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Meatball cooking — modeling and simulation
JournaIofFoodEngineerit1g24(1995)87-100 Q 1994 Elsevier Science Limited Printed in Great Britain. AU rights reserved 0260-8774/94/s7.00 ELSEVIER
Meatball Cooking - Modeling and Simulation E. Huang & G. S. Mittal” School of Engineering,
University of Guelph, Guelph, Ontario, Canada, Nl G 2W 1
(Received 16 November 1992; revised version received 7 September accepted 20 December 1993)
1993;
ABSTRACT Mathematical models were developed to predict the temperature and mass histories of meatballs (47 cm diameter, 60 g) during forced convection baking, natural convection baking (broiling), and boiling. The finite difference method was used to solve the simultaneous heat and moisture transfer equations using constant transport properties and the continuous system modeling program (CSMP). Surface heat transfer coeficients were determined by the transient temperature measurement method, with values of 230, 96, and 4518 W/m2 K, respectively, for forced convection baking, natural convection baking (broiling), and boiling. Thermal and moisture difisivity values for meatball cooking processes were estimated from the experimental data by minimizing the root-meansquare of deviations between the observed and predicted temperature and moisture histories. The thermal diffusivitiy values were 1.8~ JO- ‘, 1.9~ IO- 7, and 16 x IO- 7 m’/s for meatball cooking during forced convection baking, natural convection baking, and boiling, respectively. Moisture difisivities of the meatball for the first two processes were 39 x IO-” and 2.5 x lo-” m2/s, respectively. The average root-meansquare of deviations between the observed and predicted temperature histories (390°C) ranged from 30 to 51°C for the cooking processes, and between the observed and predicted meatball mass, accounted for the moisture Ioss only, rangedfrom 064 to OI9g.
NOTATION Bi c
Biot number Non-dimensional
*To whom correspondence
moisture
content
should be addressed. 87
E. Huang, G. S. Mittal Moisture diffusivity (m’/s) Fourier number Surface heat transfer coefficient (W/m’ K) Thermal conductivity of the meatball (W/m K) Latent heat of vaporization (J/kg) Moisture concentration, dry basis Number of data points Radial position (m) Meatball radius (m) Gas constant (1.978 g caI/g mol K) Relative humidity (decimal) Time (s) Meatball temperature at any time during cooking (“C) Absolute temperature (K) Thermal diffusivity (m’/s) Non-dimensional temperature Density of dry matter of the meatball (kg/m3) Root-mean-square of deviations for moisture Root-mean-square of deviations for temperature Non-dimensional radial distance Subscripts a Ambient Equilibrium, experimental e 0 Initial, center predicted P Surface i-9 Nodes INTRODUCTION Forced convection baking and natural convection baking (broiling) are classified as dry cooking processes, categorized by the low humidity and high processing temperatures, which typically result in crust formation on the food surface. Boiling, however, is considered a wet cooking process, at relatively lower processing temperatures ( < 1OOC) and high humidities, and promotes a soft and moist food texture, generally with minimal mass loss. Each cooking process involves simultaneous heat and mass transport phenomena, physical and chemical reactions such as protein denaturation, the Maillard group of reactions, and microbiological destruction (Scher et al., 1991). Holtz et al. ( 1984) in their study on meat loaf baking showed that the combination of high oven temperature and air velocity promotes moisture loss at the surface, resulting in thick crust formation in a relatively short time period. Natural convection baking (broiling) is realized primarily by radiative heat transfer from heating elements and the oven walls. Bengtsson et al. (1976) studied the heat and mass transfer in beef roasts and exhibited an inverse relationship between the moisture and temperature histories. Incidentally, no significant fat migration was reported.
Meatball cooking modeling
89
Boiling is performed at temperatures below or around 100°C in either water or soups (Skjoldebrand, 1984). The properties of liquid and vapor, and the heating surface conditions, such as roughness and wettability, are the main parameters that influence the heat and mass transfer coefficients in boiling (ASHRAE, 1985). Burfoot and Self (1988) reported that the dimensional changes of beef cubes during boiling substantially reduce the heating time. Holtz and Skjoldebrand (1986) developed a heat and mass transfer model to simulate the temperature histories in a meat loaf during forced convection baking. Burfoot and James ( 1984) modeled the roasting of a meat joint by considering radiative heat transfer and air convection as the main heat transfer mechanisms. Burfoot and Self ( 1988) developed a numerical model to predict the heating time of beef cubes (20 and 30 mm3) in water (65,75,85, and 95°C). Mass loss increased with water temperature up to 64°C. Burfoot et al. (1990) compared the heating/cooling times and mass losses of meat joints by convective, immersion, and pressure/vacuum processes. The heating/cooling times, in the order of increasing speed, were due to immersion, convection, and pressure. Thus, the objective of this study is to model, simulate and validate the heat and moisture transfer phenomena within a meatball during different cooking processes to predict temperature and mass of the meatball.
MODELING Heat is transferred mainly by convection (forced convection baking, boiling) and radiation (natural convection baking or broiling) from the heating media to the meatball surface, followed by conduction towards the geometric center. Meanwhile, moisture diffuses outward to the meatball surface, where it is in turn vaporized and lost to the surroundings through convection. The necessary assumptions were (i) the meatballs were homogeneous, isotropic, and spherical in geometry; (ii) initial temperature and moisture distributions in meatballs were uniform; (iii) ambient temperature and moisture were step functions of time; (iv) vaporization of water was restricted to the meatball surface only; (v) negligible meatball shrinkage; (vi) negligible effect of crust formation on physical properties; (vii) constant a and D,; and (viii) fat transport was neglected. Based on the above assumptions, the mathematical models characterizing simultaneous pseudo-one-dimensional heat and mass transport in a meatball during forced convection baking, natural convection baking, and boiling could be represented as follows (symbols are defined in the Notation section). Heat transfer aT
g=a
(
2aT
;z+z
a2T
1
(1)
Moisture transfer
(2)
E. Huang, G. S. Mittal
90
Initial and boundary conditions The initial temperature
and moisture distribution are assumed to be uniform: T(r,O)= T,;
Temperature
m(r,O)= m,
(3)
and moisture gradients at the meatball center are depicted by -i3T
= 0;
am ar r=I,=o
ar r=O
(4)
Energy balance at the meatball surface, accounting for convective heat gain at the surface from the heating medium, heat conduction from the surface into the meatball, and the latent heat of vaporization which removes heat from the meatball surface:
(5) The boundary condition of instantaneous moisture meatball surface with the environment is expressed by
of
m 1r= R= m, at t > 0, for boiling
at t > 0, for baking;
+,=m,
content equilibrium
(6)
Non-dimensional analysis: dimensionless temperature (8), moisture content (C), radial length (V) To simplifv the numerical calculations, temperature, length b ion-dimensional forms are d&ineh as: T- T
(+--...2 Z-T,’
Cc_. m-m,
moisture content and radial
qJu=I R
m,-m,’
Subsequently, the model (eqns ( 1)-( 6)) in non-dimensional
(7)
form becomes
!$;($~+2J
(8)
(9)
C(Y,O)=l;
e(Y,o)=o; kae,(cRaY
T,)
=h(T,--
C, = 0 at t> 0, for baking;
E
T,)+ ,,yLv
= =o; Y 0 $
g
= =o ‘y 0
(m, _ m,)
C, = 1 at t> 0, for boiling
(10)
(11) (12)
Meatball cooking modeling
91
Finite difference equations development A one-dimensional spherical finite difference framework, consisting of 10 concentric sheiks of equal thickness, was developed to model the heat and moisture concentrations in a meatball during processing. Eleven nodes in total, one at the center of each shell element, and the 1 lth on the outer surface, were assigned. Temperature and moisture content at each of these nodes were assumed to be representative of the entire element. Node 0 (geometric center) Using the boundary condition and central difference: (13)
(14) Nodes l-8 de. 2dt dC. I dt
izI-8
icI-8
-
8,+,-28,+-
2i2i+ 1 f&l
-
C~+~-2Ci+- 2i+ 2i- 1
(15)
(16)
ci-l
Node 9 (17)
(18) Node S (surface) By backward difference: *,=Bi.8,+208,+20D;p,;L, s
(C,-C,),
20+Bi
es= c,= e
= 0, 0
e
..
