4.1. Experiment Analysis
First, the experimental O
2 concentration data were measured by the height of the enclosure in order to obtain the interface slope
and interface descending velocity
. Therefore, O
2 concentration was measured for a total of 10 points at 0.1 m intervals from 1 to 1.9 m in height of the enclosure. The O
2 concentration data according to the measured height are shown in
Figure 8. As explained in Chapter 2,
Figure 8 shows the
y-axis as the enclosure height (independent variable) and the
x-axis as the O
2 concentration (dependent variable) to express the O
2 concentration data measured at the specified height.
Figure 8a–c present the conditions with 8 leakage holes, and
Figure 8d–f present the conditions with 12 leakage holes. The O
2 concentration measurement with an increase in the leakage hole was measured to observe how species diffusion (diffusion flux
) changes with an increase in outflow volumetric flow
.
Figure 8 shows how much O
2 introduced from the outside diffuses into the z-direction over time. In this figure, it can be seen that the interface slope
saturates after a certain period of time (approximately 200 s). In other words, it means that the O
2 introduced from the outside diffuse through the z-direction with a constant diffusion flux
(i.e., diffusion velocity
).
The interface slope
appeared to differ depending on the type of inert agent used in the experiment. The interface slope
of IG-541 was the smallest and
of IG-55 was the largest. It can be seen that this is the same as the results of a previous study [
3], that is, because the interface slope
represents the z-direction diffusion flux
, it was confirmed that the
of each inert agent presents different values.
In addition, the interface slope
, that is, the diffusion flux
, does not change significantly with the outflow volumetric flow
. This means that the diffusion flux
is an independent factor that is not affected by advection flux
. Moreover, because the theoretical models for the retention time do not consider changes in temperature, the binary diffusion coefficient
can be seen as a fixed value [
24,
25]. This means that by Fick’s first law, the diffusion flux
can also be seen as a fixed value. Therefore, in the theoretical equation for retention time, the diffusion flux
can be calculated as a constant.
The O
2 concentration near the ceiling of the enclosure showed a tendency to increase rapidly. This is also the same as the results of a previous study [
3]. This can be inferred from the turbulent mixing formed by the inflow volumetric flow
. Therefore, this part will be analyzed in detail through numerical analysis. Therefore, the analysis of the interface slope
of this part was excluded.
Figure 9 shows the interface slope
and the maximum interface thickness
for each inert agent. That is, this figure shows how, when the amount of O
2 introduced from the outside is changed, the interface slope
and the interface thickness
change when the O
2 concentration threshold value is reached at the specified height
and ceiling of the enclosure
. Therefore, the theoretical equation of the newly proposed model can be completed in
Figure 9.
Figure 9a shows a comparison of the measured O
2 concentration for each height of the enclosure for each agent when the specified height
(1.5 m) has an O
2 concentration of 0.15. The O
2 concentration value is the average value of the data measured 5 times. In this figure, the difference in the interface slope
according to each agent is shown, and it can be seen that the interface slope
(i.e., diffusion flux
) is not significantly affected by the outflow volumetric flow
. The interface slope
of each agent was measured as
0.062,
0.071, and
0.083.
Figure 9b shows the maximum interface thickness
and the leading edge of the interface
for each agent. From this figure, it can be seen that if the interface slope
is known, the interface thickness
and the leading edge of the interface
can be obtained.
The relationship between the interface slope
and the maximum interface thickness
can be inferred by
Figure 9b as follows:
The leading edge of the interface
can also be calculated using the above equation. First, if the thickness from the specified height
to
is
,
can be expressed as follows:
As shown in Equation (13), if the specified height
and the O
2 threshold concentration
are determined to measure the retention time,
would have a fixed value. Therefore, if the interface slope
and the z-direction velocity
can be inferred, the retention time including the species diffusion can be obtained using Equation (14).
To obtain the retention time using Equation (14), it is necessary to know the information about the remaining factors. Here, and are design factors, and the outflow volumetric flow can be obtained through calculation or EIT. Therefore, if the interface slope and diffusion velocity are known, a theoretical calculation of the retention time is possible.
