750
Ind. Eng. Chem. Res. 2007, 46, 750-760
MATERIALS AND INTERFACES Detailed Material-Invariant Analysis on Spatial Resonances of Power Absorption for Microwave-Assisted Material Processing with Distributed Sources Madhuchhanda Bhattacharya† Department of Chemical Engineering, Central Leather Research Institute, Adyar, Chennai 600 020, India
Tanmay Basak* Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India
This paper presents a comprehensive, closed-form-based material-invariant analysis on the occurrence and characterization of spatial resonances in microwave-induced absorbed-power distribution within a material. We have shown that occurrence of resonance depends on competitive interactions between sinusoidal and exponential position dependencies characterized by wave number (Nw) and penetration number (Np) confining the resonating regime of absorbed power within thick- (Np . 1) and thin-sample (Nw , 0.5) asymptotic limits, where it has been shown for the first time that the thick-sample limit depends on distribution of microwave source (φ0) while the thin-sample limit is invariant of φ0. Within the resonating regime, distribution of microwave source (φ0) and wave number of surrounding free space (Nw,0) have shown to play an important role in deciding resonating features of absorbed power in addition to Nw and Np, which in the literature have been considered to be the only governing factors for resonance. For example, absorbed power for the case of one-side incidence (φ0 ) 0 or 1) shows resonance only if Nw,0 ≈ e xNw2+Np2/4π2, where locations of resonating peaks are strong function of Nw,0 in addition to Nw and Np. On the other hand, for both-side incidence with equal power input from left and right sides (φ0 ) 1/2), occurrence of resonance as well as locations of resonating peaks are independent of Nw,0. For intermediate φ0, the absorbed power shows resonance if Nw, Np, Nw,0, and φ0 satisfy the condition Cn,3 * 0 as given in this work. We have performed a detail analysis for all the cases in order to quantify various resonating features of absorbed power and derived correlations for predicting the locations of resonating peaks, which are shown to be in good accordance with actual positions. 1. Introduction Volumetric heating effects associated with microwave-power absorption have found promising applications in various processes such as drying, heating, and food processing, among many others (starting from Weil,1 Ohlsson and Risman,2 Massoudi et al.,3 and Rupin4 to more recently Ratanadecho et al.5,6 and Wang and Chen7). Increasing use of microwaves in material processing is mainly due to faster processing time, which corresponds to the occurrence of maxima in average absorbed power (often termed as resonance). Thus, a series of articles has been devoted to reveal strong dependencies of average absorbed power on characteristic sample dimensions determined by wavelength (λm) and penetration depth (Dp) of the material.8-13 Another important aspect of microwave-assisted material processing is nonuniformity in temperature distribution within the sample, which may lead to hot-spot formation.11,14,15 Because nonuniformity in temperature arises from nonuniformity in absorbed-power distribution (termed as spatial resonances), which may not have one-to-one correspondence with average absorbed power, efficient use of microwaves not only depends on average absorbed power but also needs a priori knowledge * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +91-44-2257-4173. Fax: 91-44-2257-0509. † E-mail:
[email protected]. Tel.: +91-442491-6706.
about spatial patterns of power distribution, which forms the basis of the current research. Because of the lack of mathematically amenable closed-form solutions, most of the previous theoretical studies are based on numerical investigations and, thus, restrict themselves to specific materials and a narrow parameter space.8-12 Recently, the work by Basak and Kumaran13 on the occurrence of local maxima in microwave power initiated a material-invariant analysis in terms of wave number Nw (ratio of sample dimension to wavelength), penetration number Np (ratio of sample dimension to penetration depth), and wave number for free space Nw,0. However, because of the limitations of numerical solutions, they could not establish any correlation for either occurrence of spatial resonance or resonating patterns of absorbed power. Moreover, this work could not quantify the effect of Nw,0 as well as distribution of microwave source on spatial distribution of microwave power. In this work, we start with Maxwell’s equation, which takes into account the presence of surrounding free space having wavelength λ0 and distributed microwave source (0 e φ0 e 1), and present a rigorous mathematical analysis of the closed-form solution in terms of material-invariant characteristic lengths Nw ) 2L/λm, Np ) 2L/Dp, and Nw,0 ) 2L/λ0, unleashing the restrictions associated with numerical analysis including specific material. The analysis revealed that resonating features of spatial absorbed power are governed by a sinusoidal function of
10.1021/ie060408q CCC: $37.00 © 2007 American Chemical Society Published on Web 01/03/2007
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 751
Maxwell’s equation within the surrounding free space in conjunction with eq 1,
d2Ex,0 2
dz′
+ κ20Ex,0 ) 0, z′ < -L and z′ > L
(5)
where the incident electric field intensities (El,0 and Er,0) in the free space can be related to the known incident flux of radiations (Il,0 and Ir,0) as
El,0 ) Figure 1. Schematic representation of a one-dimensional sample exposed to microwaves from left and right sides.
