Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів
A method of mathematical modeling of multidimensional quasi-static processes, a generalization of quasi-static processes of equilibrium thermodynamics, is proposed and substantiated. The authors obtain a generalization of the first and the second law of thermodynamics in the form of Carathéodory to...
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System research and information technologies| _version_ | 1867334433394130944 |
|---|---|
| author | Kutsenko, Alexander Kovalenko, Sergii Kovalenko, Svitlana |
| author_facet | Kutsenko, Alexander Kovalenko, Sergii Kovalenko, Svitlana |
| author_institution_txt_mv | [
{
"author": "Alexander Kutsenko",
"institution": "National Technical University “Kharkiv Polytechnic Institute”, Kharkiv"
},
{
"author": "Sergii Kovalenko",
"institution": "National Technical University “Kharkiv Polytechnic Institute”, Kharkiv"
},
{
"author": "Svitlana Kovalenko",
"institution": "National Technical University “Kharkiv Polytechnic Institute”, Kharkiv"
}
] |
| author_sort | Kutsenko, Alexander |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2023-05-24T21:28:17Z |
| description | A method of mathematical modeling of multidimensional quasi-static processes, a generalization of quasi-static processes of equilibrium thermodynamics, is proposed and substantiated. The authors obtain a generalization of the first and the second law of thermodynamics in the form of Carathéodory to multidimensional quasi-static processes. The idea of generalization is to construct an orthogonal system of functionals similar to the work and heat functionals of classical thermodynamics along families of phase trajectories corresponding to different types of influences on a multidimensional quasi-static system. The representation of quasi-static processes by systems of ordinary differential equations containing control variables is substantiated. The obtained results make it possible to use a wide arsenal of methods of the theory of control of dynamical systems to solve problems of control of quasi-static processes. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.1.05 |
| first_indexed | 2025-07-17T10:28:08Z |
| format | Article |
| fulltext |
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko, 2023
62 ISSN 1681–6048 System Research & Information Technologies, 2023, № 1
TIДC
ПРОБЛЕМНО- І ФУНКЦІОНАЛЬНО-
ОРІЄНТОВАНІ КОМП’ЮТЕРНІ СИСТЕМИ
ТА МЕРЕЖІ
UDC 51-74
DOI: 10.20535/SRIT.2308-8893.2023.1.05
GENERALIZATION OF THE THERMODYNAMIC APPROACH
TO MULTI-DIMENSIONAL QUASISTATIC PROCESSES
A.S. KUTSENKO, S.V. KOVALENKO, S.M. KOVALENKO
Abstract. A method of mathematical modeling of multidimensional quasi-static
processes, a generalization of quasi-static processes of equilibrium thermodynamics,
is proposed and substantiated. The authors obtain a generalization of the first and the
second law of thermodynamics in the form of Carathéodory to multidimensional
quasi-static processes. The idea of generalization is to construct an orthogonal sys-
tem of functionals similar to the work and heat functionals of classical thermody-
namics along families of phase trajectories corresponding to different types of influ-
ences on a multidimensional quasi-static system. The representation of quasi-static
processes by systems of ordinary differential equations containing control variables
is substantiated. The obtained results make it possible to use a wide arsenal of meth-
ods of the theory of control of dynamical systems to solve problems of control of
quasi-static processes.
Keywords: quasi-static processes, equilibrium thermodynamics, mathematical mod-
eling, work, energy, controllability, entropy.
INTRODUCTION
The concepts of “quasi-static system” and “quasi-static process” owe their ap-
pearance to the fundamental branch of physics – thermodynamics. Thermody-
namics originated and was further developed on the basis of the phenomenologi-
cal approach, i.e. considering only the observed properties of thermodynamic
systems (TS) without a detailed analysis of the mechanisms of their manifesta-
tion. As a result of this approach to the formation of thermodynamics, its main
provisions: terminology, axiomatics and logical constructions in the form of
physical laws have not changed significantly over the past few centuries. In the
same time real practical results confirming theoretical backgrounds with a suffi-
ciently high degree of accuracy have made thermodynamics, along with mechanics,
a model of applied science. The quasi-static nature of processes in TS is one of
the main postulates of thermodynamics, along with such concepts as reversibility
and equilibrium. From the standpoint of thermodynamics, a quasi-static process is
an infinitely slow process consisting of a sequence of equilibrium states infinitely
close to each other. An essential limitation of the thermodynamics of equilibrium
processes is the exclusion from consideration of the time coordinate and the rep-
resentation of processes in the form of segments of trajectories in the phase space.
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 63
The mathematical support of thermodynamics is based on two postulates: the first
and second laws, on the basis of which mathematical models of the main thermo-
dynamic processes are introduced.
Despite the apparent limitation of the object of study in the form of quasi-
static TS, a number of concepts and approaches of thermodynamics can be
adapted to a relatively wide class of technological systems similar in external
properties to TS. In this regard, the problem of constructing a mathematical the-
ory of quasi-static processes arises, covering both the TS and other systems with
the quasi-static property. In this article, an attempt to propose the mathematical
foundations of the theory of quasi-static processes with a focus on the control
problem is made.
