250
500
I
I
I
I
I
750
I
I
1000
Time (s) Fig. 13. Effect
of oil temperature on the center temperature profile. ture = -2O”C, food slab thickness = 25.4 mm.
Initial
tempera-
312
J. Vijayan and R. P Singh
RESULTS
AND DISCUSSION
To validate the results from the predictive model, the simulations were run using the process conditions given in Table 1. The composition of the model food system was used to generate the property files required by the program. For simulation purposes, the core region was divided into 10 nodes and a time step of 0.1 s was used. Time steps ~0.1 s or divisions into > 10 nodes did not alter the predicted temperature or crust thickness. A larger time step was selected because it reduces the number of calculations involved during a simulation. Experimental results of center temperature at oil temperatures of 170, 180 and 190°C were compared with the model results. The predicted and experimental temperature profiles at the center of the frozen product, when the oil temperature was 180°C are shown in Fig. 8. Similarly, results were compared for oil temperatures of 170 and 190°C. Statistical analysis revealed that the predicted values agreed well with the experimental values (Table 2). The ? values of ca 0.98 were obtained. Only small deviations from ideality, slope of 1 and intercept of 0, were observed. The small differences between the predicted and the experimental values can be due to positioning of the thermocouples, thermophysical properties or volumetric changes of the material during frying. Similarly, moisture migration was not considered in the model. Despite the simplifying assumptions used, the predicted results were very satisfactory. The predicted crust thickness values for frying of an unfrozen sample were compared with experimentally measured values given by Farkas (1994) (Fig. 9). A linear regression analysis was performed on the data to determine the correlation. For a frying oil temperature at 180°C the slope = 0.93, intercept = 0.075 and ? = 0.995. The high ? value obtained validates the use of the mathematical model for predicting crust thickness. For a frozen material, the development of the crust, at an oil temperature of 180°C is shown in Fig. 10. The initial lag seen in the figure represents the surface heat-up period before the actual crust growth. The movement of unfrozen-frozen interface and crust-core interface, during frying, is shown in Fig. 11. Solid lines in the figure indicate the temperature profiles within the product. The position of an interface is decided by its temperature. From the figure, it is evident that the unfrozen-frozen interface moves much faster than the crust-core interface due to the higher thermal conductivity of ice. A sudden change in the temperature at the interface would be observed only if a node is present at the interface at all time steps. This will occur only if methods like variable time step or space step are used. In Fig. 11, a discontinuity is not observed exactly at an interface because nodes do not fall exactly at the interface at every time step. Figure 12 shows the simulated temperature profiles within the model food system during the frying process. Similar to Farkas (1994) it was observed that the surface temperature of a food material rises quickly and goes above the boiling point of water. After a while, the rise in surface temperature slows down and it tends towards the frying oil temperature. Temperatures at different locations within the core rise slowly to the boiling point of water. If the frying process is continued for a longer duration, the crust thickness will increase, and the core temperature will reach the boiling point of water. The model can also be used to study the effect of various process conditions on the overall immersion frying process. The model showed that the temperature
Heat transfer during immersion ftying of frozen foods
313
profile at the center of the food material was affected by the oil temperature (Fig. 13). At all three oil temperatures, the temperature profile during the thawing phase (below 0°C) did not change significantly. The heating rate decreases as the center temperature approaches the boiling point of water. With increasing oil temperature, the crust-core interface moves towards the center at a faster rate. Therefore, at any given time, the position of the interface will be different for different oil temperatures. The position of the interface in turn influences the center temperature.
CONCLUSIONS (1) A predictive
mathematical model was developed for simulating frying process when a food product is initially in frozen state. Predicted values were in close agreements with the experimental results. Statistical analysis showed high correlation of ca 0.98 between the experimental and predicted results. (2) The developed model can be used for simulating frying of both frozen and unfrozen food products. (3) The computer-aided mathematical model developed in this study can be used for predicting the effect of various process conditions, such as, the frying oil temperature, on the overall frying process.
