Multiphysics analysis for unusual heat convection in microwave heating liquid

Heating with microwaves (the thermal effect) is widely used for food, thermal processing of materials,^1,2 chemical engineering,^3–5 and agricultural and mineral processing in environmental and biomedical applications. In contrast to conventional heating (where heat is transferred from the surface to the interior of the product), microwave heating is a volumetric heating, which leads to faster heat transfer rates and shorter processing times than conventional processes. However, the significant problem associated with microwave heating is the non-uniform temperature distribution.⁶ Uneven temperature will cause out of control heat, thus damaging the heating chamber.⁷ 

Heat convection is a mode of heat transfer by the mass motion of a fluid such as liquid and gas. Usually, convective heat transfer occurs when the temperature of the surface is lower than that of the surrounding fluid and causes more uniform heating, but there is an unusual heat convection in liquids when heated using microwaves as a source of energy. According to the feedback from Midea (microwave appliance makers), when users use the microwave oven to heat liquids such as milk or water, the temperature at the top of the liquid will be significantly higher than the temperature at the bottom. Some researchers have conducted experimental research studies on microwave-heated liquids, by measuring the transient and spatial temperature profiles of liquids (water and corn oil) inside a cylindrical container during batch microwave heating at 2450 MHz. The results show that axial temperature values were increasing rapidly with height due to the deposition of the warmer liquid at the top.⁸ During microwave heating, the electric field distribution is often closely related to the final temperature distribution. The complex dielectric constants (real and imaginary parts) and the position, number, and geometry of the sample all have a large effect on the electric field distribution.⁹ Natural convection originates when a body force acts on a fluid in which there are density gradients. The net effect is a buoyancy force, which induces free convection currents. In the most common case, the density gradient is due to a temperature gradient and the body force is due to the gravitational field.¹⁰ When the microwave heats the liquid, convection also plays an important role in temperature distribution during heating.¹¹ 

Several scholars have done a lot of research studies on the uniformity of microwave heating. There are two ways to improve the uniformity of microwave heating: one is improving the uniformity of the electromagnetic field in the microwave cavity and the other is improving the uniformity of microwave energy absorption in materials. The former can improve the uniformity of microwave heating by using a mode stirrer or rotary table.^12,13 The latter depends on the material configuration and its size or moisture content and distribution.¹⁴ In optimizing microwave heating of liquids, due to the convection of liquids, we must also consider the coupling of several physical fields when we simulate the microwave heating of liquids. Some scholars carry out multi-physical field coupling simulation and experiment on microwave heated liquids.^15,16

The main aim of this paper is to predict the temperature distribution of microwave-heated alcohol and water and microwave reheated rice using a common household flat-panel microwave oven as the object to perform multiphysics coupling modeling simulation. This paper explains the difference between using a heat source to heat a liquid at the bottom and using a microwave to heat a liquid. At the same time, we consider from the perspective of improving the heating container; the ordinary glass is modified in a simple and effective way by using the cut-off-waveguide theory. In addition, the accuracy of the simulation results is verified through experiments.

1. Maxwell’s equations are solved to determine the electric field distribution in a microwave cavity. Here, $E⇀$ is the electric field intensity (V/m), ε′ is the real part of the complex dielectric of a material, $ε″$ is the imaginary part of complex dielectric of a material, ω is the angular wave frequency (rad/s), μ_r is the relative permeability of the material, and c is the speed of light in free space (3 × 10⁸ m/s).

2. Power loss per unit volume of the high-frequency electromagnetic field. Here, P_eav being the heat source in transient heat transfer is the dissipated power (W/m³).

3. Fourier’s energy balance equation to calculate the temperature distribution due to conduction and convection. Here, ρ is the material density (kg/m³), Cp is the specific heat (J/kg K), k is the thermal conductivity (W/m K), T is the temperature (K), and $v⇀$ is the velocity vector (m/s).

4. The equation describing the motion of a fluid is the Navier–Stokes equation (describing the momentum balance and continuity). Here, ∇P is the pressure, which is the perpendicular force per unit area (N/m²), g is the acceleration due to gravity (m/s²), and μ is viscosity (Pa s) for Newtonian fluids.

