- Email: [email protected]
Maximum principle and open systems including two-phase flows
Rev.
0
GLn.
Therm.
Elsevier,
(1998)
37, 813-817
Paris
Maximum
principle and open systems including two-phase flows Umberto
Applicative
Network,
INFM,
lstituto
Nazionale
Lucia
per /a Fisica de//u Materia, Office Strada Savonese, 15057 Tortone,
(Received
11 September
1997,
accepted
at the Science Italy
1 April
and
Technical
Park
in Valle Scrivia,
1998)
an extension of the principle of maximum, for the variation of the entropy due to irreversibility, for the open systems. This result is analysed and a comparision between this one and other thermodynamical of stability is developed. An application to two-phase flows is done to obtain the pressure drop. 0 Elsevier,
has been variational Paris
Abstract-Recently
obtained principles entropy pressure
/
open drop
systems
/
maximum
entropy
variation
/
irreversible
processes
- Une extension de la loi de variation entropique maximale cas des systemes ouverts. Ce resultat est analyse, et sa comparaison de la stabilitk a et6 dkveloppee. Une application aux kcoulements 0 Elsevier, Paris /
systemes
ouverts
/ variation
Nomenclature surface area ........................ D diameter ........................... f friction factor G mass flow .......................... L interference coefficients. ............. L length of the pipe. .................. LT thermodynamical Lagrangian ........ L Lagrangian ......................... P pressure drop. ...................... R pressure function ................... specific entropy. .................... s S entropy ............................ T temperature. ....................... V specific volume ..................... V volume ............................ x = mvap/mliq quality velocity ............................ W
d’entropie
maximale
Greek
/ processus
m2 m
A
kg.s-’ K+.kg-’ m J J.m-3.K-1.s-1 Pa Pa J.kg-l.K-l
J.K-l K m3.kg-’ m3
thermodynamic
stability
due aux irrkversibilitk a rkcemment avec d’autres lois thermodynamiques diphasiques permet de determiner
R&urn6
entropie charge
/
g L
A P ’ *
irrkversibles
/
stabilite
/
two-phase
flows
/
pu Otre ktablie dans le variationnelles traitant leurs pertes de charge.
thermodynamique
scalar value of generalized forces local dissipation potential . generalized velocity non-linear interference coefficients mass density time............................... enerahzed Lagrangian thermodynam$a1 codrdinates non-linear dissipative potential. .
/
pertes
de
J.kg-‘.s-l.K-’
kg.rne3 S
J.kg-l.s-’
K-l
Subscripts
in
quantity which goes into the control volume irr quantity generated by irreversibility liq liquid out quantity which goes out from the control volume vap vapour
m.s-’
symbols
* Permanent postal address: via S. G. Bosco 43, 15100 Alessandria, Italy
1. INTRODUCTION Stationary states and their evolution thermodynamical paths can be studied
on reversible by the de&
813
U. Lucia
nitions of wall and of compound systems [l--4]. The physical conditions for their equilibrium and stability can be obtained by the principle of the maximum for the entropy [l, 3-51. In the scientific literature it has been pointed out that a principle of maximum was never introduced as a general principle of investigation for the stability of the open systems [l-3, 5-71. In the physical-mathematical analysis of the Gyarmati’s Principle, the maximum for the variation of the entropy due to irreversibility was demonstrated and introduced in the thermodynamical analysis of the stability for the open systems [8]. In this paper some physical considerations about this mathematical result will be expressed, comparing this principle with other thermodynamical variational principles of stability for the open systems to try to depict the state of the art in thermodynamical analysis of stability today.
2. MAXIMUM SYSTEMS
PRINCIPLE
FOR THE OPEN
To obtain the maximum principle for the open systems, a global analysis was developed [8]. The initial and the final states are assumed uniquely defined and stable, and the system has a response time 7 after which it goes from the initial to the final state. Thus the state functions satisfy the conditions of the calculus of variations [8%10]. Following Gyarmati [8, 101 a continuum general system was examined and its partition in its subsystems with mass dm and volume dV = dm/p, with p the density, has been made. For every sub-system the density of the thermodynamical Lagrangian per unit of time and temperature L has been defined as:
where LT is the thermodynamical analytical expression of L is [8, lo]:
Lagrangian.
