- Email: [email protected]
IR temperature measurements in microwave heating
Infrared Physics & Technology 43 (2002) 145–150 www.elsevier.com/locate/infrared
IR temperature measurements in microwave heating G. Cuccurullo a
a,*
, P.G. Berardi a, R. Carfagna a, V. Pierro
b
Dipartimento di Ingegneria Meccanica, Universit a di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy b Ingegneria delle Telecomunicazioni, Universit a del Sannio Corso Garibaldi, 107, 82100 Benevento, Italy
Abstract In this paper a technique for the evaluation of the dielectric constant of a sample placed inside a microwave oven and confined in a cylindrical box is proposed. The box acts as a waveguide so that a simple model for the propagating wave can be assumed. Since traditional techniques for temperature measurements cannot be applied in microwave heating, the IR thermography shows to be an useful tool for measuring the sample surface temperature. The measure of the surface temperature evolution in the sample along with application of a simple analytical model allows to obtain the dielectric constant of the sample as a function of chemical composition, temperature and frequency. Preliminary results are presented and discussed with reference to pure water. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Dielectric constant; Thermography; Microwave
1. Introduction The potential benefit of microwave heating led several research groups to investigate the subject in the last 40 years [1]. Microwave energy is converted directly into heat only within the material, so that little or no thermal lag occurs (diffuse volume heating) and almost no heating of the container take place. This results in a more uniform temperature distribution within the material (provided the thickness of the sample does not exceed the specific penetration depth of the microwave field), together with reduced operating and maintenance costs. Moreover, the absence of thermal lag allows in principle some real time control of the microwave power, in view of process optimization.
*
Corresponding author. Fax: +39-8-996-4037. E-mail address: [email protected] (G. Cuccurullo).
These nice features, however, depend considerably on the dielectric properties of the materials [2], thus detailed knowledge of the dielectric properties as a function of chemical composition, temperature, frequency is required to exploit the full potential of microwave heating. In this connection the measure of dielectric constant of materials is a crucial problem in the microwave heating application field. There are several classical methods currently used to measure the dielectric constant at microwave frequencies, depending on the properties of materials (solid, liquid, gas, polymer resin, etc.) [3– 6]. Generally each method is tailored for a specific class of materials, and there is not a universal one. In this paper a technique for measuring the dielectric constant is proposed. The method is based on a simple analytical model and on the application of the IR thermography for measuring the surface temperature of the sample under test. Both the ability of thermographic techniques to record
1350-4495/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 5 0 - 4 4 9 5 ( 0 2 ) 0 0 1 3 3 - 0
146
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
temperature map evolution of the target and a suitable data processing helps in deriving the dielectric constant dependence on temperature. Experimental tests have been carried out on a cylindrical metallic test-tube plugged on the bottom and acting as short circuited waveguide; the tube is filled with dielectric homogeneous material. The measure of surface temperatures allows to recover the distribution of the EM field and therefore to infer the dielectric constant of the media. The IR thermography plays an essential role in experimental tests since (due to the EM field and the metal probes interactions) traditional techniques for temperature measurements cannot be applied inside microwave ovens. The measure of the dielectric constant is accomplished provided that the sample is homogeneous with respect to both the range of the electromagnetic field (for the commercial ovens the wavelength is about 10 cm) and to the geometry of the short circuited waveguide. The technique being not invasive seems to be particularly suitable for solid or liquid materials for which it is difficult to insert the probe either when the media sticks to the instrumentations (e.g. polymeric resins at high temperatures) or when the media can damage the probe or perturb the measure. Preliminary experimental tests have been carried out on distilled water to verify the feasibility of the proposed procedure.
