# Computer Modeling in the Application to Geothermal Engineering

Authors: Karpenko Vasily Nikolaevich; Yuriy Starodub; Andrii Havrys
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

In the article, investigation is given of the developed mathematical models of nonequilibrium in time and distributed in space thermodynamic state of Earth’s matter from its center to its surface depending on the cases of the presence and absence of an internal source of thermal energy concentrated in the center of mass taking into account known geophysical data about the nucleus, mantle, lithosphere and atmosphere, and endogenous and exogenous heat fluxes.The objects of research are as follows: mathematical models of geothermal energy of the Earth, its internal source, and heat balance of endogenous and exogenous heat fluxes on the Earth’s surface. Research methods used are as follows: thermometry in deep wells, ground and remote sensing of heat fluxes of the Earth and the planets of the Solar System, mathematical modeling of heat exchange and thermoelastic processes from compression of Earth’s matter by gravitational field energy information and classical physical and mathematical methods, and computer modeling. The aim of research: in computer modeling to provide new mathematical models that determine the geophysical parameters of geothermal energy, which are aimed on solving problems of energy, environmental and economic security of society, using modern technical means of calculating ground and remote sensing data development of geothermal resources, and regulation of the heat balance of the ecosystem, namely: (i) study of the geological structure of the lithosphere to a depth of 10 km by remote sensing to determine the physical parameters of its layers more accurately than ground methods; (ii) development of projects of geothermal power plants on the basis of single isolated wells of a given depth with a capacity of up to 2 ÷ 3 mW of electricity on continents of the globe; (iii) real-time monitoring and forecasting of the temperature field of the atmosphere according to its physical and chemical composition. The novelty of the obtained research results: (i) developed the mathematical model of the physical process of origin and distribution in the bowels of the density of geothermal energy of the Earth from the surface to its center, which is the density of internal energy of an elementary geological object, and which increases when approaching the center of the planet; (ii) developed the mathematical model of the thermal energy source of infrared (IR) waves of the elementary geophysical object of the Earth’s interior depending on the depth of its occurrence, which allows to determine the stable generation of geothermal energy by rocks in a deep well for extraction and conversion into electricity and to study the geological structure and physical properties of the Earth’s interior; (iii) the mathematical model of heat exchange between the layers of the Earth’s subsoil with the thermal energy of infrared waves according to the laws of Fourier thermal conductivity and Stefan–Boltzmann heat transfer, which together with the geothermal energy source model allows to determine a thermal capacity of rocks in a deep well; (iv) developed the mathematical model of stable action of a source of thermal energy in the center of mass of the Earth, in the absence of which it is hard to explain the power of its endogenous infrared heat flux, parameters of geothermal energy distribution in the Earth, and the current thermodynamic state of the atmosphere, and the change in temperature of which depends on the thermophysical parameters of the physical-chemical composition of the atmosphere more than on changes in the thermal activity of the Sun; (v) determination of new numerical values: thermophysical parameters of the Earth’s interior; kinetic, potential and own gravitational energy of the Earth and own gravitational energy of the planets of the Solar System.


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## Body

## 1. Introduction

The process of development of geothermal resources (GTRs) for energy, environmental and social security of local society is growing in many countries around the world. The experience (of the Organic Rankine Cycle World) of using GTR shows that the most promising technology for the use and conversion of geothermal energy into thermal and electrical energy consumption is electric geothermal station technology based on an isolated single well (EGS-ISW). However, the current scientific and technical level of means of development of GTR does not allow to make this technology economically attractive. The reason is the lack of an adequate unified physical idea of the nature of geothermal energy. Existing physical ideas about the origin of geothermal energy are based on two hypotheses: “cold” and “hot” Earth [1], which do not answer a number of important questions for the development of society based on geothermal energy, which arise in the analysis of endogenous and exogenous thermal flows of the Earth.At the same time, the experience [1] of drilling deep, more than 3500 m, oil and gas wells and special experimental studies [2] of wells with a depth of more than 4000 m show that single deep wells have significant potential for stable heat generation up to 15 MW, which increases with the depth of the well. This GTR can be widely used.In [3], an analysis of 94 electric geothermal stations (EGSs) located around the world was performed. The analysis showed that the scientific and technical level created by EGS for the use of geothermal waters and the conversion of their thermal energy into electricity provides different efficiencies of 1 ÷ 27% of their work. The economic efficiency of EGS is less than 5 years and depends on the generation of absolute power of thermal energy for consumption. The efficiency of EGS-ISW technology [4, 5] has a similar technical efficiency of converting thermal energy into electricity, but the economic efficiency has a payback period of more than 5 years.The task of the article is to provide and prove a physical representation of the hypothesis of “hot” Earth about the nature of geothermal energy, the conversion of gravitational energy into thermal energy, and to investigate the developed complex mathematical substantiation of this representation on known ground and remote sensing data tasks of development of geothermal resources, ecological monitoring, and forecasting of thermodynamic state of the atmosphere.For heat and mass transfer problems, in particular, for EGS-ISW technology, studies of the thermal conductivity of nanofluids play an important role [6]. In addition to thermal conductivity, nanofluids have lubricating, anticorrosive, absorbing properties of thermal energy of IR waves. The results obtained in this work on the complexation of conductive and radiative thermal conductivity allow us to study the energy-absorbing properties of infrared waves by nanofluids of different composition.


