determination of thermal diffusivity of foods using

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Determination of thermal diffusivity of foods using1D Fourier cylindrical solution

A. Baıri a,*, N. Laraqi a, J.M. Garcıa de Marıa b

a University of Paris 10, GTE, LEEE, EA 387, 1, Chemin Desvallie res, F-92410 Ville d’Avray, Franceb Universidad Polite cnica de Madrid, Departamento de Fı sica Aplicada, Ronda de Valencia, 3, E-28012 Madrid, Spain

Received 3 July 2005; accepted 9 November 2005Available online 28 December 2005

Abstract

A simple method for the determination of the thermal diffusivity of foods by using a simple test bench is presented. The method isbased on the analytical solution of the 1D Fourier equation applied to a cylinder. The estimation of its associated error for a broad rangeof temperatures is systematically accomplished. Thermal diffusivity is obtained with a precision of about 4%, acceptable for most engi-neering applications, in particular in food industry.Ó 2005 Elsevier Ltd. All rights reserved.

Keywords: Heat transfer; Foods; Thermal diffusivity; Measurement; Fourier equation; Analytical solution; Error

1. Introduction

In the food industry, the control of the temperature dur-ing the elaboration, storage, destocking, transport andcommercialisation of foods is very important. During thecooking and transformation, thermal control makes it pos-sible to limit the consumption of energy and consequentlythe production cost, but more especially to reduce the com-bustion products. That led to a better safeguarding of theenvironment and allows the manufacturing units to obeythe standards of pollution which are increasingly severe.Besides that, a correct thermal regulation is extremely

important in the microbiological mechanisms whichdirectly intervene on the qualities and characteristics of the final product as it is exposed by Wang and Sun(2001) and Tijskens, Schijvens, and Biekman (2001).

It is also known that freezing and the defrosting pro-cesses of the foodstuffs must be done according to a verystrict procedure respecting, among other, the biological

aspects. The quality of frozen foods is greatly affected bythe rate of freezing. These thermal controls can only beput in practice by the precise knowledge of the thermo-physical characteristics of the food stuffs. One of the mostimportant is the thermal diffusivity of the substance, whichintervenes in a crucial way during the transitory processes.This physical property must be necessarily determinedthrough experimental measurements.

The products and substances handled in the agro-alimentary field are very numerous and are often charac-terized by a heterogeneous composition. Appropriaterefrigeration, cooling and freezing is a challenge for the

engineers since their thermal properties and compositionmay change considerably with time and temperature.According to the needs of this industry sector, a suitableexperimental procedure must be developed for the determi-nation of the diffusivity and its dependence with respect tothe temperature.

There are numerous methods to measure the diffusivityproposed in the specialized literature. Nevertheless, mostof them need relatively complex instrumentation or exper-imental assemblies and demand an expertise of the thermalphenomena that is not within the reach of many users.

0260-8774/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfoodeng.2005.11.004

* Corresponding author.E-mail address: [email protected] (A. Baıri).

www.elsevier.com/locate/jfoodeng

Journal of Food Engineering 78 (2007) 669–675

mailto:[email protected]

mailto:[email protected]



Several papers present results of thermal characterisa-tion of different materials. Weidenfeller, Hofer, and Schil-ling (2004) propose a method of measurement of thethermal diffusivity for filler materials like magnetite, barite,talc, copper and glass fibres. Osako, Ito, and Yoneda(2004) present the evolution of the diffusivity for garnetand olivine single crystals, measured under high pressure

up to 8.3 GPa and temperatures up to 1100 K by usingheating method. Xu et al. (2004) used the Angstrommethod on cylindrical samples in multianvil bench at highpressures up to 20 GPa and temperatures up to 1373 K, tomeasure the thermal diffusivity of olivine, wadsleyite andringwoodite. The experimental data obtained for thecashew juice in the work of Azoubel, Cipriani, El-Aouar,Antonio, and Murr (2005) have been compared to thoseobtained by mathematical models found in the literature.