The
[8, lo]:
and P, the non-linear
P-3:
814
dissipative
LT =/dtJ’dT/dVL
potential
density;
is
= J’ ASirrdT
(5)
where ASi,, is the variation of the entropy due to irreversibility. Thus, as a consequence of the principle of the last action [2. 8 -lo], the following result was obtained [8]: b(ASirr)
= 0
(6)
So the required principle of stability for the open systems’ stationary states was deduced and demonstrated. This principle can be expressed as: the condition of stability for the open systems’ stationary states consists of the maximum for the variation in the entropy due to irreversibility.
3. OTHER VARIATIONAL OF STABILITY In the analysis of Prigogine introduced a To do so, the differential be considered [2: 8: lo-
PRINCIPLE
the stability, Glansdorff and universal criterion of evolution. of the entropy production must 121:
dS=-T.dd-7;-dT.T
(7)
and then two different situations must be considered. 1) The linear one, in which are verified the Onsager relations, so the variations of the velocity 7 and of the forces 2 are connected to the phenomenological relations [2] by which the following condition can be obtained [2: 8, 11, 121: dS=2xA
L=iLp dV where
To obtain the thermodynamical Lagrangian from the relation (1) the integration on the total volume V. on the temperature and on time was done, obtaining by the analytical expression (a)! with equations (3) and (4): and by the Gouy-Stodola theorem, which states that; when a system operates irreversibly, it destroys work at the rate that is proportional to the system’s rate entropy generation [6], i. e. the following expression [8]:
t3LzdL3 =2(-?.d-i’)
(8)
2) The non-linear one in which the Onsager relations are not valid; in this case the sign of the differential component 2 d-i’ cannot be defined in a general way, while the sign of the other differential component can be deduced mathematically. Thus, as a consequence of the assumption of the local equilibrium, mathematically expressed by the condition of convexity for the entropy surface, the relation follows [2, 8, 11. la]:
Maximum
principle
and open systems including
Equations (8) and the (9) give the general criterion of evolution, for which it is sufficient that the entropy production be minimum at the stability. Lavenda developed a Lagrangian theory using the principle of stability and demonstrated, by the principle of least action, that the linear dissipation function 4 = (1/2) c Lij Ei ej is always minimum at the stability [2, 61. As 1 consequence the Gibbs function and the thermodynamical Lagrangian L are minimum [2], as Gyarmati has deduced by a variational analysis of the thermodynamical systems [8, lo].
4. CONSIDERATIONS ABOUT THE MAXIMUM PRINCIPLE THE OPEN SYSTEMS
FOR
In this way in rational thermodynamics a general formulation of the principle of stability for the stationary states of the open systems has been deduced. This consists of the maximum for the variation of the entropy, due to the irreversibility [S]. It is the extension of the maximum entropy principle, up today known only for the isolated systems, to the open ones. This result is important in thermodynamics and in statistical mechanics because it gives us a global theoretical principle for the analysis of the stability of the open systems’ states. This result not only agrees with the other principle, but represents an extension of them to a general analysis. In fact we can notice that the least entropy production principle, expressed by relation (8), is referred by Prigogine to the first derivative, on time, of the specific entropy: this is one order of differential more than the entropy. So the two quantities must have opposite signs in their variation [13, 141. Moreover the Onsager conditions, fundamentals of this principle, must be verified [ll, 121 and it happens only in linear physics [2, 11, 121, while this restriction is not necessary for the maximum principle [8]. The maximum principle agrees also with the Lavenda one, which states that the dissipation function must be minimum at the stability [6]; in fact when the entropy is maximum the Gibbs potential is minimum [l-6], and it implies also the minimum for the dissipation function
[6,81. Moreover the maximum principle has been deduced starting from Gyarmati’s principle, which states that the thermodynamic Lagrangian is always minimum at the stability, so it agrees with this too [8, 151. The maximum principle obtained agrees with the Curie principle, which states that any dissymmetry in effects must be found in causes and any symmetry in these latter ones belongs to the former ones. The
two-phase
flows
maximum principle conditions of stability are obtained starting from the causes of the stability connected with the entropy variation due to irreversibility. In fact, by a phenomenological analysis of the Rayleigh-BBnard instability, the maximum variation of the entropy due to irreversibility has been recently observed always present only in the stable states of the open systems [6]. The importance of the principle obtained will be pointed out when it will be applied in physical analysis of phase transition, of hydrodynamic cavitation [15], of the behaviour of the materials with ‘memory’, of cybernetics and other physical and engineering problems.