## 2. Materials and Methods

We study the geothermal model of the Earth with radiate and conductive endogenous and exogenous heat fluxes, taking into account well-known geophysical parameters, namely: the distribution of average temperatures in its subsoil and on the surface and its external general gravitational, internal potential, and kinetic energy using an energy method. The statement involves an interpretation of the joint action in every point of the physical space of the conservation, change, and transfer energy laws [7].Consider the known energy balance of radiated exogenous and endogenous heat fluxes on the Earth’s surface.The proposed geothermal model of the Earth (GTME) with thermal flows has the form(1)geothermal Earth model=Sun⟶+atmosphere+lithosphere⟵0–50km+mantle+external core⟵50–2890–5150km+core⟵5150–6371km+ISECore centre.Thus, the geothermal model of the Earth includes energies of the Sun + atmosphere + lithosphere + (Earth mantle + outer core) + core + internal source of the Earth (ISE) based on the laws of thermodynamics, in which the ISE has a constant power of 1.859 ⋅ 1017 W (364.46 W/m2), which is observed at a depth of 50 m. The model works as follows: ISE heats the inner core, which cools, heating the outer core and mantle. Cooling of the mantle is accompanied by the heating of the lithosphere. And the cooling of the lithosphere takes place also due to the influence of the atmosphere, which cools with the density of the radiation flux of 239.9 W/m2 of infrared waves. The analysis of the thermal balance of the Earth shows that the atmosphere passes 66% of the IR waves. The heat flux of short IR waves penetrates on the Earth’s surface from the Sun and is absorbed at the level 0.41∙1017 W (161 W/m2) [8] and (163.3 W/m2) [9, 10].According to the authors of this article, the density of the thermal flux of 398.2 W/m2 from the medium and long infrared waves corresponds to an average temperature of the Earth’s surface, which is 16.3°C according to the Stefan–Boltzmann law. At a depth of 50 m, where the typical temperature is 10°C, exogenous and endogenous heat fluxes are formed with opposite gradients. Therefore, the conductive cooling of the Earth at this depth is almost stopped. The Earth is in an adiabatic thermodynamic state. The temperature of the lithosphere at a depth of 50 m remains unchanged at an average temperature of 10°C (283.15 K) with a corresponding heat flux density of 364.46 W/m2.A similar adiabatic situation is experimentally established on the Mercury and the Moon [11].The research of GTME is carried out by the method of energy analysis (MEA) [12] using the laws of thermodynamics and the function of deterministic probability (FDP) [6]. The main ideas of the MEA are the idea of a coherent action at each point of the physical space of laws: conservation, modification, transport, and packing of energy in physical points.The adiabatic distribution of temperatures in the middle of the planets is considered in the work [13].The average conductive heat flux for the Earth model assumed spherical is ofη3=η⋅4πR32 = 4.42∙1013 W (η = 0.087 W/m2), where R3 is the average radius of the Earth. The heat flux existence on the Earth’s surface (at an average depth of 50 m) and a stable generation of significant thermal energy continued by deep wells requires a much greater density of thermal energy. Thus, the heat flux cannot be considered the main thermal flux of cooling the Earth and heating the energy in the well. Then, the question about the physical cause of the origin of an endogenous heat flux in the geological environment (GE) arises.The authors of this article explain the physical fact of the thermogradient in the GE with difference between the stationary thermoplastic states of the rocks at the depth of their occurrence, the main factor. In addition, we consider additional anomalies: different horizontal tectonic stresses and faults, different fluid saturation, physic-chemical transformations, different composition of rocks absorbing the particle, the main heat flux of the IR wave of the ISE, and the proximity of the magmatic activity. The accumulated gravitational energy in the unit volume of rocks in depth increases their internal energy, which leads to an increase in their temperature and volume modulus of elasticity. That is, the internal energy of rocks at each depth is in thermodynamic equilibrium with gravitational energy. The Stefan–Boltzmann law determines temperature of rocks at each point in the vertical direction. The temperature of rocks supposedly forms the heat exchange determined by the Fourier thermal conductivity law. In fact, there is no such heat exchange, but there is a fixation of various stationary thermodynamic states of rocks. This fact proves the method of thermometry in deep wells, when the washer fluid, after a day’s carrying, on the face is 130°C and at the mouth is 10°C.For example, for rocks with a blackness factorε=0.68, temperature T3=283.15K at a depth of 50 m, coefficient of thermal conductivityλf=3.5 W/m/K, gradT3=0.03 K/m, coefficient of radiation absorption αR=−dη/ηdz=1 m−1 for large thicknesses, and normalized on z0=1m, when the absorption is equal to radiation, the equation holds(2)η=λf⋅gradT3=z0⋅αRεσT3+z0⋅gradT34−T34,(3)η=3.5⋅0.03≈1⋅1⋅0.68⋅5.67⋅10−8⋅10+273.15+1⋅0.034−10+273.154=0.105W/m2.Formula (2) reflects the equality of the heat flux density according to the law of Fourier thermal conductivity in the left part and the Stefan–Boltzmann law in the right-hand side. In the right-hand side here, the difference between the density of the heat flux emitted in the linear approximation, which increases with depth, and the density of the heat flux that occurs for the temperature of the Earth T3 is recorded. That is, a differential equation for the distribution of temperature at depth for a stationary thermodynamic state of the form is fair in GE:(4)λf⋅gradT3=z0⋅αRεσT3+z0⋅gradT34−T34,or(5)T˜4z−aT˜z+b=0.Here,a=λf/z02⋅αRεσ,b=a⋅T3−T34 are coefficients of layers of the GE at a given depth, which vary with depth and composition of rocks, and T˜z=z0dT3z/dz+T3z is the temperature of the GTME, K. The temperature depends on two main factors, namely, on the comprehensive compression and absorption of a portion of the energy of electromagnetic waves emitting by internal Earth’s source (ISE) from the depths under the law of Stefan–Boltzmann at the surface temperature of the nucleus 6273 K, σ=5.67⋅10−8 W/m2/K4, Stefan–Boltzmann constant.The energy of the waves of the ISE has the density 364.46 W/m2 on the surface of the lithosphere, which forms the stationary temperature according to Stefan–Boltzmann’s law T30=10+273.15 K. As the depth increases, the temperature of the rocks increases; that is, the Earth’s temperature T3 is a function of the density of gravitational energy T3EG, where EG is the gravitational energy of the Earth, which increases to the center to the final value.The temperature of the GTME depends on the gravitational energy. Experimental and theoretical geophysical studies of temperature dependence and the stress state of rocks on the depth of their occurrence, performed by the authors of this article, allowed to develop a functionT3EG.The authors of this paper use the theory of Debye’s solids [14, 15], which determines the volume density of the thermal energy eT of a solid, and the energy approach to modeling the change of the volume elastic modulus in the GTME from the change in gravitational energy eEG. The authors propose and investigate the relationship between the density of elastic energy, which depends on its own gravitational energy, and the density of thermal energy in the Earth’s lithosphere by the equation(6)eT=eEG⟶ρ3π45μ⋅R⋅T34EGTD3EG=ρVp2EGJ/m3.From (6) for a temperature dependent on gravitational energy, we have(7)T3EG=5μ33RπℏkB2λmin3Vp5EG4K,where R=8.31451 J/moles/K which is the universal gas constant; ℏ=6.626⋅10−34 J·s which is Planck’s constant; kB=1.380648⋅10−23 J/K which is the Boltzman constant; μ3,ρ are the molar mass and density of matter of the planet, respectively, kg/mol, kg/m3; TEG,TD=ℏΩmax/k are the inner temperature and temperature Debye matter of the planet, K; Ωmax=2πVpEG/λmin which is the planet’s matter maximum frequency of oscillations of elementary objects, depending on its gravitational energy, s−1; VPEG=VP01+ln1+EGz/w0egz which is the P-wave propagation velocity at depth z in the GE, m/s [16]; VP0 is the average velocity of the P-wave on the surface of the planet, m/s; egz=ρzVP2z which is the volumetric density of elastic energy, J/m3; w0 = 1 which is the unit volume of GE, m3; λmin=ℏ/kB⋅Vs−in⋅Vp−in/TD_Fe⋅12π2/Vs−in3+Vp−in33 = 7.368·10−10 m which is the actual minimum length of the acoustic wave in the core of the Earth, m; Vs−in=3474,Vp−in=11260 which is the interval speeds of S- and P-waves, respectively, m/s; TD_Fe = 1096.68 K which is the temperature of Debye for iron in the core (Table 1); and EGz=−γ∫RR−zmmz/R−zdmz which is the gravitational energy of the planet, J.Table 1
Physical parameters of the crust of the geological medium (GM) according to the main physical-chemical elements [17].
Parameters of physical-chemical elementsSiOAlGMFeHeat capacity (J/kg/°C)760919.32902.47816.61Heat capacity (J/mol/K)20.1629.424.3524.55Molar mass (g/mol)28.0815.9926.9819.4255.845Debye temperature∗ (K)625155394364.44478.46/1096.68Number of moles per 1 kg35.6162.5437.0651.5The composition of the elements in the GM (%)2957894∗Debye iron temperature is given for normal and core conditions [18].The Earth’s own total gravitational energy, provided byρ=ρ0+χ⋅z, is determined by the integral(8)EGz=−γ∫R3R3−zmmzR3−zdmz=mz=4π3ρ0+zχz3;dmz=3ρ0+4zχ4π3z2dz==−γ∫R3R3−zm3ρ02z5R3−zdz+∫R3R3−zm7χρ0z6R3−zdz+∫R3R3−zm4χ2z7R3−zdz=−3γρ024π3213760+lnσR35−−7γχρ04π32882360+lnσR36−4γχ24π3265342520+lnσR37,where σ=R3−zm/R3, zm = 6301000 m, which is the depth of integration, R3 = 6371008.8 m which is the radius of the Earth, and ρm−ρ0/R3=χ which is the increasing density to the center of the Earth; ρm = 13290 kg/m3, and ρ0 = 2850 kg/m3, being equal to 2.6275·1033 J. The known [19–21] gravitational energy of the Earth is defined at the level of EGz=γ∫0Rm3z/zdm3z=3/5γm32z/z = 2.23945∙1032 J, m3z=ρ3W3z,ρ3 = const.Figure1 shows the characteristics of the Earth’s own gravitational energy determined according to Table 1 equations of the form(9)eVP−teor2z/V02−1−1ρfactzVP−fact2z⋅w0︸EGz−teor EIM=EGz=γ∫RR−zmmzR−zdmz︸EGz−teor Newton's method;eteorz=ρ0+χzVP−teor2,VP−teor2=V021+ln1+EGzρfactzVP−fact2z⋅w0;efactz=ρfactzVP−fact2z;eteor+factz=ρfactzVP−teor2z.Figure 1
Characteristics of the Earth’s own gravitational energy depending on the depthR3 from the surface, determined by the energy information method (EIM) and Newton’s method (NM).Figure2 shows the characteristics of the distribution of the determined energy density of gravitational energy (Table 2) in 1 m3 and the known temperature [22].Figure 2
Characteristics of the distribution: gravitational energy densities in the middle of the Earth, determined by the method of the theory of elasticity and EIM and temperature [22].Table 2
Earth parameters are known [22, 23] and determined by the methods of the theory of thermoelasticity and EIM.
z (km)ρ3 (z) (g/сm3)Т3 (z) (K)P (z) (кbar)G (sm/s2)Vp (z) (km/s)Vs (z) (km/s)Vp (G3) (km/s)Eg-teor (p-var) (J)Λmin (m)TD (z) (K)T3 (z)-teor (K)Μ (g/mol)C (z, μ) (J/kg/°C)λSB_fact (W/m/°C)λf (z) g-const (W/m/°C)λf (z) g-var (W/m/°C)grad (T) (K/m)аR/n2 (1/m)Known parametersDefined parameters0.051.022880981.51.601.606.95E + 054.99E − 10410.30550.300.019421284.4238.942.682.685.25E + 001.2911.020.2982.21.604.452.23E + 124.99E − 10410.30550.300.019421284.4237.852.682.672.62E − 011.3352.60.3982.25.83.25.617.06E + 151.50E − 09493.001180.350.018031383.23373.0980.4080.301.78E − 011.33152.63.3983.35.93.26.751.77E + 181.53E − 09493.001190.480.018031383.23382.7083.1983.006.02E − 021.33252.96983.96.83.97.152.34E + 191.70E − 09510.491348.230.020111240.13555.85106.72106.414.24E − 021.33403.3711.2984.38.14.487.482.57E + 201.92E − 09539.141591.660.023371067.18914.55143.38142.903.26E − 021.33603.3717.8984.98.084.477.832.07E + 211.92E − 09539.141589.690.023371067.18911.16142.67142.112.17E − 021.33803.3724.5985.58.074.468.079.22E + 211.91E − 09539.141588.710.023371067.18909.46142.32141.671.63E − 021.33903.37−4.5985.58.074.468.171.71E + 221.91E − 09539.141588.710.023371067.18909.46142.32141.671.45E − 021.331153.3736.1986.68.054.458.376.22E + 221.91E − 09539.141586.740.023371067.18906.08141.61140.811.13E − 021.331503.3647.8987.88.034.448.592.57E + 231.91E − 09538.521582.210.023301070.35898.34141.08140.108.63E − 031.331853.36177059.4989.18.014.438.757.99E + 231.90E − 09538.521580.240.023301070.35894.98140.37139.226.99E − 031.332203.4371.1990.48.554.648.872.06E + 242.01E − 09542.921651.120.023791048.511020.89158.64157.146.20E − 031.332653.4686.49928.644.679.015.79E + 242.03E − 09544.811667.760.024001039.421052.07161.44159.655.21E − 031.333103.48102993.68.734.79.131.39E + 252.04E − 09546.081681.770.024141033.441078.80164.43162.354.50E − 031.333553.51117.7995.28.814.739.243.01E + 252.06E − 09547.991697.520.024341024.611109.39166.88164.503.97E − 031.334003.72133.5996.89.134.939.315.96E + 252.08E − 09561.521785.740.02580966.771291.52174.90172.133.74E − 031.334503.782010152.2997.99.385.079.401.18E + 262.12E − 09565.451826.810.02622951.421382.69183.33180.223.42E − 031.335003.84171.3998.89.645.229.472.18E + 262.16E − 09569.411869.010.02663936.561480.74192.29188.863.16E − 031.335503.91190.7999.69.95.379.543.82E + 262.21E − 09574.061914.260.02712919.791590.92201.16197.412.96E − 031.336003.972130210.41000.310.155.519.606.39E + 262.25E − 09578.081955.880.02753905.891696.95209.98205.922.78E − 031.336353.98224.31000.810.215.549.648.97E + 262.26E − 09578.751964.590.02760903.611719.74212.22208.022.64E − 031.336704.382170238.31001.410.755.949.661.24E + 272.27E − 09606.292137.970.03038821.092216.42224.57220.002.76E − 031.337214.41260.71000.610.916.099.711.92E + 272.29E − 09608.402163.150.03059815.512295.65230.50225.992.60E − 031.337714.442180282.9999.811.066.249.762.89E + 272.32E − 09610.532187.380.03079810.002373.65236.06231.622.46E − 031.338714.5327.6998.311.246.319.856.07E + 272.34E − 09614.802224.120.03121799.202495.27242.11237.922.22E − 031.339714.56372.8996.911.416.379.931.19E + 282.36E − 09619.102260.100.03163788.682618.33247.76243.812.03E − 031.3310714.622260418.6995.711.576.4410.002.18E + 282.37E − 09623.432295.310.03204778.442742.63252.99249.251.87E − 031.3311714.67464.8994.611.736.510.073.81E + 282.39E − 09627.072327.470.03239770.102859.55258.53254.991.74E − 031.3312714.732360511.6993.811.886.5610.136.37E + 282.41E − 09631.452362.120.03281760.342989.15263.34259.951.63E − 031.3313714.78558.9993.212.026.6110.191.03E + 292.42E − 09635.132392.650.03315752.383106.58268.02264.731.54E − 031.3314714.842450606.899312.166.6710.241.61E + 292.43E − 09639.582426.720.03357743.053241.17272.39269.101.45E − 031.3315714.89655.2993.112.296.7210.292.45E + 292.44E − 09643.302456.610.03392735.463362.44276.64273.271.38E − 031.3316714.952540704.1993.612.426.7710.333.64E + 292.45E − 09647.812490.100.03433726.543501.83280.56277.001.32E − 031.3317715753.5994.712.546.8210.385.28E + 292.46E − 09651.582519.350.03468719.283626.69284.35280.431.26E − 031.3318715.052640803.6996.312.666.8710.427.53E + 292.47E − 09655.382548.770.03503712.163755.20288.14283.711.21E − 031.3319715.1854.3998.512.786.9110.451.06E + 302.48E − 09659.202578.340.03537705.173887.45291.92286.811.16E − 031.3320715.15905.61001.512.96.9610.491.46E + 302.49E − 09663.042608.090.03572698.334023.55295.71289.651.12E − 031.3321715.22820957.61005.313.017.0110.521.98E + 302.49E − 09666.902636.980.03607691.614158.77299.03291.801.08E − 031.3322715.251010.3101013.137.0510.562.66E + 302.50E − 09670.792667.060.03641685.034302.70302.81294.111.05E − 031.3323715.31063.81015.813.247.0910.593.52E + 302.51E − 09674.702696.290.03676678.564445.71306.12295.631.02E − 031.3324715.351118.21022.713.367.1410.624.62E + 302.52E − 09678.632726.700.03711672.224597.85309.89297.259.87E − 041.3325715.41173.41030.913.477.1810.645.99E + 302.52E − 09682.582756.260.03745666.004749.02313.18298.039.60E − 041.3326715.4530101229.71040.613.597.2310.677.70E + 302.53E − 09686.562787.020.03780659.894909.78316.94298.799.36E − 041.3327415.491269.71048.413.687.2610.699.14E + 302.54E − 09689.762811.130.03808655.085038.32319.67299.119.21E − 041.3327515.491269.71048.413.697.2610.699.36E + 302.54E − 09689.762812.160.03808655.085043.85320.13299.559.18E − 041.3327715.51287105213.77.2610.709.82E + 302.54E − 09690.562816.920.03815653.895069.51320.23298.629.13E − 041.3328715.5531301345.61065.313.717.2610.721.24E + 312.52E − 09694.592836.660.03849648.005176.86318.84293.618.88E − 041.3328915.561357.51068.213.717.2610.731.30E + 312.52E − 09695.392840.410.03856646.835197.43318.47292.478.83E − 041.3329009.91357.51068.28.06010.791.33E + 318.95E − 101151.503672.360.06866363.2711232.5966.4761.041.17E − 031.33297110.0233101441.91050.68.19010.801.56E + 318.97E − 101167.673752.060.06950358.9211979.9267.6863.201.17E − 031.33307110.181546.910288.36010.821.94E + 318.99E − 101189.583859.290.07061353.2813036.6669.2266.061.16E − 031.33317110.321651.21004.68.51010.842.40E + 319.00E − 101209.103955.050.07158348.4914031.2970.5768.911.16E − 031.33327110.461754.4980.58.65010.862.95E + 319.00E − 101228.934050.020.07255343.8215066.5671.7371.771.15E − 031.33337110.61856.4955.78.79010.883.60E + 319.00E − 101249.084146.540.07352339.2816169.6872.8874.811.14E − 031.33347110.7338801956.9930.28.92010.904.38E + 318.99E − 101268.104237.580.07442335.1717258.2973.9377.961.14E − 031.33357110.852055.9904.19.05010.915.29E + 319.00E − 101285.914325.230.07525331.4618351.5375.0481.431.13E − 031.33367110.972153.1877.49.16010.936.36E + 318.98E − 101303.964409.300.07608327.8419442.4975.8184.771.12E − 031.33377111.082248.4850.29.27010.957.62E + 318.98E − 101320.744489.610.07685324.5820524.3376.6688.451.11E − 031.33387111.1944002341.6822.49.38010.979.08E + 318.97E − 101337.734570.950.07761321.3921660.1977.4992.441.11E − 031.33397111.292432.4794.29.48010.981.08E + 328.96E − 101353.374645.800.07830318.5522741.7578.2496.641.10E − 031.33407111.392520.9765.59.57011.001.27E + 328.94E − 101369.184719.050.07900315.7523834.5578.81101.001.09E − 031.33417111.4848702606.8736.49.66011.011.50E + 328.93E − 101383.584787.940.07962313.2724893.6379.46105.861.08E − 031.33427111.572690706.99.75011.031.76E + 328.92E − 101398.134857.540.08025310.8425995.1980.11111.171.07E − 031.33437111.652770.4677.19.83011.042.05E + 328.91E − 101411.184920.020.08080308.7027011.2880.68116.891.06E − 031.33447111.732847.86479.91011.062.39E + 328.90E − 101424.364983.090.08136306.6028063.3081.24123.171.05E − 031.33457111.852802922.2616.69.98011.072.77E + 328.89E − 101436.005038.750.08184304.7829014.2181.72130.021.04E − 031.33467111.872993.4586.110.05011.093.21E + 328.88E − 101447.735094.860.08233302.9829994.3582.20137.581.03E − 031.33477111.943061.4555.410.12011.103.70E + 328.87E − 101459.555151.420.08281301.2031004.5282.67146.021.02E − 031.334871123126.1524.710.18011.124.26E + 328.86E − 101469.765200.270.08323299.7031894.9583.07155.321.01E − 031.33497112.0656203187.4494.110.24011.134.88E + 328.85E − 101480.055249.460.08364298.2132808.6283.47165.739.98E − 041.33507112.123245.4463.610.3011.155.59E + 328.84E − 101490.405298.990.08406296.7333746.1083.87177.479.88E − 041.335149.512.1657103288.544010.35011.166.20E + 328.84E − 101497.355334.780.08434295.7634434.4984.29187.939.80E − 041.33515012.763288.544011.023.511.146.20E + 328.78E − 101605.485870.530.08850281.8545885.5389.12198.701.08E − 031.33517112.773300.4432.511.033.5111.146.38E + 328.78E − 101607.345879.470.08857281.6346095.3789.18202.281.08E − 031.33527112.823353.5397.311.073.5311.157.26E + 328.76E − 101616.715921.630.08892280.5347094.0589.31220.511.07E − 031.33537112.8758903402.336211.13.5511.178.24E + 328.73E − 101626.135961.340.08926279.4448047.9389.27241.921.06E − 031.33547112.913446.7326.411.133.5711.189.34E + 328.71E − 101633.715994.890.08954278.5748863.6589.34268.511.04E − 031.33557112.943486.6290.611.163.5911.191.06E + 338.70E − 101639.416022.170.08975277.9349533.8589.51302.161.03E − 031.33567112.983522254.711.183.6111.211.19E + 338.68E − 101647.056053.280.09003277.0750305.5689.41344.381.02E − 031.33577113.0160603552.7218.611.23.6211.221.34E + 338.67E − 101652.806078.060.09023276.4350925.9289.42401.291.00E − 031.33587113.033578.8182.311.223.6411.231.51E + 338.66E − 101656.656096.440.09037276.0151389.2889.53481.799.89E − 041.33597113.0561103600.314611.233.6511.241.69E + 338.65E − 101660.506112.140.09051275.5951787.3389.48601.259.75E − 041.33607113.063617109.611.243.6511.261.89E + 338.65E − 101662.436121.370.09058275.3752022.1689.54801.449.61E − 041.33617113.0761403628.973.111.253.6611.272.11E + 338.64E − 101664.376130.600.09065275.1652257.9789.591202.359.47E − 041.33627113.083636.136.511.263.6611.282.36E + 338.64E − 101666.306139.850.09072274.9552494.7789.652409.479.33E − 041.33637113.0861403638.5011.263.6611.292.63E + 338.64E − 101666.306139.850.09072274.9552494.7789.659.19E − 041.33The Earth’s temperature at a depthz = 50 m is(10)T3z=5μ3z3πR2ℏkBλ3minz3Vp035z4=285.3K,where μ3=0.001942 kg/mol which is the average molar mass of the Earth’s substance (Table 1);λ3min=2πℏVp03/TD_GRkB = 1.5254·10−9 m which is the minimum P-wavelength, TD_GR = 364.44 K which is the average Debye temperature of the geological environment (Table 1); and Vp03 = 1850 m/s which is the average speed of the P-wave at a depth of 50 m.The temperature of the Earth’s core,z = 6371 km, is(11)T3z=5μ3z3πR2ℏkВλminz3VP5z4=6128.7K.In addition,TD_Fe=T0D_Feexpκ/qρ0Fe/ρmaxq=478.46⋅exp1.4⋅7.868/13.29 = 1096.68 K which is Debye temperature of iron in the core (Table 1); κ=1.4 which is the first parameter of Gruneisen; q=1 which is the second parameter of Gruneisen; λ3min = 7.368·10−10 m is the minimum P-wavelength in the nucleus; ρ0Fe = 7.868 which is the density of iron [18], g/m3; and Vpz=6371km = 11260 m/s which is the average velocity of the P-wave in the nucleus [24].Figure3 shows the actual characteristics in the GE: (a) distribution of temperature and velocity of the P-wave; (b) the distribution of the density of elastic energy ρzi⋅Vp2zi, dotted lines, defined by seismic data.Figure 3
Characteristics: (a) the average velocityVz of the P-wave and temperature T (z); (b) volume density ρVp2z, no elastic and thermal energy ρVp2T, according to equation (6) in the GM (geological medium). According to the VSP (vertical seismic profiling) and thermometry, geophysical studies of well number 10 of Sentyanivska Square.
(a)(b)Figure4 shows the characteristics of the temperature distribution in the Earth’s interior, determined by the theory of elasticity, 1 (Table 2), and the method of the energy analysis, 2 by equation (7), where λminz=ℏVs−inVp−in/TDkB12π2/Vs−in3+Vp−in33=2πℏVPz/TDkBc,Vs−in3=Vp−in;c=0.5⋅12/π1+333for a homogeneous isotropic medium, TDz=TD0expκ/qρ3z/ρmaxq,k=1.52,q=1,TD0=364.44K;VPz=VPG3 according to (9); μz=μ0ρ3z/ρ30, μ0=0.01942g/mol, ρ30=2.85g/m3, ρ3z, Table 2.Figure 4
Characteristics of the temperature distribution in the Earth are determined by 1, authors in [22], and 2, by equations (7), and 3, by equation (7), for average density ρz=ρ0+χ⋅z.In Table2 for the calculation of gravitational energy at z = 0 m, the velocity of the P-wave Vp = 1600 m/s is taken for water. To calculate the parameters λmin (z = 0), TD (z = 0), and TZ (z = 0), the P-wave velocity Vp (z = 0) = 1850 m/s is taken for the crust on the Earth’s surface. For z > 0, the given physical parameters of the Earth are used.