In the present work, we propose a simple method of measurement based on the 1D solution of Fourier’s equa-tion of heat. The reliability of the method has been estab-lished by means of consistent repetitive measurementswith an acceptable margin of error. It derives from the sim-plification of the same method based on the 2D solution of Fourier’s equation. This simplification only introduces adifference of about 4% in the diffusivity values of commonsubstances in the agro-alimentary filed, which is insignifi-cant in practical terms.

The method proposed can be easily applicable to fruitsand vegetables mixtures, which cooling requirements andmethods vary greatly depending on its perishability. Theknowledge of diffusivity and other thermal characteristicsof this products is not only important at the stage of pre-serving them after the harvest but also in the previous ones,

helping to foresee and prevent irreversible damages caused

by large temperature gradients due to adverse weatherconditions.

2. Analytical solution

The measurement of the thermal diffusivity of the sub-stances is based on a cylindrical sample of diameter 2R

and length 2L, thermally homogeneous, with conductivityk, as represented in Fig. 1.

Initially isotherm at a temperature T 0, the cylinder issuddenly subjected to a thermal environment at constanttemperature T e, where the surface conductance h isimposed. This conductance is supposed to be constantthroughout the thermal process and applies in a homoge-neous way to the entire heat-transferring surface. Such the-oretical assumptions must obviously be checked during the

Nomenclature

a thermal diffusivity of the substance, m2/sBi c modified Biot Number; Bi c = hR/k, dimension-

less

D diameter of the cylinder, mFoc modified Fourier Number Foc = at/R2, dimen-sionless

h surface conductance, W/m2 KJ 0 Bessel function, first kind, order 0J 1 Bessel function, first kind, order 1L length of the cylinder, mR radius of the cylinder, mR2 correlation coefficientR radial variable, mr* reduced radius, r* = r/R 0 6 r* 6 1, dimension-

lesst time, s

T (r,t) instantaneous local temperature of the sample,°C

T 0 initial temperature of the sample, °CT 1,T 2 particular values of T , °CT e temperature of the environment, °C

T m average temperature of the environment,°

Cx axial variable, m

Greek symbols

Da/a relative error of the thermal diffusivity, dimen-sionless

DT = T À T e temperature difference sample-bath, °Cnm

positive roots of the characteristic equationn1 first positive root of the characteristic equationk thermal conductivity, W/m Ks time constant, sh(r, t) local dimensionless temperature, dimensionlessh1,h2 particular values of h, dimensionless

L = 2 L

’

2 R

r

x

0

h

h

h

Fig. 1. Considered cylinder.

670 A. Baıri et al. / Journal of Food Engineering 78 (2007) 669–675



experiments by means of an adequate assembly. Two casescan arise:

(i) If the dimensions of the cylinder make significant theheat exchange with the environment through thebasis, the problem should then be considered in 2D

(axial and radial). In this case, the transient thermalsolution T (x, r, t) is based on resolving the 2DFourier’s equation:

r2T ¼

1

r

o

or r oT

or

þo

2T

o x2¼

1

a

oT

ot ð1Þ

Several works faced this problem by means of differ-ent mathematical approaches to estimate the thermaldiffusivity of foods by using transfer functions (Gla-vina, Di Scala, Ansorena, & del Valle, in press).Hakayawa and Ball (1971) treated this question infor the case of a very high surface conductance h,leading to a negligible surface resistance, while the

study by Magee and Bransburg (1995) is based onthe exploitation of the 2D solution on the axis of the cylinder.

(ii) On the other hand, if the cylinder is long enough toconsider that the exchanges are made quasi exclu-sively through the lateral surface, the problem is sim-plified and becomes 1D. Being the determination of diffusivity more laborious in 2D, we have checkedthe limitations of the 1D method to propose it as asimpler method. After several tests carried out withdifferent products in a broad range of temperature,we noticed that the 2D model is necessary in certain

fields such as building. Conversely, for typical agro-alimentary substances, the difference between thetwo methods is of about 4%. This result aims toretain the procedure presented in this work which isbased on the analytical solution of the 1D Fourier’sequation in cylindrical co-ordinates:

r2T ¼

1

r

o

or r oT

or

¼

1

a

oT

ot ð2Þ

solved by applying it to the infinite cylinder with sym-metric contour conditions respect to the axis r = 0.Using the method of separation of variables for r

