50
Ind. Eng. Chem. Res. 1987, 26, 50-56
Jacobson, J. Ph.D. Dissertation, Yale University, New Haven, CT, 1984. Jacobson, J.; Frenz, J.; Horvlth, Cs. J . Chromatogr. 1984, 316, 53. Langmuir, I. J . Am. Chem. SOC.1916, 38, 2221. LeVan, M. D.; Vermeulen, T. J . Phys. Chem. 1981,85, 3247. Melander, W. R.; Erard, J. F.; Horviith, Cs. J.Chromatogr. 1983,282, 229. Rhee, H.-K.; Ark, R.; Amundson, N. R. Phil. Trans. R. SOC.(London) 1970, 267, 419.
Rhee, H.-K.;Amundson, N. R. Am. Inst. Chem. Eng. J . 1982, 28, 423. Tien, C.; Hsieh, J. S. C.; Turian, R. M. Am. Inst. Chem. Eng. J . 1976, 22, 498. Tiselius, A. Ark. Kemi, Mineral. Geol. 1943, 16A(18).1.
Received for review April 2, 1985 Accepted March 17, 1986
Equipartition of Entropy Production. An Optimality Criterion for Transfer and Separation Processes Daniel Tondeur* and Eric Kvaalent Laboratoire des Sciences du G&nieChimique, CNRS-ENSIC, Nancy, France
In a contacting or separation device involving a given transfer area and achieving a specified transfer duty, the total entropy produced is minimal when the local rate of entropy production is uniformly distributed (equipartitioned) along the space and/or time variables. This property is demonstrated under the conditions of classical nonequilibrium thermodynamics: linear flux-force relations and Onsager’s reciprocity relations. It is shown that so-called “equipartitioned” processes are optimal in some economic sense; that is, they correspond to the minimum of some cost function. The equipartition criterion can be extended to a cascade of uncoupled transfer units, such as a “diabatic” distillation column with heat exchange all along its length. There exists an optimal size distribution of the transfer units, such that the cost of the investment made on any unit is simply related to the cost of energy dissipated through irreversible processes in the same unit. The purpose of the present work is to propose that the optimal configuration (in some economic sense) of transfer or separation processes is one which satisfies or approaches equipartition (uniform distribution) of quantities related to entropy production, such as the driving forces. This approach builds on the numerous investigations concerned with reversibility analysis as a method for process analysis and improvement (Gouy, 1889; Stodola, 1910; Benedict, 1947; Mah et al., 1977; Franklin and Wilkinson, 1982; Sieniutycz, 1984; Bejan, 1982; Le Goff, 1982a,b). However, the point of view developed here is somewhat different: we are less concerned with diminishing the entropy production than with its optimal distribution along a process. Under certain sets of constraints, the two points of view do coincide: the optimal distribution leads to a relative minimum of entropy production. The reason for this line of thinking is that, usually, minimizing entropy production amounts to increasing the size of the equipment, for example, adding plates to a distillation column or adding transfer area to a heat exchanger or, alternatively, slowing down all the flow processes, thus diminishing the throughput per unit time. This approach cannot lead to an economic optimum, that is, to an optimum compromise between operating and investment expenses. We suggest that accepting a certain nonminimal amount of entropy production and emphasizing its distribution may lead to such an optimal compromise. Optimal here is meant in some intrinsic economic sense (not necessarily financial). Two distinct problems will be addressed. The first problem is that of the optimal configuration of a single unit process or operation in which the distribution of the fluxes, driving forces, and thus entropy production is entirely determined by the “boundary conditions”, for example,the
* Author to whom correspondence should be addressed. Presently with Israel Mining Industry, Haifa, Israel.
0888-5885/87/2626-0050$01.50/0
temperatures and flow rates at the entrance of a heat exchanger. In other words, once the flow configurations and boundary conditions are set, we cannot alter the distribution of irreversibilities inside the exchanger. The optimization is then the choice of the “best” flow configuration among several. The second problem concerns the optimal size distribution of different partially uncoupled parts of a process: for example, the optimal heights of transfer units in a distillation column with heat exchange all along the column.
