Determination of Cold Spot Location for Conduction-Heated Canned Foods Using an Inverse Approach

10 International Journal of Food Processing Technology, 015,, 10-17 Deterination of Cold Spot Location for Conduction-Heated Canned Foods Using an Inverse Approach Ibrahi O. Mohaed * Food Engineering Consultant, P.O.Box 87950 Al-Ain, United Arab Eirates Abstract: Cold spot location is a focal point in theral process lethality calculations for conduction heated canned foods. An inverse heat conduction proble (IHCP) approach was used to odel heat transfer for canned food with headspace using a one diensional heat conduction odel. Sequential function specification algorith was used to solve for heat fluxes history at the can headspace side using known internal transient teperature easureents. The estiated heat fluxes were then used to solve the direct heat conduction proble for the teperature profiles in the axial direction including at the sensor position. Deviations between estiated teperatures and easured teperatures at the sensor position were calculated using root ean square error. Fro all the treatents used, a axiu error of 0.9 o C was obtained for the whole easureent period of the treatent which is well within therocouple easureent error. The excellent agreeent between the easured and calculated teperatures at the sensor position is an indication of an accurate estiation of heat fluxes and subsequent location of the cold spot. The results revealed that the cold spot is located at about 59 % of the odel food height fro the botto of the can for the three levels of headspace investigated (10%, 14% and 0%). The result of this research provides a useful guide for accurate location of cold spot when collecting heat penetration data for conduction heated foods with headspace predoinantly occupied by water vapor. This will aid in assuring the safety of canned foods based on accurate calculation of theral process lethality. Keywords: Headspace, Lethality, Cold spot, IHCP, Theral sterilization. INTRODUCTION The classical ethod of evaluating the lethal effect of theral sterilization process is based on the heat penetration paraeters (f h, f c, j h, j c ) proposed by Ball [1]. The heat penetration paraeters are often deterined fro tie-teperature easureents at the geoetric center of a can which is often assued to be the slowest heating point (cold spot). The tie-teperature at the cold spot can also be deterined fro nuerical solutions of the governing heat conduction equation in cylindrical or other geoetries subjected to prescribed boundary conditions. The accuracy of the prediction of the tie-teperature history at the cold spot depends on how accurate the prescribed boundary conditions describe the actual theral process conditions. Different boundary conditions were used by various researchers to solve the heat conduction equation in different geoetries for teperatures at any position and tie. The boundary conditions of the first type are frequently used for siulating the theral sterilization process [-7]. The boundary conditions of the third type are also used in siulating theral sterilization [8-10]. However, the boundary conditions used by these researchers did not take into consideration the insulating effect of headspace due to lack of inforation on heat transfer coefficients at the headspace side of a can. However, the data provided by Mohaed [11] for heat transfer coefficients for can * Address correspondence to this author at the Food Engineering Consultant, P.O.Box 87950 Al-Ain, United Arab Eirates; Tel: (971)-3-767479; Fax: (971)-37675336; E-ail: io106@yahoo.co E-ISSN: 408-986/15 headspace could be useful in describing realistic boundary condition. Other reported studies related to headspace focused on experiental investigation of the effect of headspace on heat penetration paraeters and cold spot location for cans [1-14], or for flexible packages and sei rigid containers [15-17]. The cold spot location was reported to shift upward fro the geoetric center towards the top surface of the food for flexible packages with entrapped air [15, 16] and for sei rigid container [17]. Varga, Oliveira [18] used atheatical odeling to investigate the effect of four processing factors including headspace volue variability on theral process lethality (Fvalue). They found that variability of.5 % in headspace did