### 2.1. Notes

(1) Important value in these studies isρmVmP2W3=1.8252⋅1033 J which is the elastic energy in the center of the Earth, where W3=4/3πR33=1.08691⋅1021 m3 which is the volume of the Earth; ρm=13290 kg/m3 which is the density in the center of the Earth; VmP=11260 m/s which is the velocity of the P-wave in the center of the Earth; and R3 is the average radius of the Earth, m.The temperature in the Earth is a consequence of its compression by a gravitational field, which proves the characteristics in Figures1–4. If in (7), we substitute the value VPEG with the substitution in (9) of the form EGz/ρfactzVP−fact2z⋅w0=W/w0, we obtain qualitative logarithmic characteristics of the maximum temperature in the center of the nuclei depending on the gravitational energy U (volume W) of the planet (Figure 5).Figure 5
Qualitative logarithmic characteristics of functions (polynomial approximations) of the temperature of the nucleus of the Solar System planets. Depending on their mass:fTmax=lnTmax, according to the known hypothetical values of the temperature of the nucleus;fU, analogous to equation (7).lnТ≡fU=ε⋅ln1+ln1+U⋅1+ln1+U4≃ε⋅ln1+ln1+W⋅1+ln1+W4, on the y-axis, as an analogue of equation (7), where U=6⋅E˜Gz≈EGz, the planet’s own gravitational energy (Table 3), where the column numbers indicate the following: 1 is the name of the planet, 2 is the mass of the planet, 3 is the gravitational energy of the planet, 4 is the solar constant of the planet, 5 is the logarithm of the planet’s mass, 6 is the albedo of the planet, 7 is the total power of endogenous heat flow of the planet, 8 is the night total heat flow on the planet’s surface, 9 is the day total heat flow on the planet’s surface, 10 is the distance of the planet from the Sun, 11 is the absolute average daytime temperature of the planet’s surface, 12 is the absolute average nighttime temperature of the planet’s surface, 13 is the absolute temperature of the planet’s core), J; M is the mass of planets, kg; R is the radius of the planet, m; ε=3.21 is the coefficient of scale of normalization of the characteristic; temperatures of the nuclei of the planets fTmax=lnTmax, where Tmax is the known temperatures of the nuclei of the planets [26], K, and masses of planets lnM, on the abscissa.Table 3
Geophysical parameters of the planets of the Solar System [25].
ObjectМ (kg)U (J)F0∗ (W/m2)ln (М)A (%)ln (J)ln (Nmin)ln (Nmax)R (km)Tmax (K)Tmin (K)Tcore∗∗ (K)12345678910111213Pluto1.30E + 224.51E + 270.8750.920.529.4528.2729.77150048331190Moon7.36E + 221.25E + 29136052.650.1237.1034.7038.771737390951427Mercury3.28E + 231.77E + 30922854.150.14239.6933.2641.462439700902204Mars6.40E + 234.83E + 3058654.820.1737.5936.4638.6433932931703315Venus4.88E + 241.58E + 32258656.850.6740.2343.4843.8260518007355537Land5.98E + 242.25E + 32136057.050.36739.7038.0240.5363783431836273Uranus8.70E + 251.25E + 343.759.730.5136.4735.8137.182430076548315Neptune1.03E + 261.70E + 341.559.900.4135.6236.3037.022505072608400Saturn5.68E + 262.14E + 351561.600.4739.6939.4541.27604001348516650Jupiter1.90E + 272.02E + 3650.362.810.5241.2341.3342.197140015512527760Sonce1.99E + 302.28E + 4169.7769570015000000∗F0 is the solar constant and A is the albedo of the planet [25]. ∗∗Data obtained from [26].In Figure5, the characteristics of the planet’s core temperature functions indicate a qualitative physical relationship between the maximum temperature in the center of the core and the planet’s gravitational energy. The analysis of the characteristic fU points to the hypothesis, a constant temperature in the center of the nuclei of the planets. Provided EGz=ρfactzVP−fact2zW, the characteristic fU=const (approximation fU, f (Tmax) in EXCEL).


