Influence of the Shape Factor on the Heating of an Aqueous Solution by Microwave

Abstract

In this study, the microwave heating efficiency of a water body is investigated with different shape factors. In particular, the same water volume was deposited in cylindrical containers with different diameters. Here, “shape factor” refers to the ratio between the surface fluid layer, which strongly absorbs microwave energy, and the inner layer, which is heated largely via conduction. For a liquid in a cylindrical container, the shape factor is characterised as the ratio between the depth and diameter of the air/water surface area. The heating efficiency is characterised by relating the energy absorbed in the outer fluid layer with the energy gained in the bulk and monitoring the temperature in the fluid bulk at the point that the outer layer commences boiling. A correlation equation for the uniformity of the sample heating (with stirring) provided a simple link between the physical factors and microwave (MW) parameters. It was found that a depth/diameter ratio approaching 1:1 provided the most uniform heating. The correlations between the fitting parameters and physical conditions provide a simple yet effective method to characterise the thermal homogeneity of microwave heating that can assist with practical parameterisation of the design of microwave reactors.

1. Introduction

Microwave (MW) reactors are used in various applications, such as organic synthesis [1], food processing [2,3,4,5,6], materials science [7,8], chemical conversion [9], distillation [9] and chemical engineering [10]. One of the advantages of microwave heating is its ability to heat suitable reaction mixtures quickly, resulting in improved reaction efficiency and selectivity. However, the heating efficiency of a microwave reactor depends on several factors, such as the reactor design and the power input. While higher microwave power inputs can increase the heating rate, they can potentially generate overheating or thermal runaway. In addition to localised “hot spots” [11], non-uniform temperature distribution can reduce reaction conversion and generate undesirable by-products [12]. Thus, predicting and improving the uniformity of temperature distribution is one of the critical issues of microwave applications [13,14].

Microwaves heat response materials via molecular vibration while remaining largely unabsorbed by materials with low dielectric constants. As microwaves travel through the bulk of dielectric materials, the penetrating energy is reduced exponentially along its path as the incident energy is absorbed. As a result, the outer layer can absorb microwave energy and shield the interior region. A material’s capacity to absorb microwave radiation is often defined by a penetration depth, the depth at which microwave power is reduced by Euler’s number [15]. The penetration depth varies with the temperature and nature of the materials. For most aqueous solutions, the penetration depth is in the order of a few centimetres from the surface [16]. Due to the short penetration depth, microwaved aqueous fluids might experience an overheated outer region and an unheated inner region at the same time.

The food industry often designs microwave containers with a shallow depth and, thus, a large surface area. The free surface area, which is the air/water interface of an aqueous solution, creates another significant heat transfer: the evaporation of water and other volatile compounds [17]. While the value of the water evaporation coefficient remains debated in the literature [18], it is generally accepted that evaporation is governed by the dynamics of H-bond breaking [19]. A previous study has reported the surface boiling of microwaved water due to concentrated energy at the surface [16]. Consequently, it is expected that microwave-induced evaporation and resulting heat loss on the free surface are significant factors for temperature control. It is also noteworthy that the heat transfer at the free surface plays an important role in the combined microwave-distillation process [2,20]. Sol-gel MW synthesis also favours a wide over a tall vessel [8]. On the other hand, a large surface area can enhance material and heat losses via evaporation [21].

It is common for local temperature gradients to form during microwave heating, both due to the absorption of a large part of the incident radiation within the outer portion of the sample, and through the potential for the incident energy to localise in specific areas of the reactor due to constructive and destructive interference in the incident waves. Although natural convection promotes heat transfer to some extent [22], it is not sufficient to realise a uniform temperature distribution within the liquid during MW irradiation. Consequently, thermal zones are quickly formed during MW heating, as demonstrated in Figure 1. Mechanical mixing can be leveraged to minimise localised temperature regions during microwave heating. Similarly, most household microwaves are operated with an automatically rotating turntable to keep the contents moving around stable nodes in the irradiating wave: some studies have shown that the position of the sample, either in the centre or at the edge of the turntable, can significantly change the final temperature of the sample [23]. The systematic stirring can reduce thermal variation during MW by up to 81% [14].

