Guilin rice noodles, with the radius of approximately R = 1.5 mm and the length of nearly 50 cm [14] , are initially immersed in a vast pot that maintains a gentle heat. Inside the pot, boiling water serves as a constant heat source wth high temperature T_H. Meanwhile, the cloddy rice noodles convolved each other for the convenience of transportation are stirred to scatter strip by strip with a pair of long chopsticks. Therefore, it is assumed that the noodles are preheated in an elongated cylinder at a constant heat source, as shown in Figure 1 . The radial distance, the azimuth and the column length of the arbitrary point M inside the noodles is represented by r, θ, z in cylindrical coordinate system, respectively [15] . During the heating process, the internal temperature of rice noodles with the heating time is expressed as the function of temperature, i.e., $U\left(r\mathrm{,}\theta \mathrm{,}z\mathrm{,}t\right)$. By combining the boundary conditions and the initial conditions of the preheating process, the three-dimensional heat conduction equation can be expressed as follows

$(\begin{array}{l}{\partial }_{t}U\left(r\mathrm{,}\theta \mathrm{,}z\mathrm{,}t\right)=\alpha \cdot {\nabla }^{2}U\left(r\mathrm{,}\theta \mathrm{,}z\mathrm{,}t\right)\mathrm{,}\\ U\left(\mathrm{0,}\theta \mathrm{,}z\mathrm{,}t\right)\text{is}\text{bounded}\mathrm{,}\\ U\left(R\mathrm{,}\theta \mathrm{,}z\mathrm{,}t\right)=T_H\mathrm{,}\\ U\left(r\mathrm{,}\theta \mathrm{,}z\mathrm{,0}\right)=T_L\mathrm{.}\end{array}$(1)

where α is the thermal diffusion coefficient of Guilin rice noodles, and T_L is the temperature of Guilin rice noodles before cooking, demonstrating the environmental temperature.

Through the analysis of Guilin rice noodles’ properties and cooking process, we deduce the following: 1) Rice noodles are much longer than its radius (i.e., $z\ll R$), so the heat conduction during the heating process mainly comes from the cylindrical side of the rice noodles, and the heat conduction through both cross sections of the ends can be neglected. That is the variable z can be also neglected. 2) The rice noodles are completely immersed in boiling water during the cooking process. From the isotropy as shown in Figure 1 , it can be seen that the azimuthal angle θ has no effect on the internal temperature variation of rice noodles, and thus Equation (1) can be simplified as [16] [17]

$(\begin{array}{l}{\partial }_{t}U\left(r\mathrm{,}t\right)=\alpha \cdot {\nabla }^{2}U\left(r\mathrm{,}t\right)\mathrm{,}\\ U\left(\mathrm{0,}t\right)\text{is}\text{bounded}\mathrm{,}\\ U\left(R\mathrm{,}t\right)=T_H\mathrm{,}\\ U\left(r\mathrm{,0}\right)=T_L\mathrm{.}\end{array}$(2)

Thus the heat conduction equation is transformed into only two variables of the radial and the time in the cylindrical coordinate system.

Equation (1) is the inhomogeneous boundary conditions. Therefore, the transformation is performed as follows

$U\left(r\mathrm{,}t\right)=u\left(r\mathrm{,}t\right)+T_H\mathrm{.}$(3)

Thus, the equation of the homogeneous boundary conditions can be expressed as

$(\begin{array}{l}{\partial }_{t}u\left(r\mathrm{,}t\right)=\alpha \cdot {\nabla }^{2}u\left(r\mathrm{,}t\right)\mathrm{,}\\ u\left(\mathrm{0,}t\right)\text{is}\text{bounded}\mathrm{,}\\ u\left(R\mathrm{,}t\right)=\mathrm{0,}\\ u\left(r\mathrm{,0}\right)=T_L-T_H=u_0\mathrm{,}\end{array}$(4)

where u₀ is the temperature difference between the environmental temperature and the heat source temperature.