Bi*&+208, 20+Bi
for baking;
forbaking
,
forbohg c,=-- m” - me - 1, m,-m,
(19)
(20) for boiling
(21)
E. Huang, G. S. Mittal
92
METHODS
AND MATERIALS
Meatball preparation Beef portions from chuck and shoulder, obtained from the university’s abattoir, were ground through a plate with l-cm diameter orifices. A commercial minced meat recipe, which required 12.3 kg of coarsely grounded beef, 1575 kg of added water, and 1.125 kg of the hamburger binder/spice mixture (Griffith Laboratories, Scarborough, Ontario) for every 15 kg of batch was used for making meatballs. The binder/spice mixture contained toasted breadcrumbs, salt, flour, skim milk powder, spices, hydrolyzed plant protein, onion powder, and monosodium glutamate. These ingredients were mixed manually for 3 min, and ground through a plate with 0*5-cm diameter orifices. Samples of the prepared minced beef were withheld for chemical analyses. Subsequent chemical analysis (AOAC, 1990) results showed the composition of the minced meat as 64.50% water, 15.86% protein, 13.34% fat, 3.94% carbohydrate, and 2.36% ash. The prepared minced meat was portioned into packages of 5 kg each, and frozen at - 40°C for 15 h prior to storage at - 18’C until required for experiments. Frozen minced beef packages were thawed at 2°C for about 48 h prior to meatball preparation. A pair of commercial meatball shapers (4.7-cm inner diameter) was utilized to form meatballs weighing 60 f 0.05 g each. To equalize the temperature of the meatballs, all prepared meatballs were stored at 2°C for at least 2 h prior to each experiment. Temperature measurement A data acquisition and control system, which consisted of a data-logger, a CPU module, a serial interface (Labmate, CPU module Model 901; Sciemetric Instruments Inc., Nepean, Ontario), and a portable computer (Tandy 200, Radio Shack Inc., Barrie, Ontario), was assembled for the recording of data during cooking. A BASIC program was used to interface with the data acquisition system. Temperature and mass data were retrieved and stored at predetermined intervals (every 10 s during the 1st min, and every 30 s thereafter) until the completion of each experiment. Temperatures were measured with high-temperature rated copperconstantan thermocouples measuring 0.5 mm diameter at the junctions (Kapton Insulated Duowrap Parallel Duplex thermocouple wire, Therm0 Electric Canada Ltd, Brampton, Ontario). Each thermocouple wire was encased in a locm length of Teflon tubing for reinforcement. A thermocouple alignment stand was constructed to ensure accurate placement of thermocouples within a meatball. This alignment stand allowed precise vertical positioning of thermocouples. To ensure proper heat and moisture convection around each meatball, the meatballs were suspended by a specially constructed fine wire mesh rack. For meatball temperature measurements, thermocouples were placed at the center and 1.175 cm from the center of the meatball. A thermocouple was also placed in the proximity of the meatballs to monitor heating medium temperature. A meatball was considered cooked when its center reached 70°C. After cooking, the meatballs were sealed in plastic pouches and weighed prior to storage at - 18°C.
Meatball cooking modeling
93
Mass measurement Continuous mass losses during forced and natural convection bakings were recorded by two cantilever load-cells, which were constructed in full-bridge configuration to compensate for the temperature effect. No noticeable hysteresis within the experimental range (0 to 200 g) was observed during their calibration (R 2 = 0.999). Manual weighing of meatballs with an electronic scale during boiling was necessitated by the high buoyancy forces. Cooking procedures Forced convection baking
A commercial kitchen size multi-mode oven (400 cmX 380 cmx 600 cm, Deacor ‘Convection Plus’ Self Cleaning Wall Oven, Model W305C, Pasadena, California) was used. The convection heating mode (mode 5), in which the oven fan convects the heat generated by a heating element at the back of the oven cavity at a velocity of O+0.9 m/s, was used to achieve an oven temperature of 140+ 10°C. The temperature of the oven was set according to the measurements by a thermocouple located 24 cm in front of the oven fan. Natural convection baking or broiling
The same oven was utilized, with the cooking mode set to the baking mode (mode 2, heated top element only). The oven temperature of 140°C fluctuated ( f 1YC) due to the action of the built-in on-off oven temperature controller. Natural air flow induced by the temperature gradient in the oven was the sole means of convection assisted by radiative heat. Boiling
A thermostatically controlled bath circulator (Model E8, Haake, Frankfurt, Germany) was used. This circulator has a heating capacity of 1500 W, circulation rate of 15 changes/m& and fluid temperature control to within f 0-02°C. The bath vessel, measured 3 10 cm X 290 cm X 130 cm, has a capacity of 12 litres. For each experimental run, nine meatballs for heat and mass transport studies were simultaneously immersed in water at 90°C. Meatballs were retrieved at various times during boiling. Excess water on the meatball surfaces was removed by gently rolling on a large paper towel immediately after retrieval for approximately 2 s; then the meatballs were sealed in plastic pouches to prevent further moisture loss through evaporation. The mass of each meatball group was then recorded prior to storage in a freezer. Heat transfer coefficient determination The heat transfer coefficient was estimated by the transient temperature measurement method, according to the convective heat transfer equation in the integrated form, as shown by Kreith and Black ( 1980),
T,- T(t) T
_
e
T
=
I
exp(- (Bi)(Fo))
(22)
94
E. Huang, G. S. Mittal
A solid aluminum sphere of 4.7 cm in diameter was constructed and used to experimentally estimate the effective surface heat transfer coefficients of the cooking processes. Dents, about 0*4-cm diameter and O-2-cm deep, were drilled over the entire sphere surface to simulate the roughness of the meatball surface. A thermocouple was located at the center of the model sphere to monitor the temperature changes. Simulation The finite difference equations with appropriate inputs were solved by using continuous system modeling program (CSMP) simulation language on a mainframe computer. The root-mean-squares of deviations between predicted and observed values of average moisture (a,,,) and central temperature (a, ) of the meatball at various times during cooking were calculated by j=N
C (G,p,j- c0.e,i)2 j=l
a,=
N
(23)
j=N
C(G,p.j- %e,j)* j=l
a,=
N
(24)
Thermal and moisture diffusitivies for the meatball undergoing various cooking processes were determined by minimizing or and a,. For minirnization the pattern search algorithm of Hooke and Jeeves ( 1961) was used. Only one replication for each cooking method was used for this purpose. Four other replications were used to validate the models.
