Calculating water column height in the aboveground part of a low-altitude tank at abnormally low ambient temperatures

. The relevance of the study is confirmed by the widespread use of water tanks in rural settlements and farms in which ice forms inside the tank at abnormally low ambient temperatures, leading to a failure in the water supply system. The mathematical model is based on the heat balance equation, which takes into account the heat exchange of water with the ground and air through the separating metal wall, as well as heat transfer from water to air under the tank roof. Taking into account the low-potential energy of the ground, the paper presents a method for calculating the temperature of water in a non-height type tank of a given volume and radius. It allows to select the minimum height of its underground part to ensure uninterrupted water supply at average statistical values of ambient temperature and wind speed, registered in the region of research. The functions of the tank underground and above-ground part’s heights from the argument of its total volume to prevent the appearance of ice on the inner surface of the tank are graphically illustrated. The possibility of applying the method to determine the maximum water level in the above-ground part when the period of zero consumption is passed is indicated.


Introduction
At present, most of the centralised water supply systems are represented by local systems with water intake from wells, water tower and water supply networks [1].Regardless of the design, all pressure-regulating tanks fulfil the same function of compensating for the mismatch of water consumption and supply regimes at different hours.Rozhnovsky towers are widely used in rural areas due to their simple construction and quick erection.Due to the lack of insulation and heating, they often freeze in winter, which completely disrupts water supply.
Improving the serviceability of water storage tanks is an urgent task, especially in the aspect of uninterrupted water supply to rural settlements and farms [2 -7].The use of renewable energy sources [8 -9], including the low-potential energy of the Earth, is considered to be one of the promising directions for improving the water supply system in agriculture.For this purpose, it is possible to apply water storage tanks, part of which is located below the supporting ground (see Fig. 1) [10 -12].However, it remains unknown how effective such a measure is in different operating conditions.
The considered design [13] assumes the presence of a compressor for air injection into the above-water space, providing creation of the necessary head at water distribution.By increasing the height of the above-ground part of the tank it is possible to reduce the compressor power and energy consumption.At the same time, at a given volume, increasing the height of the cylindrical tank leads to a decrease in its radius, which increases heat loss from the sides and can lead to the formation of ice on the side walls of the tank and on the surface of the water.The task of the study is to determine the minimum height of the underground part of the tank of a given volume and radius, at which the water will remain in the liquid state in winter time.

Fig. 1. Schematic diagram of a low-height tank consisting of above-ground and underground parts
Icing leads to malfunctioning of the level sensors in the first place, and in case of significant ice build-up the water storage tank itself stops functioning normally.In order to avoid these preconditions for technological failure, the water level in the aboveground part can be lowered to the functionally necessary level during the zero-consumption period.
Purpose of the study -to substantiate constructive and mode parameters of operation of non-height type tanks, providing its operability at negative ambient air temperatures.

Material and methods
The basis for modelling the process of water cooling in a tank of non-height type is the heat balance equation of the system under study [14]: where:  Q change in heat content of water at temperature change by the value of dT ; Side Q  absolute value of heat outflow from water to air through the side walls of the aboveground part of the tank during the time interval dt ;  Calculation of the change in the intrinsic heat of water is carried out by the formula: where: The heat transfer from one liquid medium to another through the separating solid wall is calculated by the formula: where: air k  heat transfer coefficient through the side walls of the aboveground part of the tank, The formula for determining the convective heat transfer between the water surface and air is [14]: where:   heat transfer coefficient from water surface to air in the airspace zone, Then: In the study, the heat of evaporation of water into the air space under the roof of the tank is not taken into account.
The heat exchange of water with the ground through the side walls of the underground part of the tank can be calculated by the formula: )  () where: base area of the underground part of the cylindrical tank, 2 m .
Thus, the heat balance equation of the tank of non-height type takes the following form: This formula is also valid when the temperature of the water in the tank will be less than the temperatures .

