Generalized Analysis on Microwave-Assisted Material Processing for

Ind. Eng. Chem. Res. 2005, 44, 3075-3085

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Generalized Analysis on Microwave-Assisted Material Processing for One-Dimensional Slabs: Metallic Support versus Free Space Tanmay Basak† Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India

The dimensionless analysis has been carried out to predict the generalized trend in distribution of microwave power and temperature irrespective of thermal and dielectric properties of materials. The analysis is mainly based on three numbers: wavenumber, Nw; free-space wavenumber, Nw0; and penetration number, Np, where Nw is the ratio of sample thickness to wavelength of microwaves within a material, Nw0 is based on the wavelength within free space, and Np is the ratio of sample thickness to penetration depth. The spatial distributions of microwave power for uniform plane waves can be obtained from the combination of transmitted and reflected waves within a material. The generalized trends of microwave power absorption are illustrated via average power plots as a function of Nw, Np, and Nw0. The average power contours exhibit oscillatory behavior, with Nw corresponding to smaller Np. The spatial distributions of dimensionless electric fields and power are obtained for various Nw and Np. The spatial resonance or maxima on microwave power are represented by zero phase difference between transmitted and reflected waves. It is observed that the number of spatial resonances increases with Nw for smaller Np regimes based on relationships Nw ) 0.5n - 0.25 (for metallic support) and Nw ) 0.5n (for free space), where n is any integer. We have also investigated the spatial resonance patterns, with n being a noninteger, and it is observed that the location of the spatial resonance varies with n, but the number of resonance regimes is increased when n increases to an integer value. The spatial power follows an exponential decay law for higher Np regimes irrespective of Nw and Nw0. The heating characteristics are shown for various materials, and generalized heating patterns are shown as functions of Nw, Np, and Nw0. The generalized heating characteristics involve either spatial temperature distributions or uniform temperature profiles based on both thermal parameters and dimensionless numbers (Nw, Nw0, and Np). Microwaves can be optimally used with/without metallic support for the Np e 2 regime. The metallic support corresponds to greater heating rates for Nw ) 0.25 and 0.75, whereas Nw ) 0.5 is optimal for samples without any support. Introduction Microwaves (MWs) are electromagnetic waves in the frequency range 300 MHz to 300 GHz. MWs are largely used in food and chemical processing industries because of associated volumetric heat generation due to transport of electromagnetic waves throughout the volume. Most of the food and other substances in chemical processing industries correspond to large dielectric loss, which is responsible for the frictional dissipation of electric energy of dipoles into heat. Because of the volumetric heating effect, MWs offer more enhanced thermal processing rates than conventional burner heating, which is due to surface radiation. Therefore, MWs are largely used in chemical processing industries such as polymer processing, food processing, and many more. The volumetric heat generation due to MWs within a material is a function of the intensity of radiation, sample geometry, wavelength of the MWs, and dielectric loss of the material. A large number of experimental and theoretical investigations on applications of MWs for material processing mainly revolve around complex features on heating characteristics that involve various case studies including heating of samples, resonance or maxima in power for † Tel.: +91-44-2257-8216. Fax: 91-44-2257-0509. E-mail: [email protected].

one-dimensional (1D) and two-dimensional (2D) enclosures, enhanced heating rates for specific sample dimensions, multilayered material processing, multiphase heating, and material processing with phase changes.1-11 The MW propagation and associated heat transport are based on the interaction between material and MWs, and theoretical foundations on MW heating of 1D and 2D samples were established by Ayappa et al.5,6 The mathematical modeling on MW heating is based on detailed theoretical analysis on coupled MW and heat transport for pure and multiphase samples typically used in the food industry.5-15 Maxwell’s equation or Lambert’s exponential law for MW propagation and energy balance with a volumetric source term due to propagation on MWs form the basis of theoretical models on MW-assisted transport processes. Ayappa et al.5 analyzed the heating characteristics for a multilayered food sandwich, and nonuniform or local heating was observed for 1D bread-beef food slabs. The analysis was later extended for 2D samples because of transverse electromagnetic modes on heating, and localized nonuniform heating was still observed for samples with specific radii.6 Barringer et al.14 and Zhang et al.15 analyzed the MW heating and processing for multiphase systems that include oil, water, and food substances. MW heating and transport models were studied for thawing and heating of multiphase systems in recent investigations.7-11 MW heating applications in three-

10.1021/ie049035o CCC: $30.25 © 2005 American Chemical Society Published on Web 03/11/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005