RESULTS AND DISCUSSION Surface heat transfer coefficient Table 1 summarizes the surface heat transfer coefficients (h) for the aluminum spherical model, which were averaged from the results of five replications for each process. These experimental heat transfer coefficients are in good agreement with the calculated results by empirical equations according to Bird et al. (1960). Equilibrium moisture content The experimental minced meat moisture contents reported by Hallstrom ( 1990) were used to establish a model for the moisture isotherms. The Halsey equation (Halsey, 1948), among few models attempted, was selected on the
140 f 10 140*15 90 + 0.3
Temperature (“C)
0.5 0.5 100
RH PO)
0.7 f 0.2 -0 lf
Velocity (mls)
Processing conditions
23.0 9.0 4518
(W/m2 K)
18.8 11.5 4134
(W/m'K)
uh,, = surface heat transfer coefficient, experimental. bhca,c= surface heat transfer coefficient, calculated. ‘Pinal meatball masses based on moisture loss only. *Baking I = forced convection baking. ‘Baking II = natural convection baking. fValue estimated based on circulator manufacturer specification. gND = not determined.
Baking Id Baking II’ Boiling
Process
TABLE 1
1.8 x lo-’ 1.9 x lo-’ 1.6 x lo-’
(m;/s)
0.39~ 1O-y 0.25 x 1O-u NDs
D, (m ‘Is)
1620 2600 766
Cooking time (s)
Transport Properties and Simulation Results (See Notation Section for Symbols)
57-44 57.41 60.48
Final mass‘ (@
1.28-4.79 3.95-6.37 2.73-3.59
T
of
0.03-0.05 0.17-0.20 ND
Mass
Root-mean-square deviations
8 3 0%
8 5 00 3
g g
E. Huang, G. S. Mittal
96
basis of the highest coefficient of determination (R ’ = 0.995) using the NLIN procedure of the Statistical Analysis System (SAS, 1988): RH=exp
- 522241 RT g A
_ 1.0983 m,
(25)
Thermal diffusivity The results (Table l), show good agreement with the thermal diffusivity value for meat emulsion reported by Agrawal (1976) (152-X 10e7 m2/s at 93°C). In general, the addition of binder to raw meat decreased the specific heat of the mixture due to lower water content, which increased thermal diffisivity values (1.6 x 10e7 to 1.9 x 10m7 m*/s). The influence of processing temperature on the magnitude of thermal diffusivity is also shown, where the thermal diffusivities were higher in forced and natural convection baking processes (performed at 140 + 10°C and 140 f 15”C, respectively) than in boiling (90 + 0.3”C). Moisture diffusivity Incidentally, the descending magnitudes of heat transfer coefficients (23.0 and 9-O W/m* K for forced and natural convection bakings, respectively) were alsa reflected in the moisture diffusivity values (Table 1). Since negligible moisture change was observed during meatball boiling in water, the moisture diffusivity for meatball during boiling was not determined. Nevertheless, it was believed that bi-directional water diffusion could have occurred between the water and the meatballs, and have contributed to the slight gain ( < 1%) in the mass. Temperature and moisture histories simulations Forced convection baking
The rates of temperature change were initially sluggish due to heat transfer lag (Fig. l), but gradually increased as the cooking progressed. When the temperatures at the nodes were raised above the wet-bulb temperature, the rate of change declined slightly yet continued the rising trend toward the dry-bulb temperature. An average baking time of 1600 s was required to raise the meatball center temperature to 70°C (Table 1). The simulated temperature histories agreed well with the experimental data. An average slope of 0.996 for the relationship between predicted and observed temperatures was calculated. The average standard deviation varied between 1.28 and 4*79”C for four replications. These deviations were believed to be partly due to the constant thermal diffusivity and latent heat of vaporization values used in the model. Variations in environmental conditions, such as oven temperature and relative humidity, during experiments could also be responsible for the discrepancy. Dislocation of thermocouples was a possible source of experimental error. Holtz and Skjoldebrand (1986) likewise reported similar temperature-distribution histories in their meat loaf forced convection baking study. The observed mass history showed the combined moisture and fat losses during the baking process (Fig. 2). The average observed final meatball mass was 53.2 g, while the calculated value based on the final moisture content was
Meatball cooking modeling
97
200 Boiling
BOiliUg
simulated
observed *
BW ambiit
BpLing
Baking
Broiling
Broiling
Broiling
simulated
observed
ambient
simulated
ObscNed
- - ____
0
H
1.50
50
0
0
2ooo
1500
1000
500
2500
3ooo
Time, s
Fig. 1.
Observed and predicted meatball temperature histories during cooking (“C).
Baking simulated -
57 1 0
Fig. 2.
Baking Broiling observed simulated . -
Broiling observed .
I
I
I
I
I
I
5co
loo0
1500 Time, s
zoo0
2500
3ooo
Observed and predicted meatball mass histories during cooking.
