Ind. Eng. Chem. Res. 1989,28, 873-875
873
Chem. Eng. Data. 1988,33,29-34.
Sharma, M. M. Kinetics of Gas Absorption: Absorption of C02 and COS in Alkaline and Amine Solutions. Ph.D. Thesis, The University of Cambridge, Cambridge, U.K., 1964. Sharma, M. M. Kinetics of Reactions of Carbonyl Sulfide and Carbon Dioxide with Amine and Catalysis by Brbnsted Bases of the Hydrolysis of COS. Trans. Faraday SOC.1966,61,681-687. Sharma, M. M. Perspectives in Gas-Liquid Reactions. Chem. Eng. Sci. 1983,38,21-28. Versteeg, G.F.; Van Swaaij, W. P. M. Solubility and Diffusivity of Acid Gases (COz, NzO) in Aqueous Alkanolamine Solutions. J.
Swades Kumar Chaudhuri, Man Mohan Sharma* Department of Chemical Technology University of Bombay Matunga, Bombay 400 019, India Received for review June 13, 1988 Accepted January 17, 1989
Optimal Thermodynamic Heat Transfer This paper presents a set of heuristics useful for optimal design of heat-exchange networks and integrated heat and power systems. These heuristics consolidate other approaches reported in the literature in a formalism useful for design purposes. Introduction For a given fixed transfer area, minimization of the system entropy generation is an interesting approach because it yields optimal solutions in some economic sense. The general problem is a minimization of entropy generation under certain set of constraints. The question is which are the general rules or the general principles to achieve this objective. This problem was analyzed for generalized transfer operations (mass, momentum, and heat transfer) by Tondeur and Kvaalen (1987). For heat-exchange network synthesis, a systematic procedure was developed in Irazoqui (1986) using this same basic idea. This methodology was also utilized for thermal desalination system synthesis (Scenna, 1987; Aguirre and Scenna, 1988) and for power cycle design (Aguirre, 1987). This paper presents a set of heuristics useful for optimal design of heat-exchange networks and integrated heat and power generation systems. These heuristics consolidate the approaches reported in the literature in a formalism useful for design purposes. Optimal Criteria for Minimum Entropy Generation If we want to develop a criterion to minimize entropy generation in any heat-transfer operation, we need to adopt a set of assumptions and a set of decision variables such as gradient or temperature field. Nevertheless, care must be taken with the assumptions in order to find adequate solutions for conventional problems of heat-exchange systems design. In fact, one criterion for the functional construction is to utilize linear nonequilibrium thermodynamic analysis and Onsager’s reciprocity relations. That is, to utilize a set of flux or driving force definitions to express the usual thermodynamic relationships (Onsager, 1931a,b; Gyamarti, 1970; Wisniewski et al., 1976). Following this line of thinking, the local rate of entropy production is expressed as u = Lu,
du = X L f 2 dV
(1)
Then, for a system with a finite size V, the optimization problem is min
Q
= min I L L
f2
dV1
(2)
If our interest is a driving force or a gradient field, f, and we assume that L is not a function off, the solution for this problem is the homogeneous distribution of the gradient field, f , throughout the system. That is, f is a con-
stant value. But for design purposes, our interest is to clearly define f. So, if we want to find an expression of f through the determination of the temperature field, expression 2 must be written as a functional of this field. If we assume that the basic assumptions of continuum mechanics are valid in our system (Slattery, 1972), assume stationary state, and omit the dissipations due to momentum- and mass-transfer fluxes (because our attention is focalized only on heat-transfer phenomena), the expression of the tqtal entropy generation is u
=
u, dV =
-( $
( J V T )dV
1
=
where we assume that the heat flux (Fourier law) is
J = -XVT
(4)
where X is assumed to be a constant. It must be remarked that this assumption does not necessarily agree with the assumption of L as a constant value. Then, the problem to be solved is min
Q
= min
I-( LA(7) VTVT dV))
(5)
where u is a function of the temperature field. In the Appendix section, it is demonstrated that
V T / V l r c r=, constant
(6)
is the condition for the minimization of entropy generation. { ( x ) is a vector normal to the heat-exchange area. This is in the direction in which temperature distribution becomes outstanding. The condition given by eq 6 must be understood as the goal to be achieved by adequate designs. It must be remarked that this solution implies that the driving force must be uniformly distributed and then, from eq 3, a, too. Therefore, the term “driving force” is strictly defined as V T / T. On the other hand, if we consider that V T f T is a constant value, from eq 4, J / T (the entropy flux) is a constant value too (if h is a constant). Moreover, from the previous conclusions and eq 3, it is easy to show that a, (the entropy generation per unit of area) must be a constant value throughout the system. Then, a,, us, VTf T , or Jf T uniformly distributed can be used as the thermodynamic optimality criterion. The relationship between thermodynamic optimal solutions and economical optimal ones is analyzed according to all
0888-5885f 8912628-0873$01.50 f 0 0 1989 American Chemical Society
874 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
the works cited a t the Introduction section. The Optimal Design Problem A useful formulation of this heuristics for design purposes is as follows. If we consider VT/T = AT/T = constant (7)
with A T = T - t, we can conclude that
T = Pt
with P > 1 (8) where T and t are the temperatures of hot and cold streams, respectively, and P is a constant value. The same conclusion can be obtained if we consider the expression
U ( T - tI2 = C = constant (9) Tt where 6 does not represent a variational symbol but a small quantity, U is a total heat-transfer coefficient, and A is the heat-exchange area 6u
aa=A=
6a =
6Q(
-
$)
6A =
6Q U(T - t)
From eq 8 and 9, we can conclude that
P=
(2 + C / U ) + [(2 + C / U ) ’ - 4]’/’
