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A Method for Indirect Measurements of Thermal Properties of Foods

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A Method for Indirect Measurements of Thermal Properties of Foods SIK-rapport 2002, No. 703 A Method for Indirect Measurements of Thermal Properties of Foods Eva Olsson, Hans Janestad, Lilia Ahrné SIK-rapport 2002, No. 703 A Method for Indirect Measurements of Thermal Properties of Foods Eva Olsson, Hans Janestad, Lilia Ahrné Göteborg, 2002 ISBN 91-7290-223-X ABSTRACT A method has been developed for the simultaneous estimation of thermal diffusivity, thermal conductivity, volumetric heat capacity and the heat transfer coefficient for foods heated/cooled by convection. The method uses experimental time-temperature data of the food and the medium, which are used in a parameter estimation procedure based on the conduction heat transfer equation in one dimension for an infinite cylinder. To increase the accuracy in the results the estimations are performed in a number of sub-steps under varied model conditions, for example is an inner fictive cylinder used. The suggested method estimated the thermal diffusivity, the heat conduction and the volumetric heat capacity with a maximal relative error in the interval 6-9%. This study shows the well known difficulty of separating the thermal conductivity and the volumetric heat capacity in the expression for the thermal diffusivity, caused among other things by the uncertainty in the estimation of the heat transfer coefficient. TABLE OF CONTENTS INTRODUCTION.............................................................................................................................. 2 MATERIALS AND METHODS........................................................................................................... 3 The Experimental Set-up.......................................................................................................... 3 Mathematical Modelling ..........................................................................................................3 Parameter Estimation ............................................................................................................... 4 Step (i). Estimation of constant α ........................................................................................ 4 Step (ii). Estimation of happ .................................................................................................. 5 Step (iii). Estimation of α, k, ρcp and h ................................................................................ 5 RESULTS AND DISCUSSIONS .......................................................................................................... 6 CONCLUSIONS ............................................................................................................................. 10 FUTURE WORK............................................................................................................................ 11 ACKNOWLEDGEMENTS ................................................................................................................ 12 NOMENCLATURE ......................................................................................................................... 13 REFERENCES ...............................................................................................................................14 INTRODUCTION Heating and cooling of food products are common in food processing; heat transfer is one of the most important phenomena in the food industry. The mathematical basis of thermal processes is well established, frequently by use of the conduction heat transfer equation with a convective boundary condition. Improvement in the design or optimisation of thermal processes requires reliable thermo-physical data. The food material is often inhomogeneous and in some cases composed by pieces of different sizes and density. Accurate predictions of the thermo-physical data are not available for many semisolid and multiphase food, therefore are experimental determinations required. In a thermal process the necessary thermal properties are the density, the specific heat and the thermal conductivity, or the thermal diffusivity. The thermal properties are often affected by water content, temperature, composition, porosity and kinetic reactions. There are several different methods available for measuring the thermal diffusivity of foods. Critical reviews have been written by Nesvadba (1) and Singh (2) among others. An experimental apparatus was designed by Dickerson (3) to measure the thermal diffusivity of foods. This experimental apparatus has been used and adapted by many researchers. Magee and Bransburg (4) and Gordon and Thorne (5) estimated α using a thermal diffusivity tube under transient heat transfer conditions by the lag method and the slope method. Kee et