Abstract

Because of Earth’s motion, periodic temperature fluctuations occur in soil on daily and yearly time scales, which inevitably lead to pulsation of the air pressure in the soil. The Earth–air pressure monitoring technique called “hou-qi” is the basis of a calendar formulated by the ancient Chinese. However, the daily/yearly variation of air pressure in the soil is very weak, according to practical monitoring experiments, so hou-qi has long been considered a pseudoscience. To determine the potential maximum change of Earth–air pressure, identify what is causing the underlying Earth–air pressure variation, and reveal the mechanism, we use a closed system, which is a more appropriate system test of the possible validity of hou-qi, to monitor Earth–air pressure variation in this paper. The results show that there are potential air pressure fluctuations of about 120–190 hPa in closed soil over the daily/yearly temperature range of 5°–40°C. This provides ample magnitude for hou-qi. The largest contribution made to Earth–air pressure variation was pressurization due to air warming, with vapor due to water evaporation and desorbed gas having smaller effects. The soil’s water content has a significant effect on Earth–air pressure amplitude. Dry soil contributes almost no water vapor, but the adsorbed gases from dry soil have up to a 38-hPa influence on air pressure. The soil’s salt content also has an important regulating role on Earth–air pressure and can reduce the influence of vapor pressure. This paper provides a scientific basis for the ancient Earth–air monitoring system of hou-qi and also has important significance for hydrometeorology and research on the ancient Chinese civilization.

1. Introduction

Solar radiation is the main source of Earth’s surface energy. In the Sun–Earth system, the revolution of Earth and its tilted rotational plane relative to the Sun causes seasonal changes. This leads to the temperatures on Earth exhibiting regular seasonal changes, so it is Earth’s rotation that is responsible for the observed diurnal temperature variation. The variation in the incident solar radiation (on yearly and daily time scales) results in (yearly and daily) periodic fluctuations in soil temperature. Thus, there is a yearly heterothermozone of about 30-m depth and a daily heterothermozone, 1 m or so in depth, at Earth’s midlatitudes (Wu and Nofziger 1999; Kang et al. 2000; Walsh et al. 1991). The regular temperature changes drive moisture migration in the soil so that the soil’s water content (WC) undergoes cyclical fluctuations involving condensation and evaporation. The process of heat conduction in the soil (due to solar radiation fluctuation) also causes the WC of the soil to change; it can also cause the water vapor concentration within the air contained in the pore spaces of the soil to fluctuate (Li et al. 2014).

Water vapor fluctuations are bound to cause fluctuations in the vapor pressure. However, whether it can cause Earth–air pressure (EP) to fluctuate has not been reported. This heterothermozone exists as a continuum with the outside atmosphere, and it is generally thought that the air pressure in the soil of this zone is controlled by the atmospheric pressure (Buckingham 1904), so the ground surface can be thought of as undergoing “passive respiration” (Elberling et al. 1998; He 2009).

In ancient China, an EP-monitoring technique named “hou-qi (候气)” was used as the foundation of an ancient calendar and used to forecast weather. Indeed, the expression “earth–air (地气)” is one that is still familiar to Chinese people. The ancient Chinese believed in the existence of air pressure changes within Earth, and hou-qi theory is a foundation of the Chinese civilization (Fan 1975). Unfortunately, the monitoring methods used to validate hou-qi have been lost. As a result, doubts have gradually appeared since the Ming dynasty about the existence and authenticity of hou-qi (Wang 2016). In the eighth year of the rule of Kangxi (AD 1669), Earth–air theory was declared fake (after much effort had been made to try to verify it). This meant that Earth–air theory that had been popular for thousands of years in China had been officially denounced, fundamentally shaking the foundation of the Chinese civilization (Huang and Chang 1996; Needham 1962; Dai 2015).