### 2.2. Development of Geothermal Model of the Earth with Heat Fluxes

The task is to investigate the nonstationary thermodynamic state of the Earth depending on the power of the internal source of the Earth and the external heat flux from the Sun according to the known temperature distribution in the Earth’s interiorТ3 (z) from Table 2.Equation (2) is considered in the form στz+zmdτz/dz4=λSBdτz/dz+στ4z, the radiant transfer of thermal energy in a unit volume of the geological environment, in which the term τz+zmdτz/dz=Тm has heating to a temperature Тm at an unknown coordinate zm at known parameters z,τz,Тm,T0, namely,(12)zm=−z⋅lnТm−τzТm−T0−1⟶τz=Тm−Тm−T0exp−zzm.ISE is a source of electromagnetic waves, and an energy of which is partially absorbed by matter of the Earth and partly radiated into the open physical space. Simulating the data of physical processes for the Earth, using the results of work [20], we perform a system of equations of the formecmath;(13)σTm4s0dtinternal source of the medium=sizλSBdτzdzdtheat transfer of the medium+smAzτzdtheat transfer of medium space into an open spaceJ,nf=λfdτzdz,nSB=λSBdτzdzW/m2,where z is the coordinate from the Earth’s surface to its center, m; Tm is the radiant source temperature (for example, ISE of internal core), K; σ is the Stefan–Boltzmann coefficient, W/m2/K4; s0=si=sm is the single surface, 1 m2; nSBT0=σТ04 is the heat flux density of infrared waves at temperature T0=288K,Tm=6140K, W/m2; λSB=16n2σ/3αRτ3 is the coefficient of radiation thermal conductivity of the environment [15, 27], W/(m·K); αR=16σ/3n2λSBτ3=16σ/3λfnf/nSBτ3 [28, 29] is the average Rossland coefficient of energy absorption of radiation rays through which the given rays penetrate, m−1; n is the refractive index of the medium by electromagnetic waves; τ is the temperature of the substance of the Earth, K; mz is the mass of environment, and dτ/dz=gradτ is the conductive temperature gradient in the GE, K/m;(14)λfz=∂3U∂τ⋅∂z⋅∂t=∂3Cz⋅mz⋅τz∂τ⋅∂z⋅∂t=Cρ∂3∂τ⋅∂z⋅∂twz⋅τ=8πCρz∂z∂t=8πCGgz⋅w0zdzdt=8πCρzVp=8π3Rμ3Ggz⋅w0λminzVpz,which is coefficient of conductive thermal conductivity of the environment W/(m·K); U is the internal energy of the environment, J; C=3R which is the heat capacity of the environment, J/(mol·K), R=8.314510 J/Mole/K which is the universal gas constant; dz/dt=Vp is the velocity of the P-wave in the environment, m/s; ρ=G/g⋅w0=m/w0 which is the environment density, N/(m/с2)/m3 = kg/m3 (mol/kg/m3); Cρ=3Rμ which is the entropy density of the environment [30], J/(kg·K·m3), where μ is the number of moles in 1 m3 (or 3R/μ3=С=5/T34zπ2ℏ/kBλ3minz3Vp35z for μ3 kg/mol); Az=στ3z which is the coefficient of heat transfer of the surface of the environment, W/m2/K; z=λ30=2πℏVp3/TDkB which is the minimum wavelength of the acoustic phonon in the environment, m, where TD=290K is Debye’s temperature SiOi and Vp3=8000m/s at a depth of 50 km; and s0=4πRC2,siz=4πz2,sm=4πR32. The problem is considered for s0=si=sm=1m2; Vm = 22.4139695 which is the molar volume, l/mol.The solution of the system of equations (13) will be performed for the following initial and boundary conditions:τ0z0=0m=T0=288K and τmzm=6371008m=Tm=6140K.The first equation of the system of equations (13) has [29] an integral function of the form(15)∫dz=λSBσ∫dτzTm4−τ4z+С0⟶z=λSBσ4Tm3lnTm+τzTm−τz+2arctgτzTm+C0.The solution (15) of the system of equations (13) is the exact solution integral function for the dependence of z from the temperature distribution in the depths of the Earth in the model.ForlnTm+τz/Tm−τz≫1/4Tm3arctgτz/Tm, we obtain a simplified analytical solution of the temperature function from depth:(16)expz−C0zσ4Tm3/λSBz−1expz−C0zσ4Tm3/λSBz+1Tm=τz,C0z=z−λSBzσ4Tm3lnTm+τzTm−τz,where C0z, the constant, is defined as the analytical depth for each pair of parameters τz,z; the radial thermal conductivity parameter λSBz of the Earth’s geological environment was determined by the following equation:(17)λSBz=στz+zmdτzdz4−τ4zdτzdz−1≅στz+zmΔτzΔz4−τ4zΔτzΔz−1,τz=Δτzz=Δzzm=1m⟶λSB=στ3zz1+zm1z4−1⟶αRzn2z=16σ3λSBτ3z=163z1+zm1/z4−1.In Figures6–11, the results of research equation (16) are presented. Characteristics of distributions of radiation thermal conductivity λSB and absorption αR are determined by known data on the distribution of Earth’s temperature τz [14] by formulas (17).Figure 6
Characteristics of temperature profiles in the bowels of the Earth, whereτz is constructed by equation (16), and αRz/n2z by equation (17). In [31] for aluminum and iron, the defined parameter was 1.3. αRz is the coefficient of Rosseland absorption of electromagnetic waves by the medium; nz is the refractive index of electromagnetic waves in the medium.Figure 7
Characteristics of the profiles of radialλSBz and conductive λfz thermal permeability of the Earth’s matter from the center to its surface, λSBz_fact is the thermal conductivity of EM waves; λfz_g=var is the conductive thermal conductivity with variable acceleration of gravity; λfz_g=const is the conductive thermal conductivity with constant acceleration of gravity; λSB/λf_g=var is the ratio of the thermal conductivity of EM waves to the conductive thermal conductivity at variable acceleration of gravity.Figure 8
High temperature transport properties of Fe. Thermal conductivity temperature dependence at 112 GPa. 33 experiments obtained model fit and a 20% uncertainty envelope are in blue; matching model without a linear term: dashed line. Present data are solid circles, and data derived from prior electrical resistivity measurements are open symbols (see Figure3). Red band is the minimum thermal conductivity assuming resistivity saturation [32].Figure 9
Characteristics of thermal conductivity of the Earth’s matter for different characteristics of gravitational accelerations:gz=const,z∈0÷R3, gz=var,z∈0÷R3, and heat capacity, determined by different formulas: Cz,μ=3R/μ3; Cz,T,Vp,λmin=5/T34zπ2ℏ/kBλ3minz3Vp35z.Figure 10
Characteristics of the length distribution of acoustic waves and the heat capacity of the Earth’s matter.Figure 11
Characteristics of the values of the temperature of the nucleus, mantle, crust, and Earth’s atmosphere, depending on the time: (a) pure cooling of the Earth; (b) cooling of the Earth with heating from the exogenous heat source; (c) heating-cooling of the Earth from an endogenous source and heating from an exogenous thermal source. (a)N3−F=0NC−3=0, (b) N3−F=0NC−3≠0, and (c) N3−F≠0NC−3≠0.
(a)(b)(c)Figure6 shows the characteristics of the temperature profiles known [22] and determined by equation (7), Tz−teor, and equation (16), as well as the Rosseland coefficient with the refractive index determined by equation (17).Figure7 shows the characteristics of the radial, λSBz, and conductive, λfz, heat transfer profiles from the center of the Earth to its surface, as well as their relationship. Moreover, the parameter λfz is defined for constant acceleration gz=const and variable actual gz=var acceleration of gravity.Experimental laboratory studies of iron conductivity for geobaric conditions of the Earth at a depth of 3000 m are given in the paper [32], where, for conditions 112 GPa, 3000 K, the authors determined the value of heat conductivity 20 ÷ 80 W/m/K, shown in Figure 8.The thermal conductivity in Figure7 has a max value 67.7 W/m/K. The graph of the distribution of the heat content of the Earth’s substance is shown in Figure 9.Figure10 shows the characteristics of the acoustic wavelength distribution and the heat capacity of the Earth’s substance. The frequency of oscillations of elementary objects of the Earth’s substance was determined by the equation ω3z=Vp3z/λ3minz [16]. The characteristic of the frequency of oscillations of atoms and molecules in the propagation of the Earth’s P-waves is shown in Figure 10.The temperature on the Earth’s surface due to the heat flux of the Sun.To build the temperature characteristics of the Earth’s surface, taking into account the thermal energy of the Sun, the authors determined the total thermal conductive power in the atmosphere of sunlight rays and the temperature of their absorbed photons, by the following equation [27]:(18)NC−3=0.5S3n3−CkR=0.5S316σ3TC−33dTAdz=2.766⋅1013W,with density (0.108 W/m2), where TC=273.15+5870 which is the temperature of the Sun surface, K, and dTA/dz is the temperature gradient of the Earth’s atmosphere, K/m (0.0056 K/m) [33]. TC−3=TCRC21/LC−324 = 400 K which is the determined temperature of the solar photons on the Earth’s surface. RC=0.695 million km which is the radius of the Sun; LC−3=149.6 million km which is the average distance from the Sun to Earth; 0.5S3 is the Earth’s surface, m2, which absorbs the thermal energy of the Sun; n3−C=163.3 W/m2 which is the density of the heat flux from the Sun on the Earth’s surface; and kR is an unknown coefficient of absorption of short infrared waves by the Earth’s atmosphere, which can be determined from equation (18).Model of the Earth with exogenous and endogenous radiative heat fluxes represents the model of the process of sequential heating-cooling of the Earth’s layers [34].The authors propose a geothermal model of the Earth (1) in a thermodynamic formulation by a system of equations for successive processes of heating and cooling of its phases (inner core, outer core and mantle, crust, and atmosphere) in the form internal:(19)N3−Fdt=CGdτG+AGτGdt,−internal core;AGτGdt=CMdτM+AMτMdt,−external core+mantle;AMτMdt=CKdτK+AKτKdt,−crust;AKτK+NC−3dt=CAdτA+AAτAdt,−atmosphere;where N3−Fdt is the total thermal energy of ISE, J; NC−3dt is the total thermal energy of electromagnetic waves from the Sun on the Earth’s surface, J; CGdτG is the energy of heating the inner core of the Earth, J; τG is the current excess temperature of the inner core above the temperature of the magma, K; AGτGdt is the energy of cooling the surface of the inner core, J; CMdτM is the energy of heating magma, J; CKdτK is the energy of heating the core, J; τM is the current excess temperature of the magma on its surface above the crust temperature, K; AMτMdt is the cooling energy by a magmatic phase for the heating of the atmosphere, J; τK is the current excess temperature of the crust on the surface of the Earth above the temperature of the atmosphere, K; AKτK+NCdt is the energy that heats the atmosphere from the Earth and the Sun, J; AKτKdt is the energy of cooling the crust, J; CAdτA is the energy of heating the atmosphere, J; τA is the current high temperature of the atmosphere on the Earth’s surface above the temperature of the open physical space, K; AAτAdt is the energy that is devoted by the atmosphere to the open physical space, J; AG=N3−F/TG0 is the coefficient of heat transfer of the surface of the inner core Earth, J/°С/sec; AM=N3−F/T¯M is the coefficient of heat transfer of the surface of the mantle, J/°С/sec; AK=N3−F/T¯K is the coefficient of heat transfer of the Earth’s crust surface, J/°С/sec; AA=N3−F/T¯A is the coefficient of heat transfer of atmosphere, J/°С/sec; CG=сGmG+сMmM+сKmK+сAmA is the entropy of all phase states of the Earth substance, J/°С; CM=сMmM+сKmK+сAmA is the entropy of magma, crust, and atmosphere, J/°С; CK=сKmK+сAmA is the entropy of the crust and atmosphere, J/°С; CA=сAmA is the entropy of the Earth’s atmosphere, J/°С; TG0=6000 is the temperature of the surface of the inner core, °С; T¯K=0.5TK+TK0=0.5⋅1320+10=665 which is the average temperature of the crust between the temperatures of the inner and outer surfaces, °С; T¯M=0.5⋅TR+TM=0.5⋅1320+6000=3660 which is the average temperature of the magmatic phase, °С; T¯A=14.2 which is the average temperature of the Earth’s atmosphere, °С; E3=N3−F⋅t3 which is the total heat energy spent on heating the Earth during its geological existence time, t3; mG=ρG⋅4/3πRG3 which is the mass of internal core, kg; mM=ρM⋅4/3πR3−ΔLK3−RG3 is the mass of outer core and mantle, kg; mK=ρK⋅4/3πR33−R3−ΔLK3 which is the mass of Earth crust, kg; mA=5.3⋅1018 which is the given mass of atmosphere [35, 36], kg; ρG,ρM,ρA are the average densities of the inner core, the outer core and the magma, the crust, and the Earth’s atmosphere, respectively, kg/m3; N3−F=AKτK=η3−f⋅S3=η3−f⋅4πR32 which is the given total thermal power of conductive heating-cooling on the surface of the Earth’s crust, W; η3−f=0.087 which is the average square density of the thermal conductive flux on the surface of the crus, W/m2.The solution of the system of differential equations (19) is a function of the form(20)τGt=TG0−N3−FAGe−ωGt+N3−FAG,τMt=TG0−AGAM⋅τGte−ωMt+τGtAGAM,τKt=TG0−AMAK⋅τMte−ωKt+τMtAMAK,τAt=TG0−NC−3AA−AKAA⋅τKte−ωAt+NC−3AA+AKAA⋅τKt,where ωG=AG⋅CGmG+CMmM+CKmK+CAmA−1;ωM=AM⋅CMmM+CKmK+CAmA−1;ωK=AK⋅CKmK+CAmA−1;ωA=AA⋅CAmA−1,which is the heat exchange frequency, and s−1 is found by the method of successive integration of temperature functions for stationary mode of heat transfer. The solution of the system of differential equations (19) is executed for the given initial and boundary conditions of the stationary regime, given in Table 4.Table 4
Specified initial, boundary, and defined geophysical parameters of the Earth.
Geophysical parametersMarking formulaValueUnit of measureKnown parametersEarth’s existence time (4.5 billion years)t31.42·1017sThe Earth’s own gravitational energyЕ˜GЗz2.23945⋅103JTotal power of radiation ISEN3−SB1.86·1017WTotal power of conductive heating of the EarthN3−F4.44·1013WThe total power of conductive heating of the atmosphere by short infrared waves from the SunNC−32.77·1013WMaximum density in the center of the Earthρ3−m13290kg/m3The average radius of the EarthR36371008.8mAverage density of conductive heat flux on the surface of the lithosphere at the present timeη3−f0.087W/m2Average density of heat flux of infrared waves on the surface of the lithosphere (depth of 50 m) at presentη3−SB364.46W/m2Earth mass (average density 5510 kg/m3)M35.969·1024kgThe mass of the atmospheremA5.3·1018kgAverage atmospheric temperature near the Earth’s surfaceT014.2°СAverage temperature of the lithosphere at a depth of 50 mTK010°СThe surface temperature of the mantle at a depth of 50 kmTK1320°СTemperature of the surface of the inner core of the EarthТG06000°СThe average specific heat of the atmosphereCA1005J/kg/°СDefined parametersThe Earth’s own gravitational energyЕG3z2.65·1033JThe mass of the inner core of the Earth (radius 1455 km, average density 12657.8 kg/m3)mG1.633·1023kgMass of mantle and outer core (average density 5876.4 kg/m3, 1455÷6321 km)mM3.26·1024kgThe mass of crust (thickness 50 km, density 2850 kg/m3)mK8.64·1022kgAverage temperature of the Earth’s magmatic phaseТ¯M3660°KAverage temperature of the solid phase of the EarthТ¯K665°KAverage atmospheric temperature on the Earth’s surfaceТ¯А14.2°KAverage specific heat of internal coreCG275J/kg/°СAverage specific heat of the outer core and mantle of the EarthCM630J/kg/°СAverage specific heat of measlesCK1285J/kg/°СTotal internal energy of heating the EarthE3=t3N3−F6.297·1030JThe total energy dissipated by the Earth in an open physical spaceEE3−SB=t3N3−SB2.635·1034JGeneral gravitational energy of the EarthEGP=UG=0.5E3−С2.661·1033JTotal internal energy of the inner coreEG1.652·1029JTotal internal energy of the mantle and the outer coreEM6.075·1030JTotal internal energy of crustEK8.01·1028JTotal internal energy of the atmosphereEA1.64·1029JThe coefficient of heat transfer on the surface of the inner core to the inner core and mantleAG7.396·109J/s/°СCoefficient of heat transfer of the mantle to its solid phaseAM1.212·1010J/s/°СCoefficient of heat transfer of solid phase to atmosphereAK6.673·1010J/s/°СCoefficient of heat transfer of the atmosphere into spaceAA3.125·1012J/s/°СTemperature of physical spaceTSB−269 (4.15)°С (K)Core cooling frequencyωg3.34·10−18с−1The cooling frequency of the external core and mantleωM5.59·10−18s−1Crust cooling frequencyωK6.00·10−16s−1Atmosphere cooling frequencyωа1.198·10−9s−1


