Numerical simulation of hot-spot effects in microwave heating due to the existence of strong microwave-absorbing media

Abstract

Hot spots can occur in microwave heating when the heated materials have different microwave absorbing properties, resulting in non-uniform temperature distribution. Understanding the features and extent of hot-spot effects can be essential in microwave-assisted processes, but little has been reported quantitatively due to the difficulty in direct determination. The issues are measured experimentally and numerically simulated using silicon carbide (SiC) particles dispersed in paraffin oil as a representative case here. Hot spots are definitively shown to exist and may trigger temperature gaps between surrounding substances at the magnitude of several hundred degrees Celsius or even higher in certain cases. The temperature gaps are enhanced for larger sized SiC particles, with a higher heat generation rate and increasing heating time. The extent of hot-spot effects substantially depends on how much and how quickly heat generated by the strong microwave absorbing media can be transferred to the weak ones. The findings have great practical value. By choosing materials with strong microwave absorption, or where discharges occur due to microwave–metal interactions, prominent hot spots can be intentionally forged and the temperature gradient may be tailored to enhance chemical reactions and catalytic processes for specific scientific and engineering applications.

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1. Introduction

Microwave heating is a process of energy transformation from an electromagnetic wave to heat as microwaves are absorbed or dissipated by various media. Microwave heating technologies have developed rapidly and play increasingly important roles in many fields, such as food,¹ sanitation,² medication,³ electronics,⁴ and chemical,^5–8 materials,⁹ and environmental engineering.¹⁰ In contrast to conventional heating, microwave heating is caused by dielectric losses when motion within dipoles, ions, or electrons are driven and reversed by the high frequency oscillating electric field. Different materials have different coupling with the microwaves, i.e., reflecting, transmitting, or absorbing. The microwave absorption capacity of a certain material largely depends on its dielectric dissipation. Composite materials can be selectively heated and non-uniform temperature distribution may occur during microwave heating due to these absorption differences. For example, water (a polar molecule) has higher dielectric dissipation than non-polar protein or carbohydrates, and therefore has a stronger microwave absorption capacity.

Strongly absorbing materials may sometimes be added to weak absorbing media to improve microwave heating performances. Harfi et al.¹¹ showed that the addition of carbon particles could shorten reaction time and increase pyrolysis rate for microwave pyrolysis of oil shale. Menéndez et al.^12–17 found that pure sludge could only be dried but not be pyrolyzed due to limited temperature rise with microwave heating, but the sludge could be heated to 900 °C and pyrolyzed efficiently with the addition of coke as an enhancing agent. Our working group¹⁸ found that nano ⁰Fe could greatly enhance degradation of organic compounds in dye wastewater under microwave irradiation, possibly due to its particularly strong microwave absorption. Other reports have also showed that the addition of strong microwave absorbing materials enhanced microwave heating of waste oil,¹⁹ biomass,²⁰ electronic waste,²¹ etc.

It seems reasonable to expect some temperature gradient between media with different microwave absorptions. This temperature gradient is often referred to as the hot-spot effect, and is partly regarded as the source of the so-called non-thermal effects of microwave heating. It is also linked to improved chemical reaction rates, reduction of reaction time, or catalytic effects in microwave chemistry. Zhang et al.²² attributed the observed apparent equilibrium shifts for both hydrogen sulfide decomposition and thiophene hydrodesulfurization to the formation of spatial hot spots in catalyst bed under microwave irradiation and examined the possible location of remarkable temperature gradients experimentally and theoretically. Horikoshi et al.^23,24 recorded hot-spot phenomena using a high-speed camera in real time and confirmed the existence of hot spots when adding Pd/active carbon carrier to accelerate chemical synthesis. However, the previous studies did not unveil what extent of the temperature gradient could reach and the hot-spot effects remain largely unknown in quantitative scales.

