Microwave heating of water-rich solvents is a widely used processing technique in research and applications. High-quality outcome requires a uniform temperature environment; which, in turn, depends on the balancing of a variety of effects taking place during the heating. Here, we show that two inherent effects, namely, polarization-charge shielding and electromagnetic resonances, play a critical role in shaping up the field pattern in the heated water sample. Polarization-charge shielding produces an internal electric field sensitive to the sample size, shape, and orientation. Internal electromagnetic resonances result in a widely varying electric field, while also allowing much deeper field penetration than the attenuation length to allow large-scale treatment. The key to temperature uniformity, thus, lies in an optimized thermal flow to balance the non-uniform energy deposition. These complicated processes are examined in simulation and interpreted physically. It is shown that a spherical sample is most favorable for obtaining a high temperature uniformity mainly because of its rotational symmetry. This conclusion is significant in that prevailing sample vessels are mostly non-spherical.

Water is one of the most abundant substances, both in existence and in use. Its properties are of general interest. In particular, water behavior under microwave heating is of importance to scientific research and applications. For example, with the advantages of volumetric heating and much shorter treatment time, microwave-assisted synthesis has grown to an active field of research,1–3 for which water is environmentally the cleanest solvent.4,5 As another example, microwave heating has been explored as a method for wastewater treatment in recent years.6 The dielectric properties of water have been well documented.7–9 There have also been a number of numerical and experimental studies on microwave heating of liquid water ranging from molecular dynamics10 to performance enhancement in a specific microwave structure.11–15 These studies reveal the complexity of the water heating processes. In particular, heat convection plays a key role in the resulting temperature distribution. As discussed below, more issues still merit further investigations. The current study, based on a simplified model, is complementary to earlier studies by focusing on the underlying physics and general trends.

Although microwave heating is volumetric in nature, temperature non-uniformity has been a major hurdle to numerous applications. The exposure to a non-uniform field is a clear cause for this difficulty. There are, however, some less-understood causes, which are inherent in nature in that they persist even in a perfectly uniform field. Two major causes are addressed in the current study: polarization-charge shielding and electromagnetic resonances.

Exposed to an electromagnetic wave with E-field amplitude E0, a dielectric object reacts with slight microscopic displacements of its bond electrons to form macroscopic polarization charges, which partially shield the incident E0. As a result, the dielectric object's internal E-field can be significantly smaller than E0 (Ref. 16, Chap. 4). Furthermore, for a non-spherical object, the shielding effect can be highly dependent on its orientation with respect to E0,17,18 which constitutes a significant cause of the heating non-uniformity.

Independently, electromagnetic resonances can readily take place in a dielectric object of dimensions comparable to or greater than half of a wavelength in the dielectric. As an electromagnetic wave enters such a dielectric object, multiple reflections off the inside walls can constructively superpose into a resonant standing wave at discrete frequencies. While this can significantly boost the field strength, it also leads to a complicated field pattern and, therefore, a much greater field non-uniformity.

In Sec. II, we examine in detail the effects of field shielding and electromagnetic resonances in water in the presence of an incident plane wave at 2.45 GHz, a predominantly used frequency for microwave heating. The field pattern (hence the heating rate) is shown to be characterized by three distinct regimes of the sample size, while an elongated sample is subject to the additional complication of orientation. Section III incorporates the effects of thermal flows and compares the relative merits of spherical and elongated shapes over a range of sizes. The optimum shape is found to be spherical and two radius ranges are identified for stable operation. Realistic issues concerning water sphere heating are discussed in Sec. IV, followed by the summary, conclusion, and significance in Sec. V.

Let ex, ey, and ez be unit vectors of the Cartesian coordinates. Consider the model of an x-polarized plane wave with frequency ω, propagation constant kz, and E-field,
E = E 0 e i ω t + i k z z e x
(1)
incident from free space along the positive z axis onto a non-permeable, uniform dielectric object of complex permittivity ε = ε′+iε′′. Let Ed(x) be the position-dependent E-field amplitude inside the object, which is the sum of the incident E0 and all other E-fields induced by E0. The dielectric is heated by Ed(x). The power absorption (or heating) rate per unit volume (pd) is (Ref. 16, Chap. 7),
p d ( x ) = 1 2 ε ω | E d ( x ) | 2 = 1 2 ε ω h ( x ) | E 0 | 2 ,
(2)
where
h ( x ) | E d ( x ) | 2 | E 0 | 2 .
(3)

In Eq. (3), h(x) is the ratio of the actual heating rate normalized to the heating rate under E0. In a linear medium where ε is independent of E0, Ed(x) is proportional to E0. Thus, h(x) is independent of E0 and it gives the spatial profile of the heating rate for any E0. Given E0, ω, and ε′′, h(x) determines the actual heating rate through Eq. (2).