not produce any significant effect on the F-vale. Recently, Khakbaz Heshati et al. [19] carried out research work to investigate the effect of headspace on cold spot location using nuerical solution with convective heat transfer coefficient at the headspace side. They found that cold spot is shifted fro the center to the top of the jar for the two level of headspace investigated naely 6% and 10%. The ais of this research are to present a systeatic approach for odeling heat transfer for can with headspace coprise predoinantly of water vapor and to deterine the location of the cold spot for different headspace sizes. INVERSE HEAT CONDUCTION ALGORITHM Estiation of teperature at any position and tie fro a governing heat conduction differential equation 015 Cosos Scholars Publishing House

Cold Spot Location for Conduction-Heated Canned Foods International Journal of Food Processing Technology, 015, Vol., No. 11 and prescribed boundary and initial conditions are known as the direct ethod. While deterination of the boundary conditions, initial condition or theral properties fro transient teperature easureents is known as an inverse heat conduction proble (IHCP). There are several pioneer algoriths proposed for the solution of the IHCP including, the exact atching algorith [0] the function specification algorith [1-3] and the regularization algorith [4]. In the present study the function specification algorith will be used to solve for the unknown surface heat flux at the headspace side of a can and then the direct proble will be solved for teperature as function of tie along the axial direction to locate the cold spot. Since cold spot for cylindrical can is located along the central axis due to syetry in the radial direction, it is sufficient to consider one-diensional proble with teperature variation in the axial direction to locate cold spot as the radial direction has no influence on the location of the cold spot. To accoplish the requireent of a unidirectional heat conduction in the axial direction for cylindrical can geoetry, the boundary surface noral to the radial direction should be well insulated, this will help in eliinating the ter that contain teperature gradient in the radial direction in the Fourier's heat conduction equation. Once the ter that contains teperature variation in the radial direction is excluded, the proble will becoe siilar to unidirectional heat transfer in slab geoetry. For a can with no headspace, the boundary condition at the top and at the botto of the can ay be identical leading to cold spot at the geoetric center of the can. When headspace is present, the boundary condition at the top and the botto of the can will be different leading to shift in cold spot location fro the geoetric center along the central axis. The agnitude of the shift depends on the size of the headspace and the coposition of gases and water vapor in the headspace. The heat transfer odel to conduction canned food with headspace and negligible teperature variation in the radial direction can be expressed as: at z = L T k = q(t) z Estiation of the surface heat flux q can be obtained fro iniization of the following su of squares function: S = r i= 1 ( ) Y T i1 i1 Where, is index for discrete tie and r nuber of future tie intervals for which heat flux is assued to have certain functional for. Miniization of the above su of squares function can be easily perfored using different functional for assuption for heat flux for tie t to t r-1 such as constant, linear, cubic, parabolic or other for. Using the assuption of constant heat flux Beck, Blackwell [5] developed the following function specification algorith for calculating heat flux: qˆ r i1 i= 1 = qˆ 1 + r ( Y T ) i= 1 X i1 i1, X i1, Where X i-1, is the sensitivity coefficient defined by X i, T = q i1, () (3) 1 (4) Fro the governing heat conduction equation and the prescribed boundary conditions the following are the equations for the sensitivity coefficients. X t X = k z c With initial and boundary conditions: at t = 0 X(0,z) = 0 (5) T t T = k z c With initial and boundary conditions: at t = 0 T(0,z) = T o at z = 0 T(t,0) = T h (botto of the can) (1) at z = 0 X(t,0) = 0 at z = L X k z = 1 COMPUTER PROGRAM A FORTRAN coputer progra was written to solve nuerically equations (1) and (5) based on Crank-Nicolson iplicit finite difference discretization.