The present author recently monitored EP using modern instruments. The results obtained showed that the air pressure within the soil undergoes periodical pulsations on yearly and daily time scales, results that appear to support the traditional Chinese view on the subject. However, the yearly air pressure difference between soil (0–6-m depth) and atmosphere is only about 3 hPa, and the daily difference is only about 2 hPa. These are large, compared to instrument error (±0.4 hPa at 20°C), but are very small changes compared to the annual average atmospheric pressure of 860 hPa in the geographical area studied. The small amplitudes of the changes observed may be a result of the openness of Earth’s surface to free air exchange at the study point. Certainly, such a small variation is very difficult to monitor, even using modern instruments. Of course, in ancient times, science and technology were not as developed as they are today, which makes it rather doubtful that such small changes could have been successfully monitored in ancient times. But if the ancient Chinese inhibited exchange of air with the soil, a much larger EP may have been produced. Therefore, it is very important to determine the EP potential (i.e., the maximum change in EP amplitude) to hou-qi. If substantial EP amplitude is found, we need to identify the root causes underlying EP variation and to elucidate the mechanisms responsible for its variation.

It is well known that an ideal gas is governed by the expression PV = nRT, where P is the pressure of the gas, which is closely related to its temperature (T), the molar quantity (n), the gas constant (R), and volume (V). In soil, the presence of water means that the partial pressure due to water vapor is one of the major constituents of the EP (Buck 1981). Many factors affect the vapor pressure in the soil. Some are due to the external environment itself (e.g., the intensity of solar radiation, temperature, precipitation, and humidity). In addition, there are internal factors (e.g., soil composition, WC, salt content, and particle size). These also have a significant effect on the activity of soil water (Li and Wang 2014; Li et al. 2016).

In this paper, a closed system research method is used in the laboratory to study how the soil’s WC, salt content, and adsorption ability affects the water vapor pressure and air pressure in the soil. The data are analyzed to investigate the mechanism responsible for the air pressure and its potential. We aim to elucidate the basic rules governing such activity, both to understand hou-qi and to use for broader application in geophysical research, such as to explain how deep-buried mineral elements can penetrate to the surface (Wan et al. 2017). The work also provides scientific context for those researching the origins of the Chinese civilization and their methods.

2. Materials and methods

In this work, we tested nine soil samples and three controls. Seven of the soil samples were clay with varying WCs, and two were saline soils. The three controls were air, water, and NaCl saturated solution. The main subject materials were clay obtained from the sediment of the Daquan River (near the Mogao Grottoes) and saline soil from the Gobi region near the laboratory. These were used to compare and analyze the potential influence of temperature on the EP in soils with different WCs, mechanical composition, and chemical composition.

Clay samples with WC values of 0.11% (oven dried), 1.8% (air dried), 4.3%, 8.2%, 14.2%, 17.3%, and 21.4% were investigated. The clay was desalinated using pure water. It was then dried naturally in air, finely ground, and sieved (1.0-mm mesh). Oven-dried soil was produced by oven-drying samples at 105°C. Samples were placed in a room for one week at 18°C and with a relative humidity (RH) of 26%, which led to some moisture being regained (0.11%). Soils with other WCs were formed from the air-dried clay by directly adding water. In addition, a bottle of pure water (1.30 kg) was used as a control to represent high-humidity soil containing free water. Another bottle of saturated NaCl solution (2.95 kg) was used as another control to preliminarily analyze the effect of salt on the EP.

The saline soil was sampled from under the surface of the Gobi (20–30-cm depth) at the Mogao Grottoes. After sieving (1.0-mm mesh) and air-drying, the saline soil’s WC was found to be 5.5%. Samples of the saline soil were oven-dried (at 105°C) and then left in the room, whereupon they regained a WC of 1.8%. The composition and content of the salts in the Gobi saline soil were determined using ion chromatography (ICS-90, Diane, United States). The results are shown in Table 1.

Table 1.

The composition and content of the saline soil.

The composition and content of the saline soil.
The composition and content of the saline soil.

The distribution of particle sizes in the saline soil and clay were determined using a laser particle analyzer (Winner 2308), giving the results shown in Table 2.

Table 2.

The distribution of particle sizes in the saline soil and clay.

The distribution of particle sizes in the saline soil and clay.
The distribution of particle sizes in the saline soil and clay.

In this experiment, we used closed systems. Clay and saline soil samples with different WCs were put into 2.5-L suction filter bottles (about 2500 g of clay and 3400 g of saline soil were used so that they each had the same volume in the bottle). At the same time, an empty suction filter bottle was used to represent “air” for comparison purposes. Microthermohygrometers (HOBO U23-001; United States) were buried in the soil at the center of each bottle: for the control of air, the HOBO hung in the center of the bottle, and for the controls of water and NaCl saturated solution, the HOBO hung above the liquid. The temperature and RH were recorded every 10 min. The devices used are accurate to ±0.18°C over the range from 0° to 50°C; RH readings are accurate to ±2.5% over 10%–90% and to ±4.0% over 0%–10% and 90%–100%.