Equipartition of Irreversibilities in a Single-Unit Process Optimality Conjecture. We propose the following conjecture: For a given duty, the best configuration of an exchanger, contactor, or separator is that in which the entropy production rate is most uniformly distributed. In the following, we shall first illustrate the meaning of this statement by using a simple example and then show that it can be rigorously demonstrated under certain sets of assumptions. Let us consider the example of cocurrent and countercurrent heat exchangers. Figure 1 shows schematically the shapes of the temperature profiles in these two “configurations”. Clearly, the driving force, AT (or more strictly A ( l / T ) ) , is more uniformly distributed in countercurrent than in cocurrent flow. The driving force being directly related to the entropy production, as we shall see below, the latter is also more uniformly distributed. Our statement means that this uniform distribution is the basic thermodynamic reason for which countercurrent contactors are “better” than cocurrent contactors, as is well-known. The notion of better and the bases of comparison have to be carefully defined, however. For that purpose, we specify the “duty” of the contactor; in the present example, this will be done by specifying the flow rate, Q, and the inlet and outlet temperatures, T,and T,, of the cold stream. The total heat flux transferred from 0 1987 American Chemical Society
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 51 2
2 in
out
VI
7
I
obvirro 2
counter current
VI
J c~current
Figure 1. Temperature and driving force distributions in countercurrent and cocurrent heat exchangers.
the hot stream to the other stream is then the given duty. If, in addition, we specify that the two contactors are geometrically identical except for the flow direction, then the cocurrent contactor will require a higher flow rate of the hot fluid and/or a higher temperature of this fluid. In other words, the operating costs are higher than that of the countercurrent contactor. Alternatively, if we specify the flow rate and inlet temperature of the hot fluid, then the cocurren t contactor will require a larger exchange surface area, in other words, higher investment. The countercurrent exchanger is thus able to minimize either operating costs or investment costs, with respect to the cocurrent. It will actually also minimize some linear combination of these two costs, representing an economic objective function; for a thorough study of this problem, see Le Goff (1982a,b). This example is somewhat trivial, but in many engineering mass-, heat-, or momentum-transfer systems, the flow configurations cannot be characterized by a simple notion such as cocurrent or countercurrent. Figure 2 illustrates this problem on the case of different arrangements of plates in a distillation column. It has been known since Lewis (1936) that the parallel-flow plates (liquid flowing in the same direction in two successive plates) have a higher efficiency than the classical “anti-parallel” plates. This can be related to the more uniform distribution of composition difference (driving force) between liquid and vapor along the plate, assuming the vapor is not perfectly mixed between plates. Note that the observation of the flow configuration and of the gradients of composition is done on a rather macroscopic scale, in both the heat exchanger and the distillation examples. Similar considerations could be applied either on a microscopic scale or on an even larger plant scale. The following section attempts to show two results. The first is that the “quality” of a flow configuration can be characterized by the uniformity of the distribution of the irreversibilities, measured in the simplest case by the driving forces. The second is that making this distribution uniform leads to an “economically”optimum design. Let us emphasize that what we want to minimize is the nonuniformity of the irreversibilities, not necessarily their average magnitude. In other words, we are not trying to make processes globally less irreversible (as this approach does not lead to economic optimization) but rather to make the best use of irreversible dissipation by locating it at the best place and moment. To reach this goal, we shall nevertheless need to relate the distribution of irreversibilities to the minimum rate of entropy production. Equipartition of Driving Forces and Minimum Entropy Production. We shall demonstrate the following theorem. Theorem of Minimal Dissipation. Consider a class of unit transfer processes of specified size, duration, and
Figure 2. Flow configurations and driving force distribution Ar for volatile component on distillation plates with parallel and antiparallel liquid flow (local equilibrium is assumed between liquid and vapor a t any position on the plate).