In summary, heat distribution during microwaves involves molecular processes, such as vibration and surface evaporation, and micro-scale processes, such as fluid motions. Quantifying the interactive influence of these processes on the temperature uniformity of a microwaved liquid body remains challenging. Theoretically, a finite element method can be applied to describe the microwave heating distribution without fluid motion [24] or in a simple rotational motion [25]. Extension of the numerical models to fluid mixing is mathematically challenging and computationally costly. Previously, we proposed a characterisation technique to quantify the influence of microwave surface heating and fluid mixing on temperature uniformity [26]. This method obtained two heating values for different microwave and mixing conditions. Consequently, the heating uniformity factor is characterised by the ratio between the two heating values. The ratio was found to correspond to a combination of two dimensionless numbers, Reynolds and Asakuma numbers. While the first number accounts for the mechanical mixing in a cylindrical container [27], the second number accounts for the microwave energy concentration on the surface layer [28]. The mathematical equation is a simple tool to predict the required mixing conditions for different microwave powers, without a computationally expensive modelling framework. However, the ratio between the surface area and liquid volume, which is determined by the container shape, has not been quantified. This study investigates the influence of the container’s different shapes on the heating homogeneity. Ultimately, we aim to optimise the interactive effect of mechanical mixing and microwave heating on heated aqueous solutions.

2. Materials and Methods

2.1. Microwave Reactor

A boiling experiment by microwave heating was carried out in a reactor (Shikoku Instrumentation CO., LTD., Kagawa, Japan, 54.6 mm high and 109.2 mm wide). The microwave frequency was fixed at 2.45 GHz. A glass container was placed in the centre of the reactor, with the shape and size of the container as detailed below. The container was filled with a predetermined volume of solution, and the interface was monitored using a camera on the side of the reactor. The temperature of the solution was recorded during the MW irradiation by an optical probe located in the middle of the water body. The probe was placed approximately 1 mm under the water surface. Our initial test with different depths, within 2 mm from the surface, indicated negligible variations in temperature gradients. An electromagnetic induction-type magnetic stirrer (length: 24 mm, diameter: 6 mm) coated with polytetrafluoroethylene (PTFE) was placed at the bottom of the container and rotated by a magnetic mixer.

2.2. Container Size and Solution Volume

The heating rate and time to reach boiling for different mixing/microwave combinations were experimentally measured. The temperature in the centre of the bulk solution was continuously recorded to the point that boiling was visually observed at the surface. The effect of the vessel dimensions is quantified by altering the diameter of the borosilicate glass vessel positioned at the centre of the microwave reactor [29]. In the first series of experiments, four cylindrical flasks with different diameters (28, 35, 47 and 60 mm) and the same height (65 mm) were used. In this experiment, the aqueous volume was kept at 30 mL. The corresponding diameter/depth ratio varied from 0.57 to 5.7.

In addition to 30 mL, two different volumes (35 and 40 mL) of solution were used for the vessel with a 35 mm diameter. These two sets of experiments investigated the influence of the ratio between the depth and the water surface area for this cylinder. In addition to shape variations, the standard sample (35 mm diameter cylindrical vessel) was investigated with increasing microwave power and under an insulated condition. In this case, the container was wrapped with a 5 mm layer of glass wool. The insulation layer reduces heat loss from the walls to the surrounding environment.

2.3. Operating Conditions

In addition to the MW power in

Table 1. MW power for investigating the shape factor.

	Power	Diameter, D [mm]	Volume, [mL]
No. 1	100 W	28, 35, 47, 60	30
No. 2	100 W	35	35, 40
No. 3	200 to 800 W	35	30

, two operating factors were systematically varied: the rotation speed and salt concentration (Figure 2). The cylindrical magnetic stirrer was controlled via a controller at adjustable speeds. The rotation speed was varied from 100 to 400 rpm. The stirrer length and rotational velocity are directly input into the mixing Reynolds number as follows [27]:

$$Re=d_s\left(\pi d_s{n}_s\right)/\nu$$

(1)

where $n_s$ [1/s] is the rotational speed, $d_s$ [m] is the length of the magnetic stirrer and ν is the kinetic viscosity of a liquid [m²/s].