REFERENCES Alexiades, V. Br Solomon, A. D. (1993). Numerical methods - enthalpy formulation. In Mathematical modeling qf’Melting and Freezing Processes. Hemisphere, Washington, DC. Califano, A. N. & Calvelo, A. (1991). Thermal conductivity of potato between SO and 100°C. J. Food Sci., 56(2), 586-587,589. Carslaw, H. S. & Jaeger, J. C. (lYS9). Conduction of Heat in Solids, 2nd cdn. Oxford University Press, New York. Farkas, B. E. (1994). Modeling immersion frying as a moving boundary problem. Ph.D. Dissertation, University of California, Davis. Griffith, R. & Nassersharif, B. (1990). Comparison of one-dimensional interface-following and enthalpy methods for the numerical solution of phase change. Numerical Heat 7km$v Part B, 18, 169-187. Keller, Ch. & Escher, F. (1989). Heat and mass transfer during deep-fat frying of potato products. In Engineering and Food at ICEF 5, cds Spciss W. E. L. & Schubert H. Elsevicr Applied Science, New York. Mannapperuma, J. D. (1988). Thawing of foods in humid air. Ph.D. Dissertation. University of California, Davis. Mannapperuma. J. D. & Singh, R. P. (1988). Prediction of freezing and thawing times of foods using a numerical method based on enthalpy formulation. J. Food Sci.. 53(2), 626-630. Mannapperuma, J. D. Br Singh, R. P. (1989). A computer-aided method for the prediction of properties and freezing/thawing times of foods. Journal qf Food Engineering, 9(4), 275-304. ijzisik, M. N. (1993). Heat Conduction, 2nd edn. Wiley, New York. Robertson, C. J. (1967). The practice of deep-fat frying. Food Technol., 24, 34-36. Shukla, T. P. (1994). Future snacks and snack food technology. Cereal Foods World. 39(Y), 704-705.
Singh, R. P. (1995). Heat and mass transfer 49(4),
134-137.
in foods during deep-fat
frying. Food Tkhrwl..
314
.I. Ujayan and R. l? Singh
Singh, R. P. & Heldman, D. R. (1993). Introduction to Food Engineering, 2nd Edn. Academic Press, San Diego, CA, USA. Singh, R. P. & Mannapperuma, J. D. (1994). Industrial-scale Food Freezing - Simulation and Process. The Refrigeration Research Foundation, Suite 1200 North, Bethesda, MD, USA. Stinson, C. T. & Tomassetti, S. J. (1995). Effect of natural corn flavor on the sensory quality of low-fat tortilla chips. Food Technology, 49(5), 84-89. Vijayan, J. & Singh, R. P. (1997). Sensitivity to processing conditions during frying of frozen foods. In Engineering and Food at ICEF 7, ed. Jowitt R. Sheffield Academic Press, Sheffield, UK, pp. H49-H52. Voller, V. R. (1985). Implicit finite-difference solutions of the enthalpy formulation of Stefan problems. IM4 J. Numerical Anal., 5, 201-214. Voller, V. & Cross, M. (1981). Accurate Solutions of Moving Boundary Problems Using the Enthalpy Method. Int. J. Heat Mass Transfer, 24, 545-556.
PII:
SO260-8774(97)00087-3
Journal of Food Engineering 34 ( lYY7) 2Y3--313 lYY7 Elscvier Science Limited. All rights reserved Printed in Great Brrtain 0260-8774107 $19.00 +O.l10
ELSEVIER
Heat Bansfer
during Immersion Frying of Frozen Foods J. Vijayan & R. P. Singh*
University
of California,
(Received
Department of Biological and Agricultural Davis, CA 95616, USA 9 August 1997; accepted
13 October
Engineering,
1997)
ABSTRACT A model was developed to predict heat transfer during frying of foods that are initially in a frozen state. The model involves two moving boundaries, one between j?o.zen and unfrozen regions, and the other between the dry crust and the moist core region. Heat transfer within the crust region was predicted by Fourier’s law, while heat transfer within the core region was formulated with the enthalpy method. Food composition and thermophysical properties were used to predict heat transfer within the food. The predicted results were experimentally validated. 0 1998 Elsevier Science Limited. All rights reserved
NOMENCLATURE
9 .\J+
t
At
Tt,
’
Area (m’) Apparent specific heat at node (J m ~’ K) Surface heat transfer coefficient (W rn-’ “C) Enthalpy (J m-“) Thermal conductivity at the interface (W rn~ ’ “C) Gross thermal conductivity of the crust (W m - ’ “C) Thermal conductivity of node m (W rn- “C) Half thickness of the product (m) Moisture content (decimal) Heat flux (W m ~ “) Position of the crust/core interface from the product surface (m) Position of the interface from the surface at the present time (m) Position of the interface after time At (m) Time (s) Time step (s) Temperature at the interface (“C)
*To whom correspondence
should be addressed 293
294
J. Vijayan and R. l? Singh
Temperature at the node next to the interface on the core side (“C) Temperature of the frying oil (“C) Temperature of the crust surface (“C) Distance measured along x-axis (m) Thickness of the crust (m) Distance between the interface and node m at timej (m) Density of water (kg rn-“) Latent heat of vaporization of water (J kg-‘)