frequency Nw and phase lag γ(Nw, Np, Nw,0, φ0), while the occurrence of resonance is based on the competition between sinusoidal and exponential position dependencies of absorbed power. On the basis of these factors, we have carried out a detailed analysis of absorbed power and derived materialinvariant conditions for the occurrence of spatial resonance and correlations for predicting the position of resonating peaks. We have also presented the dependency of absorbed power on Nw, Np, Nw,0, and φ0, revealing various interesting features, which can be a future guideline for the efficient use of microwave heating. 2. Expressions for Electric Field and Power Distribution Distribution of electric field, Ex, within a one-dimensional slab of thickness 2L exposed to uniform-plane microwave radiations of intensities Il,0 and Ir,0 from left and right sides, respectively (see Figure 1), can be determined from the following Maxwell’s equation16,17
d2Ex dz′2
+ κ2Ex ) 0
-L e z′ e L
(1)
with associated power absorption, q(z′), obeying the following relation:
q(z′) )
πf0κ′′Ex(z′)E/x (z′)
2π 1 κ) +i λm Dp
(3)
where λm and Dp are the wavelength and penetration depth within the sample, respectively, and are related to dielectric constant (κ′) and dielectric loss (κ′′) as
λm )
cx2 f[xκ′ + κ′′2 + κ′]1/2
(4a)
(6)
Ex,0 ) Ex,
dEx,0 dEx ) at z′ ) -L and L dz′ dz′
(7)
In terms of the following normalized variables,
Ex,0 Ex z′ , φx ) ; E0 ) El,0 + Er,0 z ) , φx,0 ) L E0 E0
(8)
and the following dimensionless numbers,
Nw )
2L 2L 2L , Np ) , Nw,0 ) λm Dp λ0
(9)
a general solution of eqs 1 and 5 can be written as a linear combination of transmitted and reflected waves
φx,0 ) φ1,l eiπNw,0z + φ2,l e-iπNw,0z, z < -1 φx ) φ1 e(iπNw-Np/2)z + φ2 e(Np/2-iπNw)z, -1 e z e 1 φx,0 ) φ1,r e
iπNw,0z
+ φ2,r e
-iπNw,0z
(10)
, z>1
where the coefficients of transmission (φ1,l, φ1, and φ1,r) and reflection (φ2,l, φ2, and φ2,r) can be determined by solving eq 7 along with
φ1,l )
El,0 Er,0 ≡ φ0 and φ2,r ) ≡ 1 - φ0; 0 e φ0 e 1 E0 E0 (11)
Similarly, the absorbed power can be rewritten in terms of the above dimensionless variables as
c0E02 NwNp (12) φx(z)φ/x (z); Q0 ) Q(z) ≡ q(Lz) ) Q0 Nw,0 2L
2
and
Dp )
x
2Ir,0 c0
Here, Ex,0 are the induced electric field distributions within the free space on left and right sides and κ0 is the free-space propagation constant with κ0′ ) 1 and κ0′′ ) 0, i.e., κ0 ) 2π/λ0 with λ0 ) c/f, where λ0 is the wavelength within the free space. At the sample boundaries, eqs 1 and 5 are coupled through the following interfacial electric and magnetic field continuities
(2)
In the above equations, Ex lies in the x-y plane and varies only in the z′ direction, E/x denotes the complex conjugate of Ex, f is the frequency of incident radiation, 0 is the free-space permittivity, and κ is the propagation constant given by
x
2Il,0 , Er,0 ) c0
c
x2πf[xκ′ + κ′′2 - κ′]1/2
(4b)
2
with c as the velocity of light. Transmission and reflection at sample boundaries (z′ ) -L and z′ ) L) necessitate solving
Note that φ0 ) 0 and 1 correspond to one-side incidence from right and left sides, respectively, while φ0 ) 1/2 represents bothside incidence with equal power input from left and right sides. Any intermediate φ0 between 0 to 1/2 and 1/2 to 1 resembles both-side incidence with unequal power input from left and right sides, where the sample is exposed to higher incident power from right and left side, respectively. Therefore, φ0 is a measure of distribution of microwave source.
752
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
It may be noted that, in eq 9, Nw and Np, which are the ratios of sample thickness with respect to wavelength and penetration depth within the sample, are material-invariant measures of sample thickness and are termed as wave number and penetration number of the sample, respectively, where
Np e 2πNw
(13)
following eqs 4a and 4b. The presence of surrounding free space is reflected in eq 10 by Nw,0, which is the ratio of sample length to the wavelength within the free space and will be referred to as the wave number for the free space. By solving eq 7 for φ2,l, φ1, φ2, and φ1,r and defining
c1 ) fw (1 + fp - fw )
(14a)
c2 ) fw2(1 + fp2 + fw2)
(14b)
c3 ) -2fw3fp
(14c)
c4 ) 2fw3
(14d)
fw ) Nw,0/Nw and fp ) Np/2πNw
(15)
2
2
2
with
the expression for dimensionless absorbed-power distribution within the sample (p(z) ) Q(z)/Q0) can be algebraically manipulated to be expressed as
p(z) )
Np [C cosh Npz + Cn,2 sinh Npz + xC 2n,3 + C 2n,4sin Cd n,1 (2πNwz + γ)] (16)
where the coefficients Cn,1, Cn,2, Cn,3, Cn,4, and Cd are given in Appendix A. Since Cn,2 ) Cn,4 ) 0 and Cn,1 ) Cn,3 for φ0 ) 1/2 (see eq A.1), γ ) π/2 for both-side incidence with equal power input from left and right sides, which simplifies eq 16 to
p(z) )
NpCn (cosh Npz + cos 2πNwz) 2Cd
(17)
where
Cn ) c1 cos 2πNw + c2 cosh Np + c3 sin 2πNw + c4 sinh Np (18) Therefore, absorbed-power distribution within the sample for φ0 ) 1/2 will always be symmetric around the center, i.e., p(z)|φ0)1/2 ) p(-z)|φ0)1/2. It may also be noted from eq 16 that p(z) is not symmetric for any other φ0. The symmetric (for φ0 ) 1/2) and unsymmetric (for φ0 * 1/2) nature of power distributions are inherent features of eq 16 and can be seen in any of forthcoming figures (Figures 2-8). Also note that p(z ) 0) ) NpCn/Cd. Thus, absorbed power at the center of the sample is independent of the distribution of microwave source. 3. Resonance in Absorbed-Power Distribution 3.1. Bounds on Resonating Regime: Thick- and ThinSample Limits of Np and Nw. The presence of maxima in absorbed-power distribution within the sample is termed as resonance, which is governed by the sine term in eq 16. As eq 16 has both exponential and sinusoidal dependencies on z characterized by Np and Nw, respectively, the occurrence of resonance depends on relative contribution from the exponential
