- Email: [email protected]
A fuzzy set approach to non-equilibrium thermodynamics
Fuzzy Sets and Systems 47 (1992) 39-48 North-Holland
39
A fuzzy set approach to non-equilibrium thermodynamics Dennis Singer Computer and Automation Institute, HAS, Kende u. 13-17., H-1518 Budapest, Hungary Received March 1990 Revised November 1990 Abstract: The investigation of the fuzziness of a nonequilibrium thermodynamic system is justified by the structural and parametric uncertainties of such systems. The paper gives a fuzzy set formulation of the phenomenological equations and shows a realistic approach for studying the entropy production in physical systems, the time trajectories of chemical reactions, etc. Using algorithms derived for special reaction systems, bundles of time trajectories with prescribed boundary possibility measures are calculated. Keywords: Fuzzy sets; measure of fuzziness; non-equilibrium thermodynamics; fuzzy dynamic systems.
1. Introduction The modern development of thermodynamics initiated by Casimir, Prigogine, de Groot, Meixner, Kalchalsky and others made it possible to replace the inequalities of classical equilibrium thermodynamics by equalities and this led to the consistent discipline of nonequilibrium thermodynamics (NETH). This discipline is able to interpret all spontaneous dynamic processes of macroscopic nature. The universality of NETH laws justifies that this discipline should be considered as system theory of animate and inanimate nature. From the viewpoint of NETH, the division of physics into mechanics, hydraulics, electrodynamics, thermics, etc. is partly for historical, partly for practical reasons. On the other hand, many natural phenomena do not belong to any of these physical domains and must be comprehended as combined from basic phenomena belonging to the classic ones. It should be noticed that just the study of these 'cross-effects' has shown the usefulness of the non-equilibrium
thermodynamic approach. Before forming the NETH concept, these cross-effects had been treated in an ad hoc manner. Non-equilibrium thermodynamics is, first of all, a model building tool giving structural information from the studied phenomena. For numerical calculations, especially for obtaining the process trajectories, the basic parameters must be gained from the special discipline and/or by measurement. It is nearly always presupposed that the parameter values used are crisp quantities and therefore the results must be crisp quantities, as well. In reality measured quantities are burdened with uncertainties due not only to the limited accuracy of measuring devices and persons performing the measurements, but also to reasons inherent in the problem formulation and translating this into a formalized form. Only a part of the measurement errors can be eliminated by better devices, more careful work and repeating the measurement. The inherent errors still remain! This statement becomes evident from the fact that the possible achievable correctness of measurement is different in single physical domains. Mechanical quantities, such as distance and velocity can be measured to a relative error of 1 0 - 3 % . Electrical measuring devices such as those for current or potential allow a relative precision of 0.1%. The achievable limit in entropy and enthalpy measurements is about 1%. Even higher are the errors in chemical processes, due to autocatalysis and reaction mechanism, not defined sharply, and last, but not least, due to the fact that the velocity constants can be measured in general only indirectly, these constants are uncertain to 5% and more. In accordance with these facts, it seems justified to presuppose some unsharpness in all physical processes a priori in the sense of fuzzy set theory. This unsharpness is very small in simple mechanic and electric processes but must be
0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
4o
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
taken into account in complex phenomena governed by non-equilibrium thermodynamics. According to Zadeh's definition [11]: A fuzzy quantity x is a subset of real quantities, xi, i = 1, 2 . . . . , n, of the set x and a membership function p belonging to it measuring the grade of membership of the x;'s in X,
x={xieXI/t(xi)},
i=l, 2,...,n.
In the sense of this definition, to the variables (parameters) of physical systems belong 'membership functions' measuring the possibility that the value of the variable is different from the optimal, presupposed one. This paper investigates the consequences of replacing the crisp variables and operators in non-equilibrium relations by fuzzy ones. Such a fuzzy approach should be more realistic in investigations of natural processes. Besides formal relations, methods will be shown for constructing fuzzy algorithms for system trajectories and some numerical results will be presented. In addition to a broad epistemological view, the paper tries to show the practical usefulness of the possibilistic approach to physical problems.
2. Some preliminaries concerning the formal treatment of non-equilibrium thermodynamics The principles of classical non-equilibrium thermodynamics are well presented in a series of textbooks, e.g., [2, 4, 7]. From the viewpoint of NETH, the parameters of physical systems belong to three groups, to that of flow, forces and material constants. In the scalar case the flow is defined as the amount of a physical entity m passing a unit area A perpendicular to the flow direction I = ( l / A ) d m / d r Flows are the mass flow in hydrology, the flow of electricity, the flow of entropy in thermodynamics, the material flow in the theory of diffusion and chemical kinetics, etc.
Note. The notion of flow needs no further definitions, only the notion of entropy flow, having a special role in classical thermodynamics, must be explained to some extent. Entropy is a synthetic notion in physics introduced relatively late. Its extensive character
was recognized only with the advent of NETH. A full understanding of the notion was rendered difficult by the multiplicity of possible explanations having seemingly no common basis. Such explanations are: the concept of the ideal Carnot machine, considerations of probabilities of physical states and consideration of the absolute temperature as a multiplicator for integrating non-total differentials in the thermic state space. For the purpose of the paper, the entropy S (more correctly, the entropy differences of two physical states S~ and $2) will be considered as the amount of work which can be gained from, or must be delivered to the system if the state transition is considered as reversible. In general, flow is a vector and must be defined for all vector components separately:
Ik = 1/Ak dmk/dt,
k = x, y, z.
In the vectorial case instead of Ik, it is more appropriate to operate with the amount of entity passing an elementary area dA forming any angle with the direction of flow, d~ = I D A . Considering a volume V enclosed by the surface of area A, the total amount of the physical entity, the flux ~ and the flux per unit area, the divergence of the flow div I will be f (0 = I f . dA,
d i v l = lim v---,oA '
J
(1)
respectively. In Cartesian coordinates,
a].
aIx + a]z
div I = -~x + Oy
Oz
(2)
where Ix, Iy, I~ are the components of • in the directions x, y, z. Analogously to I in the scalar case, div I is a scalar measure of the flow. All points of the continuum with a positive div • are sources, those with negative div • are sinks of the physical entity. If div ! = 0, there is neither accumulation nor removal of the entity. An integral of div I over a volume V is equal to the total flow of the entity through the surface surrounding V or formally
~vdiVldV=fAldA
(3)
which is the well-known relation of Gauss. From
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
(3) it can be shown that for any arbitrary volume, the relation 9__p_p= - d i v l (4) at holds, which means that the entity flowing out of a point can be realized only at the expense of local changes of the entity density p. Equation (4) is valid only for so-called conservative entities which cannot be created or annihilated as mass, energy, electric charge, etc. Non-conservative entities are the mole numbers n i in chemical processes and the entropy produced in irreversible processes. For these (4) must be extended by source terms: for chemical reactions by viIch, where Ich is the rate of chemical changes, vi is the stoichiometric number of component i. For the entropy flow and for the rate of chemical changes one has: aps= - d i v I~ + o, at
ap,
(s)
- - = - d i v / i + vil~h, at where ps is the entropy density p~ = S / V , and p~ is the density of the i-th reaction c o m p o n e n t . The moving agents of flows are thermodynamic forces, such as hydrostatic pressure differences, electrical potential differences, the affinities in chemical reactions, etc. In most of the cases the thermodynamic forces are vectors derived from a scalar potential field. The driving force in the potential field is defined as the negative value of the potential change at the given point of the potential space, grad p, X = - g r a d p ( x , y, z).