INTRODUCTION Fried foods constitute a large percentage of the snack food industry. In 1993, snack food sales in the USA were valued at ca $60 billion (Stinson and Tomassetti, 1995). In the same year, 3.48 billion pounds of salty snacks were consumed by Americans, including potato chips, corn and tortilla chips, extruded and fabricated snacks, and multigrain chips (Shukla, 1994). Immersion frying or deep-fat frying is an important and growing food processing operation. This process results in cooking of the plant or animal tissues, extensive inactivation of enzymes and microorganisms, and partial or complete removal of moisture (Keller and Escher, 1989). Both fresh and processed foods are fried to achieve a desired end product. Frying of frozen food is common in both industrial and domestic sectors. Between 1970 and 1974, the per capita consumption of fresh potatoes (retail weight) declined 19%, while consumption of frozen potatoes more than doubled. This practice makes it possible to store food in a frozen state, which maintains a high level of quality for a longer duration, making the products available throughout the year. Information on the engineering aspects of frying is meager, although in the recent past, research on frying has been getting more attention (Singh, 1995). Typically a fried food will have an inner moist core and a dry outer crust such as a French fried potato strip. While the inner core is cooked by the heat conduction, the outer crust becomes crisp due to dehydration (Robertson, 1967). Frying of frozen food materials involves two moving boundaries, viz. the crust-core interface and the frozen-unfrozen interface. Problems involving moving boundaries are also referred as the classical Stefan problems. The solution of these types of problems are intrinsically difficult, because the interfaces between solid-liqui&vapor phases move as the latent heat is absorbed or released at the interface (Ozisik, 1993). This makes the position of the interface an unknown quantity, which also becomes a part of the solution. Carslaw and Jaeger (1959) have shown that these kinds of problems are non-linear in nature and analytical solutions are difficult to obtain for all the required conditions. When exact solutions are not available, approximate, semianalytic, and numerical methods can be used to solve the phase-change problems (Ozisik, 1993). I n many real life problems, explicit tracking of the interface is a difficult method to use. Enthalpy method bypasses explicit tracking of the interface and the jump condition (Stefan condition) is not forced into the solution, but it is obeyed automatically by its natural boundary conditions (Alexiades and Solomon, 1993). The objective of this study was to formulate a predictive model using fundamental physical laws governing heat transfer in immersion frying when the product undergoing frying is initially in frozen state.
Heat transfer during immersion frying of frozen foods
29s
THEORY Several changes occur in a food material during frying, the important one being the development of a crust on the surface of the food. In this section, heat transfer models for crust and core were developed separately and were combined with the model for crust growth to fully represent the immersion frying process. Model development
In a typical immersion frying process, at time = 0, a frozen food product is introduced into the frying oil. As the frying process continues, the temperature of the product increases and a crust layer is formed along the surface of the product due to dehydration. The thickness of the crust layer increases with frying time since the crust-core interface (evaporation front) moves towards the center of the product. The temperature within the frozen region rises during frying and the unfrozenfrozen interface (melting front) and the evaporation front move towards the center of the product. The melting front will be at ca 0°C and the evaporation front will be at the boiling point of water present in the food. In order to simplify a complex system, the following assumptions were made on the basis of observations: (1) one-dimensional heat transfer and energy fluxes are normal to the surface of the material; (2) thermophysical properties change only along streamlines parallel to the energy fluxes; (3) heating medium (oil) is at constant temperature; (4) heat transfer from the oil to the product surface is by convection; (5) heat required for chemical changes during frying, viz. starch gelation, and protein denaturation is negligible compared to that required for the vaporization of water (Farkas, 1994); (6) latent heat of evaporation is a constant; (7) the crust-core interface is sharp and of zero thickness; (8) at the crust-core interface, evaporation takes place at a constant temperature; (9) surface tension and curvature effects at the interface are insignificant; (10) at time = 0, the food material is homogenous and isothermal. The system that was considered for modeling one-dimensional heat transfer during frying is shown in Fig. 1. ABCD is the cylindrical food sample used for frying such that the axis of the cylinder coincides with the x-axis of a right handed Cartesian coordinate system xyz. Circular faces AD and BC are perpendicular to the x-axis. The circumferential surface of the sample was well insulated so that heat transfer takes place only along the x-axis. When this food sample is immersed in the frying oil, the circular surfaces AD and BC are subjected to the frying oil temperature. This results in symmetric distribution of temperature within the food material with point 0 as the center of symmetry. A system with two distinct regions, crust and core separated by an interface is shown in Fig. 2. This occurs after a short duration of frying. The crust is defined by the region 0 ix
J. Vijayan and R. I? Singh
296
Fig. 1. Schematic of the food system used for frying.
center kaT -0
&-v-
ax-
s(t + At) I
oil
TOil
unfrozen
/ frozen region
core 9
t
I x= 0
x = s(t)
-A%Fig. 2. Schematic of the system used for modeling.
