Eurotherm Seminar 77 – Heat and Mass Transfer in Food Processing June 20-22, 2005 Parma, Italy
EXPERIMENTAL DETERMINATION OF THERMAL DIFFUSIVITY USING A NUMERICAL TECHNIQUE TO SOLVE HEAT TRANSFER EQUATION IN CONDUCTIVE FOODS Massimiliano Rinaldi*, Giampaolo Betta, Emma Chiavaro, Davide Barbanti, Roberto Massini. *Dipartimento di Ingegneria Industriale, Università degli Studi di Parma Parco Area delle Scienze 181/A – 43100 PARMA Phone +39 0521 906234 Fax +39 0521 905705 e-mail
[email protected] ABSTRACT This work describes the comparison between three different methods to estimate the thermal diffusivity value of conductively foods, canned or cooked, processed respectively in thermostatic bath and air convection oven. The methods for thermal diffusivity calculation were the Ball & Olson graphical method, the Singh’s predictive method and a new developed Matlab® language finite differences resolution of unsteady-state heat transfer. Samples of tomato puree filled in four different container sizes, samples of pre-gelatinized starch suspension filled in glass jars, and fresh potatoes cylindrically shaped were used. Canned samples were treated at 80°C in well-stirred water bath; while potato samples were heated by an electrical oven at 180°C and 5 different relative humidity levels. The new proposed method allowed us to estimate thermal diffusivity values of the canned products excluding the container contribution. For potato samples apparent thermal diffusivity values were calculated comprehending the surface layer. These values didn’t agree with literature data, but presented a good correlation with relative humidity values recorded in oven chamber.
INTRODUCTION
MATERIALS AND METHODS
Thermal processing is the most common way used to destroy pathogenic an alterative microorganisms in food. The scientific design of the thermal treatment allows to couple the necessary lethal effect with the maximum retention of nutritional and organoleptic properties. The best way for carrying out this optimization is the use of mathematical models of the heat transfer applied to the kinetics of thermal destroy of microorganisms and thermal changes of food components. This way of optimization is fast and not expensive, but thermal properties related to the simulated system must be well known. Thermal properties could be founded from literature, but their calculation from experimental tests is strongly preferred because of their dependence on sample composition and processing conditions. The most important thermal parameter we need to know is thermal diffusivity, given from:
Samples
α=
κ ρ ⋅cp
(1)
with κ thermal conductivity ,cp specific heat and ρ density of the product. Generally, in foodstuffs the value of thermal diffusivity strictly depends on cell structure, intracellular air content, water content, operating pressure and temperature; it varies also along system directions cause of physical anisotropy and composition heterogeneity. Sweat [1] recommended the calculation of thermal diffusivity value by inserting experimental thermal conductivity, specific heat and density values in equation (1). More recently Carbonera [2] and Ferruh [3] reported numerical simulation as the best way to obtain thermal diffusivity value from experimental temperature data.
Commercial tomato puree, packed in aseptic bricks and purchased on the market, was used as representative of conductive heating liquid foodstuff. The product was characterized by the following composition: water 94.5%, protein 1.3%, carbohydrate 4%, fat 0.2%. The tomato puree, alternatively filled in four different container types (height x diameter): 80 x 60 (200 ml) and 110 x 75 mm (400 ml) glass jars; 80 x 50 mm (150 ml) aluminium can, and 130 x 80 mm (650 ml) tin can, were processed in a well-stirred thermostatic bath (Julabo Circulators VC, JulaboItalia, Milano, Italy) at 80°C. For every test two samples were used at each time. Samples of 20% pre-gelatinized starch (Cerestar S.p.A. Ferrara, Italy) water suspension were filled in glass jar 80 x 60 mm (200 ml) and were heated for 60 minutes at 80°C in the same thermostatic bath used for tomato puree. Samples were subsequently cooled at room temperature for at least 24 hours to obtain the complete gelatinization, before their processing at 80°C into the thermostatic bath. Samples of fresh potatoes (cv. Agata) peeled and cylindrically shaped (25 x 30 mm) were cooked by using a commercial electrical forced convection oven equipped with a steam injection apparatus. In all tests, air temperature was set at 180°C and cooking treatment was stopped when potato core temperature reached 75°C, but the air relative humidity (RH) was changed by changing the steam injection rate. The following 5 different RH levels were used: 5.6%, 7.4%, 14.3%, 20.0%, and 31.3%. An empirical equation was used for calculating RH% values from dry and wet bulb temperature difference [4].
