Meat cooking simulation by finite elements

Abstract

The production of ready-to-serve meals has been in continuous increase in the last years. Notwithstanding, there is a lacking of accurate methods to predict cooking times to ensure final safe products of optimal quality. Mathematically, cooking can be considered as simultaneous heat and mass transfers between food product and oven ambient. Modelling of cooking, including radiation, convection, food surface water evaporation with crust formation and volume shrinkage, may contribute to a better understanding of the overall process. Therefore, it is important to simulate heat and mass transfer during food cooking, considering different operative conditions (e.g. oven temperature, air circulation, air humidity, etc.). The objective of this work is to solve and to evaluate the dependence of temperature and water content on process time, during cooking of meat pieces of different regular and irregular shapes. The process is simulated using finite elements software (FEMLAB). The three-dimensional model was created extruding the scanned image of a two-dimensional transversal cut of the real piece. The software generated the finite element mesh automatically. The proposed model considers two regions: core and crust, with variable physical properties and convective boundary conditions. The numerical results are validated against experimental ones, obtained in a discontinuous convection oven.

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MEAT COOKING SIMULATION BY FINITE ELEMENTS
E. Purlis1, V. O. Salvadori12*
1CIDCA – Fac. de Cs. Exactas, UNLP y CONICET
2MODIAL – Fac. de Ingeniería, UNLP
Abstract. The production of ready-to-serve meals has been in continuous increase in the last years.
Notwithstanding, there is a lacking of accurate methods to predict cooking times to ensure final safe products
of optimal quality. Mathematically, cooking can be considered as simultaneous heat and mass transfers
between food product and oven ambient. Modelling of cooking, including radiation, convection, food surface
water evaporation with crust formation and volume shrinkage, may contribute to a better understanding of the
overall process. Therefore, it is important to simulate heat and mass transfer during food cooking,
considering different operative conditions (e.g. oven temperature, air circulation, air humidity, etc.). The
objective of this work is to solve and to evaluate the dependence of temperature and water content on process
time, during cooking of meat pieces of different regular and irregular shapes. The process is simulated using
finite elements software (FEMLAB). The three-dimensional model was created extruding the scanned image
of a two-dimensional transversal cut of the real piece. The software generated the finite element mesh
automatically. The proposed model considers two regions: core and crust, with variable physical properties
and convective boundary conditions. The numerical results are validated against experimental ones, obtained
in a discontinuous convection oven.
Keywords: Meat cooking, Finite elements and Simulation.
1. Introduction
Meat cooking is a common operation not only in the industrial production of ready-to-serve meals but also in
the catering industry. From a quality point of view, the cooking process must provide a final product with some
specific characteristics (sensory properties, microbiological safety). The most important cooking requirement is
to achieve a final temperature of 71ºC or 15 sec at 68ºC in the thermal centre to ensure safety from
contamination by Escherichia coli O157:H7 (FDA, 1997). The cooking time can only be predicted with a
complete knowledge of the thermal histories of the product.
Food water content also plays an important role in the final characteristics of the cooked piece of meat.
Therefore, a complete mathematical model must take into account the mass transfer between the food and the
environment.
Most of the published models refer to regular shapes: hamburger patties as an infinite slab (Zorrilla and
Singh, 2000) or a finite cylinder (Ikediala et al., 1996), both of them without shrinkage; patties with radial
shrinkage (Zorrilla and Singh, 2003), meat loaves (Holtz and Skjöldebrand, 1986), meatballs (Hung and Mittal,
1995). Ngadi et al. (1997) worked with a real geometry modelling mass transfer of chicken drum during deep-fat
frying. Finite difference method (FDM) and finite element method (FEM) were used to solve the models. But
FEM is more appropriate to solve models involving irregular shapes.
The reported models show different difficulties in relation with the simplifications considered in the
resolutions: heat or mass transfer only, radiation contribution to heat transfer in the oven, loss of water because
of evaporation, porosity and volume variation during cooking, crust formation, etc. Chen et al. (1999) developed
* To whom all correspondence should be addressed.
Address: CIDCA, UNLP Calle 47 esq. 116 (B1900AJJ) La Plata, Argentina. E-mail: vosalvad@ing.unlp.edu.ar