Heat transfer during immersion ,fryingof frozen foods
297
center at a rate of dsldt. The core is defined by the region s(t)
(1) where, k,. is the gross thermal conductivity of the crust (W m lo C), A is the area (m’), h is the surface heat transfer coefficient (W mP2 “C), 7’<,ilis the temperature of the frying oil (“C) and T, is the temperature of the crust surface (“C). Because the crust layer gets filled with hot oil during frying, its porosity was neglected and a gross thermal conductivity value was used in the model. The temperature at the crust surface was determined using eqn (1) by approximating the temperature gradient within the crust by the relationship,
(2)
298
J. Vijuyun and R. P Singh
where, Ax,, is the thickness interface (“C).
of the crust (m) and 7’,, is the temperature
at the
Initial crust thickness Until the product surface comes to the boiling point of water, the thickness of the crust will be zero (AXE,= 0). Once the surface comes to the boiling point of water, the crust begins to form. The Stefan condition was used for calculating the increase in crust thickness at any given frying time. For using this method, an initial value for crust thickness is necessary. This initial crust thickness value was obtained by introducing a function used by Farkas (1994). This function [eqn (3)J was derived by conducting an order of magnitude analysis. The function calculates an initial value of crust thickness just greater than zero. The initial guess of the crust thickness was at least one order of magnitude smaller than the distance traveled by the crust-core interface in one time step At used in the numerical solution. k,,hAt
h[Ax]*+k,.[Ax] = -
(Toil
-
Tb)
PA
Crust thickness The Stefan condition was used to locate the position of crust-core interface at any given time. Let & be the distance moved by the interface during the time interval At (Fig. 3). The quantity (&vpmAAs) is the heat required to evaporate the water
sj
,j+l
x=L
Fig. 3. Stefan condition
at the interface
to calculate crust-core
position.
Heat transfer during immersion frying of frozen foods
299
present in the area, between sj and sl+‘, perpendicular to the x direction. (qc,.AAt) is the amount of heat entering into the shaded element per unit area and (qOAAt) is the amount of heat escaping into the core region per unit area during the time At. Assuming there is no heat source at the interface, the heat balance for the crust increment As was written as: q,,AAt - qoAAt = &pmAAs
(4)
where m is the moisture content (decimal), p is the density of water (kg m-‘) and EL,is the latent heat of vaporization of water (J kg-‘). Dividing both sides of eqn (4) by (AAt) and letting At approach zero gives:
Solving the above equation by Euler’s method gave the following equation, which was used for calculating the position of the crust-core interface at any given time:
where si is the position of the interface and sj+ is the position of the interface were:
(Toil- Tb) qdr= kCT s
from the surface at the present time (m), after time At (m). The values of heat flux
(Tb- Tk) Ax;
(7)
where, s is the distance between the interface and the surface node, T,,, is the temperature at the node next to the interface on the core side (“C), k,, is the thermal conductivity at the interface (W m-’ “C), k,,, is the thermal conductivity of node m (W m-’ “C) and AX/ distance between the interface and node m at time j (m). Heat transfer in the core region Heat transfer in the core region is accompanied by a phase change process that involves thawing of a food material. Density and thermal conductivity of the product become temperature dependent during the phase change. These temperature dependencies are nonlinear in nature and exhibit discontinuities at the phase change temperatures (Mannapperuma, 1988). Strong discontinuity cause unstable solutions during the implementation of numerical techniques. In this research, the enthalpy method was used for the development of the model. In the enthalpy method, enthalpy is introduced into the problem as a dependent variable along with the temperature. Enthalpy becomes the primary dependent variable and temperature becomes the secondary dependent variable. The advantage
300
J. Vijayan and R. I? Singh
of using enthalpy method is that the continuously moving boundary need not be tracked over a discrete numerical grid (Voller, 1985). This method is relatively stable even when the phase change occurs over a range of temperatures (Voller and Cross, 1981). Enthalpy has a monotonic relationship with temperature from sub-freezing temperatures to ca 100°C. Once water starts to boil, the enthalpy of the system of interest will vary because of the vaporization process and in turn vapor leaves the system. Therefore, this part of analysis is restricted to the boiling point of water present in the food. The immersion frying process of a food material, when it is initially in frozen state, was formulated by an explicit enthalpy method by defining the equation that governs the process, initial and boundary conditions. In addition, a data file consisting of the thermophysical properties of the food material was used for interpolating between temperature and thermal conductivity at any given time for a given node. The governing equation was obtained by performing a heat balance over a small element in the core region. The equation in one dimension was of the following form (Mannapperuma, 1988):