transfer duty. I n the range of validity of linear nonequilibrium thermodynamics and of Onsager’s relations, the less dissipative configuration is that i n which all driving forces are uniform i n space and time (equipartition). Let us first present the demonstration in the case of transfer of a single “species” (mass, heat, or momentum) before generalizing to multicomponent coupled systems. The assumptions that prevail are the ones that define the range of validity of linear flux-force relations and Onsager’s reciprocity relations (DeGroot and Mazur, 1962; Onsager, 1931). Consider a unit process in which one species is transferred from one region to another, for instance, from one phase to another. The instantaneous and local rate of entropy production, p , is expressed as P =jf
(1)
where j is the local instantaneous flux of the entity considered and f is the conjugate driving force. Assuming a linear relation between flux and driving force, we write
j = Lf (2) where L is a phenomenological transfer coefficient, assumed constant and positive over the process. The total entropy production, P , in the process is the integral of p over the time and space variables P = s v s t p dVdt = L S V S t f z d V d t
(3)
The integral over V symbolizes integration over the diferent pertinent space variables. The total flux transferred, J (the duty of the process), is the integral over time and space of the local flux
J =
svstj
dV d t = L S v S t f dV d t = LVtf (4)
where f is the average driving force over the process, defined by the last equality of eq 4. Suppose now that the driving force, f , is constant over the whole process and equal to f. The total flux, J , is unchanged by this assumption, since it is determined by the average value, f. The total entropy production, P,,in this case is then, from eq 3 P, = L m 2 V t = J f
(5)
52
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987
Let us compare the general case and this special case by calculating the difference I) - I), €’ - Pc = L [ l $ f P dV dt - (fPVt] = L V t [ p -
a*] (6)
The bracket on the right-hand side is the difference between the mean square and the square mean of the distribution, f, which by Cauchy-Schwarz’s inequality (Bass, 1968) is a positive quantity and is nothing but the variance, 2,off. We thus write I) - I)3c
Vt
= L.Z(fl
>0
P - I), = l ” l * P . [ L ] . f dV dt - %.[L].fVt =
(7)
implying
$v$t(P.[L].f
- p - [ L ] . f ) dV dt (14)
To show that this quantity is positive, we shall make use of the fact that [L], being a positive matrix, may always be decomposed nonuniquely into a product of the form [LI = [RIT.[Rl
(15)
(Actually, owing to the symmetry of [L], a unique symmetric square root exists, such that [RIT= [R]). The quadratic form fT.[L]-f may then be written P * [ L ] * f= fT-[RlT-[R]-f = (R*f)T*(R*f) (16) Let
Pc(equipartitioned)< “(arbitrary)
(8)
The entropy production, I),, of a process with uniform driving force is thus smaller than that of a nonuniform situation with the same size, V, and duration, t, of the same average driving force, implying the same overall transfer flux, J. This is the result sought after. Note that owing to eq 1 and 2 and to the assumed constancy of L , when f is constant so is the local flux, j , and the local rate of entropy production, p . Therefore, the theorem may also be stated in terms of equipartition of the fluxes or of the entropy production. The theorem may now be generalized to the simultaneous coupled transfer of several species (mass, momentum, and heat), using the linear phenomenological relations.” The vector products used here are all “dot products”, that is “contractednor “internal” products. The order of such a product is the sum of the orders of the factors minus twice the number of dots. The dot product of two vectors is the common scalar product. The product of a vector by a second-order tensor (a matrix) is a vector. The bilinear or quadratic forms such as fr-IL1.f are scalars. The instantaneous local flux and entropy production are thus written j = [L].f
(9)
p = P.j = P - [ L ] . f
(10)
[L] is the matrix of phenomenological coefficients, a symmetric matrix (owing to Onsager’s relations) with positive diagonal elements and construction such that the quadratic form on the right-hand side of eq 10 is positive definite. The total entropy production is then
P
= l V S f f T . [ L ] - f dV dt
and with constant [L],the total flux vector (the specified duty) is J = l ” I [ L ] . f d V d t = Vt[L].f
(12)
where f i s the average driving force vector, defined by eq 12, such that its components, fi,are the averages of the individual driving forces, f,. Consider the special case where f is a constant equal to fjust defined. The corresponding entropy production is then
PC = %.[L].FVt
= %.J
(13)
Let us express the difference P - P,, which we call the “excess entropy production”, by extending the meaning of this term introduced by Glansdorff and Prigogine (1971)
e = [R].f
Then the difference I) - I), becomes