The mouths of the suction filter bottles were sealed and then placed in a controlled temperature–humidity chamber (SETH-Z-040; China). The temperature was controlled to be in the range of 5°–40°C to coincide with the daily/yearly temperature variation found in the soil at depths of 10–30 cm in the Dunhuang area. The temperature was increased by 5°C at a time, and the system was left to equilibrate for 12.5 or 24.5 h at each temperature stage to ensure time adequacy for the experiments. Hard plastic conduits connected the bottles to an atmospheric pressure transmitter (HD9408T), and a paperless recorder (BT805) placed outside the controlled temperature–humidity chamber was used to record the air pressure every 10 min (Figure 1). The effects of temperature, salt content, and WC on the air pressure were subsequently analyzed according to the measured temperatures and pressures.

Figure 1.

The experimental equipment used and installation arrangement.

Figure 1.

The experimental equipment used and installation arrangement.

3. Results and analysis

After many experiments, the air pressure and absolute humidity (AH) were recorded for bottles containing samples of the different clays and saline soils, pure water, and saturated NaCl solution, and they varied, as shown in Figure 2 (left panels). The corresponding temperature and RH variation is shown in Figure 2 (right panels).

Figure 2.

Graphs showing the observed variation in (left) air pressure and AH and (right) temperature and RH for samples of clay, saline soil, pure water, air, and saturated NaCl solution as a function of time.

Figure 2.

Graphs showing the observed variation in (left) air pressure and AH and (right) temperature and RH for samples of clay, saline soil, pure water, air, and saturated NaCl solution as a function of time.

The air pressures [Figure 2 (left panels)] can be seen to increase stepwise as the temperature increases in the closed bottle. The maximum differential pressures over the 5°–40°C temperature range can thus be determined, as shown in Table 3.

Table 3.

The maximum differential air pressures for soils of different WC, pure water, NaCl solution, and air over the range 5°–40°C.

The maximum differential air pressures for soils of different WC, pure water, NaCl solution, and air over the range 5°–40°C.
The maximum differential air pressures for soils of different WC, pure water, NaCl solution, and air over the range 5°–40°C.

Overall, ignoring the exact nature of the “soil” (i.e., be it soil of different WCs, water, saturated NaCl solution, or air) there is a 120–190-hPa air pressure variation when the temperature changes by 35°C. Therefore, there is a 3.4–5.4 hPa °C−1 potential change in soil pressure. Thus, the daily or yearly temperature changes that occur in the closed system soil will cause the air pressure inside the soil to undergo obvious fluctuations.

However, as soil is a “half open” system, the soil pressure will be affected by this openness. The closer one is to the surface, the more open the soil is. Therefore, even though the temperature changes are more intense near the surface, the greater openness of the soil in the region greatly reduces the air pressure fluctuations caused by the temperature changes. Thus, the pressure variation recorded in actual monitoring experiments will be very small. On the other hand, the air pressure deeper within the soil may have sufficient potential to enable the fluctuations to be monitored on a daily and yearly time scale. This may have been the case even in ancient China. The important point is that large air pressure differences in closed soil can exist, and this means there is the possibility that they can be monitored and utilized by hou-qi.

The AH varied, as shown in Figure 2 (right panels). As can be seen, the AH rises with temperature, even in the air bottle [Figure 2 (bottom left panel)]. This may be due to the bottle’s walls and plug releasing a small amount of adsorbed moisture.

A special note must be made of an interesting phenomenon. Initially, as the temperature of the samples dropped from the laboratory temperature (about 20°C) down to 5°C, an abrupt increase in pressure appeared in the 8.4% WC clay data [see the yellow box in Figure 2 (top left panel)]. For the others, the pressure declined with temperature. After the abrupt rise, the form of the pressure variation was the same in all samples, but the 8.4% WC clay remained consistently higher than the others by about 10–50 hPa. While repeating this specific experiment, it was found that if the sample is kept in the bottle, then once the air pressure in the bottle is balanced with the atmospheric pressure again and the experiment is repeated, this phenomenon disappears. This phenomenon may occur because when the soil sample is drying or undergoing WC modulation, the dry soil particles make good contact with the air, so a certain number of air molecules are adsorbed. When the temperature drops, the air pressure drops, and the weakly adsorbed air is fully released. Generally, this kind of phenomenon does not exist in the relatively humid soil, such as the 17.3% and 21.4% WC clays.