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a 2-D axisymmetric finite element model to simulate coupled heat and mass transfer during convection cooking
of regularly shaped chicken patties, given actual transient oven air conditions and using variable thermal
properties. But a model including simultaneous heat and mass transfer with variable properties and considering
irregular shapes is not developed yet.
Very few models proposed the optimization of cooking process depending on different criteria, such as
product safety from a microbiological point of view and the maximization of overall quality (Erdoğdu et al.,
2003; Zorrilla et al., 2003).
The aim of this research is to develop a mathematical model that simulates simultaneous heat and mass
transfer in three-dimensional pieces of meat. The three-dimensional geometry is created by extruding an image
of a two-dimensional transversal cut of the real piece. Heat and mass conservation balances are solved using
FEM software (FEMLAB). The proposed model considers two regions: core and crust, with variable physical
properties and variable boundary conditions.
The numerical results are validated against experimental ones obtained in a discontinuous electrical
convection oven.
2. Theory
An appropriate mathematical model for cooking process must solve simultaneous mass and heat transfer
balances:
wD
t
w
m
2
∇=
∂
∂ (1)
Tk
t
T
Cp 2
∇=
∂
∂
ρ
(2)
Both balances were coupled according to physical properties, which depend on water content, as follows
(Miles et al., 1983; Sanz et al., 1987):
pw
ww
ρρ
ρ
−
+
=1
1 (3)
where ρw = 1000 kg/m3 and ρp = 1380 kg/m3.
wk 52.0080.0 += (4)
wCp 27391448 += (5)
The diffusion coefficient was considered constant, equal to 1 10-10 m2/s.
Experimental work had indicated that while cooking the pieces of meat suffered a considerable weight loss,
which is not taken into account in this model.

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During cooking, two zones were distinguished in the food: the core and the crust, being the principal
difference between them their water content. Since physical properties of both zones depend on water content,
both balances (Eqs. 1 and 2) were coupled. Therefore, the two zones were automatically considered in the
model.
Uniform initial conditions were considered:
0,,
0
=
∀= tzyxww (6)
0,,
0
=
∀= tzyxTT (7)
Boundary conditions were established according to results of previous experimental work, using the same
equipment under forced convection conditions (Purlis and Salvadori, 2005). As the oven was not humidified its
relative humidity was very low (0.5 – 1 %). In spite of this, previous experimental work had indicated that food
dripped water since the first minutes of the cooking process until the last one, probably due to protein
denaturalization. This water loss produced weight loss in the sample but it didn’t allow the surface drying.
Therefore, boundary condition for the mass balance during the whole process was constant:
s
ww = (8)
Three stages were considered for heat balance boundary conditions:
()
()
isoisos
isors
rs
ttforTT
tttforTThTk
RttTwherettforTtThTk
>=
<<-=∇
)9894.0=(595.33+25.0=)(,<-)(=∇
∞
2
∞∞
(9)
The first stage was the starting period, in which the oven temperature (T∞) varied in time with a linear
relationship. During the second stage, the temperature in the oven was assumed to be constant. In both stages a
constant value of the heat transfer coefficient was considered.
Finally, according to each experiment’s operative conditions (oven temperature, air speed, relative humidity,
shape and size of the sample) the surface of the food reached a temperature (Tiso), which remained constant for a
certain period. This temperature, close to 100ºC (Tiso = 100.5 ºC), corresponds to evaporation of water at the
surface. During this stage part of the water at the surface evaporated but the major part was dripped. When the
diffusion of liquid water from inside of the food to the surface is lower than the evaporation front advance the
boundary condition should be changed for a convective type one. Besides, a moving boundary condition which
describes the dynamic of the evaporation front must be applied. As a result, a continuous variation in the other
two zones (crust and core) should be observed. This situation was not registered in any of the experiments at
standard cooking times.
3. Materials and Methods
3.1. Samples
Whole commercial meat pieces (semitendinous muscle) obtained from a local supermarket were used to
perform the experimental cooking runs.