The polarization-charge shielding and resonance effects can drastically affect the behavior of h(x) with respect to the size and orientation of the dielectric object. To quantify these effects, we define the mean value ( h ¯) of h(x) and its standard deviation (σh) as
h ¯ v h ( x ) d 3 x total volume ,
(4)
σ h v [ h ( x ) h ¯ ] 2 d 3 x total volume ,
(5)
where each integral is over the total volume of the sample under study. The vessel housing of the sample is neglected, assuming that it is made of essentially microwave transparent materials such as glass or teflon. As a useful figure of merit, σh/ h ¯ gives the spread of the heating rates relative to the mean value. It will be referred to as the “percentage spread” of the heating rate.

We now approximate the ε of water-rich solvent by the ε of pure water. For microwave heating, key factors to consider are the temperature dependence of ε (= ε′+iε′′), the wavelength in the dielectric (λd), and the power attenuation length (δ). The temperature (T) dependence of ε′/ε0 and ε′′/ε0 of water, given by Ref. 9, is plotted in Fig. 1(a) for the household microwave frequency of 2.45 GHz.

FIG. 1.

Dielectric properties of water at 2.45 GHz. (a) ε′/ε0 and ε′′/ε0 of water as a functions of T (based on Ref. 9). (b) λd and δ in water as functions of T [based on Eq. (6)]. In particular, λd = 1.39 and δ = 1.8 cm at T = 25 °C; λd = 1.57 and δ = 5.8 cm at T = 80 °C.

FIG. 1.

Dielectric properties of water at 2.45 GHz. (a) ε′/ε0 and ε′′/ε0 of water as a functions of T (based on Ref. 9). (b) λd and δ in water as functions of T [based on Eq. (6)]. In particular, λd = 1.39 and δ = 1.8 cm at T = 25 °C; λd = 1.57 and δ = 5.8 cm at T = 80 °C.

Close modal

λd and δ are given by (Ref. 16, p. 314)

(6)
λ d = λ free Re ε / ε 0 λ free ε / ε 0 ( if ε ε ) ,
(6a)
δ = c 2 Im ε / ε 0 ω ε ε 0 c ε ω ( if ε ε ) ,
(6b)
where λfree is the free-space wavelength and ε0 is the permittivity of free space.

Based on the data in Fig. 1(a) and Eq. (6), we obtain λd and δ in water as functions of T [Fig. 1(b)] with the following features relevant to the current study.

Figure 1(b) shows a significant reduction of wavelength in water (λd) from λfree (=12.25 cm) due to the relatively large ε′ of water [Fig. 1(a)]. Importantly, λd varies from λd = 1.39 cm at T = 25 °C to λd = 1.57 cm at T = 80 °C. The 11% increase will cause a significant upward shift of all resonant frequencies as T rises from 25 to 80 °C.

It can be seen in Fig. 1(b) that δ increases sharply with a higher T due to a much decreased ε′′ [Fig. 1(a)]. For example, δ = 1.8 cm at T = 25 °C and δ = 5.8 cm at and T = 80 °C. The 3.2-fold increase in δ implies a much increased attenuation length at a higher T, hence beneficial to the heating of large-size objects.

The λd values in Fig. 1(b) suggest that, for a water sample a few centimeters in dimensions, very high order modes will be excited at 2.45 GHz. The numerically evaluated field patterns will thus be too complex to interpret. On the other hand, Mie in 1908 developed a rigorous theory for the interaction of a plane electromagnetic wave with a dielectric sphere,19 including dielectric loss, wave scattering, sphere resonances, and polarization-charge buildup. Hence, for a detailed understanding of the physical processes, we apply Mie's theory to a water sphere of radius R irradiated by the plane wave in Eq. (1). As an immediate benefit of the spherical geometry, we find in the literature that, in the quasi-static limit, Rλd, h(x) is independent of x and R, given by (Ref. 16, Chap. 4),
h ( x ) | 3 ε 0 ε + 2 ε 0 | 2 , for R λ d ,
(7)
which shows a reduction in h for ε′ > ε0 due to the polarization-charge shielding. As R approaches λd/2, Mie resonances start to set in. By the theory of dielectric resonators,20 at T = 25 °C, the first resonance peaks at R = 0.68 cm, while the 127th resonance peaks at R = 10 cm.