The interface descending velocity
can be calculated using the measured O
2 concentration data according to the enclosure height. This can be seen in
Figure 10.
Figure 10 shows the time to reach an O
2 concentration of 0.15 per 0.1 m from 1.8 to 1 m of enclosure height for the IG-55, IG-01, and IG-541. In other words, the slope of this figure shows the interface descending velocity
.
From this figure, it can be seen that the interface descending velocity
descends in the z-direction at a constant velocity. This shows the same result regardless of the type of inert agent used. However, the interface descending velocity
is different for each agent. This can be attributed to the difference between the outflow volumetric flow
and diffusion flux
. The outflow volume flow
is expected to be similar because the densities of IG-55 and IG-541 are almost the same. Therefore, the advection velocity
is expected to be similar according to Equation (3). However, as the interface slope
α is larger for IG-55 than for IG-541, the diffusion velocity
is expected to be larger for IG-55. Therefore, the interface descending velocity
for IG-55 is greater than that for IG-541. In addition, because IG-01 has a higher density than IG-55, the advection velocity
will be greater for IG-01. However, because the diffusion velocity
of IG-55 is large, the interface descending velocities
for IG-01 and IG-55 show small differences. As shown in
Figure 10b, this tendency is also the same when the outflow volume flow
increases.
The retention time for each agent was measured. It was measured at a specified height of 1.5 m, based on an O
2 concentration of 0.15. This is shown in
Figure 11.
Figure 11 shows a comparison of O
2 concentrations for the three agents with 8 and 12 leakage areas. In
Figure 11, the retention time is the fastest for IG-01 and the slowest for IG-541. When compared, the retention time of IG-55 was about 260 s faster than that of IG-541. However, IG-55 and IG-541, on the other hands, have almost the same density. That is, there is no difference in outflow volume flow
, i.e., advection flow
. Therefore, it is determined that the difference in retention time between the two agents is ultimately caused by the difference in diffusion flux
. As IG-55 has a larger
than IG-541, even though
is the same, the retention time appears faster.
Moreover, there is a minor difference between the retention times of IG-01 and IG-55. The diffusion flux is greater for IG55 than for IG-01, but the retention time is faster for IG-01. This can be explained by the difference in the advection flux due to the difference in density between the agents. The density of IG-01 is 1.399 kg/m3, which is approximately 8.3% greater than the density of IG-55, 1.292 kg/m3. As a result, it can be seen from Equation (4) that the outflow volumetric flow is larger for IG-01 than for IG-55. Therefore, the difference in retention time between the two agents is reduced, and it is determined to have similar values.
The experimental data measured and the data calculated using the proposed equation are summarized in
Table 5. In
Table 5, the retention time theoretically calculated using the measured and calculated factors is compared to the measured retention time.
The experiment was performed using three types of inert agents. In this experiment, the interface slope was the smallest with IG-541 ( 0.062), it was 0.071 for IG-01, and it was the largest at 0.083 for IG-55. The retention time of IG-541 with the smallest interface slope was the longest. The edge of the interface at an O2 concentration of 0.15 of the specified height can be calculated using the interface slope (Equation (13)). Therefore, the retention time can be calculated from the interface descending velocity and the edge of the interface (Equation (14)).
However, based on the aforementioned data, there was a difference between the retention time calculated using Equation (14) and that measured in an actual experiment. The retention time calculated in theory is approximately 3% slower than the retention time measured in the experiment. This part is considered as due to the turbulence mixing zone caused by the inflow volumetric flow , along with the average error of the calculated factors. Therefore, further analysis of this error is conducted through numerical analysis.
A summary of the experimental results performed so far is as follows. First, it was found that the interface slope , that is, the diffusion flux , had a constant size after a certain time and had a unique value that was different for each agent. Second, through the interface slope , the edge of the interface , which is the interface between air and air–agent mixture gas, could be calculated. Finally, the retention time reflecting the diffusion velocity yielded similar results to the measured retention time.
