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


7.62.54 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 TW TW T T T x—— ky——Cp–—W]dV[kx–—nxky—–ny]WdS 0 XX YY 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.




Credit:ivythesis.typepad.com


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