(6)
X is the local force driving the flow I. In the case of Cartesian coordinates, . ap , . ap , a p gradp(x, y, z) = ! ~x ~-J ~y -v k ~z "
the system from the equilibrium. For systems relatively near to equilibrium, the flows are simply proportional to the forces. So in isotropic media the electric flow, the thermal flow, and the diffusion flow are governed by Ohm's, Fourier's and Fick's laws, respectively: i=GAe,
q=KAT,
I=DAc,
(7)
where i, q and I are the electric current, the thermal and the diffusional flow in the usual notation, Ae, AT and Ac are the differences of electric potential, temperature and concentration, G, K and D are the appropriate proportionality factors, the conductivities. It was discovered experimentally early on that flows of a special physical entity are also affected by forces governing other entity flows. So the gradient of temperature in a bimetal establishes a gradient of electric potential (the Seebeck effect) or, the electric current establishes a transport of heat from one electric junction to another (the Pellier effect). In cases not too far from equilibrium, these cross-effects are also proportional to the nonconjugated forces. Such linear dependences led Onsager to the recognition of representing non-equilibrium thermodynamic systems with n flows and n conjugate thermodynamic forces with a system of linear equations [i = ~ LikXi, k=l
i = 1, 2 , . . . ,
n.
(8)
The Lik'S a r e the phenomenological (coupling) coefficients of the system, those with k = i for coupling the conjugate I and X quantities, the others for coupling the non-conjugate ones. The coefficients Lik are not fully independent from each other: Lik = Lki,
(6a)
41
i :/: k.
(9)
3. Correlations between thermodynamic forces and flows
By taking this into account, the number of independent phenomenological coefficients is reduced from n 2 to ½n2 + n. However, it must be noted that Onsager's law (9) is valid only as long as the thermodynamic flows and forces are defined in the following way (which will be presupposed):
The form of the correlations between forces and flows is highly dependent on the distance of
i=1
i, j, k are the unit vectors in the directions x, y, z.
L,X, = o
(10)
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
42
where a is the entropy production of the system. According to the Second Law of thermodynamics, o must be a positive quantity for all natural (irreversible) processes. The coefficients Lik have the dimension [flow. (force)-1].
4. Fuzzy approach to non-equilibrium thermodynamics The phenomenological coefficients are determined partly by the geometric dimensions, partly by the material constants, derived in most cases indirectly from measurable quantities. The functional relationships between the Li~'s and the measurable quantities are in the most practical cases not sharp. It is, therefore straightforward to consider the Lik's as fuzzy quantities and to start the fuzzy interpretation of non-equilibrium thermodynamics with the fuzzification of the phenomenological equations. The new approach will not alter the definitions of the thermodynamic quantities. It is not the aim of this paper to treat the principles of fuzzy set theory in more detail. The reader can find the necessary information in the fundamental papers of Zadeh [12, 13], Kaufman [8], Dubois and Prade [3], Zimmermann [4] and others. The phenomenological equations were given by (8). To extend these for the fuzzy domain, the force and flow variables Xi and /,. are considered to be fuzzy quantities Xi, ~. The phenomenological coefficients, in general functions of measured quantities and physical constants, are considered as fuzzy entities, too. The algebraic operators + and x are replaced by the appropriate fuzzy operators ~ and Q. The fuzzy form of the phenomenological equations becomes therefore: :
i. =/;.,
where /~e are the appropriate membership functions. It should be assumed that the symmetricity relation (9) holds in the fuzzy domain as well, thus £ik = Zki. Besides the phenomenological equations, the most relevant law of NETH is that of the entropy production given by (10). This becomes in the fuzzy domain:
or according to (11), O = L,10~'x
Oz~I ~) L220 z~r20 2 2
~ ) . . . @ £.. 02(,, OA'..
(13)
The algorithmic definitions of the fuzzy operators in (11) are naturally dependent on the form of the variable's membership functions. For simplifying further considerations it is assumed that the membership functions are of triangular form. In reality, the membership functions can in many cases be approximated with sufficient accuracy with triangles. A triangular fuzzy number, TFN, A can be characterised by only three (crisp) parameters A = (a, b, c). The membership function of A is
#ZA(X) =
=0,
x
= (X -- a)/(b - a), =(c-x)/(c-b),
a ~ x < b, b<~x
=0,
X>C.
Alternatively the TFN can be defined by the possibility level or and by the interval of confidence at level or, A o~; see Figure 1. ore [0, 1].
A~= [(a+(b-a)or), (c-(c-b)or)],
The definitions of the elementary algebraic operators for TFN variables are given in Table 1
[8].
(11)
1.
@ g.2 E). 2 @-.. @/5..
According to the definition, the fuzzy quantities
Zm{~,Xi,£ik},
i , k = 1,2 . . . . . n,
are of the form: z7= {z e Z I/*~(x)}
Cl
(lla)
(14)
C
x
Fig. 1. Membership function of the TFN variables.
43
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
Table 1. The resulting possibilitylevels of algebraic operations on TFN variables A = [A1, B1] = [(a I + (b 1 - a l ) a ) , B = [B 1, B2] = [(a2 +
Addition Subtraction Multiplication Division
(b2 -
component. According to (4) and (5), (15a) can be written
( c l - ( c l -- b l ) a']
a z ) O 0 , (c2 - (c2 - b2)ot)]
i=1
A(~B = [(A, + B,), (A2 + B2)] A 0 B = [(A, - B2), (A2- B,)] AOB = [ ( A , " B,), (Az- B2)]
where the entropy and compound flows Is and L are considered as vector quantities. Using the vector analysis relation
A Q B = [(AI/B2), (A2/B1) ]
(17)
div(ab) = a div b + b grad a, with a scalar, b vector, (16) can be written
Note. The expressions for the ~ and Q operations are approximations which are most valid when the differences Icd-lad and Ic21-1ad are least (thus the smaller the 'fuzziness' of the variables becomes). From Table 1 one can see that the crisp algebraic operators are only special cases of the appropriate fuzzy operators for Ic,I- lall = 0 and Ic21-1a21=0. The formulae of the fuzzy non-equilibrium thermodynamics must therefore lead, for small spreads of the physical constants, to the appropriate relations of classical NETH.
o = lq grad(l/T) + i=, ~ li grad(-zi/T) + lch A -~
where A = -F, 'gi.7~iis the affinity of the reaction. For a vertical tubular reactor (in the z-direction), with uniform temperature and concentration distribution in all cross-sections, (18) becomes a = lq.
d(1/T) dz
+ ~, I~. d ( - : r i / T ) / d z i=1
5. Fuzzy thermodynamics relations for special systems In the following the fuzzy relations (11) and (12) will be examined for two special cases in more detail.