REVIEW AND ANALYSIS OF INFORMATION SOURCES
The classical substantiation of thermodynamics as a system of knowledge is al-
most completely described in the works of Carnot, Clausius, Boltzmann, Planck
[1–3] and other “classics” of thermodynamics [4, 5]. This branch of thermody-
namics has survived without fundamental changes to the present day.
One of the principal ways to involve mathematical methods in thermody-
namics and generalize the latter to other physical and technical processes can be
considered the approach outlined in [6]. A feature of Gukhman’s ideas is the sub-
stantiation of the uniformity of the components of the differential equation of the
first law of thermodynamics in the form of products of certain potentials and a
change in the corresponding coordinate of the state of the TS. At the same time,
the thermal effect was included as a particular type of the set of effects on the TS,
and the concepts of energy and amount of effect received a generalized interpreta-
tion suitable for any similar processes. The versatility of the ideas presented in [6]
is clearly demonstrated in studies in the field of ecosystems [7].
Summarizing the results of thermodynamics related to the first law or the
law of conservation of energy, we can conclude that the mathematical methods
that allow adapting classical equilibrium thermodynamics to a wide range of simi-
lar processes in nature, technology and society are limited. The fundamental dis-
advantage of the considered approach is the absence of any time dependences
linking changes in the state of the TS with the intensity (power) of actions.
In this regard, the authors of this work previously obtained some results of
the “dynamization” of quasi-static processes. This made it possible to obtain
mathematical models in the form of ordinary differential equations relating the
change in time of the state vector of a quasi-static system to changes in the power
of control actions. The results obtained in [8–10] require further generalization.
The greatest difficulty in the formation of thermodynamics was caused by its
second law, and in particular the concept of entropy. For instance, in [4] there are
18 formulations of the 2nd law, each of which ultimately has the same meaning of
the impossibility of obtaining mechanical work using only one source of heat.
Unfortunately, the abundance of formulations and interpretations of the 2nd law
of thermodynamics has led to some conservatism in the mathematization and
formalization of thermodynamic methods and their adaptation to quasi-static
processes of a different nature. The works of Schiller, Carathéodory, Afanasieva-
Ehrenfest, Belokon were a qualitative shift in the direction of axiomatization in
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 64
thermodynamics, where a generalized theory of quasi-static processes was formu-
lated in the language of mathematical formulations. Among the studies in this di-
rection, the article by Carathéodory [11] should be especially singled out, where
he elegantly formulated the 2nd law of thermodynamics in the language of Pfaf-
fian forms. This article gave impetus to the wide application of the axiomatic ap-
proach in the thermodynamics of multidimensional processes [12–15].
MATHEMATICAL MODEL OF A CONTROLLED QUASI-STATIC PROCESS
We will consider multidimensional quasi-static processes in terms similar to the
terms of classical thermodynamics: state, action, phase trajectory, work, energy,
etc. The fundamental difference from thermodynamics will be the inclusion of the
concept of heat in the category of works, as proposed in [6]. Thus, in our further
constructions, heat or other effects will not take any priority position among the
many possible effects on the system under consideration.
Let the state of the system under consideration is given by the state vector
nx R . State change over time ( )x t represents a trajectory in ( 1)n -
dimensional space. Any change of the vector x in time is associated with the
presence of one or more actions from a given set W . In the absence of actions on
a quasi-static system, the latter remains in the state in which all actions were inter-
rupted. That is any condition nRx of a quasi-static system is an equilibrium
position, in contrast to dynamical systems, for which the set of equilibrium posi-
tions is a subset of the space nR .
Associate with each m n of the types of actions kW a vector field )(xk ,
mk ,1 . We will assume that the system of vector fields )(),...,(),( 21 xxx m at
every point nRx is linearly independent. For k -th action the trajectory in phase
space can be represented as a solution to the system of differential equations
)()( tux
dt
dx
kk , (1)
where ( )ku t is a scalar function of the parameter t , which has the meaning of the
action intensity kW . Absolute value ( )ku t determines the speed of movement of
the representative point along the phase trajectory, and its sign determines the di-
rection of movement.
Let now several actions be simultaneously applied to the system. They cor-
respond to the vector field )(x . Let’s decompose the vector )(x at some point
x along linearly independent vectors )(),...,(),( 21 xxx m . We get
m
k
kk uxx
1
)()( ,
where ku are decomposition coefficients.
Then the movement of the system along an arbitrary phase trajectory corre-
sponding to the complex action will correspond to the system of differential equations
m
k
kk tux
dt
dx
1
)()( , (2)
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 65
and the coefficients ku can be interpreted as the intensity of the corresponding
actions.
If time t is chosen as a parameter, and values ( )ku t are considered as con-
trols, then the mathematical model of the quasi-static process in the form (2) is a
controlled system linear in controls. Since the vectors are linearly independent,
then the equilibrium of system (2) is )(),...,(),( 21 xxx m achieved only in the
case 0...21 muuu . That is, the effects cannot compensate each other.