dimensional (3D) enclosures include processing of food and sintering of ceramics in multimode cavities.15-17 Common to all of these studies is the counterintuitive, enhanced, or optimal heating effect, which may play an important role in efficient material processing. The heating characteristics due to MWs are highly nontrivial, and a generalized guideline on materialinvariant heating characteristics is important in material processing industries. Ayappa et al.12 and Ayappa13 studied MW heating characteristics for a series of materials, and they observed the resonance or maxima in average power that correspond to a suitable relationship between the material dimension and wavelength of MWs within the material. Their analyses are limited within the predictions of resonances that are materialinvariant. However, the detailed analysis on generalized power absorption and associated heating characteristics are yet to appear in the literature. Here we carry out a detailed dimensionless analysis on the generalized characteristics of MW power and the heating characteristics for two cases that involve 1D samples exposed to free space and with one face of the sample attached with metallic support. Note that the metallic support completely reflects the incident MWs, and processing of samples either with metallic support (at one face) or exposed to free space may be important for material-invariant heating studies. The materialinvariant power absorption and various electric fields are expressed as functions of three dimensionless numbers: wavenumber (Nw), penetration number (Np), and free-space wavenumber (Nw0), where the wavenumbers and penetration number are the ratios of sample thickness and wavelength/penetration depth of MWs within the sample. We have shown that these three numbers uniquely govern the electric field and MW power irrespective of materials. A mathematical analysis on spatial power distributions has been carried out for 1D slabs either exposed to free space or with one face attached with metallic support. MW propagation within a sample is governed by Maxwell’s equations, in which the solution is a linear combination of the traveling waves due to transmission and reflection. An analytical solution has been developed to study the influence of traveling waves on MW power distributions within a medium. The average power is evaluated as a function of Nw, Np, and Nw0, and the oscillatory behavior of average power and corresponding spatial distributions is illustrated. These oscillatory distributions demonstrate the generalized features of spatial resonances of power and resonances in average power. We have also established the criteria for exponential variation of power based on Lambert’s law. MW heating is modeled with an energy balance equation, where heat generation due MW power is a function of thermal parameters and the dimensionless numbers (Nw, Np, and Nw0). We have illustrated the generalized heating characteristics for samples with metallic support and samples exposed completely in free space. We have established that the metallic support at one face of a sample or samples exposed to free space would be advantageous depending on Nw irrespective of materials. Theory Electromagnetic Field and Power Distributions: Generalized Dimensionless Analysis. We

Figure 1. Schematic illustration of 1D slabs exposed to plane electromagnetic waves: (a) slab with metallic support; (b) slab exposed to free space.

assume that the sample thickness is much smaller compared to the lateral dimensions and, hence, a 1D slab (Figure 1) is a reasonable representation for the current study. A similar modeling assumption can be found in earlier literature.5,7,10,11,13 The electromagnetic wave propagation due to a uniform electric field may be obtained from Maxwell’s equation as

d2Ex/dZ2 + k2Ex ) 0

(1)

where k ) (ω/c)xk′+ik′′ is the propagation constant, which depends on κ′, the dielectric constant, and κ′′, the dielectric loss. Here, ω ) 2πf, where f is the frequency of the electromagnetic radiation and c is the velocity of light. The complex dielectric properties (κ′ and κ′′) are associated with two fundamental length scales of MW propagation, wavelength, λm, and penetration depth, Dp, as

{

}

{

}

-1/2

2 c κ′[x1 + (κ′′/κ′) + 1] λm ) f 2

and

2 c κ′[x1 + (κ′′/κ′) - 1] Dp ) 2πf 2

(2)

-1/2

(3)

Dp is defined as the distance at which the field intensity decreases to 1/e of its incident value. Therefore, the propagation constant in eq 1 can be written as8,9

k)

( )

1 2p +i λm Dp

(4)

The relation between the flux of incident radiation I0 in free space and the incident electric field intensity, E0, is

1 I0 ) c0E02 2

(5)

The MW power, which is a function of the uniform electric field, Ex, can be expressed as

1 p ) ω0κ′′Ex2 2

(6)

The dimensionless variables

ux )

Ex E0

and

z)

Z+L 2L

are employed to obtain the dimensionless form of

Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3077

Maxwell’s equation (eq 1)

d2ux

+ γ2ux ) 0

(7)

γ ) 2πNw + iNp

(8)

dz

2

where

The dimensionless propagation constant, γ, is a function of two length scales: wavenumber (Nw ) 2L/λm) and penetration number (Np ) 2L/Dp). Note that Nw denotes the number of waves within the sample and Np denotes the strength of penetration of MWs within the sample. Smaller Np corresponds to greater penetration of MWs within a sample. The analytical solution of Maxwell’s equation (eq 7) for a slab, as shown in Figure 1, can be expressed as a linear combination of traveling waves propagating in opposite directions

ux ) Aeiγz + Be-iγz

Because the incident field intensities from the left- and right-hand sides are known, i.e., A1 ) 1 and Bn ) 0 (for air) or An ) Bn ) 0 (for metallic support), eq 13 is solved for the remaining 2n - 2 coefficients using MATLAB. For the lth layer, the transmitted and reflected waves are δx,lt

t ux,l ) Aleiγlz ) Atl ei

δx,lr

r ux,l ) Ble-iγlz ) Arl ei

For waves in the lth layer, the corresponding amplitudes t t ux,l * Atl ) xux,l r r Brl ) xux,l ux,l *

and the phase states

t δx,l ) tan-1

where A and B are the transmitted and reflected field coefficients, respectively. In an n-multilayered sample, the electric field for the lth layer obtained from eq 7 is

r δx,l ) tan-1

dz2

+ γl2ux,l ) 0

(10)

where zl-1 e z e zl and l ) 1, ..., n. Here zl denotes the coordinate of the phase interface between the lth and (l + 1)th media. Note that l ) 1 denotes free space, l ) 2 denotes the sample, and the metallic support or free space may be denoted as l ) 3. We assume that the dielectric properties are independent of temperature and constant in each layer. Hence, the general solution to eq 10 represented as a linear combination of transmitted and reflected waves propagating in opposite directions is

ux,1 ) A1eiγ1z + B1e-iγ1z ux,l ) Aleiγlz + Ble-iγlz ux,n )

z e z1 z1 e z e z2

Aneiγnz + Bne-iγnz z g z2 (air) z ) z2 (metallic support) 0 (11)

}

l ) 2, 3; z ) z1, z2

r Re(ux,l )

(16)

(17)

and the difference in the phase angle t r - δx,l δx,l ) δx,l

(18)

where the quantities ux,l and ux,l* appearing in eq 17 are evaluated using eqs 11 and 14. At the resonance, the difference in the phase angle is zero, i.e., δx,l ) 0. The expression for power as a function of dimensionless numbers can be obtained from eqs 5 and 6, and writing κ′′ as a function of λm and Dp