5.1. The entropy production of chemical reactions occurring in a reactor with a linear temperature gradient The entropy density, the density of heat and that of the i-th chemical component at a point of the system are Ps, Pq, Pi. According to the First Law of thermodynamics,
T dPs = dpq - ~ ati dpi
(18)
+ I¢hA.
(19)
1
For fuzzifying (19), the derivatives will be written in difference form: d£ 1 dz ~ Az 0 (2i 0 £j+,),
(20)
where j is the index of the z-interval, Az is a small crisp quantity equal for all j's. The fuzzy entropy production Oj in the j-th section of the reactor will be:
i =1
(15)
i=1 or
TO..Ps= OPq
at
Ot
2., ati
i=1
api Ot '
(15a)
where z~ is the chemical potential of the i-th
Comparing (21) and (12), one can see that the conjugated fuzzy thermodynamic forces of the heat and diffusion flows are represented by the terms in the first and second { } brackets of (21). The fuzzy thermodynamic force of the reaction is represented by the term (A (~ ~).
44
D. Singer ] Fuzzy set approach to non-equilibrium thermodynamics
5.2. The c h e m i c a l f l o w a n d the f u z z y p h e n o m e n o l o g i c a l coefficient o f s i m p l e c h e m i c a l reactions
the concentrations. In the neighbourhood of the equilibrium, if c B / c s , and p c / C c e < < l , the In-functions can be approximated by p B / c e e and pMcc~:
Let the chemical reaction be of the type
A = R T [ ( p B / c B ~ ) - (pc/Cc~)].
B,
kl k-1
' C.
(22)
For isothermal conditions and constant pressure, the phenomenological equation for the ideal crisp case is: Ich = L - X = L . A,
(23)
where A is the chemical affinity. For a reaction of type (22), A = :re - arc
(24)
where arB and arc are the chemical potentials of the substances B and C. The physical chemistry of ideal solutions furnishes the following relation between the chemical potential zi and the concentration ci (by constant temperature T) [111: ari = :rio + R T In ci
(25)
with R the gas constant. The reversible reaction (22) tends to an equilibrium for which the equilibrium potentials arB~ and arc~ become equal: (26)
arBe = arCe"
The driving forces of the system in nonequilibrium states are in reality the potential differences z z - arne and a r c - arc¢. Introducing, therefore, new variables into (25), Pe = ce - CBe
and
P c = Cc - Cc~,
(27)
affinity of the reaction (22) becomes: A = arzo + R T In Cze + R T In(1 + Pz/CB¢) - arco - R T In Cce - R T In(1 + pc/Cc¢)
where cz~ and cc~ are the equilibrium concentrations. In equilibrium A vanishes. This is only possible if areo + R T In cz~ - arce - R T In c = 0.
(29)
Ich = L R T [ ( p B / c B e ) - (pc/Cc¢)].
(30)
As can be seen from (30) the driving force of the chemical reaction is not a linear function of
(32)
The phenomenological coefficient L of the chemical reaction can be obtained on the basis of kinetic considerations. For a reaction of type (22), one has: __ = dCSdt - - k l C B + k - l C c ,
dcc . ~ = KlCB -- k - i c e ,
(33)
where kl and k - i are the velocity constants for the forward and backward reaction, respectively. By a substitution from (27), dcB
d--7 = - k l ( P B + cBe) + k - l ( P c + Cco).
(34)
Since for equilibrium d c s / d t = 0 and PB = P c = O, cn~/Cc~ = k - 1 / k l = K ,
k-1 = Kkl.
(35)
K is the equilibrium constant of the reaction. Since there is no exchange of matter with the environment and the volume is assumed to be constant the law of the conservation of mass requires CB + CC = C8~ + CC~
CB -- Cn~ = --(Cc -- CCn),
(36)
PB = - P c .
(see (22)). According to (30), (31) and (32), the chemical flow Ich will be dcB dt
= c e ( k , + k - l ) = k~pn(1 + K ) ,
(37)
leh = L rL~ pB(1 + K). CBe
Since l~h from (37) and (32) must be equal, the phenomenological coefficient L becomes L = (RT)-lklCBe .
Equation (28) can therefore be simplified to A = R T [ 2 In(1 + p s / C B , ) -- In(1 + p c / c c e ) ] .
Inserting this into (23), one has:
leh = -
(28)
(31)
(38)
The thermodynamic driving force of the reaction is X = R T ( 1 + K ) ( - 1 + CB/CB¢).
(39)
45
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics In most cases the appropriate fuzzy expressions can be simplified, considering only a part of the variables as fuzzy. It is in all cases reasonable to consider the rate constant kl as a fuzzy quantity. kt is measured only indirectly and its 'true' value is affected in many cases by badly definable autocatalytic effects, as well. K, ca, and ca are results of relatively precise measurements and therefore these can be regarded in most cases as crisp quantities. Under these assumptions, the fuzzy phenomenological equation will be:
solution of (42) is of the form
ca(t) = ca, + (CBo -- Ca,)(1 + K ) e x p ( - ( 1 + K)kt). (43) similarly for kll :# k12 one has for the solution of (42),
Ca(l) = [CBl(t), cB2(t)] = cae + (ca0 - c s , ) e x p ( - ( 1 + K)[kll,
k121t)
= [(cae + (ca0 - ca,)exp(-(1 + g)kl2t)) , (cae + (ca0 - ca,)exp(-.(1 + K ) k l d ) ) ] (43a)
For reactions approaching the equilibrium state slowly, K and cBe are fuzzy quantities as well. In such cases one has for L and X:
where cm(t) and cs2(t) represent the boundaries of the possibility (uncertainty) region of the Ca(t) trajectory for the possibility level 0¢. In a similar manner the possibility region of ~c(t) can be calculated. According to (36) this becomes
L = (RT)-IQeBeC)k,
ec(t) = [ccl(t), cc2(/)]
L = (g T)- 1CBeQ kl,
(40)
f~ = R T ( 1 + K ) ( - 1 + cB/cs,).
2 = RT(1 +/~) Q ( - 1 • (1 (~)eB,) (~) Ca).
(41)
The fuzzy phenomenological equations for other types of reaction can be evaluated in a similar way.