## 2.1. Notes

(1) Important value in these studies isρmVmP2W3=1.8252⋅1033 J which is the elastic energy in the center of the Earth, where W3=4/3πR33=1.08691⋅1021 m3 which is the volume of the Earth; ρm=13290 kg/m3 which is the density in the center of the Earth; VmP=11260 m/s which is the velocity of the P-wave in the center of the Earth; and R3 is the average radius of the Earth, m.The temperature in the Earth is a consequence of its compression by a gravitational field, which proves the characteristics in Figures1–4. If in (7), we substitute the value VPEG with the substitution in (9) of the form EGz/ρfactzVP−fact2z⋅w0=W/w0, we obtain qualitative logarithmic characteristics of the maximum temperature in the center of the nuclei depending on the gravitational energy U (volume W) of the planet (Figure 5).Figure 5
Qualitative logarithmic characteristics of functions (polynomial approximations) of the temperature of the nucleus of the Solar System planets. Depending on their mass:fTmax=lnTmax, according to the known hypothetical values of the temperature of the nucleus;fU, analogous to equation (7).lnТ≡fU=ε⋅ln1+ln1+U⋅1+ln1+U4≃ε⋅ln1+ln1+W⋅1+ln1+W4, on the y-axis, as an analogue of equation (7), where U=6⋅E˜Gz≈EGz, the planet’s own gravitational energy (Table 3), where the column numbers indicate the following: 1 is the name of the planet, 2 is the mass of the planet, 3 is the gravitational energy of the planet, 4 is the solar constant of the planet, 5 is the logarithm of the planet’s mass, 6 is the albedo of the planet, 7 is the total power of endogenous heat flow of the planet, 8 is the night total heat flow on the planet’s surface, 9 is the day total heat flow on the planet’s surface, 10 is the distance of the planet from the Sun, 11 is the absolute average daytime temperature of the planet’s surface, 12 is the absolute average nighttime temperature of the planet’s surface, 13 is the absolute temperature of the planet’s core), J; M is the mass of planets, kg; R is the radius of the planet, m; ε=3.21 is the coefficient of scale of normalization of the characteristic; temperatures of the nuclei of the planets fTmax=lnTmax, where Tmax is the known temperatures of the nuclei of the planets [26], K, and masses of planets lnM, on the abscissa.Table 3
Geophysical parameters of the planets of the Solar System [25].
ObjectМ (kg)U (J)F0∗ (W/m2)ln (М)A (%)ln (J)ln (Nmin)ln (Nmax)R (km)Tmax (K)Tmin (K)Tcore∗∗ (K)12345678910111213Pluto1.30E + 224.51E + 270.8750.920.529.4528.2729.77150048331190Moon7.36E + 221.25E + 29136052.650.1237.1034.7038.771737390951427Mercury3.28E + 231.77E + 30922854.150.14239.6933.2641.462439700902204Mars6.40E + 234.83E + 3058654.820.1737.5936.4638.6433932931703315Venus4.88E + 241.58E + 32258656.850.6740.2343.4843.8260518007355537Land5.98E + 242.25E + 32136057.050.36739.7038.0240.5363783431836273Uranus8.70E + 251.25E + 343.759.730.5136.4735.8137.182430076548315Neptune1.03E + 261.70E + 341.559.900.4135.6236.3037.022505072608400Saturn5.68E + 262.14E + 351561.600.4739.6939.4541.27604001348516650Jupiter1.90E + 272.02E + 3650.362.810.5241.2341.3342.197140015512527760Sonce1.99E + 302.28E + 4169.7769570015000000∗F0 is the solar constant and A is the albedo of the planet [25]. ∗∗Data obtained from [26].In Figure5, the characteristics of the planet’s core temperature functions indicate a qualitative physical relationship between the maximum temperature in the center of the core and the planet’s gravitational energy. The analysis of the characteristic fU points to the hypothesis, a constant temperature in the center of the nuclei of the planets. Provided EGz=ρfactzVP−fact2zW, the characteristic fU=const (approximation fU, f (Tmax) in EXCEL).