The proposed mathematical models of quasi-static processes with single (1)
and complex action (2) give a qualitative model of behavior . The point is that
the vector fields )(xk ( 1, )k m specify only the directions of the corresponding
fields and can be normalized arbitrarily. In this case, the form of phase trajectories
will remain unchanged. At the same time, the trajectory in time will depend
both on the normalization of the vector )(xk , as well as on the size ( )ku t , form-
ing the speed of movement along the phase trajectory defined by the vectors
)(xk and some starting point 0x of the specific process.
ENERGY MEASURE OF THE ACTION
The previously introduced concept of the action intensity, which has the meaning
of controlling a quasi-static process, requires a quantitative measurement. To do
this, as an analogy, we use the classical thermodynamic approach for thermal de-
formation systems, i.e., TS subject only to mechanical and thermal effects, the
phase trajectories of which have the form constpV and constp respec-
tively, where ,p V are the pressure and volume of the gas in relative units and
is adiabatic exponent )5.13.1( , depending on the chemical composition of
the substance.
It is known from thermodynamics [5] that the first law of thermodynamics is
valid for a thermal deformation system, which is formulated as follows:
dU L Q , (3)
where dU is an increment of internal energy, L is a deformation work, Q is a
supplied heart. In (3) internal energy ( )U x is a function of state and quantities
L and Q are functions of trajectories in a state space.
For ideal gases [5] the internal energy U has the form
pVU
1
1
, (4)
and the mechanical work and heart are calculated as curvilinear integrals along
the phase trajectory l :
( )l
L pdV ; (5)
)( 11
1
l
pdVVdpQ . (6)
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko
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Thus, ratio (5) and (6) can be interpreted as a quantitative measure of the ac-
tion on the TS in the form of the mechanical work (5) and heat (6). For fixed ini-
tial 0x and final fx process points 0( ) ( ) constfU x U x for any trajectories
connecting 0x and fx , the sum L and Q in accordance with (3) will be con-
stant for various ratios L and Q .
Let us generalize the thermodynamic approach (4) as applied to finding simi-
lar quantitative measures of actions for an arbitrary n -dimensional quasi-static
process with m n actions. Let us introduce nR the scalar continuous function in
static space – the energy:
1 2( ) ( , , , )nU x U x x x . (7)
The energy increment (7) in some point nx R will look like:
1 2
1 2
n
n
U U U
dU dx dx dx
x x x
. (8)
Denoting )(xg
x
U
— the gradient ( )U x , write down (8) as
( , )dU g dx , (9)
and along the trajectory of the solution (1), or along the phase trajectory of the
k -th action, the (8) takes the form
dtgdU k ),( . (10)
Ratio (10) will also be valid for any phase trajectory generated by an arbi-
trary vector field )(x
dtgdU ),( . (11)
Let us denote the set of possible phase trajectory kl corresponding to k -th
type of action as kL , and the set LLk . The set of all feasible trajectories cor-
responding to an arbitrary composition of actions is denoted by *L . Then there is
the system of inclusions
*, 1,kL L L k m .
We will call the energy measure ( )J l of any phase trajectory l the value
( )
( ) ( , )
l
J l g dx . (12)
Due to (9) the energy measure (12) depends only on the initial 0x and final
fx points of the curve l :
0 0( ) ( ) ( ) ( , )f fJ l U x U x J x x . (13)
The energy measure (13) satisfies the following axioms:
1) ( , ) 0J x x ;
2) ( , ) ( , )J x y J y x ; (14)
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
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3) ( , ) ( , ) ( , )J x y J x z J z y .
Axioms 2 and 3 in (14) differ from the symmetry and triangle axioms known
from the theory of metric spaces:
( , ) ( , )J x y J y x ;
( , ) ( , ) ( , )J x y J x z J z y .
By analogy with first law of thermodynamics, we will look for the quantities
of each type of actions in the form of functionals ( )kA l , calculated along phase
trajectories similar to the mechanical work and heat for the simplest thermody-
namic system (5), (6):
( )
( ) ( ( ), ), 1,k k
l
A l q x dx k n . (15)
Functionals ( )kA l we will call the works of action kW . Vectors ( )kq x we
will be chosen so that following conditions are satisfied:
,if,0
,if),(
)(
i
k
k Ll
LllJ
lA ., 1, , kk n ii (16)
Requirements (16) will by met under the following conditions to choose
from ( )kq x :
nkigq iikik ,1, ),,(),( , (17)
where ik is the Kronecker symbol.
The conditions (17) are essentially conditions for the quasi-biorthogonality
of vector systems ),( 21 n and 1 2( , )nq q q .
Rewrite the last equation in matrix form:
diag TQ g , (18)
where the rows of matrix Q are desired vectors 1 2( , )nq q q and the columns of
the matrix are vector fields ),( 21 n of actions on the system.