The lack of quantitative study on microwave hot-spot effects is mainly due to the difficulty of direct measurement in temporal and spatial scales. Macroscopic temperature measurements can only provide an average and lagged temperature profile of the heating media, and it is difficult to focus on a specific isolated region or to capture the instantaneous temperature differences due to various heat transfer conditions. Although optical fiber and infrared sensors have been adopted for direct temperature collection in electromagnetic fields,^25–29 the lagging problem cannot be completely avoided and the practical temperature profiles inside millimeter or even smaller level of catalyst particles cannot be obtained. Hence, numerical simulation should be the ideal method to explore the hot-spot effects with instantaneity or in micro scales.

This work used silicon carbide (SiC) as a representative of strongly microwave absorbing media, and simulated hot-spot effects when SiC was put into and surrounded by weak microwave absorbers under simultaneous microwave irradiation. Investigation of heat transfer issues related to hot-spot effects was facilitated by measurements of actual thermal generation. Hot-spot effects were shown quantitatively in micro and instantaneous scales. Understanding hot spot intensity and other effects will be beneficial for its proactive utilization and the optimization of microwave heating processes.

2. Experiments and simulation

SiC particulates dispersed in a homogeneous solvent with weak microwave absorption and relatively high stability is an ideal design to study hot-spot effects in microwave heating, since the heat generated in SiC can be absorbed and stored in the solvent. Different SiC particle sizes were selected as representative hot-spot generators, and paraffin oil was adopted as the solvent, for its relatively weak microwave absorption and volatility. Their relevant physical parameters at room temperature are shown in Table 1.

Table 1 Physical parameters of silicon carbide (SiC) and paraffin oil

Physical parameters	Density (kg m^−3)	Specific heat capacity (J (kg °C)^−1)	Thermal conductivity (W (m °C)^−1)	Relative permittivity (at 298 K)
SiC	3220	445	83.6	9.72
Paraffin oil	880	2800	0.12	2–2.5

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The formation of hot spots during the microwave heating of SiC dispersed in paraffin oil includes two processes: the dissipation of microwave energy generates in the SiC particles, and heat transfer between SiC particles and the paraffin oil. This work does not acquire heat generation through calculation because the electric field intensity of the interactions between microwaves and heating media is complex and accuracy of the electromagnetic field intensity, SiC particle movements, and actual microwave absorption all influence its validity. The media dielectric properties are also affected by many factors, including the electric field frequency, temperature, medium homogeneity, etc.

Therefore, it is difficult to obtain a reliable direct calculation of heat generation for SiC particles. Thus, the overall heating of SiC particles was experimentally measured and only temperature gradients inside and outside the particles were simulated. The ANSYS software package—a large computer-aided engineering tool, integrating fluid dynamics with electronics, magnetism, and temperature—was employed. The influences of particle size, irradiation time, and heat generation, as well as some extreme cases, are discussed.

2.1. Experimental set-up

The microwave absorption and heating of SiC particles were measured experimentally in a modified domestic microwave oven, as shown in Fig. 1. The rated microwave frequency was 2450 MHz and the output power could be continuously adjusted from 0–800 W. A 200 ml quartz beaker was employed, containing approximately 120 g paraffin oil with 1 g SiC particles dispersed at the bottom. A PTFE board, 4 cm high, with good microwave transparency and heat insulation, was placed between the base of the cavity and the quartz beaker to locate the container at the center of the cavity, and reduce heat loss through conduction. The room temperature was stable at 20 ± 1 °C. Before the start of each test, the initial temperature of paraffin oil was measured by digital K-type thermocouple with accuracy ±0.1 °C and quick response. The beaker was irradiated for 120 s at microwave output power 800 W, and the final paraffin oil temperature was measured immediately after rapid agitation with a PTFE stirrer rod. The thermocouple was only inserted when the power was off. The microwave cavity was kept open and idle to cool down for at least 20 min before any new round of experiment.

Fig. 1 Experimental microwave oven: 1 = microwave cavity; 2 = PTFE board; 3 = quartz beaker; 4 = K type thermocouple.