In terms of the separation of variables method, we can define as

$u\left(r\mathrm{,}t\right)=f\left(r\right)\cdot g\left(t\right)\mathrm{.}$(5)

By substituting into Equation (4), one can get

$(\begin{array}{l}{g}^{\prime }\left(t\right)+\alpha {\lambda }^{2}g\left(t\right)=\mathrm{0,}\\ \frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}f}{\text{d}r}\right)+{\lambda }^{2}f=0\left(R>r\ge 0\right)\mathrm{,}\\ f\left(0\right)\text{is}\text{bounded}\mathrm{,}\\ f\left(R\right)=0.\end{array}$(6)

The general solution of Equation (6) can be expressed as

$u\left(r\mathrm{,}t\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{C}_n{\text{e}}^{-\alpha {\lambda }_n^{2}t}{J}_0\left({\lambda }_{n}r\right)\mathrm{,}$(7)

where $J_0\left(\lambda r\right)$ is the zero order Bessel function, and ${\lambda }_n$ is the eigenvalue derived from $f\left(R\right)=J_0\left(\lambda R\right)=0$ as shown in Equation (6). According to the initial condition $u\left(r\mathrm{,0}\right)=u_0$, the coefficient $C_n$ can be determined as

$C_n=\frac{2u_0}{R^2{J}_1^2\left({\lambda }_{n}R\right)}{\displaystyle {\int }_0^R}{J}_0\left({\lambda }_{n}r\right)r\text{d}r\mathrm{.}$(8)

According to the recurrence formula $\frac{\text{d}}{\text{d}t}\left[r{J}_1\left(r\right)\right]=r{J}_0\left(r\right)$ of Bessel function [13] , Equation (8) can be rewritten as following

$C_n=\frac{2u_0}{{\mu }_n^{\left(0\right)}{J}_1\left({\mu }_n^{\left(0\right)}\right)}\mathrm{,}$(9)

where ${\mu }_n^{\left(0\right)}$ represents the n-th zero value of the zero order Bessel function and $J_1\left({\mu }_n^{\left(0\right)}\right)$ represents the value of the one order Bessel function at ${\mu }_n^{\left(0\right)}$, as shown in Table 1 .

Table 1 shows the values of the first 10 zero points of the zero order Bessel function and the values of the function of the one order Bessel function at these zero points. When the number of summation terms is taken as $n_{\max }=10$, the accuracy of the zero order and one order Bessel functions is sufficiently high. After substituting ${\mu }_n^{\left(0\right)}$ and $J_1\left({\mu }_n^{\left(0\right)}\right)$ in Table 1 into Equation (9), we obtain the value of $C_n$ [18] .

The special solution of the internal temperature function of Guilin rice noodles is obtained by combining Equations (3), (7) and (9) as follows

$U\left(r,t\right)=T_H+2u_0{\displaystyle \underset{n=1}{\overset{n_{\max }}{\sum }}}\frac{1}{{\mu }_n^{\left(0\right)}{J}_1\left({\mu }_n^{\left(0\right)}\right)}{J}_0\left(\frac{{\mu }_n^{\left(0\right)}}{R}r\right){\text{e}}^{-\alpha {\left(\frac{{\mu }_n^{\left(0\right)}}{R}\right)}^{2}t}.$(10)

In order to measure the inactivation of Pseudomonas cocovenenans subsp. farinofermentans in rice noodles, the sterilisation efficiency $\eta$ is defined as the effect of sterilisation per unit of time within a certain temperature range, i.e.,

${\eta }_0=\frac{1}{\delta t}\times \mathrm{100\; <mi>\; \%\; </mi>.}$(11)

where $\delta t$ represents the time required to reach 100% sterilisation at the given temperature range.

In actual problems, the Pseudomonas cocovenenans subsp. farinofermentans in contaminated rice noodles is usually divided into low, medium and high concentrations. It is worth mentioning that the contaminated rice noodles fail to meet the explicit requirements of the Guilin Market Supervision Bureau, so that they cannot be used for consumption [11] - [12] . In an effort to ensure the generalizability of the study results, the medium concentration is selected for investigation. It is evident that the findings in this case are also obviously suitable for the low concentration. Experimental studies have shown that Pseudomonas cocovenenans subsp. farinofermentans can be effectively inactivated when the environmental temperature is above 60˚C. When the rice noodles are moderately contaminated, the time required for complete inactivation of the strain is about 90 s after heating at 60˚C. When the heating temperature is 70˚C - 90˚C, the time required for complete inactivation of the strain is reduced to about 60 s. As the heating temperature increases to 100˚C, the time for complete inactivation of the strain is reduced to about 25 s [8] . Based on the above results, the relationship between the sterilisation efficiency ${\eta }_0$ and the duration time $\delta t$ at different temperature range T during the preheating process of Guilin rice noodles can be summarized as shown in Table 2 . In this Table, the labels, i.e., I, II, III, and IV, represent different regions where the temperature ranges are corresponding to 60˚C - 70˚C, 70˚C - 90˚C, 90˚C - 100˚C, and constant 100˚C, respectively.