()

I0 NwNp u (z) ux*(z) L Nw0 x

(19)

where Nw0 is the wavenumber of free space and is given by

Nw0 ) 2L/λ0

(12) q(z) )

}

r Im(ux,l )

(20)

where λ0 ) c/f is the wavelength of the radiation in free space. The dimensionless power can be given as

The coefficients Al and Bl can be obtained by solving the set of algebraic equations given by the interface conditions (eq 12) and the general solutions (eq 11):

Aleiγlzl + Ble-iγlzl )0 Al+1eiγl+1zl - Bl+1e-iγl+1zl γlAleiγlzl - γlBle-iγlzl )0 γl+1Al+1eiγl+1zl + γl+1Bl+1e-iγl+1zl

t Re(ux,l )

Al ) xux,lux,l*

The boundary conditions at the interface are

ux,l-1 ) ux,l dux,l-1 dux,l ) dz dz

t Im(ux,l )

where the superscript denotes the complex conjugate. For a stationary wave in the lth layer, the amplitude is

p)

{

(15)

[ ] [ ]

(9)

d2ux,l

(14)

l ) 1, 2 (13)

2NwNp u (z) ux*(z) Nw0 x

(21)

The average power obtained by integrating the power across the slab is

qav )

1

1

n

+L q(z) dz ≈ ∑q(zi) ∫ -L 2L n

(22)

i)1

Here, -L and L denote the left and right faces of the slab, respectively, 0 e zi e 2L, n denotes the total

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number of data sets, and q(zi) denotes the power as a function of zi, where zi may be measured from the left edge of the slab or sample. Modeling of Heat Transport: Dimensionless Analysis and Numerical Simulations. The heat transport due to a volumetric heat source in a 1D slab can be given by

∂2T ∂T )k 2+p FC ∂t ∂Z

(23)

where k is the thermal conductivity of the material, FC is the specific heat per unit volume, and p is the heat source due to MW propagation within the material, as given in eq 19. Here we consider MW heating of a slab of uniform thickness 2L. Initially, the sample is at a uniform temperature of T0 and the ambient temperature is constant at T∞. The thermal properties in the slab are assumed to be isotropic, and the sample boundaries are assumed to be insulated. As seen in eq 19, the volumetric heat source, p, is a function of the electric field, ux. The electric field variable ux is a complex quantity, and the dimensionless electric field ux ) vx + iwx is substituted, the real and imaginary components are obtained from eq 7 as 2

d vx

+ (4π2Nw2 - Np2)vx - 4πNwNpwx ) 0 (24)

dz2 and

d 2 wx 2

dz

+ (4π2Nw2 - Np2)wx + 4πNwNpvx ) 0 (25)

The boundary conditions for the real and imaginary components are5

}

dvx - 2πNw0wx ) 4πNw0 sin(πNw0) dz at z ) 0 dwx + 2πNw0vx ) 4πNw0 cos(πNw0) dz

}

dvx + 2πNw0wx ) 0 dz at z ) 1 dwx - 2πNw0vx ) 0 dz

(26)

(27)

The electric field distributions within a sample in the presence of reflective support need special mention. For a 1D slab, the reflective support is placed at the right end where z ) 1; hence, the boundary conditions are vx ) 0 and wx ) 0. Using the dimensionless variables

θ)

I0 R0t (T - T∞)k0 , p0 ) , and τ ) 2 2 2L 4p0L 4L

(28)

where R0 is a reference diffusivity, eq 23 for a 1D slab can be written as

FC

∂θ ∂2θ 2NwNp 2 (vx + wx2) )k h 2+ ∂τ N ∂z w0

The initial condition is

(29)

θ(τ)0) )

(T0 - T∞)k0

for 0 e z e 1

4p0L2

(30)

and the boundary conditions are

∂θ/∂z ) 0

z)0

∂θ/∂z ) 0

z)1

(31)

The electric field and energy balance equations associated with corresponding boundary conditions are functions of Nw, Np, and Nw0. The energy balance equation also consists of the dimensionless terms as FC and k h , which are scaled with respect to a reference substance, e.g., water. The energy balance equation and the electric field equations with the appropriate boundary conditions are solved using the Galerkin finite-element method as discussed by Reddy.18 To discretize the time domain, the Crank-Nicholson method is used, and the nonlinear residual equations are solved using a Newton-Raphson method.7,10,11 Results and Discussion Generalized Characteristics: MW Power and Resonances. A generalized trend for MW power distributions is illustrated with dimensionless average power distributions obtained from eq 22 as a function of Nw, Np, and Nw0 based on eq 21. The dimensionless power, as defined in eq 21, is obtained based on a scaling with respect to the intensity of the incident wave and sample length. The incident frequency and sample length are typically characterized by Nw0, as defined in eq 20. The electric field and power distributions are based on the analytical solution, as discussed in eqs 11 and 21. Parts a-f of Figure 2 illustrate qav contours in Nw Np domains for various Nw0 (Nw0 ) 0.1, 0.4, and 1). Note that smaller values of Nw0 indicate small sample thicknesses at a fixed frequency based on eq 20. Here, Nw0 ) 0.1 corresponds to the sample length 2L ) 0.012 m for MWs with frequency 2450 MHz. Parts a, c, and e of Figure 2 represent qav contours for slabs with metallic support. As seen in Figure 2a, for 0 < Nw < 3, the average power distribution is highly oscillatory, and during this regime, efficient heating of the sample would depend on Np as well. It is interesting to note that, for Nw ) 0.5n - 0.25 (n being an integer), the enhanced MW power absorption would be possible for a range of Np values because all of the contour lines seem to emanate from Nw ) 0.5n - 0.25 and Np ) 0, as seen in Figure 2a. Note that, for Nw ) 0.5n - 0.25, the maxima in average power occur at very small Np values. For larger Np, the average power seems to decrease monotonically with Nw for fixed Np. It is also interesting to note that the oscillatory features of the average power during smaller Np would suggest the choice of typical thicknesses for various materials for a maximal heating. Parts c and e of Figure 2 illustrate qav contours for Nw0 ) 0.4 and 1, respectively. Oscillations in the average power still exist with Nw. It is interesting to note that, as Nw0 increases, the contour values of the average power also increase. At Nw0 ) 1, the oscillations in the average power still exist; however, lower average power corresponds to a smaller Np regime for all Nw values. Note that Nw0 ) 1 corresponds to sample thickness 2L ) 0.12 m.