---
where, H is the enthalpy (J rn-‘). Since enthalpy is used as a dependent variable along with temperature, the problem becomes equivalent to nonlinear heat conduction without phase change (Griffith and Nassersharif, 1990). Incorporation of enthalpy into the equation allows the use of the same governing equation for both phases, frozen and unfrozen, by implicitly incorporating the phase change boundary into the equation. Both enthalpy and thermal conductivity exhibits temperature dependencies. For using enthalpy as the primary dependent variable, temperature and thermal conductivity are expressed as functions of enthalpy as T = T(H) and k =k{H}. To solve eqn (8) two boundary conditions and one initial condition were required. The first boundary condition was written for the crust-core interface. B.C. 1
T = Tb at x = s(t), t > 0
(9)
The latent heat of vaporization for water is obtained from the sensible heat of oil present in the crust that is in contact with the crust-core interface. There may be a cooling effect or slight drop in the temperature at the interface due to the evaporation of water from the interface. It was assumed that the existence of vapor at the interface serves as a source term and maintains the interface temperature at the boiling point of water. Due to the symmetric nature of the problem (Fig. 2) heat flux at the center is zero. This condition was used as the second boundary condition. B.C.2
kz=Oatx=L,t>O
(10)
301
Heat transfer during immersion ftying of frozen foods
At the beginning of the process, product was assumed to be at uniform temperature, Tir1,hence, I.C. 1
T=Ti”
at t=O, O
(11)
To solve the above problem numerically, heat balance equations were written for each of the nodes. A schematic of the nodal representation followed in this problem is shown in Fig. 4. The product was divided into (n - 1) segments of Ax thickness. At time t = 0, enthalpy method was applied over the entire product. The heat balance equation for the segment of thickness Ax at node i between time intervals j and j+ 1 (Mannapperuma, 1988) was written as:
where A is the area Equation (12) was dimensional nature Therefore, eqn (12)
(m2), At is the time step (s) and Ax is the spatial increment (m). used to calculate enthalpy at all the interior nodes. The oneof the uroblem allowed the area terms to be canceled out. becomes’:
At HI+’ = H,‘+ (c’&(T:_, (Ax)’
-T;)-kkj+$T/-T,‘,,)}
( 1.3)
center
surface -7
i-l
i i_-
1 2
Fig. 4. Nodal representation
i+-
used for enthalpy
i+l 1 2
formulation
P-l
P P-i
at time = 0.
J. Vgayan and R. P Singh
302
Thermal conductivity, k, at half nodes is the average of the values at the two nodes on either side, obtained from an interpolation table created from a property prediction program as described later in this paper. Calculation of enthalpy at core node nearest to the interface Fig. 5(a) shows the nodal representation used in the model at time = 0, when there is no crust region. As the interface starts to move towards the center of the food product during frying, the position of the interface crosses the first core node from 1 to l,, [Fig. 5(b)]. This changes the nodal distance between the interface and the next node to AXE from its original value of AX. The enthalpy at the interface is always a constant because the interface temperature is assumed to be a constant, Tb. During every time step, the interface moves towards the center and takes a new position [Fig. 5(b)]. Depending upon the position of the interface, the nodal distance AX,, changes. This change must be incorporated into the equation, which calculates enthalpy of the core node next to the interface. Depending upon the position of the interface between the nodes m-l and m, the equation that calculates enthalpy at the node m also changes. Figure 6(a) shows the position of the interface between nodes m -1 and m in a generalized manner. m-i and m+h represent the half nodes. In Fig. 6(a), the element in which heat balance is carried
(a)
oil
center *, 2
1
-4
*s
Ax*
3
-----
P-l
L-
P
center
I -I I
Oil 1
Fig. 5. (a) Nodal
’ 1,
2
representation representation
3
--
-
-
-
P-l
P
used in the computer program at time = 0. (b) Nodal when the interface crosses the first node.
303
Heat transfer during immersion fying of frozen foods
out lies between half nodes m-i and m+$ . When the interface was between m - 1 and half node m-i, eqn (14) was used to calculate enthalpy at node m.
H,‘,:’ = H,‘,,+ s
k;,
CT:, - Tl;,) _ k,i
Ax,,
node
CT:,, - Tl,,+, 1
‘I’+’
(14)
Ax
where, H,,,is the enthalpy at the node next to the interface (J m -‘), AX,, is the distance between the interface and the node next to the interface (m), and k,, is the average of the thermal conductivities at the interface and at the node m ~’ “C). The thermal conductivity at node m+$ is the average of the thermal (Wm conductivities at nodes m and m+l. When the interface crosses the half node m-i [Fig. 6(b)], eqn (15) was used to calculate enthalpy at node m:
(a)
interface
m-l
m--
m
1
m+-
2
1
m+l
2
interface
(h)
+
j++&++ m-l
Fig. 6. (a) Position
of the interface
m--
m
1 2
m+I
1
m+l
2
before the half node. (b) Position the half node,
of the interface
after
304
J. Vijayan and R. l? Singh
0 81.5 mm
1
- Thermocouple
-
,-
Nylon rod
0 0.51 mm
Fig. 7. Sample holder (mold) used for experimental
validation.