is expected due to the sample EM properties temperature dependence. This latter effect is less significant than the former and, at first glance, will be neglected. In the limits of the present study, focused on the analysis of the thermal aspects of the problem at hand, a simplified model is introduced to synthesize the character of the propagation of an electromagnetic wave inside the sample and the interface phenomenon. The tube is a homogeneous (dielectric-dissipative) waveguide in which a single electromagnetic wave propagates (monomodal propagation). The dissipation phenomena can be taken into account using a complex propagating constant resulting into an exponentially decreasing amplitude of EM field intensity along the propagation direction [7]. This, in turn, originates an exponentially decreasing production term, proportional to the square of EM field intensity, in the energy balance inside the media. The simple propagation model is appropriate provided that reflected waves (from the short circuit) are negligible. Furthermore, having to consider small times, the radial terms in the Laplace operator are assumed to be small with respect to the longitudinal term. Thus the problem can be conveniently sketched with reference to a semi-infinite flat plate (Fig. 1) heated by the absorption of EM energy and cooled by radiative–convective heat transfer. Note that the assumption regarding the convective motions inside the sample is realistic since energy
2. Basic equations and analytical solution The computation of the dielectric constant from temperature–time profiles needs a model for the heat diffusion provided that, for liquid media, MW heating inside the sample is such that convective motions and evaporation the sample are negligible. This condition is needed to simplify the analytical model and requires only early stages of the heating process to be considered and small ambient to media temperature differences. Considering that the electromagnetic field inside the sample is the result of the interaction between the present load and the produced field, the production term turns out to be varying in space. In addition a time variation of the production term
Fig. 1. Sketch of the analytical model.
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
generation decreases from the top to the bottom of the waveguide. For such a problem, the energy balance equation and the boundary conditions can be written in dimensionless form as:
147
hðn; sÞ ¼ Rm ½Em Xm ðnÞ=ðk2m Þ Rm ½Em expðk2m sÞXm ðnÞ=ðk2m Þ
ð8Þ
hnn hs ¼ ð/ L =dÞ exp½ð1 nÞ=d
ð1Þ
hðn; s ¼ 0Þ ¼ 0
ð2Þ
where the first term on the right hand side is recognized as the steady solution. To show the temperature field behaviour, it is interesting to examine two limiting cases for the solution, Figs. 2 and 3:
hn ðn ¼ 0; sÞ ¼ 0
ð3Þ
1. d ! 0, slight attenuation of microwave field
hn ð1; sÞ ¼ Bi½hð1; sÞ 1
ð4Þ
where h ¼ ðT Ti Þ=ðTa Ti Þ, is the temperature, Ti being the initial and Ta the environment temperatures; n ¼ x=L, is the spatial coordinate, L being the slab thickness; s ¼ at=L2 , is the time, a being the thermal diffusivity; Bi ¼ hL=k is the Biot number, h and k being the heat transfer coefficient and the thermal conductivity; / L ¼ ðqL LÞ=½kðTa Ti Þ, qL being the absorbed wall radiative intensity; d ¼ D=L is the penetration depth generally depending on the media, the frequency and the thermodynamic state, e.g. temperature, water content. The unsteady thermal field in the slab has been solved analytically in closed form by the method of separation of variables in terms of infinite series assuming: X hðn; sÞ ¼ Xn ðnÞAn ðsÞ ð5Þ n
hðn; s ! 1Þ ¼ 1 þ / L =Bi
ð9Þ
the steady temperature profile attains an uniform value inside the slab, thus the dissipation
Fig. 2. Temperature field for slight microwave attenuation.
where the eigenfunctions Xn ðnÞ ¼ cosðkn nÞ are built from the corresponding homogeneous and the eigenvalues, kn , are related to the Biot number by the characteristic equation kn ¼ Bi=tgðkn Þ. Recalling the orthogonality of the eigenfunctions, an ordinary differential equation can be set up for the coefficients Am ðsÞ [8]: dAm ðsÞ=ds þ k2m Am ðsÞ ¼ Em ;
Am ð0Þ ¼ 0
ð6Þ
where Em ¼ 2Bi cosðkm Þ Z 1 =dÞ exp½ð1 nÞ=dXm ðnÞdn þ 2ð/ L
ð7Þ
0
Finally, the unsteady temperature field is given by:
Fig. 3. Temperature field for strong microwave attenuation.