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The average initial weight of the samples was 600 g (length: 0.1 m; cross section: 5.6 10-4 m2) and the water
content was 0.75 (w. b.).
3.2. Cooking Experiments
Several cooking experiments were performed in order to validate numerical results. An electrical domestic
oven ARISTON model FM87-FC was used. This oven has seven different cooking modes. Two of them were
employed:
- Conventional (natural convection): The temperature can be set between 60ºC and 230ºC (maximum
temperature). Two heating elements are turned on and the flux of heat is uniform from the top and the bottom.
This mode of cooking is recommended for slow cooking. The results obtained from using this cooking mode are
not shown in this work.
- Ventilated (forced convection): The temperature can be set between 60ºC and the maximum temperature,
the heating elements and the fan are activated. As the heat flux is uniform in the entire oven, this mode is good
for roasted foods that require long cooking times.
In our experiments, the temperature of the oven was set to 180ºC. During the first 636 seconds the oven
temperature increases with a linear relationship until it reaches the established value (the average final value was
182.37ºC). The oven air was not humidified.
Samples with a uniform initial temperature, equal to 20ºC, were placed over a grill pan, placed in the centre
of the oven (Figure 1). The cooking time was established as the necessary time for the thermal centre to reach
71ºC according to the microbiological criteria applied in these situations.
Rigid (0.7 mm of diameter) and flexible (1 mm of diameter) T-thermocouples were placed in different
positions inside the food and in the oven, and connected to a data acquisition DASTC system (Keithley, USA) in
order to record the thermal histories. Heat transfer coefficient and surface temperature were measured with a
heat flux sensor Omega HFS23, which was adhered to the food’s surface. The average h value was 25 W/m2 ºC.
Figure 1. Piece of meat in the oven during cooking process.
The water content (before and after cooking) was measured determining the samples’ dry weight, drying
them in a vacuum stove at 80ºC. After cooking, several samples corresponding to different zones of the meat

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piece were selected to measure the water content profile. The weight loss was calculated by difference between
the raw and cooked samples.
3.3. Geometry modelling
First, an image of the meat piece transversal cut was captured. This image was then processed in a CAD
software (AutoCAD) with NURBS mathematics. The irregular contour was reproduced by a B-spline curve.
Secondly, the obtained curve was imported from AutoCAD to FEMLAB, where it was scaled up according to
the real dimensions of the food (Figure 2).
Finally, the curve is converted into a two-dimensional solid in FEMLAB and extruded to recover the real
three-dimensional geometry (Figure 3).
Figure 2. 2-D contour of the piece.
Figure 3. 3-D model.

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3.4. Mesh generation
In 3-D, the domain (meat piece) is divided into tetrahedrons (mesh elements) whose faces, edges and corners
are called mesh faces, mesh edges, and mesh vertices, respectively. The mesh vertices are sometimes called node
points. The mesh was created with the FEMLAB internal function meshinit. In this work, 2111 elements were
generated, with 621 nodes and 962 edges. Figure 4 shows the resulting mesh.
3.5. FEMLAB solving
Once the finite element mesh was generated, physical properties and operative parameters were set to solve
the problem. The solution was obtained in three stages according to the boundary conditions described in
Section 2. Both balances were solved simultaneously using a time step equal to 1 second. Shorter time steps
were not used because the time solution turned too high, even though the process was carried out in a PC AMD
Athlon 1.67 GHz 1 GB RAM.
Figure 4. Finite elements mesh
4. Results and Discussion
4.1. Mass transfer
Meat water content was measured at different cooking times. The initial value was 0.75 (w.b.). Experimental
values of water content corresponding to a middle transversal section of the piece of meat at the end of the
cooking are shown in Table 1.
These values indicate that the crust was more dried than the core, but it was not completely dried. The water
content profile in the core is flat. This indicates that dripping is uniform along the whole meat piece during
cooking. The weight loss also confirms the dripping theory. The final weight of the meat samples were 21% to
29% lower than the initial values. Volume contraction was also observed. Average reduction of 17% was
registered in the length of the principal axis.