5. AN APPLICATION: THERMODYNAMICAL ANALYSIS OF TWO-PHASE FLOWS In this section an application of the previous results and consideration is shown: the global analysis of the ideal two-phase flows will be developed by applying the principle of maximum of irreversible entropy variation. To do that, we consider that the pressure drop Ed of an ideal flow satisfies the Bernoulli’s law (for horizontal pipe Pid = pw2/2) and, as a consequence, in a similar way, the pressure drop of an ideal two-phase flows can be considered as a function ?I?of the only velocity w of the flow in the pipe considered, that is: P = R(w)
(10)
from which its differential dP results: dP=~dw=&-j
W
ClW
(11)
Now, we consider the first and second laws of thermodynamics for the open systems [15-171: out out T -zz ASir, GvdP+ GwdW 02) A’ s in s in where T is the temperature, G the msss flow, ASi,, is the irreversible entropy variation, At the time in which the irreversible entropy variation occurs, v the specific volume, P the pressure drop, w the velocity of the fluid and ‘in’ and ‘out’ represent the entrance and the discharge sections respectively. By using relations (10) and (11) in relation (12), considering the mass ilow G to be constant, the following functional expression can be obtained:
TASin =G Out At sin(w!R’+w)dw
(13)
Considering temperature T constant and applying the principle of maximum for the irreversible entropy variation (6), it results: S(w%‘$w)
=o
(14)
815
U. Lucia
from which the following differential equation. in function of the quality x = mvap/mliq, with mvap the vapour mass and mliq the liquid mass, can be obtained:
that the authors’ Rohsenow results drop.
formula is in accordance with the and does not overvalue the pressure
(15)
6. CONCLUSIONS Now, considering
that
[15-171:
fdw
G
dz
=
2
dv z
=
udiff
vdiff
and using these relations in equation tial equation (15) itself results: dR’
(15)> the differenG
(17)
by using where udiff = Vliq - ‘u,,~,. This equation, relations (11) and (16), can be written as: G2
Vdiff
(Uliq + Vdiff X) E
the
v2.
= - Az(Vliq +dci,
considered with the following boundary obtained by the relations which describe behaviour without vapour 117, 181:
TX)
(I’)
condition, the fluid
The principle of maximum for the variation of the entropy due to irreversibility has been analysed and compared with other thermodynamical variational principles of stability. This result not only agrees with them. but represents also their extension and completion. It follows that it represents a general principle of investigation for the stability of the open systems. In fact this principle states that: in a general nonlinear thermodynamical transformation, the condition of the stability for the open systems’ equilibrium states consists of the maximum for t,he variation in the entropy due to the irreversibility. This statement represents an important result in thermodynamics and statistical mechanics because it gives us a global theoretical principle for the analysis of the stability of the open systems’ states. A simple and ideal application has been done to evaluate the pressure drop for the ideal two-phase flow: the result obtained agrees with the ones in literature (figure).
where A is the area of pipe section, D its diameter and L its length. Solving the differential equation (18), the pressure drop of two-phase ilows can be obtained when P is converted into AP:
(20) This result must be compared with the phenomenological relation proposed by Rohsenow et al. [18], and usually used in engineering applications [18-201: AP=4f
g
(21)
where vout is the specific volume at the discharge, win is the specific volume at the entrance of the pipe, D the diameter of the pipe, L its length and f is the friction factor [18--201. This formula, as Rohsenow himself has pointed out, is a phenomenological expression which overvalues the real pressure drop. The comparision between the phenomenological model and the authors’ is done in the figure 1, evaluated for a case of definite physical condition. It can be noted
816
Figure. Comparison between the authors’ analytical (20) and the Rohsenow’s one (21) for a pipe with 0.05 m, length 1 m and mass flow 2 kg.s-‘.
formula diameter
The results here obtained must be compared with the ones recently obtained by Bejan [al]. In fact Be,jan has developed the ‘constructal’ theory, a theory by which it is possible to predict some macroscopic shapes, originated by the spatial organisation, in nature. both in living and in non-living. The Bejan result is an optimization principle. In particular, in fluid flow analysis. he has shown that the pressure difference between the furthest point of the volume considered and the sink point is minimum. The result has been expressed as a function of the geometrical parameters of the system considered. In a similar way it was
Maximum
principle
and
open
systems
obtained an optimization of the building design [22]. Bejan himself has pointed out that the theoretical bases of the architecture of many living and non-living systems rest on an unknown design principle [21]. Our results, here obtained, represent the theoretical fundamental of this design principle. In this way our results agree with the Bejan ones and represent the theoretical basis of the constructal theory itself.
including
two-phase
[lo] Gyarmati Springer-Verlag,
flows
I., Non-equilibrium Berlin, 1970.