term compared to the sinusoidal one, which in turn depends on Np. For example, in the limit of sample length much greater than penetration depth, i.e., Np . 1, exponential dependency of p(z) dominates over sinusoidal dependency, eliminating resonating features of absorbed-power distribution. This limit will be referred to as thick sample, where eq 16 can be simplified to
p(z) )
4fwNp fp + (1 + fw) 2
(φ02 e-Np(1+z) + (1 - φ0)2 e-Np(1-z))
2
(19)
Equation 19 with φ0 ) 0 or 1 (one-side incidence) is known as Lambert’s law, where p(z) decays exponentially within the sample, attaining minimum power absorption at the opposite face of incidence as shown in parts a-d of Figure 2 for rightside incidence (φ0 ) 0) with Nw ) 2, fw ) 0.1, and Np ) 3, 4, 5, and 6 in subplots along the first, second, third, and fourth columns, respectively. In Figure 2, power distributions corresponding to the exact solution (eq 16) and the thick-sample asymptote (eq 19) are shown by continuous and dotted lines, respectively, and the results are shown for φ0 ) 0, 1/4, 1/3, and 1/ in subplots along the first, second, third, and fourth rows, 2 respectively. Because p(z) is mirror symmetric with respect to left- and right-side incidence, i.e., p(z)|φ0)j ) p(-z)|φ0)1-j, 0 e j e 1 (see eq 16), absorbed-power distributions for 1/2 e φ0 e 1 will be exact mirror reflections of those for 1/2 g φ0 g 0 and are not shown in either Figure 2 or any of the forthcoming illustrations. As can be seen from Figure 2, the location of minimum absorption, which occurs at z ) -1 for φ0 ) 0, moves toward the center of the sample as φ0 varies from 0 (one-side incidence) to 1/2 (both-side incidence with equal power input from left and right sides) because of exponential dominance from both the faces for φ0 * 0 (or 1). However, note that the critical value of Np above which eq 16 can be approximated by eq 19 increases as φ0 varies from 0 to 1/2. This can be explained by expressing the error associated with eq 19 (denoted by err(z)) as
err(z) ≈
[
8Np e-2Np
2φ0(1 - φ0)xc12 + c32 cos
fw(fp2 + (1 + fw)2)2
(2πNw - γ1) cosh Npz + xCn,32 + Cn,42 sin (2πNwz + γ) + (c2 - c4) 2 Npz (φ0 e + (1 - φ0)2 e-Npz) e-Np (20) 2
]
which simplifies to
err(z) ≈ [(c2 - c4) e-Np(1+(-1)φ0z) + 2xc12 + c32
cos(2πNw(1 + (-1)φ0z) - γ1)] 4Npe-2Np] e-2Np fw(fp2 + (1 + fw)2)2
(21a)
for φ0 ) 0 or 1 and to
[
4Np e-Np
c 2 + c4 cos 2πNwz + 2 fw(fp + (1 + fw) ) c -c xc12 + c32 cos(2πNw - γ1) e-Np + 2 2 4 e-2Np
err(z) ≈
(
2
2 2
]
)
(cosh Npz + cos 2πNwz) (21b) for φ0 ) 1/2. Thus, the order of maximum error changes from
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 753
Figure 2. Dimensionless power distribution and its dependence on φ0 in the thick-sample regime. The results are shown for Nw ) 2; fw ) 0.1; Np ) 3, 4, 5, and 6 in subplots along the first, second, third, and fourth columns, respectively; and φ0 ) 0, 1/4, 1/3, and 1/2.
e-2Np to e-Np as φ0 varies from 0 or 1 to 1/2, requiring critical Np for φ0 ) 1/2 to be almost double than that for φ0 ) 0 or 1, as can also be seen from Figure 2. It may be further noted from eq 17 that dominances of exponential terms are much stronger at z ) -1 and z ) 1 compared to z ) 0, leading to the elimination of resonating peaks of p(z) near the faces of the sample at much lower Np compared to those around the center for both-side incidence with equal power input from left and right sides. Consequently, for φ0 ) 1/2, the decay of resonating peaks starts from both the faces and penetrates toward the center as Np increases, which is shown in Figure 3d by increasing Np from 1 (continuous lines) to 2 (dashed lines), 2.5 (dashed-dotted lines), and 3 (dotted lines) while keeping Nw and fw fixed at 2.2 and 0.1, respectively. It may be observed from Figure 3d that the peaks of p(z)|φ0)1/2 near z ) -1 and z ) 1 vanish for Np g 2.5, while the peaks around z ) -0.5, 0, and 0.5 are still present at Np ) 3. On the other hand, for one-side incidence, where eq 16 can be simplified to
p(z) )
Np [c cosh Np(1 + (-1)σ z) + c4 sinh Np(1 + Cd 2
(-1)σ z) + xc12 + c32 cos(2πNw(1 + (-1)σ)z - γ1)] (22) with σ ) 0 and 1 for φ0 ) 0 and 1, respectively, exponential terms dominate most strongly at z ) (-1)σ, resulting in elimination of peaks near the face of incidence at much lower Np compared to the opposite face of incidence as shown in Figure 3a for φ0 ) 0. In this case, the decay of resonating peaks
Figure 3. Decrease in number of resonating peaks of dimensionless absorbed power with increasing Np near the thick-sample limit. The results are shown for Nw ) 2.2; fw ) 0.1; φ0 ) 0, 1/4, 1/2.5, and 1/2; and Np ) 1 (continuous lines), 2 (dashed lines), 2.5 (dashed-dotted lines), and 3 (dotted lines).
starts from the face of incidence and approaches the other face as Np f 3, where absorbed-power distribution follows Lambert’s law. For intermediate φ0, decay of resonating peaks may start either from both the faces or from the face exposed to higher
754
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 Table 1. Comparison of zc (eq 24a) and zp (eq 24b) with Exact Location of Resonating Peaks Corresponding to Equation 16 for Nw ) 1.2, fw ) 0.1, and Various Np and O0 Values Np 0.1 0.5 1 1.2
0.1 0.5 1.0 Figure 4. Attenuation of uniform power distribution as Nw approaches the thin-sample regime. Results are shown for fw ) 0.5; fp ) 0.1; Nw ) 0.5 (continuous lines), 0.25 (dashed lines), 0.1 (dashed-dotted lines), and 0.05 (dotted lines); and φ0 ) 0, 1/4, 1/3, and 1/2 in subplots a, b, c, and d, respectively.