Figure 2. Average power (qav) contours in the Nw-Np domain for Nw0 ) 0.1 (a and b), 0.4 (c and d), and 1 (e and f). Cases a, c, and e correspond to slabs with metallic support, and cases b, d, and f correspond to slabs exposed to free space. The oscillations in average power are observed for a smaller Np regime, and as Nw0 increases, the oscillations become less, especially for slabs exposed to free space.

Parts b, d, and f of Figure 2 illustrate the average power distributions for slabs exposed to free space for similar Nw0 regimes. Similar to Figure 2a, the oscillatory features in the average power are observed, and it is found that Nw ) 0.5n (n being an integer) corresponds to the local maxima in qav, which contrast with the cases with metallic support. It is interesting to note that, as Nw0 increases, the degree of oscillations tends to decrease within Np < 1, as seen in Figure 2d,f, which contrast with Figure 2c,e. The oscillation free regime leads to almost uniform average power for a fixed Np during lower Nw regime corresponding to Nw0 ) 1. For Nw0 ) 1, the average power is seen to increase monotonically with Np for a fixed Nw value (Figure 2d), whereas the average power is still oscillatory, however small, with metallic support. The maxima in average power, characterized by the local maxima in the NwNp domain are observed for Nw0 e 0.4, and some of these features would provide guidelines for an efficient and faster MW processing of materials. The generalized analysis on local maxima or resonances in average power for MW heating needs special attention. The oscillatory distribution of the average power leads to local maxima within Np < 1 for samples in free space. The set of local maxima would occur at Nw ) 0.5n (n being an integer) for samples in free space. For samples with metallic support, the oscillations occur during all Nw0 regimes, and for smaller Nw0 regimes,

t r Figure 3. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs with the unexposed face attached with metallic support: (a) Nw0 ) 0.1, Nw ) 0.25, and Np ) 0.08; (b) Nw0 ) 0.1, Nw ) 0.75, and Np ) 0.775; (c) Nw0 ) 0.1, Nw ) 1.25, and Np ) 0.426; (d) Nw0 ) 0.1, Nw ) 1.75, and Np ) 0.3. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. The number of spatial resonances in electric fields and power increase proportionally with Nw, and the number of maxima corresponds to Nw ) 0.5n - 0.25 (n is an integer).

the oscillations would lead to a set of local maxima at Nw ) 0.5n - 0.25. Note that the occurrence of local maxima is restricted within a lower Np limit. Our generalized analysis is a consequence of the earlier studies by Ayappa et al.12 and Ayappa,13 and their analysis also established the similar relationship between the occurrence of resonance and Nw irrespective of Np values. However, their analysis fails to highlight the enhanced or optimal MW power absorption for intermediate Nw and Np values. In general, the maxima in average power do occur for Nw ) 0.5n - 0.25 or Nw ) 0.5n and the local maxima in average power also vary with Np, and this information was overlooked by earlier researchers.12,13 We observe that, for greater Nw0, the oscillatory behavior of power is less pronounced (especially for samples exposed in free space) and the monotonic increase in the average power with Np is observed. Our generalized predictions on the average power are invariant of materials, MW frequencies, and incident intensities. Figure 3 illustrates the spatial distributions of amt r plitudes (Ax,l, Ax,l , and Ax,l ), phase difference of the t r ), and the dimensionless stationary wave (δx,l - δx,l power (q) for Nw0 ) 0.1 corresponding to samples with metallic support. The detailed spatial distributions correspond to the maxima in average power occurring at Nw ) 0.5n - 0.25 (n being an integer) for smaller Np (see Figure 2a,c,e). As seen in Figure 3a (Nw ) 0.25), the amplitudes of the transmitted and reflected waves

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t r Figure 4. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs with the unexposed face attached with metallic support: (a) Nw0 ) 0.1, Nw ) 0.35, and Np ) 0.148; (b) Nw0 ) 0.1, Nw ) 0.45, and Np ) 0.585; (c) Nw0 ) 0.1, Nw ) 0.55, and Np ) 1.0; (d) Nw0 ) 0.1, Nw ) 0.65, and Np ) 0.52. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. The symbol b denotes the regime with spatial resonances of the electric field and power. The regime with spatial resonance shifts toward the unexposed face as Nw increases. Note that the number of spatial resonances remains constant, which corresponds to 1 < n < 2 for Nw ) 0.5n - 0.25.