[l 11 Clansdorff P., Prigogine criterion in macroscopic Physics,
Thermodynamics, I., On a general evolution Physica 30 (1964) 35 1.
[12] Clansdorff P., Prigogine I., ThermodynamicTheory of Structure. Stabilitv and Fluctuations. _lohn Wilev New York, 1’971. ’ [13] York,
Apostol 1969.
T.M.,
Calculus,
Principles [14] Rudin W., McGraw-Hill, New York, 1964.
REFERENCES
John of
Wiley
& Sons.
& Sons,
Mathematical
New
Analysis,
[l 51 Lucia U., Analisi termodinamica della cavitazione con transizione di fase, Thesis, Dipartimento di Energetica ‘5. Stecco’, Universita di Firenze, Firenze, Italy, 1995 (in italian).
[l]
Callen H., ThermodynamicsJohn Wiley& Sons, New 1960. [2] Lavenda B.H., Thermodynamics of Irreversible Processes, Macmillan Press, London, 1978. [3] Landau L.D., E.M. Lifshitz, Statistical Physics, Part 1, Pergamon Press, Oxford, 1980 [4] Huang K., Statistical Mechanics, John Wiley & Sons, New York, 1987. [S] Geskin ES., Application of Thermodynamical Formalism for the Description of Process Kinetics in Flowers’ 94: Energy for the 2 i th Century: Conversion, Utilisation and Evironmental Oualitv. Florence. Italv. lulv 6-8. 1994. “’ Circus, Roma, 1994, p. 923. [6] Lucia U., Crazzini C., Global analysis of dissipations due to irreversibility, Rev. Cert. Therm. 36 (1997) 605. [7] Olah K., The entropy production: new aspects, in: Second Law Analysis of Energy Systems: towards the 2 1 st Century, Roma,July 5-7, 1995, Circus, Roma, 1995, p. 165. [8] Lucia U., Mathematical Consequences of Cyarmati’s Principle in Rational Thermodynamics, II Nuovo Cimento BllO 10 (1995) 1227. [9] Goldstein H., Classical Mechanics, Addison Wesley, New York, 1970.
York,
John
[16] Bejan A., Advanced Engineering Wiley & Sons, New York, 1988. [17]
Fluid
Bejan A., Entropy Generation through Flow, John Wiley & Sons, New York, 1982.
[18] Rohsenow W.M., Hartnett J.P., book of Heat Transfer Fundamentals, Company, New York, 1985. [19] Hammitt F.G., Phenomena, McGraw-Hill, [20] mentum
Thermodynamics,
Rohsenow Transfer,
Heat
Ganic E.N., McGraw-Hill
HandBook
Cavitation and Multiphase New York, 1980.
W.M., Choi Prentice-Hall,
H.Y., Heat, Mass London, 1961.
and
Flow and
Mo-
[21] Bejan A., Contructal theory: from thermodynamic and geometric optimization to predicting organization in nature. in: Flowers’ 97. Florence World Enerav Research Symposium ‘Clean Energy for the New Century’, 30 July1 August 1997, Gianpaolo Manfrida ed., Firenze, Italy, SGE, Padova, 1997, p. 15-28. [22] Lucia U., Geometrical fer in building’s protruding D18 (1996) 41-46.
characteristic structure,
and heat transII Nuovo Cimento
cl
817
0
GLn.
Therm.