The above-described equations are given as follows:

An iterative numerical model was developed in COMSOL 5.3. The electromagnetic wave, heat transfer, and fluid flow modules were coupled and calculated. Finally, the electric field and temperature distribution of the heated object were obtained.

As shown in Fig. 1, the numerical model of a 20L flatbed microwave oven is established, including the chamber, agitator, glass, antenna head of the magnetron, and transmission waveguide. The glass is placed in the center of the bottom plate of the microwave oven. The top diameter of the glass is 86 mm, the bottom diameter is 58 mm, and the height is 150 mm. At the antenna port of the magnetron, the input power is 1000W and the frequency is 2450 MHz. To reduce the mesh, we used stainless steel impedance boundary conditions inside the wall. The glass is placed in the center of the ceramic plate.

In the simulation of microwave heating, we do not consider the convection of water; fill the glass with water and heat it in the microwave oven for 90 s. The dielectric properties of water are shown in Table I,¹⁷ and for other physical properties, refer to the built-in data of COMSOL Multiphysics. As shown in Fig. 2(a), the electric field distribution in water is not uniform. The final temperature distribution is shown in Fig. 2(b), and the temperature distribution and electric field strength are very similar.

If we consider the convection of water, the final temperature distribution is as shown in Fig. 3(a) and the temperature appears to be stratified up and down. However, we set the heat source at the bottom of the glass, simulating the traditional heating method of the heat source at the bottom. The final temperature distribution is shown in Fig. 3(b). We can see that the water temperature distribution is quite uniform.

Next, we simulated microwave heating of alcohol and rice reheating. The dielectric constant of alcohol at 2450 MHz and 20 °C is as follows: ε′ = 8.94 and ${\relax \special {t4ht=ε}}″$ = 7.9817. Other physical properties of alcohol come from the built-in data of COMSOL Multiphysics. The dielectric constant of rice at 2450 MHz is as follows: ε′ = 51.4 and ${\relax \special {t4ht=ε}}″$ = 0.001T² − 0.224T + 17.133. The specific heat of rice is C_P = 2.930 − 1.469e^(−T/3.1), and the thermal conductivity is 0.694 W/(m °C). The density of rice is ρ = 1036 kg/m³. T is the temperature (°C).

The simulation results show that the temperature distribution of ethanol and rice is as shown in Figs. 4(b) and 4(d), respectively. Figures 4(a) and 4(c) show the electric field distribution inside alcohol and rice, respectively. The temperature distribution of alcohol shows the phenomenon of upper and lower temperature stratification, and the temperature distribution of rice is consistent with that of the electric field. The temperature distribution of alcohol shows a stratification phenomenon in which the upper temperature is higher than the bottom temperature, and the rice temperature distribution showed consistency with its electric field distribution.

In general, since alcohol and water are both liquids, convection has a greater influence on their temperature distribution. To suppress the upward flow of heated water during microwave heating, we envisage adding a partition in the long glass to isolate the upper and lower parts of the water during simulation. For the convenience of experimenting, as shown in Fig. 5, we put a small beaker filled with water on top of another long glass and the lower part of the long glass is also filled with the same volume of water.

As shown in Fig. 6(a), it can be seen that the electric field of the lower part of the water is stronger than the electric field of the upper part of the water. From the simulated temperature distribution, as shown in Fig. 6(b), the average temperature of the lower half of the water is also higher than the average temperature of the upper part of the water. When we look at it separately, it is interesting to see that the temperature of the water in the upper and lower parts also appears to be stratified.

To make the temperature distribution of the microwave heated liquid more uniform, we have improved the ordinary glass. In waveguide theory, electromagnetic waves below the cut-off frequency will produce attenuation in the waveguide. The main mode of the circular waveguide is TE11 mode, and the cut-off frequency is 2.93 GHz for a circular waveguide with a diameter of 6 cm. Because of the good conductivity of silver, we plated a layer of metallic silver on the top of the glass, as shown in Fig. 7, and the cup is placed in the center of the ceramic plate; since the wall of the glass is very thin, the silvered part of the glass can be considered as a circular waveguide. The diameter of the glass is 6 cm, the height of the glass is 15 cm, and the length of the cut-off waveguide is 7 cm. The water surface is 1 cm above the cut-off waveguide. At present, there are already microwave rice cookers made of metal in the market, which have good safety. Because the cup is placed in the center of the microwave oven, the distance between the cup and the cavity wall is far and the possibility of ignition is also small. Besides this, we also simulated the temperature distribution of heated water after adding a metal cover to the glass. The lid is also 7 cm in length and slightly larger in diameter than the glass.