1 International Journal of Food Processing Technology, 015, Vol., No. Mohaed Equation (3) is also incorporated into the progra for calculating the heat flux at each tie step. Once the heat flux is known, the proble becoes a direct proble and the progra then solves the direct proble for the teperature distribution in the axial direction. The nubers of nodes used were 100 and the tie increents were one second. The progra can handle different values for the nuber of future tie steps (r). MATERIAL AND METHOD Material Bentonite suspension was used extensively as a odel food due to its stability and possibility of preparation of suspensions with different therophysical properties. In this research 8% (w/w) bentonite suspension was prepared according to the procedure proposed by Josef et al (1996). The theral conductivity of the suspension was deterined using the KD theral conductivity probe ((Labcell Ltd., UK) using three replicates is 0.61 ± 0.01 W/(- o C). The theral diffusivity was calculated using the equation proposed by Martens [6] = [0.057363W + 0.00088( T + 73)] x10 Equation (6) was developed based on ore than 00 experiental data and was reported to be ore accurate copared to Riedel [7] correlation (Dinçer [8]). The voluetric heat capacity (c) was calculated fro the theral conductivity and theral diffusivity. Experiental Setup Can with an effective diaeter of 74 and height 50 was used in this study. The can inner side wall was coated with a thick layer of Silicon gel to retard heat transfer fro the radial direction. Three levels of headspace were used ainly, 5 (10%), 7 (14%) and 10 (0%), the percentage refer to the volue of the headspace relative to the volue of the can. These levels of headspace cover the range of 9% to 16.4% reported in previous published work [14, 18, 9]. Once the headspace is known the height of the odel food is also known then the cans were punctured and special receptacle was fitted to allow insertion of a hypoderic needle therocouple type-j at a position of 10 below the surface of the odel food for all the headspace levels used. Following the preparation of the cans, the prepared bentonite suspension was heated in a kettle using boiling water 6 (6) bath until the teperature of the bentonite about 80 o C, then poured into the prepared cans, and filled to the desired arked level and carefully vacuu sealed to eliinate all gases using a vacuu seaing achine. Further insulation to the outside of the cans was done by placing the prepared can inside a hollow Styrofoa cylinder with special cut for the receptacle that holds the therocouple, then all the interfaces between the can and the Styrofoa at the edges were sealed carefully with a silicone gel to prevent water entry to the side of the can. Furtherore, the outer side of the Styrofoa cylinder was wrapped with a shrink polyeric fil. The prepared cans were then attached to two Aluinu blocks coated with silicon gel. The purpose of the Aluinu blocks is to provide support for the cans to be in the upright position during iersion in a boiling water bath and to allow the botto of the can to be exposed to the boiling water. The coating is to iniize the drop in the boiling water upon iersion, details of the experiental setup is shown in Figure 1. The experients were carried out in duplicate for each headspace with separate run for each headspace size. The short can with sall height/diaeter ratio was selected for this study because such can will be ost sensitive to error in axial position of therocouple copared to cans with large height/diaeter ratio. Data Collection For data collection, a hypoderic therocouple was first attached to a data logger OPTO (Solution Engineering SdnBhd, Kuala Lupur, Malaysia). The data logger was then connected to a coputer equipped with a LABTECH version 1.1 software which anages and controls the data collection process. The therocouple was calibrated using glass theroeter and ice-water ixture. The calibration was verified using boiling water yielding teperature reading of 100 ± 0.1 o C. The therocouple was then inserted carefully into the can at the set position. A stainless steel cylindrical container was used to heat about six liters of water using a gas stove. When the water reached agitated boiling stage the prepared can was suberged into the water bath and the data logger was activated to start collecting tieteperature data. The collection of the data was set at 30 second intervals for a period of 15 inutes. The data was stored in the hard disc of the coputer for further processing.