57.4 g. Since fat transport was not considered in the model, the meatball mass history was simulated to account for moisture lost through evaporation only. The average standard deviation between the observed and calculated mass histories was about 0.04 g (Table 1). More work is needed to include the fat diffusion in the present models. The release of water due to protein denatura-
98
E. Huang, G. S. Mittal
tion became prominent as the temperature of the meatball increased above 50°C (Hung et al., 1978). However, minimal drip loss was observed. This observation suggests that the water released from the denaturated protein was absorbed by the water binder. A similar observation on the water-retaining effect of potato starch in meat loaf was also reported by Skjoldebrand and Hallstrom (1980) in their forced convection baking experiments. Nahlral convection baking
The average time required for the meatball center to reach 70°C was 2600 s (Fig. 1). The average standard deviation ranged from 3.95 to 6*37”C between the observed and predicted temperature histories. The oscillatory oven temperature, which fluctuated (10.7%) around the 140°C set point, could be the cause of high standard deviations. Compared with the time required for a meatball to be cooked by forced convection baking, natural convection baking required approximately 60% more time. This difference in cooking time was due to the lack of induced air during natural convection, which is also reflected in the low surface heat transfer coefficient of 9.0 W/m2 K, as compared with 23.0 W/m* K for the forced convection. The observed mass history (Fig. 2) indicates a slight increase in the short period immediately after the initiation of the baking. Up to 0.0 15% of the initial mass was gained due to the condensation of moisture on the meatball surface, during the preheating period. As the meatball surface temperature rose above the dew point temperature (39”(Z), the condensed moisture was soon evaporated. No constant moisture loss rate period was observed, which conforms to similar observations reported by Skjoldebrand (1980) and Mittal and Blaisdell ( 1982). The simulated mass history was successfully predicted, with an average standard deviation between 0.17 and 020 g (Table 1). Boiling
The temperature history followed an S-shaped response (Fig. 1). An average meatball boiling time was 770 s. The average standard deviation ranged from 2.73 to 3*59”C between the observed and predicted temperature histories (Table 1). The experimental moisture history established by moisture content analysis showed slightly ( < 1%) negative moisture loss. This observation suggested minimal water movement into the meatball during this process. Consequently, no simulation of the mass history of a meatball during boiling was conducted. It is believed that the binder in the meatball was responsible for retaining most of the water that would otherwise be lost as a result of the cooking. Bi-directional moisture movements might have taken place at the meatball surface, and hence contributed to a slight negative moisture loss ( < 1%). In general, it was observed that the initial predicted temperature histories tend to lag the experimental values. This phenomenon, common to all cooking processes, is believed to be caused by the relatively high heat transfer coefficient at the beginning of each process (Skjoldebrand, 1980). Shrinkage To assess the modeling assumption of negligible meatball shrinkage during cooking, measurements of cooked meatball diameters were taken for all cooking
Meatball cooking modeling
99
processes. Changes in meatbaIl diameter ranging from 0 to 9% were recorded. Despite these dimensional changes, however, negligible effect on the accuracy of the predicted cooking times was observed. Agrawal (1976) also did not find much difference in the predicted temperatures of meat emulsion products using shrink versus no shrink assumptions in his model for liquid diffusion. It seems that at shorter times the product did not shrink enough, and at longer times the gradients were not large enough to show significant differences. It appears that such a minimal shrinkage effect is restricted only to a limited group of foods with high water-holding capacity due to water-binding materials in the recipe (i.e. meat emulsion, minced meat), and is not widely observed in the cooking of other foods (Burfoot & James, 1984; Holtz & Skjoldebrand, 1986; Burfoot & Self, 1988). CONCLUSIONS The average h-values determined for meatballs during forced and natural convection baking and boiling were 23.0, 9.0, and 4518 W/m’ K, respectively. The h-values, which were determined by the transient temperature measurement method, were within 20% of the h-values calculated from empirical equations for baking and boiling. The cooking processes were modeled and solution techniques developed. The average thermal diffisivity values were l-8 x lo-‘, 1.9 x lo-‘, and 1.6 x 10e7 m2/s, respectively, for meatball cooking during forced convection baking, natural convection baking, and boiling. The average moisture diffusivity values during forced convection baking and natural convection baking were 3.9 X 10 - * and 25 X 10 -8 m2/s, respectively. The good agreement achieved between the observed and predicted results demonstrated the feasibility in predicting meatball cooking time and moisture loss during cooking.
ACKNOWLEDGEMENT This research was supported Council of Canada.
by the Natural Science and Engineering
Research
REFERENCES Agrawal, Y. ( 1976). Modelling of experimental analysis of moisture and heat in emulsion products during smokehouse thermal processing. PhD dissertation, Ohio State University, Columbus, Ohio. AOAC (1990). OficiuZ Methods of Analysis. Assoc. Official Analytical Chemists, Washington, DC. ASHRAE (1985). Handbook of Fundamentals. Amer. Sot. of Heating, Refrigeration and Air Conditioning Engineers, Atlanta, Georgia. Bengtsson, N. E., Jakobsson, B. & Dagerskog, M. (1976). Cooking of beef by oven roasting: a study of heat and mass transfer. J. Food Sci., 41,1047-53. Bird, R., Stewart, W. & Lightfoot, E. (1960). Transport Phenomena. John Wiley and Sons, New York, p. 413.
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Burfoot, D. & James, S. (1984). Problems in mathematically modelling the cooking of a joint of meat. In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Anolied Science Publishers, London, DD.467-72. Bu&&, D. & Self, K. P. (1988). Predidtion of heating times for cubes of beef during water cooking. J. Food Sci. Technol., 23 (3) 247-57. Burfoot, D., Self, K. P., Hudson, W. R., Wilkins, T. J. & James, S. J. (1990). Effect of cooking and cooling method on the processing times, mass losses and bacterial condition of large meat joints. ht. J. Food Sci. Technol., 25,657-67. Hallstrom, B. ( 1990). Mass transport of water in foods - a consideration of engineering aspects. J. Food Engng, 12,45-52. Halsey, G. (1948). Physical adsorption on non-uniform surfaces. J. Chem. Phys., 16, 931. Holtz, E. & Skjoldebrand, C. (1986). Simulation of the temperature of a meat loaf during the cooking process. J. Food Engng, 5,109-2 1. Holtz, E., Skjoldebrand, C., Bognar, A. & Piekarski, J. (1984). Modelling the baking process of meat products using convective ovens. In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Applied Science Publishers, London, pp. 329-38. Hooke, R. & Jeeves, T. A. ( 196 1). Direct search solution of numerical and statistical problems. J. Ass. Comp. Mach., 8,212-29. Hung, G. C., Gordon, J. & Davis, H. T. (1978). Mechanisms of water loss of bovine semitendinous muscles dry cooked from frozen state. J. Food Sci., 43,119 l-5. Kreith, F. & Black, W. (1980). Basic Heat Transfer. Harper and Row, New York, pp. 137-9,464-93. Mittal, G. S. & Blaisdell, J. (1982). Moisture mobility in frankfurters. J. Food Proc. Preserv., 6,11 l-26.
SAS (1988). SAS Procedure Guide, Release 6.03. Statistical Analysis Institute, Cary, North Carolina. Scher, L. I., Fazio, P. & Hsieh, M. W. ( 199 1). Process design and analysis of dry cooking operations. Presented at CoFE’91, Chicago, Illinois, lo-12 March. Skjoldebrand, C. (1980). Convection oven frying: heat and mass transfer between air and product, J. Food Sci., 45,1354-g. Skjoldebrand, C. (1984). Introduction to process group A (frying, grilling, boiling). In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Applied Science Publishers, London, pp. 3 13-l 7. Skjoldebrand, C. & Hallstrom, B. (1980). Convection oven frying: heat and mass transport in the product. J. Food Sci., 45.1347-53.
