The radius of thermal influence of the chambers of underground structures of the cryolithozone

Galkin Aleksandr ORCID: 0000-0002-5924-876X Doctor of Technical Science Professor, Chief Researcher of the Melnikov Permafrost Institute SB RAS 677010, Russia, Yakutsk, ul. Merzlotnaya, 36, IMZ SO RAN. Laboratoriya geotermii kriolitozony afgalkin@yandex.ru Other publications by this author     Pankov Vladimir Yur'evich PhD in Geology and Mineralogy Associate Professor, Department of Construction of Roads and Airfields, North-Eastern Federal University 677027, Russia, respublika Sakha(Yakutiya), g. Yakutsk, ul. Belinskogo, 58 pankov1956@gmail.ru Other publications by this author

Fedorov Yan Vasil'evich Student, Department of Construction of Roads and Airfields, North-Eastern Federal University 58 Belinsky str., Yakutsk, 677000, Russia pankov1956@inbox.ru Other publications by this author

Introduction. The underground space of the cryolithozone has been widely used in human economic activity since ancient times to the present, mainly as underground storages and refrigerators ^[1,2,3,4,5]. First of all, this is due to the energy efficiency of underground structures, compared with similar ground facilities. Earlier, we carried out a comprehensive comparative assessment of the energy efficiency of the type of placement of various facilities and concluded that, according to operational energy costs, storage facilities in the cryolithozone, operated under both positive and negative thermal conditions, it is advisable to place underground ^[6,7]. At the same time, many researchers note that an additional effective way to manage energy costs in underground structures is the use of thermal insulation or special thermal protection ^[8,9,10,11] The design of underground structures in the cryolithozone, in particular the choice of spatial planning solutions, is carried out taking into account three factors determining efficiency and safety: technological, geomechanical and thermal (Territorial building codes. Underground objects in the mine workings of the cryolithozone of Yakutia. TSN-31-323-2002 The Republic of Sakha (Yakutia). The publication is official. Yakutsk: Ministry of Construction of the RS(Ya). 2002. 24s. ). Moreover, often these factors are not just interrelated, but according to the requirements they may contradict each other, or significantly improve or worsen the technical solutions used for reliable and safe operation of underground structures ^[12,13,14]. Therefore, proper consideration of the relationship and quantitative assessment of the degree of influence of individual determining factors in the design of underground structures of the cryolithozone is an urgent task.

The purpose of this work was to determine the zone of thermal influence of a single chamber of an underground structure, depending on the type of fastening used (the presence of a thermal protective layer)  and the duration of the operational period, using various calculation formulas.

Method. To achieve this goal, we will use the methodology described in ^[15,16], which is based on the use of dimensionless parameters of the desired quantities. The calculation formulas have the following form ^[15]:

The following notation is used in these formulas.  Criterion (number) Fourier - ; criterion (number) The bio - dimensionless radius of the thermal influence of the camera is R = .  Where: a is the coefficient of thermal conductivity of rocks, m2/s; ? is time, s; x ₀ is the characteristic size (linear scale), m; ? is the coefficient of heat transfer from air to rocks, W/(m2 K); ? is the coefficient of thermal conductivity of rocks, W/(m K); ? is the depth of the zone of thermal influence, m.

       The range of changes in the Fourier numbers for various operating conditions of technical facilities in the cryolithozone is given in ^[17].

       Note that formula (3) for determining the radius of thermal influence provides for its independence from the number of Bio, and can be obtained from formula (1) provided that the number of Bio tends to infinity. That is, under boundary conditions of the first kind. The value of the Bio number is mainly determined by the heat transfer coefficient, which depends on the thermal resistance of the support (special protective coating) and the air velocity in the chamber and can be found by the following known dependence

Where R _T is the thermal resistance of the thermal insulation layer or the support layer, ^m2/(W ° C); ? ₀ is the coefficient of convective heat exchange of air with the surface, W/(^m2 ° C), which can be found by the dependencies given, for example, in ^[13,18,19].

 It follows from formula (4) that the lower the Bio number, the greater the thermal resistance of the support layer. For example, a Bio number equal to one corresponds to a thermal resistance equal to 1.0 ^m2/WK, and a Bio equal to two corresponds to 0.5 ^m2/WK. According to a known number of Bio, it is possible to quickly determine the thermal resistance and select the appropriate material for an additional thermal insulation structural layer of the chamber support to limit the zone of its thermal influence.