The measurement results including standard deviations are shown in Fig. 2, averaged over 10 measurements so as to avoid accidental errors. Paraffin oil without SiC shows an average temperature rise of approximately 3.6 °C. When different SiC particle sizes were added, the temperature rise of the oil increased to over 7.0 °C. Although there is some small variation for different SiC size, this is not significant, indicating that SiC size, within certain range, makes little difference of microwave absorption, and the average temperature rise may be assumed to be 7.2 °C for all sizes.

Fig. 2 Temperature rise in paraffin oil with SiC.

2.2. Simulation

It is of great difficulty to measure temperature gradients inside SiC particles and among paraffin oil directly and accurately without any delay, and it is even harder to directly measure the SiC particle temperature at the moment when the heat has been generated but not yet transferred into the surrounding paraffin oil. Therefore, numerical simulation is the best method to probe the depth and extent of hot-spot effects.

The simulation was focused on heat transfer in the SiC–paraffin system. Heat generation rate was assumed to be the same among the different sized SiC particles, and the simulation focused on heat transfer of a single SiC particle with the surrounding paraffin oil. This simplification particularly highlights hot spots.

The simplified model is shown in Fig. 3. The SiC particle is assumed to be spherical and located in the center of the surrounding paraffin oil. The ratio of SiC particle and the paraffin oil ball radii was set to 1 : 10. Relative motion between SiC and paraffin oil was initially assumed to be zero, with more complicated cases discussed later. Hence, the simulation is strongly focused on heat transfer within the SiC particle and between SiC and paraffin. The initial temperatures of SiC and paraffin oil were both set to 20.0 °C. When the SiC particle starts to convert microwave energy into heat, it acts like an internal power source, and the key simulation parameter is to obtain a reasonable value for this source.

Fig. 3 Simplified model.

Usually, heat generation in microwave heating may be calculated from

P = 2πfε₀ε′′_effE_rms² (1)

where P is the average heat generation per unit volume; f is the radiation frequency; ε₀ is the permittivity of free space; ε′′_eff is the complex component of the relative permittivity of the dielectric, also known as the effective relative dielectric loss factor; and E_rms is the root mean square of the electric field. However, the difficulty in obtaining practical SiC parameters and electric field intensity renders the calculation meaningless, and direct measurement of heat generation should be superior and more reliable.

During the process of heat generation and transmission, microwave energy absorbed by SiC particles is converted to heat within the particle, which subsequently causes warming of the paraffin oil and quartz container by heat transfer. Thus, the SiC particle heating can be derived by directly measuring the oil temperature rise (Section 2.1).

The paraffin and beaker temperature rise are assumed to be identical. When no SiC was added, the heat generated by microwave absorption of the beaker and paraffin is, Q₀. The heat generated in the case of SiC addition is Q₁, and includes the additional heat accumulation of the SiC particles, and the difference of Q₁ and Q₀, Q₂, is the heat generated from SiC. The heat generation power of SiC by volume, P_v, may then be directly calculated,

Q₀ = (c₁m₁ + c₂m₂)Δt₀ (2)

Q₁ = (c₁m₁ + c₂m₂ + c₃m₃)Δt₁ (3)

Q₂ = Q₁ − Q₀ (4)

P_v = Q₂/(Vt) (5)

where Q₀ is the heat without SiC addition; Q₁ is the heat with SiC addition (J); Q₂ is the heat generated by SiC particles (J); c₁ is the specific heat capacity of SiC (0.445 J (g °C)⁻¹); m₁ is the mass of SiC (1 g); c₂ is the specific heat capacity of paraffin oil (2.8 J (g °C)⁻¹); m₂ is the mass of paraffin oil (120 g); c₃ is the specific heat capacity of the container (0.8 J (g °C)⁻¹); m₃ is the mass of the container (95.28 g); Δt₀ is the temperature rise without SiC addition (3.6 °C); Δt₁ is the temperature rise with SiC addition (7.2 °C), P_v is the heat generation power of SiC by volume (W m⁻³), V is the volume of SiC (m³), and t is the heating time (120 s).