Since vectors )(xk are assumed to be linearly independent for nx R ,
matrix is non-degenerate, therefore the solution (18) can be written as
1)(diag gQ T . (19)
Let us now show that for the system of works 1 2, nA A A with vector fields,
defined in accordance with (19), *l L the following relation in true:
)()(
1
lAlJ k
n
k
. (20)
To prove (20), we sum relations (17) over k . As a result, we get:
nigq iik
n
k
ik
n
k
,1),,(),(
11
;
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 68
nigq iik
n
k
,1),,(,
1
.
It follows directly from the latter:
1
n
k
k
U
q g
x
. (21)
Let us sum over k the values ( )kA l , determined in accordance with (15). As
a result, we get
11
( ) ( , )
n n
k k
kk l
A l q dx
. (22)
Taking into account relations (12) and (21), and from (22) we get (20), i.e.
the sum of the work calculated along an arbitrary trajectory is equal to the incre-
ment in the energy of the system at the ends of this trajectory.
As an example consider thermodynamic system with actions in the form of
mass transfer, deformation, and heat transfer The vector fields of such actions are
well known [5] and are represented by the matrix ( , , )p V m , looking like
001
010
1
V
p
m
p
, (23)
where mVp ,, are pressure, volume and mass of the system. The columns of the
matrix (22) are vector fields of mass transfer, deformation and thermal effects,
respectively.
Let us introduce the energy of system in the form of a known relation for
ideal gases (3), then
),,(0,
1
1
,
1
1
321 gggpV
x
U
. (24)
Substituting the (23) and (24) in the formula (19), we get a matrix
m
pVpV
p
m
pV
Q
111
00
1
00
,
the rows of which are the components of the vectors of the quantities of the cor-
responding actions
)(
1 1l
dm
m
pV
A ;
)(
2
l
pdVA ;
,
111)(
3
l
dm
m
pV
pdVdp
V
A
where 1A is a work of mass transfer, 2A is a mechanical work, 3A is a heat.
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 69
It is easy to verify that sum of the rows of the matrix Q coincides with the
internal energy gradient, and
)()()()( 321 lJlAlAlA .
The considered approach to constructing a system of functionals satisfying
the additivity condition (20) can also be extended to the case when the number of
actions m is less than the dimension of the system n . In this case, the matrix
in the system of equations (18) is rectangular, and, consequently, constructions
(19)–(24) lose their meaning. A way out of this situation can be obtained as a re-
sult of redefining the system of actions with the missing n m dummy actions.
These dummy actions are chosen so that the square matrix )( 21 , where
1 ( )n m – the matrix given actions, and ))(( 2 mnn – the matrix of
dummy actions will be nondegenerate. Substituting the matrix obtained this
way in relation (19) we get the matrix of works, which will depend on an arbitrary
matrix 2 .
INVERSE PROBLEM
The performed analysis showed that if the laws of change of the phase coordi-
nates of the thermodynamic system depending on the type of influences are
known, then for an arbitrarily given state function ( )U x , a system of functionals
can be chosen that satisfies the structure of the energy conservation law. The
question arises, is it possible to solve the inverse problem: for a given functional
corresponding to the work of some action, construct the corresponding function
( )U x , which has the meaning of the energy of the thermodynamic system. This
approach is quite pragmatic, since it allows to “join” various physical processes
that have the same types of external influences. For example, in mechanics and
thermodynamics, such an action is deformation, and the corresponding work in
both cases is calculated by similar functionals:
)(
l
fdsA is mechanics;
)(
l
pdVL is thermodynamics.
So, let’s formulate the problem as follows. Let some reference action be
given, which corresponds to the phase trajectories of the system of differential
equations
nRxxx ),( ,
where )( x is a known vector function.
Let the known functional correspond to the standard action, with the help of
which the work of the standard action is calculated:
1 1 2 2
( ) ( )
( , )( ) ( ) ( )n n
l l
L q dx q dx q dx q dxx x x .
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Then, due to relation (17), the vector functions )( x , ( )q x and ( )g x must
satisfy the condition
),() ,( gq .
The last equation can be interpreted as a linear differential equation in partial
derivatives with respect to the function of the energy of the thermodynamic sys-
tem ( )U x :
),(,
q
x
U
.
To solve it, we use the method of characteristics. The system of differential
equations of characteristics in symmetrical form can be written as
))(),(()()()( 2
2
1
1
xxq
dU
x
dx
x
dx
x
dx
n
n
. (25)
Since all )(xk included in (25) do not depend on the desired function
( )U x , then the set of first integrals of the subsystem of differential equations (25)
)()()( 2
2
1
1
x
dx
x
dx
x
dx
n
n
(26)
will also be a subset of the set of integrals of the complete system (25).
Let
kk Cx )( , 1, 1k n (27)
be the first integrals of the system (26). Based on (27), we express the first
1 2 1, , , nx x x through nx and 1 2 1, , , nC C C :
),,,,,(
............................................
