Thermodynamico-kinetic equation of the universe

Dr. Ivan M.Kolesnikov,Dr/ Kolesnikov S.I., Vinokurov V.A.

Gubkin Russian State University of Oil and Gas, Moscow, Leninsky prospectus 65, Tel/Fax:8-499-135-87-25

This article presents a new approach to the description of the thermodynamics of equilibrium-none-quilibrium processes in the annex to the universe. The equation takes into account spontaneously and non-spontaneously occurring processes, the equations for the flow of functions and functionals.

The universe may be presented as extended in space and time material objective reality. It is filled with phase-open systems. Between these systems occurs exchange of energy and matter through occurring simultaneously, continuously and interconnected equilibrium-none-quilibrium processes. This approach to development of the equation the universe was not addressed properly in scientific studies of classic thermodynamics of equilibrium-none-quilibrium processes.

One of basic principles as respects to the universe states that overall reserve of internal energy in the universe is constant. Redistribution of energy between objects in the universe may be connected to the heat and the work, according to the first law of thermodynamics, videlicet:

(1)

(2)

Based on classical thermodynamics, these equations determine that in the universe heat of non-spontaneously occurring processes increases the supply of internal energy for the phase-open system. Unfortunately in these equations don’t take into account type of the process in the phase-open system. But it cannot be generalized for the universe, because in accordance to stated above principle in the universe we have the equality:

(3)

The second concept for creation the equation of the universe came to be the new thermodynamic function, introduced in thermodynamics as the entropy by Rudolf Clausius. Entropy is a thermodynamic function that interconnects the structure in phase-open system and possible direction of occurring processes. Clausius defined the Principle of existence of entropy of the concept of increasing entropy, analytically expressed in the following form:

(4)

(5)

Analytical expression of the concept of existence of entropy reflects the following principles:

- The system is finde not far from the equilibrium state.

- The system transgresses to the equilibrium by the equilibrium way (* this state is a science fiction)

- A plus sign before a sign of entropy change defines the spontaneously occurring process.

The Рrinciple of increasing entropy, defined for isolated systems, reflects capability of exclusively spontaneously occurring system state changes, under constant volume and intraatomic energy of the system. This assumption is also a science fiction because an isolated system does not interact with the universe.

During system transition to equilibrium state with U,v=const energy change vanishing due to following criteria:

(6)

With this equation Clausius hypothesized the heat death of the universe. However, assumption (5) was defined for isolated system and cannot be applied to the open system, such as the universe, without additional criteria. It can only be extended to the universe based on the fourth law of thermodynamics( I.M.Kolesnikov), which will be formulated later. Deficiencies of criterion (6) in the annex to the universe are as follows:

- It does not take into account the type of occurring process, with which the system transcends to the equilibrium state.

- Constancy of U and V is inapplicable to the universe since it determines finiteness of their values in conditions of the unlimited universe.

- The fact that this criterion can only be used for locally isolated system is not accounted for.

According to Clausius both criteria (5) and (6) is supplement each other:

(7)

(8)

These criteria do not fully reflect the behavior of the local phase-open systems interacting with the environment.An isolated system is a science fiction but only for isolated systems the total change in entropy leads to its increase to achieve the equilibrium.One may not extend conclusions obtained for isolated systems onto the universe since in the universe spontaneous and –non-spontaneous processes occurring simultaneously, continuously and interconnected, which was not taken into account by Clausius. According to Clausius, only spontaneous occurring in the universe [1], which is absurd. As criterion allowing to determine the direction of occurring processes for the universe may be used the following expression:

(9)

Where the signs:

>defines the concept of increasing entropy of Clausius

= the concept of existence of the entropy of Clausius,

<the concept of decreasing entropy of Kolesnikov.

During the development of thermodynamics of irreversible processes in the middle of the twentieth century, N.Belokon [2], implicitly shared heat and entropy into two types:

- Q¹,S¹ - the processes occurring within the system

- Q¹¹, S¹¹- heat-exchange and entropy to the environment.

N.Belokonexpressedthe heat and entropy change in the following form:

δQ=δQ¹+δQ¹¹,

dS=dS1+dS11, δW=δW1+δW11 (10)

The equation of the first law of thermodynamics is:

δQ =dU+δW (11)

Accounting expressions (10)it can be represented in a more specific form, although not entirely adequate for real systems:

δQ^¹+δQ^¹¹=dU+δW¹+δW¹¹ (12)

And the fundamental equation of thermodynamics will take the following form:

dS¹+dS¹¹= dU+δW¹+δW¹¹ (13)

These equations do not reflect types of occurring processes (spontaneous and non-spontaneous) nor division of internal energy. Thoughthey take into account interaction of the system with the environment but not the loss, they already partially reflect the thermodynamics of irreversible processes.