As the incident wave penetrates into the sphere, the fields inside are radically different with and without resonances. Assuming that the water sphere is centered at the origin of coordinates, the Mie solution is in the form of an infinite series. Based on the documented formulas in Ref. 21, Fig. 2 displays h ¯ [the mean heating rate, Eq. (4)] and σh/ h ¯ (the percentage spread of the heating rate) as functions of R for T = 25 and T = 80 °C. Figure 2 is characterized by three distinct regimes:

FIG. 2.

Analytically evaluated h ¯ (left figure) and σh/ h ¯ (right figure) as functions of R for T = 25 °C (dashed line) and T = 80 °C (solid line). (a) shows a transition from the quasi-static regime (Rλd), to the sharp-resonance regime (λd/2 < R < 2λd), and finally to the multiple-resonance regime (Rλd).

FIG. 2.

Analytically evaluated h ¯ (left figure) and σh/ h ¯ (right figure) as functions of R for T = 25 °C (dashed line) and T = 80 °C (solid line). (a) shows a transition from the quasi-static regime (Rλd), to the sharp-resonance regime (λd/2 < R < 2λd), and finally to the multiple-resonance regime (Rλd).

Close modal
  1. The quasi-static regime (Rλd). In this regime, ω at 2.45 GHz is well below the lowest resonant frequency; hence, polarization-charge shielding plays the dominant role. As R → 0, h ¯→1.4 × 10−3 at T = 25 °C and h ¯ → 2.2 × 10−3 at T = 80 °C with a negligible σh/ h ¯, indicating a much-reduced, but nearly uniform E-field inside the sphere due to polarization-charge shielding. This is in good agreement with Eq. (7).

    As R rises to values close to or above λd, electromagnetic resonances start to set in, each with a finite resonant width. Hence, at a given radius, the modes are more and less overlapping in the sphere. While there is no sharp boundary, we refer to those modes with >3 dB separation (i.e., a trough falling more than one half of the neighboring peaks) as strong resonances and those with a shallower trough as overlapping resonances. The former is dominated by a single mode with a strong sensitivity to input frequency fluctuations, while the latter by multiple modes with a reduced frequency sensitivity. The associated stability issue in regard to the input frequency fluctuations will be discussed at the end of Sec. III.

  2. The sharp-resonance regime (λd/2 < R < 2λd). The sphere interior is now dominated by sharp electromagnetic resonances, with the mode spectrum given by Fig. 2(a). It can be seen that, as T increases from 25 to 80 °C, the resonances become sharper due to the reduction of ε′′ [Fig. 1(a)]. More significantly, this is accompanied by a shift of the resonant peaks to a higher R due to the reduction of ε′. The amount of the shift (∼11%) is predicted by Fig. 1(a) and Eq. (6a). At a fixed R, the shift is large enough to cause a mode switching as T rises, suggesting that this is a relatively unstable regime during the heating.

  3. The multiple-resonance regime (R≫λd). As R increases to much beyond λd, the mode spectrum becomes so crowded that a few densely populated weak resonances overlap to result in a smaller h ¯ and smoother variations of h ¯ and σh/ h ¯ with R. This implies a return to stable operation.

Overall, the combined polarization-charge shielding and resonance effects result in variations of h ¯ and σh/ h ¯ by orders of magnitude over the range of R under consideration. However, as will be shown in Sec. IV, this can be mostly compensated by optimized thermal flows and multiple wave paths.

Figure 2 indicates that, at the same frequency (2.45 GHz), the h ¯ and σh/ h ¯ profiles are highly dependent on the relative values of R and λd. This is also reflected in the E-field patterns under the plane wave in Eq. (1). The E-field patterns, representative of the three regimes, are plotted on the x-z plane (Fig. 3) at 25 °C for three values of R. In a linear medium, the E-field amplitude can be normalized to E0. The normalized value is given by the color code in logarithmic scale.

FIG. 3.

Analytically evaluated E-field patterns on the x-z plane in and around the water sphere representative of the three regimes, for T = 25 °C under a 2.45 GHz incident plane wave in the z-direction. (a) R = 0.1; (b) R = 0.68; and (c) R = 6 cm. The results agree with HFSS simulations.