al. (6) used a similar thermal diffusivity tube and measured the thermal diffusivity using the log method. The drawbacks with these methods are that they are limited to homogeneous, isotropic foods of certain geometry. Jaramillo-Flores and Hernandez-Sanchez (7) used heat penetration curves to estimate the thermal diffusivity. Most methods estimate only one parameter under constant temperature. The temperature dependency and the mutual interactions between the parameters is then lost. Nahor et al. (8) measure the volumetric heat capacity and the thermal conductivity of heated foods simultaneously and uniquely using the hot wire probe method. The objective of this study is to demonstrate a method where thermal properties (k, ρcp, α and h) of foods are estimated simultaneously by use of experimental data and a theoretical model. Primary, semisolid and bulk foods are considered. The optimisation routines are based on the least squares method, where the parameters are estimated sequentially in a number of sub- steps. 2 MATERIALS AND METHODS The Experimental Set-up The experimental set-up is shown in figure 1. The food is put in a copper cylinder with a diameter of 35 mm and a length of 120 mm, the cylinder has rubber plugs in both ends. Through the rubber plug at the top two bayonet thermocouples are stuck, one at the centre of the cylinder and one near the surface (6 mm from the inside of the cylinder). In addition, two thermocouples are placed in the water at different depth and one at the surface of the cylinder. The heating media is water and a mixer is used to create a uniform heat distribution. The thermocouples are connected to a logger and a personal computer, EasyView 5 is used to record the data. The temperatures are measured every 30 s for 20-30 minutes, during the temperature is increased from 20-85 °C. The model food tested was pancake dough and a meat filling of a spring roll. Figure 1. The experimental set-up. Mathematical Modelling When heat is transferred by conduction through a solid material it can be described by the Fourier equation with a Neumann convective boundary condition. ∂T ρc = ∇(k∇T ), x∈Ω (1) p ∂t ∂T k = −h(T (t, x) −T∞ (t)), x∈δΩ ∂n T(0, x) = T (x), x∈Ω, with T = T(t, x) 0 The thermal diffusivity can be expressed in terms of the thermal conductivity, the density and the specific heat as α=k/(ρcp). By assuming no temperature gradients parallel to the axis of the cylinder and in absence of a temperature difference around the circumference, equation (1) reduces to (in cylinder coordinates) along with convective boundary conditions. 3 11∂∂TT2 ∂ T =+, tandrr ≥00 ≤≤ (2) α ∂∂trrr2 ∂ c ∂∂ TT==−− 0((,)())and k h T t rc T∞ t ∂∂== rrrrr0 c TrTrr(0, )=∈Ω ( ), 0 Parameter Estimation The estimation procedure is formulated as a non-linear constrained optimisation problem: ≤≤ minFu ( ), umin u u max (3) nn =−+−cc22 dd FuTT()()()∑∑mi,, ci TT mi ,, ci == ii11 c c T m,i/ T c,i are the measured/calculated temperatures in the centre of the cylinder at time ti for d d i=1,2,…n , analogous T m,i/ T c,i are the temperatures in a desired point near the surface in the cylinder. The vector u consists of the unknown variables to be estimated in three sub-steps. In the first step (i), the thermal diffusivity is estimated, that is u=[α]. In the second step (ii), the apparent heat transfer coefficient is estimated, u=[happ]. In the third step (iii), the thermal conductivity, the volumetric heat capacity and the heat transfer coefficient are estimated, u=[k ρcp h]. The heat equation (2) is solved numerically by the difference method “Euler backwards” in the software Matlab. The method is unconditionally stable and the discretisation is chosen so small that the error is negligible compared to the exact solution (Schneider (9)). The optimisation problem (3) is solved by the least square method, using the Optimization Toolbox in Matlab. The sub-steps (i)-(iii) of the simultaneous determination (using the same set of data) of the volumetric heat capacity, the thermal conductivity and the heat transfer coefficient are: Step (i). Estimation of constant α Equation (2) is used with the radius of inner fictive cylinder (rd), see figure 2. The boundary condition is then T(t,rd)=T∞(t). This is a Dirichlet boundary condition at the surface of the inner fictive cylinder. The optimisation problem (3) is solved for u=[α] by use of the target function, with realistic upper and lower bounds for α. In step (i) the second summation in the target function F is zero because the desired point is used as a boundary condition. rd rc Figure 2. The inner fictive cylinder giving Dirichlet boundary conditions. 4 Step (ii). Estimation of happ The radius of the cylinder (rc), see figure 2 is used with known α from step (i). Equation (2) is solved for u=[happ] analogous as in step (i). Step (iii). Estimation of α, k, ρcp and h The radius of
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