.105
Heat transfer during immersion ,frying qf,frozen foods
TABLE 1 Property Values and Other Parameters
used in the Model
Property
Source(s)
Density of water (p), 1000 kg m ’ Thermal conductivity of the crust (restructured potato), 0.219 W m ~’ “C Heat transfer coefficient, h, 500 W m 2 “C Initial moisture content (decimal wet basis) O-73 Oil temperature, T<>il170, 180, 190°C Boiling point of water, T,,, 102°C Composition of the food used for generating the property file. Water (73.00/o), carbohydrate (21.8%), protein (3.0%), fat (c).5%), fiber (0.2%) and ash (15%)
Singh and Heldman Estimated value
Porosity of the food material,
H,‘,:’ = H,‘,,+
Farkas ( 1994) Actual measurement Actual measurement Estimated value Estimated value
0.0
Estimated
At k!
AX i A41+ 2
(Ti, - T:,,,
A-r,,
( 1993)
_k,
value
_ CT!,-T:,,,,)
“‘+’
(1.s)
AX
i
Once the interface crosses node m, node m+l in Fig. 6(b) becomes to the interface. The steps were then repeated as explained above.
the node next
Calculation of enthalpy at the center node At the center, the heat balance was carried out in the half element, of thickness Ax/Z, to determine the enthalpy at nodep. Due to the symmetry boundary condition, heat flux at the center node is equal to zero. In that case, the equation for enthalpy at the center node becomes:
H
k;,
+
CT:,
I - T;,)
AX
l<](7)
Stability criterion The stability criterion [eqn (18)] used for the apparent adapted from Mannapperuma and Singh (1988).
specific heat formulation
was
C/,1,/+1 = ST;_ ,+( 1 - 2S)T;+ST,!+, Cl
(17)
.I. VTayan and R. I? Singh
306
g
50
0
-25
. ..................................... ......................................
c
t ........................___............ ................... ...........
_______.......__.___.................. / .._.._.__.__.___..........
1 0
I 250
I
500
1000
750
Time (s) Fig. 8. Predicted
and experimentally
determined temperature frozen food material.
profiles at the center
of the
TABLE 2
Slope and Intercept Oil temperature (“C) 170 180 190
Values Obtained
from the Regression
Analysis
?
Slope
Intercept
0986 0.982 0,983
0.987 1.065 1.189
- 2.162 - 1.935 -0492
307
Heat transfer during immersion frying of frozen foods
where Ci is the apparent
specific heat at node i (J mP3 K), and s=
kibt (18)
(Ax)‘Cl’
The stability of the problem is preserved if the right-hand side of the eqn (17) is always positive (Mannapperuma, 1988). Hence the stability criteria is expressed by the following function.
S
(19)
L
At the center node p, the denominator in eqn (18) becomes [O~.~(AX)~C/]because only a half element is available for heat balance. The value of AX at the crust-core
/
............................
.................................
l-
2
3
4
Model crust thickness, mm Fig. 9. Linear regression analysis between the predicted and experimental crust thickness (Farkas, 1994) of the unfrozen food material. Initial temperature = 24”C, frying oil temperature = 18O”C, food slab thickness = 25.4 mm.
J. Vzjayanand R. I? Singh
308
interface varies depending upon the interface position at any given time. The solution to the model equations was stable if the stability criterion was satisfied at the boundaries of the core region since maximum and minimum apparent specific heat occur at the core boundary. Using eqn (18), depending upon the value of AX, the value of At could be calculated to ensure stability. Food properties Thermophysical properties such as, enthalpy, density, thermal conductivity and specific heat, of a food product were required for the simulation of the immersion frying process by the enthalpy method. In this study, property data files for the frying simulation were created using Industrial-Scale Food Freezing - Simulation and ProcessTM (PROPSIMTM), a commercial simulation program developed by Singh and Mannapperuma (1994). This simulation program requires following inputs: composition of the food product (%); unfreezable water content (%); initial freezing point (OF); porosity (%); and upper and lower temperature limit (“F). Using the input values, the program generates an interpolation table where thermal con-
.. .._ __ ___.
._ __
_
/
_ .. .._.. .
I--
..
..I
L
500 Time (s)
Fig. 10. Predicted
crust thickness values. Initial temperature = -2O”C, ature = 18O”C, food slab thickness = 25.4 mm.
frying oil temper-
Heat transfer during immersion fying of frozen foods
ductivity and temperature are arranged (Mannapperuma and Singh, 1989).
in equal increments
of volumetric
309
enthalpy
Computer simulation To simulate the frying process, a computer program was written in Microsoft Quick Basic. The input items for the simulation model included: total thickness of the slab; initial temperature of the product; temperature of the frying oil; thermal conductivity of the crust; surface heat transfer coefficient; moisture content of the food; boiling point of water; time at which the program should stop; name of the property input file; how many times the results must be printed; and name of the output file.