FIG. 3.

Analytically evaluated E-field patterns on the x-z plane in and around the water sphere representative of the three regimes, for T = 25 °C under a 2.45 GHz incident plane wave in the z-direction. (a) R = 0.1; (b) R = 0.68; and (c) R = 6 cm. The results agree with HFSS simulations.

Close modal

Figure 3(a) is in the quasi-static regime (Rλd) showing a strong polarization-charge shielding effect. In the sharp-resonance regime, the first resonance at R = 0.68 cm is displayed in Fig. 3(b). Its field pattern (a pure TE10 mode) is in good agreement with the analytical prediction of Ref. 20. Figure 3(c) plots the field patterns for R = 6 cm in the multiple-resonance regime. Although the incident field decreases exponentially toward the center because R ≫ δ (=1.8 cm), the resonant mode the incident field has excited on its path to the center can still build up to a significant field near the center, as shown in Fig. 3(c). This enables large-scale heating.

Previous theoretical and experimental studies23 indicate that the scattering cross section of a plane wave by a water sphere is relatively weak in the Rayleigh limit [Rλfree, Figs. 3(a) and 3(b)], then increases sharply as R approaches λfree. This explains the scattered field pattern outside the sphere in Fig. 3(c).

Figures 2 and 3 apply to a water sphere. In reality, the heated sample is usually non-spherical in shape. In chemical synthesis, for example, the reaction vessel is typically cylindrical. Here we consider a plane wave incident onto a water cylinder of radius a and length L. This is a configuration not amenable to an analytical solution, so the 3D high-frequency structure simulator (HFSS, see  Appendix, part 1) is used for numerical solutions.

Assume that the water cylinder, with a fixed length-to-radius ratio of L/a = 10, is under the incident plane wave in Eq. (1). The cylindrical case (and non-spherical case in general,) is complicated by the orientation angle (θ) of the cylinder axis with the x-directed wave polarization. Figure 4 plots h ¯ and σh/ h ¯ at T = 25 °C (dashed lines) and T = 80 °C (solid lines) as functions of a for a cylinder lying on the x axis [θ = 0, Fig. 4(a)] and another one on the z axis [θ = 90°. Figure 4(b)]. Similar to the case of a water sphere, Fig. 4 displays three distinct regimes: the quasi-static regime (aλd), sharp-resonance regime (λd/4 < a < λd) of high-Q modes, and multiple-resonance regime (aλd) of densely populated low-Q modes. Also, as T increases from 25 to 80 °C, there is a ∼11% shift of the resonant peaks to a higher R due to the reduction of ε′.

FIG. 4.

h ¯ and σh/ h ¯ as functions of a for a water cylinder with L/a = 10 at T = 25 °C (dashed line) and 80 °C (solid line) under the 2.45 GHz incident plane wave in Eq. (1). (a) θ = 0°; (b) θ = 90°. Simulated by HFSS software. Because of the complexity of overmoding in large structures, our computational capability only allows an evaluation up to a = 3.2 cm.

FIG. 4.

h ¯ and σh/ h ¯ as functions of a for a water cylinder with L/a = 10 at T = 25 °C (dashed line) and 80 °C (solid line) under the 2.45 GHz incident plane wave in Eq. (1). (a) θ = 0°; (b) θ = 90°. Simulated by HFSS software. Because of the complexity of overmoding in large structures, our computational capability only allows an evaluation up to a = 3.2 cm.

Close modal

However, as a qualitative difference from the water sphere case, the water cylinder lacks the rotational symmetry. As a result, h ¯ is highly dependent on θ in the quasi-static regime, where h ¯(θ = 0) is ∼40 times greater than h ¯(θ = 90°). As discussed below, this is due to the strong orientational dependence of the polarization-charge shielding effect (see Fig. 5, a = 0.1 cm data).

FIG. 5.

Overall E-field patterns on the x-z plane in and around the water cylinder representative of the three regimes. T = 25 °C, L/a = 10, and the 2.45 GHz incident plane wave is in the z-direction. Row (a): cylinder on the x axis (θ = 0). Row (b): cylinder on the z axis (θ = 90°). Simulated by HFSS software.

FIG. 5.

Overall E-field patterns on the x-z plane in and around the water cylinder representative of the three regimes. T = 25 °C, L/a = 10, and the 2.45 GHz incident plane wave is in the z-direction. Row (a): cylinder on the x axis (θ = 0). Row (b): cylinder on the z axis (θ = 90°). Simulated by HFSS software.