Although the SiC heat effect was not as large as the blank test (Table 2), its microwave absorption was still considerable given it's the small amount of material.

Table 2 Calculated heat effects

	Q0	Q1	Q2	Pv
Value	1502 J	2641 J	1139 J	4.0 × 10^7 W m^−3

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Errors may arise due to heat losses during microwave heating via conduction, convection, and radiation. Conduction loss only occurred between the quartz beaker bottom and the PTFE board and may be neglected due to the excellent heat insulation of PTFE. Radiation loss from the beaker and its load was less than 0.003% of the total generated heat (from the Stefan–Boltzmann law), and may also be neglected. Convection loss occurred on the outer wall of the quartz beaker and on the upper surface of the paraffin oil, and may amount to at most 0.4% of the total generated heat.³⁰ Since this heat loss may be assumed constant with or without SiC addition, Q₂ (the difference of Q₁ and Q₀), with which we are most concerned, would be little affected. Therefore, convection loss may also be neglected.

Thus, substituting the known and measured parameters,

P_v = 4.0 × 10⁷ W m⁻³. (6)

3. Results and discussion

3.1. Hot spot occurrence

Using the simulation and experimentally obtained heat generation power of SiC, the temperature profile of the SiC–paraffin model can be calculated under microwave irradiation. The SiC particle can be regarded as an internal heat source having the heat generation rate of 4.0 × 10⁷ W m⁻³ (eqn (6)). Fig. 4a shows the temperature distribution for SiC particle diameter 0.4 mm and paraffin oil diameter 4 mm, after microwave irradiation for 300 s.

Fig. 4 Temperature distribution for SiC particle diameter 0.4 mm and paraffin oil diameter 4 mm, after microwave irradiation for 300 s. (a) Temperature field cloud of a quarter of the whole model. (b) Temperature distribution of the SiC particle.

Fig. 4a shows significant, but non-homogenous, temperature rise for the whole model from the initial 20.0 °C. Temperature rise in the center, the SiC particle, significantly exceeds 8.0 °C, distinguished clearly by the red color. Within the SiC particle itself, the temperature is reasonably uniform (Fig. 4b), whereas in the outer paraffin zone, the temperature rise is only approximately 4.7 °C, a difference of approximately 3.3 °C. Thus, the model supports the occurrence of hot spots.

Note that the simulation only reflects the equilibrium state. If the SiC generated heat cannot be transferred to the surrounding paraffin oil in time, the outcome would be entirely different. In the equilibrium, the SiC particle contains less than 1% of the total heat generated, with paraffin containing the balance. Prior to equilibrium, a much higher proportion of the heat will remain within the SiC particle, and so the hot-spot would be more apparent.

It is reasonable to consider a larger share of generated heat is retained by the SiC particle at earlier times. For 90%, 50%, 10% and 5% heat transfer, the difference between the average temperature of the SiC particle (diameter 0.4 mm and heat generation rate 4.0 × 10⁷ W m⁻³) and paraffin can be calculated as

Q = P_vVt (7)

Δt₂ = Q(1 − μ)/(c₁m₁) (8)

Δt₃ = Qμ/(c₂m₂) (9)

Δt = Δt₂ − Δt₃ (10)

as shown in Fig. 5, where, Q is the total heat generated (J), Δt₂ is the average temperature rise of SiC particles (°C); Δt₃ is the average temperature rise of paraffin oil (°C), Δt is the temperature gap, and μ is the heat transfer share.

Fig. 5 Temperature difference between SiC particle and paraffin oil for different heat transfer share.

Lower heat transfer share causes higher inner temperature rise. When the heat transfer share is 5% or 10%, i.e. 95% or 90% is retained within the SiC particle, and its average temperature may exceed 100 °C after only 5 s microwave irradiation. Hence, hot spots will be prominent at lower heat transfer frames.