Elsevier,
(1998)
37, 813-817
Paris
Maximum
principle and open systems including two-phase flows Umberto
Applicative
Network,
INFM,
lstituto
Nazionale
Lucia
per /a Fisica de//u Materia, Office Strada Savonese, 15057 Tortone,
(Received
11 September
1997,
accepted
at the Science Italy
1 April
and
Technical
Park
in Valle Scrivia,
1998)
an extension of the principle of maximum, for the variation of the entropy due to irreversibility, for the open systems. This result is analysed and a comparision between this one and other thermodynamical of stability is developed. An application to two-phase flows is done to obtain the pressure drop. 0 Elsevier,
has been variational Paris
Abstract-Recently
obtained principles entropy pressure
/
open drop
systems
/
maximum
entropy
variation
/
irreversible
processes
- Une extension de la loi de variation entropique maximale cas des systemes ouverts. Ce resultat est analyse, et sa comparaison de la stabilitk a et6 dkveloppee. Une application aux kcoulements 0 Elsevier, Paris /
systemes
ouverts
/ variation
Nomenclature surface area ........................ D diameter ........................... f friction factor G mass flow .......................... L interference coefficients. ............. L length of the pipe. .................. LT thermodynamical Lagrangian ........ L Lagrangian ......................... P pressure drop. ...................... R pressure function ................... specific entropy. .................... s S entropy ............................ T temperature. ....................... V specific volume ..................... V volume ............................ x = mvap/mliq quality velocity ............................ W
d’entropie
maximale
Greek
/ processus
m2 m
A
kg.s-’ K+.kg-’ m J J.m-3.K-1.s-1 Pa Pa J.kg-l.K-l
J.K-l K m3.kg-’ m3
thermodynamic
stability
due aux irrkversibilitk a rkcemment avec d’autres lois thermodynamiques diphasiques permet de determiner
R&urn6
entropie charge
/
g L
A P ’ *
irrkversibles
/
stabilite
/
two-phase
flows
/
pu Otre ktablie dans le variationnelles traitant leurs pertes de charge.
thermodynamique
scalar value of generalized forces local dissipation potential . generalized velocity non-linear interference coefficients mass density time............................... enerahzed Lagrangian thermodynam$a1 codrdinates non-linear dissipative potential. .
/
pertes
de
J.kg-‘.s-l.K-’
kg.rne3 S
J.kg-l.s-’
K-l
Subscripts
in
quantity which goes into the control volume irr quantity generated by irreversibility liq liquid out quantity which goes out from the control volume vap vapour
m.s-’
symbols
* Permanent postal address: via S. G. Bosco 43, 15100 Alessandria, Italy
1. INTRODUCTION Stationary states and their evolution thermodynamical paths can be studied
on reversible by the de&
813
U. Lucia
nitions of wall and of compound systems [l--4]. The physical conditions for their equilibrium and stability can be obtained by the principle of the maximum for the entropy [l, 3-51. In the scientific literature it has been pointed out that a principle of maximum was never introduced as a general principle of investigation for the stability of the open systems [l-3, 5-71. In the physical-mathematical analysis of the Gyarmati’s Principle, the maximum for the variation of the entropy due to irreversibility was demonstrated and introduced in the thermodynamical analysis of the stability for the open systems [8]. In this paper some physical considerations about this mathematical result will be expressed, comparing this principle with other thermodynamical variational principles of stability for the open systems to try to depict the state of the art in thermodynamical analysis of stability today.
2. MAXIMUM SYSTEMS
PRINCIPLE
FOR THE OPEN
To obtain the maximum principle for the open systems, a global analysis was developed [8]. The initial and the final states are assumed uniquely defined and stable, and the system has a response time 7 after which it goes from the initial to the final state. Thus the state functions satisfy the conditions of the calculus of variations [8%10]. Following Gyarmati [8, 101 a continuum general system was examined and its partition in its subsystems with mass dm and volume dV = dm/p, with p the density, has been made. For every sub-system the density of the thermodynamical Lagrangian per unit of time and temperature L has been defined as:
where LT is the thermodynamical analytical expression of L is [8, lo]:
Lagrangian.
The
[8, lo]:
and P, the non-linear
P-3:
814
dissipative
LT =/dtJ’dT/dVL
potential
density;
is
= J’ ASirrdT
(5)
where ASi,, is the variation of the entropy due to irreversibility. Thus, as a consequence of the principle of the last action [2. 8 -lo], the following result was obtained [8]: b(ASirr)
= 0
(6)
So the required principle of stability for the open systems’ stationary states was deduced and demonstrated. This principle can be expressed as: the condition of stability for the open systems’ stationary states consists of the maximum for the variation in the entropy due to irreversibility.