M edge T
of the ground at a depth of 2 h or edge Т , which corresponds to the heat flux from the ground to the water in the tank.After substituting the formulas for area and volume, equation ( 8) is reduced to the form: By opening the brackets and regrouping the summands, we obtain the equality: After introducing additional notations: equation is reduced to the form: Equation ( 13) is a differential equation with separable variables (time  t argument,  T time-dependent function), which after their separation is reduced to the equality: In order to uniquely determine the dependence between these variables, it is necessary to set the initial conditions.Let at the initial moment of time the temperature of water in the tank was the same everywhere and equal to 0 Potentiation of both parts of the equality and expression of water temperature at any moment of time gives the relation: Thus, the dependence between the temperature of water in the vessel and the temperatures of the environment and water at the initial moment of time, the heat transfer coefficients through a thin cylindrical wall from water to air and ground and the heat transfer coefficient from water to air under the roof of the vessel, the dimensions of the cylinder and the cooling time is obtained.

Results and discussion
Formula (18) establishes the relationship between the heights of the above-ground and underground parts of a cylindrical tank of a given size and the environmental parameters, as well as the initial water temperature, provided that the temperature of water in the tank does not fall below zero after the interval of zero consumption.
The calculation method is formalised in the form of a block diagram shown in Figure 2.
During the winter period there are few abnormally cold days, so in order to save the used electricity, calculations can be carried out by the minimum of the average temperature for the coldest winter month and the average wind velocity.The heat transfer coefficient from water to air and ground through the side walls was determined using an approximate method [15,16]; the heat transfer coefficient from water to air under the tank roof was calculated using the criterion formulae for describing heat transfer under free convection [17].
However, there is still a question about uninterrupted functioning of the hydraulic structure in case of high wind speed or abnormally low ambient air temperature registered in the considered region.
To prevent ice from forming on the inner surface of the vessel, it was suggested that the height be reduced 1 h The water column in the aboveground part of the tank, thereby achieving slower cooling of the tank.Figure 4 allows us to determine the height of the water fill in the elevated part of a tank of a given volume at a predicted ambient temperature to prevent in-vessel ice formation.Calculations are performed for a tank with a total volume of 3 5m and radius 0, 7m , buried at 0, 96m .Taking into account the low-potential energy of the ground, a method of calculating the temperature of water in a cylindrical tank for any moment of time is proposed.
2. Typical dimensions of above-ground and underground parts of the tank of given volume and radius are found, at which water will remain in liquid state at the minimum of average temperature for the coldest winter month and average wind velocity.
3. This method allowed us to determine the maximum water level in the elevated section to prevent ice from forming on the inside of the vessel when the zero-consumption period passed.

1 h and underground 2 h
k Q heat lost by water due to convective heat exchange with air in the airspace zone per dt ; .S edge Q  the value of heat exchange of water with the ground through the side walls of the underground part of the tank during the same time interval; E3S Web of Conferences 458, 08020 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345808020 .M edge Q absolute value of water-soil heat exchange through the base of the underground part of the tank during the time interval dt .Standardised containers of fixed diameter are often used for the water supply process, so the study assumes that the volume of the V and radius R of the cylindrical tank are considered to be set.The task of the study is to determine the elevations of the aboveground tank.tank parts for uninterrupted functioning of the water supply.
surface area of the aboveground part of the tank, 2 m ;  T water temperature at a given time, C  ; av T  ambient temperature, C  ;  t time, s.


surface area of the above-ground part of the reservoir, air temperature above the water surface in the tank, С  .

E3S
Web of Conferences 458, 08020 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345808020where: edge k  heat transfer from water to soil in the underground part of the tank, average ground temperature from surface to depth 2 h , С  .To find the absolute value of the heat exchange of water with the ground through the base of the underground part of the tank, the following formula is used:

1 .
After transformations by the property of logarithms, the equality takes the form:

E3SFig. 2 .
Fig. 2. Algorithm for calculating tank measures of low-altitude Fig. 3 shows theoretical graphs of heights 1 h and 2 h parts of the water storage tank at different values of the total volume of the tank and constancy of its radius.

Fig. 3 .-
Fig. 3. Graphs of height dependencies of the above-ground and underground parts of the tank on the total volume for uninterrupted water supply operation So, the initial parameters are: -minimum average temperature for the coldest winter month

Fig. 4 .
Fig. 4. Graph of the dependence of the water filling height in the above-ground part of the tank on the ambient temperature for uninterrupted water supply operation Ambient temperature for uninterrupted water supply operation