3.2. Effect of SiC particle size

Fig. 6 shows the temperature field clouds for different SiC particle sizes with fixed heat generation rate (4.0 × 10⁷ W m⁻³) after 300 s microwave irradiation. The diameter ratio of the paraffin and SiC particle was fixed at 10 : 1 for all four cases. SiC particle size has a significant influence on the profile and intensity of hot spots. Larger particle sizes produce more heat generation and hence higher temperature rise. For 0.04 mm SiC particles, the hot spots are obscure and the temperature difference may be less than 0.1 °C, whereas 8 mm SiC particles produce a difference exceeding 800 °C. The larger temperature gradients for larger particles are because the heat generation rate is larger than the heat transfer rate, and more heat is retained in the particle. Hence, heat transfer is another key factor for hot spots. If a strong microwave absorbing medium is surrounded by materials with weak heat transfer, hot spots should be more prominent.

Fig. 6 Temperature field clouds of different SiC particle size after 300 s microwave irradiation. (a) Particle diameter 0.04 mm, paraffin diameter 0.4 mm. (b) Particle diameter 0.4 mm, paraffin diameter 4 mm. (c) Particle diameter 4 mm, paraffin diameter 40 mm. (d) Particle diameter 8 mm, paraffin oil diameter 80 mm.

3.3. Effect of heat generation rate

The microwave absorption of a material depends on its dielectric properties, particularly on its dielectric loss factor. Consequently, different materials have varying heat generation rates. Consider just the case of 99% heat transfer, for 0.4 mm diameter particle and 4 mm outer paraffin diameter. Eqn (7)–(10), show that the temperature gap is proportional to the heat generation rate, P_v. Fig. 7 shows the differences between the average temperatures of the inner particle and the paraffin oil for the cases P_v = 100, 10 and 1/10 that of SiC, i.e., P_v = 4.0 × 10⁹, 4.0 × 10⁸ and 4.0 × 10⁶ W m⁻³, respectively. This represents a reasonable P_v range, relative to SiC microwave absorption.

Fig. 7 Average temperature difference between SiC particle and paraffin oil for different heat generation rates (P_v).

Heat generation rate has a significant influence on hot-spot effects, with large P_v producing wider temperature differences. When P_v is 10 times that of SiC, the temperature difference reaches several hundreds of degrees, and for 100 times SiC, the difference reaches 1000 °C with less than 50 s irradiation. These are reasonable outcomes, especially for the initial period when heat transfer is far from the equilibrium state.

Hot-spot effects will be substantial if the heated material includes components with strong microwave absorption or if generated heat can be retained within the absorbing material. For the case where P_v = 4 × 10⁹ W m⁻³, 100 times that of SiC, and 10% heat transfer share, the temperature difference can reach 200 °C within 0.1 s irradiation, as shown in Fig. 8.

Fig. 8 Average temperature difference between SiC particle and paraffin oil for different heat transfer share, with heat generation rate P_v = 4.0 × 10⁹ W m⁻³.

3.4. Effect of heating time

Consider the case, as previously described, with a SiC particle diameter of 0.4 mm, an outer paraffin oil diameter of 4 mm, a heat generation rate 4.0 × 10⁷ W m⁻³, and 99% heat transfer. Fig. 7 shows differences between the average temperature of the SiC particle and paraffin oil (the line P_v = 4.0 × 10⁷ W m⁻³) for 0–300 s microwave irradiation.

The temperature difference is directly proportional to the heating time. The difference is initially a few degrees and increases to approximately 90 °C after 300 s irradiation. The average temperatures for both are rising, but the temperature gap continues to widen, indicating that hot spots may be enhanced with longer microwave irradiation. On the other hand, a larger temperature difference will also enhance heat transfer, and the actual difference may be somewhat lower, but hot spot occurrence is definitely confirmed. The temperature different would be very prominent if the heat was generated over a very short period with no significant heat transfer.