Meatball Cooking - Modeling and Simulation E. Huang & G. S. Mittal” School of Engineering,
University of Guelph, Guelph, Ontario, Canada, Nl G 2W 1
(Received 16 November 1992; revised version received 7 September accepted 20 December 1993)
1993;
ABSTRACT Mathematical models were developed to predict the temperature and mass histories of meatballs (47 cm diameter, 60 g) during forced convection baking, natural convection baking (broiling), and boiling. The finite difference method was used to solve the simultaneous heat and moisture transfer equations using constant transport properties and the continuous system modeling program (CSMP). Surface heat transfer coeficients were determined by the transient temperature measurement method, with values of 230, 96, and 4518 W/m2 K, respectively, for forced convection baking, natural convection baking (broiling), and boiling. Thermal and moisture difisivity values for meatball cooking processes were estimated from the experimental data by minimizing the root-meansquare of deviations between the observed and predicted temperature and moisture histories. The thermal diffusivitiy values were 1.8~ JO- ‘, 1.9~ IO- 7, and 16 x IO- 7 m’/s for meatball cooking during forced convection baking, natural convection baking, and boiling, respectively. Moisture difisivities of the meatball for the first two processes were 39 x IO-” and 2.5 x lo-” m2/s, respectively. The average root-meansquare of deviations between the observed and predicted temperature histories (390°C) ranged from 30 to 51°C for the cooking processes, and between the observed and predicted meatball mass, accounted for the moisture Ioss only, rangedfrom 064 to OI9g.
NOTATION Bi c
Biot number Non-dimensional
*To whom correspondence
moisture
content
should be addressed. 87
E. Huang, G. S. Mittal Moisture diffusivity (m’/s) Fourier number Surface heat transfer coefficient (W/m’ K) Thermal conductivity of the meatball (W/m K) Latent heat of vaporization (J/kg) Moisture concentration, dry basis Number of data points Radial position (m) Meatball radius (m) Gas constant (1.978 g caI/g mol K) Relative humidity (decimal) Time (s) Meatball temperature at any time during cooking (“C) Absolute temperature (K) Thermal diffusivity (m’/s) Non-dimensional temperature Density of dry matter of the meatball (kg/m3) Root-mean-square of deviations for moisture Root-mean-square of deviations for temperature Non-dimensional radial distance Subscripts a Ambient Equilibrium, experimental e 0 Initial, center predicted P Surface i-9 Nodes INTRODUCTION Forced convection baking and natural convection baking (broiling) are classified as dry cooking processes, categorized by the low humidity and high processing temperatures, which typically result in crust formation on the food surface. Boiling, however, is considered a wet cooking process, at relatively lower processing temperatures ( < 1OOC) and high humidities, and promotes a soft and moist food texture, generally with minimal mass loss. Each cooking process involves simultaneous heat and mass transport phenomena, physical and chemical reactions such as protein denaturation, the Maillard group of reactions, and microbiological destruction (Scher et al., 1991). Holtz et al. ( 1984) in their study on meat loaf baking showed that the combination of high oven temperature and air velocity promotes moisture loss at the surface, resulting in thick crust formation in a relatively short time period. Natural convection baking (broiling) is realized primarily by radiative heat transfer from heating elements and the oven walls. Bengtsson et al. (1976) studied the heat and mass transfer in beef roasts and exhibited an inverse relationship between the moisture and temperature histories. Incidentally, no significant fat migration was reported.
Meatball cooking modeling
89
Boiling is performed at temperatures below or around 100°C in either water or soups (Skjoldebrand, 1984). The properties of liquid and vapor, and the heating surface conditions, such as roughness and wettability, are the main parameters that influence the heat and mass transfer coefficients in boiling (ASHRAE, 1985). Burfoot and Self (1988) reported that the dimensional changes of beef cubes during boiling substantially reduce the heating time. Holtz and Skjoldebrand (1986) developed a heat and mass transfer model to simulate the temperature histories in a meat loaf during forced convection baking. Burfoot and James ( 1984) modeled the roasting of a meat joint by considering radiative heat transfer and air convection as the main heat transfer mechanisms. Burfoot and Self ( 1988) developed a numerical model to predict the heating time of beef cubes (20 and 30 mm3) in water (65,75,85, and 95°C). Mass loss increased with water temperature up to 64°C. Burfoot et al. (1990) compared the heating/cooling times and mass losses of meat joints by convective, immersion, and pressure/vacuum processes. The heating/cooling times, in the order of increasing speed, were due to immersion, convection, and pressure. Thus, the objective of this study is to model, simulate and validate the heat and moisture transfer phenomena within a meatball during different cooking processes to predict temperature and mass of the meatball.
MODELING Heat is transferred mainly by convection (forced convection baking, boiling) and radiation (natural convection baking or broiling) from the heating media to the meatball surface, followed by conduction towards the geometric center. Meanwhile, moisture diffuses outward to the meatball surface, where it is in turn vaporized and lost to the surroundings through convection. The necessary assumptions were (i) the meatballs were homogeneous, isotropic, and spherical in geometry; (ii) initial temperature and moisture distributions in meatballs were uniform; (iii) ambient temperature and moisture were step functions of time; (iv) vaporization of water was restricted to the meatball surface only; (v) negligible meatball shrinkage; (vi) negligible effect of crust formation on physical properties; (vii) constant a and D,; and (viii) fat transport was neglected. Based on the above assumptions, the mathematical models characterizing simultaneous pseudo-one-dimensional heat and mass transport in a meatball during forced convection baking, natural convection baking, and boiling could be represented as follows (symbols are defined in the Notation section). Heat transfer aT
g=a
(
2aT
;z+z
a2T
1
(1)
Moisture transfer
(2)
E. Huang, G. S. Mittal
90
Initial and boundary conditions The initial temperature
and moisture distribution are assumed to be uniform: T(r,O)= T,;
Temperature
m(r,O)= m,
(3)
and moisture gradients at the meatball center are depicted by -i3T
= 0;
am ar r=I,=o
ar r=O
(4)
Energy balance at the meatball surface, accounting for convective heat gain at the surface from the heating medium, heat conduction from the surface into the meatball, and the latent heat of vaporization which removes heat from the meatball surface:
(5) The boundary condition of instantaneous moisture meatball surface with the environment is expressed by
of
m 1r= R= m, at t > 0, for boiling
at t > 0, for baking;
+,=m,
content equilibrium
(6)
Non-dimensional analysis: dimensionless temperature (8), moisture content (C), radial length (V) To simplifv the numerical calculations, temperature, length b ion-dimensional forms are d&ineh as: T- T
(+--...2 Z-T,’
Cc_. m-m,
moisture content and radial
qJu=I R
m,-m,’
Subsequently, the model (eqns ( 1)-( 6)) in non-dimensional
(7)
form becomes
!$;($~+2J
(8)
(9)
C(Y,O)=l;
e(Y,o)=o; kae,(cRaY
T,)
=h(T,--
C, = 0 at t> 0, for baking;
E
T,)+ ,,yLv
= =o; Y 0 $
g
= =o ‘y 0
(m, _ m,)
C, = 1 at t> 0, for boiling
(10)
(11) (12)
Meatball cooking modeling
91
Finite difference equations development A one-dimensional spherical finite difference framework, consisting of 10 concentric sheiks of equal thickness, was developed to model the heat and moisture concentrations in a meatball during processing. Eleven nodes in total, one at the center of each shell element, and the 1 lth on the outer surface, were assigned. Temperature and moisture content at each of these nodes were assumed to be representative of the entire element. Node 0 (geometric center) Using the boundary condition and central difference: (13)
(14) Nodes l-8 de. 2dt dC. I dt
izI-8
icI-8
-
8,+,-28,+-
2i2i+ 1 f&l
-
C~+~-2Ci+- 2i+ 2i- 1
(15)
(16)
ci-l
Node 9 (17)
(18) Node S (surface) By backward difference: *,=Bi.8,+208,+20D;p,;L, s
(C,-C,),
20+Bi
es= c,= e
= 0, 0
e
..