At the same time in Belgium Ilya Prigoginecommittedly begins to develop thermodynamics of irreversible spontaneously occurring processes, [3] excluding thermodynamics of non-spontaneously occurring processes.In his works, he also identifies internal processes and the processes occurring in the environment by using the following expressions:

δQ=δ_iQ+δ_eQ, dS=d_iS+d_eS (14)

Where δ_iQ, d_iS – flow of heat and entropy, due to the processes occurring inside the system. And δ_eQ – flow of heat and entropy, due to the interaction with the environment.

We may note that Prigogine’s statement, that entropy difference d_iS,due to changes within the system, never has a negative value[3, 4, 5, 6], is only valid for isolated systems and may have a negative value for open systems due to external influence.

According to Prigogine the fundamental equation of thermodynamics will take the following form:

d_iS+d_eS=dU+δ_iW+δeW (15)

This equation is set for a single phase interacting with the environment.

Let the interacting system have two phases (1 and 11). Then for these phases heat exchange and entropy change may be presented by the following equation:

δ¹Q=δ¹_iQ+δ¹eQ, δ¹¹Q=δ¹¹_iQ+δ¹¹eQ (16)

Dividing by corresponding temperatures derives following expressions:

δ¹Q/Т=δ¹_iQ/Т¹+δ¹eQ/Т¹, δ¹¹Q/Т=δ¹¹_iQ/Т¹¹+δ¹¹eQ/Т¹¹ (17)

Considering the concept of entropy, these equations convers to the following form:

dS=d¹S+d¹¹S (18)

Substitute (17) to (18) and get following equation:

dS=δ¹_iQ/Т¹+δ¹eQ/Т¹+δ¹¹_iQ/Т¹¹+δ¹¹eQ/Т¹¹ (19)

Merge terms with the same subscript:

dS=δ¹_iQ/Т¹+δ¹¹_iQ/Т¹¹+δ¹eQ/Т¹+δ¹eQ/Т¹ (20)

Or

dS=d_i¹S+d_i¹¹S+d_e¹S+d_e¹¹S (21)

This equation reflects only thermodynamic processes but not chemical and other processes.

We already can express the fundamental equation of thermodynamics for two-phase subsystem within the system which interacts with the environment.

T¹_edi¹S+T_i¹¹di¹¹S+T_e¹de¹S+T_e¹¹de¹¹S=dU+δ_i¹W+δe¹W+δ_i¹¹W+δe¹¹W (22)

But it still does not reflect types of occurring processesnor division of internal energy and the loss. The equation can be transformed furthermore based on the forth law of thermodynamics, which reflects the new thermodynamics of equilibrium-nonequilibrium processes.

The forth law of thermodynamics was enunciated in the following form [7,8]: Spontaneous and nonspontaneous processes occurring simultaneously, continuously and interconnected in phase-open systems and the environment.Moreover, spontaneously occurring processes pass with increasing entropy change (pos-entropy) and decreasing of free internal energy, and non-spontaneously occurring processes pass with increasing free internal energy’s reserve and decreasing entropy change (neg-entropy).

For a single phase-open system the fundamental equation of thermodynamics will take the following form:

T_i¹di¹S^SOP+T_i¹di¹S^NSOP+T_e¹de¹S^SOP+T_e¹de¹S^NSOP=diU^SOP+d_iU^NSOP+d_eU^SOP+d_eU^NSOP+δi¹W^SOP+δ_i¹W^NSOP+δe¹W^SOP+δe¹W^NSOP (22)

This expression still does not take into account the loss in each occasion of entropy change, internal energy change and work change. Supplementing the loss to each term of (22) we will get the following expression

T_i¹di¹S^SOP+(T_i¹di¹S^SOP)^loss+T_i¹di¹S^NSOP+(T_i¹di¹S^NSOP)^loss+T_e¹de¹S^SOP+(T_e¹de¹S^SOP)^loss+T_e¹de¹S^NSOP+(T_e¹de¹S^NSOP)^loss=diU^SOP+(diU^SOP)^loss+d_iU^NSOP+(diU^NSOP)^loss+d_eU^SOP+(d_eU^SOP))^loss+d_eU^NSOP+δi¹W^SOP+(δi¹W^SOP)^loss+δ_i¹W^NSOP+(δ_i¹W^NSOP)^loss+δe¹W^SOP+(δe¹W^SOP)^loss+δe¹W^NSOP+(δe¹W^NSOP)^loss (23)

This equation determines the stationary state of the subsystem of a single phase-open system which interacts with the environment. It can be converted to the form applicable to the universe by summation, within its boundary, from 1 to ∞. As a result, receiving following expression for non-flow systems:

{T_i^jS^SOP+(T_i^jdi^jS^SOP)^loss+T_I^jdi^jS^NSOP+(T_i^jdi^jS^NSOP)^loss+T_e^jde^jS^SOP(T^jede^jS^SOP)^loss+T_e^jde^jS^NSOP+(T_e^jde^jS^NSOP)^loss}=

{di^jU^SOP+(di^jU^SOP)^loss+d_i^jU^NSOP+(di^jU^NSOP)^loss+d_e^jU^SOP+(de^jU^SOP)^loss+d_e^jU^NSOP+(de^jU^NSOP)^NSOP}+

{δ_e^jW^SOP+(δ_i^jW^SOP)^loss+δ_i^jW^NSOP+(δ_i^jW^NSOP)^loss+δe^jW^SOP+(δe^jW^SOP)^loss+δe^jW ^NSOP+(δe^jW^NSOP)^loss} (24)

Numerical value of the sum varieties form j=1 to j=∞. Though, this form does not take into account the energy flow and the functional flow. To allow this for the flows in the universe, multiplying each part of the equation by dρ,dv,dτ (parameters thatdetermine the density, linear velocity and time). The resulting expression is the equation of the universe that includes the mass flow and energy flow:

{T_i^jS^SOP+(T_i^jdi^jS^SOP)^loss+T_I^jdi^jS^NSOP+(T_i^jdi^jS^NSOP)^loss+T_e^jde^jS^SOP(T_e^jde^jS^SOP)^loss+T_e^jde^jS^NSOP+(T_e^jde^jS^NSOP)^loss}dρ,dv,dτ=

{di^jU^SOP+(di^jU^SOP)^loss+d_i^jU^NSOP+(di^jU^NSOP)^loss+d_e^jU^SOP+(de^jU^SOP)^loss+d_e^jU^NSOP+(de^jU^NSOP)^NSOP}dρ,dv,dτ +

{δ_e^jW^SOP+(δ_i^jW^SOP)^loss+δ_i^jW^NSOP+(δ_i^jW^NSOP)^loss+δe^jW^SOP+(δe^jW^SOP)^loss+δe^jW ^NSOP+(δe^jW^NSOP)^loss}dρ,dv,dτ (25)

According to Prigogine thermodynamic functions and functional change in time, and considering the concepts of entropy and fluctuations of thermodynamic functions can be expressed as follows:

etc. (26)

Analytical equation of the forth law of thermodynamics for non-stationary processes can be expressed in the following form:

(27)

Where:

dS^смпп=diS^смпп+deS^смппand dS^нсмпп=d_iS^нсмпп+d_eS^нсмпп (28)

It was previously assumed that only one-way spontaneous processes may occur in the system and the universe, but now we can see that above equations represents two-way processes. Spontaneously occurring process may dominate over non-spontaneously occurring process and vice versa. This statement is confirmed by definition of the forth law of thermodynamics and substantively occurring processes in phase-open systems and the universe. Spontaneously occurring processes lead to degradation of the system while non-spontaneously occurring processes result in its evolution.

Expression (25) defines thechanges within the system, and in the local environment, occurring over time dτ. To determine the changes in the system and the universe over a single unit of time the equation (25) differentiated with respect to time. The resulting form is the general equation of the universe.

{T_i^jS^SOP+(T_i^jdi^jS^SOP)^loss+T_I^jdi^jS^NSOP+(T_i^jdi^jS^NSOP)^loss+T_e^jde^jS^SOP(T_e^jde^jS^SOP)^loss+T_e^jde^jS^NSOP+(T_e^jde^jS^NSOP)^loss}dρ,dv,dτ=

{di^jU^SOP+(di^jU^SOP)^loss+d_i^jU^NSOP+(di^jU^NSOP)^loss+d_e^jU^SOP+(de^jU^SOP)^loss+d_e^jU^NSOP+(de^jU^NSOP)^NSOP}dρ,dv,dτ +

{δ_e^jW^SOP+(δ_i^jW^SOP)^loss+δ_i^jW^NSOP+(δ_i^jW^NSOP)^loss+δe^jW^SOP+(δe^jW^SOP)^loss+δe^jW ^NSOP+(δe^jW^NSOP)^loss}dρ,dv,dτ (29)

It seems to me that this is the classical expression for the universe, which so far has phenomenological content. To localize the equation it is necessary to identify the mechanism of changes for each of its terms in both sides. For resolving local problems of the universe we may turn to some equations, such as equations for the mass flow, heat flow (entropy) and momentum transfer motion flow. At the moment we have worked out the main goal – spontaneous and non-spontaneous processes accounting losses for functions and functionals are united in a single equation. Derivations for specific functions and functionals are represented in our tutorial [8]. There is indirect statement that mass and energy in above equations allowed as infinity (m=∞and U=∞). Similar equations may be introduced to the global economy and biology.The equations for mass flow, heat flow and entropy account the mechanism of various processes in the system and the environment. For the flow system (such as the Gulf Stream, for instance) these equations can be presented in different form as shown further in this article.