;),,,,(
;),,,,(
12111
12122
12111
nnnn
nn
nn
CCCxx
CCCxx
CCCxx
(28)
and substitute the obtained relations into the last of equations (25)
))(),(()( xxq
dU
x
dx
n
n
.
As a result of such a substitution, we obtain a differential equation with
separable variables nx and U
)),(),,((),( CxCx
dU
Cx
dx
nnnn
n
,
where n , , q are the functions of the vector nx and the vector of constants
1 2 1, ,..., nC C C C obtained after substituting (28) into the expressions for
n , , q respectively.
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 71
Integrating the last differential equation, we obtain the n -th first integral of
the original system of differential equations of characteristics (25) in the form
nnn CCxU ),( . (29)
Substituting into (29) the expressions for the components of the vector of
constants C in the form (27), we finally obtain
( )n nU x C .
As is known from the theory of partial differential equations, the general so-
lution ( )U x satisfies the equation
1 2 1( ), ( ), ( ), , ( ) 0n nU x x x x ,
where is an arbitrary function of its arguments.
Solving the last equation with respect to the first argument, we finally obtain
,)](,),(),([)( 1210 xxxxUU n (30)
where )()(0 xxU n , and is an arbitrary function of its arguments.
Let’s analyze the result. As can be seen from (30), the function ( )U x that
satisfies the differential equation for the operation of a given action consists of
two terms. The first term 0( )U x is determined both by the type of the phase tra-
jectory of the system subject to the reference action and by the type of the stan-
dard functional (vector ( )q x ). The second term depends only on the phase trajec-
tories and is an arbitrary function of the first integrals of the system of differential
equations that model the behavior of the thermodynamic system under the
reference action.
Thus, all ( )i x 1, 1i n that are arguments of an arbitrary function are
constants, and therefore ],,,[ 121 n also an arbitrary constant. Thus,
CxUxU )()( 0 ,
i.e. the energy of a quasi-static system is determined up to some constant C .
Consider an example of finding the internal energy ),( VpU for the simplest
thermodynamic system subjected to a standard deformation effect, the work of
which is defined as
2
1
V
V
L pdV .
The system of differential equations modeling the deformation effect, as fol-
lows from [5], has the form
,
V
p
dt
dp
,1
dt
dV
and differential equation (24) for ( , )U p V can be written in the form
.p
V
U
p
U
V
p
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The corresponding system of ordinary differential equations of characteris-
tics can be written as
p
dUdV
V
p
dp
1
. (31)
The first integral 1( , )p V , which is the equation of the adiabatic phase
curve, is obtained by integrating the differential equation
1
dV
V
p
dp
and has a well-known form
1CpV . (32)
Expressing p through 1C and V , and substituting the result into (31), we
obtain a differential equation for determining the energy ( , )U p V . Thus
VCp 1 , dVVCdU 1 , dVVCU 1 , and C
V
CU
1
1
1 .
Substituting expression (32) for 1C into the last relation, we finally obtain
the expression, well known from thermodynamics for the internal energy of an
ideal gas,
CpVU
1
1
.
It should be noted that this result was obtained only on the basis of the
known experimental adiabatic curves without involving the results of the well-
known experiment of Gay–Lussac and Joule, which showed the invariance of the
gas temperature during its expansion into a vacuum.
Let us now return to the original mathematical model of the thermodynamic
system (2) and find the physical meaning of the quantities ku – an action intensi-
ties of kW . To do this, we proceed as follows: let the work jA be given by an
integral of the form:
( , )j j
l
A q dx , (33)
where jq is previously defined vector of j -th work. Multiply the left and right
parts of (2) scalarly by the vector jq , as a result we get
n
k
kkjj uqxq
1
),(),( . (34)
Taking into account the conditions of orthogonality of works (12), from rela-
tions (33)–(34) we get
jjj
j uq
dt
dA
),( .
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 73
From the last expression, and also from (17) it follows
.,1,
),(
1
),(
1
njA
g
A
q
u j
j
j
jj
j
(35)
Let us substitute (35) into original differential equation (2). As result we get
k
n
k k
k A
g
x
dt
dx
1 ),(
)(
. (36)
From (36) it follows that if the column vectors )(xk of matrix ( )x divide
accordingly into dot products ),( kg , then each of the controls ku will match the
intensity of the corresponding work kA , i. е., using electromechanical terminol-
ogy, ku with this normalization corresponds to the action power kW . In addition,
the original system of equations (2) can be immediately normalized in accordance
with (36), which will in no way affect the form of phase trajectories correspond-
ing to certain types of action and the corresponding work vector ( )kq x .
Thus, without loss of generality, we can assume that the right side of the sys-
tem of equations (2) is normalized in accordance with (34), and the control vector
ju corresponds to the power vector jA . Therefore, any of the works can be cal-
culated both in the form of a curvilinear integral (33) and in the form of an inte-
gral over the time
dttuA
t
t
jj )(
1
0
.
QUASI-STATIC CONTROL SYSTEM OF GENERAL VIEW
Let us consider an empirical approach to the construction of mathematical models
of quasi-static systems, given by families of phase trajectories 1 2, ,..., nL L L , corre-
sponding to the impacts 1 2, ,..., nW W W . Let kL it look like
,),...,,(
..................................
