Measurement of thermal diffusivity of granular food materials

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Int. Comm. Heat Mass Transfer, Vol. 24, No. 7, pp. 945-953, 1997 Copyright © 1997 Elsevier Science Ltd Printed in the USA. All fights reserved 0735-1933/97 $17.00 + .00

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MEASUREMENT OF THERMAL DIFFUSIVITY OF GRANULAR FOOD MATERIALS

S. Tavman Food Engineering Department Ege University, 35100 lzmir - Turkey I.H.Tavman Mechanical Engineering Department Dokuz Eylill University, 35100 Izmir - Turkey S. Evcin Unilever, P.K.41, 59870 ~orlu - Tekirdag - Turkey

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT In this study thermal diffusivity of two varieties of Triticum durum wheat and a wheat product, bulgur, is determined at different moisture contents and the results are compared with predictive models as well as with literature values. Thermal diffusivity is measured with transient heat conduction method which consists of placing a cylindrical sample in a constant temperature environment and recording the temperature at the center of the smnple with respect to time. If the surthce convective resistance is negligible, thermal diffusivity is determined from the timetemperature history and the dimensions of the sample. The measured thermal diffusivity values ranges between 8.92×I0 -s and l l.43×10-Sm2/s for Eregli, 8.76×10-Sand 10.78×10 -s m2/s for Saruhan and 8.28×10 -s m2/s for Bulgur. The results are in good agreement with literature values and close to Riedel and Martens models. © 1997 Elsevier Science Ltd

Introduction Knowledge of thermophysical properties of food substances is essential to researchers and designers in the field of food engineering for a variety purposes, e.g. predicting the drying rate or temperature distribution within the foods of various compositions and geometric shapes when subjected to different drying, heating and cooling conditions; or to allow optimum design of heat transfer equipment, dehydrating and sterilizing apparatus. Thermophysical properties playing an 945

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important role in the design and analysis of food processes and processing equipment are thermal diffusivity, thermal conductivity, specific heat and density. For liquid foods, these properties are available in the literature or they can be predicted with a good accuracy[l]. However, the properties of paniculate foods (powders, granular or porous products) are more difficult to predict, due to their variable heterogeneous structure [2]. Therefore, experimental measurements are especially important for this class of food products. Particulate foods, containing no free (liquid) water, can be considered two-phase systems of food panicles containing sorbed water, and air containing water vapor in equilibrium with the panicles. In such systems the measured or estimated thermal transport properties are essentially the effective thermal conductivity and the effective thermal diffusivity, which represents the overall heat conduction through the panicles and the gas phase. In situations where heat transfer occurs at unsteady state, thermal diffusivity is most useful. The value of thermal diffusivity determines how fast heat propagates or diffuses through a material. Thermal diffusivity of a material is affected by both its water content and temperature, as well as composition and porosity. Since in many processes, the water content and temperature of the product may change considerably, one can expect a variable thermal diffusivity value within a given process. Several empirical models useful in predicting thermal diffusivity of foods have appeared in the literature. Most of these models are specific to the product studied. An expression which encompasses a wider range of food products has been developed by Riedel [3] and allows prediction of thermal diffusivity as a function of water content (W) and thermal diffusivity of water (ctw) at the desired temperature: ct = 0.088 x 10 -6 + (ct w - 0.088 × 10-6)W

(1)

Martens [4] investigated the influence of water, fat, protein, carbohydrate, and temperature on thermal diffusivity. Using statistical analysis, he found that temperature and water content are the major factors affecting thermal diffusivity. Variation within the solid fraction of fat, protein, and carbohydrate had a small influence on thermal diffusivity. Martens performed multiple regression analysis on 246 published values on thermal diffusivity of a variety of food products and obtained the following regression equation: ct = [0.057363W + 0.000288(T + 273)] x 10 -6 where W is the water content, % by weight and T is the temperature in (°C).

(2)

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Increasing mechanization and the introduction of new processes in the drying and storage of cereal grains has stressed the importance of calculation of heat and mass transfer in bulk grain in order to be able to optimize drying and storage facilities. The requirement that the quality of the grain be retained, limits the increase of dryer capacity and reduction of specific energy consumption through higher drying air temperatures. Many of the problems encountered in drying and storing may be analyzed by using heat transfer principles e.g. temperature changes in a grain bin due to external or internal temperature changes may be calculated by use of the basic heat transfer equations. The use of these equations requires a knowledge of the thermal properties of the grains. Many studies are encountered in the literature about the thermal conductivity of granular foods [5-9], a linear increase of the thermal conductivity with moisture content has been determined by all the authors. Whereas studies on thermal diffusivity of granular foods [8-1 I] reveal that thermal diffusivity is not necessarily a linear function of moisture content, an increase or a decrease of thermal diffusivity with moisture content are both encountered in the literature, as thermal diffusivity is strongly dependent on bulk density and its variation with moisture content. Measurement Princiale

One-dimensional heat transfer in an infinite slab under unsteady-state conditions can be expressed as : 02T 1 tgT 0x 2 - ct /gt

(3)

Similarly the expression for heat conduction in the radial direction in an infinite cylinder is: t92T l tTF l c3T tgr2 + r & - cto~

(4)

Using Newman's rule, one can combine the solutions of equations 3 and 4 to obtain temperature profiles for a finite cylinder or a finite slab. For a finite cylinder at a uniform initial temperature, Ti, exposed to a different constant temperature environment, T s, and having negligible surface convective resistance, the solution is given by Carslaw and Jaeger [12]:

Ts-T ~ ~. 2(-l)m+l (-~) Ts-Ti - m=ln=l 13m cos

2J0(13n r/R)ex p [ ( 132 4132/ ] 13nJl(13n) -[.R2 + ~ - ) t~t

where R is the radius of the cylinder and I its length.