6. The fuzzy time trajectories of nonequilibrium thermodynamic systems
= [((cB0 - Ca,)(1 -- exp(--(1 + K)k,2t), ((ca0 - ca,)(1 - e x p ( - ( 1 + K)k,,t)]. (43b) Figure 2 shows the computer plot for the reaction (22). The numeric values of the constants are as follows: the equilibrium constant K = 0.05; the initial and equilibrium concentration of the component B and C are CBo= 1.130 mol •dm -3,
Considering the rate constant kl as a triangular fuzzy n u m b e r / ~ = [kH, kl2], (40) can be written
= --[((Ca -- CB¢)(1 + K)kH), (42)
where kH and k12 are functions of the possibility level or, k11=a+(b-a)ce,
respectively. The TFN parameters of the rate constant are /~ = (a, b, c) = (1.17, 1.30, 1.43). 10-2 min.
dea/dt = -(Ca - CB¢)(1 + K)[k,,, k~2] ((ca - ca,)(1 + K)k12)]
ca, = 0.029 m o l . dm -3
k12=c-(c-b)te;
see (15). Equation (42) represents a bundle of linear differential equations furnishing for each value a¢ two time trajectories Ca1 and cs2. (For tr = 1, both trajectories are identical.) For the crisp case, kH = k~2 = kl, assuming, that the initial concentrations of B and C at time t = 0 are cs = C8o and cc = 0 respectively, the
The curves 1 and 2 in the figure are the lower and upper boundaries of the possibility region of Ca(t) for tr = 0 respectively. Curve 3 is. the fiB(t) trajectory for o~= 1. As a second example the reaction scheme 2n,
kl , c k-1
(44)
will be studied, and concretely the fuzzy time trajectories of the dimerization of cyclohexane, following this reaction scheme, will be computed. B and C denote cyclohexane C6Hlo and dicyclohexane C12H20 respectively. The phenomenological equation of the reaction in fuzzified
46
D. Singer / Fuzzy set approach to non-equifibrium thermodynamics I 1.0
% "0
"6 E
0.5
i
200
36o
46o
t min
5c;o
Fig. 2. Fuzzy time trajectories of a reaction of type B ~- C.
form is
written
Lh = d62/ dt =/~1 (~) (Clo O 262) 2 O/~-t (S) 62.
d62/dt = 4/~1C) {622@ (clo + 0.25K-~)62 (45)
The phenomenological equation is in this case a non-linear one. 61 and 62 are the concentrations of C6Hlo and C12H2o respectively, clo is the (crisp) initial concentration of C6Hlo,/~1 and/~_~ are the rate constants of the forward and backward reactions respectively. Using the relation /~_1=/¢1(~)K, where K is the crisp equilibrium constant of the reaction, (45) can be
+ 0.25C~o} or with/~a = [k11, k12] and c2 = [c21, c22], d62/dt = 4[ku, klE]
Q {[C21, C2212~) (C10-{-0.25K-1)[c21, c22] + 0.25C2o}.
(47)
k11, k~2, c21 and c22 are functions of the form (15) of the possibility level or.
i01 E /
(46)
1
0.5
Fig. 3. Fuzzy time trajectories of the dimerization of cyclohexane.
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
For the crisp case, k n = k12 and c21 = c22 and the integrated form of (47) is [1] ln((r2 - c2)/(rl - c2)) = ln(r2/rl) + 4(r2 - rl)kHt. (48) rl and r2 are the roots of the quadratic equation r = 0, where r denotes the expression in the { ) brackets in (47). rl, r2 = (Cl0 + 0.25K -1) + ((clo + 0.25K-~) 2 - c20)1/2. For k11~k12 and C21=/~C22 one solution of (47) similarly
(49) has for the
ln((r2 O [c2,, c221)(~) (rl t~ [c21, c22])) = ln(r2/rl) • 4(r2 -- rl)[kll, k12]t.
(50)
(As can be seen from (47), r~ and r2 are crisp quantities.) Performing the fuzzy operations in (50) one has [ln((r2 - c22)/(rl - c21)), ln((r2 - c21)/(rl - c22))] -- [(ln(r2/rl) + 4(r2 - rl)kllt), (ln(rE/r~) + 4(r2 - rl)klEt)].
(51)
The time trajectories for the dicyclohexane computed by (51) are shown in Figure 3. The numeric values of the constants are: The equilibrium constant of the reaction is K - 0.810mol. dm -3. The TFN-parameters of the rate constant are /~1 = (a, b, c) = (1.26, 1.40, 1.54) • 10 -4 dm 3- mo1-1 • s -l. The initial concentration of C6H10 and C~2H20 are C~o=2 mol. dm -3 and C2o = 0 respectively. The curves 1 and 2 in Figure 3 are the lower and upper boundaries of the possibility region of 82(t) for tr = 0. Curve 3 is the time trajectory for o~=1. Other reaction schemes are analysed and demonstrated with numerical examples, using the fuzzy approach, in the paper [10].
7. Conclusion Non-equilibrium thermodynamics is a universal tool for constructing a dynamic model of macroscopic physical systems from basic principles. Its paradigmatic significance can be
47
heavily overestimated! On the other hand, its practical use is limited by the fact that some parameter values of N E T H models are charged with considerable errors or are even uncertain. This is mainly the case for systems far f r o m equilibrium. The paper presents an attempt to extend the applicability of N E T H also to such cases by considering the system parameters as fuzzy quantities. Models obtained by the fuzzy NETH furnish possibilistic information about the system. The paper gives numerical examples for calculating the possibility region of the time trajectories for typical N E T H systems such as chemical reactions with uncertain velocity constants. Inherent in the fuzzy approach to N E T H is the problem of measurement of the membership functions. Exhaustive information on these topics can be found in Zimmermann [14]. A relevant contribution to the question is that of Klir [9]. In the author's opinion, the choice of the membership function is above all a task for the expert, having wide practical experiences with the investigated problem.
Acknowledgement The author is indebted to Professor P. M6ritz and Mr. Jan Stroud for his aid in clarifying some concepts and for improving the stylistic level of the paper. Thanks are also due to Mrs. Judith Varga for typing the manuscript carefully.
References [1] H.-J. Bittrich, D. Haberlend und G. Just, Methoden Chemisch-Kinetischer Berechnungen (VEB Deutscher Verlag fiir Grundstotiindustrie,Leipzig, 1979). [2] S.R. de Groot and P. Mazur, Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1962). [3] D. Dubois and H. Prade, Fuzzy real algebra: Some results, Fuzzy Sets and Systems 2 (1979) 329-348. [4] D.D. Fitts, Non-equilibrium Thermodynamics (McGraw-Hill, New York, 1962). [5] C.S. Hsu, A theory of cell-to-cell mapping of dynamic systems, ASME J. Appl. Mechanics 47 (1980) 193-939. [6] A. Kandel and W.J, Byatt, Fuzzy processes, Fuzzy Sets and Systems 34 (1980) 117-152. [7] A. Katchalsky and P.F. Curran, Non-equilibrium Thermodynamics in Biophysics (Harvard University Press, Cambridge, MA, 1965).
48
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
[8] A. Kaufmann and M.M. Gupta, Fuzzy Mathematical Models (North-Holland, Amsterdam, 1988). [9] G.J. Klir, Where do we stand on measures of uncertainty, ambiguity, fuzziness and the like, Fuzzy Sets and Systems 24 (1987) 141-160. [10] D. Singer, Fuzzy chemical kinetics, Internat. J. Systems Sci. (to appear). [11] P. Stevens, Chemical Kinetics (Chapman and Hall, London, 1961). [12] L.A. Zadeh, Outline of a new approach to the analysis
of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28-44. [13] L.A. Zadeh, The role of fuzzy logic in the management of uncertainty in expert systems. Fuzzy Sets and Systems l l (1983) 199-127. [14] H.J. Zimmermann, Fuzzy Set Theory and its Application (Kluwer-Nijhoff, Dordrecht, Boston, MA, 1986). [15] H.J. Zimmermann, Fuzzy Sets, Decision Making and Expert Systems (Kluwer-Nijhoff, Dordrecht-Boston, MA, 1987).