;),...,,(
;),...,,(
1211
2212
1211
nnn
n
n
Cxxxf
Cxxxf
Cxxxf
(37)
of ( 1)n -parametric family of lines in n -dimensional phase space. The set of
constants 1 2 1, ,..., nC C C set a specific phase trajectory k kl L , and a change in one
of the variables (for example, nx ) determines all the coordinates of the points of
the selected phase trajectory as a result of solving system (37). Let’s associate the
change nx with some parameter 1t R , which can be taken as time. Then all kx
will also be functions of time due to equations (37). To find the vector field corre-
sponding to trajectories from kL , we differentiate (37) with respect to t . As a re-
sult, we get
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 74
.0...
....................................................
;0...
1
2
2
1
1
1
1
1
2
2
1
1
1
1
n
n
nnn
n
n
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
(38)
Complementing (38) with the equation
1nx ,
we obtain n n a linear system of algebraic equations with respect to 1 2, ,..., nx x x ,
the solution of which gives the vector field of the k -th action or the correspond-
ing system of differential equations
),...,,( 21 nk
k xxx
dt
dx
, 1,k n . (39)
Thus, from a family of phase trajectories of some process (37) we can get a
quasi-static model of the same process, which has the structure of a system of dif-
ferential equations (39).
The transition to the environment of differential equations allows us to use a
wide arsenal of modern methods of control theory for solving all kinds of control
problems for technical systems.
REACHABILITY, CONTROLLABILITY, AND ENTROPY OF QUASI-STATIC
SYSTEMS
In the previous sections, the concepts of energy and work of various action were
formalized and substantiated, which made it possible to interpret quasi-static sys-
tems as controlled dynamic ones. One of the central concepts of control theory is
the concepts of reachability and controllability [16], which qualitatively charac-
terize the fundamental possibility of solving a control problem, i.e., the existence
of a certain control law that implements the transition between two reassigned
points of the phase space. Let us extend these concepts to controlled multidimen-
sional quasi-static processes.
If the system of actions on the TS is complete, i.e. m n , then it is obvious
that it is always possible to choose such a sequence of actions that the phase tra-
jectory will pass through any predetermined point of the state space. So the whole
space nR is achievable, and the TS is fully controllable. In another limiting case,
when all controls except ku are identically equal to zero, i.e. 1m , the reachable
set is some isolated phase trajectory k kl L , on which the represented point is
currently located, and the system becomes uncontrollable. Bellow the obtain a
criterion for the controllability of quasi-static systems for the general case
1 m n , relying, as before, on the methods of equilibrium thermodynamics.
It is easy to see that the concept of controllability of quasi-static systems is
closely related to the concept of adiabatic inaccessibility introduced by
Carathéodory [11] when formulating the second law of thermodynamics. Accord-
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 75
ing to Carathéodory, in the vicinity of an arbitrary state of the TS, there are adia-
batically inaccessible states. In other words, in the absence of thermal action, the
phase trajectories of the TS belong to some hypersurfaces, which are a one-
parameter family
( )f x c .
To curry out to the transition of the representative point from a hypersurface
1( )f x c on a hypersurface 2( )f x c it is necessary to involve thermal action.
Carathéodory proved that in this case the differential form included in the inte-
gral (15) for the elementary amount of heat dQ is holonomic, i.e must have an
integrating factor )(x :
)()()),()(()),(( xdSxdxxqxdxxqdQ .
Function ( )S x , which remains constant in the absence of thermal interac-
tion, corresponds to the entropy of the TS. In [17], general criteria for the holon-
omy of differential Pfaffian forms are formulated.
Thus, the second law of thermodynamics in the formulation of Carathéodory
from the point of view of control theory is a criterion for the controllability of a
n –dimensional dynamical system of the form (1) under a 1n control action.
Since this paper presents an abstract approach to equilibrium thermodynamics, the
authors consider it fundamental to extend the Carathéodory method to an arbitrary
number of control actions. In other words, below we will analyze the controllability
of dynamic systems of the form (1) with the ( 1)m n numbers of actions.
To solve the formulated problem, we study the structure of the set of phase
trajectories of solution of differential equations corresponding to m action
)(),...,(1 x
dt
dx
x
dt
dx
m . (40)
Regarding the set of phase trajectories of m systems (40), we assume that it
belongs to some invariant one-parameter manifold
( )S x c , (41)
i.e. all points ( )x t of phase trajectories of the set of systems (32), satisfy the con-
dition
( ( ))S x t c , [0, )t ,
where the constant c is determined from the initial condition
0( ( ))c S x t .
In this case, the set of states, that do not satisfy condition (41) will be inac-
cessible for a given set of actions, and from the point of view of control theory,
such a system will be uncontrollable.