Figures 8(a) and 8(b) show the electric field distribution and temperature distribution of the water heated in the long glass. It can be seen that the temperature distribution is more in the upper part of the glass and the difference between the temperatures at the upper and lower part is close to 8 °C (microwave heating for 90 s). Figures 8(c) and 8(d) show the electric field distribution and temperature distribution of the water heated in the silver-plated glass (microwave heating for 90 s). It can be seen that the electric field in the upper part of the glass is much weaker than that of the ordinary glass and the overall electric field is also weakened. In addition, the temperature distribution is quite uniform.

Figures 8(e) and 8(f) show the electric field distribution and temperature distribution of water heated in a glass with a metal lid (microwave heating for 90 s). It can be seen that the electric field at the upper part of the glass is much weaker than that of an ordinary glass and the electric field at the bottom is enhanced. However, there are localized hot spots at the bottom of the final temperature distribution, but the temperature distribution is more uniform than that before the lid was added.

As shown in Fig. 9(a), the traditional electric heating method is used to heat the liquid. Because the heat source is only at the bottom, the density of the water at the bottom becomes smaller and flows upward after being heated, the water at the top of the glass transfers the heat to the air and evaporates away the heat, and the water at the top becomes colder and denser and flows downward. Finally, top–down global convection is formed. When heating the liquid using microwaves, because of the uneven distribution of the electric field, there will be many local hot spots in the water, which only causes unusual convection in the water. At the same time, because the water on the top of the glass is also in the state of being heated, the hot water will gather on the top of the glass. Finally, the phenomenon that the temperature in the upper part is higher than that in the lower part is caused.

As shown in Fig. 9(b), the upper part of the glass is equivalent to the cut-off waveguide. The height of the water in the waveguide is up to 1 cm; the electric field in this part of the water is weak, so the heating temperature is lower, and water with a height of 1 cm at the top is in contact with the metal, which is more conducive to heat dissipation. Finally, the convection of the water in the glass is closer to the top-to-bottom convection and the final temperature distribution is more uniform.

Figure 10(a) shows the temperature distribution of alcohol. It can be seen that the upper and lower temperatures are stratified. Figure 10(b) shows the temperature distribution of water. It can be seen that the upper and lower temperatures are stratified. Figure 10(c) shows the temperature distribution of rice reheating, and the hot spots are mainly at the bottom of the rice. Figure 10(d) shows the temperature distribution of the water heated by overlapping two glasses. On the whole, the water temperature in the lower half is higher than that in the upper half. If only one glass is concerned, the temperature is still higher in the upper part. Figure 10(e) shows the temperature distribution of water heated by the ordinary long glass. It can be seen that the temperature difference between the top and bottom is close to 8 °C. As shown in Fig. 10(f), when heating the same volume of water with the improved glass, the temperature difference between the upper and lower part reaches 0.5 °C. The experimental results are roughly consistent with the simulation results, verifying the accuracy of the simulation.

The coupling model of multi-physical fields (the electromagnetic field, flow field, and temperature field) of microwave heated liquids is established. The electric field and temperature distribution of microwave heated water and ethanol and microwave reheated rice are simulated. When microwaves are used to heat solids, there is no convection in the solids, so the temperature distribution trend of the solids is closer to the distribution trend of the electric field. However, when microwaves are used to heat liquids, there have to be some local hot spots in the liquid, which may be present anywhere in the liquid. This unusual convection keeps the liquid always heated at the top and does not drop to the bottom. In order to suppress the unusual convection in the liquid heated by microwaves, we use the cut-off waveguide, and the metal plated part of the upper part of the glass can be equivalent to a circular cut-off waveguide. The microwave rapidly decays in the cut-off waveguide, and the water on the top of the glass is not heated by the microwave, thereby forming a global convection from top to bottom, making the water temperature distribution more uniform.

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

This work was supported by the NSFC (Grant No. 61921002).