Results and discussion. Multivariate calculations were performed using formulas (1)-(3), which are shown in the form of 3D graphs in Figures 1 and 2. Analysis of the calculations performed showed that calculations using all three formulas give similar results in a fairly wide range of changes in the initial parameters. Moreover, formula (3), which does not take into account the influence of the Bio number on the radius of thermal influence, gives a certain calculated margin. In general, it is obvious that the higher the value of the Bio number, the less its effect on the depth of the thermal influence zone of the underground chamber. Small values of the Bio number (up to 5-6) are typical for chambers that are fixed with sprayed concrete or have special heat-protective coatings ^[15]. Figure 1 shows a 3D graph for calculating the zone of thermal influence of chambers fixed with concrete spray and chambers with thermal insulation.

      Fig.1. Change in the radius of thermal influence of a fixed underground chamber depending on the Bio and Fourier criteria:

1 – calculation by formula (1); 2 – calculation by formula (2); 3 – calculation by formula (3).

    As can be seen from the graphs shown in the figure, for short periods of camera operation, all three formulas give approximately the same values of the desired parameter. However, the area of this is not very large and decreases sharply with an increase in the operating period. Moreover, the lower the value of the Bio criterion (the greater the thermal resistance of the thermal protective layer), the greater this discrepancy. For values of the Bio criterion greater than three, formulas (1) and (2) give similar results. Formula (3) in the entire considered range of changes in the Bio and Fourier numbers gives overestimated results, however, the degree of their excess and the possibility of attributing the results to the calculated reserve needs additional verification to minimize errors in the selection of certain technical solutions in the design of underground structures.   Figure 2 shows a graph of the change in the depth of the thermal influence zone of a single unfastened chamber for different operating periods.

Fig.2. Change in the radius of thermal influence of an unfixed underground chamber depending on the Bio and Fourier criteria:

1 – calculation by formula (1); 2 – calculation by formula (2); 3 – calculation by formula (3).

As can be seen from the graphs, all three formulas give approximately the same values of the dimensionless radius of thermal influence over the entire considered range of changes in the Bio and Fourier criteria. A comparison of the graphs in Fig. 1 and 2 shows that for loose cameras it is quite acceptable to determine the zone of thermal influence of the camera without taking into account its dependence on the number of Bio. That is, do not take into account the actual conditions of heat exchange on the surface of rocks (assume the equality of air temperature and rock surface temperature during the entire operational period). For fixed cameras and cameras with a special heat-protective coating, it is necessary to use calculated dependencies that take into account the influence of the Bio number on the depth of the thermal influence zone of the camera.

Conclusion. A comparison of numerical calculations for determining the zone of thermal influence of a single chamber of an underground structure, performed according to various formulas, presented in a criterion form, as a function of the Bio and Fourier criteria. It is shown that there is a range of changes in the initial data characteristic of loose chambers or when using a rod support, when it is acceptable to assume that the depth of the zone of thermal influence does not depend on the Bio criterion. At the same time, for chambers fixed with spray-concrete fasteners or when using thermal insulation on the surface of rocks, such an assumption is in most typical cases unacceptable or requires a special additional assessment of the error that occurs. The results of variant numerical calculations are presented in the form of 3D graphs, which allows you to visually determine the degree of discrepancy between the calculation results according to different formulas. The article has both applied and methodological significance and can be useful for both design engineers of various underground facilities in the cryolithozone, as well as graduate students and students studying in areas 1.6. and 2.1. In the future, additional research should be conducted to determine the rational area of use of the proposed methodological approach in determining the zone of thermal influence of closely located chambers of underground structures. And, in particular, when calculating chambers located in the zone of mutual thermal influence, but operated under different thermal conditions. For example, positive and negative, which is typical for the operation of cameras during natural or man-made emergencies. It is necessary to determine the range of changes in the initial data, in which all formulas give similar results, when the deviations of the calculated values do not exceed the values permissible in engineering practice. In addition, it is advisable to conduct similar studies for chambers with sharply varying thermal conditions for a short time in the presence of dependence of the thermophysical properties of rocks on temperature and phase transitions of moisture.

The work was carried out according to the state assignment on the topic: "Thermal field and cryogenic strata of the North-East of Russia. Features of formation and dynamics" (No. 122011800062-5).