Ind. Eng. Chem. Res. 2004, 43, 7671-7675
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RESEARCH NOTES Analysis of Microwave Propagation for Multilayered Material Processing: Lambert’s Law versus Exact Solution Tanmay Basak† Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India
Microwave heating of a multilayered system has wide applications in thawing, drying, hyperthermia treatment, and many more. A typical multilayered system can be viewed as a combination of low and high dielectric materials. The propagation of microwaves within a multilayered system can be represented either by exponential Lambert’s law or by an exact solution of Maxwell’s equation. The exact solution for microwave propagation is based on both the transmitted and reflected waves, whereas Lambert’s law is based purely on transmission. Lambert’s law is a good approximation for a semi-infinite sample, and the validity of Lambert’s law for a multilayered assembly is limited to orientation of various dielectrics. Two specific cases for a multilayered sample are considered, and the detailed scattering effects during microwave propagation have been analyzed via Lambert’s law and the exact solution. Introduction Because of volumetric heating effects, microwaves are extensively used for faster thermal processing.1-5 For an efficient microwave heating process, deep and localized heating techniques are sometimes necessary. During microwave propagation, the depth of penetration is generally shallow and it is difficult to heat a deep-lying layer and relatively large layered volumes without excessive surface heating. On the basis of recent investigations, it is observed that absorption of microwave energy and distribution of temperature within the sample are enhanced when a layer of low dielectric material is attached in front of the sample (termed as a high dielectric material).6 The microwave heating of multilayered materials has wide applications in thawing, drying, hyperthermia treatment, and many more.5,7,8 The thawing of ice sometimes is being carried out with a combination of frozen and unfrozen layers, and on the basis of the direction of microwaves, the enhanced thawing may be observed.7 The concept of microwave heating of multilayered materials was extended for drying.5 During drying of the porous body, the drying takes place on a front retreating from the surface into the interior of the sample, dividing it into dry and wet layers. Inside the drying front, the sample is wet, whereas outside the drying front, no liquid water exists; all water is in a vapor state, and the dry layer acts as the lower dielectric material. The design of an electromagnetic hyperthermia system for the cancer treatment is based on the microwave heating of multilayered materials.8 The low dielectric material can cause greater microwave energy to deposit deep inside the human tissue. Various applications for heating of multilayered materials need a theoretical understanding of microwave propagations within layered samples. Tel.: +91-44-2257-8216, Fax: 91-44-2257-0509. E-mail:
[email protected].
The current work stems from a recent investigation for microwave thawing of the frozen layer where the effects of layered configurations and layer thickness were investigated.6 This study finds that when the unfrozen layer is under the frozen layer, the microwaves can penetrate deeper inside the unfrozen layer. Consequently, the heating takes place at the leading edge of the unfrozen layer, resulting in greater thawing rates. This study is based on thawing of a 1D multilayered slab, and the microwaves are assumed to propagate based on Lambert’s exponential decay law throughout the layers.6 Although this simplified model predicts qualitatively the experimental data, the exact mechanism of the microwave transport within multilayered systems is not properly understood. Lambert’s law may be an approximation for microwave transport within layers where the reflection within the phase interface is negligible. Most of the situations involve a combination of low and high dielectric materials, and complex interaction between the dielectrics may not result in exponential decay of electric fields within the layers. The present work attempts to analyze the electric field propagation within multilayers based on Maxwell’s equation and would estimate the microwave power absorption in each phase. The suitability of Lambert’s formulation would be addressed for various situations upon microwave heating of multilayered slabs. Theory: Microwave Transport in a Multilayered Slab Let us assume a 1D composite slab with two dielectric materials surrounded by the air and water media as shown in Figure 1. The wave propagation due to uniform electric field Ex within a medium is governed by1
