Literature review of mathematical modeling of the process of heating of the turkey in the oven

 

Modeling Heat Transfer During Oven

Roasting of Unstuffed Turkeys

ABSTRACT

A finite element method was used to solve the unsteady state

heat transfer equations for heating of turkeys in a conventional

electric oven. Breast, and thigh and wing joint temperature

in 5.9, 6.8, 8.6, 9.5, and 10.4 kg turkeys were simulated.

A surface heat transfer coefficient of 19.252 W/m2K

determined by transient temperature measurements in the

same oven, was used. Thermal conductivity measured using

a line heat source probe from 0 to 80

°C was 0.464 W/mK.

Simulated temperatures were within 1.33, 1.47, and 1.22

°C

of experimental values of temperature in the breast, thigh,

and wing joint, respectively. Initial temperature 1 , 2, and 3

°C

lower than 4

°C required additional baking time of 16, 22, and

27 min., respectively for the thigh joint to reach the target

endpoint temperature.

Key Words: turkey, heat transfer, mathematical modeling,

thermal conductivity, finite element

INTRODUCTION

A CLEAR UNDERSTANDING OF THE DYNAMICS OF CHANGES IN

food product properties during processing treatments is required to

maintain the desired quality, texture, and sterility. The finite-element

method has been successfully applied for understanding, describing,

and analyzing food-processing operations. Many works have been

reported using mathematical modeling to simulate roasting of meat.

 (1976) modeled beef roasting and compared the results

with experimental measurements. They found that varying partial

pressure of water vapor during the roasting process affected rate

of moisture loss by evaporation. Similar results were also reported

by  (1977a, b), . (1978) and Singh et al.

(1983).  (1977) showed how heat transfer problems

may be solved using the finite element method in roasting beef

and chicken.

The unstuffed turkey can be a potential carrier of foodborne illnesses

associated with infections by Salmonella. The best means of

control is ensuring that temperatures attained are adequate to inactivate

pathogens. Our objective was to simulate heat transfer of unstuffed

turkeys in order to identify sources of variation in cooking

times needed to achieve a specified endpoint temperature at critical

points in baked whole turkeys.

MATERIALS & METHODS

Heat transfer coefficient estimation

Intact turkey muscles were made into 10.2

∞7.6∞2.54 cm brickshaped

samples. The simple geometry simplified calculations of the

surface heat transfer coefficient. The sample was introduced into the

oven (Model 130, Daycor, Pasadena, CA) at 162.8

°C, with convection

fan disabled. The meat temperature was measured with a copper-

constantan thermocouple in the center of the sample block. Thermocouple

output was acquired and recorded through a Model DASTC

interface card () and a personal computer.

Temperature data were stored at 30 sec intervals for 1.5 h. The experiment

was repeated four times. Recorded data were imported into

a spreadsheet and analyzed for the slope of the semilogarithmic heating

curve, from which, the effective mean surface heat transfer coefficient

was calculated ( 1991).

Thermal conductivity determination

Thermal conductivity was measured using the line heat source

method ( 1991). Thermal conductivity of turkey

muscle was measured at 0, 20, 40, 60, and 80

°C. Turkey breasts

were cored into cylindrical samples 2.54 cm dia.

∞ 15 cm. The thermal

conductivity probe was inserted into the center of the sample.

When initial temperatures between samples and surrounding environments

had equilibrated, the probe was energized for 25 sec while

temperature in the probe was continuously monitored using the DASTC

temperature acquisition card. Data were retrieved and imported

into a spreadsheet. The slope of the linear portion of a plot of temperature

rise (T) vs ln(t) and the level of energy input were used to

calculate thermal conductivity of the samples as follows

1976):

k = C q / 4 (slope)

where q, the rate of energy input, is:

q = 2 I 2 R

We used a current (I) of 0.130 A , and the heater wire resistance

(R) was 106.9 Ohms. The instrument was calibrated using glycerin

at room temperature (23

°C) , and the calibration constant, C, was

0.948.