microwave-power incidence, depending on whether φ0 is closer to 1/2 or 0 (or 1), respectively, as shown in parts b and c of Figure 3 for φ0 ) 1/4 and 1/2.5, respectively. Note that, for φ0 ) 1/2.5, decay of resonating peaks starts from both the faces, while it starts from the right face (exposed to higher-incident microwave power) for φ0 ) 1/4. It may be noted ∀φ0 that the number of peaks in absorbed-power distribution decreases with increasing Np and vanishes as Np approaches the thick-sample limit, where absorbed power varies exponentially within the sample. It follows from the above discussion that absorbed power does not exhibit any spatial oscillations in thick samples confining the resonating regime of p(z) within Np e 3 - 6 for φ0 ) 0 or 1 - 1/2. Within the above resonating limit of Np, it may be further noted that absorbed-power distribution exhibits resonance if the sine term in eq 16 completes at least one cycle within the sample. Since the sine term in eq 16 can complete 2Nw cycles within the sample thickness, p(z) does not show any resonance if Nw , 0.5, where almost uniform power distribution is attained within the sample. This limit of Nw , 0.5 will be referred to as thin sample, where sample length is much smaller than the wavelength within the sample and p(z) becomes a weak function of position and distribution of microwave source. The weak dependence of p(z) on position as well as φ0 vanishes in the limit of Nw f 0, where eq 16 can be simplified to
lim p(z) f
Nwf0
Np (c cos 2πNw + c2 cosh Np + c3 sin 2πNw + Cd 1 c4 sinh Np) (23)
as shown in Figure 4 by plotting p(z) vs z for various Nw in decreasing order from 0.5 to 0.05. In Figure 4, p(z) has been determined for fp ) 0.1, fw ) 0.5 and φ0 ) 0, 1/4, 1/3, and 1/2 in subplots a, b, c, and d, respectively, and continuous, dashed, dashed-dotted, and dotted lines correspond to Nw ) 0.5, 0.25, 0.1, and 0.05, respectively. It may be observed that the oscillations in p(z) decrease with decreasing Nw and almost
1.5
zc
zp
φ0 ) 0 -1.0004 -1.0000 -0.1670 -0.1665 0.6663 0.6670 -1.0018 -1.0000 -0.1684 -0.1626 0.6649 0.6758 -1.0035 -1.0000 -0.1702 -0.1481 0.6632 0.7217 -1.0042 -1.0000 -0.1708 -0.1378 0.6625 0.7678 φ0 ) 1/3 -0.8948 -0.8947 -0.0615 -0.0612 0.7719 0.7723 -0.8884 -0.8895 -0.0551 -0.0525 0.7783 0.7850 -0.8759 -0.8833 -0.0426 -0.0346 0.7907 0.8205 -0.8632 -0.8849 -0.0299 -0.0145 0.8035 0.8901
exact
Np
-1.0000 -0.1665 0.6670 -1.0000 -0.1626 0.6758 -1.0000 -0.1480 0.7241 -1.0000 -0.1374 0.7905
0.1
-0.8947 -0.0612 0.7723 -0.8895 -0.0525 0.7850 -0.8833 -0.0346 0.8208 -0.8850 -0.0144 0.9007
0.1
0.5 1 1.4
1.0 1.5 2.0
zc
zp
φ0 ) 1/4 -0.9241 -0.9239 -0.0907 -0.0904 0.7426 0.7431 -0.9168 -0.9169 -0.0835 -0.0798 0.7499 0.7580 -0.9011 -0.9046 -0.0678 -0.0555 0.7655 0.8034 -0.8866 -0.8968 -0.0532 -0.0311 0.7801 0.8762 φ0 ) 1/2 -0.8333 -0.8335 0.0000 0.0000 0.8333 0.8335 -0.8333 -0.8371 -0.0000 -0.0000 0.8333 0.8371 -0.8333 -0.8790 0.0000 0.0000 0.8333 0.8790 -0.8333 -0.9446 0.0000 0.0000 0.8333 0.9446
exact -0.9239 -0.0904 0.7431 -0.9169 -0.0798 0.7580 -0.9046 -0.0555 0.8040 -0.8968 -0.0310 0.8920 -0.8335 0.0000 0.8335 -0.8371 -0.0000 0.8371 -0.8802 0.0000 0.8802 -1.0000 0.0000 1.0000
uniform power distribution of p(z) ≈ 0.05 is attained within the sample for Nw ) 0.05, which remains constant for φ0 ) 0, 1/ , 1/ , and 1/ (see subplots a-d). 4 3 2 3.2. Location of Resonating Peaks. It follows from Section 3.1 that resonating regime of p(z) is confined within thick(exponential distribution) and thin-sample (uniform distribution) limits characterized by Np and Nw, respectively. Within the resonating regime of Np and Nw, absorbed-power distribution follows the sine function in eq 16 and exhibits spatial resonances if xCn,32+Cn,42 ≈ e 0 with peaks appearing around z ) zc given by
zc )
(1 ( 4j)π - 2γ , j ) 0, 1, 2, ... such that -1 e zc e 1 4πNw (24a)
The exact locations of resonating peaks (zp) depend on the hyperbolic functions and can be predicted by the following relation obtained via asymptotic expansion,
z p ) zc +
Np(Cn,2 cosh Npzc + Cn,1 sinh Npzc) 4π2Nw2xCn,32 + Cn,42 2 Np (Cn,1 cosh Npzc + Cn,2 sinh Npzc)
≡ zc + δc (24b)
with -1 e zp e 1. The predictions of eqs 24a and 24b are compared with the exact locations of resonating peaks corresponding to eq 16 in Table 1 for Nw ) 1.2, fw ) 0.1, 0 e φ0 e 1/ , and various N values within the resonating regime. Since 2 p the correction term δc f 0 as Np f 0 (see eq 24b), eqs 24a and 24b predict the same value for the location of resonating peaks in the limit of Np , 1, which matches with the exact value corresponding to eq 16 as can be seen from Table 1 for Np ) 0.1. However, as Np increases, the correction term contributes significantly, resulting in eqs 24a and 24b deviating from each other, where eq 24b predicts very close to the exact location of resonating peaks ∀Np while eq 24a breaks down for Np g 1 (see Table 1).