Before consumption, Guilin rice noodles should be put into a pot with boiling water. The bottom of the pot is heated continuously by a gentle fire to ensure that the water in the pot continues to boil, so the boiling water is regarded as a heat source with temperature T_H. The altitude and air pressure of a given area exert an influence on the boiling point of water. Under a standard atmospheric pressure, the relationship between the boiling point of water and the local altitude is as follows [19]

$y=1.13\times {10}^5\left(\frac{100-T_H}{273+T_H}\right)\mathrm{,}$(12)

where y is the altitude. The average altitude of Guilin is 150 m [20] . According to Equation (12), the average temperature of boiling water in Guilin is 99.5˚C, and this is the high temperature T_H of the heat source for preheating the rice noodles.

The thermal diffusion coefficient is an important physical quantity that describes how quickly the thermal energy propagates in a substance. In 1969, Riedel [21] proposed an empirical formula to describe the thermal diffusion coefficient of foodstuffs with the temperature above the freezing point, i.e.,

$\alpha =0.088\times {10}^{-6}+\left({\alpha }_w-0.088\times {10}^{-6}\right)X_w,$(13)

where $X_w$ represents the content of water in food. Guilin rice noodles are usually made of wet rice noodles, which have a high content of water about 40% - 50% [14] . And ${\alpha }_w=1.4\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}$ represents the thermal diffusion coefficient of water. By taking the average content of water in Guilin rice noodles ( $X_w=45\%$) and substituting it into Equation (13), the average thermal diffusion coefficient of Guilin rice noodles can be obtained as

$\alpha =1.114\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}\mathrm{.}$(14)

For one of the most common environmental temperatures in the urban area of Guilin, the temperature of the boiling water with a open cover is taken as $T_H=99.5˚\text{C}$, and the initial temperature of rice noodles is taken as room temperature $T_L=25.0˚\text{C}$. By substituting the radius R of rice noodles’ cross-section and the heat diffusion coefficient α into Equation (10), we get the relationship between the temperature and the radial distance r of rice noodles and the time t. Figure 2 shows the temperature distribution of rice noodles with radius r for different preheating times, where the horizontal coordinate is the radial distance r of the rice noodles in the cylindrical coordinate system, and the vertical coordinate is the temperature of the rice noodles U. The vertical dashed line is corresponding to the location of the radius R of the cross-section of the rice noodles, and the red, green, blue, light blue, purple and dark blue lines indicate the internal temperature distributions of the rice noodles when the time of preheating is t = 1 s, 3 s, 5 s, 10 s, 15 s and 20 s, respectively. As shown in Figure 2 , the temperature at each point inside the rice noodles increases with the increase of preheating time. Meanwhile, the temperature at the core of the rice noodles is always the lowest, and the radial temperature of the rice noodles gradually converges while the preheating time is enough prolonged.

When Guilin rice noodles are taken out from the pot, we set the temperature at the core of the noodles to be the same as the temperature of the boiling water, i.e., T_H, which is the initial temperature of the rice noodles. And the rice noodles

are subjected to room temperature, meaning that the temperature of outside surface is T_L. Therefore, after the rice noodles are taken out from the boiling water, the three-dimensional thermal conduction equation can be reexpressed as follows

$(\begin{array}{l}{\partial }_{t}U\left(r\mathrm{,}t\right)=\alpha \cdot {\nabla }^{2}U\left(r\mathrm{,}t\right)\mathrm{,}\\ U\left(\mathrm{0,}t\right)\text{is}\text{bounded}\mathrm{,}\\ U\left(R\mathrm{,}t\right)=T_L\mathrm{,}\\ U\left(r\mathrm{,0}\right)=T_H\mathrm{.}\end{array}$(15)