Close modal

Figure 5 plots the overall E-field patterns in and around the water cylinder on the x-z plane representative of the three regimes. Under the x-polarized, incident E-field, the polarization charges are separated in opposite E-directions. When the cylinder axis is aligned along the E-field (θ = 0), the ±polarization charges reside on opposite ends [e.g., Fig. 5(a)], hence farthest apart and producing the least shielding. When the cylinder axis is aligned perpendicular to E0 (θ = 90°), the ±polarization charges reside on opposite sidewalls [e.g., Fig. 5(b)], hence much closer and producing stronger shielding.

The shielding effect is most pronounced for small samples (e.g., Fig. 5, a = 0.1 cm). This is because, along with the incident E-field [Eq. (1)], the induced polarization charge varies sinusoidally in z with a period of λf. If the polarization charge is distributed over an area (at the end or on the sidewall) with a z-dimension comparable to λf [e.g., Fig. 5(a), a = 3 cm] or greater than λf [e.g., Fig. 5(b), a = 3 cm], its sign variation along z will, thus, offset the overall shielding effect.

The parameter h(x) examined so far reveals much on the instantaneous heating rate through Eq. (2). However, the additional information of primary interest is the temperature distribution [T(x)] in the equilibrium state (steady-state), in which the total absorbed power equals the power loss on the surface (by radiation and convection). Equilibration processes involve thermal conduction, heat convection, and surface radiation. They are further complicated by the variations of ε′ and ε′′ as the temperature rises. Similar to Eqs. (4) and (5). we define the mean temperature ( T ¯) and its standard deviation (σT) as follows:
T ¯ v T ( x ) d 3 x total volume ,
(8)
σ T v [ T ( x ) T ¯ ] 2 d 3 x total volume .
(9)

T(x) (hence T ¯ and σT) can be evaluated with the coupled software of HFSS and Icepak (described in  Appendix), a suitable tool for the present purpose. Icepak returns the equilibrium solutions upon the specification of the geometrical configuration, the incident wave E-field amplitude E0 (or power density pinc = | E 0 | 2/754 in SI unit), and the documented water properties listed in Table I. The initial temperature of both water and ambient air is assumed to be 25 °C.

TABLE I.

Water properties used for Icepak simulation.

Property Value Reference
Thermal conductivity (25 °C)  6.07 × 10−1 W/m K  Ref. 8  
Mass density (25 °C)  9.97 × 102 kg/m3  Ref. 8  
Specific heat (25 °C)  4.18 × 103 J/kg K  Ref. 8  
Thermal expansion coefficient (volume)  2.56 × 10−4/K  Ref. 6  
Radiation emissivity  9.50 × 10−1  Ref. 22  
Temperature-dependent ε/ε0  Fig. 1   Ref. 9  
Thermal diffusivity (25 °C)  1.46 × 10−7 m2/s  Ref. 8  
Molecular mass  1.80 × 10−2 kg/mol  Ref. 8  
Dynamic viscosity (25 °C)  8.90 × 10−4 kg/m s  Ref. 8  
Property Value Reference
Thermal conductivity (25 °C)  6.07 × 10−1 W/m K  Ref. 8  
Mass density (25 °C)  9.97 × 102 kg/m3  Ref. 8  
Specific heat (25 °C)  4.18 × 103 J/kg K  Ref. 8  
Thermal expansion coefficient (volume)  2.56 × 10−4/K  Ref. 6  
Radiation emissivity  9.50 × 10−1  Ref. 22  
Temperature-dependent ε/ε0  Fig. 1   Ref. 9  
Thermal diffusivity (25 °C)  1.46 × 10−7 m2/s  Ref. 8  
Molecular mass  1.80 × 10−2 kg/mol  Ref. 8  
Dynamic viscosity (25 °C)  8.90 × 10−4 kg/m s  Ref. 8  

We first consider the case of a water sphere by assuming a desired steady-state temperature of T ¯ = 80 °C for a certain application. This can be accomplished by adjusting the incident power (i.e., E0). Figure 6 plots the required E0 to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of the water sphere radius R. As expected, the required E0 is the greatest in the quasi-static regime (Rλd) because of polarization charge shielding. It drops smoothly and significantly into the sharp-resonance regime (λd/2 < R < 2λd), in which E0 fluctuates by a factor of ∼2 with R. The fluctuation is caused by a large change of the mode structure due to a small change in R. As R increases further into the multiple-resonance regime (R ≫ λd), E0 is characterized again by a smooth variation with R.