and t developed by Baıri and Laraqi (2003), it is ob-

tained that the solution can be written as

hðr ; t Þ ¼T ðr ; t ÞÀT e

T 0ÀT e

¼ 2X1m¼1

exp Àn2m Foc

À Á 1

nm

J 1ðnmÞ

J 20ðnmÞþ J

21ðnmÞ

J 0ðnmr ÃÞ

ð3Þ

where J 0 and J 1 are the Bessel function of first kind of order 0 and 1, respectively, n

mthe positive roots of

the characteristic equation:

nm J 1ðnmÞ À Bic J 0ðnmÞ ¼ 0; ð4Þ

r Ã ¼

r

R

ð5Þ

is the reduced radius,

Bic ¼hR

kð6Þ

the modified Biot number, and

Foc ¼at

R2 ð7Þ

the modified Fourier number.

When the film coefficient h is very high, it can be shownthat the previous infinite series can be approximated by thefirst term. Taking then the precaution in the experimentalprocess of submerging the cylindrical sample in a fluidcharacterized by a high surface conductance h, the solutionis written as

hðr ; t Þ ¼ 21

n1

J 1ðn1Þ

J 2

0

ðn1Þ þ J 2

1

ðn1Þ

J 0ðn1r ÃÞ

& 'exp Àðn2

1 FocÞÂ Ãð8Þ

Calling K to the constant part of this equation, itbecomes

hðr ; t Þ ¼ K exp Àðn21 FocÞ

Â Ã¼ K exp Àa

n21

R2

t

!ð9Þ

The first root of the characteristic equation (with h verylarge) is n1 2.405. The measurements clearly show (Figs. 3,6 and 9) that after some time t, the temperature T evolves,for any radius r, following a pure exponential. The mini-mum time t corresponds to the limit value of Foc that

allows the convergence of the explicit form of the discreteFourier equation. The time constant s would be in thiscase:

s ¼ an

21

R2

!À1

ð10Þ

Measuring carefully the slope of the curve ln(h) = f (t)through a normal statistical study, it is easy to obtain thevalue of s and from it the thermal diffusivity a by meansof the simple relation:

a ¼1

s

R

2:405

2

ð11Þ

We also evaluate the precision corresponding to thismethod of measurement. It must be stand out that themethod does not require knowing the exact place in whichthe temperature is measured within the cylindrical sample.Otherwise it would be very difficult to obtain it with theadequate precision. This constitutes the greater advantageof the present method for determining the thermaldiffusivity.

However, the experiment shows that better results areobtained when the thermocouple is placed in the farthestpoints from the surface subjected to the environment, i.e.

on the axis of the cylinder. In this case, measurements

A. Baıri et al. / Journal of Food Engineering 78 (2007) 669–675 671



are less sensitive to the difficulties and errors in handlingthe sample during the short experiment. This simplifiesthe examination of the data and their statistical analysisand makes this experimental approach accessible to peoplenot necessarily qualified in sciences of heat.

3. Proving bench

The proving stand, displayed in Fig. 2, consists of:

(i) A thermo-regulated bath able to maintain the tem-perature with a precision of less than 0.1 °C. Therange of temperatures of reference extends fromÀ150 to 300 °C. A cryogenic fluid bath (cryostat withdouble compression) is used for À150 < T e < +20 °Cwhile the classic oil bath is used for 10 < T e < 300 °C.For a more narrow range (10 °C < T e < 80 °C), a sim-ple water bath is adequate. The temperature of thefluid bath is kept uniform by means of a stirrer,

imposing in addition a high surface conductance hwhich is essential for our method.

(ii) A cylindrical sample-holder of aluminium that con-tains the substance for which we want to measurethe thermal diffusivity. The wall of the cylinder is verythin and with a high thermal conductivity, leading toa very low thermal resistance to conduction. Sample-holders of different sizes were used in this work: thelength varies between 100 and 150 mm and the diam-eter 2R between 9 and 16 mm depending on theexperiment and the sample tested.

(iii) Three K-type thermocouples of 0.1 mm of diameter.