(18) This quantity is positive by the Cauchy-Schwarz inequality (Bass, 1968), and we have the general result I)(arbitrary) > I),(equipartitioned) (19) The multicomponent equivalent of eq 7 can be shown to be I) - PC - CCL, cov (fJJ
(20) Vt 1 1 where cov (fL,fl)is the covariance of the distributions of f, and fl (the terms i = j give the variance off,). Optimal Design and Minimal Entropy Production. The fact of minimizing entropy production is not a priori an economic criterion, inasmuch as this result is achieved, for example, at the cost of large investments. It is therefore necessary to discuss the relationship between the overall entropy production, its distribution through the process, and the economy of the process. Let us do this with the sample of a simple contactor for solvent extraction with a single chemical solute being extracted. We first have to give careful attention to the degrees of freedom of the process, in other words, which variables are specified and which are “free”. In the statment of the minimal dissipation theorem, the size, V , the duration, t , and the duty are specified, the latter corresponding to the total flux transferred, J . From eq 12, this is equivalent to specifying the average driving force, 7. In addition, we may specify the flow rate, Q, and the inlet concentration of the solute in the phase to be treated so that, a t steady state, the outlet concentration is determined. The only undetermined variables then, for a given configuration, are the flow rate and composition of the second phase, designated here as “solvent”. Only one of these two variables is independent and is a decision variable. If the inlet composition of the solvent is specified, then the flow rate is determined (through the specified fluxes and the equations governing the process). Reciprocally, specifying the flow rate will impose the solvent composition. Comparing Solvent Costs. Keeping this in mind, we can now proceed to compare two configurations, for example, cocurrent and countercurrent flow, with the same initial specifications, ( V , t , J , Q, co), leaving one decision variable on the solvent. We know the cocurrent configuration will have a larger entropy production, P 2 ,than the countercurrent configuration, P I , and we ask what im-
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 53
p
c
2
j_
j I ACI
I
I
m
Solute concentration
c
Figure 3. Schematic entropy-composition diagram for the solvent. 1: countercurrent process. 2: cocurrent process.
plication this has on the decision variable, for a fixed duty. Let us write an entropy balance on each process, assuming steady-state and adiabatic operation in such a way that no entropy is transferred to or from the environment. For process 1 FlAsl = -AS + 7'1 (21) and for process 2
FzAs2 = -AS
+ P2
(24)
where Ac is the concentration change of the solute in the solvent across the process. Using eq 24, we can eliminate F1 and F2 in eq 23 to get
The specific entropy of a solvent is an increasing and convex function of the solute concentration; therefore as shown on Figure 3, if the inlet solvent is the same, inequality 25 implies
>Ac~
(28)
and for process 2, accounting for inequality 19, P2 =
1s L - f 2d V d t >
Pc2= L ( f 2 ) 2 ( V t ) 2 (29)
with, by hypothesis
P1 = P2 Combining eq 28-30, we obtain
In addition, since the amount of mass transfer between the treated stream and solvent is specified to have the same J value in both processes, we have
AC1
P, = Pcl = L(f1)2(Vt),
(22)
where AS is the total entropy change of the treated stream across the process per unit time, determined by the specification on that stream, and thus is the same in the two processes; Fl and F2 are the solvent flow rates and As, and As2 the changes in specific entropy of the solvent. Subtracting eq 22 from eq 21, we obtain FlAs1- FzAs2 = PI - P2 < 0 (23)
FlAcl = F ~ A = c ~J
technical reasons to build a countercurrent contactor than a cocurrent one of equal size and, if so, whether this additional cost offsets the savings in operating costs. This question cannot, of course, be answered in thermodynamic terms. Comparing Transfer Duties. A similar analysis may be done with other specifications. Let us, for example, compare two processes that have the same total entropy production, the same size and duration, and the same transfer coefficient, L. Process 1 is assumed to have equipartitioned driving forces, process 2, not. Then, the overall fluxes transferred will be different in the two processes. In order to show this, we write for process 1
(26)
(Vt)J12> ( v t ) J 2 2
Since by hypothesis the size and durations are the same
( V t ) ,= 0 7 t h Inequality 31 then requires that dll
(32)
> d2l
(33)
> lJ2l
(34)
and, therefore, from eq 4 lJll
T h e transfer flux of equipartitioned process 1 is thus globally larger t h a n that of process 2, at given size, duration, and entropy production. Comparing Size and Duration. Let us consider yet another set of constraints: given the duty and entropy production, compare the respective sizes and/or durations, in other words, the products, Vat. Since the entropy productions are the same in equipartitioned process 1and arbitrary process 2, eq 28-31 remain valid P, = Lf12(Vt),> Pc2 = Lf22(Vt),