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G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
due to microwaves is negligible inside the slab and its effect is felt only at the interface, i.e. on the boundary condition. Traditional IR heating in the classical ovens is recovered. 2. d > 1, strong attenuation of microwave field Qgener ¼ ð/ L =dÞ exp½ð1 nÞ=d ð/L =dÞ
ð10Þ generation is uniform inside the slab while, as expected, developing the solution in Taylor series gives: 2 hðnÞ ¼ 1 þ / L =ðBidÞ þ /L ð1 n Þ=ð2dÞ
ð11Þ
a parabolic temperature distribution. Fig. 4. Sketch of the experimental apparatus.
The limiting cases show that the effects of the microwave heating can be divided in two steps: in the early stage the sample is heated both by the environment and by the EM field; later, a particular time location can be expected such as the cooling rate and the heat generation are balanced: a zero wall slope curve can be singled out after which the heating due to microwave still holds while the environment acts as cooling source. 3. Experimental set-up The above model may be used for setting up an experimental procedure to measure the penetration depth and, in turn, the dielectric constant; in fact, for dissipative materials (e00eff =e0 1), these parameters are related by [1]: d ¼ kðe0 Þ1=2 =ð2pe00eff Þ
ð12Þ 0
e00eff
where k is the wavelength, e and are respectively the real and imaginary part of complex dielectric constant. Eq. (12) shows that it is possible to recover e0 and e00eff from several measurements of d at different frequencies, provided that an appropriate model for dielectric constant is available. Thus, the aim of the procedure is shifted in measuring the penetration depth. The experimental setup, shown in Fig. 4, is essentially composed of an Infrared Thermography System (Inframetrics SC1000) for the surface temperature detection, a commercial microwave oven (230 V, 50 Hz, max MICRO OUTPUT 900 W a
2450 MHz), a can containing the sample to test and a suitable SW for data reduction. Particular care was taken about the insulation of the oven walls with polystyrene sheets in order to avoid the effect of the magnetron cooling system. Aluminium cans (thickness 0.5 mm, diameter 67 mm, height of 50, 100 and 150 mm) and steel cans (thickness 0.5 mm, diameter 85 mm, height of 200 mm) were tested with different heating powers. The use of different heights of the can is required to control the effect of the reflected wave. Measures of temperature profiles along the sample were taken quickly opening and closing the oven door at prescribed time intervals. According to the model, the dimensionless temperature profile depends on the following parameters: T Ti x at hL ; ¼ f n ¼ ; s ¼ 2 ; Bi ¼ L L k Ta Ti D d ¼ ; /L ¼ ðqL LÞ=½kðTa Ti Þ L ð13Þ Assuming the measurements errors to be independent and Gaussian distributed, the parameter estimator can be chosen as the chi-square: v2 ¼ 2 R½Ti T ðxi , a; bÞ=ri =n, where Ti and xi (i ¼ 1–n) are the experimental data for a fixed time with their standard deviation ri ; a is the unknown adjustable parameters vector with three components, that is the heat transfer coefficient, h, the
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
penetration depth and the absorbed wall heat flux, q L ; b is the known data vector (Ta , Ti , L, t, k, a). Data reduction is obtained by minimizing the chi-square evaluated with temperature profiles at fixed times. The model exhibits non-linear dependence on the unknown parameters except for wall heat flux; thus, the numerical Marquardt method [9] for non-linear parameter estimation, an extension of linear scheme with a proper technique to calculate the Hessian matrix, is used. The merit function of the problem at hand being not unimodal, the Marquardt technique is preceded by a random sampling of unknown parameter values within fixed ranges in order, to estimate the region in which the absolute minimum can occur.