Cold Spot Location for Conduction-Heated Canned Foods International Journal of Food Processing Technology, 015, Vol., No. 13 Figure 1: Scheatic showing the can experiental setup. RESULTS AND DISCUSSION Surface Heat Flux Estiation The input data to the developed coputer progra are teperature history data, the position of the sensor, the height of the saple, the therophysical properties of the food aterial, tie and space step sizes, the known boundary conditions, the teperature easureent intervals and the nuber of future tie intervals ( r ). Based on this inforation, the progra calculates the heat flux in sequential anner at each tie (). Once the heat flux at the headspace side is known the proble becoes a direct proble. The sae coputer progra then calculates the teperature profile in the axial direction using a subroutine for the solution of the direct heat conduction proble. The accuracy of the calculated heat flux was checked by coparing the calculated teperature at the sensor position with the easured value using the root ean squares error (RMSE) given as: 1/ N 1 RMSE = ( ˆ Y i T i ) (7) N i= 1 Restivo and Sousa [30] indicated that by using a root su square ethod all the coponents are cobined for producing the cobined uncertainty. The value of the nuber of future tie easureents found to yield iniu RMSE value based on preliinary investigation was found to be r = 4 for all the treatents. Table 1 shows the RMSE for the different cans used in this study. A axiu value of 0.9 o C which is within the allowable error encountered during teperature easureents by therocouples was obtained. This represents the cobined errors fro teperature easureents and nuerical coputation over the whole easureent tie doain indicating the accuracy of the calculated heat flux and the subsequent solution of the direct proble for the teperature distribution in the axial direction. Figure shows coparison between the calculated teperature fro the estiated heat flux and the easured teperature at the sensor position for the 14% headspace. It is evident that there is an excellent agreeent between these two teperature histories indicating the reliability of the algorith and the accuracy of the calculated heat flux. Beck, Blackwell [5] investigated different types of function fors for heat flux including cubic, parabolic, linear and constant for the function specification ethod the authors found that the constant heat flux for resulted in excellent and efficient estiation of the heat flux copared to the experiental heat flux data and the other functional fors. Table 1: Root Mean Square errors for Different Experiental Run Run Headspace () RMSE ( o C) 1 5 0.11 5 0.3 3 7 0.1 4 7 0.9 5 10 0.10 6 10 0.16

14 International Journal of Food Processing Technology, 015, Vol., No. Mohaed Figure : Experiental and calculated teperature profiles at the sensor position for 14% headspace. Cold Spot Location Figure 3 to 5 show the teperature profile in the diensionless axial direction relative to the height of the odel food (L). It is clear that the teperature profile is not syetric for all the three levels (10%, 14% and 0% ) of headspace resulting in cold spot at position 59%, 58% and 59% of the odel food height, respectively. The unsyetrical teperature profile is due to the difference in theral resistance at the botto and the top of the can. It is iportant to ephasize that the position of the cold spot for the cans replicates at each headspace levels are identical with slight difference in agnitudes of the teperatures due to the differences in the initial teperature for the different replicates. The reason for the cold spot position being at about a constant value of 59% relative to the odel food height irrespective of the size of the headspace indicates that the heat transfer coefficients at the headspace side are close for the three levels. Mohaed [11] found that the effective heat coefficients for headspace of 10%, 14% and 0% consist ainly of water vapor and were 54., 53. and 50.8 W/( - o C), respectively with an average value of 5.73 and standard deviation of ±1.75. This indicates that the effective heat transfer coefficients for can headspace coprised ainly of water vapor were not significantly different. The author attributed that to water vapor equilibriu considerations that ay result into the sae water density and hence sae convective effects in the headspace. However, presence of gases and water vapor ixture ay produce different results as phase equilibriu in the headspace will depend on pressure, teperature and coposition resulting in different thero-physical property and hence different convective effect in the headspace. Due to the fact that the effective