39
A fuzzy set approach to non-equilibrium thermodynamics Dennis Singer Computer and Automation Institute, HAS, Kende u. 13-17., H-1518 Budapest, Hungary Received March 1990 Revised November 1990 Abstract: The investigation of the fuzziness of a nonequilibrium thermodynamic system is justified by the structural and parametric uncertainties of such systems. The paper gives a fuzzy set formulation of the phenomenological equations and shows a realistic approach for studying the entropy production in physical systems, the time trajectories of chemical reactions, etc. Using algorithms derived for special reaction systems, bundles of time trajectories with prescribed boundary possibility measures are calculated. Keywords: Fuzzy sets; measure of fuzziness; non-equilibrium thermodynamics; fuzzy dynamic systems.
1. Introduction The modern development of thermodynamics initiated by Casimir, Prigogine, de Groot, Meixner, Kalchalsky and others made it possible to replace the inequalities of classical equilibrium thermodynamics by equalities and this led to the consistent discipline of nonequilibrium thermodynamics (NETH). This discipline is able to interpret all spontaneous dynamic processes of macroscopic nature. The universality of NETH laws justifies that this discipline should be considered as system theory of animate and inanimate nature. From the viewpoint of NETH, the division of physics into mechanics, hydraulics, electrodynamics, thermics, etc. is partly for historical, partly for practical reasons. On the other hand, many natural phenomena do not belong to any of these physical domains and must be comprehended as combined from basic phenomena belonging to the classic ones. It should be noticed that just the study of these 'cross-effects' has shown the usefulness of the non-equilibrium
thermodynamic approach. Before forming the NETH concept, these cross-effects had been treated in an ad hoc manner. Non-equilibrium thermodynamics is, first of all, a model building tool giving structural information from the studied phenomena. For numerical calculations, especially for obtaining the process trajectories, the basic parameters must be gained from the special discipline and/or by measurement. It is nearly always presupposed that the parameter values used are crisp quantities and therefore the results must be crisp quantities, as well. In reality measured quantities are burdened with uncertainties due not only to the limited accuracy of measuring devices and persons performing the measurements, but also to reasons inherent in the problem formulation and translating this into a formalized form. Only a part of the measurement errors can be eliminated by better devices, more careful work and repeating the measurement. The inherent errors still remain! This statement becomes evident from the fact that the possible achievable correctness of measurement is different in single physical domains. Mechanical quantities, such as distance and velocity can be measured to a relative error of 1 0 - 3 % . Electrical measuring devices such as those for current or potential allow a relative precision of 0.1%. The achievable limit in entropy and enthalpy measurements is about 1%. Even higher are the errors in chemical processes, due to autocatalysis and reaction mechanism, not defined sharply, and last, but not least, due to the fact that the velocity constants can be measured in general only indirectly, these constants are uncertain to 5% and more. In accordance with these facts, it seems justified to presuppose some unsharpness in all physical processes a priori in the sense of fuzzy set theory. This unsharpness is very small in simple mechanic and electric processes but must be
0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
4o
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
taken into account in complex phenomena governed by non-equilibrium thermodynamics. According to Zadeh's definition [11]: A fuzzy quantity x is a subset of real quantities, xi, i = 1, 2 . . . . , n, of the set x and a membership function p belonging to it measuring the grade of membership of the x;'s in X,
x={xieXI/t(xi)},
i=l, 2,...,n.
In the sense of this definition, to the variables (parameters) of physical systems belong 'membership functions' measuring the possibility that the value of the variable is different from the optimal, presupposed one. This paper investigates the consequences of replacing the crisp variables and operators in non-equilibrium relations by fuzzy ones. Such a fuzzy approach should be more realistic in investigations of natural processes. Besides formal relations, methods will be shown for constructing fuzzy algorithms for system trajectories and some numerical results will be presented. In addition to a broad epistemological view, the paper tries to show the practical usefulness of the possibilistic approach to physical problems.
2. Some preliminaries concerning the formal treatment of non-equilibrium thermodynamics The principles of classical non-equilibrium thermodynamics are well presented in a series of textbooks, e.g., [2, 4, 7]. From the viewpoint of NETH, the parameters of physical systems belong to three groups, to that of flow, forces and material constants. In the scalar case the flow is defined as the amount of a physical entity m passing a unit area A perpendicular to the flow direction I = ( l / A ) d m / d r Flows are the mass flow in hydrology, the flow of electricity, the flow of entropy in thermodynamics, the material flow in the theory of diffusion and chemical kinetics, etc.
Note. The notion of flow needs no further definitions, only the notion of entropy flow, having a special role in classical thermodynamics, must be explained to some extent. Entropy is a synthetic notion in physics introduced relatively late. Its extensive character
was recognized only with the advent of NETH. A full understanding of the notion was rendered difficult by the multiplicity of possible explanations having seemingly no common basis. Such explanations are: the concept of the ideal Carnot machine, considerations of probabilities of physical states and consideration of the absolute temperature as a multiplicator for integrating non-total differentials in the thermic state space. For the purpose of the paper, the entropy S (more correctly, the entropy differences of two physical states S~ and $2) will be considered as the amount of work which can be gained from, or must be delivered to the system if the state transition is considered as reversible. In general, flow is a vector and must be defined for all vector components separately:
Ik = 1/Ak dmk/dt,
k = x, y, z.
In the vectorial case instead of Ik, it is more appropriate to operate with the amount of entity passing an elementary area dA forming any angle with the direction of flow, d~ = I D A . Considering a volume V enclosed by the surface of area A, the total amount of the physical entity, the flux ~ and the flux per unit area, the divergence of the flow div I will be f (0 = I f . dA,
d i v l = lim v---,oA '
J
(1)
respectively. In Cartesian coordinates,
a].
aIx + a]z
div I = -~x + Oy
Oz
(2)
where Ix, Iy, I~ are the components of • in the directions x, y, z. Analogously to I in the scalar case, div I is a scalar measure of the flow. All points of the continuum with a positive div • are sources, those with negative div • are sinks of the physical entity. If div ! = 0, there is neither accumulation nor removal of the entity. An integral of div I over a volume V is equal to the total flow of the entity through the surface surrounding V or formally
~vdiVldV=fAldA
(3)
which is the well-known relation of Gauss. From
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
(3) it can be shown that for any arbitrary volume, the relation 9__p_p= - d i v l (4) at holds, which means that the entity flowing out of a point can be realized only at the expense of local changes of the entity density p. Equation (4) is valid only for so-called conservative entities which cannot be created or annihilated as mass, energy, electric charge, etc. Non-conservative entities are the mole numbers n i in chemical processes and the entropy produced in irreversible processes. For these (4) must be extended by source terms: for chemical reactions by viIch, where Ich is the rate of chemical changes, vi is the stoichiometric number of component i. For the entropy flow and for the rate of chemical changes one has: aps= - d i v I~ + o, at
ap,
(s)
- - = - d i v / i + vil~h, at where ps is the entropy density p~ = S / V , and p~ is the density of the i-th reaction c o m p o n e n t . The moving agents of flows are thermodynamic forces, such as hydrostatic pressure differences, electrical potential differences, the affinities in chemical reactions, etc. In most of the cases the thermodynamic forces are vectors derived from a scalar potential field. The driving force in the potential field is defined as the negative value of the potential change at the given point of the potential space, grad p, X = - g r a d p ( x , y, z).