One of them is differential for measuring directlythe temperature difference DT = T À T e between thesample and the bath. For safety’s sake, we disposetwo additional separate thermocouples for measuringeach one of the temperatures T and T e. The weldslocated inside the sample are fixed relatively far fromthe wall, although its precise situation is not veryimportant with our method.

(iv) A computer-controlled data acquisition system towhich we connected the thermocouples. Consideringthe constant of time of the measurement chain, thereading of channels is made with a cadence of 200 ms. In general, a few minutes of measurementin transient state are sufficient ($4 min).

4. Results

To illustrate our method, we present next the resultsobtained in the measurement of the diffusivity of differentsubstances. We firstly analyse the water which, apart of being the basic content of most foods and subsequentlydominates their thermal properties, will be useful as a ref-erence. We have studied also foodstuffs such as the olive oiland the chopped red meat. The encountered values havebeen successfully confronted with the data found in theliterature.

4.1. Water

The sample-holder of length L = 150 mm and diameter2R = 10 mm, initially at T 0 = 60.0 °C, is cooled by meansof a bath at temperature T e = 10.0 °C ± 0.1 °C. The evolu-tion of the temperature of the sample T , registered each200 ms, is plotted in Fig. 3. It can be seen that, after a cer-tain time of immersion of the sample in the bath (of about5 s in this case), the evolution of T is exponential, in accor-dance to what has been pointed out in the previous section.Fig. 4 corresponds to the temporal evolution of the dimen-

sionless temperature h for the time interval 0 < t < 100 s.A least squares fitting of the logarithm of h vs. time

yields a correlation coefficient R2 higher than 99.5%, whatproves that the experimental data perfectly fit into a linearlaw. The slope p obtained in this way allows us to calculate,for an average temperature T m of 35.0 °C, a value of thediffusivity a = 1.498 · 10À7 m2 sÀ1. This value coincideswith those given in the basic literature.

Further measurements were carried out at different butrelatively close temperatures T 0 and T e (differencejT 0 À T ej % 5–7 °C) and in both senses (heating and cool-ing baths). We present in Fig. 5 the evolution of the diffu-sivity of water for an average temperature T m in the range

Fig. 2. Elements of the breadboarding: (1) fluid bath; (2) stirrer; (3)

sample-holder cylinder; (4) computer; (5) data-acquisition system.

00 20 40 60 80 100

t (s)

20

40

60

Water T : T 0 = 60˚C to T e=10˚C

T

Fig. 3. Evolution of the temperature T for the sample of water.




35.0–73.2 °C. The values of diffusivity used in that figureare the averaged values corresponding to different test inheating and cooling at the considered temperature. Theresults confirm that the hotter the water the more diffusive.

This proportion obeys a linear law which can be written asawater ¼ ð3:36 Â 10À10

T m þ 1:38 Â 10À7Þ m2=s;

where T m is expressed in C. ð12Þ

4.2. Olive oil and minced red meat

Numerous tests have been made with a cylindrical sam-ple-holder of length L = 130 mm and diameter2R = 14 mm filled with olive oil, plunged into coolingand heating baths, with different values of T 0 and T e andthen of T m. Fig. 6 presents the temporary evolution of T

for a cooling test from 30 to 0 °C. Details of the evolutionof h are given in Figs. 7 and 8. We wanted to examine theimportance of the period of sampling on the determinationof diffusivity. It can be noticed that the results obtained byconsidering only the first 140 s of the test are very close tothose obtained by taking 350 s, i.e, including the distur-

bances corresponding to the end of the test, clearlyobserved in Fig. 8.Similar tests were carried out with minced red meat

(sample of length 150 mm and diameter 2R = 11 mm).We present in Fig. 9 the evolution of T for a test between27 and 0 °C. The evolution of the dimensionless tempera-ture h for this case is presented in Fig. 10. It can be seenin Fig. 11 that the diffusivity of this product, as for the

-40 20 40 60 80 100

t (s)

-3

-2

-1

0

L n ( )

Water T : T 0= 60˚C to T e=10˚C

Ln( ) = -0.0349 t ; R2=0.9955

awater=1.498.10-7 m2 /s

θ

θ

Fig. 4. Evolution of the dimensionless temperature h for the sample of water.