(5)

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For situations where the cylindrical sample is exposed to temperature gradient for a long time, that is for Fourier number Fo=c~t/R2>-_0.2, the first term of the series solution alone may be sufficient. Therefore, the solution expressed in Eq.(5) can be simplified retaining only the terms with m=n=l; i.e., tim= n/2, Bn=2.405 and J1(2.405)=0.519. At the center of the cylindrical sample, x--0, r=0 and J0(0)=l.0 . Thus, the approximation of Eq.5 for long times can be expressed as : - 2.0396.exp Ts-Ti

- ( (2.405)2 + ~ ~, R 2

at

(6)

or calculating the natural log: [(2"405)2 n 2 1 ln(Ts - T c ) = in(2.0396(T s - T i ) ) - ~ ~ ~-~- at

(7)

From this equation it can be seen that a plot of ln(T~-Tc) versus time, is a straight line in the form: ln(T s - Tc) = A - Bt

(8)

The thermal diffusivity can be calculated from the dimensions of the cylindrical sample and the slope B:

Experimental

The thermal diffusivity measuring apparatus which is shown schematically in Fig. 1, consists of a water bath with a capacity of 15.8 liters, a cylindrical sample holder, thermocouples and temperature measuring equipment. The water bath which is equipped with a thermostat and 1.5 kW heating element, is stirred and maintained at constant temperature. The sample holder tube consists of a chromium plated brass cylinder, with a radius of 2.5 cm and length 18.8 cm and two Teflon end caps of 1.5 cm length each. Chromium plated brass was chosen for the combined requirements of high thermal conductivity and rigidity, the chromium plating has excellent food particle release characteristics without impairing heat transfer and also affords protection from corrosive effect of acidic foods. The radius of the tube was chosen to be large enough to yield a measurable temperature difference between the center and the surface of the sample and small enough to eliminate the initial temperature transient. Teflon was used for the

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THERMAL DIFFUSIVITY OF GRANULARFOOD MATERIALS

Polystyrenecover

•

I°° ii

II

949

Heater/

]

Datalogger

Fig. 1 Thermal diffusivity measuring apparatus.

end caps, as it has a thermal diffusivity of 1.075x 10-7 m=/s which approximates that of the foods that are being evaluated. Thus, the length "/" subjected to constant temperature boundary condition should include the length of the two end caps and be taken as 21.8 cm. T-type thermocouples, soldered to the outside surface of the cylinder, monitor the temperature of the sample at radius R. Another T-type thermocouple of 0.2 mm in diameter, consists of a thin hypodermic probe protruding into the food sample and is located at the center of the tube. 60

50

,dr** o&.. °~..

°&° ~ * ' A ° ° ' b

**'~

• Ao.., k*''

P

°°~'*

40

°* ,*

e~

A°

E

°

°"

temperature,at the center temperature a._ttthe surface

20

I

0

20

40 Time (rain)

60

80

Fig. 2 Time - temperature history of the surface and center thermocouples.

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Regression line: y=- 0.000792x + 3.265

3 ~

;-,.2 e2 i

0

'

0

I

1,000

'

I

'

2,000 Time (S)

I

3,000

'

I

•

4,000

Fig. 3 Ln of temperature difference between center and surface temperatures versus time for bulgur.

The assembly procedure of the thermal diffusivity tube consists of inserting an end cap and filling the tube with the food sample. Then, the other cap is positioned in place and the thermocouple probe is inserted to full immersion before screwing the tube tightly shut. This ensures the proper radial positioning of the center thermocouple which is sufficiently long for proper longitudinal positioning of the temperature sensitive part of the probe. With the sample and caps, surface and center thermocouples in place, the entire assembly is positioned in the constant temperature water bath. All the thermocouples are connected to a datalogger which was set up to simultaneously read and record the temperatures every 5 minutes with a precision of 0.1°C, see Fig.2. Each experimental run takes about between one to two hours. The natural log of the temperature difference between the constant surface temperature and temperature at the center is plotted versus time and the slope B of this graph is determined using linear regression, see Fig.3. Thermal diffusivity is then calculated from the slope B according to the equation: et = 1.057 x 10-4B

(10)

Each measurement is repeated five times and the average values and standard deviations are reported.

