Research Articles Base: Mathematical Modeling

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. Credit:ivythesis.typepad.com

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 ...

Linear Programming in Mathematics

Linear Programming in Mathematics   The focus will be linear programming which fulfill the requirements of International/global market. The difficulty in designing and implementing global markets is oftentimes interdependence of technical and economic objectives ( 2006). From economic viewpoint, the global market must encompass common economic performance desiderata such as allocative efficiency. Thus, relying on existing market mechanisms known from other contexts when constructing new markets may result in poor efficiency (2005). The mechanism designer also has to account for the technical conditions of the target domain. Thus, for an example, in case of market for allocating Grid computing resources several conditions comprises of underlying environment in terms of Grid middleware and requiring potential Grid users and applications pointing to linear programs. The market should act as resource allocation manager, fulfilling requirements upon such manager as there allows introduction of precondition that international market apt for linear Grid has to be realized as global market, which the market cannot fulfill automated resource allocation as required by Grid resource management system. Different requirements from technical and economic viewpoints may lead to different and conflicting objectives as (2005) pointed out that “pure mechanism designer is likely to design an economic mechanism with high economic efficiency”. Aside, when constructing electronic markets, it is essential to consider different influences that arise from technical fundamentals, potential user requirements, business constraints, economic objectives. Each of these influences has profound impact on the outcome on the acceptance of the market ( 2003).   Furthermore, methodology procedure is developed for representing competitive and noncompetitive market structures in linear programming models. Arbitrarily close approximations to nonlinear forms in both the objective function and constraint set can be made without much loss of the computational efficiency of the simplex algorithm. The noncompetitive market structure may be used for measuring income at endogenous prices in competitive model and may serve as a constraint on that measure of income to represent certain classes of economic policies. Product substitution effects in demand can be approximated by a linear program. The demand structure can be transformed to take account of any shift in demand which can be represented by a rotation of the demand function.   The research will be applied for market engineering and economic analysis, the need to formulate multi regional single time period linear goal programming model for agricultural planning in developing economy. In addition to specifying different levels of input and output for each activity, the need to describe explicit crop interdependencies which account for rotational requirements. Constraints are developed for land, labor, water, machinery, fertilizer and capital resources. Economic aspects are addressed through the use of different production functions and the incorporation of several objectives from conventional models which stress maximization of “economic welfare” defined as efficiency through maximum social product without consideration for income distribution. The emphasis on income distribution is reflected in our model in delineation of regional employment goals. In addition, foreign exchange expenditures and regional demand satisfaction goals reflect the importance of limiting foreign trade deficit and providing basic nutrition for the population, efficient and rigorous technique will be based on linear programming as readily accessible with the advent of computers. For this research, basic principles of linear programming will be briefly examined and some practical applications for formulating sound market economics recommendations in different contexts will be explained, the research should facilitate the adoption of the linear technique by international market and economic professionals. The proposed approach uses the strategy of simultaneously minimizing the most possible value of the imprecise total costs, maximizing the possibility of obtaining lower total costs, and minimizing the risk of obtaining higher total costs. Consequently, proposed linear programming approach yields an efficient solution and overall degree of decision maker satisfaction with determined goal values.         Particularly, several significant management implications and characteristics of linear programming approach that distinguish it from other decision models will be presented, has become increasingly common to use mathematical programming methods for deriving economic equilibrium of supply and demand. Well-defined approaches exist for the case of single firm and for the case of many firms leading to mathematical programming-based algorithm for determining market equilibrium. To present market engineering as holistic approach for designing, implementing and introducing electronic markets, services can then be composed to customized market processes. The market itself can then be provided as service and integrated into existing enterprise applications, explore several domains where markets should be increasingly employed in the future for resource allocation, to provide incentives for market participants or for risk management.   Credit:ivythesis.typepad.com

Linear Programming in Mathematics

Linear Programming in Mathematics The focus will be linear programming which fulfill the requirements of International/global mar...