(35)
In addition, the equality of the duties is expressed by
J 1 = Lf,(Vt), = J 2 = L f 2 ( V t ) , Dividing inequality 35 by equality 36, we obtain
which in turn entails, from eq 24 (27) Fl < F2 The solvent flow-rate is thus smaller in the less dissipative configuration, and the solvent at the outlet is more concentrated. Alternatively, if the solvent flow rate is the same in the two processes, the same duty can only be achieved if the inlet solvent is purer in the most dissipative configuration. In summary, the economic price to pay for a more dissipative flow configuration, with the set of specifications considered on size, duration, and duty, lies in operating costs related to solvent flow rate and/or purity. I n other words, the least dissipative configuration is optimal with respect to solvent operating costs. Whether this optimum is an overall financial optimum will depend on the cost of technology. The size (volume, number of stages, etc.) of the contactors considered to be the same, the question is whether it is more costly for
(31)
-P1> -
Pc2
J1
J2
(36)
(37)
and thus 71
> 72
Combined with eq 36, inequality 38 leads to
(Vt),< ( V t ) , (39) and we may state for given flux transferred and entropy production, the equipartitioned configuration is "smaller", that is, it requires less transfer area t h a n other configurations, for a given contact time. Alternatively, it requires less contact time for a given transfer area and has, thus, a higher throughput. Comparing Overall Economics. The equipartitioned configuration appears capable of minimizing expenses tied to the quality and quantity of solvent, related to operating costs on one hand, and of minimizing the size of the
54
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987
equipment, related to investment costs on the other hand. We may therefore expect it to minimize some linear combinations of these factors and determine an economic optimum. This can be shown with some assumptions. Let us assume the operating costs are a linear function of the solvent entropy change and, therefore, a linear function of entropy production (see eq 21). Next, assume the investment costs are a linear function of the space time of the process, Vt. We then use the variational approach and seek to minimize the total cost, expressed as the integral Q = aP
+ b + cTVt = l V l t ( a L f + c7) dV d t + b (40)
subject to the constraint of a specified flux
J =
s”s; dV dt
where T is the amortization rate and a , b, and c are cost constants. The Euler equation for this variational problem in terms of the variable f is
a af
-[aLf2
+ c7 + A L f ] = O
(41
where A is a Lagrange multiplier. Equation 41 is developed into 2aLf + .IL = 0 and i f =-= constant (42) 2a The distribution off that minimizes f2 subject to J is thus a uniform distribution. The demonstration is easily extended to multicomponent transfer: each variable f , entering the cost and flux integrals and J gives rise to an Euler equation which yields an equation similar to eq 42. Optimal Size Distribution of Tl’ansfer Units Along a Process Example of Diabatic Distillation. In the previous section, we have considered systems or processes in which the distribution of entropy production was globally determined by the flow configuration chosen. In the present section, we shall consider systems in which this distribution may be continuously or discretely adjusted along the process, by design or by operating decisions. The meaning and significance of this problem is best illustrated by the example of “diabatic” or “diathermal” distillation, that is, distillation with heat exchange all along the column, or at some discrete levels in the column (Benedict, 1947; King, 1971). Such distributed heat exchange makes it possible to adjust the flow ratio of the two phases and thus the shape of the operating line and the driving force all along the column, as illustrated on Figure 4 for binary distillation. Let us consider this distillation column as a cascade of ”transfer units”. Transfer units may be defined in the usual chemical engineering way as a part of the process in which one phase undergoes a given composition change. Alternate and perhaps more appropriate definitions in the present context can be based on a unit change of free enthalpy of one of the phases or on a unit amount of thermodynamic work of separation. Whatever definition is preferred, the important point is that in diathermal distillation, the height of any transfer unit of the cascade may be adjusted independently of the others, by adjusting the amount of heat added or subtracted to the corre-
Figure 4. Binary distillation with distributed heat input or removal: flowsheet and operating lines (the discontinuous operating lines are that of a classical configuration with all heat input at the reboiler and achieving the same separation; the continuous operating line is that of “diabatic” distillation).