are almost equal and uniform throughout the sample. It is observed that the spatial amplitude of the stationary wave has one maxima occurring at the exposed face of the slab. Note that the maxima in amplitudes are illustrated with the difference between the phase angle t r - δx,l of zero, as shown in the spatial distribution of δx,l vs z plot. The power distribution follows a trend similar to that of the spatial distribution of stationary fields, and the spatial dimensionless power reaches zero at the unexposed end attached with the metallic support. The spatial distribution of the amplitude of the stationary wave shows two maxima at Nw ) 0.75, as seen in Figure 3b, and the maxima occur at the exposed face and near the unexposed end of the sample. It is interesting to observe that the number of maxima in the spatial amplitude of the stationary wave or power increase proportionally with Nw. We have found that three maxima occur at Nw ) 1.25 and four maxima at Nw ) 1.75 (Figure 3c,d). For all of the cases (Figure 3b-d), we observe that the amplitudes of transmitted waves are greater than those of reflected waves except at the unexposed face. Note that the total electric field at the unexposed face is zero because of metallic support and the greater amplitude of the reflected field at the unexposed face is due to the presence of metallic support. Parts a-d of Figure 4 illustrate the distributions of fields and power for 0.25 < Nw < 0.75 when the samples are processed with metallic support. Note that Figure

t r Figure 5. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs with the unexposed face attached with metallic support: (a) Nw0 ) 0.1, Nw ) 0.25, and Np ) 2.5; (b) Nw0 ) 0.1, Nw ) 0.75, and Np ) 2.5; (c) Nw0 ) 1, Nw ) 0.25, and Np ) 2.5; (d) Nw0 ) 1, Nw ) 0.75, and Np ) 2.5. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. For all of the cases, the amplitudes of the stationary electric field and power decay exponentially from the left face irrespective of Nw.

3 illustrates spatial resonances when n is an integer and Figure 4 corresponds to 1 < n < 2. Note that one spatial resonance corresponds to Nw ) 0.25 and two spatial resonances correspond to Nw ) 0.75. For Nw ) 0.35, the spatial resonance occurs at z ) 0.29, and the spatial resonances would occur at z ) 0.445, 0.55, and 0.62 for Nw ) 0.45, 0.55, and 0.65, respectively. Here Np values are selected to illustrate the greater average power for a fixed Nw, as seen in Figure 2a. The resonances are illustrated with zero-phase difference, as seen in the t r - δx,l vs z plot. Note that only one spatial resonance δx,l occurs at z ) 0 for Nw ) 0.25, and this single resonance is found to be shifted with increasing values of Nw. In addition, the power deposition at the exposed face would initially decrease with Nw, and later the power deposition at the exposed face tends to attain a maximum as Nw tends to 0.75. At Nw ) 0.75, the secondary power maxima also occur at the exposed face, as seen in Figure 3b. Therefore, this analysis establishes that the spatial resonance does exist at the intermediate Nw values, with n being a noninteger, and the number of spatial maxima change, with n being an integer. This analysis would also be useful to study the local hot spots or power maxima for all ranges of Nw, whereas an earlier theoretical investigation by Ayappa13 was limited to finding maxima in average power at Nw ) 0.5n - 0.25, with n being an integer only. Parts a-d of Figure 5 illustrate the distributions of fields and power for greater Np values (Np ∼ 2.5) corresponding to various Nw values (Nw ) 0.25 and 0.75)


t r Figure 6. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs exposed in free space: (a) Nw0 ) 0.1, Nw ) 0.5, and Np ) 0.13; (b) Nw0 ) 0.1, Nw ) 1, and Np ) 0.695; (c) Nw0 ) 0.1, Nw ) 1.5, and Np ) 0.38; (d) Nw0 ) 0.1, Nw ) 2, and Np ) 0.25. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. The number of spatial resonances in electric fields and power increases proportionally with Nw, and the number of maxima corresponds to Nw ) 0.5n (n is an integer).

for samples with metallic supports. Parts a and b of Figure 5 denote the distributions for Nw0 ) 0.1, and for both of the cases, it is observed that the amplitude of the transmitted field varies exponentially while the reflected field is almost zero throughout the domain. As a result, the amplitude of the stationary wave follows a trend similar to the amplitude of the transmitted field. It is interesting to observe that, because of a weak reflected field, the interference of the wave is negligible and the local spatial maxima in power or the stationary field does not occur for zero-phase differences. The weak interference is also attributed to the fact that the phase t r difference δx,l - δx,l shows two zeros occurring at the exposed face and at a point within a sample for Nw ) 0.75 and the maxima always occurs at the exposed face. For all of the cases, the transmitted and stationary fields decay exponentially, which is also termed Lambert’s exponential distribution, as described by Ayappa et al.5 Parts c and d of Figure 5 correspond to Nw0 ) 1, and the multiple zero-phase differences are observed; however, weak interference between traveling waves results in exponential decay of the stationary electric field and power. Note that the maxima in spatial power for Nw0 ) 1 are greater than those for Nw0 ) 0.1. Figure 6 illustrates the spatial distributions of amt r plitudes (Ax,l, Ax,l , and Ax,l ), phase difference of the t r stationary wave (δx,l - δx,l), and dimensionless power (q) for Nw0 ) 0.1 corresponding to samples exposed to free space. The detailed spatial distributions correspond to the maxima in average power occurring at Nw ) 0.5n

t r Figure 7. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs exposed in free space: (a) Nw0 ) 0.1, Nw ) 0.6, and Np ) 0.24; (b) Nw0 ) 0.1, Nw ) 0.7, and Np ) 0.67; (c) Nw0 ) 0.1, Nw ) 0.8, and Np ) 0.9; (d) Nw0 ) 0.1, Nw ) 0.9, and Np ) 0.58. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. The symbol b denotes the regime with spatial resonances of the electric field and power. The regime with spatial resonance shifts toward the unexposed face as Nw increases, while the other regime of resonance is fixed at the unexposed face. Note that the number of spatial resonances remains constant, which corresponds to 1 < n < 2 for Nw ) 0.5n.