Bi*&+208, 20+Bi
for baking;
forbaking
,
forbohg c,=-- m” - me - 1, m,-m,
(19)
(20) for boiling
(21)
E. Huang, G. S. Mittal
92
METHODS
AND MATERIALS
Meatball preparation Beef portions from chuck and shoulder, obtained from the university’s abattoir, were ground through a plate with l-cm diameter orifices. A commercial minced meat recipe, which required 12.3 kg of coarsely grounded beef, 1575 kg of added water, and 1.125 kg of the hamburger binder/spice mixture (Griffith Laboratories, Scarborough, Ontario) for every 15 kg of batch was used for making meatballs. The binder/spice mixture contained toasted breadcrumbs, salt, flour, skim milk powder, spices, hydrolyzed plant protein, onion powder, and monosodium glutamate. These ingredients were mixed manually for 3 min, and ground through a plate with 0*5-cm diameter orifices. Samples of the prepared minced beef were withheld for chemical analyses. Subsequent chemical analysis (AOAC, 1990) results showed the composition of the minced meat as 64.50% water, 15.86% protein, 13.34% fat, 3.94% carbohydrate, and 2.36% ash. The prepared minced meat was portioned into packages of 5 kg each, and frozen at - 40°C for 15 h prior to storage at - 18’C until required for experiments. Frozen minced beef packages were thawed at 2°C for about 48 h prior to meatball preparation. A pair of commercial meatball shapers (4.7-cm inner diameter) was utilized to form meatballs weighing 60 f 0.05 g each. To equalize the temperature of the meatballs, all prepared meatballs were stored at 2°C for at least 2 h prior to each experiment. Temperature measurement A data acquisition and control system, which consisted of a data-logger, a CPU module, a serial interface (Labmate, CPU module Model 901; Sciemetric Instruments Inc., Nepean, Ontario), and a portable computer (Tandy 200, Radio Shack Inc., Barrie, Ontario), was assembled for the recording of data during cooking. A BASIC program was used to interface with the data acquisition system. Temperature and mass data were retrieved and stored at predetermined intervals (every 10 s during the 1st min, and every 30 s thereafter) until the completion of each experiment. Temperatures were measured with high-temperature rated copperconstantan thermocouples measuring 0.5 mm diameter at the junctions (Kapton Insulated Duowrap Parallel Duplex thermocouple wire, Therm0 Electric Canada Ltd, Brampton, Ontario). Each thermocouple wire was encased in a locm length of Teflon tubing for reinforcement. A thermocouple alignment stand was constructed to ensure accurate placement of thermocouples within a meatball. This alignment stand allowed precise vertical positioning of thermocouples. To ensure proper heat and moisture convection around each meatball, the meatballs were suspended by a specially constructed fine wire mesh rack. For meatball temperature measurements, thermocouples were placed at the center and 1.175 cm from the center of the meatball. A thermocouple was also placed in the proximity of the meatballs to monitor heating medium temperature. A meatball was considered cooked when its center reached 70°C. After cooking, the meatballs were sealed in plastic pouches and weighed prior to storage at - 18°C.
Meatball cooking modeling
93
Mass measurement Continuous mass losses during forced and natural convection bakings were recorded by two cantilever load-cells, which were constructed in full-bridge configuration to compensate for the temperature effect. No noticeable hysteresis within the experimental range (0 to 200 g) was observed during their calibration (R 2 = 0.999). Manual weighing of meatballs with an electronic scale during boiling was necessitated by the high buoyancy forces. Cooking procedures Forced convection baking
A commercial kitchen size multi-mode oven (400 cmX 380 cmx 600 cm, Deacor ‘Convection Plus’ Self Cleaning Wall Oven, Model W305C, Pasadena, California) was used. The convection heating mode (mode 5), in which the oven fan convects the heat generated by a heating element at the back of the oven cavity at a velocity of O+0.9 m/s, was used to achieve an oven temperature of 140+ 10°C. The temperature of the oven was set according to the measurements by a thermocouple located 24 cm in front of the oven fan. Natural convection baking or broiling
The same oven was utilized, with the cooking mode set to the baking mode (mode 2, heated top element only). The oven temperature of 140°C fluctuated ( f 1YC) due to the action of the built-in on-off oven temperature controller. Natural air flow induced by the temperature gradient in the oven was the sole means of convection assisted by radiative heat. Boiling
A thermostatically controlled bath circulator (Model E8, Haake, Frankfurt, Germany) was used. This circulator has a heating capacity of 1500 W, circulation rate of 15 changes/m& and fluid temperature control to within f 0-02°C. The bath vessel, measured 3 10 cm X 290 cm X 130 cm, has a capacity of 12 litres. For each experimental run, nine meatballs for heat and mass transport studies were simultaneously immersed in water at 90°C. Meatballs were retrieved at various times during boiling. Excess water on the meatball surfaces was removed by gently rolling on a large paper towel immediately after retrieval for approximately 2 s; then the meatballs were sealed in plastic pouches to prevent further moisture loss through evaporation. The mass of each meatball group was then recorded prior to storage in a freezer. Heat transfer coefficient determination The heat transfer coefficient was estimated by the transient temperature measurement method, according to the convective heat transfer equation in the integrated form, as shown by Kreith and Black ( 1980),
T,- T(t) T
_
e
T
=
I
exp(- (Bi)(Fo))
(22)
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E. Huang, G. S. Mittal
A solid aluminum sphere of 4.7 cm in diameter was constructed and used to experimentally estimate the effective surface heat transfer coefficients of the cooking processes. Dents, about 0*4-cm diameter and O-2-cm deep, were drilled over the entire sphere surface to simulate the roughness of the meatball surface. A thermocouple was located at the center of the model sphere to monitor the temperature changes. Simulation The finite difference equations with appropriate inputs were solved by using continuous system modeling program (CSMP) simulation language on a mainframe computer. The root-mean-squares of deviations between predicted and observed values of average moisture (a,,,) and central temperature (a, ) of the meatball at various times during cooking were calculated by j=N
C (G,p,j- c0.e,i)2 j=l
a,=
N
(23)
j=N
C(G,p.j- %e,j)* j=l
a,=
N
(24)
Thermal and moisture diffusitivies for the meatball undergoing various cooking processes were determined by minimizing or and a,. For minirnization the pattern search algorithm of Hooke and Jeeves ( 1961) was used. Only one replication for each cooking method was used for this purpose. Four other replications were used to validate the models.