sponding part of the column. This change in heat flux affects the driving force distribution and thus the entropy production of the process. To make a transfer unit more reversible will save free enthalpy but will require more column height and heat-transfer area. The question that arises then is whether there exists an optimal distribution of the height of the transfer units, which minimizes a combination of operating costs related to energy and of investment costs related to the column height. Optimal Size Distribution of Transfer Units. We shall indeed show that such an optimal distribution exists and establish its properties, under a certain number of assumptions (Kvaalen, 1981). First, assume steady-state operation and that equipartition of the driving forces is closely approached within any one transfer unit, so that eq 5 applies. This can be reasonably approached by taking the transfer units small enough (that is, by taking the free enthalpy change that defines it small enough); alternately, this condition is met irrespective of the size of the transfer unit if the whole process is equipartitioned and/or close to reversible. Next, we assume that the investment cost, Ci, of a transfer unit is linearly related to the size, V, of this transfer unit Ci = Cif + A-V (43) where Cif is a fixed investment cost. Finally, let us assume that the operating costs Co in a transfer unit are linearly related to the “exergy consumption” in this unit Co = Cof + BAEx (44) where Cof is a fixed operating cost. We are using here the term exergy introduced by Rant (1956) and widely accepted in Europe (Le Goff, 1982). The concept of exergy was first used by Gouy (1889) and called the “availability function” by Keenan (1932). It is defined by EX = H - TOS where Tois a reference temperature, usually 298 K. Let us write that the exergy consumption, AEx, is the sum of a thermodynamic minimum value plus the exergy transformed into entropy through irreversible processes
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 55 AEx = AEx,~,
+ TOP,
(45)
When eq 5 is recalled,
P, = J*f
(5) which expresses that the overall entropy produc_tionin the transfer unit is the product of the driving force, f , assumed constant, and the total flux transferred. Finally, eq 4 relates the flux to the driving force J =L V ~ (4) Expressing V from eq 43 and substituting into eq 4, we have
LKV A where Cv is the difference, Ci - C8, that is the variable part of the investment cost. Eliminating f in eq 5 by using eq 46, we obtain an expression for the entropy production ASZ Pc = (47) LC" Substituting eq 47 into eq 45 and the latter into eq 44,we obtain a relation between operating and investment consts J = -
K1 Co = CV
+ K2
irreversible dissipation and the second is the variable part of the amortized investment cost, we may restate the final result in the following way. Optimal Size Distribution Theorem. Assuming operating costs are linearly related to exergy consumption and investment costs are linearly related to the size of the equipment, the optimal size distribution of the transfer units is such that the amortized variable investment cost in any transfer unit is equal to the cost of irreversibly dissipated exergy in that unit. Let us emphasize the fact that the cost parameters A and B need not be the same in all transfer units: the contacting area may be more expsensive in certain parts of the process, and similarly, the cost of exergy generally depends on the temperature level. Also, the transfer coefficients which determine the entropy production may be variable, provided the initial assumption of equipartition of driving forces within a transfer unit is satisfied. Actually, in the special case where A and B are the same all along the process, then PC/Vop, is a constant, and the optimal size distribution simply reduces to equipartition of the local rate of entropy production. This establishes the connection with the result of the previous section. An explicit expression for the optimal size of a transfer unit defined by a specified duty J is obtained from eq 50
(48)
where and then from eq 43 Vop, = J( K1 and K2are, in principle, known constants (provided one can calculate the minimum thermodynamic work and evaluate the transfer coefficient, L, assumed constant). The optimal size for the section at hand is defined as that which minimizes the sum fl of operating and investment costs, the latter being linearly amortized
n(cJ = 7ci + co = 7ci + ___ ci - Cif + K2 fil
(49)
where 7 is the amortization rate. The minimum of Cl satisfies (50) and d2Cl -=.-
2Ki
dC:
C,3
>O
Combining eq 47 and 50 and the definition of K1 yields, after some rearrangement,