∇2Ex + γ2Ex ) 0
(1)
†
where Ex lies in the x-y plane and varies along the
10.1021/ie040152t CCC: $27.50 © 2004 American Chemical Society Published on Web 10/14/2004
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direction of propagation, and the propagation constant
ω γ ) xκ′ + jκ′′ c
(2)
depends on the dielectric constant, κ′, and the dielectricloss, κ′′. Here ω ) 2πf, where f is the frequency of electromagnetic radiation and c is the velocity of light. Equation 1 is an alternative representation of Maxwell’s equation with time harmonic variation, e-jωt. Rattanadecho6 assumed a solution of eq 1 as -Rz
Ex ) Eine
(3)
which is also known as Lambert’s exponential law of an electric field. In eq 3, R-1 denotes the penetration depth (Dp), which is related to κ′ and κ′′ in the following manner:
Dp )
c
{ [x ( ) ]}
x2πf κ′
κ′′ 2 1+ -1 κ′
1/2
(4)
dz2
E1 ) Et,1ejγ1z + Er,1e-jγ1z
z e z1
El ) Et,lejγlz + Er,lejγlz
zl-1 e z e zl l ) 2, ... n - 1
En ) Et,nejγnz + Er,nejγnz
z g zn,
where Et,l and Er,l are the coefficients due to transmission and reflection, respectively. The boundary conditions at the interface are
El-1 ) El dEl dEl-1 ) dz dz
}
l ) 2, ..., n z ) z1, ..., zn-1
+ γl El ) 0
(7)
transmitted field intensity for the first medium and the reflected field intensity for the nth medium are known, i.e., Et,1 ) E0 and Er,n ) 0 (no reflection due to the dummy water load). For the multilayered slab consisting of the low and high dielectric materials, the entire system has three interfaces, with the first layer being air and the fourth layer being the water load (Figure 1). The transmission and reflection coefficients are obtained via solution of the system of equations derived from eq 8
My ) b 2
(6)
The general solutions on the electric field for each layer obtained from eqs 6 and 7 are because the
The field solution based on the exponential law (eq 3) is valid if there is no reflected field within a medium, and this may be the case for an infinite medium as assumed by Rattanadecho.6 However, a significant amount of reflected field can alter the entire electric field pattern, which is far from Lambert’s exponential law. A typical microwave waveguide setup consists of an air medium, low/high dielectric materials, and dummy water load at the end, as seen in Figure 1. Figure 1a represents the state (a) where the high dielectric material is at the top, and Figure 1b corresponds to the low dielectric material at the top, which represents state b. The electric field in an n-multilayered sample for the lth layer obtained from eq 1 is
d2El
dielectric properties and, hence, the general solution to eq 5 represented as a linear combination of traveling waves propagating in opposite directions is4
(5)
where zl-1 e z e zl and l ) 1, ..., n. Here, zl denotes the position of the lth interface and γl is the propagation constant in the lth medium. Note that the distance zl is measured with respect to the center of the multilayered slab. The first interface at z1 ) -L and the last interface at zn ) L denote the boundaries for the slab thickness of 2L. We may assume that each layer has constant
(9)
where M, y, and b are given in Chart I. For the lth layer, the transmitted and reflected waves are t
Etl ) Et,le jγlz ) Atl e jθl
r
Erl ) Er,le-jγlz ) Arl e jθl
(10)
with corresponding amplitudes
Atl ) xEtl Etl * Arl ) xErl Erl * and the phase states
θtl ) tan-1
Figure 1. Schematic illustrations of state a, where the unfrozen layer is at the top, and state b, where the unfrozen layer is at the bottom. For both states, the multilayered system is incident with the uniform plane wave normal at the top. The shaded region denotes the unfrozen (water) layer.