FIG. 6.

The required incident E0 for the water sphere of radius R to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of R. The results are simulated by coupled software of HFSS and Icepak.

FIG. 6.

The required incident E0 for the water sphere of radius R to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of R. The results are simulated by coupled software of HFSS and Icepak.

Close modal

Significantly, despite up to 100% heating rate spread [σh/ h ¯ in Fig. 2(b)], the steady-state temperature spread relative to the mean temperature [σT/ T ¯ in Fig. 6(b)] at T ¯ = 80 °C is rather low (0.2–1.1%) in all three regimes [Fig. 6(b)]. This is mainly due to heat convection and partly due to thermal conduction within the water sphere.

Similar to Fig. 6, we assume a desired steady-state temperature of T ¯ = 80 °C for a water cylinder with L/a = 10. Figure 7 plots the required E0 to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of the cylinder radius a. For clarity, the simulated data points for a cylinder lying on the x axis (θ = 0) and z axis (θ = 90°) are connected by dashed and solid straight lines, respectively. The following differences are notable in comparison with the water sphere case (Fig. 6): (1) The results for a water cylinder rather strongly depend on the cylinder orientation. (2) The temperature spread (σT/ T ¯) is in general, greater than that for the water sphere.

FIG. 7.

The required incident E0 for the water cylinder of radius a and L/a = 10 to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of a. The results are simulated by the coupled software of HFSS and Icepak. Because of the complexity of overmoding in large structures, our computational capability only allows an evaluation up to a = 3 cm.

FIG. 7.

The required incident E0 for the water cylinder of radius a and L/a = 10 to reach T ¯ = 80 °C and the resulting σT/ T ¯ as functions of a. The results are simulated by the coupled software of HFSS and Icepak. Because of the complexity of overmoding in large structures, our computational capability only allows an evaluation up to a = 3 cm.

Close modal

For a comparison of the characters and merits of spherical and cylindrical samples, portions of the data in Fig. 6 (spherical) and Fig. 7 (cylindrical) are superposed and plotted as functions of the reaction volume (V) in the range of 0.1–10 mL (Fig. 8). The range of V is chosen because most examples of microwave-assisted syntheses were performed on a less than 1 g scale (typical V = 1–5 mL).1  Figure 8 suggests a critical role of polarization-charge shielding and indicates a significant advantage for the spherical geometry in terms of the temperature spread for reasons discussed below.

FIG. 8.

Superposition of portions of the data in Figs. 6 and 7 corresponding to the reaction volume range of V = 0.1–10 mL.

FIG. 8.

Superposition of portions of the data in Figs. 6 and 7 corresponding to the reaction volume range of V = 0.1–10 mL.

Close modal
  1. The E-field in cylindrical samples has a rather strong orientational dependence (Fig. 5). Furthermore, their sizes also significantly affect the temperature equilibration. As discussed in connection with Fig. 5, the shielding effect is most pronounced for small samples (e.g., Fig. 5, a = 0.1 cm). The surface area enclosing a unit volume is larger for smaller samples, especially near the end surface of the cylinder, where more heat (per unit volume) leaks into the air, hence resulting in a cooler spot. This is illustrated in the temperature profiles in Fig. 9 for a small (a = 0.1 cm) and a large (a = 0.5 cm) cylinder oriented along E0 (θ = 0), both at T ¯ = 80 °C. The smaller cylinder [Fig. 9(a)] has a center-to-end temperature ratio of 1.20 while the larger cylinder has a smaller ratio of 1.08. Similarly (but not illustrated in figure), when the same two cylinders are oriented perpendicular to E0 (θ = 90°), the smaller one has a center-to-end temperature ratio of 1.11 while the larger one has a smaller ratio of 1.01. This explains the rise of the temperature spread toward the small-V region in Fig. 8.

  2. In comparison, for the (small) spherical samples in Fig. 8, the internal E-field is uniform [Eq. (7)]; hence, their equilibrium states feature a lower (or much lower) temperature spread than the cylindrical samples. Furthermore, the orientational independence of a sphere also makes the heating process much less sensitive to the applicator structure, hence more reproducible (discussed further in Sec. IV 1).

FIG. 9.