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 755 Table 2. Solution of Cn,3 ) 0 for Nw ) 1.5, 1.875, and 2; fw ) 0.1 and 10; and O0 ) 1/4 and 1/3 Nw ) 1.5 fw ) 0.1 fw ) 10 fw ) 0.1 fw ) 10
Nw ) 1.875
Np ) 0.8996
Np ) 0.4929
Nw ) 2
φ0 ) 1/4 Np ) 0.3759 φ0 )
1/
Np ) 0.8978
3
Np ) 0.4924
It follows from the following simplification of eq 24b for φ0 ) 1/2
zp ) (
(
Np sinh
jNp Nw
jNp 4π2Nw2 - Np2 cosh Nw
+
j Nw
)
(25)
that absorbed power always exhibits a peak at the center of the slab (z ) 0) for both-side incidence with equal power input from left and right sides. It may also be noted that peaks of p(z)|φ0)1/2 appear at both the faces for integer Nw’s (i.e., Nw ) j, j ) 1, 2, ...), while they appear inside the sample for Nw as odd multiples of 1/2 (i.e., Nw ) (2j - 1)/2, j ) 1, 2, ...). Therefore, p(z) with φ0 ) 1/2 exhibits 2Nw + 1 peaks for integer Nw’s, while it shows 2Nw peaks for Nw as odd multiples of 1/2 (i.e., Nw ) (2j - 1)/2 with j ) 1, 2, ...), as shown in Figure 5 for Nw ) 0.5, 1, 1.5, and 2; fp ) 0.05; and φ0 ) 0 (continuous lines), 1/4 (dashed lines), 1/3 (dashed-dotted lines), and 1/2
(dotted lines). In Figure 5, distributions of p(z) are shown for fw ) 0.5, 1, and 5 in subplots along the first, second and third columns, respectively. It may be mentioned that both fw and fp are invariant of sample length as well as frequency of incident radiation and, thus, remain constant for a particular material while varying from material to material. Thus, varying Nw from 0.5 to 2 with constant fp and fw in subplots along a vertical line in Figure 5 corresponds to increasing sample length by a factor of 4, keeping the material and frequency of incident radiation constant, whereas variation of fw from 0.5 to 5 in subplots along a horizontal line in Figure 5 corresponds to different materials with constant Nw. It may be interesting to note from eq 25 that the locations of resonating peaks do not depend on Nw,0 (or fw for constant Nw) for both-side incidence with equal power input from left and right sides, while they depend on fw for one-side incidence, as follows from the following simplification of eq 24b for φ0 ) 0 or 1:
[
zp ) (-1)φ0 zc - 1 +
Np(c2 sinh Npzc + c4 cosh Npzc)
|
4π2Nw2xc12 + c32 2 Np (c2 cosh Npzc + c4 sinh Npzc)
]
(26)
zc)(γ1(2jπ)/2πNw
This may or may not impose a phase difference between power distributions corresponding to φ0 ) 1/2 and φ0 ) 0 or 1
Figure 5. Location of resonating peaks of absorbed power as a function of fw and φ0 for odd and even Nw with constant fp. Results are shown for Nw ) 0.5, 1, 1.5, 2; fp ) 0.05; fw ) 0.5, 1, and 5 in subplots along the first, second, and third columns, respectively; and φ0 ) 0 (continuous lines), 1/4 (dashed lines), 1/ (dashed-dotted lines), and 1/ (dotted lines). 3 2
756
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
depending on fw, as reflected in Figure 5 for fw ) 0.5, 1, and 5. It may be noted from Figure 5 that, for fw ) 0.5, power distributions corresponding to φ0 ) 0 and φ0 ) 1/2 remain in the same phase for Nw ) 1 and 2, while they are in the opposite phase for Nw ) 0.5 and 1.5. Whereas a completely opposite behavior is observed for fw ) 5, where p(z)|φ0)0 and p(z)|φ0)1/2 are in the same phase for Nw ) 0.5 and 1.5 and are in the opposite phase for Nw ) 1 and 2. This can be explained by expressing the phase difference between power distributions corresponding to one-side incidence and both-side incidence with equal power input from left and right sides from eq A.1f as
φd ≡ γ|φ0)0 or 1 - γ|φ0)1/2 ) (-1) (2πNw - γ1) (27) σ
where σ ) 0 and 1 for φ0 ) 0 and 1, respectively. Thus, φd is a function of Nw as well as γ1, where γ1 depends on fw and fp (eq A.1g). Since 0 e fp e 1 according to eq 13, γ1 f -π and 0 in the limit of fw . 1 and fw , 1, respectively (refer to eqs A.1g and 14). Thus, for integer Nw’s (i.e., Nw ) j, j ) 1, 2), φd f (-1)σ(2j + 1)π and φd f (-1)σ2πj in the limit of fw . 1 and fw , 1, respectively. Consequently, power distributions corresponding to φ0 ) 1/2 and φ0 ) 0 or 1 with integer Nw’s will be in the same phase for fw , 1 and in the opposite phase for fw . 1, as can be seen from Figure 5 for fw ) 0.5 and 5. Whereas for Nw, as odd multiples of 1/2 (i.e., Nw ) (2j - 1)/2, j ) 1, 2), φd f (-1)σ2πj and (-1)σ(2j - 1)π for fw . 1 and fw , 1, respectively, resulting in p(z)|φ0)0 or 1 and p(z)|φ0)1/2 being in the same phase for fw . 1 and in the opposite phase for fw , 1. For intermediate φ0, power distribution depends on fw, fp, and φ0 in addition to Nw, where the location of resonating peaks can be predicted by eq 24b. In contrast with the previous discussion for fw . 1 or fw , 1, it follows from the second column of Figure 5 that resonating peaks for the case of fw ) 1 appear around the same positions for all φ0 except φ0 ) 0, where absorbed-power distribution does not show any resonance. This can be attributed to the fact that c1 f 0 in the limit of fw f x1+fp2 (or fw f 1 since fp , 1 is within the resonating regime; see eq 14a), which results in γ1 f -π/2 simplifying eq 16 to
p(z) ) Np{(x1 + fp2 cosh Np + sinh Np)[2φ02 cosh Npz +
on distribution of microwave source and Np, as can be predicted by eq 24b. On the other hand, for φ0 ≈ 0 or 1, exponential terms dominate ∀Np (see eq 28b) and p(z) does not show
resonance for fw ≈ x1+fp2 but shows plateaus at zp ) (-1)σ((j/Nw - 1) (j ) 1, 2, ... such that -1 e zp e 1) due to the presence of sin 2πNw(1 + (-1)σz) in eq 28b (as can be seen from Figure 5). 3.3. Amplitude of Resonating Peaks. Since fp , 1 within the resonating regime, it follows from eq 28 that the order of magnitude of either resonating peaks for φ0 ≈ e 0 or 1 (where cos 2πNwz ) 1) or absorbed power for φ0 ≈ 0 or 1 are invariant