## 2.2. Development of Geothermal Model of the Earth with Heat Fluxes

The task is to investigate the nonstationary thermodynamic state of the Earth depending on the power of the internal source of the Earth and the external heat flux from the Sun according to the known temperature distribution in the Earth’s interiorТ3 (z) from Table 2.Equation (2) is considered in the form στz+zmdτz/dz4=λSBdτz/dz+στ4z, the radiant transfer of thermal energy in a unit volume of the geological environment, in which the term τz+zmdτz/dz=Тm has heating to a temperature Тm at an unknown coordinate zm at known parameters z,τz,Тm,T0, namely,(12)zm=−z⋅lnТm−τzТm−T0−1⟶τz=Тm−Тm−T0exp−zzm.ISE is a source of electromagnetic waves, and an energy of which is partially absorbed by matter of the Earth and partly radiated into the open physical space. Simulating the data of physical processes for the Earth, using the results of work [20], we perform a system of equations of the formecmath;(13)σTm4s0dtinternal source of the medium=sizλSBdτzdzdtheat transfer of the medium+smAzτzdtheat transfer of medium space into an open spaceJ,nf=λfdτzdz,nSB=λSBdτzdzW/m2,where z is the coordinate from the Earth’s surface to its center, m; Tm is the radiant source temperature (for example, ISE of internal core), K; σ is the Stefan–Boltzmann coefficient, W/m2/K4; s0=si=sm is the single surface, 1 m2; nSBT0=σТ04 is the heat flux density of infrared waves at temperature T0=288K,Tm=6140K, W/m2; λSB=16n2σ/3αRτ3 is the coefficient of radiation thermal conductivity of the environment [15, 27], W/(m·K); αR=16σ/3n2λSBτ3=16σ/3λfnf/nSBτ3 [28, 29] is the average Rossland coefficient of energy absorption of radiation rays through which the given rays penetrate, m−1; n is the refractive index of the medium by electromagnetic waves; τ is the temperature of the substance of the Earth, K; mz is the mass of environment, and dτ/dz=gradτ is the conductive temperature gradient in the GE, K/m;(14)λfz=∂3U∂τ⋅∂z⋅∂t=∂3Cz⋅mz⋅τz∂τ⋅∂z⋅∂t=Cρ∂3∂τ⋅∂z⋅∂twz⋅τ=8πCρz∂z∂t=8πCGgz⋅w0zdzdt=8πCρzVp=8π3Rμ3Ggz⋅w0λminzVpz,which is coefficient of conductive thermal conductivity of the environment W/(m·K); U is the internal energy of the environment, J; C=3R which is the heat capacity of the environment, J/(mol·K), R=8.314510 J/Mole/K which is the universal gas constant; dz/dt=Vp is the velocity of the P-wave in the environment, m/s; ρ=G/g⋅w0=m/w0 which is the environment density, N/(m/с2)/m3 = kg/m3 (mol/kg/m3); Cρ=3Rμ which is the entropy density of the environment [30], J/(kg·K·m3), where μ is the number of moles in 1 m3 (or 3R/μ3=С=5/T34zπ2ℏ/kBλ3minz3Vp35z for μ3 kg/mol); Az=στ3z which is the coefficient of heat transfer of the surface of the environment, W/m2/K; z=λ30=2πℏVp3/TDkB which is the minimum wavelength of the acoustic phonon in the environment, m, where TD=290K is Debye’s temperature SiOi and Vp3=8000m/s at a depth of 50 km; and s0=4πRC2,siz=4πz2,sm=4πR32. The problem is considered for s0=si=sm=1m2; Vm = 22.4139695 which is the molar volume, l/mol.The solution of the system of equations (13) will be performed for the following initial and boundary conditions:τ0z0=0m=T0=288K and τmzm=6371008m=Tm=6140K.The first equation of the system of equations (13) has [29] an integral function of the form(15)∫dz=λSBσ∫dτzTm4−τ4z+С0⟶z=λSBσ4Tm3lnTm+τzTm−τz+2arctgτzTm+C0.The solution (15) of the system of equations (13) is the exact solution integral function for the dependence of z from the temperature distribution in the depths of the Earth in the model.ForlnTm+τz/Tm−τz≫1/4Tm3arctgτz/Tm, we obtain a simplified analytical solution of the temperature function from depth:(16)expz−C0zσ4Tm3/λSBz−1expz−C0zσ4Tm3/λSBz+1Tm=τz,C0z=z−λSBzσ4Tm3lnTm+τzTm−τz,where C0z, the constant, is defined as the analytical depth for each pair of parameters τz,z; the radial thermal conductivity parameter λSBz of the Earth’s geological environment was determined by the following equation:(17)λSBz=στz+zmdτzdz4−τ4zdτzdz−1≅στz+zmΔτzΔz4−τ4zΔτzΔz−1,τz=Δτzz=Δzzm=1m⟶λSB=στ3zz1+zm1z4−1⟶αRzn2z=16σ3λSBτ3z=163z1+zm1/z4−1.In Figures6–11, the results of research equation (16) are presented. Characteristics of distributions of radiation thermal conductivity λSB and absorption αR are determined by known data on the distribution of Earth’s temperature τz [14] by formulas (17).Figure 6
Characteristics of temperature profiles in the bowels of the Earth, whereτz is constructed by equation (16), and αRz/n2z by equation (17). In [31] for aluminum and iron, the defined parameter was 1.3. αRz is the coefficient of Rosseland absorption of electromagnetic waves by the medium; nz is the refractive index of electromagnetic waves in the medium.Figure 7
Characteristics of the profiles of radialλSBz and conductive λfz thermal permeability of the Earth’s matter from the center to its surface, λSBz_fact is the thermal conductivity of EM waves; λfz_g=var is the conductive thermal conductivity with variable acceleration of gravity; λfz_g=const is the conductive thermal conductivity with constant acceleration of gravity; λSB/λf_g=var is the ratio of the thermal conductivity of EM waves to the conductive thermal conductivity at variable acceleration of gravity.Figure 8
High temperature transport properties of Fe. Thermal conductivity temperature dependence at 112 GPa. 33 experiments obtained model fit and a 20% uncertainty envelope are in blue; matching model without a linear term: dashed line. Present data are solid circles, and data derived from prior electrical resistivity measurements are open symbols (see Figure3). Red band is the minimum thermal conductivity assuming resistivity saturation [32].Figure 9
Characteristics of thermal conductivity of the Earth’s matter for different characteristics of gravitational accelerations:gz=const,z∈0÷R3, gz=var,z∈0÷R3, and heat capacity, determined by different formulas: Cz,μ=3R/μ3; Cz,T,Vp,λmin=5/T34zπ2ℏ/kBλ3minz3Vp35z.Figure 10
Characteristics of the length distribution of acoustic waves and the heat capacity of the Earth’s matter.Figure 11
Characteristics of the values of the temperature of the nucleus, mantle, crust, and Earth’s atmosphere, depending on the time: (a) pure cooling of the Earth; (b) cooling of the Earth with heating from the exogenous heat source; (c) heating-cooling of the Earth from an endogenous source and heating from an exogenous thermal source. (a)N3−F=0NC−3=0, (b) N3−F=0NC−3≠0, and (c) N3−F≠0NC−3≠0.
(a)(b)(c)Figure6 shows the characteristics of the temperature profiles known [22] and determined by equation (7), Tz−teor, and equation (16), as well as the Rosseland coefficient with the refractive index determined by equation (17).Figure7 shows the characteristics of the radial, λSBz, and conductive, λfz, heat transfer profiles from the center of the Earth to its surface, as well as their relationship. Moreover, the parameter λfz is defined for constant acceleration gz=const and variable actual gz=var acceleration of gravity.Experimental laboratory studies of iron conductivity for geobaric conditions of the Earth at a depth of 3000 m are given in the paper [32], where, for conditions 112 GPa, 3000 K, the authors determined the value of heat conductivity 20 ÷ 80 W/m/K, shown in Figure 8.The thermal conductivity in Figure7 has a max value 67.7 W/m/K. The graph of the distribution of the heat content of the Earth’s substance is shown in Figure 9.Figure10 shows the characteristics of the acoustic wavelength distribution and the heat capacity of the Earth’s substance. The frequency of oscillations of elementary objects of the Earth’s substance was determined by the equation ω3z=Vp3z/λ3minz [16]. The characteristic of the frequency of oscillations of atoms and molecules in the propagation of the Earth’s P-waves is shown in Figure 10.The temperature on the Earth’s surface due to the heat flux of the Sun.To build the temperature characteristics of the Earth’s surface, taking into account the thermal energy of the Sun, the authors determined the total thermal conductive power in the atmosphere of sunlight rays and the temperature of their absorbed photons, by the following equation [27]:(18)NC−3=0.5S3n3−CkR=0.5S316σ3TC−33dTAdz=2.766⋅1013W,with density (0.108 W/m2), where TC=273.15+5870 which is the temperature of the Sun surface, K, and dTA/dz is the temperature gradient of the Earth’s atmosphere, K/m (0.0056 K/m) [33]. TC−3=TCRC21/LC−324 = 400 K which is the determined temperature of the solar photons on the Earth’s surface. RC=0.695 million km which is the radius of the Sun; LC−3=149.6 million km which is the average distance from the Sun to Earth; 0.5S3 is the Earth’s surface, m2, which absorbs the thermal energy of the Sun; n3−C=163.3 W/m2 which is the density of the heat flux from the Sun on the Earth’s surface; and kR is an unknown coefficient of absorption of short infrared waves by the Earth’s atmosphere, which can be determined from equation (18).Model of the Earth with exogenous and endogenous radiative heat fluxes represents the model of the process of sequential heating-cooling of the Earth’s layers [34].The authors propose a geothermal model of the Earth (1) in a thermodynamic formulation by a system of equations for successive processes of heating and cooling of its phases (inner core, outer core and mantle, crust, and atmosphere) in the form internal:(19)N3−Fdt=CGdτG+AGτGdt,−internal core;AGτGdt=CMdτM+AMτMdt,−external core+mantle;AMτMdt=CKdτK+AKτKdt,−crust;AKτK+NC−3dt=CAdτA+AAτAdt,−atmosphere;where N3−Fdt is the total thermal energy of ISE, J; NC−3dt is the total thermal energy of electromagnetic waves from the Sun on the Earth’s surface, J; CGdτG is the energy of heating the inner core of the Earth, J; τG is the current excess temperature of the inner core above the temperature of the magma, K; AGτGdt is the energy of cooling the surface of the inner core, J; CMdτM is the energy of heating magma, J; CKdτK is the energy of heating the core, J; τM is the current excess temperature of the magma on its surface above the crust temperature, K; AMτMdt is the cooling energy by a magmatic phase for the heating of the atmosphere, J; τK is the current excess temperature of the crust on the surface of the Earth above the temperature of the atmosphere, K; AKτK+NCdt is the energy that heats the atmosphere from the Earth and the Sun, J; AKτKdt is the energy of cooling the crust, J; CAdτA is the energy of heating the atmosphere, J; τA is the current high temperature of the atmosphere on the Earth’s surface above the temperature of the open physical space, K; AAτAdt is the energy that is devoted by the atmosphere to the open physical space, J; AG=N3−F/TG0 is the coefficient of heat transfer of the surface of the inner core Earth, J/°С/sec; AM=N3−F/T¯M is the coefficient of heat transfer of the surface of the mantle, J/°С/sec; AK=N3−F/T¯K is the coefficient of heat transfer of the Earth’s crust surface, J/°С/sec; AA=N3−F/T¯A is the coefficient of heat transfer of atmosphere, J/°С/sec; CG=сGmG+сMmM+сKmK+сAmA is the entropy of all phase states of the Earth substance, J/°С; CM=сMmM+сKmK+сAmA is the entropy of magma, crust, and atmosphere, J/°С; CK=сKmK+сAmA is the entropy of the crust and atmosphere, J/°С; CA=сAmA is the entropy of the Earth’s atmosphere, J/°С; TG0=6000 is the temperature of the surface of the inner core, °С; T¯K=0.5TK+TK0=0.5⋅1320+10=665 which is the average temperature of the crust between the temperatures of the inner and outer surfaces, °С; T¯M=0.5⋅TR+TM=0.5⋅1320+6000=3660 which is the average temperature of the magmatic phase, °С; T¯A=14.2 which is the average temperature of the Earth’s atmosphere, °С; E3=N3−F⋅t3 which is the total heat energy spent on heating the Earth during its geological existence time, t3; mG=ρG⋅4/3πRG3 which is the mass of internal core, kg; mM=ρM⋅4/3πR3−ΔLK3−RG3 is the mass of outer core and mantle, kg; mK=ρK⋅4/3πR33−R3−ΔLK3 which is the mass of Earth crust, kg; mA=5.3⋅1018 which is the given mass of atmosphere [35, 36], kg; ρG,ρM,ρA are the average densities of the inner core, the outer core and the magma, the crust, and the Earth’s atmosphere, respectively, kg/m3; N3−F=AKτK=η3−f⋅S3=η3−f⋅4πR32 which is the given total thermal power of conductive heating-cooling on the surface of the Earth’s crust, W; η3−f=0.087 which is the average square density of the thermal conductive flux on the surface of the crus, W/m2.The solution of the system of differential equations (19) is a function of the form(20)τGt=TG0−N3−FAGe−ωGt+N3−FAG,τMt=TG0−AGAM⋅τGte−ωMt+τGtAGAM,τKt=TG0−AMAK⋅τMte−ωKt+τMtAMAK,τAt=TG0−NC−3AA−AKAA⋅τKte−ωAt+NC−3AA+AKAA⋅τKt,where ωG=AG⋅CGmG+CMmM+CKmK+CAmA−1;ωM=AM⋅CMmM+CKmK+CAmA−1;ωK=AK⋅CKmK+CAmA−1;ωA=AA⋅CAmA−1,which is the heat exchange frequency, and s−1 is found by the method of successive integration of temperature functions for stationary mode of heat transfer. The solution of the system of differential equations (19) is executed for the given initial and boundary conditions of the stationary regime, given in Table 4.Table 4
Specified initial, boundary, and defined geophysical parameters of the Earth.
Geophysical parametersMarking formulaValueUnit of measureKnown parametersEarth’s existence time (4.5 billion years)t31.42·1017sThe Earth’s own gravitational energyЕ˜GЗz2.23945⋅103JTotal power of radiation ISEN3−SB1.86·1017WTotal power of conductive heating of the EarthN3−F4.44·1013WThe total power of conductive heating of the atmosphere by short infrared waves from the SunNC−32.77·1013WMaximum density in the center of the Earthρ3−m13290kg/m3The average radius of the EarthR36371008.8mAverage density of conductive heat flux on the surface of the lithosphere at the present timeη3−f0.087W/m2Average density of heat flux of infrared waves on the surface of the lithosphere (depth of 50 m) at presentη3−SB364.46W/m2Earth mass (average density 5510 kg/m3)M35.969·1024kgThe mass of the atmospheremA5.3·1018kgAverage atmospheric temperature near the Earth’s surfaceT014.2°СAverage temperature of the lithosphere at a depth of 50 mTK010°СThe surface temperature of the mantle at a depth of 50 kmTK1320°СTemperature of the surface of the inner core of the EarthТG06000°СThe average specific heat of the atmosphereCA1005J/kg/°СDefined parametersThe Earth’s own gravitational energyЕG3z2.65·1033JThe mass of the inner core of the Earth (radius 1455 km, average density 12657.8 kg/m3)mG1.633·1023kgMass of mantle and outer core (average density 5876.4 kg/m3, 1455÷6321 km)mM3.26·1024kgThe mass of crust (thickness 50 km, density 2850 kg/m3)mK8.64·1022kgAverage temperature of the Earth’s magmatic phaseТ¯M3660°KAverage temperature of the solid phase of the EarthТ¯K665°KAverage atmospheric temperature on the Earth’s surfaceТ¯А14.2°KAverage specific heat of internal coreCG275J/kg/°СAverage specific heat of the outer core and mantle of the EarthCM630J/kg/°СAverage specific heat of measlesCK1285J/kg/°СTotal internal energy of heating the EarthE3=t3N3−F6.297·1030JThe total energy dissipated by the Earth in an open physical spaceEE3−SB=t3N3−SB2.635·1034JGeneral gravitational energy of the EarthEGP=UG=0.5E3−С2.661·1033JTotal internal energy of the inner coreEG1.652·1029JTotal internal energy of the mantle and the outer coreEM6.075·1030JTotal internal energy of crustEK8.01·1028JTotal internal energy of the atmosphereEA1.64·1029JThe coefficient of heat transfer on the surface of the inner core to the inner core and mantleAG7.396·109J/s/°СCoefficient of heat transfer of the mantle to its solid phaseAM1.212·1010J/s/°СCoefficient of heat transfer of solid phase to atmosphereAK6.673·1010J/s/°СCoefficient of heat transfer of the atmosphere into spaceAA3.125·1012J/s/°СTemperature of physical spaceTSB−269 (4.15)°С (K)Core cooling frequencyωg3.34·10−18с−1The cooling frequency of the external core and mantleωM5.59·10−18s−1Crust cooling frequencyωK6.00·10−16s−1Atmosphere cooling frequencyωа1.198·10−9s−1