The monotonic decay of temperature profiles shows the goodness of the electromagnetic model. In particular the absence of reflected waves on the plugged side of the can, can be noted. As expected two different regions can be identified: the first one close to the heated surface where the sample is cooled by the ambient; a second one where temperature profiles monotonically decrease deeper into the sample due to the decreasing energy production of the electromagnetic field. With reference to water, a further simplification in the computation can be obtained by noting the low values attained in principle by the heat transfer coefficient and by the group a=L2 . Thus, Eq. (9) can be written: x=d DT ¼ T ðx; tÞ Ta ffi ½/ L =ðqcp Þt e
4. Results and discussion In order to verify the feasibility of the above procedure, preliminary experimental tests have been performed on pure water, assuming that the initial sample temperature is the ambient temperature. In Fig. 5, a typical surface temperature map shows the temperature decay in response to microwave heating taken at steps of 5 up to 35 s for pure water placed in an aluminium can 100 mm tall. The quite perfect axial symmetric temperature distribution shows that the assumption of electromagnetic field intensity uniformity is fulfilled.
Fig. 5. Typical surface temperature map.
149
ð14Þ
Eq. (14) shows the exponential decay in space and a linear increase in time of the ambient excess temperature. The problem can be reduced to a single parameter regression, d, since: (i) the estimation of the term / L is inessential due to the linear dependence exhibited in Eq. (14); (ii) Eq. (14) corresponds to the physical case of adiabatic heating, thus no trace of the heat transfer coefficient can be found. In Fig. 6 experimental and analytical curves according to Eq. (14) are plotted assuming a typical value of D ¼ 17 mm for water. The linear increase in time is evident and justifies the assumptions done; a further observation can be done about the trend shown: the curves markedly differ toward the wall where temperatures are higher, while the extension of the region in which
Fig. 6. Experimental vs. analytical temperature profiles.
150
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
5. Conclusions
Fig. 7. Penetration depth temperature dependence.
The simple analytical model on which the experimental procedure is based seems to allow the evaluation of the temperature dependence of the penetration depth with good accuracy avoiding difficulties connected with probe insertion in microwave ovens. The encouraging result obtained with reference to pure water suggest further developments of the proposed experimental protocol. Future efforts should be directed to the improvement of the experimental setup for the continuous detection of temperature profiles and to improve the EM analytical model to take into account the reflected back waves observed in the case of short cans. Additional experimental tests could scan wider temperature ranges by cooling the media and/or controlling the environment temperature. References
Fig. 8. Data reduction results for penetration depth.
agreement is satisfying increases with decreasing times. This is a proof of the stronger attenuation close to the surface. This behaviour is coherent with the one outlined in literature [1], see Fig. 7, where the trend of the penetration depth vs. temperature is reported. The description given is, of course, inappropriate in the cooled zone very close to the interface, where the adiabatic slab assumption fails. A suitable data reduction for dðT Þ allowed to plot Fig. 8, where the average results (with maximum relative error equal to 7%) related to temperature profiles taken at 15 and 20 s are reported and compared with classical curves found in Ref. [1]. These values agree quite fairly with the ones found in literature [1]. The apparent trend shows an overestimation for higher temperatures, probably due to the cooling close to the interface.