3. OTHER VARIATIONAL OF STABILITY In the analysis of Prigogine introduced a To do so, the differential be considered [2: 8: lo-
PRINCIPLE
the stability, Glansdorff and universal criterion of evolution. of the entropy production must 121:
dS=-T.dd-7;-dT.T
(7)
and then two different situations must be considered. 1) The linear one, in which are verified the Onsager relations, so the variations of the velocity 7 and of the forces 2 are connected to the phenomenological relations [2] by which the following condition can be obtained [2: 8, 11, 121: dS=2xA
L=iLp dV where
To obtain the thermodynamical Lagrangian from the relation (1) the integration on the total volume V. on the temperature and on time was done, obtaining by the analytical expression (a)! with equations (3) and (4): and by the Gouy-Stodola theorem, which states that; when a system operates irreversibly, it destroys work at the rate that is proportional to the system’s rate entropy generation [6], i. e. the following expression [8]:
t3LzdL3 =2(-?.d-i’)
(8)
2) The non-linear one in which the Onsager relations are not valid; in this case the sign of the differential component 2 d-i’ cannot be defined in a general way, while the sign of the other differential component can be deduced mathematically. Thus, as a consequence of the assumption of the local equilibrium, mathematically expressed by the condition of convexity for the entropy surface, the relation follows [2, 8, 11. la]:
Maximum
principle
and open systems including
Equations (8) and the (9) give the general criterion of evolution, for which it is sufficient that the entropy production be minimum at the stability. Lavenda developed a Lagrangian theory using the principle of stability and demonstrated, by the principle of least action, that the linear dissipation function 4 = (1/2) c Lij Ei ej is always minimum at the stability [2, 61. As 1 consequence the Gibbs function and the thermodynamical Lagrangian L are minimum [2], as Gyarmati has deduced by a variational analysis of the thermodynamical systems [8, lo].
4. CONSIDERATIONS ABOUT THE MAXIMUM PRINCIPLE THE OPEN SYSTEMS
FOR
In this way in rational thermodynamics a general formulation of the principle of stability for the stationary states of the open systems has been deduced. This consists of the maximum for the variation of the entropy, due to the irreversibility [S]. It is the extension of the maximum entropy principle, up today known only for the isolated systems, to the open ones. This result is important in thermodynamics and in statistical mechanics because it gives us a global theoretical principle for the analysis of the stability of the open systems’ states. This result not only agrees with the other principle, but represents an extension of them to a general analysis. In fact we can notice that the least entropy production principle, expressed by relation (8), is referred by Prigogine to the first derivative, on time, of the specific entropy: this is one order of differential more than the entropy. So the two quantities must have opposite signs in their variation [13, 141. Moreover the Onsager conditions, fundamentals of this principle, must be verified [ll, 121 and it happens only in linear physics [2, 11, 121, while this restriction is not necessary for the maximum principle [8]. The maximum principle agrees also with the Lavenda one, which states that the dissipation function must be minimum at the stability [6]; in fact when the entropy is maximum the Gibbs potential is minimum [l-6], and it implies also the minimum for the dissipation function
[6,81. Moreover the maximum principle has been deduced starting from Gyarmati’s principle, which states that the thermodynamic Lagrangian is always minimum at the stability, so it agrees with this too [8, 151. The maximum principle obtained agrees with the Curie principle, which states that any dissymmetry in effects must be found in causes and any symmetry in these latter ones belongs to the former ones. The
two-phase
flows
maximum principle conditions of stability are obtained starting from the causes of the stability connected with the entropy variation due to irreversibility. In fact, by a phenomenological analysis of the Rayleigh-BBnard instability, the maximum variation of the entropy due to irreversibility has been recently observed always present only in the stable states of the open systems [6]. The importance of the principle obtained will be pointed out when it will be applied in physical analysis of phase transition, of hydrodynamic cavitation [15], of the behaviour of the materials with ‘memory’, of cybernetics and other physical and engineering problems.