RESULTS AND DISCUSSION Surface heat transfer coefficient Table 1 summarizes the surface heat transfer coefficients (h) for the aluminum spherical model, which were averaged from the results of five replications for each process. These experimental heat transfer coefficients are in good agreement with the calculated results by empirical equations according to Bird et al. (1960). Equilibrium moisture content The experimental minced meat moisture contents reported by Hallstrom ( 1990) were used to establish a model for the moisture isotherms. The Halsey equation (Halsey, 1948), among few models attempted, was selected on the
140 f 10 140*15 90 + 0.3
Temperature (“C)
0.5 0.5 100
RH PO)
0.7 f 0.2 -0 lf
Velocity (mls)
Processing conditions
23.0 9.0 4518
(W/m2 K)
18.8 11.5 4134
(W/m'K)
uh,, = surface heat transfer coefficient, experimental. bhca,c= surface heat transfer coefficient, calculated. ‘Pinal meatball masses based on moisture loss only. *Baking I = forced convection baking. ‘Baking II = natural convection baking. fValue estimated based on circulator manufacturer specification. gND = not determined.
Baking Id Baking II’ Boiling
Process
TABLE 1
1.8 x lo-’ 1.9 x lo-’ 1.6 x lo-’
(m;/s)
0.39~ 1O-y 0.25 x 1O-u NDs
D, (m ‘Is)
1620 2600 766
Cooking time (s)
Transport Properties and Simulation Results (See Notation Section for Symbols)
57-44 57.41 60.48
Final mass‘ (@
1.28-4.79 3.95-6.37 2.73-3.59
T
of
0.03-0.05 0.17-0.20 ND
Mass
Root-mean-square deviations
8 3 0%
8 5 00 3
g g
E. Huang, G. S. Mittal
96
basis of the highest coefficient of determination (R ’ = 0.995) using the NLIN procedure of the Statistical Analysis System (SAS, 1988): RH=exp
- 522241 RT g A
_ 1.0983 m,
(25)
Thermal diffusivity The results (Table l), show good agreement with the thermal diffusivity value for meat emulsion reported by Agrawal (1976) (152-X 10e7 m2/s at 93°C). In general, the addition of binder to raw meat decreased the specific heat of the mixture due to lower water content, which increased thermal diffisivity values (1.6 x 10e7 to 1.9 x 10m7 m*/s). The influence of processing temperature on the magnitude of thermal diffusivity is also shown, where the thermal diffusivities were higher in forced and natural convection baking processes (performed at 140 + 10°C and 140 f 15”C, respectively) than in boiling (90 + 0.3”C). Moisture diffusivity Incidentally, the descending magnitudes of heat transfer coefficients (23.0 and 9-O W/m* K for forced and natural convection bakings, respectively) were alsa reflected in the moisture diffusivity values (Table 1). Since negligible moisture change was observed during meatball boiling in water, the moisture diffusivity for meatball during boiling was not determined. Nevertheless, it was believed that bi-directional water diffusion could have occurred between the water and the meatballs, and have contributed to the slight gain ( < 1%) in the mass. Temperature and moisture histories simulations Forced convection baking
The rates of temperature change were initially sluggish due to heat transfer lag (Fig. l), but gradually increased as the cooking progressed. When the temperatures at the nodes were raised above the wet-bulb temperature, the rate of change declined slightly yet continued the rising trend toward the dry-bulb temperature. An average baking time of 1600 s was required to raise the meatball center temperature to 70°C (Table 1). The simulated temperature histories agreed well with the experimental data. An average slope of 0.996 for the relationship between predicted and observed temperatures was calculated. The average standard deviation varied between 1.28 and 4*79”C for four replications. These deviations were believed to be partly due to the constant thermal diffusivity and latent heat of vaporization values used in the model. Variations in environmental conditions, such as oven temperature and relative humidity, during experiments could also be responsible for the discrepancy. Dislocation of thermocouples was a possible source of experimental error. Holtz and Skjoldebrand (1986) likewise reported similar temperature-distribution histories in their meat loaf forced convection baking study. The observed mass history showed the combined moisture and fat losses during the baking process (Fig. 2). The average observed final meatball mass was 53.2 g, while the calculated value based on the final moisture content was
Meatball cooking modeling
97
200 Boiling
BOiliUg
simulated
observed *
BW ambiit
BpLing
Baking
Broiling
Broiling
Broiling
simulated
observed
ambient
simulated
ObscNed
- - ____
0
H
1.50
50
0
0
2ooo
1500
1000
500
2500
3ooo
Time, s
Fig. 1.
Observed and predicted meatball temperature histories during cooking (“C).
Baking simulated -
57 1 0
Fig. 2.
Baking Broiling observed simulated . -
Broiling observed .
I
I
I
I
I
I
5co
loo0
1500 Time, s
zoo0
2500
3ooo
Observed and predicted meatball mass histories during cooking.