of Nw for fw ≈ x1+fp2, as shown in Figure 6 for φ0 ) 0, 1/4, and 1/2 in subplots along the first, second, third, and fourth columns, respectively, by varying Nw from 1.5 to 2 with fw ) 1 (continuous lines) and Np ) 0.5. In contrast, it follows from the following simplification of eq 16 within the resonating regime for fw . 1 and fw , 1,
1/ , 3
p(z) )
1
2πNwz + (-1)Ψ cos 2πNw cosh Npz] +
(-1)Ψ cos 2πNw(1 + (-1)j+1z)]) (29a) with
{
0, f , 1 Ψ ) 1, fw . 1 w
)
x1 + fp2sinh 2Np
(28a)
p(z) ) Np{x1 + fp2cosh Np(1 + (-1)σz) + sinh Np(1 +
(
2 + fp2 cosh 2Np + 2
)
x1 + fp sinh 2Np 2
{
2/fw, fw . 1 2fw/(1 + fp2), fw , 1 (29b)
Ap ≈
Np|c1| (cosh Np + mf sinh Np + |Cd| (-1)Ψ cos 2πNw) cosh2
Npzp (30) 2
Thus, as Nw varies from (2j - 1)/2 to j (j ) 1, 2, ...) with φ0 ) the amplitudes of resonating peaks decrease if fw . 1 and increase if fw , 1, as shown from parts d, h, l, and p of Figure 6 for fw ) 10 (dashed lines) and 0.1 (dashed-dotted lines) by varying Nw from 1.5 to 2 (i.e., j ) 2). However, for φ0 ≈ 1/2, the locations of resonating peaks are independent of Nw,0, i.e., fw for constant Nw (see eqs 17 and 25). On the other hand, from the following simplification of eq 29a for one-side incidence, 1/ , 2
for φ0 ≈ e 0 or 1 and to
(-1)σz) - fp sin 2πNw(1 + (-1)σz)}/
and mf )
that amplitudes of resonating peaks (Ap) depend on Nw for fw ≈ e 1, where exact functionality of Ap(Nw) is determined by φ0. For example, in the limit of φ0 ≈ 1/2, Ap can be expressed from eq 29a as
2φ0(1 - φ0) cos 2πNwz] + (1 - 2φ0)[x1 + fp cosh Np(1 +
(
(j(1 - 2φ0) + ∑ j)0
φ20)[cosh Np(1 + (-1)j+1z) + mf sinh Np(1 + (-1)j+1z) +
2
2 + fp2 z) + sinh Np(1 + z)]}/ cosh 2Np + 2
Np|c1| (2φ0(1 - φ0)[(cosh Np + mf sinh Np) cos |Cd|
p(z) ) (28b)
for φ0 ≈ 0 or 1. It immediately follows that, for φ0 ≈ e 0 or 1
with fw ≈ x1+fp2 (or fw ≈ 1 within the resonating regime), locations of resonating peaks, which appear around zp ≈ (j/ Nw (j ) 1, 2, ... such that -1 e zp e 1), have weak dependence
Np|c1| (cosh Np(1 + (-1)σz) + mf sinh Np(1 + |Cd| (-1)σz) + (-1)Ψ cos 2πNw(1 + (-1)σz)) (31)
it follows that amplitudes of resonating peaks (with both fw . 1 and fw , 1) are of the same order of magnitude for Nw ) j and Nw ) j - 1/2 and vary nonmonotonically as Nw varies from
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 757
Figure 6. Amplitude of resonating peaks as a function of Nw, fw, and φ0 with constant Np. Results are shown for Nw ) 1.5, 1.75, 1.875, and 2; Np ) 0.5; fw ) 0.1 (dashed-dotted lines), 1 (continuous lines), and 10 (dotted lines); and φ0 ) 0, 1/4, 1/3, and 1/2 in subplots along the first, second, third, and fourth columns, respectively.
j - 1/2 to j reaching minima at Nw ) j - 1/4. It also follows from eq 31 that, for one-side incidence, Ap|fw,g ) Ap|fw,l if fw,g ≈ 1/fw,l (since fp , 1 is within the resonating regime), where fw,g and fw,l denote fw values for fw . 1 and fw , 1, respectively. However, because of a phase difference of π between power distributions corresponding to fw . 1 and fw , 1, maxima of p(z)|fw,g correspond to the minima of p(z)|fw,l for one-side incidence, as shown in parts a, e, i, and m of Figure 6. It can be observed from these subplots that the amplitudes of the oscillations in p(z) are the same for fw ) 0.1 and fw ) 10, while the maxima of p(z) for fw ) 0.1 correspond to minima of p(z) for fw ) 10. 3.4. Is Resonance Possible at any φ0? It follows from the above discussion that satisfying resonating bounds of Nw and Np is a necessary condition for the occurrence of resonance, while a sufficient condition is that xCn,32+Cn,42 ≈ e 0, which is the case for either φ0 ≈ 1/2 or one-side incidence with
fw ≈ e x1+fp2. As a result, if Nw and Np satisfy their resonating bounds, power distribution exhibits resonance independent of fw for both-side incidence with equal power input from left and
x1+fp2 for one-side incidence. For intermediate φ0, the magnitude of xCn,32+Cn,42 depends
right sides and ∀fw ≈ e
on fw, Nw, and Np in addition to φ0, and consequently, the occurrence of resonance depends on the above factors in spite of Nw and Np being within the resonating limits. As xCn,32+Cn,42
≈ e 0 for either fw ≈ x1+fp2 or cos 2πNw ≈ 0 within the resonating regime with intermediate φ0, p(z) always shows resonance for φ0 ≈ e 0, 1, and 1/2 and Np and Nw within the resonating regime if either fw ≈ x1+fp2 or Nw ≈ (2j - 1)/4 (j ) 1, 2, ...) as shown in the second and third columns of Figure 6 for φ0 ) 1/4 and φ0 ) 1/3, respectively. On the other hand, for
fw ≈ e x1+fp2 with |cos 2πNw| ≈ e 0, xCn,32+Cn,42 f 0 if and only if Cn,3 f 0, which can be approximated as
2φ0(1 - φ0) cos 2πNw + (-1)Ψ cosh Np ≈ 0 (32) 1 - 2φ0 + 2φ02 Thus, the solution of eq 32 gives an approximate combination of Np and Nw, which eliminates the resonance of p(z), while solution of Cn,3 ) 0 predicts a more accurate value of Np or Nw (as the power distribution is very sensitive to Np and Nw). A close observation of eq 32 reveals that there exists a possibility for satisfying eq 32 with fw , 1 if cos 2πNw is negative, while eq 32 may have a feasible root for fw . 1 if cos 2πNw is positive. Consequently, p(z) with Nw ) (2j - 1)/2 and Nw ) j does not show resonance for fw , 1 and fw . 1, respectively, if Np satisfies eq 32 or, more accurately, Cn,3 ) 0. The solutions of Cn,3 ) 0 in terms of Np are listed in Table 2 for various fw, Nw, and intermediate φ0 values considered in Figure 6. It may be noted from this table that, for φ0 ) 1/4, Cn,3
758
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
Figure 7. Variations of resonating characteristics of absorbed power with decreasing Np for even Nw. Results are shown for Nw ) 2; Np ) 2, 1, 0.5, and 0.1; fw ) 0.1, 1, and 10 in subplots along the first, second, and third columns, respectively; and φ0 ) 0 (continuous lines), 1/4 (dashed lines), 1/3 (dasheddotted lines), and 1/2 (dotted lines).