However, there was a limit to the information obtained from the experimental results. The difference in interface slope for each agent could be interpreted as a difference in diffusion flux , that is, diffusion velocity , but because the outflow volumetric flow could not be measured, the diffusion velocity of each inert agent could not be calculated. In addition, it was difficult to obtain information on the cause of the slope transients at the top of the enclosure, which are expected to cause differences in the measured and calculated retention times. Therefore, in the next chapter, more detailed information was obtained through numerical analysis.
4.2. Numerical Analysis
The interface slope for each agent was measured through the experiments and was confirmed to be unaffected by the z-direction advection flux . Then, the edge of the interface was calculated by measuring the interface slope for each agent. In addition, by measuring the interface descending velocity , it was also possible to confirm the difference between the calculated and measured retention times.
However, it was challenging to obtain all the information regarding the retention time of the inert agent through experiments. Information regarding the outflow volumetric flow and turbulence mixing zone of each agent was insufficient. Therefore, numerical analysis was performed to obtain additional information that was lacking in the experiment.
First,
Figure 12 presents the retention time at a specified height
of 1.5 m and the outflow volumetric flow
for each agent. The comparison between the experiment and numerical analysis for the retention time was optimized through the mesh-sensitive test in the previous chapter.
Figure 12a shows the numerical and experimental results of the O
2 concentration over time. In
Figure 12a, the numerical results of the O
2 concentration over time also presented the same results as those of the experiment. For the 8 holes, the retention time of IG-01 was the fastest and that of IG-541 was the slowest. For 12 holes, the results of the O
2 concentration over time also yielded similar results.
Figure 12b shows the results for the outflow volumetric flow
. As the outflow volumetric flow
is proportional to the density of the air–agent mixture gas, IG-01 showed the largest value. In addition, the increase in leakage holes resulted in an increase in the outflow volumetric flow
. It can be seen that this is the same result as the theory in the previous chapter.
In order to define the newly proposed theoretical equation for the retention time, it is necessary to determine the diffusion velocity
. The interface descending velocity
/
was measured through experiments; thus, the diffusion velocity
can be calculated using Equation (6) if the advection velocity
is known. Therefore, the advection velocity
can be calculated using the outflow volumetric flow
measured through numerical analysis. The outflow volumetric flow
and advection velocity
are listed in
Table 6.
Table 6 indicates that the z-direction advection velocity
is proportional to
because it is a function of the outflow volumetric flow
, and the z-direction diffusion velocity
is proportional to the interface slope
(i.e., diffusion flux
). Therefore, it was confirmed that the diffusion velocity
of each agent was not affected by the outflow volumetric flow
.
In addition, species diffusion is determined by the difference in species concentration and the binary diffusion coefficient
, as can be seen from Fick’s law [
18]. As all of the theoretical models for the retention time, including this study, do not consider the effect of temperature, the binary diffusion coefficient
of each agent can also be treated as a constant value [
24,
25]. That is, the interface slope
and the binary diffusion coefficient
can be kept constant regardless of the outflow volumetric flow
(i.e., advection flux
) and the geometry of the enclosure. Therefore, when the retention time is calculated using the theoretical equation, the z-direction diffusion velocity
for each inert agent can be calculated by substituting a fixed value. From
Table 6, it can be seen that the difference in species diffusion for each inert agent occurs clearly.
Based on the optimized numerical analysis results, the turbulent mixing zone of the upper part of the enclosure that could not be analyzed in the experiment was analyzed. Experiments have shown that this turbulent mixing region can cause distortion in the interface slope
and the theoretical retention time. As mentioned in the experimental study, a turbulence mixing zone is created near the ceiling of the enclosure due to incoming air. Therefore, turbulent mixing occurs between air and air–agent mixture gas in this zone by the driving force of the inlet volume flow
. This is shown in detail in
Figure 13.