By the method of Section 2.2, the temperature distribution function of Equation (15) for the cooling process can be obtained as

$U\left(r,t\right)=T_L-2u_0{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\frac{1}{{\mu }_n^{\left(0\right)}{J}_1\left({\mu }_n^{\left(0\right)}\right)}{J}_0\left(\frac{{\mu }_n^{\left(0\right)}}{R}r\right){\text{e}}^{-\alpha {\left(\frac{{\mu }_n^{\left(0\right)}}{R}\right)}^{2}t}.$(16)

Figure 3 shows the temperature distribution of rice noodles as a function of radius r for different preheating times, where the horizontal coordinate is the radial distance r of the rice noodles in the cylindrical coordinate system, and the vertical coordinate is the temperature of the rice noodles U. The vertical dashed line corresponds to the location of the rice noodle cross-section radius R. The red, green, blue, light blue, purple, and dark blue lines indicate the internal temperature distributions of the rice noodles when the preheating time is t = 1 s, 3 s, 5 s, 10 s, 15 s, and 20 s, respectively. From Figure 3 , it can be seen that the temperature at each point inside the rice noodles decreases with the increase in time. Meanwhile, the temperature at the core of the rice noodles is always at its highest, and with the prolongation of the cooling time, the radial temperature of the rice noodles gradually converges to the environmental temperature, T_L.

The rice noodles’ cooking process with sterilising effect includes the preheating and the cooling processes. From the analyses mentioned above, it can be seen that during the preheating process of Guilin rice noodles and next to the cooling process, the rate of temperature variation at the core of the rice noodles is the slowest compared to the other parts. Therefore, the complete inactivation of Pseudomonas cocovenenans subsp. farinofermentans in rice noodles can be determined by examining the sterilising effect at the core of the rice noodles. By combining Equations (10) and (16) and setting r = 0, the temperature distribution function with time at the core of Guilin rice noodles in the cooking process can be obtained, as shown in Figure 4 , where the horizontal coordinate is the cooking time t and the vertical coordinate indicates the temperature distribution function U at the core of rice noodles. The shaded area indicates the time ranges in which the sterilisation conditions are met at the core of rice noodles. The sections designated as I, II, III, and IV represent the specific time of the preheating process (beginning from the left) and the cooling process (beginning from the right), where the internal temperatures at the core of rice noodles are 60˚C - 70˚C, 70˚C - 90˚C, 90˚C - 99.5˚C, and 99.5˚C (as shown in Table 2 ), respectively. And the labels t₁~t₈ on the horizontal coordinate indicate the time required to reach each temperature range, respectively.

The total sterilisation effect is expressed as follows

$\eta ={\eta }_{0i}\times \Delta t_j={\eta }_{0i}\times \left(t_{j+1}-t_j\right)\mathrm{,}$(17)

where i = I, II, III, IV, and $j=\mathrm{1,2,}\cdots \mathrm{,8}$. The values of ${\eta }_{0i}$ are shown in Table 2 , and i and j are shown in Figure 4 . Combined with Figure 4 , the time ranges corresponding to the sections I, II, and III (beginning from the left) in the preheating process can be strictly obtained from Equation (10), so that the value of $t_j\left(j=\mathrm{1,2,3,4}\right)$ can be determined. In the same way, the time ranges corresponding to the sections I, II and III (beginning from the right) in the cooling process can be obtained by using Equation (16), so that the time ranges from t₅ to t₈ can be determined. Based on the fact that the total effect η must be greater than 100%, the IV-th time interval (i.e., the value of $t_5-t_4$) can be obtained. When the total time range $t=t_8-t_1=\text{34}.\text{57}\text{s}$, i.e., the preheating time $t_5=\text{3}0.\text{28 s}$, the sterilisation effect of the cooking process reaches 100%. The sterilisation time and effect for each temperature interval are shown in Table 3 . It can be concluded that the time of the preheating process of Guilin rice noodles should sustain at least 31 s before the possible existence of Pseudomonas cocovenenans subsp. farinofermentans in Guilin rice noodles can be considered to be completely inactivated.