5. AN APPLICATION: THERMODYNAMICAL ANALYSIS OF TWO-PHASE FLOWS In this section an application of the previous results and consideration is shown: the global analysis of the ideal two-phase flows will be developed by applying the principle of maximum of irreversible entropy variation. To do that, we consider that the pressure drop Ed of an ideal flow satisfies the Bernoulli’s law (for horizontal pipe Pid = pw2/2) and, as a consequence, in a similar way, the pressure drop of an ideal two-phase flows can be considered as a function ?I?of the only velocity w of the flow in the pipe considered, that is: P = R(w)
(10)
from which its differential dP results: dP=~dw=&-j
W
ClW
(11)
Now, we consider the first and second laws of thermodynamics for the open systems [15-171: out out T -zz ASir, GvdP+ GwdW 02) A’ s in s in where T is the temperature, G the msss flow, ASi,, is the irreversible entropy variation, At the time in which the irreversible entropy variation occurs, v the specific volume, P the pressure drop, w the velocity of the fluid and ‘in’ and ‘out’ represent the entrance and the discharge sections respectively. By using relations (10) and (11) in relation (12), considering the mass ilow G to be constant, the following functional expression can be obtained:
TASin =G Out At sin(w!R’+w)dw
(13)
Considering temperature T constant and applying the principle of maximum for the irreversible entropy variation (6), it results: S(w%‘$w)
=o
(14)
815
U. Lucia
from which the following differential equation. in function of the quality x = mvap/mliq, with mvap the vapour mass and mliq the liquid mass, can be obtained:
that the authors’ Rohsenow results drop.
formula is in accordance with the and does not overvalue the pressure
(15)
6. CONCLUSIONS Now, considering
that
[15-171:
fdw
G
dz
=
2
dv z
=
udiff
vdiff
and using these relations in equation tial equation (15) itself results: dR’
(15)> the differenG
(17)
by using where udiff = Vliq - ‘u,,~,. This equation, relations (11) and (16), can be written as: G2
Vdiff
(Uliq + Vdiff X) E
the
v2.
= - Az(Vliq +dci,
considered with the following boundary obtained by the relations which describe behaviour without vapour 117, 181:
TX)
(I’)
condition, the fluid
The principle of maximum for the variation of the entropy due to irreversibility has been analysed and compared with other thermodynamical variational principles of stability. This result not only agrees with them. but represents also their extension and completion. It follows that it represents a general principle of investigation for the stability of the open systems. In fact this principle states that: in a general nonlinear thermodynamical transformation, the condition of the stability for the open systems’ equilibrium states consists of the maximum for t,he variation in the entropy due to the irreversibility. This statement represents an important result in thermodynamics and statistical mechanics because it gives us a global theoretical principle for the analysis of the stability of the open systems’ states. A simple and ideal application has been done to evaluate the pressure drop for the ideal two-phase flow: the result obtained agrees with the ones in literature (figure).
where A is the area of pipe section, D its diameter and L its length. Solving the differential equation (18), the pressure drop of two-phase ilows can be obtained when P is converted into AP:
(20) This result must be compared with the phenomenological relation proposed by Rohsenow et al. [18], and usually used in engineering applications [18-201: AP=4f
g
(21)
where vout is the specific volume at the discharge, win is the specific volume at the entrance of the pipe, D the diameter of the pipe, L its length and f is the friction factor [18--201. This formula, as Rohsenow himself has pointed out, is a phenomenological expression which overvalues the real pressure drop. The comparision between the phenomenological model and the authors’ is done in the figure 1, evaluated for a case of definite physical condition. It can be noted
816
Figure. Comparison between the authors’ analytical (20) and the Rohsenow’s one (21) for a pipe with 0.05 m, length 1 m and mass flow 2 kg.s-‘.
formula diameter
The results here obtained must be compared with the ones recently obtained by Bejan [al]. In fact Be,jan has developed the ‘constructal’ theory, a theory by which it is possible to predict some macroscopic shapes, originated by the spatial organisation, in nature. both in living and in non-living. The Bejan result is an optimization principle. In particular, in fluid flow analysis. he has shown that the pressure difference between the furthest point of the volume considered and the sink point is minimum. The result has been expressed as a function of the geometrical parameters of the system considered. In a similar way it was
Maximum
principle
and
open
systems
obtained an optimization of the building design [22]. Bejan himself has pointed out that the theoretical bases of the architecture of many living and non-living systems rest on an unknown design principle [21]. Our results, here obtained, represent the theoretical fundamental of this design principle. In this way our results agree with the Bejan ones and represent the theoretical basis of the constructal theory itself.
including
two-phase
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[21] Bejan A., Contructal theory: from thermodynamic and geometric optimization to predicting organization in nature. in: Flowers’ 97. Florence World Enerav Research Symposium ‘Clean Energy for the New Century’, 30 July1 August 1997, Gianpaolo Manfrida ed., Firenze, Italy, SGE, Padova, 1997, p. 15-28. [22] Lucia U., Geometrical fer in building’s protruding D18 (1996) 41-46.
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817