1.48

1.52

1.56

1.60

1.62

a . 1

0 7

( m 2 / s )

30 50 70

T m (˚C)

Water

awater=(3.3556.10-10 T m+1.3840.10-7) m2 /sT m in ˚C ; R2=0.9956

Fig. 5. Dependence of the diffusivity of the water with the averagetemperature T m.

t

T

0 100-5

200 300

20

10

0

30

Olive oil

T : T 0= 30˚C to T e=0˚C

Fig. 6. Evolution of the temperature T for the sample of olive oil.

t (s)

0 60 120

-0.8

-2.0

-1.6

-1.2

-0.4

0.0

Ln( )=-0.0129 t ; R2=0.9989

T : T 0 = 30˚C to T e=0˚C

Olive oil 0<t <140 s

a=1.089.10

-7

m

2

/s

L n (

) θ

θ

Fig. 7. Evolution of the dimensionless temperature h for the olive oilsample (0 < t < 140 s).

-6

-4

-2

0

L n (

)

0 100 200 300

t (s)

Ln( )=-0.0133 t ; R2=0.9855

T : T 0= 30˚C to T e=0˚C

Olive oil 0<t <350 s

a=1.119.10-7 m2 /s

θ

θ

Fig. 8. Evolution of the dimensionless temperature h for the olive oilsample (0 < t < 350 s).

0

10

20

30

0 40 80 120 160

t (s)

Minced red meat

T : T 0= 27˚C to T e=0˚C

T

Fig. 9. Evolution of the temperature T for the sample of minced red meat.




water, varies linearly with the temperature. The correlationcoefficient is a little worse in this case, but still exceedingthe 98% in the range of temperature T

mgoing from

À6.6

to +55.8 °C. The dependence of diffusivity with T m canbe modelled by the following law:

aminced red meat ¼ ð1:68 Â 10À9T m þ 1:12 Â 10À7Þ m2=s;

with T m is expressed in C ð13Þ

It is clear that diffusivity depends on several parameters,among other the composition of the product, in particularof its water content, and the direction of the fibres in thecase of meat. We do not claim in this paper to examinethe evolution of the diffusivity according to these character-istics; that is left to the appreciation of the specialists infoodstuffs analysis. The values provided in this study wereobtained without a prior examination of all these charac-teristics, our objective being simply to validate the straight-forward method proposed. In any case, the results obtainedagree with the values found in the literature, as thosequoted by Kubasek et al. (in press) for the olive oil andby Markowski, Bialobrzewski, Cierach, and Paulo (2004)for the minced beef. Tests were also carried out for carrotsand potatoes and the results are also satisfactory.

5. Associated error and method validation

Since the reliable determination of the diffusivity is

bound with the inaccuracy of the measures, the realized

approaches and the inspection of the results, we have esti-mate the corresponding error. Keeping in mind the expres-sion of the diffusivity and assuming that the errorassociated to the determination of the roots is sufficientlysmall, the calculation of the relative error of the diffusivitya is given by

Da

a¼ 2

h2

ln h1=h2

1T 1 À T e

þ 1T 2 À T e

þ 2T 0 À T e

DT

þ2

t 1 À t 2

Dt þ 2

D R

R

The error considered for the time is of 1 · 10À3 s and of 1 · 10À5 m for the diameter. The temperature is measuredwith a relative precision better than 0.5%. The most unfa-vourable calculation of the error provides us a value of the order of 7% (max(Da/a) = 0.07), taking into accountthe correlation coefficient R2 derived from the least squaresfitting. It should be pointed out that the last two terms of

the previous expression are negligible compared to the firstone.

We consider that this value of the error is sufficientlysmall as to validate the proposed method, more inasmuchas the diffusivity is also very small. Although the thermalresistance to conduction of the sample-holder (cylinder of aluminium) is insignificant, the measurements should becorrected in transient state. It is necessary to consider theheating lag carried with it and to evaluate it in a systematicway, as well as the heat capacity of the sample-holder. Thisaspect has been studied by means of the verification withnonporous solid products that can be proven with and

without the cylindrical sheath. We have checked that mostof the discrepancies between the measures are within a 3%in average. This confirms the validity of the methodexposed in this work at least for nonporous materials forwhich the influence of the external fluid (bath) is unperceiv-able (absence of diffusion).