The plots of T vs ln time had high correlation coefficients

(1<r2<0.99) for the linear portion which was between 10 and 25 sec

after energizing the heater. The same sample was used for measurement

at all temperatures. Samples were immersed in a water bath set

at the designated temperature. Two water baths were used. After measurement

was completed at one temperature, the sample was transferred

to the other water bath maintained at the next desired temperature.

All measurements were replicated 4 times at each temperature.

Model equations for heat transfer, initial and boundary

conditions

The equations were for two-dimensional unsteady state heat transfer

with surface conduction and evaporation. To reduce the complexity

of the problem, the turkey was assumed to be an infinitely

long column with a cross-section (Fig. 1). Breast temperature was

calculated in the thickest section in the upper part, wing joint temperature

was calculated in the thickest section of the lower left side,

and thigh joint temperature was calculated at the bend in the lower

middle part of the figure. The assumption of two-dimensional heat

transfer was justified by actual temperature vs time curves where

temperature recorded in the breast at positions 2.54 cm from each

side of a point on a plane perpendicular to the plane (Fig. 1), at the

same depth from the surface, were practically the same (1997).

The following assumptions were used in formulating the finiteelement

model for temperature changes at specific points during roasting:

(1) Water at the surface behaves like free, or unbound, water as

long as sufficient water is available; and (2) There is no internal

movement of water by convective flow and diffusion. Only heat transfer

was considered within the meat while evaporation was consid-

ered a surface boundary condition. The differential equation for heat

transfer in two dimensions is:

—

≤—T = a ≤ T ≤ T x—— + ay—— (1) ≤ t ≤ x ≤ y

X and Y are the spatial dimensions, t is time, T is temperature,

=k/Cp is the thermal diffusivity of the meat, and the subscripts x

and y allow the directional variation of thermal conductivity to be

incorporated into the model. k is the thermal conductivity, Cp is specific

heat, and is density. Equation (1) is valid as long as there is no

internal heat generation (1983).

The initial condition is T = T0, when the turkeys were introduced

into ovens at uniform temperature distribution of T0

°C. The surface

boundary condition was that the conductive heat transfer equaled

the sum of heat input by convection and heat removed by the latent

heat of evaporation at the meat surface. Surface heat transfer to the

meat, and convection gain or loss through the surface is given by

(

1976):

h(TT

≤ T ≤ T )q (kx——nxky——) (2)

≤

X ≤Y

where nx and ny are the direction cosines; h is the surface heattransfer

coefficient; T is the fluid (air) temperature surrounding the

body; and q is a boundary heat source, which is the latent heat of

evaporation. It was assumed that , Cp, q and h were rotationally

symmetric. The term q in Eq. (2) is

q = W Lv (

≤ m/≤ t)

where the W are weighting functions in the Galerkin finite element

formulation, using the shape functions Ni and Nj (Segerlind, 1984);

Lv is the latent heat of vaporization, and m is moisture concentration,

dry basis. The rate of moisture loss

≤ m/≤ t was modeled based

on an average evaporative moisture loss of 8.6% obtained during

cooking of 30 whole unstuffed turkeys. This moisture loss was prorated

over the total cooking time by using a mass transfer coefficient

and the vapor pressure of water at the surface temperature of the

turkey.

The Galerkin Residual Method (

 1956;

1973) was used to transform eqn. 2 into a finite element form. A trial

function for T, which satisfied the boundary conditions, was substituted

into Eq. (2) and the resulting residual was made orthogonal

with respect to a weighting function W.

≤ ≤T ≤ ≤T ≤T v[—–(kx—–)—–(ky—–)Cp—–] W d V 0 (3) ≤X ≤X ≤Y ≤Y ≤ t

Integrating Eq. (3) by parts and applying the divergence theorem

yields:

[k ≤T≤W ≤T≤W ≤T ≤ T ≤ T x—— ky——Cp–—W]dV[kx–—nxky—–ny]WdS 0 ≤X≤X ≤Y≤Y ≤t ≤ X ≤ Y (4)

The solution domain was subdivided into smaller elements to

which Eq. (4) was applied. The variable temperature Tc in each element

was approximated as a function of the temperature values at

the nodes (Segerlind, 1984). Tc = T1N1 + T2N2 + T3N3 , where T1 ,

T2 , T3 are the nodal temperatures and N1 , N2 , N3 are element shape

functions derived from the geometry of the element. The weighting

function W was made identical to the interpolating functions N1 , N2

, N3. The equation for Tc in vector form was substituted into Eq. (4)

to form a matrix for each element. The differential equation in matrix

notation is: [K}{T}[C]

≤ {T}/≤ t{F} 0.