(n being an integer) for smaller Np (see Figure 2b). For Nw ) 0.5, the amplitude of the transmitted wave is greater than that of the reflected wave, and the phase difference distributions illustrate the zero-phase difference at the faces (Figure 6a). Therefore, the stationary electric fields and power exhibit two maxima at the faces. The spatial distribution of amplitude of the stationary wave shows three maxima at Nw ) 1, as seen in Figure 6b, and the maxima occur at the center and at both of the faces of the sample. Note that the number of maxima in the spatial amplitude of the stationary wave or power increases proportionally with Nw. We have found that four maxima occur at Nw ) 1.5 and five maxima at Nw ) 2 (Figure 6c,d). Note that a similar number of power maxima occur at a greater Nw value for slabs with metallic support, as seen in Figure 3. Parts a-d of Figure 7 illustrate the distributions of fields and power for 0.5 < Nw < 1 with 1 < n < 2 for the samples exposed in free space. Note that two spatial resonances corresponding to Nw ) 0.5 and three spatial resonances corresponding to Nw ) 1 occur. The resonance at the left face is seen to shift as Nw increases (Figure 7). For Nw ) 0.6, the spatial resonance is shifted at z ) 0.17, and the spatial resonances would occur at z ) 0.29, 0.38, and 0.45 for Nw ) 0.7, 0.8, and 0.9, respectively. The resonances are illustrated with zerot r - δx,l vs z plot. Note phase difference as seen in the δx,l that the power deposition at the exposed face would

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t r Figure 8. Amplitudes of the electric field (Ax,l, Ax,l , and Ax,l ), t r phase difference (δx,l - δx,l), and power distribution (q) for slabs exposed in free space: (a) Nw0 ) 0.1, Nw ) 0.5, and Np ) 2.5; (b) Nw0 ) 0.1, Nw ) 1, and Np ) 2.5; (c) Nw0 ) 1, Nw ) 0.5, and Np ) 2.5; (d) Nw0 ) 1, Nw ) 1, and Np ) 2.5. Legend: ‚‚‚, transmitted wave; - -, reflected wave; s, stationary wave. For all of the cases, the amplitudes of the stationary electric field and power decay exponentially from the left face irrespective of Nw.

initially decrease with Nw and later the power deposition at the exposed face tends to attain a maximum as Nw tends to 1. At Nw ) 1, another power maximum also occurs at the exposed face, as seen in Figure 6b, and the other two maxima occur at the center and at the unexposed face. Similar situations on the maxima in power for slabs with metallic supports were also observed, as seen in Figure 4. Parts a-d of Figure 8 illustrate the distributions of fields and power for greater Np values (Np ∼ 2.5) corresponding to various Nw values (Nw ) 0.5 and 1) for samples in free space. Parts a and b of Figure 8 denote the distributions for Nw0 ) 0.1, and parts c and d of Figure 8 denote the distributions for Nw0 ) 1. For all of the cases, it is observed that the amplitude of the transmitted field varies exponentially while the reflected field is almost zero throughout the domain. Therefore, the amplitude of the stationary field is seen to be identical with that of the transmitted field. In contrast, for metallic support, the stationary and transmitted fields are not identical because of the reflection offered by metallic support. We may also note that Lambert’s exponential decay law on MW power distribution is observed for Np ∼ 2.5, which is similar to that of slabs with metallic support. The greater Np denotes a slab with greater thickness with a fixed MW frequency, and for greater thickness samples, the power decays exponentially from the exposed face irrespective of either metallic support or free space. Generalized Heating Characteristics: Case Studies for Various Materials. Figures 9 and 10 illustrate

Figure 9. Dimensionless temperature (θ × 102) distributions for water (FC ) 1 and k h ) 1) samples for two cases: (a, c, and e) slabs with a metallic support; (b, d, and f) slabs exposed to free space. The insets represent the spatial power distributions. The heating rates are compared for identical dimensionless quantities: (a and b) Nw0 ) 0.03, Nw ) 0.25, and Np ) 0.1; (c and d) Nw0 ) 0.06, Nw ) 0.5, and Np ) 0.2; (e and f) Nw0 ) 0.08, Nw ) 0.75, and Np ) 0.3. For all of the cases, temperature distributions are similar to power distributions. It is observed that metallic support is efficient for Nw ) 0.25 and 0.75 whereas the slabs with no support are advantageous for Nw ) 0.5.

the spatial temperature distributions for various materials (water and alumina) whose thermal and dielectric properties are enlisted in Table 1. The heating characteristics due to MW power absorption can be represented by a dimensionless temperature, θ, which is scaled with respect to the incident power, p0, as shown in eq 28. The dimensionless temperature characteristics are shown for various dimensionless time steps τ. The maximum temperatures for all of the cases are within the limit of 100 °C (Figures 9 and 10). For all materials, the inset plots illustrate the spatial power depositions, which are functions of three material-invariant dimensionless numbers (Nw, Np, and Nw0), and the heating characteristics may be strong/weak functions of spatial power depositions for various cases as discussed next. Figure 9 represents temperature distributions for water with various Nw0 values. The dimensionless h , are 1 specific heat, FC, and thermal conductivity, k because they are scaled with respect to the corresponding properties of water. As mentioned earlier, for a fixed MW intensity, a greater Nw0 represents greater sample thickness. The insets show the spatial power distributions for various Nw, Np, and Nw0. Parts a, c, and e of Figure 9 represent the distributions for slabs with metallic support, and parts b, d, and f of Figure 9


