A Method for Indirect Measurements of Thermal Properties of Foods

SIK-rapport 2002, No. 703

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 ii==11

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 the cylinder (rc) is used, and equation (2) is solved for u=[k ρcp h] with Neumann boundary condition. α is calculated from the definition α=k/(ρcp). Upper and lower bounds for the unknown variables are derived by results from step (i) and (ii). The following inequalities are obtained:

≤ ≤ kmin k kmax (a priori) ⇒ (4) = ≤ ≤ = happ kmin hmin h hmax happ kmax (happ estimated) k kmin = ρ ≤ ρ ≤ ρ = max α c p,min c p c p,max ( estimated) α α

5 RESULTS AND DISCUSSIONS

The square mean error (Råde and Westergren (10)) was used as a criterion on curve fitting. The suggested method estimates the thermal properties (except for the heat transfer coefficient) with a maximal relative error between 6-9%, see table 1. The measured temperatures in the cylinder and the calculated temperatures are shown in figure 3. The measured and calculated temperature curves in the centre of the cylinder are in well agreement. The value of α is estimated as described in step (i) - step (iii) and the temperatures in figure 3 are calculated with Dirichlet boundary condition.

Meas. No Meat filling Square Pancake dough Square [10-7 m2/s] mean error [10-7 m2/s] mean error 1 1.09 0.48 1.01 1.16 2 0.93 1.11 1.04 0.93 3 1.04 1.33 0.97 1.48 4 1.02 0.95 0.92 0.53 Estimate 1.02±0.09 0.985±0.065 Rel. error <9% <7%

Table 1. Estimated values of thermal diffusivity, α (m2s-1).

35 C] o

T [ 30

10 0 2 4 6 8 10 12 14 16 18 20 t [min]

Figure 3. Time-temperature curve of the case with Dirichlet boundary condition; – calculated T at centre; -- measured T at centre; – - – measured T at surface of the inner cylinder.

The uncertainty in the relative error generates a discrepancy in the prediction of the elapsed time-temperature profile, exemplified by figure 4. Typically the predicted temperature differs ±1 °C from the measured one, which seems to be a quite satisfactory result. In spite of the large uncertainty in the calculated heat transfer coefficients, the two different ways (step (i) and step (iii)) for calculation of the thermal diffusivity presented gave results that agreed well.

6 45

30 C] o

T [ 25

10 0 2 4 6 8 10 12 14 16 18 20 t [min]

Figure 4. The temperature at the centre of the cylinder (Dirichlet boundary condition); – measured T at centre; -- lower limit; – - – upper limit.

The method was good at predicting one parameter with Dirichlet boundary condition, where a distinct minimum always was found for the target function, see figure 5. However, in the case with Neumann boundary condition, the target function did not in most cases have a distinct minimum. This fact yields a large uncertainty, especially in the prediction of the heat transfer coefficient.

1.5

Log (Functional)Log 0.5

0 0 1 2 3 2 -7 α [m /s] x 10

Figure 5. Target function for the thermal diffusivity.

As stated above, it is a more complex problem to determine all thermo-physical parameters simultaneously. The thermal diffusivity is defined by the ratio of k and ρcp. The determination of the heat transfer coefficient simultaneously is a well-known ill-conditioned problem, illustrated by figure 6. The results of a measurement of the meat filling and the pancake dough with different initial values are shown in table 2.

7 2.5

1.5

Log(Functional) 1 8 6 6 4 x 10 0.8 2 0.6 0.4 0.2 ρ c 0 0 p k

6 x 10 7

5 p

c 4 ρ

1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k

Figure 6. Target function and contour plot of k and ρcp with constant h.

-6 Meas. No. Initial start guess k ρcp ·10 h Square -1 -1 -3 -1 -2 -1 (h/ρcp /k) [Wm K ] [Jm K ] [Wm K ] mean error Meat filling 1 300/2.66·106/0.2 0.530 4.88 250 0.48 2 700/6.39·106/0.6 0.598 5.51 283 0.48 3 500/3.20·106/0.4 0.567 5.23 268 0.48 Estimate 0.565±0.033 5.21±0.33 Rel. error <6% <7% Pancake dough 1 700/2.13·106/0.3 0.590 5.67 635 0.93 2 1500/4.85·106/0.5 0.530 5.12 600 0.93 3 1800/6.39·106/0.6 0.594 5.71 639 0.93 Estimate 0.57±0.04 5.50±0.38 Rel. error <8% <7%

Table 2. Estimated thermal properties and error estimate for the meat filling and pancake dough (one measurement of each) with different starting values.

