Dynamic absorption efficiency of paracetamol powder in microwave drying

3.1 Impact of temperature on absorption efficiency of paracetamol powder

Figure 3 presents the patterns of the reflection loss coefficient of the paracetamol powder, which changes with thickness at different temperatures (20, 30, 40, 50, 60, 70^∘C) with a certain fixed moisture content (0.2, 0.4, 0.6, 0.8, 1.0 kg kg⁻¹). Figure 3 shows that the reflection loss of the paracetamol powder fluctuates at different temperatures and there are several troughs (absorption peak) in the RL curves. (Although RL is a negative value, the larger the absolute value of RL, the smaller the energy of the electromagnetic wave reflected by the material surface, indicating more energy taken by the material and better absorption efficiency of the material.) The fluctuation can be explained by the 1/4 wavelength theory [26, 27]: the matching thickness of the corresponding material is very close to 1/4 of the wavelength of the microwave in the material and multiplied by an odd number, at which an interference phenomenon occurs, thereby forming an absorption peak.

When the moisture content is fixed, the number of absorption peaks of the paracetamol powder at different temperatures in the same thickness range is different. For example, when the water content is 0.2 kg kg⁻¹ and the thickness range is 0–0.10 m, there are five absorption peaks at 20^∘C, but when the temperature rises to 70^∘C, there are eight absorption peaks. This is because the increase in temperature results in an increase in the dielectric constant of paracetamol powder, which makes the microwave wave-length in the powder shorter. The wavelength variation law and the wavelength values under various conditions are shown in Figure 4 and Table 3. The wavelength in the medium is given by [28]:

where λ_d is the propagation wavelength of the electromagnetic wave in the medium. According to the 1/4 wavelength theory, the wavelength in the drug powder medium becomes shorter, which leads to an increase in the number

Figure 3

Temperature dependence (20, 30, 40, 50, 60, 70^∘C) of reflection loss of paracetamol as thickness varies from 0 to 0.1 m for moisture contents: (a) 0.2 kg kg⁻¹, (b) 0.4 kg kg⁻¹, (c) 0.6 kg kg⁻¹, (d) 0.8 kg kg⁻¹, (e) 1.0 kg kg⁻¹.

Figure 4

Microwave length in paracetamol at temperatures between 20 and 70^∘C and in the moisture contents range of 0.2–1 kg kg⁻¹ (2450 MHz).

of absorption peaks in the same thickness interval, and the positions of the absorption peaks also shift toward the smaller thickness direction. The shorter wavelength makes the microwave phase constant increase, because the dielectric constant varies widely, so the values of the position shift of the absorption peaks are larger as the temperature increases. As the thickness increases, the subsequent position shift accumulates an extra wavelength deviation. Thus, more absorption peaks appear later, which is the more obvious sign of this shift phenomenon. For example, when the moisture content was fixed at 0.2 kg kg⁻¹, the first absorption peak at 20^∘C was located at a thickness of 0.01078 m, while at 70^∘C, the first absorption peak was at 0.00596 m; the fifth absorption peak at 20^∘C was located at a thickness of 0.09257 m, and at 70^∘C it was at 0.05288 m. Table 4 lists the distribution of absorption peaks of paracetamol powders at different temperatures and moisture contents (values at which RLs tend to be flat are not listed).

There is a thickness for the maximum absorption efficiency in each reflection loss curve of paracetamol powder under different conditions. For example, when the moisture content was0.2 kg kg⁻¹ and the temperature was 70^∘C, the absorption peak amplitude first increased with the powder thickness, and the maximum value (RL = −32.39 dB) was obtained when the powder thickness was 0.01765 m. Above this thickness, the absorption peak amplitude began to decrease, and at higher temperatures, this reduction was more pronounced. This is because the penetration depth of the microwave in the pharmaceutical powder decreases with the increase in temperature (the pattern of D_p is shown in Figure 2). For example, the penetration depth of the microwave was 0.1075 m when the moisture content of the paracetamol powder was 0.2 kg kg⁻¹ and the temperature was 70^∘C, but it was only 0.0301 m when the temperature increased to 70^∘C. Therefore, there is still an absorption peak (RL = 47.11 dB) at 20^∘C when the powder thickness is 0.09257 m, but the RL values around the peaks are slightly over -10 dB when the temperature rises to 70^∘C.

According to the variation law of the absorption peaks in Figure 3, it can be found that under the premise of ensuring uniform heating (the powder thickness is smaller than the microwave penetration depth), the thickness values for the second absorption peak should be selected as the preferred thickness. This is because the position shifts of the absorption peaks around the second absorption peaks are smaller, and with higher absorption efficiency, and it applies under various temperature conditions. The complex dielectric constants of the pharmaceutical powders change dynamically under different temperature conditions. Therefore, under the condition of fixing the thickness of the powder layer, it is difficult to ensure the best heating efficiency for microwave drying under different temperature conditions. The best powder thickness should be based on the absorption peak positions and RL values that create the best absorbing efficiency under conditions that match the powder thickness. Table 5 lists the thickness ranges corresponding to the reflection loss coefficients of paracetamol powder less than −20 dB. The reflection loss coefficient that is less than −20 dB (99% of the microwave energy can be absorbed) is usually taken as an indicator of absorption efficiency [29, 30]. Therefore, in microwave drying of paracetamol powder, the thickness of the powder should be preferentially selected in the thickness range listed in Table 5.