Carbohydra tes
Thermal diffusivity calculation was carried out from temperature values measured by the centre thermocouple and that placed in the heating medium, or one of the other thermocouples placed within the container. By using the centre and the external medium thermocouples, the global “apparent” thermal diffusivity can be obtained, comprehending the product and the container wall and boundary layer contributes. Whereas, by using two thermocouples both placed inside the container, an estimation of the “only product” thermal diffusivity can be obtained.
α = 8.0842 ⋅ 10 − 2 + 5.3052 ⋅10 − 4 T − 2.3218 ⋅ 10 − 6 T 2
Thermal diffusivity determination – graphical method
Fats
Thermal diffusivity was graphically calculated by Ball & Olson equation [6], the traditional approach based on heat penetration curves. Time-temperature data were represented on a semi-log plot in order to obtain the heat penetration rate from the slope of the straight-line portion of the curve. The time (minutes) correspondent to the crossing of a logarithmic cycle in the ordinate axis gives the fh value which allows to calculate the thermal diffusivity by the following Ball & Olson equation:
Thermal diffusivity determination – theoretical method For the theoretical prediction of thermal diffusivity the equations proposed by Singh [5] were used. This method considered food centesimal composition for the calculation of thermal properties at a certain temperature. Pr oteins
α = 6.8714 ⋅10 − 2 + 4.7578 ⋅10 − 4 T − 1.4646 ⋅10 − 6 T 2
α = 9.877 ⋅ 10 − 1.2569 ⋅10 T − 3.8286 ⋅ 10 T −2
−4
−8
2
Water
α = 1.3168 ⋅10 −1 + 6 .2477 ⋅ 10 − 4 T − 2.4022 ⋅ 10 − 6 T 2 Heat penetration curves measurement Heat penetration curves were measured using 0.9 mm wire thermocouples K type connected with a multimeter data acquisition system (Keithley Instruments Inc., Cleveland, Ohio, U.S.). Process recordings were planned measuring temperature variation inside the samples, in the thermal centre and in a point near it, on the surface and in the surroundings during all the treatments. For tomato puree temperature measurements were carried out using the equipments shown in Figure 1. The container were almost completely immersed into the thermostatic bath.
α=
0.398 1 0.427 2 + 2 ⋅ fh h R
(2)
Thermal diffusivity determination – mathematical method We considered Fourier’s equation as the main heat exchange law for a conductive heating:
∂T = α ⋅ ∇ 2T ∂t
(3)
where α is the thermal diffusivity and ∇2 Laplace’s operator. Considering a bi-dimensional system, equation (3) becomes: ∂ 2T ∂ 2T ∂T = α ⋅ 2 + 2 ∂t ∂y ∂x Figure 1. Thermocouples placement (not to scale)
For pre-gelatinized starch only the 80 x 20 mm (200 ml) glass jar was used. Preliminary tests were carried out, both with tomato puree and pre-gelatinized starch, by placing thermocouples along parallel planes above and below the product’s half height. Results obtained confirmed the absence of convection and a radial symmetry for the experimental system. Consequently, in our tests a conductive only radial heat transfer was assumed. Figure 2 shows as centre, surface and surroundings temperature were measured for potato cylinders heated by the combined air-steam oven.
Figure 2. Temperature recording in potato cylinders
(4)
For the radial symmetry of an infinite cylinder, equation (4) becomes: 1 ∂T ∂ 2 T 1 ∂T ∂ 2 T ⋅ = + ⋅ + α ∂t ∂r 2 r ∂r ∂z 2
(5)
With explicit method resolution of Fourier’s law it’s possible to obtain temperature expression at coordinate (M,N) at moment (p+1) depending on its temperature value and temperature near it at previous time. When the target point falls on cylinder axis the resolution is represented by:
T((Mp,+11)) =
4 ⋅ α ⋅ ∆t p α ⋅ ∆t p ⋅ T( M ,2) + ⋅ (T( M +1, N ) + T( Mp −1, N ) ) + 2 2 (∆r) (∆z)
2 ⋅ α ⋅ ∆t 2 ⋅ α ⋅ ∆t + T( Mp , N ) ⋅ 1 − − (∆r )2 (∆z)2
(6)
In a previous work [7] equation (6) has been developed with Matlab® language and the obtained algorithm was validated to simulating temperature history of conductive samples considered as infinite cylinders.