MATERIALS
AND METHODS
A restructured dried potato mixture (Basic American Foods, Blackfoot, ID, US4), commercially used in making french fries, was used for experimental validation of
Location of the frozen/unfrozen Surface
interface; Center
Surface
Thickness (mm)
Fig. 11. Movement of unfrozen-frozen interface and crust-core interface towards the center of the product with frying time. Initial temperature = -2O”C, frying oil temperature = 180°C. food slab thickness = 25.4 mm.
J. Vijayan and R. P Singh
310
computer predicted results. The dry product, stored under refrigerated conditions, was rehydrated before use. The restructured dried potato granules were powdered using a blender and sieved thorough a mesh (No. 40) to obtain uniform particle size. Warm water (70-75°C) was then added to the powder (30 g of water per 70 g of potato starch powder) and mixed using a mixer. The mixture was then kneaded to obtain a uniform dough. The dough was wrapped in a polyethylene film (to avoid drying) and was allowed to stand for at least 8 h in an incubator at 5°C to allow complete rehydration. A slab configuration was used to study the one-dimensional heat transfer. For this purpose, a mold (Fig. 7) was fabricated from a 25.7 mm thick TeflonTM sheet to withstand higher temperatures encountered during frying. Thermocouples (Type T, 20 gauge) were inserted through a nylon rod (2.04 mm diameter) which was attached to the mold to place the thermocouples at the desired location. The rehydrated potato mixture was added to the cavity of the mold around the thermo-
‘-
0
~~
-1
500
II lo
Time (s) Fig. 12. Temperature profiles () obtained from the mathematical model at various locations from the surface in the unfrozen food material during frying. Initial temperature = -2O”C,
frying oil temperature = 18O”C, food slab thickness = 25.4 mm. Solid lines represent temperature profiles at various frying times.
Heat transfer during immersion frying of frozen foods
311
couples in small amounts and compacted to remove air pockets. After filling the cavity and leveling the surface using a straight edge, the mold was placed in an airblast freezer maintained at - 20°C for 3 h to insure complete freezing of the product. A commercial deep fat fryer (Capacity: 19 1; Hobart Corporation, Troy, OH, USA) was used for frying the frozen samples. Canola oil (Chefs pride, Wilsey Food, CA, USA) was used as the frying medium. A total of 21 frying experiments were conducted for the validation purpose, of which six experiments were conducted at an oil temperature of 170+ 15”C, eight experiments at 180+ 135°C and seven experiments at 190f 1.5”C. The experiments were started with fresh oil and the residual oil was retamed for all the remaining experiments at various temperatures. Temperature data were collected at the center of a given test sample during the frying process.
-
0
-25
1
0
250
500
I
I
I
I
I
750
I
I
1000
Time (s) Fig. 13. Effect
of oil temperature on the center temperature profile. ture = -2O”C, food slab thickness = 25.4 mm.
Initial
tempera-
312
J. Vijayan and R. P Singh
RESULTS
AND DISCUSSION
To validate the results from the predictive model, the simulations were run using the process conditions given in Table 1. The composition of the model food system was used to generate the property files required by the program. For simulation purposes, the core region was divided into 10 nodes and a time step of 0.1 s was used. Time steps ~0.1 s or divisions into > 10 nodes did not alter the predicted temperature or crust thickness. A larger time step was selected because it reduces the number of calculations involved during a simulation. Experimental results of center temperature at oil temperatures of 170, 180 and 190°C were compared with the model results. The predicted and experimental temperature profiles at the center of the frozen product, when the oil temperature was 180°C are shown in Fig. 8. Similarly, results were compared for oil temperatures of 170 and 190°C. Statistical analysis revealed that the predicted values agreed well with the experimental values (Table 2). The ? values of ca 0.98 were obtained. Only small deviations from ideality, slope of 1 and intercept of 0, were observed. The small differences between the predicted and the experimental values can be due to positioning of the thermocouples, thermophysical properties or volumetric changes of the material during frying. Similarly, moisture migration was not considered in the model. Despite the simplifying assumptions used, the predicted results were very satisfactory. The predicted crust thickness values for frying of an unfrozen sample were compared with experimentally measured values given by Farkas (1994) (Fig. 9). A linear regression analysis was performed on the data to determine the correlation. For a frying oil temperature at 180°C the slope = 0.93, intercept = 0.075 and ? = 0.995. The high ? value obtained validates the use of the mathematical model for predicting crust thickness. For a frozen material, the development of the crust, at an oil temperature of 180°C is shown in Fig. 10. The initial lag seen in the figure represents the surface heat-up period before the actual crust growth. The movement of unfrozen-frozen interface and crust-core interface, during frying, is shown in Fig. 11. Solid lines in the figure indicate the temperature profiles within the product. The position of an interface is decided by its temperature. From the figure, it is evident that the unfrozen-frozen interface moves much faster than the crust-core interface due to the higher thermal conductivity of ice. A sudden change in the temperature at the interface would be observed only if a node is present at the interface at all time steps. This will occur only if methods like variable time step or space step are used. In Fig. 11, a discontinuity is not observed exactly at an interface because nodes do not fall exactly at the interface at every time step. Figure 12 shows the simulated temperature profiles within the model food system during the frying process. Similar to Farkas (1994) it was observed that the surface temperature of a food material rises quickly and goes above the boiling point of water. After a while, the rise in surface temperature slows down and it tends towards the frying oil temperature. Temperatures at different locations within the core rise slowly to the boiling point of water. If the frying process is continued for a longer duration, the crust thickness will increase, and the core temperature will reach the boiling point of water. The model can also be used to study the effect of various process conditions on the overall immersion frying process. The model showed that the temperature
Heat transfer during immersion ftying of frozen foods
313
profile at the center of the food material was affected by the oil temperature (Fig. 13). At all three oil temperatures, the temperature profile during the thawing phase (below 0°C) did not change significantly. The heating rate decreases as the center temperature approaches the boiling point of water. With increasing oil temperature, the crust-core interface moves towards the center at a faster rate. Therefore, at any given time, the position of the interface will be different for different oil temperatures. The position of the interface in turn influences the center temperature.