≈ 0 if Np ≈ 0.89, while Cn,3|φ0)1/3 ≈ 0 if Np ≈ 0.49. Consequently, power distribution does not show resonance for Nw ) 1.5 with fw ) 0.1 and Nw ) 2 with fw ) 10 for the case of φ0 ) 1/3, while power distribution shows resonance independent of Nw and fw for φ0 ) 1/4 as Np ) 0.5 in Figure 6. For the same reasons, the variations of p(z) are much lower for φ0 ) 1/4 in Figure 5l and for φ0 ) 1/3 in Figure 5f. Suppression of resonating peaks for intermediate φ0 and combinations of fw ≈ e
columns, respectively, and φ0 ) 0 (continuous lines), 1/4 (dashed lines), 1/3 (dashed-dotted lines), and 1/2 (dotted lines). For either integer Nw with fw . 1 or Nw ) (2j - 1)/2 with fw , 1, where eq 33 can be simplified to
p(z) ) Np|c1|(1 - 2φ0)2(1 - cos 2πNwz)/|Cd| for φ0 ≈ e 1/2 and to
x1+fp , Nw, and Np satisfying Cn,3 ) 0 (as reported in Table 2) 2
can also be seen in Figures 7 and 8 (Figures 7c and 8a for φ0 ) 1/4 and Figures 7f and 8d for φ0 ) 1/3), which show the variations of resonating features of p(z) with decreasing Np as discussed in the following section. 3.5. Does Power Distribution Depend on φ0 in the Limit of Np f 0? In the limit of Np f 0, eq 16 can be simplified to
2Np|c1| [(Ψ sin πNw + (1 |Cd| Ψ) cos πNw) cos πNwz + (1 - 2φ0)(Ψ cos πNw - (1 -
p(z) )
Ψ) sin πNw) sin πNwz]2 (33) for fw . 1 and fw , 1. Thus, p(z)|Npf0 becomes invariant of distribution of microwave source (φ0) for fw , 1 with integer Nw and fw . 1 with Nw as odd multiples of 1/2 as shown in Figures 7 (Nw ) 2) and 8 (Nw ) 1.5) by decreasing Np from 1 to 0.1. In both figures, power distributions are shown for fw ) 0.1, 1, and 10 in subplots along the first, second, and third
(34)
p(z) ) Np|c2 - c1|(1 + cos 2πNwz)/2|Cd|
(35)
for φ0 ≈ 1/2, amplitudes of resonating peaks are functions of φ0 with resonating peaks appearing around the same positions ∀φ0 ≈ e 1/2. At φ0 ) 1/2, power distribution for either integer Nw with fw . 1 or Nw ) (2j - 1)/2 with fw , 1 remains in the opposite phase with much lower Ap compared to those corresponding to other φ0 (see Figures 7i and 8g for Nw ) 2 and Nw ) 1.5, respectively). For fw ≈ x1+fp2, the location as well as amplitude of resonating peaks become invariant of distribution of microwave source ∀Nw in the limit of Np f 0, where eq 28 simplifies to
p(z) )
2Npx1 + fp2 2 + fp
2
(1 - 2φ0 + 2φ02 + 2φ0(1 φ0) cos 2πNwz) (36)
Thus, Ap ) 2Npx1+fp2/(2 + fp2) and zp ≈ (j/Nw, j ) 0, 1, 2,
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 759
Figure 8. Variations of resonating characteristics of absorbed power with decreasing Np for odd Nw. Results are shown for Nw ) 1.5; Np ) 2, 1, 0.5, and 0.1; fw ) 0.1, 1, and 10 in subplots along the first, second, and third columns, respectively; and φ0 ) 0 (continuous lines), 1/4 (dashed lines), 1/3 (dasheddotted lines), and 1/2 (dotted lines).
..., are independent of φ0, as can be seen from Figures 7 and 8. However, the minima of p(z) given by
p(z) )
2NPx1 + fp2 (1 - 4φ0 + 4φ02) 2 + fp2
(37)
increase as φ0 varies from 1/2 to 0 or 1 with a consequent decrease in the oscillations of spatial distribution of absorbed power. Finally, the oscillations vanish for one-side incidence, where the plateaus in power distribution at zp ) (-1)σ((j/Nw - 1) (j ) 1, 2, ...) convert to inflection points as Np f 0. 4. Conclusion We have carried out a comprehensive, closed-form-based material-invariant analysis on spatial resonances of microwave power, taking into account the presence of surrounding free space and distributed microwave source. It has been shown that nonuniformity (or resonance) in power distribution results from sinusoidal position dependency of absorbed power, while occurrence of resonance depends on competition between exponential and sinusoidal functions confining the resonating regime within thick- and thin-sample limits characterized by penetration number (Np) and wave number (Nw), respectively. Within thick samples (Np . 1), absorbed power follows exponential decay, while uniform power distribution is attained in thin samples (Nw , 0.5). A detailed analysis on these two limiting regimes revealed for the first time that the thick-sample
limit depends on distribution on microwave source, while the thin-sample limit is invariant of distribution on microwave source. We have also performed a detailed analysis on the resonating regime to show that the frequency of oscillations (or number of resonating peaks) in absorbed-power distribution depends only on Nw, with the location and amplitude of resonating peaks being functions of Nw, Np, Nw,0, and distribution of microwave source, while earlier researchers considered Nw and Np to be the only factors governing the resonating characteristics. We have shown that, for the case of both-side incidence with equal power input from left and right sides (φ0 ≈ 1/2), absorbed-power distribution always shows resonance if Np and Nw satisfy the resonating regime limit, where locations of resonating peaks are independent of Nw,0 with amplitude being a function of Nw, Np, and Nw,0. In contrast, absorbed power shows resonance for one-side incidence if fw ≈ e x1+fp2. In this case, the location as well as amplitude of resonating peaks are functions of Nw,0 and Np in addition to Nw. For intermediate φ0, the occurrence of resonance depends on Np, Nw, and Nw,0 and can be predicted by the condition provided in this work. In all the cases, the positions of resonating peaks can be predicted from the correlation derived here based on an asymptotic expansion, which has been shown to be in good agreement with actual positions of resonating peaks. As a final remark, our analysis may provide a guideline to judiciously select the sample dimension with proper arrangement of microwave incidence in order to utilize maximum of input energy in an efficient manner.