heat transfer coefficients were siilar for the levels investigated here, the shape of the teperature profiles in the axial direction for the three levels of headspaces were not significantly different. Therefore, the cold spot will be at position of 0.59L fro the botto of the can and the agnitude of L depends on the size of the headspace. Figure 6 shows the cold spot position relative to the can height (H). It is obvious that for 10% headspace the cold spot is above the geoetric center. As the headspace increased to 14% the cold spot is slightly above the geoetric center. For the 0% headspace, the cold spot oved below the geoetric center. The trend shown in the figure clearly deonstrated that headspace size influence the position of the cold spot and it could be either below or above the geoetric center. However, the results of Figure 6 also showed that cold spot will be at the geoetric center for headspace in the range 14 0 %. For short cans such as the one used in this study, the height of the food (L) that results in cold spot at the geoetric center can be easily calculated fro the result of this study using the forula [L = H/(1.18)] this correspond to headspace of 15.5% for the can size used in this study. Josef et al. [14] carried out heat penetration tests at different axial positions for bentonite dispersion (5g/L) packed in 307 x 409 with different headspace levels. They found that for a headspace of 19 which correspond to 16.4 % the cold spot is at the geoetric center which is close to the value of 15.5% obtained in this study using a relatively shorter can. Varga, Oliveira [18] showed that for headspace of 10% with variability of.5% the effect on process lethality is ore relevant for short can (11 x 109 and 307 x 113) copared to tall can (11 x 304 and 307 x 51). Although they indicated that the variability range in headspace investigated did not have very iportant effects on process lethality copared to the other variables investigated, they acknowledge that headspace ight change the exact location of the least lethality point inside the package. Robertson and Miller [13] and Joseph et al. [14] found that process lethality at the geoetric center of canned bentonite increases significantly with increases of headspace, this suggests that the heat transfer coefficient at the can headspace is not a liiting factor in heat transfer for the range of headspaces investigated which is in agreeent with the results of Mohaed [11] which revealed that, the heat transfer coefficients for headspace of 10%, 14% and 0% were not significantly different (P < 0.01). This eans, that theral resistance at the can headspace did not increase with increase of the headspace size at least for the range investigated. If heat transfer

Cold Spot Location for Conduction-Heated Canned Foods International Journal of Food Processing Technology, 015, Vol., No. 15 coefficient at the headspace decreases with increase of headspace the theral process lethality at the geoetric center would be expected to decrease with increase of headspace due to increase of theral resistance at the headspace side. Therefore due to the fact that theral process lethality increases with increase of headspace, one would expect cold spot to igrate along the central axes depending on the size of the headspace. Exact location of the cold spot is very iportant in optiizing theral process schedule. Uno and Hayakawa [31] investigated the effect of errors in therocouple positioning when collecting transient teperature easureents for deterination of theral diffusivity in cylindrical can. They found that a one deviation in locations, where the teperatures were onitored, as well as siilar deviations in the height and radius of a cylinder resulted in significant variations in the therophysical property values this will subsequently affect the tie-teperature distribution. It is also shown by the previous authors that these variations ay be significantly reduced if one uses a cylinder for which height and radius are 100 or greater. This clearly indicates that teperatures distribution will be influenced by errors in position as sall as 1 especially for short cans; therefore it is very iportant to place the therocouple exactly at the cold spot when carrying out a heat penetration test to iniize the error in theral process lethality calculation. Exact positioning of the teperature easuring therocouples can be achieved using sensors with very sall tip with probe attached to a fitting holder affixed to the can wall, any design of such sensor systes are currently available fro different anufacturers. However, use of coputerized atheatical odels will provide accurate prediction of tie-teperature