The RH results [Figure 2 (right panels)] show that in the case of air, the RH becomes significantly lower as the temperature increases. In the 0.11%, 1.8%, and 4.3% WC clay samples, saturated NaCl solution, and 5.5% and 1.8% WC saline soils, owing to soil released absorbed water as the temperature increases, the RH is relatively stable at about 1%, 52%, 90%, 78%, 29%, and 8.0%, respectively. Also, the clay sample WC ≥ 8.4% has an RH close to saturation, which is consistent with the results for pure water.

The RH exhibits transient behavior when the temperature changes abruptly. Generally, the RH and AH of dry soil (≤8.4% WC) rise when the temperature increases rapidly, a very common phenomenon in extremely arid areas (Li et al. 2014). But for wetter soils, especially 21.4% WC clay, RH becomes unsaturated at the beginning of each temperature stage from 20° to 40°C [Figure 2 (right middle panel)]. This is because the wet soil reforms an aggregate structure. The evaporation happens on the surface, rather than throughout the three-dimensional soil matrix, so soils release water vapor slowly in response to abrupt air temperature changes during the warmer temperature stages. The same is true for the interface wrapped around the HOBO. In addition, we can see from Figure 2 that soil temperature generally reached equilibrium in about 6–8 h, so a temperature-stage duration of 12.5 h is sufficient, and setting a longer time, such as 24.5 h, is unnecessary.

Because of the differences in the moisture conditions and composition of the soil, the response of the air pressure to temperature differs depending on the range involved. As a result, the graphs sometimes cross over around 25°–30°C [Figure 2 (left panels)]. Considering that there are small temperature differences (Figure 2 right panels), the relationship between air pressure and temperature needs to be explored in more detail.

The molecular concentration of water vapor can be found by dividing AH by the molecular weight of water (18 g mol−1). The change in molar concentration of water vapor with temperature is shown in Figure 3 for the different bottles. The variation found with air is essentially linear. The others vary nonlinearly, in a manner similar to that shown by standard saturated water vapor, thus reflecting the basic characteristics of the water vapor present. These traces can be fitted using polynomial regression to produce the quadratic equations shown in Table 4.

Figure 3.

The relationship between temperature and water vapor concentration for the different soil samples used in the experiment.

Figure 3.

The relationship between temperature and water vapor concentration for the different soil samples used in the experiment.

Table 4.

The quantitative relationships between water vapor concentration C (mol m−3) and temperature t (°C) in the different soils.

The quantitative relationships between water vapor concentration C (mol m−3) and temperature t (°C) in the different soils.
The quantitative relationships between water vapor concentration C (mol m−3) and temperature t (°C) in the different soils.

To know how much the soil-released water vapor and the soil air itself contributed to the pressurization, and to determine whether there are other adsorbed gases released when temperature is increasing, we calculate as below.

Assume that m0 is the concentration of air (molarity) in the suction filter bottle in the absence of water vapor. Then, the molar concentration in the bottle at t = 5°C, m5°C, will be

 
formula

where mw5°C is the water molar concentration at 5°C. Using the expression m = n/V = P/RT, we also have

 
formula

so that

 
formula

The value of mw5°C can be obtained from the formulas in Table 4 for each sample. Hence, we can find the air molar concentration m0 at 5°C in each bottle. Here, P5°C is the air pressure at 5°C, which can be calculated by temperature and relative humidity (Buck 1981).

Now assume that the volume of the suction filter bottles remains constant between 5° and 40°C. According to the equation P = mRT, the calculated pressure (Pc) is

 
formula

Here, mwt = Nwt/1000 (where Nwt is the value calculated from the fitting formulas in Table 4; unit is m3). We can put m0 and mwt into Pc = (m0 + mwt) RT to calculate the air pressure at different temperatures. Then, we can compare Pc with the measured pressure Pm to know how much desorbed gas has been produced.

The air pressure differences (ΔP = PmPc) calculated according to the above method are shown in Figure 4.