760
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
Appendix A: Expressions for the Coefficients in Equation 16
Cd )
c32 - c12 c22 + c42 cos 4πN + cosh 2Np + w 4fw3 4fw3 fpc1 sin 4πNw + c2 sinh 2Np (A.1a)
Cn,1 ) 2φ0(1 - φ0)xc12 + c32cos(2πNw - γ1) + (1 -
2φ0 + 2φ02)(c2 cosh Np + c4 sinh Np) (A.1b)
Cn,2 ) (1 - 2φ0)(c2 sinh Np + c4 cosh Np)
(A.1c)
Cn,3 ) (1 + 2φ02 - 2φ0)xc12 + c32cos(2πNw - γ1) + 2φ0(1 - φ0)(c2 cosh Np + c4 sinh Np) (A.1d) Cn,4 ) -(1 - 2φ0)xc1 + c3 sin(2πNw - γ1) (A.1e) 2
2
Cn,4
γ ) sign(Cn,3) cos-1
xCn,32 + Cn,42
γ1 ) -cos-1
c1
xc12 + c32
(A.1f)
(A.1g)
Nomenclature Ap ) amplitude of resonating peaks of p(z) c ) velocity of light (m s-1) Dp ) penetration depth within the sample (m) f ) frequency of incident microwave radiation (Hz) fw ) ratio of Nw,0 and Nw fp ) ratio of Np and 2πNw El,0, Er,0 ) incident electric field from left, right sides, respectively (V m-1) E0 ) total incident electric field from left and right sides (V m-1) Ex ) microwave induced electric field distribution within the sample (V m-1) Il,0, Ir,0 ) incident flux of radiation from left, right sides (W cm-2) Nw ) dimensionless sample thickness scaled with λm Np ) dimensionless sample thickness scaled with Dp Nw,0 ) dimensionless sample thickness scaled with λ0 p(z) ) dimensionless absorbed-power distribution within the sample q(z′) ) absorbed-power distribution within the sample (W cm-3) z′ ) spatial coordinate with origin at the center of the slab z ) dimensionless spatial coordinate scaled with L zc ) approximate location of resonating peaks of p(z) zp ) exact location of resonating peaks of p(z) Greek Symbols 0 ) free space permittivity (F m-1) φ0 ) fractional incident electric field from left side φx,0 ) dimensionless electric field within the free space scaled with E0
φx ) dimensionless electric field within the sample scaled with E0 φ1 (φ2) ) dimensionless transmission (reflection) coefficient for the sample φ1,l (φ1,r) ) dimensionless transmission coefficient for the leftside (right-side) side free space φ2,l (φ2,r) ) dimensionless reflection coefficient for the leftside (right-side) side free space κ ) propagation constant of the sample (m-1) κ0 ) free space propagation constant (m-1) κ′ ) dielectric constant of the sample κ′′ ) dielectric loss of the sample λm ) wavelength within the sample (m) λ0 ) wavelength within the free space (m) Literature Cited (1) Weil, C. M. Absorption characteristics of multilayered sphere models exposed to UHF/microwave radiation. IEEE Trans. Biomed. Eng. 1975, BME-22, 468. (2) Ohlsson, T.; Risman, P. O. Temperature distribution of microwave heating-spheres and cylinders. MicrowaVe Power 1978, 13, 303. (3) Massoudi, H.; Durney, C. H.; Barber, P. W.; Iskander, M. F. Electromagnetic, absorption in multilayered cylinder models of man. IEEE Trans. MicrowaVe Theory Tech. 1979, 25, 825. (4) Rupin, R.; Electromagnetic, power deposition in a dielectric cylinderin the presence of a reflecting surface. IEEE Trans. MicrowaVe Theory Tech. 1979, 27, 910. (5) Ratanadecho, P.; Aoki, K.; Akahori, M. The characteristics ofmicrowave melting of frozen packed beds using a rectangular waveguide. IEEE Trans. MicrowaVe Theory Tech. 2002, 50, 1495. (6) Ratanadecho, P.; Aoki, K.; Akahori, M. A numerical and experimental study of microwave drying using a rectangular wave guide. Drying Technol. 2001, 19, 2209. (7) Wang, W.; Chen, G. H. Heat and mass transfer model of dielectricmaterial-assisted microwave freeze-drying of skim milk with hygroscopic effect. Chem. Eng. Sci. 2005, 60, 6542. (8) Barringer, S. A.; Ayappa, K. G.; Davis, E. A.; Davis, H. T.; Gordon, J. Power absorption during microwave-heating of emulsions and layered systems. J. Food. Sci. 1995, 60, 1132. (9) Ayappa, K. G.; Davis, H. T.; Barringer, S. A.; Davis, E. A. Resonant microwave power absorption in slabs and cylinders. AIChE J. 1997, 43, 615. (10) Ayappa, K. G. Resonant microwave power absorption in slabs. J. MicrowaVe Power Electromagnetic Energy 1999, 34, 33. (11) Basak, T.; Priya, A. S. Role of ceramic supports on microwave heating of materials. J. Appl. Phys. 2005, 97, 083537. (12) Basak, T. Generalized analysis on microwave-assisted material processing for one-dimensional slabs: Metallic support versus free space. Ind. Eng. Chem. Res. 2005, 44, 3075. (13) Basak, T.; Kumaran, S. S. A generalized analysis on material invariant characteristics for microwave heating of slabs. Chem. Eng. Sci. 2005, 60, 5480. (14) Feng, H.; Tang, J.; Cavalieri, R. P.; Plumb, O. A. Heat and mass transport in microwave drying of porous materials in a spouted bed. AIChE J. 2001, 47, 1499. (15) Reimbert, C. G.; Jorge, M. C.; Minzoni, A. A.; Vargas, C. A. A note on caustics and two-dimensional hot spots in microwave heating. J. Eng. Math. 2002, 44, 147. (16) Stratton, J. A. Electromagnetic Theory; McGraw-Hill Book Company, Inc.: New York, 1941. (17) Balanis, C. A. AdVanced Engineering Electromagnetics; John Wiley and Sons: New York, 1989.
ReceiVed for reView March 31, 2006 ReVised manuscript receiVed November 2, 2006 Accepted November 6, 2006 IE060408Q