Figure 13 presents the cross-sectional area of the enclosure for each inert agent at
t = 10 s. The arrow in the figure represents the species flow field inside the enclosure; air introduced from the outside in the turbulence mixing zone is mixed in all directions by advection. Therefore, it is not possible to interpret this part simply by the z-direction descending interface. Rather, this area can be considered as if mixing occurs in all areas of the turbulence mixing zone, similar to
the continuous mixing model. Therefore, the turbulent mixing area may be interpreted as a region having a single concentration rather than a concentration gradient in the z-direction. It can be interpreted that the starting point of the interface is not
when calculating the retention time, but
, which is the boundary between the turbulent mixing region and the interface descending region. In addition, it can be expected that
, the starting time of the retention time measurement, should be approximately 10 s instead of 0 s. Therefore, it can be seen that the distortion of the interface slope
and the theoretical retention time occur in this zone. In addition, the error between the calculated retention time and the measured retention time can be seen as being caused by this zone.
4.3. Summary of the Experimental and Numerical Analysis Results
Through experiments and numerical analysis, information regarding the interface slope
, interface thickness
, leading edge of the interface
, turbulence mixing zone, and z-direction diffusion velocity
was obtained. Therefore, the results of this study can be expressed in
Figure 14 and
Table 7.
Table 7 summarizes the results obtained through experiments and numerical analysis, and
Figure 14 shows the newly proposed model as a figure based on the experiment and numerical analyses.
First, the interface slope has a unique constant value for each agent. As presented in Equation (8), this value is eventually proportional to the z-direction diffusion flux . Therefore, the greater the interface slope , the faster the O2 diffusion and the faster the retention time. The interface slope of IG-55 is the largest with = 0.083, and it has values of = 0.071 and = 0.062. The interface slope is ultimately determined by the concentration difference between air and the agent–air gas mixture; thus, it will have a constant value regardless of the enclosure height. Therefore, if the composition of the agent–air gas mixture does not change, the interface slope has a fixed value.
Second, using the interface slope and the specified height , the leading edge of the interface can be calculated using Equation (12). Therefore, if the leading edge of the interface can be calculated, the retention time considering species diffusion can also be calculated.
Third, the diffusion velocity can also be taken as a fixed value. This is because the interface slope is proportional to the diffusion flux , and the diffusion velocity is also proportional to the diffusion flux . Therefore, if temperature is not considered, can also be taken as a fixed value when calculating the theoretical retention time.
Finally, it was confirmed that there was a turbulence mixing zone at the top of the enclosure; it cannot be interpreted as a z-direction descending interface model, because this area induces mixing of the air and the agent–air gas mixture by advection of the air introduced from the outside. Therefore, when calculating the retention time, it was confirmed that more accurate results were obtained when the turbulence mixing zone was excluded.
Based on the results thus far,
Table 7 presents the calculated retention time using the theoretical equation proposed in the previous chapter.
The diffusion velocity was = −3.40 × 10−4, = −4.40 × 10−4, and = −4.18 × 10−4. These results were the same as those for the interface slope . The measured and calculated retention times for all the inert agents present errors within 2.8%. This part is considered to be due to the turbulence mixing zone and the average error of the calculated factors.
However, the retention time considering species diffusion proposed in this study is significantly more accurate than the retention time of the existing models. This can be seen in more detail in
Figure 15. That is, this figure compares the retention time according to the height measured in the experiment with the retention time of the existing theoretical models, including the theoretical method proposed in this study. A comparison of the existing theoretical model was conducted in
the sharp descending interface model and
the wide descending interface model. From this figure, it can be seen that the calculation of the retention time through the newly proposed model can express the actual measured retention time better than the existing models.
First, as the sharp descending interface model does not consider species diffusion, the retention time is largely calculated. Therefore, it can be confirmed that this model shows optimistic results for the retention time. In addition, the wide descending interface model quickly calculates the species diffusion. Therefore, the retention time was calculated to be faster than the actual measured data. That is, it is confirmed that the results are very conservative.
In addition, the sharp descending interface model and the wide descending interface model do not reflect the unique species diffusion characteristics (the interface slope and the diffusion velocity ) of each inert agent, because the difference between IG-55 and IG-541 is not distinguished.
On the other hand, it was confirmed that the newly proposed model reflects the fixed interface slope and the z-direction diffusion velocity of each inert agent, thus representing the most accurate retention time. Therefore, when predicting the retention time of the inert agent, if the interface slope and diffusion velocity proposed in this study can be reflected, a more accurate retention time can be predicted.