Figure 10. Dimensionless temperature (θ × 102) distributions for alumina (FC ) 0.94 and k h ) 43.4) samples for two cases: (a, c, and e) slabs with a metallic support; (b, d, and f) slabs exposed to free space. The insets represent the spatial power distributions. The heating rates are compared for identical dimensionless quantities: (a and b) Nw0 ) 0.076, Nw ) 0.25, and Np ) 0.01; (c and d) Nw0 ) 0.15, Nw ) 0.5, and Np ) 0.02; (e and f) Nw0 ) 0.23, Nw ) 0.75, and Np ) 0.034. For all of the cases, a uniform temperature profile is observed irrespective of spatial power distributions. Table 1. Thermal and Dielectric Properties Given for Water and Aluminaa material property

water

alumina

heat capacity, Cp [W‚s‚kg-1‚°C-1] thermal conductivity, k [W‚m-1‚°C-1] density, F [kg‚m-3] dielectric constant, κ′ dielectric loss, κ′′

4190 0.609 1000 78.1 10.44

1046 26 3750 10.8 0.1566

a

The dielectric data correspond to 2450 MHz.5,11,19

represent the distributions for slabs in the free space. Note that the maxima in average power with the metallic support correspond to minima with slabs exposed in free space during Nw ) 0.25 and 0.75, whereas at Nw ) 0.5, the maxima occur with slabs in free space and minima for slabs with metallic support (Figure 2). During Nw ) 0.25, it is observed that the spatial power has a strong maximum at the exposed face for slabs with metallic support and a weak maximum in power occurs at the unexposed face when the slab is exposed in free space (Figure 9a,b). Because of greater power absorbed, the temperature near the exposed face rises rapidly for the slabs with the metallic support and the heating is very low for slabs in free space. For both of the cases, note that the temperature profiles follow a trend similar to that of the spatial power distributions during τ ) 0.01-0.09. During Nw ) 0.5, the heating rate

is greater for slabs in free space based on greater spatial power deposition and the slab with metallic support has very small heating rates (Figure 9c,d). For Nw ) 0.75, the metallic support exhibits greater heating effects whereas the slab with free space shows smaller heating effects (Figure 9e,f). It is interesting to note that the heating at the unexposed face is considerably greater for slabs with metallic support for Nw ) 0.25 and 0.75 whereas the power absorbed near the unexposed face for slabs with metallic support is almost zero. We have considered Nw ranging within 0.25-0.75, which corresponds to Np within 0.1-0.3, and therefore Lambert’s exponential variation in power absorption was not shown. We have also considered the case studies for oil (k h ) 0.28), and it is observed that the heating rate of oil samples is quite small at the unexposed face attached with metallic support, which contrast the greater heating effects for water samples with those for similar situations (figures not shown). Figure 10 represents the temperature distributions for an alumina sample. For all of the Nw values, it is observed that the temperature profile is always uniform throughout the sample as k h ) 43.4. On the basis of the above analysis, one may infer that the temperature profile may follow power distributions if k h e 1 during lower Np values whereas a uniform temperature profile would be observed for k h . 1 even with smaller Np as seen for alumina. Metallic support has a critical role in the heating rates for various materials, as seen in Figures 9 and 10. We have observed that, for Nw ) 0.25 and 0.75, the metallic support at the unexposed face enhances the average power in the sample whereas Nw ) 0.5 corresponds to greater heating rates for samples exposed to free space. Conclusion The generalized dimensionless analysis on MW propagation and heating for uniform plane waves has been illustrated based on three numbers: wavenumber (Nw), penetration number (Np), and free-space wavenumber (Nw0). The MW power absorption and rate of heating have been investigated for slabs with one side attached with metallic support and slabs exposed in free space. The detailed mathematical analysis on the MW propagation is based on the role of individual traveling waves, where both the transmitted and reflected waves are functions of Nw and Np. The volumetric heat source due to MWs is a function of the dimensionless numbers and dimensionless electric fields. The dimensionless forms of energy balance and electric field equations are solved using a Galerkin finite-element method. Analysis on the material-invariant average power distribution in samples has been illustrated via average power contour profiles in Nw-Np coordinate space for Nw0 ) 0.1, 0.4, and 1. It is observed that the average power shows strong oscillations for 0 < Nw e 3 and Np < 1.5 for smaller Nw0 ( 1.5 irrespective of Nw corresponding to samples exposed to free space. The generalized trend of spatial variation of power has been shown for various Nw and Np during one side incidence. It is observed that the spatial power shows a maximum occurring at the exposed face for Nw ) 0.25 corresponding to slabs with the unexposed face attached with metallic support, and the number of maxima increases proportionally with Nw ) 0.5n - 0.25 (n being an integer) for smaller values of Np. The maxima of