The global stiffness matrix [K], and capacitance matrix [C], are

square matrices which includes the thermal conductivity, specific

heat, and are dependent on the element geometry. The force vector

{F} is a column vector of values of the heat input, while {T} is a

column vector of the unknown nodal temperatures ( 1984).

The differential equation was solved using the weighted residual

method with linear time elements (1974). The procedure

calculated a temperature at the present time {T}1, from temperature

at a previous time {T}o using a variable time-step size t which

satisfied the following:

(2/3)[K](1/t)[C]{T}1 (1/3){F}o(2/3){F}1([K]/3[C]/t){T}o

where {F}o and {F}1 are heat inputs at the beginning and end of the

time step.

Generation of experimental time-temperature data

Data were extracted from results of a previous study (

1998) that recorded time-temperature histories at different parts

of the turkey on a total of 126 birds baked in a conventional oven at

162.8

°C. We used the data on temperature histories of 15 each of

fresh or previously frozen/thawed turkeys which were baked unstuffed

and unsheilded. The turkeys ranged in weight from 5.8 to

10.4 kg with 3 fresh or previously frozen birds in each of 5 weight

categories within this range. The weight categories were designated

as a range to guide the suppliers on the weight distribution of the

turkeys required. Exact weight of each bird was considered when

evaluating temperature histories and moisture loses. Procedures for

preparing the turkeys and cooking were described by

(1998). Detailed time-temperature histories at 1 min interval throughout

the cooking and hold period after cooking, for each bird at 8

positions in the bird, have been reported (1997). Temperature

data at three thermocouple locations were extracted and used as

the experimental data for model verification. These locations were

at the breast in the customary insertion point for pop-up timers, 4.13

cm deep from the surface. The thermocouples at the wing and thigh

joints were inserted in the bird perpendicular to its outside surface

1.25 cm towards the surface from the bones forming the wing and

thigh joints.

Model validation

A computer program written in C was used to solve the two-dimensional

field finite element equation with the given appropriate

inputs. To account for bird weight in the simulations, size factors

were determined using the ratio of average flesh thickness between

the 5.9 kg bird and those in the other weight categories. The heat

transfer model was validated against the average observed temperature

profiles. The root-mean-square of deviations between predicted

and observed values was calculated by:

(
j=1°N (Tp,j To,j)2 (n 1)2)°

where Tpj and Toj are predicted and observed temperatures, respectively,

at a specific time. A paired-T test was performed to determine

the difference between predicted and observed values (

1985)

RESULTS & DISCUSSION

Surface heat transfer coefficient and thermal

conductivity

The effective surface heat transfer coefficient, under the same

conditions used in the cooking of the turkeys, was 19.252 W/m2K.

This value is an effective heat transfer coefficient which combines

convection and radiation. Thus, the oven temperature, radiant heating

element positioning, and temperature cycling in the oven would

affect this value. We obtained a higher value than that reported by

 (1993), which could be attributed to differences in

the type of oven used. Thermal conductivity was 0.461, 0.464, 0.464,

0.462, and 0.468 at 0, 20, 40, 60 and 80

°C, respectively. The effect

of temperature was not significant. The average, 0.464 W/m K, was

used in the heat transfer simulation.