The right-hand side is a constant, independent of the process and of the transfer unit considered and depending only on the financial conditions and the environment temperature. Its order of magnitude for an amortization rate of 10% /year and an ambient temperature of 293 K is 0.0034 year-' K-l. The expressions in the two left-hand sides should therefore be the same in all transfer units, and this defines the optimal size distribution. More simply, the quantities BToPcand TAVopt should be equal in any transfer unit (but not the same from one unit to another). Recalling that the first of these quantities is the exergy cost related to
z)''2
Conclusions We may summarize the main results of this article as follows. The basic qualitative idea is that transfer and separation processes are improved, in some economic sense, by distributing as evenly as possible a function of entropy production along the space and time variables of the process. In single-unit processes where the driving forces (and the transfer fluxes) are entirely determined by the flow configuration, and in which linear relations between fluxes and driving forces prevail, the local rate of entropy production, p , should be uniformly distributed. The equipartition of p is equivalent to the equipartition of the driving forces and of the fluxes. The best configuration among a possible set is then the configuration in which the "variance" of the distribution of driving forces is smallest. This variance may be used as a criterion for classification. If, instead of a single-unit process, a cascade of independent transfer units is considered, then there is an economically optimum size distribution of the units, such that the ratio BI)/AV is the same in all units; the entropy production, I), the size, V, the variable investment cost factor, A , and the variable exergy cost factor, B, may be different in all units. An equivalent statement is to say that the amortized variable investment expense made on any transfer unit (7AV) should be equal to the operating expense that corresponds to irreversible energy degradation (BToI)). The following special cases are of interest: when A and B are the same in all transfer units, the property reduces to the equipartition of the local rate of entropy production, p = P / V, in other words, the result established for a single-unit process. The two results are actually two expressions of the same general property.
Ind. Eng. Chem. Res. 1987,26, 56-65
56
When the cost of exergy is the same in all sections, the ratio P/C, should be equipartitioned. In other words "...in the optimal process, the changes in apparatus price are balanced by the changes of thermodynamic irreversibility ... for expensive apparata, only very intensive processes can be optimal...". The latter statements are quoted from Sieniutycz (1984) who substantiates them by extensive studies of drying processes. Similar considerations are put forward implicitly or explicitly by Bejan (1982): "...the heat transfer area should be concentrated in that region where the heat transfer is most intense...". The idea of an optimum distribution of investment, related to the distribution of irreversibility, is in fact more or less explicit in much of the literature on exergy analysis (Szargut, 1980). We have attempted to formalize this idea and to give it a somewhat general and fundamental basis. We believe the interest of the equipartition concept is primarily qualitative, in that it helps finding new routes for process improvement and avoiding misconception in the design of new processes. The extension of some of the arguments presented to multicomponent transfer and the generalization of the equipartition concept to nonlinear flux-force relations is left for further research.
L, L, Lij = transfer coefficient, matrix of transfer coefficients, and element thereof p = local rate of entropy production, J.K-'-S-'.~-~ P,Pc = overall rate of entropy production in arbitrary and in equipartitioned configuration, respectively, J.K-'.s-' Q = flow rate of treated stream As = specific entropy change A S = total entropy change AEx = exergy consumption t = time T , TO= absolute temperature and absolute ambient temperature V = volume or size variable of process Greek Symbols
R = cost function (eq 40 and 49) A = Lagrange multiplier in eq 41 2 = variance of the driving force distribution, in eq 7 T = amortization rate of investments Literature Cited Bass, J. COUMde Muthematiques; Masson: Paris, 1968; Vol. 1. Bejan, A. Entropy Generation through Heat and Fluid Flow;Wiley Interscience: New York, 1982. Benedict, M. Trans. Am. Inst. Chem. Eng. 1947, 43, 41. De Groot, S. R.; Mazur, P. Non-equilibrium Thermodynamics; North Holland: Amsterdam, 1962. Franklin. N. L.; Wilkinson, M. B. Trans. Inst. Chem. Eng. 1982,60, 276.