Temperature profiles at T ¯ = 80 °C for (a) a small cylinder (a = 0.1 cm, V = 0.03 mL, σT/ T ¯** = 5.5%) and (b) a larger cylinder (a = 0.5 cm, V = 3.67 mL, σT/ T ¯** = 2.1%).

FIG. 9.

Temperature profiles at T ¯ = 80 °C for (a) a small cylinder (a = 0.1 cm, V = 0.03 mL, σT/ T ¯** = 5.5%) and (b) a larger cylinder (a = 0.5 cm, V = 3.67 mL, σT/ T ¯** = 2.1%).

Close modal

Focusing on the water sphere, there is still the issue of finding the optimum radius based on stability considerations. The magnetron is the predominantly used source for heating applications. As reported in Refs. 24 and 25, the measured spectrum of a household microwave oven is typically composed of a cluster of peaks spanning from 2.4 to 2.5 GHz. Furthermore, within this 4% band, it constantly changes with time and varies with the location, size, and geometry of the load.25 On the other hand, the required E0 for steady-state operation at a fixed T ¯ is sensitive to R in the sharp-resonance regime (Fig. 6). A change in the operating frequency has the same effect as a proportional change in R. From Fig. 6, we find that a change of R up to 4% can present difficulties for equilibrium temperature control in the sharp-resonance regime. This leaves two favorable radius ranges for stable operation: R < 0.7 cm (in the quasi-static regime) and R > 2 cm (in the multiple-resonance regime). The percentage temperature spread is relatively small in both ranges (σT/ T ¯<1% for R < 0.7 cm and σT/ T ¯<0.3% for R > 2 cm). The R < 0.7 cm size (with a volume <1.2 mL) is suitable for research purpose, while the R > 2 cm size (with a volume >33.5 mL) is suitable for production purpose.

  1. Applicability of the single-path, traveling-wave model to realistic situations

    A great variety of 2.45 GHz microwave applicators (single- or over-moded) are commercially available.26 The majority, if not all, employs a resonator structure, in which each standing wave is formed of two oppositely directed traveling waves bouncing back and forth between walls. Since a spherical sample is independent of the direction and polarization of the incident wave, the results obtained under the single-path, traveling-wave model (Fig. 6) apply to traveling waves from all directions in a resonator chamber. However, results for cylindrical samples under the single-path, traveling-wave model do not apply directly to an over-moded microwave cavity.

  2. Deep field penetration through sphere resonances

    The attenuation length (δ) of water at 25 °C is 1.8 cm [Fig. 1(b)]. Thus, the wave incident into a large sphere (R ≫ δ) damps sharply within a shallow outer region. However, densely populated resonant modes are excited as a result [Fig. 3(c)]. The natural structure of these normal modes extends to the whole sphere, hence allowing significant heating in the core region. Further, the attenuation length increases by a factor of 3.2 from 25 to 80 °C [Fig. 1(b)]. It is worth noting that, for these reasons, the percentage temperature spread (σT/ T ¯) is as low as 0.2% for R = 10 cm (≫δ) and shows a decreasing trend at still larger R [Fig. 6(b)].

  3. Heating efficiency

    Figure 6(a) shows that the incident E0 required for the steady-state operation at a fixed T ¯ varies widely with the sphere radius R. However, this does not preclude high-efficiency heating over a wide range of R, because each traveling wave may execute multiple paths through the sphere in actual operations. Consider, for example, a water sphere being heated in a microwave resonator. The input structure (probe, loop, or aperture) can be tuned to the condition of critical coupling,27 under which all the incident power is fed into the cavity to be shared between the water sphere and the resonator wall. Since the wall is a good conductor, the water sphere normally absorbs the vast majority of the input energy.

  4. The results reported here scale with R/λd. They can be generalized to other microwave frequencies or other solvents similar in dielectric properties.

Two basic physics effects, polarization-charge shielding and electromagnetic resonances, are incorporated into the theory of microwave heating of water at the predominantly used frequency of 2.45 GHz. The former effect greatly reduces the internal E-field in water, while the latter effect significantly boosts the field strength along with a substantial field non-uniformity. A combination of these two effects results in orders-of-magnitude variations of the heating rate with respect to the sample size and shape (Figs. 2 and 4). However, this is only a snapshot of the overall process, which also involves thermal convection and conduction to even up the temperature. These thermal flows can lead to an equilibrium state with a high temperature uniformity.