To find the surface ( )S x we compose the system of differential equations in
partial derivatives, which it must satisfy
0)(,,...,0)(, 1
x
x
S
x
x
S
m . (42)
A.S. Kutsenko, S.V. Kovalenko, S.M. Kovalenko
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 76
The system (42) the conditions of orthogonality for the gradient vector of the
function ( )S x and for all vector fields that determined the TS trajectories for a
given structures of actions. Thus the problem of inaccessibility of a TS of an arbi-
trary set of actions is reduced to the analysis of the solvability of the system of
equations (42), which is the system of linear differential equations in partial de-
rivatives with respect to the function ( )S x .
The theory of linear partial differential equations is quite well developed and
contains a solution algorithm, including an analysis of the conditions for the exis-
tence of a solution [18]. This condition can be represented as
nv mmm }...]],[,[],,[...][],,[,,...,,{rank 32112121 , (43)
where ],[ is the commutator of the vector fields and .
xx
],[ .
Thus, this fulfilment of the condition (43) indicates that there is an invariant
surface ( )S x c , to which all possible trajectories of the TS motion belong, i.e. (43)
is a criterion for the uncontrollability of the generalized thermodynamic system.
THE DISCUSSION OF THE RESULTS
The thermodynamic approach to mathematical modeling of multidimensional
quasi-static processes is generalized. The concepts of energy and work corre-
sponding to each of the feasible actions on the controlled system are introduced.
The conditions for the orthogonality of the vectors of the coefficients of the dif-
ferentials of work made it possible to formulate the law of conservation of energy,
similar to the first law of thermodynamics, in the most general form without in-
volving any physical laws.
The axiomatic approach to the formation of the 2nd law of thermodynamics
has been further developed as applied to multidimensional quasi-static systems.
The concept of controllability of a quasi-static system is introduced.
Conditions for the existence of an invariant set of states of a quasi-static sys-
tem are obtained for the number of actions less than the dimension of the system.
It is shown that the existence of an invariant set is equivalent to the criterion of
uncontrollability of a quasi-static system.
REFERENCES
1. M. Plank, Thermodynamics. Leningrad-Moscow : GIZ, 1925, 311 p .
2. S. Carnot, Reflections on the driving force of fire and machines capable of develop-
ing this force: Series “Classics of natural science”. Moscow: Kniga po Trebovaniju,
2013, 80 p.
3. R. Clauzilus, “Mechanical theory of heat. Mathematical introduction,” The second
law of thermodynamics. Moscow-Leningrad: Goztekhizgrad, 1934, pp. 73–157.
4. K.A. Putilov, Thermodynamics. Moscow: Kniga po Trebovaniju, 2013, 375 p.
5. R. Kubo, Thermodynamics. Moscow: Mir, 1970, 304 p.
6. A.A. Gukhman, On the foundations of thermodynamics. Moscow: LKI, 2010, 384p.
7. G.V. Averyn, System Dynamics. Donetsk: Donbass, 2014, 403 p.
Generalization of the thermodynamic approach to multi-dimensional quasistatic processes
Системні дослідження та інформаційні технології, 2023, № 1 77
8. A.S. Kutsenko, “Optimal control of thermodynamic processes,” Mechanics and ma-
chine building, no.1, pp. 93–97, 1999.
9. A.S. Kutsenko, “Equilibrium thermodynamics from the point of view of control the-
ory,” Integration technologies and energy saving, no.4, pp. 35–41, 2001.
10. A.S. Kutsenko, S.V. Kovalenko, and V.I. Tovazhnyanskii, “Structural and Paramet-
ric Synthesis of a Stabilization System for a Quasistatic Technological Process,”
Bulletin of the National Technical University “KhPI”. Series: System analysis, man-
agement and information technology, no. 58 (1167), pp. 24–28, 2015.
11. K. Carathéodory, “On the foundations of thermodynamics,”Development of modern
physics. Moscow, 1964, pp. 188–222.
12. P.T. Landsberg, “On Suggest Simplification of Carateodory’s Thermodynamics,”
Phys. Status Solidi, vol. 1, no. 2, pp. 120–126, 1961.
13. G.V. Averin, A.V. Zvyagintseva, and M.V. Shevtsova, “On the Question of Substan-
tiation of Thermodynamic Statements by the Method of Differential Geometry of
Multidimensional Spaces,” Scientific statements of NRU BelSU, Ser. Mathematics,
Physics, issue 45, no. 27(248), pp. 36–44, 2016.
14. G. Falk and H. Jung, Axiomatik der Thermodynamik. Handbuch der Physik, III/2,
pp. 119–175, 1959.
15. E.H. Lieb and J. Yngvason, “The phisics and mathematics of the second law of
thermodynamics,” Physics Reports, vol. 310, no. 1, pp. 1–96, 1999.
16. Yu.N. Andreev, Control of finite-dimensional linear objects. Moscow: Nauka, 1976, 424 p.
17. N.I. Belokon, Thermodynamics. Moscow–Leningrad: Gosenergoizdat, 1954, 416 p.
18. I. Gorn, Introduction to the theory of differential equations with partial derivatives.
Moscow: GONTI NKTP USSR, 1938, 274 p.