θrl )
tan
-1
[ ] [ ]
(11)
Im(Etl ) Re(Etl )
Im(Erl ) Re(Erl )
(12)
where the superscript * in eq 11 denotes the complex
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7673 Chart 1
conjugate. For a stationary wave in the lth layer, the amplitude is
Al ) xElEl*
(13)
and the difference in phase angle
θl ) θtl - θrl
(14)
where the quantities El and El* appeared in eq 13 are evaluated using eqs 6 and 10. At the resonance, the difference in the phase angle is zero, i.e., θl ) 0. For the exponential decay law or Lambert’s law, as given by eq 3, Ein can be obtained from A tl , which results in
El )
{
|Et,l-1ejγlzl-1|e-Rl(z+L), -Rl(z-zl-1)
El-1|z)zl-1e
,
l)2 lg3
(15)
microwave within the composite system. The analysis is based on the analytical solution versus Lambert’s law, where the analytical solution considers the detailed scattering effects at the phase interface and the validity of Lambert’s law would be shown for various cases. The dielectric data are taken from Rattanadecho,6 and for all of the cases, the shaded region denotes the water (high dielectric material). We assume for all cases that the material is incident with the microwave electric field intensity, 7000 V m-1. Figure 2 represents distributions of amplitudes of the electric field and microwave power for the state a, as seen in Figure 1a, where the high dielectric layer (water) is above the low dielectric layer layer (ice). The entire multilayered slab thickness is 70 mm, with the thickness of water layer being 20 mm.6 Figure 2a represents the amplitudes of the electric field based on the exact solution and Lambert’s law. The transmitted field and Lambert’s exponential field are identical within the water layer, and the amplitude of the reflected field is
Here L is the half-sample thickness. For the first interface between air and the sample, the transmitted electric field does not equal the incidence electric field originating from the source as reflection occurs at the first interface. The subsequent layers are assumed to be infinite, and hence Lambert’s law does not account for the reflected waves. Therefore, spatial resonances within the sample cannot be predicted using Lambert’s law. The transmitted electric field, Ein, can also be evaluated from volumetric heat generation due to microwave energy and a local temperature rise.6 The absorbed power in the lth layer is
1 ql ) ω0κ′′l El(z) E/l (z) 2
(16)
where El can be obtained either by exact wave solutions (eq 10) or by Lambert’s law (eq 15). Case Studies We will illustrate various cases for microwave heating of ice-water multilayered slabs. Here ice and water represent the low and high dielectric materials, respectively, and the composite systems are illustrated by Figure 1. The microwave thawing of ice (frozen layer) sometimes is being carried out with an attachment of a water layer (unfrozen layer). This multilayered system is a typical example for microwave thawing of ice,4,6 and the enhanced thawing may be observed based on the specific assembly of a multilayered system. The present work attempts to analyze the electric fields and power distribution on each layer and the penetration of
Figure 2. (a) Amplitudes of electric fields and (b) microwave power distributions for state a when the unfrozen layer is above the frozen layer. The shaded region denotes the unfrozen (water) layer.
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enhanced within the ice layer because of the water layer at the bottom. There is almost negligible reflection within the water layer, and the amplitude of the stationary electric field decays exponentially. Lambert’s electric field also decays exponentially, but Lambert’s field overestimates the electric field more than that based on the exact solution. Consequently, the power absorption based on Lambert’s law is considerably greater than that based on the exact solution in the water layer, as seen in Figure 3b. On the basis of the inset of Figure 3b, the power distribution within the ice layer is found to be qualitatively similar to that in state a and Lambert’s law produces a constant power. It is interesting to note that the power absorption in the ice layer for state b is greater than that in ice layer for state a. Therefore, to efficiently heat the ice slabs, the water layer (high dielectric material) has to be placed at the bottom. The greater power absorption in the ice layer for state b can also be validated with Lambert’s law. However, the detailed energy balance requires the exact solution of the power absorption because Lambert’s exponential law may overestimate or underestimate the power, as seen in Figures 2 and 3. Rattanadecho6 also reported the higher heating rate for state b; however, the heat transport model was purely based on Lambert’s law on power absorption. Conclusive Remarks Figure 3. (a) Amplitudes of electric fields and (b) microwave power distributions for state b when the frozen layer is above the unfrozen layer. The shaded region denotes the unfrozen (water) layer.