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spatial power, also termed resonances, correspond to constructive interference of traveling waves with the zero-phase difference. We have observed the presence of spatial resonances at intermediate Nw values and for 0.25 < Nw < 0.75 (1 < n < 2); the occurrence of maxima of power or resonance would shift from the exposed face to an interior point in a sample, and a second resonance would appear at the exposed face at Nw ) 0.75. This study would be useful to detect greater spatial heating rates at any Nw for any material. Similar studies on spatial resonances were carried out for the samples exposed in free space. We have found the proportionally increasing number of maxima in spatial power at Nw ) 0.5n (n being an integer) and intermediate resonance features for noninteger values of n. Note that one maximum in spatial power occurs for Nw ) 0.25 for samples with metallic support, and for Nw ) 0.5, two maxima occur at the faces of the slabs exposed in free space. This analysis suggests a suitable strategy of heating samples depending on Nw for smaller Np values. During greater Np (∼2), the spatial resonances are weak, resulting in an exponential variation for power following Lambert’s law irrespective of Nw and Nw0. The heating characteristics are illustrated with spatial temperature distributions. It is observed that temperature distributions follow spatial power distribution h e 1 and a uniform during smaller Np values for k temperature profile is observed for k h . 1 irrespective of Nw and Np. We have examined two cases that involve samples attached with the support and samples exposed in free space, and each case will be preferred depending on Nw values, which are material-invariant. MWs can be efficiently used with the application of metallic support. We have seen that during Np e 2 metallic support enhances the heating rates for Nw ) 0.25 and 0.75 whereas Nw ) 0.5 corresponds to enhanced heating without any support irrespective of materials. In perspective, the “dimensionless numbers” based on MW propagation adequately establish material-invariant dynamics during MW propagation. Currently, we are extending this analysis on material-invariant MW heating and dynamics for 2D enclosures. Notation Al ) amplitude of the stationary wave for the lth layer Atl ) amplitude of the transmitted wave for the lth layer Arl ) amplitude of the reflected wave for the lth layer c ) velocity of light (m‚s-1) C ) specific heat (J‚kg-1‚K-1) Dp ) penetration depth (m) Ex ) electric field intensity (V‚m-1) E0 ) incident electric field intensity (V‚m-1) f ) frequency (Hz) I0 ) intensity of incident microwave (W‚m-2) k ) thermal conductivity (W‚m-1‚K-1) L ) half-slab thickness (m) Np ) penetration number Nw ) wavenumber Nw0 ) wavenumber for free space p ) microwave source term (W‚m-3) q ) dimensionless microwave source term t ) time (s) T ) temperature (K) ux,l ) dimensionless electric field for the lth layer r ux,l ) dimensionless electric field due to reflection for the lth layer

t ) dimensionless electric field due to transmission for ux,l the lth layer vx ) dimensionless real field component wx ) dimensionless imaginary field component Z ) distance (m) z ) dimensionless distance

Greek Symbols γ ) dimensionless propagation constant 0 ) free space permittivity (F‚m-1) θ ) dimensionless temperature κ′ ) relative dielectric constant κ′′ ) relative dielectric loss λm ) wavelength in the medium (m) F ) density (kg‚m-3) τ ) dimensionless time δx,l ) phase difference in the stationary wave for the lth layer r δx,l ) phase state for the reflected wave for the lth layer t δx,l ) phase state for the transmitted wave for the lth layer ω ) angular frequency (rad‚s-1) Subscript l ) layer number Superscripts r ) reflected wave t ) transmitted wave

Literature Cited (1) Ohlsson, T.; Risman, P. O. Temperature Distribution of Microwave Heating-Spheres and Cylinders. J. Microwave Power 1978, 13, 303. (2) 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. (3) Weil, C. M. Absorption Characteristics of Multilayered Sphere Models Exposed to UHF/Microwave Radiation. IEEE Trans. Biomed. Eng. 1975, BME-22, 468. (4) Datta, A. K. Heat and Mass Transfer in the Microwave Processing of Food. Chem. Eng. Prog. 1990, 6, 47. (5) 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. (6) Ayappa, K. G.; Davis, H. T.; Davis, E. A.; Gordon, J. TwoDimensional Finite Element Analysis of Microwave Heating. AIChE J. 1992, 38, 1577. (7) Basak, T.; Ayappa, K. G. Analysis of Microwave Thawing of Slabs with Effective Heat Capacity Method. AIChE J. 1997, 43, 1662. (8) Basak, T.; Ayappa, K. G. Influence of Internal Convection during Microwave Thawing of Cylinders. AIChE J. 2001, 47, 835. (9) Basak, T.; Ayappa, K. G. Role of Length Scales on Microwave Thawing Dynamics in 2D Cylinders. Int. J. Heat Mass Transfer 2002, 45, 4543. (10) Basak, T. Analysis of Resonances during Microwave Thawing of Slabs. Int. J. Heat Mass Transfer 2003, 46, 4279. (11) Basak, T. Role of Resonances on Microwave Heating of Oil-Water Emulsions. AIChE J. 2004, 50, 2659. (12) 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. (13) Ayappa, K. G. Resonant Microwave Power Absorption in Slabs. J. Microwave Power Electromagn. Energy 1999, 34, 33. (14) 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. (15) Zhang, H.; Datta, A. K.; Taub, I. A.; Doona, C. Electromagnetics, Heat Transfer, and Thermokinetics in Microwave Sterilization. AIChE J. 2001, 47, 1957.

Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3085 (16) Bows, J. R.; Patrick, M. L.; Janes, R.; Metaxas, A. C. R.; Dibben, D. C. Microwave Phase Control Heating. Int. J. Food Sci. Technol. 1999, 34, 295. (17) Subirats, M.; Iskander, M. F.; White, M. J.; Kiggans, J. O. FDTD Simulation of Microwave Sintering in Large (500/4000 liter) Multimode Cavities. J. Microwave Power Electromagn. Energy 1997, 32, 161. (18) Reddy, J. N. An Introduction to the Finite Element Method; McGraw-Hill: New York, 1993.

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Received for review October 5, 2004 Revised manuscript received December 10, 2004 Accepted February 2, 2005 IE049035O

Generalized Analysis on Microwave-Assisted Material Processing for

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