(6)
X is the local force driving the flow I. In the case of Cartesian coordinates, . ap , . ap , a p gradp(x, y, z) = ! ~x ~-J ~y -v k ~z "
the system from the equilibrium. For systems relatively near to equilibrium, the flows are simply proportional to the forces. So in isotropic media the electric flow, the thermal flow, and the diffusion flow are governed by Ohm's, Fourier's and Fick's laws, respectively: i=GAe,
q=KAT,
I=DAc,
(7)
where i, q and I are the electric current, the thermal and the diffusional flow in the usual notation, Ae, AT and Ac are the differences of electric potential, temperature and concentration, G, K and D are the appropriate proportionality factors, the conductivities. It was discovered experimentally early on that flows of a special physical entity are also affected by forces governing other entity flows. So the gradient of temperature in a bimetal establishes a gradient of electric potential (the Seebeck effect) or, the electric current establishes a transport of heat from one electric junction to another (the Pellier effect). In cases not too far from equilibrium, these cross-effects are also proportional to the nonconjugated forces. Such linear dependences led Onsager to the recognition of representing non-equilibrium thermodynamic systems with n flows and n conjugate thermodynamic forces with a system of linear equations [i = ~ LikXi, k=l
i = 1, 2 , . . . ,
n.
(8)
The Lik'S a r e the phenomenological (coupling) coefficients of the system, those with k = i for coupling the conjugate I and X quantities, the others for coupling the non-conjugate ones. The coefficients Lik are not fully independent from each other: Lik = Lki,
(6a)
41
i :/: k.
(9)
3. Correlations between thermodynamic forces and flows
By taking this into account, the number of independent phenomenological coefficients is reduced from n 2 to ½n2 + n. However, it must be noted that Onsager's law (9) is valid only as long as the thermodynamic flows and forces are defined in the following way (which will be presupposed):
The form of the correlations between forces and flows is highly dependent on the distance of
i=1
i, j, k are the unit vectors in the directions x, y, z.
L,X, = o
(10)
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
42
where a is the entropy production of the system. According to the Second Law of thermodynamics, o must be a positive quantity for all natural (irreversible) processes. The coefficients Lik have the dimension [flow. (force)-1].
4. Fuzzy approach to non-equilibrium thermodynamics The phenomenological coefficients are determined partly by the geometric dimensions, partly by the material constants, derived in most cases indirectly from measurable quantities. The functional relationships between the Li~'s and the measurable quantities are in the most practical cases not sharp. It is, therefore straightforward to consider the Lik's as fuzzy quantities and to start the fuzzy interpretation of non-equilibrium thermodynamics with the fuzzification of the phenomenological equations. The new approach will not alter the definitions of the thermodynamic quantities. It is not the aim of this paper to treat the principles of fuzzy set theory in more detail. The reader can find the necessary information in the fundamental papers of Zadeh [12, 13], Kaufman [8], Dubois and Prade [3], Zimmermann [4] and others. The phenomenological equations were given by (8). To extend these for the fuzzy domain, the force and flow variables Xi and /,. are considered to be fuzzy quantities Xi, ~. The phenomenological coefficients, in general functions of measured quantities and physical constants, are considered as fuzzy entities, too. The algebraic operators + and x are replaced by the appropriate fuzzy operators ~ and Q. The fuzzy form of the phenomenological equations becomes therefore: :
i. =/;.,
where /~e are the appropriate membership functions. It should be assumed that the symmetricity relation (9) holds in the fuzzy domain as well, thus £ik = Zki. Besides the phenomenological equations, the most relevant law of NETH is that of the entropy production given by (10). This becomes in the fuzzy domain:
or according to (11), O = L,10~'x
Oz~I ~) L220 z~r20 2 2
~ ) . . . @ £.. 02(,, OA'..
(13)
The algorithmic definitions of the fuzzy operators in (11) are naturally dependent on the form of the variable's membership functions. For simplifying further considerations it is assumed that the membership functions are of triangular form. In reality, the membership functions can in many cases be approximated with sufficient accuracy with triangles. A triangular fuzzy number, TFN, A can be characterised by only three (crisp) parameters A = (a, b, c). The membership function of A is
#ZA(X) =
=0,
x
= (X -- a)/(b - a), =(c-x)/(c-b),
a ~ x < b, b<~x
=0,
X>C.
Alternatively the TFN can be defined by the possibility level or and by the interval of confidence at level or, A o~; see Figure 1. ore [0, 1].
A~= [(a+(b-a)or), (c-(c-b)or)],
The definitions of the elementary algebraic operators for TFN variables are given in Table 1
[8].
(11)
1.
@ g.2 E). 2 @-.. @/5..
According to the definition, the fuzzy quantities
Zm{~,Xi,£ik},
i , k = 1,2 . . . . . n,
are of the form: z7= {z e Z I/*~(x)}
Cl
(lla)
(14)
C
x
Fig. 1. Membership function of the TFN variables.
43
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
Table 1. The resulting possibilitylevels of algebraic operations on TFN variables A = [A1, B1] = [(a I + (b 1 - a l ) a ) , B = [B 1, B2] = [(a2 +
Addition Subtraction Multiplication Division
(b2 -
component. According to (4) and (5), (15a) can be written
( c l - ( c l -- b l ) a']
a z ) O 0 , (c2 - (c2 - b2)ot)]
i=1
A(~B = [(A, + B,), (A2 + B2)] A 0 B = [(A, - B2), (A2- B,)] AOB = [ ( A , " B,), (Az- B2)]
where the entropy and compound flows Is and L are considered as vector quantities. Using the vector analysis relation
A Q B = [(AI/B2), (A2/B1) ]
(17)
div(ab) = a div b + b grad a, with a scalar, b vector, (16) can be written
Note. The expressions for the ~ and Q operations are approximations which are most valid when the differences Icd-lad and Ic21-1ad are least (thus the smaller the 'fuzziness' of the variables becomes). From Table 1 one can see that the crisp algebraic operators are only special cases of the appropriate fuzzy operators for Ic,I- lall = 0 and Ic21-1a21=0. The formulae of the fuzzy non-equilibrium thermodynamics must therefore lead, for small spreads of the physical constants, to the appropriate relations of classical NETH.
o = lq grad(l/T) + i=, ~ li grad(-zi/T) + lch A -~
where A = -F, 'gi.7~iis the affinity of the reaction. For a vertical tubular reactor (in the z-direction), with uniform temperature and concentration distribution in all cross-sections, (18) becomes a = lq.
d(1/T) dz
+ ~, I~. d ( - : r i / T ) / d z i=1
5. Fuzzy thermodynamics relations for special systems In the following the fuzzy relations (11) and (12) will be examined for two special cases in more detail.