3.2 Impact of moisture content on absorption efficiency of paracetamol powder

Figure 5 shows the variation of the reflection loss coefficients of paracetamol powder with different moisture contents and increasing thickness under fixed temperature conditions. Figure 5 shows that the impact on the reflection loss of the paracetamol powder caused by moisture content is similar to that caused by temperature: the curve of the reflection loss coefficients with powder thickness also fluctuates with different moisture contents and at the same temperature. This is because the impact on the electromagnetic parameters of the pharmaceutical powder caused by moisture content is similar to that caused by temperature. As the moisture content increases, the real and imaginary parts of the relative complex permittivity of the paracetamol powder increase. Thus, the fluctuation caused by moisture content is the same as that caused by temperature.

Figure 5

Moisture dependence (0.2, 0.4, 0.6, 0.8, 1.0 kg kg⁻¹) of reflection loss of paracetamol as the thickness varies from 0 to 1.0 m for temperatures: (a) 20^∘C, (b) 30^∘C, (c) 40^∘C, (d) 50 ^∘C, (e) 60^∘C, (f) 70^∘C.

In the reflection loss curve related to the moisture content of the paracetamol powder, there is also a matching thickness corresponding to the maximum absorption efficiency. After exceeding this thickness, the fluctuation of the RL curve decreased until the RL gradually approached a constant value after reaching a certain thickness. This is because water is a strong absorbing material. Therefore, the electromagnetic wave absorption of paracetamol powder increases with moisture content, and when the moisture content reaches a certain level, the powder–water mixture consumes almost all of the microwave energy. In this case, the microwave energy reflected back from the latter interface is very rare, and the superposition effect of the wave caused by the impedance mismatch of the front interface reflected wave is not obvious. Therefore, the RL curves gradually become stable. As the moisture content increases, the relative complex permittivity changes, which causes the microwave penetration depth and the wavelength in the powder medium to decrease. This decreasing trend was most pronounced when the moisture content increased from 0.2 kg kg⁻¹ to 0.4 kg kg⁻¹. Thus, most changes in the RL curves in Figure 5 can be explained with the above analysis and the 1/4 wavelength theory. For example, the thickness position corresponding to the absorption peak of the RL curve is at an odd multiple of 1/4 wavelength under the corresponding condition; as the moisture content increases, the number of absorption peaks increases; the position of the absorption peak shifts toward a smaller thickness direction; and as the thickness increases, the position shift increases.

The above analysis found that the second absorption peaks should be selected as the preferred thickness. In different microwave drying conditions, the best thickness of paracetamol powder for microwave drying should be based on positions of relevant absorption peaks and the thickness ranges within which the RL values are less than −20 dB, as shown in Tables 4 and 5.

3.3 Impact of thickness on absorption efficiency of paracetamol powder

As displayed in Figures 3 and 5, there is a similar rule for the variations of RL curves with thickness. When the thickness exceeds the matching thickness of the maximum absorbing efficiency, and continues to increase, the fluctuation amplitudes of the absorption peaks decrease continuously and the RL gradually tends to a constant value. These phenomena were particularly evident under conditions of relatively high temperature and high moisture content. For example, there are six absorption peaks of the RL curve of paracetamol powder within the thickness range of 0–0.06 m, at 70^∘C and with a moisture content of 1.0 kg kg⁻¹. The first absorption peak is located at the thickness of 0.00444 m, with an RL value of −32.86 dB. The RL values of the third, fourth, fifth, and sixth absorption peaks are −13.26, −9.94, −8.30, and −7.73 dB, respectively, for which the thickness range is 0.022–0.04839 m. After the third absorption peak, the smoothing of the RL curve becomes more obvious. This is because the microwave penetration depth under this condition is small (D_p = 0.0288 m). When the powder is thicker than the penetration depth of the microwave under the corresponding conditions, the microwave reflected at the latter interface has been attenuated to a small extent, and has little effect on the overall power reflection. At this time, not only is the uniform heating condition not satisfied, but the drying efficiency is also low.

The impedance matching characteristics of the material are also largely influenced by its thickness parameters. The electromagnetic parameters of the material are basically fixed when the temperature, moisture content, and frequency are fixed. In this case, the electromagnetic wave attenuation properties of the material would have little effect on the RL value. The RL value would be mainly subject to whether the impedance matching characteristics of the material are appropriate, based on the equation [26, 27]:

where the thickness (d) of the absorbing material is an odd multiple of the λ_d/4 thickness of the medium, and the phase difference between the wave reflected by the former interface and the wave reflected by the latter interface is π. Therefore, when the thickness of the drug powder is larger than λ_d/4, the RL curve exhibits a fluctuation distribution, and, as the thickness increases, the absorption peak appears in sequence.

The above analysis found that the powder thickness should be smaller than the microwave penetration depth (D_p) as much as possible in the microwave drying of paracetamol powder. In addition, under the premise of uniform heating, the absorption efficiency of the powder can be improved by using the right powder thickness (Tables 4 and 5), thereby improving the drying efficiency.