[1] A.C. Metaxas, R.J. Meredith, Industrial Microwave Heating, Peter Peregrinus, Ltd., London, 1983. [2] A. von Hippel, Dielectric Materials and Applications, John Wiley & Sons, New York, 1954. [3] G. Franceshetti, A complete analysis of the reflection and transmission methods for measuring the complex permittivity of the materials at microwaves, Alta Frequenza 36 (8) (1967) 757–764. [4] J.A. Batt, R. Ruckus, M. Gilden, General purpose high temperature microwave measurement of electromagnetic properties, in: R.L. Beatty, W.S. Sutton, M.F. Iskander (Eds.), Microwave Processing of Materials III, vol. 269, Materials Research Society, Pittsburg, PA, 1992, pp. 553– 559. [5] U. Kaatze, K. Giese, Dielectric relaxation spectroscopy of liquids: Frequency domain and time domain experimental methods, J. Physics E: Sci. Instrum. 13 (1980) 133–141. [6] M.N. Afsar, J.R. Birch, R.N. Clarke, G.W. Chantry, The measurement of the properties of materials, Proc. IEEE 74 (1986) 183–199. [7] R.E. Collin, Foundations for microwave engineering, McGraw-Hill, New-York, 1992. [8] R. Carfagna, Studio Termografico del Riscaldamento a Microonde, Ph.D. dissertation, Universita degli Studi di Salerno, 2001. [9] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C., Cambridge University Press, Cambridge, 1990.
IR temperature measurements in microwave heating G. Cuccurullo a
a,*
, P.G. Berardi a, R. Carfagna a, V. Pierro
b
Dipartimento di Ingegneria Meccanica, Universit a di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy b Ingegneria delle Telecomunicazioni, Universit a del Sannio Corso Garibaldi, 107, 82100 Benevento, Italy
Abstract In this paper a technique for the evaluation of the dielectric constant of a sample placed inside a microwave oven and confined in a cylindrical box is proposed. The box acts as a waveguide so that a simple model for the propagating wave can be assumed. Since traditional techniques for temperature measurements cannot be applied in microwave heating, the IR thermography shows to be an useful tool for measuring the sample surface temperature. The measure of the surface temperature evolution in the sample along with application of a simple analytical model allows to obtain the dielectric constant of the sample as a function of chemical composition, temperature and frequency. Preliminary results are presented and discussed with reference to pure water. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Dielectric constant; Thermography; Microwave
1. Introduction The potential benefit of microwave heating led several research groups to investigate the subject in the last 40 years [1]. Microwave energy is converted directly into heat only within the material, so that little or no thermal lag occurs (diffuse volume heating) and almost no heating of the container take place. This results in a more uniform temperature distribution within the material (provided the thickness of the sample does not exceed the specific penetration depth of the microwave field), together with reduced operating and maintenance costs. Moreover, the absence of thermal lag allows in principle some real time control of the microwave power, in view of process optimization.
*
Corresponding author. Fax: +39-8-996-4037. E-mail address: [email protected] (G. Cuccurullo).
These nice features, however, depend considerably on the dielectric properties of the materials [2], thus detailed knowledge of the dielectric properties as a function of chemical composition, temperature, frequency is required to exploit the full potential of microwave heating. In this connection the measure of dielectric constant of materials is a crucial problem in the microwave heating application field. There are several classical methods currently used to measure the dielectric constant at microwave frequencies, depending on the properties of materials (solid, liquid, gas, polymer resin, etc.) [3– 6]. Generally each method is tailored for a specific class of materials, and there is not a universal one. In this paper a technique for measuring the dielectric constant is proposed. The method is based on a simple analytical model and on the application of the IR thermography for measuring the surface temperature of the sample under test. Both the ability of thermographic techniques to record
1350-4495/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 5 0 - 4 4 9 5 ( 0 2 ) 0 0 1 3 3 - 0
146
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
temperature map evolution of the target and a suitable data processing helps in deriving the dielectric constant dependence on temperature. Experimental tests have been carried out on a cylindrical metallic test-tube plugged on the bottom and acting as short circuited waveguide; the tube is filled with dielectric homogeneous material. The measure of surface temperatures allows to recover the distribution of the EM field and therefore to infer the dielectric constant of the media. The IR thermography plays an essential role in experimental tests since (due to the EM field and the metal probes interactions) traditional techniques for temperature measurements cannot be applied inside microwave ovens. The measure of the dielectric constant is accomplished provided that the sample is homogeneous with respect to both the range of the electromagnetic field (for the commercial ovens the wavelength is about 10 cm) and to the geometry of the short circuited waveguide. The technique being not invasive seems to be particularly suitable for solid or liquid materials for which it is difficult to insert the probe either when the media sticks to the instrumentations (e.g. polymeric resins at high temperatures) or when the media can damage the probe or perturb the measure. Preliminary experimental tests have been carried out on distilled water to verify the feasibility of the proposed procedure.