6. Conclusion

The method described in this work is easy to apply andit allows us to obtain the thermal diffusivity of substancesfor a broad range of temperatures. The calculation of theerrors keeping in mind the simplification of the analytical

solution and the inaccuracies of the measurements associ-ated with the present instrumentation, provides a veryacceptable margin of error as far as the applications inengineering is concerned, in particular for agro-alimentaryproducts.

References

Azoubel, P. M., Cipriani, D. C., El-Aouar, A. A., Antonio, G. C., &Murr, F. E. X. (2005). Effect of concentration on the physicalproperties of cashew juice. Journal of Food Engineering, 66 (4),413–417.

Baıri, A., & Laraqi, N. (2003). Diagrams for fast transient conduction

in sphere and long cylinder subject to sudden and violent thermal

-4

-3

-2

-1

0

0 40 80 120

t (s)

L n ( )

Minced red meat

T : T 0= 27˚C to T e=0˚C

Ln( )=-0.0259 t ; R2=0.9969

a=1.3454.10-7 m2 /sθ

θ

Fig. 10. Evolution of the dimensionless temperature h for the sample of minced red meat.

1.00

1.40

1.80

2.20

a . 1

0 7 ( m

2 / s )

-10 10 30

T m (˚C)

50

Minced red meat

a=(1.68493.10-9 T m+1.12448.10-7) m2 /s

T m in ˚C ; R2=0.98089

Fig. 11. Dependence of the diffusivity of the minced red meat with theaverage temperature T m.




effects on its surface. Applied Thermal Engineering, 23(11), 1373– 1390.

Glavina, M. Y., Di Scala, K. C., Ansorena, R., & del Valle, C. E. (inpress). Estimation of thermal diffusivity of foods using transferfunctions, LWT .

Hakayawa, K., & Ball, C. O. (1971). Theoretical formulas for tempera-tures in cans of foods and for evaluating various heat processes.Journal of Food Science, 36 , 306–310.

Kubasek, M., Houska, M., Landfeld, A, Strohalm, J., Kamarad, J., &Zitny, R. (in press). Thermal diffusivity estimation of the olive oilduring its high-pressure treatment. Journal of Food Engineering ,doi:10.1016/j.foodeng.2005.03.019.

Magee, T. R. A., & Bransburg, T. (1995). Measurement of thermaldiffusivity of potato, meat bread and wheat flour. Journal of Food

Engineering, 25, 223–232.Markowski, M., Bialobrzewski, I., Cierach, M., & Paulo, M. (2004).

Determination of thermal diffusivity of Lyoner type sausages duringwater bath cooking and cooling. Journal of Food Engineering, 65,591–598.

Osako, M., Ito, E., & Yoneda, A. (2004). Simultaneous measurements of thermal conductivity and thermal diffusivity for garnet and olivineunder high pressure. Physics of the Earth and Planetary Interiors, 143–

144, 311–320.Tijskens, L. M. M., Schijvens, E. P. H. M., & Biekman, E. S. A. (2001).

Modelling the change in color of broccoli and green beans duringblanching. Innovative Food Science & Emerging Technologies, 2(4),303–313.

Wang, L., & Sun, D. W. (2001). Rapid cooling of porous and moisturefoods by using vacuum cooling technology. Trends in Food Science and

Technology, 12, 174–184.Weidenfeller, B., Hofer, M., & Schilling, F. R. (2004). Thermal conduc-

tivity, thermal diffusivity and specific heat capacity of particle filledpolypropylene. Composites, Part A: Applied Science and Manufactur-

ing, 423–429.Xu, Y., Shankland, T. J., Linhardt, S., Rubie, D. C., Langenhorst, F., &

Klasinski, K. (2004). Thermal diffusivity and conductivity of olivine,wadsleyite and ringwoodite to 20 GPa and 1373 K. Physics of the

Earth and Planetary Interiors, 143–144, 321–336.


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determination of thermal diffusivity of foods using

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