RESULTS AND DISCUSSION Canned samples Thermal diffusivity determination – theoretical method For tomato puree samples Singh’s thermal diffusivity was 0.149 mm2s-1 referred at 40°C. For 20% pre-gelatinized starch suspension theoretical thermal diffusivity value was 0.142 mm2s-1 referred at 40°C. Thermal diffusivity determination – graphical method In Table 1 thermal diffusivity values for tomato puree in thermostatic bath at 80°C obtained with the Ball & Olson method are reported. Tomato puree (Singh’s α = 0,149 mm2s-1) Experimental “Apparent” thermal diffusivity Container type mean st.dev. Glass jar 200ml 0,149 0,004 Glass jar 400ml 0.144 0.003 Tin can 0.155 0.002 Aluminium can 0.160 0.002 Experimental “Only product” thermal diffusivity Container type mean st.dev. Glass jar 200ml 0,043 0,005 Glass jar 400ml 0.015 0.001 Tin can 0.059 0.002 Aluminium can 0.035 0.002 Table 1. α (mm2s-1) values for tomato puree – graphical method
The graphical method allowed us to calculate only the “apparent” thermal diffusivity, while “only product” values resulted absolutely incoherent compared to the Singh’s theoretical one. In fact, the graphical method can be correctly applied only having a constant reference temperature (that of an external heating or cooling fluid). The same remark apply to results obtained for 20% pregelatinized starch suspensions heated within the 200 ml glass jar (Table 2)
penetration curve (Tables 3 and 4). As output, in the command window is also showed the mean of squared absolute temperature differences between simulated and experimental curves (not reported). Tomato puree (Singh’s α = 0,149 mm2s-1) Experimental “Apparent” thermal diffusivity Container type mean st.dev. Glass jar 200ml 0,145 0,002 Glass jar 400ml 0.147 0.002 Tin can 0.154 0.004 Aluminium can 0.166 0.005 “Only product” thermal diffusivity Container type mean st.dev. Glass jar 200ml 0,144 0,007 Glass jar 400ml 0.146 0.004 Tin can 0.142 0.003 Aluminium can 0.149 0.006 Table 3. α values for tomato puree – mathematical method
Pre-gelatinized starch (Singh’s α = 0,142 mm2s-1) Experimental mean st.dev. “Apparent” thermal diffusivity 0,152 0,006 “Only product” thermal diffusivity 0.139 0.013 Table 4. α (mm2s-1) values for pregel. starch – mathematical method
It can be seen that experimental “only product” thermal diffusivity values obtained by our mathematical method well agree with the theoretical one. On the other hand, by using the same software, the thermal diffusivity values obtained with the mathematical method allowed us to simulate with good accuracy the experimental heat processing. An examples of the practical coincidence of simulated and experimental heat penetration curves is shown in Figure 2.
Pre-gelatinized starch (Singh’s α = 0,142 mm2s-1) Experimental mean st.dev. “Apparent” thermal diffusivity 0,143 0,003 “Only product ”thermal diffusivity 0.046 0.008 Table 2. α (mm2s-1) values for pregel. starch – graphical method
For potatoes samples it was not possible to determine the thermal diffusivity values, because time-temperature data did not present straight-line portion in a semilogarithmic graph.
Figure 2. Tomato puree heat processing simulation
Thermal diffusivity determination – mathematical method
Potato samples
The new developed software compares simulated and experimental curves, by using a function that interpolates experimental data by creating a vector length as defined intervals number, not depending on the acquisition rate. The command used allowed us to define as routine limits a minimum and a maximum value of expected thermal diffusivity, in order to obtain less number of iterance and shorter calculation time. At the end of iteration the software gives the thermal diffusivity value corresponding to the best simulated heat
For potato samples the thermal diffusivity was calculated by using centre and surface temperature only for centre temperature values lower than 60°C, because at higher values the starch gelatinization causes a thermal diffusivity change. However the obtained values did not agree with the literature values [8, 9] (Table 5). Experimental values did not agree also with the Singh’s thermal diffusivity values calculated both for the potato raw composition and for water contents reduced according to the weight loss measured after each cooking test and inversely correlated to the heating air RH% (Table 6).