Acknowledgment We thank Maurice Roger, ENSIC-Nancy, discussions and constructive criticism.
for fruitful
Nomenclature
Glansdorff, P.; Prigogine, I. Structure, StabilitQ et Fluctuations; Masson: Paris, 1971. Gouy, M. J . Phys. (Orsay, FF.) 1889, 8, 501. Keenan, J. H. Mech. Eng. 1932, 54, 195. King, C. J. Separation Processes; McGraw Hill: New York, 1971. Kvaalen, E. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1981.
a , b, c = cost constants in eq 40 A = cost factor for size-dependent investment cost (eq 43) B = cost factor for exergy-dependent operating cost (eq 44) c = solute concentration in solvent phase, mo1.L-l Ci, C8 = total and fixed in. estment costs, respectively (eq 43) C, = variable investment cost, equal to the difference Ci - Cif C,, COf total and fixed operating costs, respectively (eq 44) f , f,f l , f = driving force: scalar, vector, component of vector,
and average, respectively F = flow rate of solvent H = enthalpy j , j , J , J = transfer flux: local scalar, local vector, overall scalar, overall vector, respectively K,, K2 = constants defined by eq 48
Le Goff, P. Energetique Industrielle; Technique et Documentation: Paris, 1982a; Vol. 1. Le Goff, P. Energetique Industrielle; Technique et Documentation: Paris, 198213; Vol. 3. Lewis, W. K. Ind. Eng. Chem. 1936,28 (4), 399. Mah, R. S. H.; Nicholas, J. J.; Wodnik, R. B. AZChEJ 1977,23, 651. Onsager, L. Phys. Rev. 1931, 37, 405; 1931, 38, 2265. Rant, Z. Forsch. Zngenieurwes. 1956, 22, 36. Sieniutycz, S. Chem. Eng. Sei. 1984, 39(12), 1647-1659. Stodola, A. Steam and Gas Turbines; McGraw Hill: New York, 1910.
Szargut, J. Exergy 1980, 5, 709.
Received for review April 18, 1985 Accepted March 12, 1986
Modification of Supercritical Fluid Phase Behavior Using Polar Cosolvents J. M. Dobbs, J. M. Wong, R. J. Lahiere, and K.P. J o h n s t o n * Department of Chemical Engineering, T h e University of Texas, Austin, Texas 78712
T h e solubility of certain solids was increased markedly in supercritical carbon dioxide by adding small amounts of various cosolvents. For 2-aminobenzoic acid, the addition of only 3.5 mol % methanol increased the solubility 620%. By use of a modified van der Waals equation of state, over 15 new solubility isotherms were correlated within 7% and were predicted qualitatively by calculating the attraction constants using dispersion, orientation, acidic, and basic solubility parameters. These binary supercritical solvents can be highly selective for particular solutes due to specific types of intermolecular interactions. The density and likewise the solubility parameter of a supercritical solvent vary strongly with respect to pressure and temperature in the critical region. A number of experimental investigators have found that the solubility of a hydrocarbon solid varies exponentially with the density of a nonpolar supercritical solvent such as carbon dioxide, 0888-5885/87/ 2626-0056$01.50/0
ethane, or ethylene (Diepen and Scheffer, 1953;Johnston et al., 1982; Johnston and Eckert, 1981; Kurnik et al., 1981). Therefore, the solubility is an extremely strong function of pressure and temperature in the near critical region, e.g., at a reduced density below 1.3 where the magnitude of the isothermal compressibility and the 1987 American Chemical Society