Figure 4.

The EP differences between the actual monitored and calculated values.

Figure 4.

The EP differences between the actual monitored and calculated values.

Figure 4 shows that in the oven-dried clay (WC 0.11%), air-dried clay (WC 1.8%), and oven-dried saline soil (WC 1.8%) cases, a small amount of adsorbed air was released as the temperature increased from 5° to 40°C. At the beginning of each temperature stage, the rapid temperature increase results in the release of gases, and over time, much of these gases will be reabsorbed by the soil. Furthermore, the drier the sample, the greater the release of soil-adsorbed gas as the temperature increases. At 40°C, the desorption effect from the oven-dried saline soil reached 38 hPa, which is larger than that from the oven-dried clay. This shows that salt has a distinct effect on the internal pressure.

When the temperature was over 30°C, the desorbed gas from the 17.3% and 21.4% WC clays underwent an obvious increase in amount released (Figure 4b). As a result, in the region of 30°–40°C, the moisture and other adsorption gases released from these samples made their pressures increase quickly. Relative to the initial adsorption shown in Figure 2 (top left panel; yellow box), this corresponds to strong adsorption.

When the WC of the clay is higher than 4.3% (i.e., 8.4% –17.3%), the vapor pressure variation is about 60 hPa (Buck 1981). Based on the ideal gas equation to calculate the pressurization of air m0 in the bottle in the isovolumic process, the pressurization is about 105 hPa; therefore, the other gases released by the soil contribute little (<38 hPa; Figure 4). As the water vapor and air in the bottle are not absolute ideal gases, and considering the experimental error involved, some measured results are smaller than the calculated values. This is the possible main reason why negative pressure differences appear in Figure 4.

Overall, for a soil of given WC, it is the soil air itself (m0) that makes the largest contribution to the pressurization. Second, the release of water vapor occurs due to the increasing temperature. The smallest contribution is from adsorbed gases. For very dry soil (especially very dry saline soil), the contribution made by the release of water vapor is relatively small.

4. Discussion

The maximum air pressure changes found in the controlled laboratory experiment were significantly different from what recently was monitored in the natural environment. The main cause is that the latter is a closed system, while the former is an open system. The ancient Chinese people created a closed system to monitor Earth–air pressure by sealing the opening of a test bamboo pipe that was embedded in the ground with animal blood using a membrane. So, they changed an open system to a closed system. When the EP exceeds the limit that the membrane can bear, the membrane ruptures with a “blasting thud” (the so-called 地应以响, or “earth responds with sound”; Wang 2016). The monitoring method thus used the natural temperature variation occurring in nature.

By sealing the surface of the soil and burying the bamboo tubes to specific depths, a linear correlation between pressure and time could be realized. That is, a transformation between space and time was realized, so hou-qi does indeed have a scientific basis for the creation of a calendar. The soil conditions, and especially the humidity, thus have important effects on the hou-qi. The monitored pressure and temperature results can also be used to derive linear formulas (Table 5), which provide a reference for choosing a suitable soil for sealing to monitor the hou-qi. The intercepts in these fitting formulas are associated with the atmospheric pressure when the bottle was closed (i.e., the number of moles of air in the bottle at the time).

Table 5.

Linear formulas derived by fitting the air pressure P (hPa) and temperature t (°C).

Linear formulas derived by fitting the air pressure P (hPa) and temperature t (°C).
Linear formulas derived by fitting the air pressure P (hPa) and temperature t (°C).

5. Conclusions

To determine the maximum change amplitude of the EP, identify what is causing the underlying EP variation, and reveal the mechanism, we monitored the air pressure above clay and saline soil samples with different water contents under closed conditions. It was found that the air pressure has a potential variation of about 120–190 hPa over the range 5°–40°C. Besides thermal pressurization, the water vapor concentration has an important effect on the EP. The dryness and presence of salts also have important effects on the vapor pressure of the soil and regulate the EP. Dry soil can absorb more gases from the air, and as the temperature increases, the air subsequently desorbed will have up to a 38-hPa influence on the air pressure.

Influenced by water vapor, the air pressure depends on temperature quadratically. Therefore, when the temperature is high, the vapor pressure increases rapidly. Under the influence of temperature, EP undergoing “autonomous respiration” and the closed soil could generate sufficient pressure to be detected by the ancient Chinese monitoring method known as hou-qi.