at the cold spot if a reliable heat transfer coefficient at the can headspace is provided. Figure 4: Teperature profile in the axial direction at tie 10.5 inutes for 14% headspace. Figure 5: Teperature profile in the axial direction at tie 10.5 inutes for 0% headspace. Figure 6: cold spot locations relative to the can height for different percent headspace level. Figure 3: Teperature profile in the axial direction at tie 10.5 inutes for 10% headspace. The results of this research clearly indicates that it is very iportant to precisely deterine the location of the cold spot for can with headspace, as the tie teperature data collected at the geoetric center ay provide less accurate inforation regarding process

16 International Journal of Food Processing Technology, 015, Vol., No. Mohaed lethality in the presence of headspace. This ay have a serious consequences regarding product safety or quality as the geoetric center ay not be the cold spot at soe levels of headspace and that ay result in either under processing or over processing depending on the level of the headspace which ay have serious effect on consuer health or product quality. CONCLUSIONS Heat transfer through can headspace was investigated using an inverse heat conduction approach. The ethod is based on estiating the boundary heat flux at the headspace side using internal transient teperature easureents. 8 % bentonite suspension packed in cylindrical can with diaeter 74 and height 50 with three levels of headspace 10%, 14% and 0% were used in this study. The accuracy of the calculated heat flux at the can headspace side was confired by coparing calculated and easured teperature at the sensor position. Furtherore, the results reveals that the cold spot was located at about 59% of the odel food height fro the botto of the can for the three levels of headspace investigated. These results provide guideline for deterining cold spot location when collecting heat penetration data to be used for lethality calculation for theral sterilization process for cans with coparable heights. However, the presence of significant aount of gases in the headspace ay yield different results regarding cold spot location. NOMENCLATURE c = H = specific heat, J/(kg-K) height of the can, k = theral conductivity, W/(- o C) L = N = t = r = height of the odel food, nuber of easureents tie, s nuber of future tie steps for function specification S = su of squares function for function specification ethod T = teperature, o C T o = Tˆ = Initial teperature, o C estiated teperature, o C T h = heating ediu teperature (100 o C) W = Y = fractional oisture content (wet basis) easured teperature, o C q = heat flux at the headspace boundary, (W/ ) qˆ = estiated heat flux at the boundary, (W/ ) z = z c = Greek space in the axial position, cold spot position, = density, kg/ 3 = theral diffusivity /s Subscript i = = tie index tie index REFERENCES [1] Ball CO. Theral process tie for canned food: Published by the National research council of the National acadey of sciences; 193. [] Teixeira, AA, Dixon, JR, Zaradnik, JW, Zinseister, G. E. Coputer optiization of nutrient retention in theral processing of conduction-heated foods. Food Technology 1969; 3: 137-14 [3] Teixeira A, Zinseister GE, Zahradnik JW. Coputer siulation of varlable retort control and container geoetry as a possible eans of iproving thiaine retention in therally processed foods. Journal of Food Science 1975; 40: 656-659. http://dx.doi.org/10.1111/j.1365-61.1975.tb005.x [4] Datta A, Teixeira A, Manson J. Coputer based retort control logic for on line correction of process deviations. Journal of Food Science 1986; 51: 480-483. http://dx.doi.org/10.1111/j.1365-61.1986.tb11161.x [5] Bichier J, Teixeira A, Balaban M, Heyliger T. Theral process siulation of canned foods under echanical agitation1. Journal of food process engineering 1995; 18: 17-40. http://dx.doi.org/10.1111/j.1745-4530.1995.tb0035.x [6] Durance TD, DOU J, Mazza J. Selection of variable retort teperature processes for canned salon. Journal of food process engineering 1997; 0: 65-76. http://dx.doi.org/10.1111/j.1745-4530.1997.tb00411.x [7] Mohaed IO. Coputer siulation of food sterilization using an alternating direction iplicit finite difference ethod. Journal of food Engineering 003; 60: 301-306. http://dx.doi.org/10.1016/s060-8774(03)00051-7 [8] Tucker G, Clark P. Modelling the cooling phase of heat