5.1. The entropy production of chemical reactions occurring in a reactor with a linear temperature gradient The entropy density, the density of heat and that of the i-th chemical component at a point of the system are Ps, Pq, Pi. According to the First Law of thermodynamics,
T dPs = dpq - ~ ati dpi
(18)
+ I¢hA.
(19)
1
For fuzzifying (19), the derivatives will be written in difference form: d£ 1 dz ~ Az 0 (2i 0 £j+,),
(20)
where j is the index of the z-interval, Az is a small crisp quantity equal for all j's. The fuzzy entropy production Oj in the j-th section of the reactor will be:
i =1
(15)
i=1 or
TO..Ps= OPq
at
Ot
2., ati
i=1
api Ot '
(15a)
where z~ is the chemical potential of the i-th
Comparing (21) and (12), one can see that the conjugated fuzzy thermodynamic forces of the heat and diffusion flows are represented by the terms in the first and second { } brackets of (21). The fuzzy thermodynamic force of the reaction is represented by the term (A (~ ~).
44
D. Singer ] Fuzzy set approach to non-equilibrium thermodynamics
5.2. The c h e m i c a l f l o w a n d the f u z z y p h e n o m e n o l o g i c a l coefficient o f s i m p l e c h e m i c a l reactions
the concentrations. In the neighbourhood of the equilibrium, if c B / c s , and p c / C c e < < l , the In-functions can be approximated by p B / c e e and pMcc~:
Let the chemical reaction be of the type
A = R T [ ( p B / c B ~ ) - (pc/Cc~)].
B,
kl k-1
' C.
(22)
For isothermal conditions and constant pressure, the phenomenological equation for the ideal crisp case is: Ich = L - X = L . A,
(23)
where A is the chemical affinity. For a reaction of type (22), A = :re - arc
(24)
where arB and arc are the chemical potentials of the substances B and C. The physical chemistry of ideal solutions furnishes the following relation between the chemical potential zi and the concentration ci (by constant temperature T) [111: ari = :rio + R T In ci
(25)
with R the gas constant. The reversible reaction (22) tends to an equilibrium for which the equilibrium potentials arB~ and arc~ become equal: (26)
arBe = arCe"
The driving forces of the system in nonequilibrium states are in reality the potential differences z z - arne and a r c - arc¢. Introducing, therefore, new variables into (25), Pe = ce - CBe
and
P c = Cc - Cc~,
(27)
affinity of the reaction (22) becomes: A = arzo + R T In Cze + R T In(1 + Pz/CB¢) - arco - R T In Cce - R T In(1 + pc/Cc¢)
where cz~ and cc~ are the equilibrium concentrations. In equilibrium A vanishes. This is only possible if areo + R T In cz~ - arce - R T In c = 0.
(29)
Ich = L R T [ ( p B / c B e ) - (pc/Cc¢)].
(30)
As can be seen from (30) the driving force of the chemical reaction is not a linear function of
(32)
The phenomenological coefficient L of the chemical reaction can be obtained on the basis of kinetic considerations. For a reaction of type (22), one has: __ = dCSdt - - k l C B + k - l C c ,
dcc . ~ = KlCB -- k - i c e ,
(33)
where kl and k - i are the velocity constants for the forward and backward reaction, respectively. By a substitution from (27), dcB
d--7 = - k l ( P B + cBe) + k - l ( P c + Cco).
(34)
Since for equilibrium d c s / d t = 0 and PB = P c = O, cn~/Cc~ = k - 1 / k l = K ,
k-1 = Kkl.
(35)
K is the equilibrium constant of the reaction. Since there is no exchange of matter with the environment and the volume is assumed to be constant the law of the conservation of mass requires CB + CC = C8~ + CC~
CB -- Cn~ = --(Cc -- CCn),
(36)
PB = - P c .
(see (22)). According to (30), (31) and (32), the chemical flow Ich will be dcB dt
= c e ( k , + k - l ) = k~pn(1 + K ) ,
(37)
leh = L rL~ pB(1 + K). CBe
Since l~h from (37) and (32) must be equal, the phenomenological coefficient L becomes L = (RT)-lklCBe .
Equation (28) can therefore be simplified to A = R T [ 2 In(1 + p s / C B , ) -- In(1 + p c / c c e ) ] .
Inserting this into (23), one has:
leh = -
(28)
(31)
(38)
The thermodynamic driving force of the reaction is X = R T ( 1 + K ) ( - 1 + CB/CB¢).
(39)
45
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics In most cases the appropriate fuzzy expressions can be simplified, considering only a part of the variables as fuzzy. It is in all cases reasonable to consider the rate constant kl as a fuzzy quantity. kt is measured only indirectly and its 'true' value is affected in many cases by badly definable autocatalytic effects, as well. K, ca, and ca are results of relatively precise measurements and therefore these can be regarded in most cases as crisp quantities. Under these assumptions, the fuzzy phenomenological equation will be:
solution of (42) is of the form
ca(t) = ca, + (CBo -- Ca,)(1 + K ) e x p ( - ( 1 + K)kt). (43) similarly for kll :# k12 one has for the solution of (42),
Ca(l) = [CBl(t), cB2(t)] = cae + (ca0 - c s , ) e x p ( - ( 1 + K)[kll,
k121t)
= [(cae + (ca0 - ca,)exp(-(1 + g)kl2t)) , (cae + (ca0 - ca,)exp(-.(1 + K ) k l d ) ) ] (43a)
For reactions approaching the equilibrium state slowly, K and cBe are fuzzy quantities as well. In such cases one has for L and X:
where cm(t) and cs2(t) represent the boundaries of the possibility (uncertainty) region of the Ca(t) trajectory for the possibility level 0¢. In a similar manner the possibility region of ~c(t) can be calculated. According to (36) this becomes
L = (RT)-IQeBeC)k,
ec(t) = [ccl(t), cc2(/)]
L = (g T)- 1CBeQ kl,
(40)
f~ = R T ( 1 + K ) ( - 1 + cB/cs,).
2 = RT(1 +/~) Q ( - 1 • (1 (~)eB,) (~) Ca).
(41)
The fuzzy phenomenological equations for other types of reaction can be evaluated in a similar way.
6. The fuzzy time trajectories of nonequilibrium thermodynamic systems
= [((cB0 - Ca,)(1 -- exp(--(1 + K)k,2t), ((ca0 - ca,)(1 - e x p ( - ( 1 + K)k,,t)]. (43b) Figure 2 shows the computer plot for the reaction (22). The numeric values of the constants are as follows: the equilibrium constant K = 0.05; the initial and equilibrium concentration of the component B and C are CBo= 1.130 mol •dm -3,
Considering the rate constant kl as a triangular fuzzy n u m b e r / ~ = [kH, kl2], (40) can be written
= --[((Ca -- CB¢)(1 + K)kH), (42)
where kH and k12 are functions of the possibility level or, k11=a+(b-a)ce,
respectively. The TFN parameters of the rate constant are /~ = (a, b, c) = (1.17, 1.30, 1.43). 10-2 min.