Received 08.05.2022
INFORMATION ON THE ARTICLE
Alexander S. Kutsenko, ORCID: 0000-0001-6059-3694, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: kuzenko@i.ua
Sergii V. Kovalenko, ORCID: 0000-0001-8763-0862, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: adbc@ukr.net
Svitlana M. Kovalenko, ORCID: 0000-0001-6770-6778, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: svetkovalenko@gmail.com
УЗАГАЛЬНЕННЯ ТЕРМОДИНАМІЧНОГО ПІДХОДУ ДО БАГАТОВИМІРНИХ
КВАЗІСТАТИЧНИХ ПРОЦЕСІВ / О.С. Куценко, С.В. Коваленко, С.М. Коваленко
Анотація. Запропоновано і обґрунтовано метод математичного моделювання
багатовимірних квазістатичних процесів, які є узагальненням квазістатичних
процесів рівноважної термодинаміки. Отримано узагальнення першого, а та-
кож другого закону термодинаміки у формі Каратеодорі на багатовимірні ква-
зістатичні процеси. Ідея узагальнення — побудова ортогональної системи
функціоналів, аналогічних функціоналам роботи і теплоти класичної термоди-
наміки уздовж сімей фазових траєкторій, що відповідають різним видам впли-
вів на багатовимірну квазістатичну систему. Обґрунтовано подання квазіста-
тичних процесів системами звичайних диференціальних рівнянь, що містять
керувальні змінні. Отримані результати дозволяють залучити широкий арсе-
нал методів теорії керування динамічними системами до розв’язання задач
керування квазістатичними процесами.
Ключові слова: квазістатичні процеси, рівноважна термодинаміка, математи-
чне моделювання, робота, енергія, керованість, ентропія.
|
| id | journaliasakpiua-article-279760 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:08Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/1c/122d927ec2f90c4edb36faa403ee081c.pdf |
| spelling | journaliasakpiua-article-2797602023-05-24T21:28:17Z Generalization of the thermodynamic approach to multi-dimensional quasistatic processes Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів Kutsenko, Alexander Kovalenko, Sergii Kovalenko, Svitlana квазістатичні процеси рівноважна термодинаміка математичне моделювання робота енергія керованість ентропія quasi-static processes equilibrium thermodynamics mathematical modeling work energy controllability entropy A method of mathematical modeling of multidimensional quasi-static processes, a generalization of quasi-static processes of equilibrium thermodynamics, is proposed and substantiated. The authors obtain a generalization of the first and the second law of thermodynamics in the form of Carathéodory to multidimensional quasi-static processes. The idea of generalization is to construct an orthogonal system of functionals similar to the work and heat functionals of classical thermodynamics along families of phase trajectories corresponding to different types of influences on a multidimensional quasi-static system. The representation of quasi-static processes by systems of ordinary differential equations containing control variables is substantiated. The obtained results make it possible to use a wide arsenal of methods of the theory of control of dynamical systems to solve problems of control of quasi-static processes. Запропоновано і обґрунтовано метод математичного моделювання багатовимірних квазістатичних процесів, які є узагальненням квазістатичних процесів рівноважної термодинаміки. Отримано узагальнення першого, а також другого закону термодинаміки у формі Каратеодорі на багатовимірні квазістатичні процеси. Ідея узагальнення — побудова ортогональної системи функціоналів, аналогічних функціоналам роботи і теплоти класичної термодинаміки уздовж сімей фазових траєкторій, що відповідають різним видам впливів на багатовимірну квазістатичну систему. Обґрунтовано подання квазістатичних процесів системами звичайних диференціальних рівнянь, що містять керувальні змінні. Отримані результати дозволяють залучити широкий арсенал методів теорії керування динамічними системами до розв’язання задач керування квазістатичними процесами. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-03-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/279760 10.20535/SRIT.2308-8893.2023.1.05 System research and information technologies; No. 1 (2023); 62-77 Системные исследования и информационные технологии; № 1 (2023); 62-77 Системні дослідження та інформаційні технології; № 1 (2023); 62-77 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/279760/274351 |
| spellingShingle | квазістатичні процеси рівноважна термодинаміка математичне моделювання робота енергія керованість ентропія Kutsenko, Alexander Kovalenko, Sergii Kovalenko, Svitlana Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title | Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title_alt | Generalization of the thermodynamic approach to multi-dimensional quasistatic processes |
| title_full | Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title_fullStr | Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title_full_unstemmed | Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title_short | Узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| title_sort | узагальнення термодинамічного підходу до багатовимірних квазістатичних процесів |
| topic | квазістатичні процеси рівноважна термодинаміка математичне моделювання робота енергія керованість ентропія |
| topic_facet | квазістатичні процеси рівноважна термодинаміка математичне моделювання робота енергія керованість ентропія quasi-static processes equilibrium thermodynamics mathematical modeling work energy controllability entropy |
| url | https://journal.iasa.kpi.ua/article/view/279760 |
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