57.4 g. Since fat transport was not considered in the model, the meatball mass history was simulated to account for moisture lost through evaporation only. The average standard deviation between the observed and calculated mass histories was about 0.04 g (Table 1). More work is needed to include the fat diffusion in the present models. The release of water due to protein denatura-
98
E. Huang, G. S. Mittal
tion became prominent as the temperature of the meatball increased above 50°C (Hung et al., 1978). However, minimal drip loss was observed. This observation suggests that the water released from the denaturated protein was absorbed by the water binder. A similar observation on the water-retaining effect of potato starch in meat loaf was also reported by Skjoldebrand and Hallstrom (1980) in their forced convection baking experiments. Nahlral convection baking
The average time required for the meatball center to reach 70°C was 2600 s (Fig. 1). The average standard deviation ranged from 3.95 to 6*37”C between the observed and predicted temperature histories. The oscillatory oven temperature, which fluctuated (10.7%) around the 140°C set point, could be the cause of high standard deviations. Compared with the time required for a meatball to be cooked by forced convection baking, natural convection baking required approximately 60% more time. This difference in cooking time was due to the lack of induced air during natural convection, which is also reflected in the low surface heat transfer coefficient of 9.0 W/m2 K, as compared with 23.0 W/m* K for the forced convection. The observed mass history (Fig. 2) indicates a slight increase in the short period immediately after the initiation of the baking. Up to 0.0 15% of the initial mass was gained due to the condensation of moisture on the meatball surface, during the preheating period. As the meatball surface temperature rose above the dew point temperature (39”(Z), the condensed moisture was soon evaporated. No constant moisture loss rate period was observed, which conforms to similar observations reported by Skjoldebrand (1980) and Mittal and Blaisdell ( 1982). The simulated mass history was successfully predicted, with an average standard deviation between 0.17 and 020 g (Table 1). Boiling
The temperature history followed an S-shaped response (Fig. 1). An average meatball boiling time was 770 s. The average standard deviation ranged from 2.73 to 3*59”C between the observed and predicted temperature histories (Table 1). The experimental moisture history established by moisture content analysis showed slightly ( < 1%) negative moisture loss. This observation suggested minimal water movement into the meatball during this process. Consequently, no simulation of the mass history of a meatball during boiling was conducted. It is believed that the binder in the meatball was responsible for retaining most of the water that would otherwise be lost as a result of the cooking. Bi-directional moisture movements might have taken place at the meatball surface, and hence contributed to a slight negative moisture loss ( < 1%). In general, it was observed that the initial predicted temperature histories tend to lag the experimental values. This phenomenon, common to all cooking processes, is believed to be caused by the relatively high heat transfer coefficient at the beginning of each process (Skjoldebrand, 1980). Shrinkage To assess the modeling assumption of negligible meatball shrinkage during cooking, measurements of cooked meatball diameters were taken for all cooking
Meatball cooking modeling
99
processes. Changes in meatbaIl diameter ranging from 0 to 9% were recorded. Despite these dimensional changes, however, negligible effect on the accuracy of the predicted cooking times was observed. Agrawal (1976) also did not find much difference in the predicted temperatures of meat emulsion products using shrink versus no shrink assumptions in his model for liquid diffusion. It seems that at shorter times the product did not shrink enough, and at longer times the gradients were not large enough to show significant differences. It appears that such a minimal shrinkage effect is restricted only to a limited group of foods with high water-holding capacity due to water-binding materials in the recipe (i.e. meat emulsion, minced meat), and is not widely observed in the cooking of other foods (Burfoot & James, 1984; Holtz & Skjoldebrand, 1986; Burfoot & Self, 1988). CONCLUSIONS The average h-values determined for meatballs during forced and natural convection baking and boiling were 23.0, 9.0, and 4518 W/m’ K, respectively. The h-values, which were determined by the transient temperature measurement method, were within 20% of the h-values calculated from empirical equations for baking and boiling. The cooking processes were modeled and solution techniques developed. The average thermal diffisivity values were l-8 x lo-‘, 1.9 x lo-‘, and 1.6 x 10e7 m2/s, respectively, for meatball cooking during forced convection baking, natural convection baking, and boiling. The average moisture diffusivity values during forced convection baking and natural convection baking were 3.9 X 10 - * and 25 X 10 -8 m2/s, respectively. The good agreement achieved between the observed and predicted results demonstrated the feasibility in predicting meatball cooking time and moisture loss during cooking.
ACKNOWLEDGEMENT This research was supported Council of Canada.
by the Natural Science and Engineering
Research
REFERENCES Agrawal, Y. ( 1976). Modelling of experimental analysis of moisture and heat in emulsion products during smokehouse thermal processing. PhD dissertation, Ohio State University, Columbus, Ohio. AOAC (1990). OficiuZ Methods of Analysis. Assoc. Official Analytical Chemists, Washington, DC. ASHRAE (1985). Handbook of Fundamentals. Amer. Sot. of Heating, Refrigeration and Air Conditioning Engineers, Atlanta, Georgia. Bengtsson, N. E., Jakobsson, B. & Dagerskog, M. (1976). Cooking of beef by oven roasting: a study of heat and mass transfer. J. Food Sci., 41,1047-53. Bird, R., Stewart, W. & Lightfoot, E. (1960). Transport Phenomena. John Wiley and Sons, New York, p. 413.
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Burfoot, D. & James, S. (1984). Problems in mathematically modelling the cooking of a joint of meat. In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Anolied Science Publishers, London, DD.467-72. Bu&&, D. & Self, K. P. (1988). Predidtion of heating times for cubes of beef during water cooking. J. Food Sci. Technol., 23 (3) 247-57. Burfoot, D., Self, K. P., Hudson, W. R., Wilkins, T. J. & James, S. J. (1990). Effect of cooking and cooling method on the processing times, mass losses and bacterial condition of large meat joints. ht. J. Food Sci. Technol., 25,657-67. Hallstrom, B. ( 1990). Mass transport of water in foods - a consideration of engineering aspects. J. Food Engng, 12,45-52. Halsey, G. (1948). Physical adsorption on non-uniform surfaces. J. Chem. Phys., 16, 931. Holtz, E. & Skjoldebrand, C. (1986). Simulation of the temperature of a meat loaf during the cooking process. J. Food Engng, 5,109-2 1. Holtz, E., Skjoldebrand, C., Bognar, A. & Piekarski, J. (1984). Modelling the baking process of meat products using convective ovens. In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Applied Science Publishers, London, pp. 329-38. Hooke, R. & Jeeves, T. A. ( 196 1). Direct search solution of numerical and statistical problems. J. Ass. Comp. Mach., 8,212-29. Hung, G. C., Gordon, J. & Davis, H. T. (1978). Mechanisms of water loss of bovine semitendinous muscles dry cooked from frozen state. J. Food Sci., 43,119 l-5. Kreith, F. & Black, W. (1980). Basic Heat Transfer. Harper and Row, New York, pp. 137-9,464-93. Mittal, G. S. & Blaisdell, J. (1982). Moisture mobility in frankfurters. J. Food Proc. Preserv., 6,11 l-26.
SAS (1988). SAS Procedure Guide, Release 6.03. Statistical Analysis Institute, Cary, North Carolina. Scher, L. I., Fazio, P. & Hsieh, M. W. ( 199 1). Process design and analysis of dry cooking operations. Presented at CoFE’91, Chicago, Illinois, lo-12 March. Skjoldebrand, C. (1980). Convection oven frying: heat and mass transfer between air and product, J. Food Sci., 45,1354-g. Skjoldebrand, C. (1984). Introduction to process group A (frying, grilling, boiling). In Thermal Processing and Quality of Foods, ed. P. Zenthen. Elsevier Applied Science Publishers, London, pp. 3 13-l 7. Skjoldebrand, C. & Hallstrom, B. (1980). Convection oven frying: heat and mass transport in the product. J. Food Sci., 45.1347-53.