is expected due to the sample EM properties temperature dependence. This latter effect is less significant than the former and, at first glance, will be neglected. In the limits of the present study, focused on the analysis of the thermal aspects of the problem at hand, a simplified model is introduced to synthesize the character of the propagation of an electromagnetic wave inside the sample and the interface phenomenon. The tube is a homogeneous (dielectric-dissipative) waveguide in which a single electromagnetic wave propagates (monomodal propagation). The dissipation phenomena can be taken into account using a complex propagating constant resulting into an exponentially decreasing amplitude of EM field intensity along the propagation direction [7]. This, in turn, originates an exponentially decreasing production term, proportional to the square of EM field intensity, in the energy balance inside the media. The simple propagation model is appropriate provided that reflected waves (from the short circuit) are negligible. Furthermore, having to consider small times, the radial terms in the Laplace operator are assumed to be small with respect to the longitudinal term. Thus the problem can be conveniently sketched with reference to a semi-infinite flat plate (Fig. 1) heated by the absorption of EM energy and cooled by radiative–convective heat transfer. Note that the assumption regarding the convective motions inside the sample is realistic since energy
2. Basic equations and analytical solution The computation of the dielectric constant from temperature–time profiles needs a model for the heat diffusion provided that, for liquid media, MW heating inside the sample is such that convective motions and evaporation the sample are negligible. This condition is needed to simplify the analytical model and requires only early stages of the heating process to be considered and small ambient to media temperature differences. Considering that the electromagnetic field inside the sample is the result of the interaction between the present load and the produced field, the production term turns out to be varying in space. In addition a time variation of the production term
Fig. 1. Sketch of the analytical model.
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
generation decreases from the top to the bottom of the waveguide. For such a problem, the energy balance equation and the boundary conditions can be written in dimensionless form as:
147
hðn; sÞ ¼ Rm ½Em Xm ðnÞ=ðk2m Þ Rm ½Em expðk2m sÞXm ðnÞ=ðk2m Þ
ð8Þ
hnn hs ¼ ð/ L =dÞ exp½ð1 nÞ=d
ð1Þ
hðn; s ¼ 0Þ ¼ 0
ð2Þ
where the first term on the right hand side is recognized as the steady solution. To show the temperature field behaviour, it is interesting to examine two limiting cases for the solution, Figs. 2 and 3:
hn ðn ¼ 0; sÞ ¼ 0
ð3Þ
1. d ! 0, slight attenuation of microwave field
hn ð1; sÞ ¼ Bi½hð1; sÞ 1
ð4Þ
where h ¼ ðT Ti Þ=ðTa Ti Þ, is the temperature, Ti being the initial and Ta the environment temperatures; n ¼ x=L, is the spatial coordinate, L being the slab thickness; s ¼ at=L2 , is the time, a being the thermal diffusivity; Bi ¼ hL=k is the Biot number, h and k being the heat transfer coefficient and the thermal conductivity; / L ¼ ðqL LÞ=½kðTa Ti Þ, qL being the absorbed wall radiative intensity; d ¼ D=L is the penetration depth generally depending on the media, the frequency and the thermodynamic state, e.g. temperature, water content. The unsteady thermal field in the slab has been solved analytically in closed form by the method of separation of variables in terms of infinite series assuming: X hðn; sÞ ¼ Xn ðnÞAn ðsÞ ð5Þ n
hðn; s ! 1Þ ¼ 1 þ / L =Bi
ð9Þ
the steady temperature profile attains an uniform value inside the slab, thus the dissipation
Fig. 2. Temperature field for slight microwave attenuation.