The value of Re represents the net effect of stirring on MW heating [14]. It should be noted that the Reynolds number in the above equation does not include parameters for the size of the container. The second varying factor is the electrolyte concentration of the solution. This study varied NaCl concentration (C_NaCl) at 0, 0.1, 0.2, 0.5 and 1 mol/L. The electrolyte concentration directly influences the microwave penetration depth, $d_p$. The relationship between C_NaCl and $d_p$ has been reported previously [28]. The microwave heating of the solution is described by the As number:

$$As=W/\left(H{d}_p\alpha \rho \right)$$

(2)

where W, H, $\alpha$ and $\rho$ are microwave power [J/s], latent heat of vaporisation [J/kg], thermal diffusivity [m²/s] and density of a liquid [kg/m³].

The two dimensionless numbers, $Re$ and $As$, will be used to characterise the heating uniformity as discussed later.

2.4. Procedure and Calculations

This study compares the temperature rise of the mixed bulk fluid with the observation of boiling at the vessel’s surface, which provides an indication of non-uniform heating [30]. Boiling was identified visually from the recordings of the fluid surface, evident from the forceful disturbance of the free surface. The initial temperature of the solution is room temperature. The bulk temperature is measured during the irradiation by an optical probe, with the tip fixed at the centre of the vessel. The sample is irradiated at the specified power until boiling is observed.

Figure 3 shows two examples of the temperature profile at different operating conditions. It can be seen that the temperature rising rate varies with the stirring speed and NaCl concentration. In the case of high NaCl concentration (C = 1 mol/L), the heating rate is faster. Accordingly, the solution reached a boil quickly. Time until boiling, t_b, was confirmed from the movie. An induction period is observed between the start of microwave irradiation and the rise in the temperature of the bulk sample. The heating rate, r [K/s], was calculated as the slope of the temperature from induction time to boiling time.

3. Results

3.1. Characterisation of Heating Conditions

The study aims to characterise thermal uniformity [30] and heating without boiling [1]. To characterise the heating homogeneity, two aspects of thermal energy transfer were calculated. First, microwave energy concentrated in the surface layer is calculated based on the surface area of the sample, with the thickness of the outer volume being defined by the penetration depth. The total energy gained by the outer layer, q₁ [J], is calculated as follows:

$$q_1=V_p\rho C_p∆T$$

(3)

where $V_p$ is the volume of the cylindrical shell of solution in the vessel with a thickness of $d_p$, $\rho$ and $C_p$ are the density [kg/m³] and specific heat [J/(kg K)] of the solution, and ΔT is the temperature difference between the theoretical boiling temperature (100 °C) and the initial temperature [1,31].

Second, the averaged microwave absorbance energy, $q_2$ [J], is calculated. The value of $q_2$ represents the heating of the whole vessel within the irradiation time, t_b [s], is calculated as follows,

$$q_2=V_t\rho C_{p}r{t}_b$$

(4)

$V_t$ and $r$ are the liquid volume and the temperature change rate, which is determined from the temperature gradient (Figure 3), respectively.

The ratio $q_2$/$q_1$ represents the homogeneity of temperature distribution during microwave irradiation. In a previous study, we proposed a simple function:

$$q_2/q_1-1=a{\displaystyle \frac{{As}^c}{{Re}^b}}$$

(5)

In the above equation, $a$, $b$ and $c$ are empirical parameters obtained by fitting Equation (5) against the experimental data. It should be noted that the two variables in experiments, $n_s$ and $d_p$, directly determine Re and As. A large combination of $n_s$ (4 velocities) and $d_p$ (5 concentrations) can generate sufficient data to capture the overall interaction.

3.2. Influence of the Shape Factor

Figure 4 shows fitting plots for different cylindrical containers. The equation fits very well for all containers at $a$ = 0.065 and $c$ = 1.0. While the values of $a$ and $c$ are constant, the value of $b$ changes significantly with the cylinder and, consequently, the diameter/depth ratio.