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THERMAL DIFFUSIVITY OF GRANULAR FOOD MATERIALS

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Results and Discussion

Thermal diffusivity of two varieties of Triticum durum wheat, i.e. Eregli and Saruhan, and a wheat product, bulgur, is measured at different moisture contents (wet basis). The initial temperature of the samples which is around 20°C, is measured with a precision of 0.1 °C, whereas the constant water bath temperature is maintained at 50+0.2°C during the measurements, the average temperature for the measurements is therefore 35°C. The bulk density of each sample is determined by measuring the volume of a known weight in a graduated cylinder. The density is plotted as a function of moisture content in Fig. 4, the density decreases according to a power function with moisture content with a correlation coefficient of 0.80 : p = 904.286(100 x W)-°"°5683

( 11 )

Results of thermal diffusivity measurements are plotted in Fig.5 with respect to moisture content and the experimental results are compared with calculated thermal diffusivity values from Riedei and Martens models.Thermal diffusivities ranges from 8,92x 10-8 to 11.43x 10-8mVs for Eregli, from 8.76x!0 -8 to 10.78x10-SmVs for Saruhan, 8.28x10-SmVs for Buigur for moisture contents ranging from 5.9 to 39.71% w.b., it may be noticed that there is no significant difference between varieties in terms of thermal diffusivity results. The experimental values are respectively 7.8% and 7.3% in average lower with respect to Riedel and Martens models. This discrepancy is due to the existence of other variables than moisture content, that effect

1000 ~r~gli Sartjhan Bu~ur I A

90o v

,-- 800 O 700

600

i

0

i

i

i

10 20 30 40 Moisture content, (% by weight)

50

Fig. 4 Relation between bulk density and moisture content.

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"o --

8

E Eregli Saruhan Riedel model Martens model (T.~35°C) (T=35°C)

I.u

I

0

10

1

I

Butut/ I

20 30 40 Moisture content, (% by weight)

50

Fig. 5 Experimental thermal diffusivity results compared with Riedel and Martens models.

thermal diffusivity in nonhomogeneous food, such as particle size of the grains, their distribution, surface resistance between grains, etc. It will be necessary to study the influence of these variables and probably others, in order to establish a more reliable model for predicting the value of thermal diffusivity in nonhomogeneous food. The thermal diffusivity values for soft white wheat measured by Kazarian and Hall [8] range from 9.26x10-SmVs to 8.0xl0-amVs for moisture content (w.b.) from 0.68 to 20.3%, whereas, Chuma et al. [10] measured values ranging from 9.72x10-Sm2/s to 7.5xl0-Sm2/s for moisture content (w.b.) from 13 to 35.5%, both compare well with our experimental results, Nomenclature

B

Slope ofln(Ts-Tc) versus time

Fo Fourier number (dimensionless) J0 Bessel function of first kind of order zero J!

Bessel function of first kind of order one

k

Thermal conductivity (W/mK)

l

Length of finite cylinder (m)

R

Radius of cylinder (m)

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THERMAL DIFFUSIVITY OF GRANULARFOOD MATERIALS

r

Radial cylindrical coordinate

T

Temperature (°C)

953

Tc Temperature at the center of the sample (°C) Ti

Initial temperature of the sample and surrounding (°C)

T~ Temperature of surrounding media (°C) t

Time (see)

W Moisture content (%, w.b.) x

Rectangular coordinate (m)

ct

Thermal diffusivity (mVs)

13m Root of the cosine function 1~,

Root of Bessel function

p

Density (kg/m3) References

1. V.E.Sweat, Engineering Properties of Foods, ed. M.A.Rao and S.S.H.Rizvi, p. 49. Marcel Dekker Inc. NewYork (1986). 2. K.Wallapapan, V.E.Sweat, K.G.Diehl, and C.R.Engler, Physical and Chemical Properties of Foods, ed. M.R. Okos, A.S.A.E., St Joseph, Michigan, p.78. (1986). 3. L.Riedel, Kaltetechnik-Klimatisierung 21 (11), 315 (1969). 4. T.Martens, Mathematical Model of Heat Processing in Flat Containers. PhD thesis, Katholeike University of Leuven, Belgium (1980). 5: W.K.Bilanski, D.R.Fisher, ASAE Trans. 8 (1), 788 (1976). 6. C.S.Chang, ASAE Trans.29 (5), 1447 (1986). 7. S.Tavman, I.H.Tavman, Proceedings of the 1996 Engineering Systems Design and Analysis Conference - ASME PD-Vol.78-6, 123 (1996). 8. E.A.Kazarian, C.W.HalI, ASAE Trans. 8 (1), 33 (1965). 9. M.Kustermarm, R.Scherer, H.D.Kutzbach, J. of Food Proc. Eng. 4, 137 (1981). 10. Y.Chuma, S.Uchida, K.H.H.Shemsanga, J. Fac. Agr., Kyushu Univ., 26(1), 57 (1981). 11. A.E.Drouzas, Z.B.Maroulis, V.T.Karathanos, G.D.Saravacos, J.Food Eng. 13, 91 (1991). 12. H.S.Carslaw and J.C.Jaeger, Conduction of Heat in Solids, Oxford Univ. Press., Oxford, England (1959). Received April 9, 1997