Cold Spot Location for Conduction-Heated Canned Foods International Journal of Food Processing Technology, 015, Vol., No. 17 sterilization processes, using heat transfer coefficients. International Journal of Food Science & Technology 1990; 5: 668-681. http://dx.doi.org/10.1111/j.1365-61.1990.tb0118.x [9] Bhowik SR, Shin S. Theral Sterilization of Conduction Heated Foods in Plastic Cylindrical Cans Using Convective Boundary Condition. Journal of food science 1991; 56: 87-830. http://dx.doi.org/10.1111/j.1365-61.1991.tb0539.x [10] Silva C, Hendrickx M, Oliveira F, Tobback P. Optial sterilization teperatures for conduction heating foods considering finite surface heat transfer coefficients. Journal of food science 199; 57: 743-748. http://dx.doi.org/10.1111/j.1365-61.199.tb08086.x [11] Mohaed IO. Deterination of an effective heat transfer coefficients for can headspace during theral sterilization process. Journal of food engineering 007; 79: 1166-1171. http://dx.doi.org/10.1016/j.jfoodeng.006.04.015 [1] Evans H, Board P. Studies in canning processes. I. effect of headspace on heat penetration in products: heating by conduction. Food technology 1954; 8: 58-6. [13] Robertson G, Miller S. Uncertainties associated with the estiation of Fo values in cans which heat by conduction. International Journal of Food Science & Technology 1984; 19: 63-630. http://dx.doi.org/10.1111/j.1365-61.1984.tb01879.x [14] Joseph S, Speers R, Pillay V. Effect of head space variation and heat treatent on the theral and rheological properties of nonagitated, conduction-heated aterials. LWT-Food Science and Technology 1996; 9: 556-560. http://dx.doi.org/10.1006/fstl.1996.0085 [15] Berry Jr M, Kohnhorst A. Critical factors for theral processing of institutional pouches [Whole kernel corn]. Journal of Food Protection 1983. [16] Capbell S, Raasway H. Heating Rate, Lethality and Cold Spot Location in Air entrapped Retort Pouches during Over pressure Processing. Journal of food science 199; 57: 485-489. http://dx.doi.org/10.1111/j.1365-61.199.tb055.x [17] Raasway H, Grabowski S. Influence of entrapped air on the heating behavior of a odel food packaged in sei-rigid plastic containers during theral processing. LWT-Food Science and Technology 1996; 9: 8-93. http://dx.doi.org/10.1006/fstl.1996.0011 [18] Varga S, Oliveira JC, Oliveira FA. Influence of the variability of processing factors on the F-value distribution in batch retorts. Journal of Food Engineering 000; 44: 155-161. http://dx.doi.org/10.1016/s060-8774(99)00174-0 [19] Khakbaz Heshati M, Shahedi M, Hadai N, Hejazi M, Motalebi A, Nasirpour A. Matheatical Modeling of Heat Transfer and Sterilizing Value Evaluation during Caviar Pasteurization. Journal of Agricultural Science and Technology 014; 16: 87-839. [0] Stolz G. Nuerical solutions to an inverse proble of heat conduction for siple shapes. Journal of heat transfer 1960; 8: 0-5. http://dx.doi.org/10.1115/1.3679871 [1] Beck J. Calculation of surface heat flux fro an internal teperature history. ASME paper 6-HT-46, 196 [] Beck JV. Surface heat flux deterination using an integral ethod. Nuclear Engineering and Design 1968; 7: 170-178. http://dx.doi.org/10.1016/009-5493(68)90058-7 [3] Beck JV. Nonlinear estiation applied to the nonlinear inverse heat conduction proble. International Journal of Heat and Mass Transfer 1970; 13: 703-716. http://dx.doi.org/10.1016/0017-9310(70)90044-x [4] Tikhonov AN, Arsenin VIAk, John F. Solutions of ill-posed probles: Winston Washington, DC; 1977. [5] Beck J, Blackwell B, Haji-Sheikh A. Coparison of soe inverse heat conduction ethods using experiental data. International Journal of Heat and Mass Transfer 1996; 39: 3649-3657. http://dx.doi.org/10.1016/0017-9310(96)00034-8 [6] Martens T. Matheatical odel of heat processing in flat containers. 1980. [7] Riedel L. Measureents of theral diffusivity on foodstuffs rich in water. Kaltetechnik 1969; 1: 315. [8] Dinçer I. Heat transfer in food cooling applications: CRC Press; 1997. [9] Giannoni Succar EB, Hayakawa KI. Correction factor of deviant theral processes applied to packaged heat conduction food. Journal of Food Science 198; 47: 64-646. http://dx.doi.org/10.1111/j.1365-61.198.tb10140.x [30] Restivo M, Sousa C. Measureent uncertainties in the experiental field. Sensors and Transducers Journal 008; 95: 1-1. [31] Uno JI, Hayakawa KI. A ethod for estiating theral diffusivity of heat conduction food in a cylindrical can. Journal of Food Science 1980; 45: 69-695. http://dx.doi.org/10.1111/j.1365-61.1980.tb04134.x Received on 16-04-015 Accepted on 6-05-015 Published on 30-07-015 http://dx.doi.org/10.15379/408-986.015.0.0.0 015 Mohaed; Licensee Cosos Scholars Publishing House. This is an open access article licensed under the ters of the Creative Coons Attribution Non-Coercial License (http://creativecoons.org/licenses/by-nc/3.0/), which perits unrestricted, non-coercial use, distribution and reproduction in any ediu, provided the work is properly cited.