8 Food is a complex material. In this study the thermal properties of pancake dough and meat filling in a spring roll was studied. The meat filling contains pieces of meat, vegetables etc., but the filling was assumed to be homogenous. If the density and moisture content of the various components are found to be very different, then a separate thermal conductivity and specific heat could be measured for each component (Singh (2)). As in Nahor (8) it was found that the accuracy of the estimates is highly dependent on some features of the set-up and the heat transfer model.

9 CONCLUSIONS

An indirect method for estimation of thermal properties for foods has been demonstrated. The method gave satisfactory results for the thermal diffusivity, the heat conduction, and the volumetric heat capacity. However, the accuracy of the result is highly dependent of the experimental set-up. Therefore, a number of trials have to be carried out so that average values and error estimates could be calculated. The models can of course also be used for straight forward calculation of elapsed time/temperature profiles for different settings of the product and the process parameters, easily generalised to arbitrary product geometries. Advantages with the concept are the simplicity in the experimental set-up and in the mathematical modelling. Drawback is the ill- conditioned nature of the problem, formulated in terms of the conduction heat transfer equation with a convective boundary condition.

10 FUTURE WORK

To be able to increase the accuracy in the estimates with an indirect method, more research is needed on estimating the heat transfer coefficient. A more accurate value helps separating the thermal diffusivity into thermal conductivity and volumetric heat capacity. A natural proceeding is also to supply terms for phase changes and kinetic reactions to the basic heat transfer equations, some extra parameters that need to be estimated are then added. Different models to calculate the temperature dependency of the thermal properties could be further investigated and also the influence of other parameters, such as the water content.

11 ACKNOWLEDGEMENTS

This project is a part of a Ph.D. project, which is financed by the Swedish Knowledge Foundation and Lecora AB in Vadstena, Sweden.

12 NOMENCLATURE

2 α=k/(ρ cp) Thermal diffusivity (m /s) cp Specific heat (J/kg°C) h Heat transfer coefficient (W/m2°C ) -1 happ=h/k Apparent heat transfer coefficient (m ) k Thermal conductivity, (W/m°C ) n Number of samples rc Radius of the cylinder (m) rd Radius of the inner fictive cylinder (m) ρ Density (kg/m3) 3 ρcp Volumetric heat capacity (J/m °C) T=T(t,r) Temperature at point r at time t in the cylinder (°C) T0 Initial temperature (assumed uniform) (°C) T∞= T∞(t) Ambient temperature at time t (°C) c T c,i Calculated temperature in the centre of the cylinder (°C) c T m,i Measured temperature in the centre of the cylinder (°C) d T c,i Calculated temperature at a point near the surface of the cylinder (°C) d T m,i Measured temperature at a point near the surface of the cylinder (°C) ∂/∂n Normal derivative overbar Average value

13 REFERENCES

1. Nesvadba, P. (1982). Methods for the Measurement of Thermal Conductivity and Diffusivity of Foodstuff. J. Food Eng., 1, 93-113 2. Singh, R.P. (1982). Thermal Diffusivity in Food Processing. Food Technol., 36, 87-93 3. Dickerson, R.W. (1965). An Apparatus for the Measurement of the Thermal Diffusivity of Foods. Food Technol., 19, 198-204 4. Magee, T.R.A.; Bransburg, T. (1995). Measurement of the Thermal Diffusivity of Potato, Malt Bread and Wheat Flour. J. Food Eng., 25, 223-232 5. Gordon, C.; Thorne, S. (1990). Determination of the Thermal Diffusivity of Foods from Temperature Measurement during Cooling. J. Food Eng., 11, 135-45 6. Kee, W.L.; Ma, S.; Wilson, D.I. (2002). Thermal Diffusivity Measurements of Petfood. J. Food Prop., 5 (1), 145-151 7. Jaramillo-Flores, M.E.; Hernandez-Sanchez, H. (2000). Thermal Diffusivity of Soursop (Annona muricata L.) Pulp. J. Food Eng., 46, 139-143 8. Nahor, H.B.; Scheerlinck, N.; Verniest, R.; De Baerdemaeker, J.; Nicolai, B.M. (2001). Optimal Design for the Parameter Estimation of Conducted Heated Foods. J. Food Eng., 48, 109-119 9. Schneider, P.J. (1957). Conduction Heat Transfer, 1st Ed.; Addison-Wesley; Reading, MA 10. Råde L;Westergren B. (1995). Mathematics Handbook for Science and Engineering, 3rd Ed.; Studentlitteratur; Lund, Sweden