## 3. Results and Discussion

The investigation of the geothermal model of the Earth with endogenous and exogenous heat fluxes by equations (20) is fulfilled for the following conditions:(1)
Lack of ISE and heat flux from the Sun (pure cooling)(2)
Lack of ISE and the presence of heat flux from the Sun (cooling the Earth and heating the atmosphere and the Earth’s surface from the Sun)(3)
Presence of ISE and heat flux from the Sun (real-heating-cooling)(4)
The individual influence of random variations in the increase of the activity of the Sun and the reduction of the heat transfer of the atmosphere to 50%Substitution of values from Table4 into the system of equations (20) allowed to construct temporal characteristics of the temperature values of the inner core, Tgt, the inner core and the mantle, the crust, and the Earth’s atmosphere, which are shown in Figure 11.The value of temperature on the Earth’s surface, as shown in Figure12, shows the individual influence of arbitrary variations (increase) in the solar power and (decrease) the heat transfer of the atmosphere by 50%.Figure 12
Characteristics of the values of atmospheric temperature, depending on the increase in the activity of the Sun and the accidental decrease in the heat transfer of the atmosphere to 50%: (a) variations in the increase of the heat flux of the Sun on the Earth’s surface to 50%; (b) variations in the reduction of the heat transfer of the atmosphere to 50%.
(a)(b)


### 3.1. Analysis of Results

Function of deterministic probability (FDP) also suggests a hypothesis about ISE [12, 37], which is based on two principles: first, there is a joint action of the laws of conservation, modification, transfer, and packaging of energy in a physical point (PP); second, there is physical space, as a system of physical points (SPP), at each point of which the joint action of the laws of conservation, modification, transfer, and packaging of energy is carried out.The dimensionless FDP of space is represented in the following form:(21)E=K+U=A2x,KUE2=lnA−2x,where Еx/E0=A2x is the value of the total energy at a point, which is distributed in space, х, of a particular physical system with given energy E0=const; K=Kx/E0. U=Ux/E0 is the relative units of kinetic and potential energies at a point that are also distributed in the physical system (furthermore, we accept const=1).The system of equations (21) defines the values of potential and kinetic energy at each point of the physical system by the formulas(22)Ux=A2x⋅12±14−ln1A2x,Kx=A2x⋅12∓14−ln1A2x.For the Solar System, the energy characteristics are shown in Figures13 and 14.Figure 13
Characteristics of the function of the determined probability (FDP), planet Earth, and planets of the Solar System, as systems of physical points: (a) physical parameters of the Earth; (b) the energy characteristics of the FDP states of internal and external points of the physical system, depending on the ratioU/E where K is the kinetic (calorific) energy of the system, U is a potential (elastic) energy of the system, E is the total energy of the system, KU is the transfer of the energy invariant in the system, KU/E2 = Ψ2 is the phase of the energy state of the system, E0 = 1 is the initial given energy in the system of physical points; F > 0 is the gravitational compression force; F < 0 is the gravitational tensile force.Figure 14
Characteristics of energy correlations of planets of the Solar System by formulasξ=e−ψ2,ψ2=KU/E2,ξ=Е/E0.According to Zone III, where the external influence allows an increase in potential or kinetic energy, the Earth’s own gravitational potential energy is determined from the system of equations (22).


### 3.2. Analysis of Energy Relations for Planets of the Solar System

Figure14 testifies that (а) all planets of the Solar System are located at point B in Figure 13 (energy packaging law); (b) the total energy of each planet is determined by the system of equations (21) in its center and in orbit (24); and (c) the own total gravitational energy of each planet is divided into potential gravitational and thermoelastic in Zones I and II.Example 1.
(1) SPP-«Sun-Earth», (2) SPP-«Earth».(23)EC−3=γMCm3LC−3=5.3⋅1033J;K3=m3V22=0.5EC−3=2.65⋅1033J;U3=EC−3−K3=0.5EC−3,EG=K˜3=U˜3;E˜3=K˜3+U˜3=U3=2.65⋅1033J.
The Earth’s own gravitational total energy provided byρ=ρ0+χ⋅z is determined by the integral: U3=2.65⋅1033J.
From Figure13, it is evident that the internal kinetic (thermal) energy of the planet reaches the total gravitational energy in the center, and the potential energy disappears, as is the acceleration of gravity.
The geothermal model of the Earth with endogenous and exogenous heat fluxes suggests an increase in the temperature of the bowels of the planet with an unchanging total gravitational energy with increasing depth, as well as the existence of the following geophysical sequential mechanisms triggered by the ISE thermal density when approaching the center of the Earth.
The boundary of Zone III and the whole Zone II:(1)
The existence of a solid lithosphere on the Earth’s surface.(2)
Destruction-formation of a solid body of the lithosphere on the border with the mantle.(3)
Formation of the liquid phase of the Earth from different atoms and molecules, various compounds from chemical elements.(4)
Dissociation and synthesis of atoms and molecules. Atoms of hydrogen are destroyed by the latter with increasing depth, after which sharply increases the density of matter, and the nuclei of light chemical elements (the middle of the radius of the Earth) accumulate. The nucleus of heavier elements moves to Zone I.(5)
Dissociation and synthesis of the nucleus of chemical elements (the latter, closer to the center of the Earth, the nuclei of iron break down into protons and neutrons).(6)
At the center of the Earth on the surface of a sphere with a radius of proton, the density of thermal energy reachesηP=η3−f−maxR32/RP2=0.35⋅63710002/8.78⋅10−162=1.84287⋅1043 W/m2, and for the destruction of the proton, density of thermal energy on its surface should be equal to 1.5923·1043 W/m2 (the corresponding temperature of electromagnetic waves 7.26212·1012 K, and the frequency of oscillations of the De Broglie wave of the proton 2.27766·1023 Hz [20]). The destroyed proton will create a density of heat flux 0.3 W/m2 on the Earth’s surface. That is, we can assume that physical conditions in the center of the Earth’s core for the destruction of protons and neutrons exist. The maximum density of the heat flux recorded on the Earth’s surface is η3−f−max=0.35 W/m2 [38]:(24)E=γMmR,K=mV22,U=E−K,EK=γM/RV2/2=2γMω2R3.


### 3.3. Analysis of Results

Formulas (2), (4), and (5) are proposed by the author Karpenko for modeling the energy equilibrium of the joint action of heat fluxes of conductive and radiant EM waves in a geological environment.Formula (5) for the input data provides the following solutions of temperature:(25)T3=273.15+10=283.15K;λf=3.5W/m⋅K;ε=1;z0=1m;αR=1m−1;a=λfz02⋅αR⋅ε⋅σ=617283950.06K3;b=11050537213K4;σ≃5.6704×10−8W/m2⋅K4;⟹T˜1=283.15K;T˜2=211.331K;T˜3=−247.24−j351.24K;T˜4=−247.24+j351.24K;⟹T˜1+T˜2=T˜3+T˜4,where the temperature T˜1=283.15K characterizes the heating temperature of the geological environment by electromagnetic IR waves, which have an absorption coefficient αR=1m−1. Temperatures T˜2,T˜3,T˜4 provide additional physical information, but the interpretation of these temperatures is currently unknown and requires further research.Formula (6) is proposed by the author Karpenko to identify the energy density of Debye theory and elasticity theory, which provides important geophysical information about the physical relationships of elastic and thermodynamic parameters of the geological environment with the kinematic parameters of the P-wave and thermometry parameters obtained in terrestrial and downhole geophysics.Formula (7) is derived from formula (6) for physical identification: T3EG=TDEG. As a result, we obtain information about the frequency of oscillations Ωmax of the thermal field of the geological environment, which transmits infrared waves from the depths of the Earth to the Earth’s surface and is observed by satellite equipment. And seismic prospecting information about Vs−in,Vp−in, allows you to determine the parameter λmin. Known parameters Ωmax and λmin allow from equation (7) to determine the depth parameters μ3,ρ of the geological environment.Formula (8), obtained by the author Karpenko, provides information about the increasing temperature of the geological environment from the growth of gravitational (total) energy of the geological environment of the Earth depending on the depth, which is physically related to the average P-wave velocity at a given depth. This information is the basis for determining the total thermal capacity of a geothermal well to select the stable operation of a geothermal station by the technology of an isolated single well.A new value of the Earth’s own gravitational energy is 2.6275·1033 J, which is an order of magnitude greater than the known value 2.23945·1032 J, and which is equal to the kinetic energy of the Earth’s motion in orbit around the Sun, as proved by formula (24). An important parameter in formula (8) is the parameter σ=R3−zm/R3, which indicates the empty space in the center of mass of the Earth.Formula (9) is obtained by the author Karpenko in [16].The characteristicfU in Figure 5 indicates that the maximum temperature in the center of mass of the planets has the same value, which suggests the hypothesis of a single mechanism for converting the total gravitational energy of the planet into its thermal energy, which depends on the mass of the planet. This hypothesis is tested by equation (20), which takes into account the effective capacities of endogenous and exogenous heat fluxes on the Earth’s surface and shows that the presence of a heat source in the center of the Earth allows to maintain a temperature of 6000°C.The formulalnT=fU=ε1+ln1+U⋅1+ln1+U4≃1+ln1+W⋅1+ln1+W4 is the identification formula from equation (9), which is used to study the internal maximum temperatures of the planets of the Solar System depending on their own gravitational energy, as shown by the characteristic fTmax=lnTmax in Figure 5. This characteristic suggests that the center of mass of each planet’s maximum temperatures is almost identical. This result suggests the following hypothesis: “In the center of mass of each space object, the gravitational energy is converted into thermal energy of EM waves, the power of which depends on the mass of the object, and the maximum temperature in the center of mass of objects is the same.”The formulasλfz=∂3U/∂τ⋅∂z⋅∂t and Cz=5/T34zπ2ℏ/kBλ3minz3Vp35z are obtained by the author Karpenko.Figures6–10 using solution (16) of the system of equations (13) and (17) obtained from equation (2) and known experimental geophysical data on the temperature distribution in the center of the Earth show the characteristics of the distribution in the middle. The Earth has its thermodynamic parameters: λfz, λSBz, ω3z, λ3minz, Cz, αRz/n2z.In the system of equations (19), first developed by the author Karpenko in [38], for heat transfer processes in time in multilayer media, the author investigates the thermodynamic state of the Earth in the presence and absence of a heat source in the center of the Earth and on its surface from the Sun. Studies of equation (20) have shown that the presence of an internal heat source in the center of the Earth provides a stable surface temperature of the inner core at the level of 6000°C. And the change in heat transfer of the atmosphere significantly affects the average temperature of the Earth’s surface of Figure 12(a), and to a greater extent than the fluctuations of heat from the Sun in Figure 12(b).The system of equations (21), first developed by the authors in [34, 37, 39], models the infinite physical space of Gauss, in which the laws act together and simultaneously: conservation, change, transfer, and packaging of energy. From this space, various micro- and macrophysical objects are formed, in particular, space objects, which have an internal energy structure, which is shown in Figure 13 with identification to the various geophysical parameters of the Earth. The energy structure can explain the jumps, fractures, and the very characteristics of the distribution of geophysical parameters in the middle of the Earth. The main consequence of the energy analysis of gravitational energy in the middle of the Earth is the following hypothesis: “the increase in the body’s own gravitational energy, E, from the surface to the center of mass, masses (from 3000 m to 6378 m) of the inner masses of the electronic shells of molecules, atoms, nuclei of physicochemical elements, and quark shells of free protons and neutrons in the center of the masses, releasing their internal energy.This hypothesis explains the origin and dissociation of nuclei, physicochemical atoms, molecules, elements, and substances of the lithosphere, and the generation of endogenous heat by planets and stars, accompanied by a change in the gravitational force from the change in potential energy by the depth of compression by internal masses (to a depth of 3000 m) to the gravitational tensile force by external masses (from 3000 m to 6378 m) of internal masses of electronic shells of molecules, atoms, nuclei of physicochemical elements, and quark shells of free protons and neutrons in the very center of the masses, releasing their inner energy.”This hypothesis explains the origin and dissociation of nuclei, physicochemical atoms, molecules, elements, and substances of the lithosphere, and the generation of endogenous heat by planets and stars.