where the eigenfunctions Xn ðnÞ ¼ cosðkn nÞ are built from the corresponding homogeneous and the eigenvalues, kn , are related to the Biot number by the characteristic equation kn ¼ Bi=tgðkn Þ. Recalling the orthogonality of the eigenfunctions, an ordinary differential equation can be set up for the coefficients Am ðsÞ [8]: dAm ðsÞ=ds þ k2m Am ðsÞ ¼ Em ;
Am ð0Þ ¼ 0
ð6Þ
where Em ¼ 2Bi cosðkm Þ Z 1 =dÞ exp½ð1 nÞ=dXm ðnÞdn þ 2ð/ L
ð7Þ
0
Finally, the unsteady temperature field is given by:
Fig. 3. Temperature field for strong microwave attenuation.
148
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
due to microwaves is negligible inside the slab and its effect is felt only at the interface, i.e. on the boundary condition. Traditional IR heating in the classical ovens is recovered. 2. d > 1, strong attenuation of microwave field Qgener ¼ ð/ L =dÞ exp½ð1 nÞ=d ð/L =dÞ
ð10Þ generation is uniform inside the slab while, as expected, developing the solution in Taylor series gives: 2 hðnÞ ¼ 1 þ / L =ðBidÞ þ /L ð1 n Þ=ð2dÞ
ð11Þ
a parabolic temperature distribution. Fig. 4. Sketch of the experimental apparatus.
The limiting cases show that the effects of the microwave heating can be divided in two steps: in the early stage the sample is heated both by the environment and by the EM field; later, a particular time location can be expected such as the cooling rate and the heat generation are balanced: a zero wall slope curve can be singled out after which the heating due to microwave still holds while the environment acts as cooling source. 3. Experimental set-up The above model may be used for setting up an experimental procedure to measure the penetration depth and, in turn, the dielectric constant; in fact, for dissipative materials (e00eff =e0 1), these parameters are related by [1]: d ¼ kðe0 Þ1=2 =ð2pe00eff Þ
ð12Þ 0
e00eff
where k is the wavelength, e and are respectively the real and imaginary part of complex dielectric constant. Eq. (12) shows that it is possible to recover e0 and e00eff from several measurements of d at different frequencies, provided that an appropriate model for dielectric constant is available. Thus, the aim of the procedure is shifted in measuring the penetration depth. The experimental setup, shown in Fig. 4, is essentially composed of an Infrared Thermography System (Inframetrics SC1000) for the surface temperature detection, a commercial microwave oven (230 V, 50 Hz, max MICRO OUTPUT 900 W a
2450 MHz), a can containing the sample to test and a suitable SW for data reduction. Particular care was taken about the insulation of the oven walls with polystyrene sheets in order to avoid the effect of the magnetron cooling system. Aluminium cans (thickness 0.5 mm, diameter 67 mm, height of 50, 100 and 150 mm) and steel cans (thickness 0.5 mm, diameter 85 mm, height of 200 mm) were tested with different heating powers. The use of different heights of the can is required to control the effect of the reflected wave. Measures of temperature profiles along the sample were taken quickly opening and closing the oven door at prescribed time intervals. According to the model, the dimensionless temperature profile depends on the following parameters: T Ti x at hL ; ¼ f n ¼ ; s ¼ 2 ; Bi ¼ L L k Ta Ti D d ¼ ; /L ¼ ðqL LÞ=½kðTa Ti Þ L ð13Þ Assuming the measurements errors to be independent and Gaussian distributed, the parameter estimator can be chosen as the chi-square: v2 ¼ 2 R½Ti T ðxi , a; bÞ=ri =n, where Ti and xi (i ¼ 1–n) are the experimental data for a fixed time with their standard deviation ri ; a is the unknown adjustable parameters vector with three components, that is the heat transfer coefficient, h, the
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
penetration depth and the absorbed wall heat flux, q L ; b is the known data vector (Ta , Ti , L, t, k, a). Data reduction is obtained by minimizing the chi-square evaluated with temperature profiles at fixed times. The model exhibits non-linear dependence on the unknown parameters except for wall heat flux; thus, the numerical Marquardt method [9] for non-linear parameter estimation, an extension of linear scheme with a proper technique to calculate the Hessian matrix, is used. The merit function of the problem at hand being not unimodal, the Marquardt technique is preceded by a random sampling of unknown parameter values within fixed ranges in order, to estimate the region in which the absolute minimum can occur.