A uniform temperature distribution requires measures to improve the heating uniformity. Unlike a sphere, an elongated object has a heating rate highly dependent on its orientation relative to the wave polarization (Fig. 4), which constitutes a major source for non-uniform heating. This is the principal reason for our conclusion that a spherical sample is most favorable for temperature uniformity (cf. Figs. 6 and 7). On the other hand, the incident wave can be much-attenuated in the core region of a large sphere. However, this can be compensated by the deep penetration of excited electromagnetic resonances [Fig. 4(c)].

Sharp electromagnetic resonances in the water sphere may, however, present a stability problem for a microwave source with poor frequency control (typically a magnetron). A quantitative examination has identified two radius ranges for stable operation: R < 0.7 and R > 2 cm (Sec. IV) suitable for research and large-scale treatment, respectively.

The current study sheds light on key details, such as the thermal flow and internal temperature distribution, that are difficult to measure experimentally. The conclusion on the desirability of spherical sample for temperature uniformity is significant in that non-spherical (particularly cylindrical) reaction vessels are typically employed for laboratory tests. It is expected that the change to a spherical vessel may significantly improve the quality and reproducibility of the acquired data through a higher temperature uniformity.

As a final note, we have argued in Sec. III C about the relative advantages of a spherical sample as compared with a L/a = 10 cylinder. We have not studied other non-spherical samples. Since the sphere is unique in its orientational independence, these advantages are expected to carry over, to a greater or lesser extent, to other non-spherical shapes.

The authors are grateful to ANSYS, Inc. for generously providing the HFSS and Icepak package as well as the technical support from Cybernet Systems Taiwan Co., Ltd. on its use. We are also grateful to Dr. H. H. Teng for helpful assistance. This work was funded by the National Science and Technology Council, Taiwan, under Grant No. MOST111-2112-M-002-016.

The authors have no conflicts to disclose.

Li Chung Liu: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Funding acquisition (supporting); Investigation (lead); Methodology (lead); Project administration (equal); Resources (lead); Software (lead); Supervision (equal); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Ju Ching Liang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (lead); Supervision (supporting); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). Kwan Wen Chen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (lead); Supervision (supporting); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Kwo Ray Chu: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1. Ansys HFSS

Electromagnetic field simulations were performed using the Ansys HFSS (high-frequency structure simulator) software, which solves the full set of Maxwell equations by the finite element method (http://anlage.umd.edu/HFSSv10UserGuide.pdf). In our simulations, PMLs (perfectly matched layers) are assumed on a closed boundary, which encloses the entire simulated structure. PMLs are fictitious materials that fully absorb the electromagnetic fields acting upon them.

2. Ansys ICEPAK

Thermal simulations were performed using the Ansys Icepak software. According to its manual (https://www.ansys.com/products/electronics/ansys-icepak), the Icepak uses the computational fluid dynamics (CFD) solver engine for thermal and fluid-flow calculations. Heat transfers through conduction, convection, and radiation. Laminar flow (assumed in the current study) is calculated by solving the Navier–Stokes equations for transport of mass, momentum, species, and energy to yield a fully coupled conduction and convection heat transfer prediction.

For the natural convection in a fluid which varies in temperature from one place to another, the gravity (through buoyant forces) drives a flow of fluid and heat transfer. In Icepak, the Boussinesq approximation is used to model the effect of density variations due to temperature while the fluid is assumed to be incompressible. The density is given by ρ = ρ0αρ0(TT0), where ρ is the local density, ρ0 is the reference density, α is the coefficient of thermal expansion, T is the local temperature, and T0 is the reference temperature.

Radiation loss is calculated with the discrete ordinates (DO) radiation model, which solves the radiative transfer equation (RTE) for a finite number of discrete solid angles. Moreover, EM mapping and two-way coupling are used to perform two-way loss information between HFSS and Icepak. EM mapping allows Icepak to calculate the heat loss from electromagnetic fields in HFSS; two-way coupling allows HFSS to re-simulate EM field with the varying temperature and Icepak to re-simulate heat flow. HFSS and Icepak will do a coupled simulation for the steady state of either solid or fluid materials. The required input parameters are thermal conductivity, mass density, specific heat, thermal expansion coefficient, radiation emissivity, temperature-dependent electric properties (e.g., the dielectric constant) and, for fluid materials, the additional input parameters of thermal diffusivity, molecular mass, dynamic viscosity. Free openings are assumed on the boundary, which encloses the entire simulated structure, implying that the structure is exposed to free space.

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