LINEAR PROGRAMMING IN LAYMAN’S TERM

LINEAR PROGRAMMING IN LAYMAN’S TERM         Linear Programming is a mathematical technique and operation that can be used in administrative and economic planning in maximizing the function of bigger and wide range of variables that signifies constrain. This function is widely used for complicated computation of broad spectrum of calculation and precise handling of data processing just like on a high speed computer, it is also used for engineering, design, industrial, military and research operation purpose and experimentation of new products or software development programming. Usually if there are missing links of inputs linear, programming is excellent that supersede the expectation of computation using sets of symbols and formula development of any given problems that needs accurate method or solution.        This computation requires a thorough knowledge and trial and error to highly develop sets of variables in problem solving which is logical to understand that the brainchild of a computer understand and  function through this sets of given data that transform into a command to execute from. To be able to fully illustrate such function and maximize its potential we will thoroughly discuss it in layman’s term so that in itself is effective. We are also going to express some example that you may need to re read to fully understand such expression. The easiest method would be to analyze the data and put it on the table you can also use Microsoft excel as shown below or simply create an equation that is according to the situation.   TOY COMPANY         My company can produce a different kinds of toys let us say a Teddy Bear, Rabbit, Panda and Doggy. The minimum number of toys produced per day are TBmin, Rmin,Pmin and Dmin. The maximum numbers are stated as TBmax, Rmax, Pmax and Dmax. The prices of the toys are TBp, Rp, Pp and Dp: Toys Min number Max number Price, $ TB 100 200 5 R 95 190 6 P 90 180 7.0 D 95 190 7.0       I have used the following machines to produce the toys:  M1, M2 and M3. Each machine can operate from Tmin to Tmax hours per day: Machine \ Time Tmin, hrs Tmax, hrs M1 20 22 M2 20 22 M3 20 22       The time required to produce each toys on each machine (T) are given in the table (in minutes): Machine \ T TB R P D M1 10 12 10 18 M2 10 14 11 19 M3 10 10 12 20       So, time to produce toys TB on machine M1 is denote as T_TB_M1 and so on. The following amounts of toys have been required to maximize the profit. The decision variables are amount of toys of each type: TBx, Rx, Px and Dx.       Although there are several constraint we are going to focus on the following constraints:  T_TB_M1 * TBx + T_R_M1 * Rx + T_P_M1 * Px + T_D_M1 * Dx >= Tmin for M1 T_TB_M1 * TBx + T_R_M1 * Rx + T_P_M1 * Px + T_D_M1 * Dx <= Tmax for M1 and etc.       The objective function is sum of multiplications of devices amounts and their costs: we are going to measure its profitability which is equal to F and c is equal to the cost. F = TBx * TBc + Rx * Rc + Px * Pc + Dx * Dc       You can now create several questions and formula using the data above. Or using tables you can easily figure out future production and monitor its minimum or maximum output to give simple expression for example.        Let us say I am going to produce a Teddy bear in a minimum amount as ordered by the customer using machine 1 how much would I earn? The formula would be T_TB1_M1 / T_Tmin_M1 X TBp which is equal to 10 / 20 x 5 first we need to convert 20 to minutes which is 60 X 20 since 60 minutes is equal to 1 hour therefore 1200 should be in the final equation would be the following: 10/ 1200 x 5 = 0 dollars will be the minimum earnings using the minimum hours using the machine 1 in this question.       This question has been the simplest question but if you need to compute the maximum amount it would be very easy. You can also add up many constraints depending on the particular production of works involved but the table has been very useful in determining the basic formula. In a more sensitive approach which is the average linear programming the following table can be introduced using graphical interface that can be use for special training or basic presentation but the equation would be the same.   Credit:ivythesis.typepad.com

LINEAR PROGRAMMING IN LAYMAN’S TERM

LINEAR PROGRAMMING IN LAYMAN’S TERM Linear Programming is a mathematical technique and operation that can be used in administrati...

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