dea/dt = -(Ca - CB¢)(1 + K)[k,,, k~2] ((ca - ca,)(1 + K)k12)]
ca, = 0.029 m o l . dm -3
k12=c-(c-b)te;
see (15). Equation (42) represents a bundle of linear differential equations furnishing for each value a¢ two time trajectories Ca1 and cs2. (For tr = 1, both trajectories are identical.) For the crisp case, kH = k~2 = kl, assuming, that the initial concentrations of B and C at time t = 0 are cs = C8o and cc = 0 respectively, the
The curves 1 and 2 in the figure are the lower and upper boundaries of the possibility region of Ca(t) for tr = 0 respectively. Curve 3 is. the fiB(t) trajectory for o~= 1. As a second example the reaction scheme 2n,
kl , c k-1
(44)
will be studied, and concretely the fuzzy time trajectories of the dimerization of cyclohexane, following this reaction scheme, will be computed. B and C denote cyclohexane C6Hlo and dicyclohexane C12H20 respectively. The phenomenological equation of the reaction in fuzzified
46
D. Singer / Fuzzy set approach to non-equifibrium thermodynamics I 1.0
% "0
"6 E
0.5
i
200
36o
46o
t min
5c;o
Fig. 2. Fuzzy time trajectories of a reaction of type B ~- C.
form is
written
Lh = d62/ dt =/~1 (~) (Clo O 262) 2 O/~-t (S) 62.
d62/dt = 4/~1C) {622@ (clo + 0.25K-~)62 (45)
The phenomenological equation is in this case a non-linear one. 61 and 62 are the concentrations of C6Hlo and C12H2o respectively, clo is the (crisp) initial concentration of C6Hlo,/~1 and/~_~ are the rate constants of the forward and backward reactions respectively. Using the relation /~_1=/¢1(~)K, where K is the crisp equilibrium constant of the reaction, (45) can be
+ 0.25C~o} or with/~a = [k11, k12] and c2 = [c21, c22], d62/dt = 4[ku, klE]
Q {[C21, C2212~) (C10-{-0.25K-1)[c21, c22] + 0.25C2o}.
(47)
k11, k~2, c21 and c22 are functions of the form (15) of the possibility level or.
i01 E /
(46)
1
0.5
Fig. 3. Fuzzy time trajectories of the dimerization of cyclohexane.
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
For the crisp case, k n = k12 and c21 = c22 and the integrated form of (47) is [1] ln((r2 - c2)/(rl - c2)) = ln(r2/rl) + 4(r2 - rl)kHt. (48) rl and r2 are the roots of the quadratic equation r = 0, where r denotes the expression in the { ) brackets in (47). rl, r2 = (Cl0 + 0.25K -1) + ((clo + 0.25K-~) 2 - c20)1/2. For k11~k12 and C21=/~C22 one solution of (47) similarly
(49) has for the
ln((r2 O [c2,, c221)(~) (rl t~ [c21, c22])) = ln(r2/rl) • 4(r2 -- rl)[kll, k12]t.
(50)
(As can be seen from (47), r~ and r2 are crisp quantities.) Performing the fuzzy operations in (50) one has [ln((r2 - c22)/(rl - c21)), ln((r2 - c21)/(rl - c22))] -- [(ln(r2/rl) + 4(r2 - rl)kllt), (ln(rE/r~) + 4(r2 - rl)klEt)].
(51)
The time trajectories for the dicyclohexane computed by (51) are shown in Figure 3. The numeric values of the constants are: The equilibrium constant of the reaction is K - 0.810mol. dm -3. The TFN-parameters of the rate constant are /~1 = (a, b, c) = (1.26, 1.40, 1.54) • 10 -4 dm 3- mo1-1 • s -l. The initial concentration of C6H10 and C~2H20 are C~o=2 mol. dm -3 and C2o = 0 respectively. The curves 1 and 2 in Figure 3 are the lower and upper boundaries of the possibility region of 82(t) for tr = 0. Curve 3 is the time trajectory for o~=1. Other reaction schemes are analysed and demonstrated with numerical examples, using the fuzzy approach, in the paper [10].
7. Conclusion Non-equilibrium thermodynamics is a universal tool for constructing a dynamic model of macroscopic physical systems from basic principles. Its paradigmatic significance can be
47
heavily overestimated! On the other hand, its practical use is limited by the fact that some parameter values of N E T H models are charged with considerable errors or are even uncertain. This is mainly the case for systems far f r o m equilibrium. The paper presents an attempt to extend the applicability of N E T H also to such cases by considering the system parameters as fuzzy quantities. Models obtained by the fuzzy NETH furnish possibilistic information about the system. The paper gives numerical examples for calculating the possibility region of the time trajectories for typical N E T H systems such as chemical reactions with uncertain velocity constants. Inherent in the fuzzy approach to N E T H is the problem of measurement of the membership functions. Exhaustive information on these topics can be found in Zimmermann [14]. A relevant contribution to the question is that of Klir [9]. In the author's opinion, the choice of the membership function is above all a task for the expert, having wide practical experiences with the investigated problem.
Acknowledgement The author is indebted to Professor P. M6ritz and Mr. Jan Stroud for his aid in clarifying some concepts and for improving the stylistic level of the paper. Thanks are also due to Mrs. Judith Varga for typing the manuscript carefully.
References [1] H.-J. Bittrich, D. Haberlend und G. Just, Methoden Chemisch-Kinetischer Berechnungen (VEB Deutscher Verlag fiir Grundstotiindustrie,Leipzig, 1979). [2] S.R. de Groot and P. Mazur, Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1962). [3] D. Dubois and H. Prade, Fuzzy real algebra: Some results, Fuzzy Sets and Systems 2 (1979) 329-348. [4] D.D. Fitts, Non-equilibrium Thermodynamics (McGraw-Hill, New York, 1962). [5] C.S. Hsu, A theory of cell-to-cell mapping of dynamic systems, ASME J. Appl. Mechanics 47 (1980) 193-939. [6] A. Kandel and W.J, Byatt, Fuzzy processes, Fuzzy Sets and Systems 34 (1980) 117-152. [7] A. Katchalsky and P.F. Curran, Non-equilibrium Thermodynamics in Biophysics (Harvard University Press, Cambridge, MA, 1965).
48
D. Singer / Fuzzy set approach to non-equilibrium thermodynamics
[8] A. Kaufmann and M.M. Gupta, Fuzzy Mathematical Models (North-Holland, Amsterdam, 1988). [9] G.J. Klir, Where do we stand on measures of uncertainty, ambiguity, fuzziness and the like, Fuzzy Sets and Systems 24 (1987) 141-160. [10] D. Singer, Fuzzy chemical kinetics, Internat. J. Systems Sci. (to appear). [11] P. Stevens, Chemical Kinetics (Chapman and Hall, London, 1961). [12] L.A. Zadeh, Outline of a new approach to the analysis
of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28-44. [13] L.A. Zadeh, The role of fuzzy logic in the management of uncertainty in expert systems. Fuzzy Sets and Systems l l (1983) 199-127. [14] H.J. Zimmermann, Fuzzy Set Theory and its Application (Kluwer-Nijhoff, Dordrecht, Boston, MA, 1986). [15] H.J. Zimmermann, Fuzzy Sets, Decision Making and Expert Systems (Kluwer-Nijhoff, Dordrecht-Boston, MA, 1987).