3.5. Application values

From the above discussion, it can be concluded that hot spots definitely exist in some microwave heating processes due to selective heating principle, and can produce temperature differences with surrounding substances up to several hundred degrees or even larger for materials with different absorption and heat transfer properties.

Before the hot-spot effects were unveiled to the extent of this paper, they are already linked to microwave-assisted catalytic reactions and considered to trigger acceleration of reaction rates or improvement energy efficiency.^8,25,31 However, this is an unintentional use of hot spots, and later may be proactively utilized for a variety of applications. For example, some materials have very strong microwave absorption, such as Fe, Co, or Ni crystals or non-crystals, and some novel carbon materials.^32–34 If these materials were manufactured in appropriate sizes and used as microwave absorption enhancing agents, hot spots could be intentionally produced in gaseous or liquid media to create temperature gradients to enhance chemical reactions, such as macromolecule decomposition,^35–37 pollutant removal,^38–42 gas reforming,²⁷ etc.

Another hot-spot consequence is microwave discharge caused by interaction with metals.^43,44 When metals with sharp edges or tips, or submicroscopic irregularities are subjected to microwave irradiation, intense discharge phenomenon, usually an electric spark or arc, may occur and considerable heat is generated. Since the discharge process is transient, the concentrated heat release can lead to the formation of high local temperature, i.e., the hot-spot effect. Approximately 40% of the input microwave energy can be transformed into heat by discharge. If the transformed microwave energy is released into a 1 cubic millimeter volume in less than 1 s, the heat generation rate can easily reach 10⁹ to 10¹² W m⁻³, and the corresponding hot spot temperatures can exceed 1000 °C. This outcome is practically manifested by the phenomenon that metal terminals can be melted by microwave discharge.

Intensive generation of hot spots may have other special applications. In a previous study, we found microwave-metal discharge had the potential to greatly affect chemical reactions and product composition. For example, there is potential to decrease activation energy related to pyrolysis and increase H₂ yield. Besides, selective heating, which is the source of hot-spot effects, have the potential to intensify some industrial processes and guide the ongoing microwave equipment developments,⁷ since the efficiency of generating hot spots depends on the nature of the microwave generator.²³ Thus, intense hot spots have scientific and engineering value, but, specific applications require further development.

In addition, the research approach of combining thermal effect measurement and simulation may be referred to in studies of other electromagnetic heating cases. In another angle, a method to determine the microwave absorption properties of some certain materials is also provided. Since the direct temperature measurement of specific micro regions with instantaneity is really hard, this approach is effective and efficient for researchers focusing on microwave-assisted catalysis, heating, pyrolysis, and so on.

4. Conclusions

Hot spots occurring in microwave heating were studied experimentally and by numerical simulation using SiC particles dispersed in paraffin oil as a representative case of materials with significantly different microwave absorbances. We have shown that hot spots will occur in such a mixture, and even at equilibrium, the SiC particles can have significantly higher temperature profile than the surrounding paraffin oil.

Hot-spot profiles vary for different SiC particle geometry, heat generation, or heat transfer. Larger particle sizes, higher heat generation rate, and smaller heat transfer share all highlight the temperature difference of the hot spot from surrounding media. The temperature difference can exceed several hundred degrees Celsius under certain conditions. The magnitude of the effect depends on how quickly the heat generated in the strongly absorbing medium can be transferred to the weakly absorbing media.

Hot spots may have significant application potentials in scientific and engineering fields. Materials with strong microwave absorption, such as Fe, Co, or Ni crystals, or novel carbon materials, could be used to enhance microwave heating. Prominent hot spots could be intentionally created, including the use of microwave discharges, and the temperature differences employed to enhance chemical reactions, such as macromolecule decomposition, pollutant removal, etc. Consequently, hot-spot effects in microwave heating deserve further research and application development.

Acknowledgements

The authors thank the support of National Natural Science Foundation of China (Grant 51376112 and Grant 51506116), National Key Technologies R&D Program of China (Grant No. 2014BAC26B03) and Shandong Science Fund for Distinguished Young Scholars (JQ201514).

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