Potato thermal diffusivity Magee et al. (1995) Martens (1980) Our experimental values
0.130-0.144 0,136 0.171-0.254
Table 5. α (mm2s-1) values for potatoes – literature references
Potato thermal diffusivity Singh’s value for raw product: 0.142 Sing’s values Experimental corrected for weight loss values 0.085 0.171 0.081 0.175 0.114 0.199 0.116 0.225 0.114 0.254
RH% 5.5 7.4 14.3 20.0 31.3
Table 6. α (mm2s-1) predicted and estimated values for potatoes
However, by using our thermal diffusivity values, a well agreement was found between simulated and experimental heat penetration curves. Moreover, thermal diffusivity values showed a linear dependence with relative humidity levels of the air within the oven chamber (Figure 3). Fresh potatoes
Thermal diffusivity
0,3 0,25 0,2 0,15
y = 0,0033x + 0,1524
0,1
2
R = 0,9902
0,05 0 0
5
10
15
20
25
30
35
RH% (wet bulb)
Figure 3. Potatoes thermal diffusivity vs. oven RH%
We obtained experimental thermal diffusivity values higher than the Singh’s ones, because the last don’t consider the potato surface water evaporation and the dependence of their rate on the air relative humidity. For oven cooked meat samples, Carciofi [10] defined the thermal diffusivity calculated from experimental test as “effective thermal diffusion coefficient” because it considers also the contribution of water migration and of water evaporation that directly depends from relative humidity. So, using a single value it is possible to describe the global mass and energy transfer mechanism. Study of method limits During this work also limits of the proposed method were investigated. Even if this method has a very simple utilisation, great critical point is the heat penetration curve acquisition. In fact, the distance between the thermocouples positioned at the product’s geometrical centre and the others must be well known and constant during the thermal treatment. Since we worked with a small dimensions system, even little error in thermocouples positioning could generate great error in the thermal diffusivity determination. Table 7 presents the influence of small thermocouple positioning errors on the calculated thermal diffusivity value.
TC distance [mm] 15 16 17
Distance error [%] 0 6.7 13.4
α [mm2s-1] 0.103 0.117 0.132
Table 7. Influence of thermocouple distance on α
α error [%] 0 13.6 28.2 calculation
It is evident that even for a relatively small thermocouple distance error, the amplified error in thermal diffusivity calculation can’t be considered acceptable for the simulation and optimisation of thermal treatments, with particular reference to heat processes which require the application of a well defined sterilizing effect and/or cook value. For this reason, the proposed method needs technical developments that can guarantees the exact and unchanging thermocouples positioning. Conclusions An hardware-software apparatus for the calculation of the thermal diffusivity of conductive foods based on experimental data was developed. The proposed method allows to calculate the α value even in cases for which Ball & Olson equation is not applicable. We were able to determine “only product” thermal diffusion values for a tomato puree and starch suspension in agree to the Singh’s theoretical ones. For potatoes samples it was been possible to obtain an “effective diffusion coefficient” that considers also relative humidity and is an useful tool for cooking simulation. The proposed method is simple and could be an alternative of traditional methods, but the accuracy of the calculation strongly depends on the thermocouples positioning errors along system radius. ACKNOWLEDGMENTS Carlo Dall’Asta, Andrea Rossi and Luigi Volgarino are acknowledged for technical support and great help in prototypes development. NOMENCLATURE Symbol
Quantity
SI Unit
Latin letters specific heat capacity cp fh heating rate factor H can height R can radius r any position in radial direction Z any position in axial direction T temperature t time
JKg-1°C minutes mm mm mm mm °C minutes
Greek letters thermal diffusivity α thermal conductivity κ density ρ
mm2s-1 Wm-1°C-1 Kgm-3
Subscripts M surface node in the axial direction N surface node in the axial direction Superscripts p generic time moment
minutes
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[6] C.O. Ball and F.C.W. Olson, Sterilization in food technology, McGraw-Hill Book Co., New York, 1957. [7] L. Zurla, Modello alle differenze finite per simulare trattamenti termici applicati a prodotti alimentari. Mechanical Engineering thesis, University of Parma, , Italy, 2003. [8] T.R.A. Magee and T. Bransburg, Measurement of thermal diffusivity of potato, malt bread and wheat flour, Journal of Food Engineering, vol. 25, pp. 223-232, 1995. [9]
T. Martens, Mathematical Model of Heat Processing in Flat Containers, Ph.D. thesis, Katholeike University of Leuven, Belgium, 1980.
[10] A.M. Carciofi, J.Faistel, Gláucia M.F. Aragão and J.B. Laurindo, Determination of thermal diffusivity of mortadella using actual cooking process data, Journal of Food Engineering, vol. 55, pp. 89-94, 2002.