The expression (10) does not account mechanism of occurrence of thermodynamic processes. Phenomenologically, mechanism of occurrence of thermodynamic processes in the mass flow reflected in equations of the flow of energy in a form of mass flow, energy flow and entropy, which includes convective flow, thermal conductivity, heat transfer and energy accumulation within the flow due to internal physicochemical processes.

The equations for mass flow, energy in a form of heat and momentum for non-stationary and stationary conditions are presented further (accounting works of Dr. Usov). The equation of the mass flowfor non-stationary conditions in the3D-space, defined by x, y, z coordinates may be presented in the form of five components: I – convection, II – diffusion, III – mass transition, IV – physicochemical process and V – derivative, defining the flow state change with respect to time.

,

I II III IV V

(30)

where: c_i – concentration of matter in the flow, x; y; z – coordinates, D – diffusion constant, β – mass transition constant, V_i – liner velocity of the flow, τ – time, W_i – rate of passing of physicochemical process

Under stationary conditions the derivative and the equation (???) reduced to following expression:

div (C _I V)–DΔ²C_i+βΔC_i+W_i=0 (31)

These equations are accurate for the matter flow in a pipe with rigid walls and without interaction with the environment.

Expressions (30) and (31) are the mathematical basis for the development of kinetic models of any physicochemical processes occurring in the flow, in stationary and non-stationary conditions, the hydrodynamic modes of ideal displacement and ideal mixing and composite models, as well as for fixed systems.

The equations of mass flow supplemented with the kinetic equations, the dependencies of the diffusion constants from T, the equations of mass transition, velocity constants with respect to temperature and other terms.

The equation of energy flow in the form of heat and entropy

Heat flow is induced to move by convection, heat conduction, heat transfer, the occurrence of physicochemical process. Change of the state of flow occurring for non-stationary conditions, which is represented in the equation of the first derivative by heat over time.

The driving force of the flow is the temperature difference over the area and its value. Higher the temperature difference within the flow and between the flow and the environment, higher the value of the first derivative and higher the change of the state of flow.

General equation, defining the change of the state of heat flow with respect to change of temperature and time presented in the following form:

(32)

where: ρ – density of matter flow, C_p – heat capacity with p=const, x;y;z – flow coordinates, T – temperature, λ – heat conduction constant, α – heat emission constant, W_i – rate of passing of physicochemical process, ΔH_i – enthalpy of physicochemical process, τ – time, Vi – liner velocity of the flow.

This expression is accurate for non-stationary conditions of transition of energy flow in the form of heat through the specified device. For stationary conditions the derivative with respect to time vanishing.

(33)

Thus the equation (???) is reduced to the following form:

div(VρC_pT)–²T+αΔT+W_iΔН=0 (34)

or

div(VρC_pT)–a²(ρC_pT +b(ρC_pТ₂–ρС_pT₁)+W_i ΔH_i=0, (35)

where : a=, b=, assuming that the flow section is 1m²

Introducing a substitution to the expression (32):

С_рT=Q_p (36)

The resulting form is:

(37)

Dividing this by T will get us the following:

(38)

Including the concept of entropy, specifically

, (39)

derive the equation of the entropy flow in the following form:

(40)

This is only accurate for entropy change within the matter flow.

Above equations is the basis for thespeculative study of the technology of physicochemical processes. Pairing corresponding terms of these equations may play important role in the development of specific mathematical models for physical, chemical, physicochemical and hydrodynamic processes.

Exposed matter flows, energy in the form of heat flows and entropy flows can be described by the equations, indispensably including the characteristics of internal and external nature. It is essential to account the interaction of the flow (internal system) and the environment (external system). These flows are studied in thermodynamics of irreversible processes and thermodynamics of spontaneous-nonspontaneous processes.

The momentum equations

The momentum equation for the flow (in particular theNavier–Stokes equations) defines the displacement of particles within the flow of an incompressible fluid under the influence of differences of velocities of shifting particles, gravity, pressure drop and friction.In the projections of forces on the axes x, y, z motion of an incompressible fluid can be presented for the non-stationary flow through the following equations:

(???)

(???)

(41)

For stationary conditions the partial derivatives of the velocity of the fluid particles with respect to corresponding time coordinateare vanishing:

, =0, =0. (42)

Where: ρ – density, μ – mediumviscosity, V – velocity, g – gravity, P – pressure, τ – time.

The equations presented in this chapterare the basis for theoretical models of chemical and technological processes, processes for oil and gas, transportation of oil and gas, the study of migration processes in the lithosphere, water and air environments, as well as in biological systems. These kinetic-thermodynamic equations can be used for local systems in the universe, given the mechanism of the processes.

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