The monotonic decay of temperature profiles shows the goodness of the electromagnetic model. In particular the absence of reflected waves on the plugged side of the can, can be noted. As expected two different regions can be identified: the first one close to the heated surface where the sample is cooled by the ambient; a second one where temperature profiles monotonically decrease deeper into the sample due to the decreasing energy production of the electromagnetic field. With reference to water, a further simplification in the computation can be obtained by noting the low values attained in principle by the heat transfer coefficient and by the group a=L2 . Thus, Eq. (9) can be written: x=d DT ¼ T ðx; tÞ Ta ffi ½/ L =ðqcp Þt e
4. Results and discussion In order to verify the feasibility of the above procedure, preliminary experimental tests have been performed on pure water, assuming that the initial sample temperature is the ambient temperature. In Fig. 5, a typical surface temperature map shows the temperature decay in response to microwave heating taken at steps of 5 up to 35 s for pure water placed in an aluminium can 100 mm tall. The quite perfect axial symmetric temperature distribution shows that the assumption of electromagnetic field intensity uniformity is fulfilled.
Fig. 5. Typical surface temperature map.
149
ð14Þ
Eq. (14) shows the exponential decay in space and a linear increase in time of the ambient excess temperature. The problem can be reduced to a single parameter regression, d, since: (i) the estimation of the term / L is inessential due to the linear dependence exhibited in Eq. (14); (ii) Eq. (14) corresponds to the physical case of adiabatic heating, thus no trace of the heat transfer coefficient can be found. In Fig. 6 experimental and analytical curves according to Eq. (14) are plotted assuming a typical value of D ¼ 17 mm for water. The linear increase in time is evident and justifies the assumptions done; a further observation can be done about the trend shown: the curves markedly differ toward the wall where temperatures are higher, while the extension of the region in which
Fig. 6. Experimental vs. analytical temperature profiles.
150
G. Cuccurullo et al. / Infrared Physics & Technology 43 (2002) 145–150
5. Conclusions
Fig. 7. Penetration depth temperature dependence.
The simple analytical model on which the experimental procedure is based seems to allow the evaluation of the temperature dependence of the penetration depth with good accuracy avoiding difficulties connected with probe insertion in microwave ovens. The encouraging result obtained with reference to pure water suggest further developments of the proposed experimental protocol. Future efforts should be directed to the improvement of the experimental setup for the continuous detection of temperature profiles and to improve the EM analytical model to take into account the reflected back waves observed in the case of short cans. Additional experimental tests could scan wider temperature ranges by cooling the media and/or controlling the environment temperature. References
Fig. 8. Data reduction results for penetration depth.
agreement is satisfying increases with decreasing times. This is a proof of the stronger attenuation close to the surface. This behaviour is coherent with the one outlined in literature [1], see Fig. 7, where the trend of the penetration depth vs. temperature is reported. The description given is, of course, inappropriate in the cooled zone very close to the interface, where the adiabatic slab assumption fails. A suitable data reduction for dðT Þ allowed to plot Fig. 8, where the average results (with maximum relative error equal to 7%) related to temperature profiles taken at 15 and 20 s are reported and compared with classical curves found in Ref. [1]. These values agree quite fairly with the ones found in literature [1]. The apparent trend shows an overestimation for higher temperatures, probably due to the cooling close to the interface.
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