## 3.1. Analysis of Results

Function of deterministic probability (FDP) also suggests a hypothesis about ISE [12, 37], which is based on two principles: first, there is a joint action of the laws of conservation, modification, transfer, and packaging of energy in a physical point (PP); second, there is physical space, as a system of physical points (SPP), at each point of which the joint action of the laws of conservation, modification, transfer, and packaging of energy is carried out.The dimensionless FDP of space is represented in the following form:(21)E=K+U=A2x,KUE2=lnA−2x,where Еx/E0=A2x is the value of the total energy at a point, which is distributed in space, х, of a particular physical system with given energy E0=const; K=Kx/E0. U=Ux/E0 is the relative units of kinetic and potential energies at a point that are also distributed in the physical system (furthermore, we accept const=1).The system of equations (21) defines the values of potential and kinetic energy at each point of the physical system by the formulas(22)Ux=A2x⋅12±14−ln1A2x,Kx=A2x⋅12∓14−ln1A2x.For the Solar System, the energy characteristics are shown in Figures13 and 14.Figure 13
Characteristics of the function of the determined probability (FDP), planet Earth, and planets of the Solar System, as systems of physical points: (a) physical parameters of the Earth; (b) the energy characteristics of the FDP states of internal and external points of the physical system, depending on the ratioU/E where K is the kinetic (calorific) energy of the system, U is a potential (elastic) energy of the system, E is the total energy of the system, KU is the transfer of the energy invariant in the system, KU/E2 = Ψ2 is the phase of the energy state of the system, E0 = 1 is the initial given energy in the system of physical points; F > 0 is the gravitational compression force; F < 0 is the gravitational tensile force.Figure 14
Characteristics of energy correlations of planets of the Solar System by formulasξ=e−ψ2,ψ2=KU/E2,ξ=Е/E0.According to Zone III, where the external influence allows an increase in potential or kinetic energy, the Earth’s own gravitational potential energy is determined from the system of equations (22).


## 3.2. Analysis of Energy Relations for Planets of the Solar System

Figure14 testifies that (а) all planets of the Solar System are located at point B in Figure 13 (energy packaging law); (b) the total energy of each planet is determined by the system of equations (21) in its center and in orbit (24); and (c) the own total gravitational energy of each planet is divided into potential gravitational and thermoelastic in Zones I and II.Example 1.
(1) SPP-«Sun-Earth», (2) SPP-«Earth».(23)EC−3=γMCm3LC−3=5.3⋅1033J;K3=m3V22=0.5EC−3=2.65⋅1033J;U3=EC−3−K3=0.5EC−3,EG=K˜3=U˜3;E˜3=K˜3+U˜3=U3=2.65⋅1033J.
The Earth’s own gravitational total energy provided byρ=ρ0+χ⋅z is determined by the integral: U3=2.65⋅1033J.
From Figure13, it is evident that the internal kinetic (thermal) energy of the planet reaches the total gravitational energy in the center, and the potential energy disappears, as is the acceleration of gravity.
The geothermal model of the Earth with endogenous and exogenous heat fluxes suggests an increase in the temperature of the bowels of the planet with an unchanging total gravitational energy with increasing depth, as well as the existence of the following geophysical sequential mechanisms triggered by the ISE thermal density when approaching the center of the Earth.
The boundary of Zone III and the whole Zone II:(1)
The existence of a solid lithosphere on the Earth’s surface.(2)
Destruction-formation of a solid body of the lithosphere on the border with the mantle.(3)
Formation of the liquid phase of the Earth from different atoms and molecules, various compounds from chemical elements.(4)
Dissociation and synthesis of atoms and molecules. Atoms of hydrogen are destroyed by the latter with increasing depth, after which sharply increases the density of matter, and the nuclei of light chemical elements (the middle of the radius of the Earth) accumulate. The nucleus of heavier elements moves to Zone I.(5)
Dissociation and synthesis of the nucleus of chemical elements (the latter, closer to the center of the Earth, the nuclei of iron break down into protons and neutrons).(6)
At the center of the Earth on the surface of a sphere with a radius of proton, the density of thermal energy reachesηP=η3−f−maxR32/RP2=0.35⋅63710002/8.78⋅10−162=1.84287⋅1043 W/m2, and for the destruction of the proton, density of thermal energy on its surface should be equal to 1.5923·1043 W/m2 (the corresponding temperature of electromagnetic waves 7.26212·1012 K, and the frequency of oscillations of the De Broglie wave of the proton 2.27766·1023 Hz [20]). The destroyed proton will create a density of heat flux 0.3 W/m2 on the Earth’s surface. That is, we can assume that physical conditions in the center of the Earth’s core for the destruction of protons and neutrons exist. The maximum density of the heat flux recorded on the Earth’s surface is η3−f−max=0.35 W/m2 [38]:(24)E=γMmR,K=mV22,U=E−K,EK=γM/RV2/2=2γMω2R3.


## 3.3. Analysis of Results

Formulas (2), (4), and (5) are proposed by the author Karpenko for modeling the energy equilibrium of the joint action of heat fluxes of conductive and radiant EM waves in a geological environment.Formula (5) for the input data provides the following solutions of temperature:(25)T3=273.15+10=283.15K;λf=3.5W/m⋅K;ε=1;z0=1m;αR=1m−1;a=λfz02⋅αR⋅ε⋅σ=617283950.06K3;b=11050537213K4;σ≃5.6704×10−8W/m2⋅K4;⟹T˜1=283.15K;T˜2=211.331K;T˜3=−247.24−j351.24K;T˜4=−247.24+j351.24K;⟹T˜1+T˜2=T˜3+T˜4,where the temperature T˜1=283.15K characterizes the heating temperature of the geological environment by electromagnetic IR waves, which have an absorption coefficient αR=1m−1. Temperatures T˜2,T˜3,T˜4 provide additional physical information, but the interpretation of these temperatures is currently unknown and requires further research.Formula (6) is proposed by the author Karpenko to identify the energy density of Debye theory and elasticity theory, which provides important geophysical information about the physical relationships of elastic and thermodynamic parameters of the geological environment with the kinematic parameters of the P-wave and thermometry parameters obtained in terrestrial and downhole geophysics.Formula (7) is derived from formula (6) for physical identification: T3EG=TDEG. As a result, we obtain information about the frequency of oscillations Ωmax of the thermal field of the geological environment, which transmits infrared waves from the depths of the Earth to the Earth’s surface and is observed by satellite equipment. And seismic prospecting information about Vs−in,Vp−in, allows you to determine the parameter λmin. Known parameters Ωmax and λmin allow from equation (7) to determine the depth parameters μ3,ρ of the geological environment.Formula (8), obtained by the author Karpenko, provides information about the increasing temperature of the geological environment from the growth of gravitational (total) energy of the geological environment of the Earth depending on the depth, which is physically related to the average P-wave velocity at a given depth. This information is the basis for determining the total thermal capacity of a geothermal well to select the stable operation of a geothermal station by the technology of an isolated single well.A new value of the Earth’s own gravitational energy is 2.6275·1033 J, which is an order of magnitude greater than the known value 2.23945·1032 J, and which is equal to the kinetic energy of the Earth’s motion in orbit around the Sun, as proved by formula (24). An important parameter in formula (8) is the parameter σ=R3−zm/R3, which indicates the empty space in the center of mass of the Earth.Formula (9) is obtained by the author Karpenko in [16].The characteristicfU in Figure 5 indicates that the maximum temperature in the center of mass of the planets has the same value, which suggests the hypothesis of a single mechanism for converting the total gravitational energy of the planet into its thermal energy, which depends on the mass of the planet. This hypothesis is tested by equation (20), which takes into account the effective capacities of endogenous and exogenous heat fluxes on the Earth’s surface and shows that the presence of a heat source in the center of the Earth allows to maintain a temperature of 6000°C.The formulalnT=fU=ε1+ln1+U⋅1+ln1+U4≃1+ln1+W⋅1+ln1+W4 is the identification formula from equation (9), which is used to study the internal maximum temperatures of the planets of the Solar System depending on their own gravitational energy, as shown by the characteristic fTmax=lnTmax in Figure 5. This characteristic suggests that the center of mass of each planet’s maximum temperatures is almost identical. This result suggests the following hypothesis: “In the center of mass of each space object, the gravitational energy is converted into thermal energy of EM waves, the power of which depends on the mass of the object, and the maximum temperature in the center of mass of objects is the same.”The formulasλfz=∂3U/∂τ⋅∂z⋅∂t and Cz=5/T34zπ2ℏ/kBλ3minz3Vp35z are obtained by the author Karpenko.Figures6–10 using solution (16) of the system of equations (13) and (17) obtained from equation (2) and known experimental geophysical data on the temperature distribution in the center of the Earth show the characteristics of the distribution in the middle. The Earth has its thermodynamic parameters: λfz, λSBz, ω3z, λ3minz, Cz, αRz/n2z.In the system of equations (19), first developed by the author Karpenko in [38], for heat transfer processes in time in multilayer media, the author investigates the thermodynamic state of the Earth in the presence and absence of a heat source in the center of the Earth and on its surface from the Sun. Studies of equation (20) have shown that the presence of an internal heat source in the center of the Earth provides a stable surface temperature of the inner core at the level of 6000°C. And the change in heat transfer of the atmosphere significantly affects the average temperature of the Earth’s surface of Figure 12(a), and to a greater extent than the fluctuations of heat from the Sun in Figure 12(b).The system of equations (21), first developed by the authors in [34, 37, 39], models the infinite physical space of Gauss, in which the laws act together and simultaneously: conservation, change, transfer, and packaging of energy. From this space, various micro- and macrophysical objects are formed, in particular, space objects, which have an internal energy structure, which is shown in Figure 13 with identification to the various geophysical parameters of the Earth. The energy structure can explain the jumps, fractures, and the very characteristics of the distribution of geophysical parameters in the middle of the Earth. The main consequence of the energy analysis of gravitational energy in the middle of the Earth is the following hypothesis: “the increase in the body’s own gravitational energy, E, from the surface to the center of mass, masses (from 3000 m to 6378 m) of the inner masses of the electronic shells of molecules, atoms, nuclei of physicochemical elements, and quark shells of free protons and neutrons in the center of the masses, releasing their internal energy.This hypothesis explains the origin and dissociation of nuclei, physicochemical atoms, molecules, elements, and substances of the lithosphere, and the generation of endogenous heat by planets and stars, accompanied by a change in the gravitational force from the change in potential energy by the depth of compression by internal masses (to a depth of 3000 m) to the gravitational tensile force by external masses (from 3000 m to 6378 m) of internal masses of electronic shells of molecules, atoms, nuclei of physicochemical elements, and quark shells of free protons and neutrons in the very center of the masses, releasing their inner energy.”This hypothesis explains the origin and dissociation of nuclei, physicochemical atoms, molecules, elements, and substances of the lithosphere, and the generation of endogenous heat by planets and stars.


## 4. Conclusions

The developed geothermal model of the Earth with internal and external sources of thermal energy showed the following:(1)
The absence of an internal source of thermal energy of the Earth, of arbitrary physical nature, and thermal energy of the Sun on its surface leads to a decrease in core temperature to 4500 K and the average Earth surface temperature to 10 K (Figure11(a)) with existing atmospheric heat transfer.(2)
The absence of an internal source of thermal energy and the presence of thermal energy of the Sun on its surface leads to a decrease in core temperature to 4500 K and raises the temperature on the Earth’s surface to the existing value of 14.2 K (Figure11(b)).(3)
The presence of an internal source of thermal energy of the Earth with a value of 1,859·1017 W and the presence of thermal energy of the Sun on its surface leads to stabilization of the core temperature at 5986.88 K and raises the temperature on the Earth’s surface to 15°C (Figure 11(c)).(4)
The established connection of the radiation heat flux with a capacity of 1.859·1017 W and conductive heat flux with a capacity of 4.42·1013 W is the result of one physical process, namely, the transfer of electromagnetic waves through the substance of the Earth, which arises from the work of the internal heat source of the Earth. That allows to determine the basic geothermal parameters of the calorific value of rocks.(5)
The energy is determined by the energy method; that is, the proper total gravitational energy of the Earth at the level 2.65·1033 J is structured on the potential 0.5·2.65·1033 J and kinetic (thermal) 0.5·2.65·1033 J energy.(6)
The performed research studies established the following: reduction of heat transfer of the atmosphere, or increase of its specific heat capacity by 1% of emissions from dirty technologies of generation of thermal and electric energy at the level of 1.992·1013 W, which in the model increases endogenous heat of the Earth by (1.992·1013)/(4.42·1013)·100% by 45%, forms a thermal resonance with an increase in its temperature by 1.43% per year. Thermal resonance in the atmosphere: a higher temperature increases the water vapor in the atmosphere, which further independently increases the heat capacity and, accordingly, the temperature of the atmosphere.



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*Source: 6619991-2021-08-10.xml*