CONCLUSIONS (1) A predictive
mathematical model was developed for simulating frying process when a food product is initially in frozen state. Predicted values were in close agreements with the experimental results. Statistical analysis showed high correlation of ca 0.98 between the experimental and predicted results. (2) The developed model can be used for simulating frying of both frozen and unfrozen food products. (3) The computer-aided mathematical model developed in this study can be used for predicting the effect of various process conditions, such as, the frying oil temperature, on the overall frying process.
REFERENCES Alexiades, V. Br Solomon, A. D. (1993). Numerical methods - enthalpy formulation. In Mathematical modeling qf’Melting and Freezing Processes. Hemisphere, Washington, DC. Califano, A. N. & Calvelo, A. (1991). Thermal conductivity of potato between SO and 100°C. J. Food Sci., 56(2), 586-587,589. Carslaw, H. S. & Jaeger, J. C. (lYS9). Conduction of Heat in Solids, 2nd cdn. Oxford University Press, New York. Farkas, B. E. (1994). Modeling immersion frying as a moving boundary problem. Ph.D. Dissertation, University of California, Davis. Griffith, R. & Nassersharif, B. (1990). Comparison of one-dimensional interface-following and enthalpy methods for the numerical solution of phase change. Numerical Heat 7km$v Part B, 18, 169-187. Keller, Ch. & Escher, F. (1989). Heat and mass transfer during deep-fat frying of potato products. In Engineering and Food at ICEF 5, cds Spciss W. E. L. & Schubert H. Elsevicr Applied Science, New York. Mannapperuma, J. D. (1988). Thawing of foods in humid air. Ph.D. Dissertation. University of California, Davis. Mannapperuma. J. D. & Singh, R. P. (1988). Prediction of freezing and thawing times of foods using a numerical method based on enthalpy formulation. J. Food Sci.. 53(2), 626-630. Mannapperuma, J. D. Br Singh, R. P. (1989). A computer-aided method for the prediction of properties and freezing/thawing times of foods. Journal qf Food Engineering, 9(4), 275-304. ijzisik, M. N. (1993). Heat Conduction, 2nd edn. Wiley, New York. Robertson, C. J. (1967). The practice of deep-fat frying. Food Technol., 24, 34-36. Shukla, T. P. (1994). Future snacks and snack food technology. Cereal Foods World. 39(Y), 704-705.
Singh, R. P. (1995). Heat and mass transfer 49(4),
134-137.
in foods during deep-fat
frying. Food Tkhrwl..
314
.I. Ujayan and R. l? Singh
Singh, R. P. & Heldman, D. R. (1993). Introduction to Food Engineering, 2nd Edn. Academic Press, San Diego, CA, USA. Singh, R. P. & Mannapperuma, J. D. (1994). Industrial-scale Food Freezing - Simulation and Process. The Refrigeration Research Foundation, Suite 1200 North, Bethesda, MD, USA. Stinson, C. T. & Tomassetti, S. J. (1995). Effect of natural corn flavor on the sensory quality of low-fat tortilla chips. Food Technology, 49(5), 84-89. Vijayan, J. & Singh, R. P. (1997). Sensitivity to processing conditions during frying of frozen foods. In Engineering and Food at ICEF 7, ed. Jowitt R. Sheffield Academic Press, Sheffield, UK, pp. H49-H52. Voller, V. R. (1985). Implicit finite-difference solutions of the enthalpy formulation of Stefan problems. IM4 J. Numerical Anal., 5, 201-214. Voller, V. & Cross, M. (1981). Accurate Solutions of Moving Boundary Problems Using the Enthalpy Method. Int. J. Heat Mass Transfer, 24, 545-556.