Acknowledgments

We gratefully acknowledge funding from the National Natural Science Foundation of China (41363009) and the Gansu Province Science and Technology Plan (2018-0405-JCC-0456).

References

References
Buck
,
A. L.
,
1981
:
New equations for computing vapor pressure and enhancement factor
.
J. Appl. Meteor.
,
20
,
1527
1532
, https://doi.org/10.1175/1520-0450(1981)020<1527:NEFCVP>2.0.CO;2.
Buckingham
,
E.
,
1904
:
Contributions to our knowledge of the aeration of soils
.
U.S. Dep. Agric. Bur. Soils Bull.
,
25
,
5
52
.
Dai
,
N. Z.
,
2015
:
A discussion of the history of hou qi in ancient China
.
Stud. Hist. Nat. Sci.
,
34
,
26
38
.
Elberling
,
B.
,
F.
Larsen
,
S.
Christensen
, and
D.
Postma
,
1998
:
Gas transport in a confined unsaturated zone during atmospheric pressure cycles
.
Water Resour. Res.
,
34
,
2855
2862
, https://doi.org/10.1029/98WR02037.
Fan
,
Y.
,
1975
: Hou Han-Shu (The History of the Eastern Han Dynasty). Zhonghua Press, pp. 3015–3016.
He
,
F. X.
,
2009
: Atmospheric pressure fluctuation caused the Earth surface breathe in multi-layer vadose zone system belt. Liaoning Normal University, 23 pp.
Huang
,
Y. L.
, and C. C.
Chang
,
1996
: The evolution and decline of the ancient Chinese practice of watching for the ethers. Chin. Sci., 13, 82–106.
Kang
,
S.
,
S.
Kim
,
S.
Oh
, and
D.
Lee
,
2000
:
Predicting spatial and temporal patterns of soil temperature based on topography, surface cover and air temperature
.
For. Ecol. Manage.
,
136
,
173
184
, https://doi.org/10.1016/S0378-1127(99)00290-X.
Li
,
H. S.
, and
W. F.
Wang
,
2014
:
Determination and analysis of phreatic water evaporation in extra-arid dune region
.
Acta Ecol. Sin.
,
34
,
116
122
, https://doi.org/10.1016/j.chnaes.2013.06.004.
Li
,
H. S.
,
W. F.
Wang
, and
B.
Liu
,
2014
:
The daily evaporation characteristics of deeply buried phreatic water in an extremely arid region
.
J. Hydrol.
,
514
,
172
179
, https://doi.org/10.1016/j.jhydrol.2014.04.025.
Li
,
H. S.
,
F. S.
Wu
,
H. T.
Zhan
,
F.
Qiu
, and
W. F.
Wang
,
2016
:
The effect of precipitation pulses on evaporation of deeply buried phreatic water in extra-arid areas
.
Vadose Zone J.
,
15
, https://doi.org/10.2136/vzj2015.09.0127.
Needham
,
J.
,
1962
: Sound acoustics. Science and Civilisation in China: Volume 4, Physics. Cambridge University Press, 126–251.
Walsh
,
S. J.
,
J. D.
Vitek
, and
S. E.
Panciera
,
1991
:
Variability of soil temperature: A spatial and temporal analysis
.
J. Geog.
,
90
,
82
90
, https://doi.org/10.1080/00221349108979241.
Wan
,
W.
,
M. Q.
Wang
,
M. Y.
Hu
, and
Y. Y.
Gao
,
2017
:
Identification of metal sources in Geogas from the Wangjiazhuang copper deposit, Shandong, China: Evidence from lead isotopes
.
J. Geochem. Explor.
,
172
,
167
173
, https://doi.org/10.1016/j.gexplo.2016.10.008.
Wang
,
Y. M.
,
2016
: Houqishu: The Bond of Heaven, Earth and Human in the Ancients’ Concept (in Chinese). Zhongzhou Ancient Books Publishing House, pp. 132–136.
Wu
,
J.
, and
D. L.
Nofziger
,
1999
:
Incorporating temperature effects on pesticide degradation into a management model
.
J. Environ. Qual.
,
28
,
92
100
, https://doi.org/10.2134/jeq1999.00472425002800010010x.

Footnotes

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