Heat transfer simulation results

Computer simulated temperature histories in the breast, thigh joint,

and wing joint were calculated based on an initial temperature at

4

°C. The size factors for the different weight ranges were evaluated

using a ratio of average flesh thickness, which included the breast,

thigh, and wing joint muscles, in the smallest weight turkey (5.9 kg)

to that of larger birds. The turkey shape was obtained using a cross

section of the turkey in the customary timer position which was cut

parallel from 2.54 cm below the keel bone. This cross-sectional area

included the thigh joint and wing joint positions (Fig. 1).

The simulated temperature histories agreed with the experimental

data (Fig. 2 to 6). Paired-t test confirmed that there were no significant

differences between predicted values and individually observed

values (p>0.05). The root mean square deviations between

predicted and observed temperatures in the breast, thigh joint and

wing joint, respectively, were 1.33, 1.47 and 1.22

°C (Table 1).

The breast temperature always reached 82

°C in less time than the

temperatures at thigh joint and wing joint in an oven at 162.8

°C

(Fig. 7B). However, temperatures at the thigh joint and wing joint

reached 82

°C almost at the same time. Thus, if the initial temperatures

in the breast, thigh joint and wing joint were at 4

°C, when the

thigh joint reached 82

°C, the wing joint could be > 71°C, the temperature

required to kill Salmonella.

The initial turkey body temperature has a strong influence on the

total time required to bake turkeys. The initial temperature in the

breast, thigh, and wing joint muscles may not be the same because

of differences of flesh thickness in these positions. When initial breast

temperature was 4

°C, the thigh joint and wing joint temperatures

were < 4

°C. Thus, additional heating time may be needed for the

thigh and wing joint to reach the desired endpoint temperature. Heating

time needed for the thigh joint and wing joint to reach 4C from

lower initial temperatures in the first 40 min of baking, was simulated.

An average 118

°C oven temperature was used for simulations

because in the initial phase of baking, birds were introduced into a

cold oven and timing was started when the oven was energized. The

average measured oven temperature from ambient to 40 min after

being energized (set to 162.8

°C ) was 118°C. The thermal conductivity

used in the simulation was 0.461 W/m K, which was measured

at 0

°C, and close to 1 to 4°C.

Results of the simulation for different initial temperatures in the

thigh joint muscle were compared (Table 2). The time needed to increase

temperature of the thigh joint was 13 to 16 min from 1 to 4

°C,

18 to 22 min from 2 to 4

°C and 21 to 27 min from 1 to 4°C. There

were no differences between simulated and observed values by paired

T-Test (p>0.05). The time for bringing-up bird initial temperature at

the wing joint was 18 to 22 min from 3 to 4

°C, 22 to 28 min from 2

to 4

°C and 28 to 34 min from 1 to 4°C (Table 3). Paired T-Test validated

no significant differences between estimated times and observed

values (p>0.05).

Cooking times and oven temperatures

A higher oven temperature would shorten processing time (Fig.

7). There was about 50 min difference in total cooking time between

the 148.9

°C baking temperature and 176.7°C baking temperature.

However, a higher oven temperature for roasting turkeys might result

in a darker color and a dryer texture in the breast, because the

breast could be exposed to the high temperature for at least 25 min

longer after the thigh joint temperature reached 82

°C at the time of

removal from oven (Fig. 7C).

CONCLUSIONS

A TWO DIMENSIONAL FINITE ELEMENT MODEL ADEQUATELY

modeled temperature in the breast, thigh joint and wing joint during

baking of whole unstuffed turkeys. Surface heat transfer was repre-

sented by a measured heat transfer coefficient which combined convection

and radiation effects. Surface energy loss from evaporation

was quantified from average evaporative loss and prorated over the

entire baking time using the vapor pressure of water at the surface

temperature of the turkey. Simulation results revealed that increasing

oven temperature reduced baking time but resulted in breast temperature

reaching the designated endpoint much earlier than what

would be required for thigh joint temperature to reach the endpoint.

Initial temperature at critical points in the turkey had a strong influence

on baking time. Storing turkeys at 4.4

°C prior to baking did not

ensure initial temperatures of 4

°C, and initial temperatures lower

than 4

°C can increase cooking time to the 82°C endpoint temperature

at the critical points by 18 to 34 min.

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