found to increase with the water depth. It is interesting to note that the exact solution predicts a maximum in amplitude of the reflected field at the water-ice interface, which contrasts with Lambert’s exponential decay field. Because of significant transmission and reflection, the amplitude of the stationary electric field with various local maxima is observed within the water layer. Because the amplitude of the reflected field is smaller than that of the transmitted field, Lambert’s exponential law may be an approximation of the total electric field within the water layer (Figure 2a). The amplitudes of transmitted and reflected electric fields are greater in the ice regime, resulting in a stronger stationary electric field that is much greater than Lambert’s electric field. Note that the thicker ice layer (thickness ) 50 mm) would cause significant reflection, and field representation using Lambert’s law may not be appropriate. As seen in Figure 2b, the power profile with the exact solution agrees qualitatively well with Lambert’s law within the water layer, and because of lower dielectric loss of the ice layer, both the exact solution and Lambert’s law would provide power within the same order of magnitude for the ice layer. However, Lambert’s law does underestimate the power absorption within the ice layer, as seen in the inset of Figure 2b. The inset of Figure 2b also shows that the exact power distribution in the ice layer is oscillatory whereas Lambert’s law predicts a constant power distribution. Figure 3 illustrates the amplitudes of electric fields and power distributions for the state b (Figure 1b), where the frozen layer (thickness ) 50 mm) is at the top of the unfrozen layer. The electric field distribution within the ice layer shows behavior qualitatively similar to that seen in Figure 2a. In addition, the amplitudes of both transmitted and reflected electric fields are
The contrasting features between Lambert’s law and the exact analytical solution are important for the theoretical models on microwave heating of layered samples. Note that Lambert’s law is purely based on the transmitted field and a significant amount of reflection within the ice layer plays a critical role in determining power distributions for water or the unfrozen layer. Although Lambert’s exponential law predicts results qualitatively similar to those with the exact solution within the low dielectric material, the absorbed power with Lambert’s law may be overestimated within the high dielectric material. As a consequence, the temperature distribution may not be evaluated accurately if there is a strong coupling between the energy balance and electric field equations. Although, for both states a and b, Lambert’s law behavior and the analytical solution for the power distribution of the ice layer are within the same order of magnitude, the power distributions within the water layer may play an important role in determining the propagation of the thawing front. In addition, the error in predicting the moving front would be significant because of the heat of melting based on Lambert’s power absorption in the water layer, especially for state b. In contrast, for state a, Lambert’s law may be a good approximation to predict the thawing dynamics. Notation Ax,l ) amplitude of the stationary wave for the lth layer (V‚m-1) t Ax,l ) amplitude of the transmitted wave for the lth layer (V‚m-1) r Ax,l ) amplitude of the reflected wave for the lth layer (V‚m-1) c ) velocity of light (m‚s-1) Dp ) penetration depth (m) Ex,l ) electric field for the lth layer (V‚m-1)
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7675 E tx,l ) electric field due to transmission for the lth layer (eq 10) (V‚m-1) r E x,l ) electric field due to reflection for the lth layer (eq 10) (V‚m-1) Ex ) electric field intensity (V‚m-1) E0 ) incident electric field intensity (V‚m-1) f ) frequency (Hz) L ) half-slab thickness (m) q ) microwave source term (W‚m-3) z ) distance (m) Greek Symbols γ ) propagation constant 0 ) free space permittivity (F‚m-1) κ′ ) relative dielectric constant κ′′ ) relative dielectric loss δx,l ) phase difference in the stationary wave for the lth layer δtx,l ) phase state for the transmitted wave for the lth layer δrx,l ) phase state for the reflected wave for the lth layer ω ) angular frequency (rad‚s-1) Subscripts l ) layer number Superscripts t ) transmitted wave r ) reflected wave
Literature Cited (1) Ayappa, K. G.; Davis, H. T.; Crapiste, G.; Davis, E. A.; Gordon, J. Microwave Heating: An Evaluation of Power Formulations. Chem. Eng. Sci. 1991, 46, 1005. (2) Basak, T.; Ayappa, K. G. Analysis of Microwave Thawing of Slabs with Effective Heat Capacity Method. AIChE J. 1997, 43, 1662. (3) Basak, T.; Ayappa, K. G. Influence of Internal Convection During Microwave Thawing of Cylinders. AIChE J. 2001, 47, 835. (4) Basak, T. Analysis of Resonances During Microwave Thawing of Slabs. Int. J. Heat Mass Transfer 2003, 46, 4279. (5) Rattanadecho, P.; Aoki, K.; Akahori, M. Influence of Irradiation Time, Particle Sizes, and Initial Moisture Content During Microwave Drying of Multi-layered Capillary Porous Materials. J. Heat Transfer 2002, 124, 151. (6) Rattanadecho, P. Theoretical and Experimental Investigation of Microwave Thawing of Frozen Layer Using a Microwave Oven (effects of layered configurations and layer thickness). Int. J. Heat Mass Transfer 2004, 47, 937 (7) Rattanadecho, P.; Aoki, K.; Akahori, M. A Numerical and Experimental Investigation of the Modelling of Microwave Melting of Frozen Packed Beds Using a Rectangular Wave Guide. Int. Commun. Heat Mass Transfer 2001, 28, 751. (8) Nikawa, Y.; Katsumata, T.; Kikuchi, M.; Mori, S. An Electric Field Converging Applicator with Heating Pattern Controller for Microwave Hyperthermia. IEEE Trans. Microwave Theory Tech. 1986, 34, 631.
Received for review May 17, 2004 Revised manuscript received September 7, 2004 Accepted September 13, 2004 IE040152T