To highlight the relationship, the best-fitted values of $b$ are plotted versus the diameter of cylindrical vessels in Figure 5. It can be seen that the combination of 35 mm diameter and 30 mL of the solution produced the lowest value of b. In Equation (5), $q_2$ is the total heating energy of the liquid volume, and $q_1$ is the energy supplied by the microwave to the outer layer (to the penetration depth). The ratio represents the degree of mixing between the heated (outer) and non-heated (inner) regions. A smaller $q_2$/$q_1$ ratio means the heating is more uniform. Larger $q_2$/$q_1$ means most microwave-supplied energy remains in the outer layer. For a cylindrical shape, as the diameter increases, the depth-to-surface area ratio decreases. Consequently, the relative influence of mixing and heating also changes. Since the length and thickness of the stirrer were fixed (24 mm and 6 mm, respectively), the shape factor influenced the heating. When the water body is very deep, the top region of the solution is not well-mixed, as demonstrated in Figure 1a. In contrast, if the water body is very wide in comparison with the mixer, the region near the wall is not well-agitated. Consequently, an optimal ratio will exist where the heating effect is maximised [8] and closest to uniform: this state is a combination of good mixing and a large absorbing volume. For a 30 mL volume, the heat distribution is optimised with the 35 mm cylinder (the corresponding solution depth is 31.2 mm). Provided that good mixing can be achieved, the most effective diameter/depth ratio is about 1:1, where the surface area to volume ratio is optimal to maximise the absorbing area.

As the liquid height increased at larger volumes (40 and 45 mL), the fraction of the poorly mixed region increased. In these two cases, well-mixed regions were reduced and, thus, the influence of mixing. Consequently, the value of b is increased significantly. Accordingly, parameter b strongly depends on solution volume, as with vessel diameter. Since the microwave penetration depth is limited to 1.4 cm, the correlation to the surface-volume ratio is not directly scalable to other volumes.

Finally, the formula is applied to different microwave heating under various irradiation powers, with and without insulation (Figure 6) at the same vessel shape (cylindrical shape of 35 mm-diameter) and volume (30 mL). For different powers and insulation, the values of $a$ and $b$ are constant. However, the value of c varied. With increased microwave power, previous results for the relation with microwave power [26] showed a decreasing $c$ trend. In this scenario, the heating power increased within the same microwave-absorbing region. Consequently, the relative influence of the surface layer in Equation (5) is reduced. It is noteworthy that the trend is not strictly linear. With the increased power, the irradiation time is reduced, and the temperature increases faster. Consequently, the temperature gradient and thermal convection increase and $c$ is reduced.

By insulating the wall, the amount of heat loss is significantly reduced. Consequently, the available heat in the microwave-absorbing region was increased. As such, the value of $c$ is also reduced. However, the reduction in c due to insulation is relatively small since the heat loss is still significant at the free water surface [29]. The dominating heat loss at the surface indicated that the optimal condition in Figure 5 is applicable to other container shapes, such as a square container. In this case, an equivalent diameter can be calculated from the total free surface area. The significant correlation of MW power with surface parameter is consistent with the drying process [3]. Hence, the value of $c$ and the surface area are more important for energy saving [23] in industrial scale-up. A relevant industrial application is MW-assisted hydrodistillation [31], in which the aqueous reaction and vaporisation are combined within the same reactor. The method can be used to optimise MW power, irradiation periods and liquid level. The key advantage of the method is that heating uniformity can be predicted without a finite element analysis of the whole liquid body. A single temperature can be linked to the control loop to maintain the optimal MW operation. The method is applicable to the design of industrial MW reactors to optimise the impact of surface evaporation [32]. The correlation simplified heating operations to avoid a complicated element modelling [33], computer-enhanced prediction [34], extensive sensor systems [35] and elementary measurement systems [36,37]. Further studies on the non-symmetrical shape, such as a long rectangular shape with multiple stirring configurations, are recommended to extend the uniformity correlation.

4. Conclusions

In this study, the influence of the shape factor on heating uniformity was investigated. It was found that the uniformity correlation can be applied to the different liquid volumes. More importantly, the data indicated the direct correlation between the physical factors and the obtained parameters. The shape and volume of the irradiated liquid controlled the parameter $b$, which is the power of the Reynolds number. On the other hand, the microwave power and insulation condition are correlated to parameter $c$, which is related to microwave penetration depth. The insights demonstrate that the proposed formula can be extended to include a simplified shape factor (depth to surface diameter) for the optimisation and the control of temperature uniformity of microwave heating in industrial processes. The correlations between the fitting parameters, such as $b$ and $c$, and physical conditions provide a simple yet effective method to characterise the thermal uniformity of microwave heating. The method can supplement finite-element simulation to predict microwave heating effectively.