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Table 1. Experimental water content at the end of cooking in a transversal section of the piece.
Position Water content (w.b.)
Crust (1-2 mm thickness) 0.55
Superior 0.66
Central 0.68
Inferior 0.65
Figure 5 shows the numerical water content profile, at time 3530 sec. The experimental surface value (ws) of
0.55 was used in the numerical simulations. When comparing the numerical results (Figure 5) with the
experimental ones (Table 1), important differences are observed. This shows that in order to be accurate, the
model should consider the dripping mechanism as a mass loss term in the mass balance. Also, results show that
water loss by dripping is higher than water loss by superficial evaporation. The model should also consider the
volume reduction.
The difference between the minimum numerical value (0.77) and the initial one (0.75), was probably due to a
computing numerical error.
Figure 5. Numerical water content profile
4.2. Heat Transfer
Figure 6 shows the thermal histories, both experimental and simulated. All the thermocouples were positioned
in the central transversal cut of the piece but at different height. The positions during the experiments were the
following:

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Te1: 0.0312 m from surface (approximate thermal centre), Te2: 0.002 m from surface, Te3: 0.001 m from
surface, Te4: surface.
The simulated curves corresponded to the following node position:
Ts1: 0.0312 m from surface (approximate thermal centre), Ts2: 0.0137 m from surface, Ts3: 0.0087 m from
surface, Ts4: 0.0037 m from surface, Ts5: 0.0007 m from surface, Ts6: surface.
Figure 6. Experimental and simulated thermal histories. Experimental positions: Te1: 0.0312 m from surface
(approximate thermal centre), Te2: 0.002 m from surface, Te3: 0.001 m from surface, Te4: surface. Simulated positions: Ts1:
0.0312 m from surface (approximate thermal centre), Ts2: 0.0137 m from surface, Ts3: 0.0087 m from surface, Ts4: 0.0037
m from surface, Ts5: 0.0007 m from surface, Ts6: surface.
After comparing numerical and experimental results we are able to affirm that the model makes acceptable
predictions of the cooking process. The temperature in the thermal centre is accurately predicted from 0 to 2200
seconds. Above that value the predicted temperature increases in a faster way than the experimental. The causes
of this error could be:
-Dripping: since thermal properties depended on water content, the water loss mechanism may affect the
solution of the energy balance.
-Volume reduction, which also affects the inner gradients of temperature.

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-The supposition of prescribed temperature in the last stage of the cooking. Although it is adequate to
consider a real temperature at the surface, this might overestimate the temperatures which are close to the
surface. This happens because the vaporization latent heat was not included in the heat balance.
5. Conclusions
The performed work has shown that the proposed method for modelling the real geometry of a three-
dimensional food with irregular contour represents satisfactorily the shape and volume of the food. Applying
this mechanism different processes in which food engineering is interested (cooking, refrigeration, freezing,
sterilization, drying) could be modelled, simulated and optimized properly.
The model used to predict cooking of a commercial piece of beef fitted acceptably to the real process, in spite
of the simplifications considered in this work.
Still, it is necessary to keep on working in the development of a more complete model that represents more
accurately the experimental results. This model should consider variable properties (water content and
temperature), volume shrinkage, water loss by dripping and heat flux associated to surface water evaporation.
Nomenclature
Cp Specific heat, J kg-1ºC-1
Dm Diffusion coefficient, m2s-1
h Heat transfer coefficient, Wm-2ºC-1
k Thermal conductivity, Wm-1ºC-1
t Time, s
tr Time necessary to reach pseudo-constant temperature in the oven, s
tiso Time necessary to reach constant temperature at the meat surface, s
T Temperature, ºC
T∞ Oven temperature, ºC
Tiso Isotherm meat temperature, ºC
To Initial temperature, ºC
Ts Surface temperature, ºC
w Water content, kg of water/kg of food (w. b.)
ws Surface water content, kg of water/kg of food (w. b.)
Greek symbols
ρ Density, kg m-3
ρw Water density, kg m-3
ρp Solid matrix (proteins) density, kg m-3
.

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Acknowledgments
This work was supported by grants from CONICET, ANPCyT (PICT 2003/09-14677) and UNLP.