(10) 2 From eq 8, we can see that the gradient AT = (P - l ) t is a function of the heat-exchange temperatures and P > 1 is a parameter, from which we obtain a family of optimal solutions “Weighting” the entropy generation per unit of heat-exchange area (C).In fact, when C 0, P 1,and then T t (infinite heat-exchange area). On the other hand, if we consider the energy balance W d T = w dt (11)
-
- -
where w and Ware the heat capacity flow rates of cold and hot streams, respectively, we can write from eq 11 and 8 dT/dt = P = w / W = T / t (12) which are the matching conditions to minimize the entropy generation in any heat-exchange device. Then, to match two streams, some combinations are available, but in all the cases, eq 12 must hold. For example, given a heat-exchange area, for each given value of hot inlet temperature (Ti),the outlet temperature (To) and the heat capacity flow rate ( W )can be determined. Moreover, if we analyze (for this given value of the heatexchange area) the functionality u versus Ti, the minimum value for u is obtained when
_w -- -Ti= - -To - constant w to ti The optimal relationship u versus A obtained with this solution agrees with that reported by Nishitani and Kunugita (1982), obtained with other approaches. For the more general problem, that is, an arbitrary number of cold and hot streams that must be matched in a heat-exchange network, the usual case is that in which all the streams have fixed heat capacity flow rates and inlet-outlet (target) temperatures. For this problem, the same solution holds. In fact, a superstream can be defined as the “stream” where the heat capacity flow rate is obtained as the summation of hot and colds streams, respectively (Umeda et al., 1978; Linnhoff and Flower, 1978). In this form, the superstreams must be matched in a “composite heat exchanger” for which the previous rela-
tionships obviously must hold; that is, all the thermal matches must be carried out as indicated in eq 12. This conclusion was derived by Irazoqui (1986). He defines an objective function for the heat recovery system in such a way that it can preserve the shaft power production capability to a maximum. Under a certain set of constraints, it is shown that this criterion is equivalent to the minimization of the system entropy generation. Then, the function dw,T(t)) = a(T(t)) + pA(T(t)) (13) is built. The problem is to find the operating lines T(t) such that, for each given value of A, they make u a minimum. It must be remarked that when an operating line T(t) is given, the structure of the corresponding heat-exchange network can be derived from it. This is a conditioned minimum problem where 1 is a parameter weighting the heat-exchange area. With adequate restrictions, the function of eq 13 can be written as a(T(t),y) =
+ j m d t x O i ( t )Ri(t,T,dT/dt) O
x
i
[(l/To- 1 / T ) + p/(U(T - t))l (14) where a0is a constant, Tois the absolute temperature of the heating medium, and Riand e‘ are constraint functions. The solution of this function of T(t) agrees with the expressions of eq 8 and 12 (Irazoqui, 1986). But in this case, constraint P is (2 + p / U ) + [(Z + p / U I 2 - 4(1 - P / U ) ] ~ / ~ P= (15) 2 0 - P/U) with 0 < pL/U < 1. On the other hand, this family of segments (T = Pt) lies entirely in the region
Then, it is possible to display a chart of optimal operating lines which are independent of any particular problem. From eq 15 and 10, we can relate p / U with C, both parameters of the problem. Finally, eq 16 is an additional constraints to the optimal solution due to the consideration of the heating medium with temperature, To. Then we can conclude that the optimal thermodynamic solutions reported in the literature can be derived from the heuristics previously analyzed and can be summarized as follows: The adequate expression for f is VT/T and must be uniformly distributed along the heat-exchange process. J / T (the entropy flux) and o8 (the entropy generation per unit of heat-exchange area) must be uniformly distributed. uv must be uniformly distributed. To illustrate this methodology in some design problems, the reader is encouraged to try it on the literature cases cited in the Introduction section. Nomenclature A = total exchange area f = driving force L = phenomenological coefficient J = heat flux P = slope of operating line (T versus t ) Q = heat exchanged per unit time between cold and hot
streams T = absolute temperature of hot streams t = absolute temperature of cold streams U = overall heat-transfer coefficient V = volume W = hot stream heat capacity flow rate w = cold stream heat capacity flow rate
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 875 Greek Letters X = heat conductivity u = rate of internal generation of entropy p = Lagrange multiplier Subscripts i = designates either stream or temperature interval v = indicates quantities per unit of volume s = indicates quantities per unit of area
Appendix Let us consider a fixed volume, V, in which pure heat conduction in the stationary temperature distribution (time-independent temperature distribution) takes place. Along the boundary surfaces, an unknown but time-independent temperature distribution will be maintained. Let us assume that entropy generation due to momentum and mass transfer can be neglected. For such a body, the total entropy production is u =
-[ Jh(VT/T)2
dV]
Now let us determine the temperature distribution which corresponds to a minimum of u. Suppose the function u has an extremum (in a certain class of admissible functions) for T = T ( x ) satisfying a given time-independent temperature distribution along the boundary surfaces. Then, since u has an extremum for T ( x )compared to all admissible functions, it certainly has an extremum for T(x)compared to all functions that satisfied the boundary conditions. Therefore, T(x) is an extremal, i.e., a solution of the Euler-Lagrange equations (Bolza, 1973):
Thus, the function for which u has an extremum must first be an extremal and then satisfy additional conditions according to particular boundary conditions. Equation 2a leads to
= constant T T(X) However, (1/T) V T = K (constant vector) is also a solution for eq 2a, and we can say that by keeping this driving force constant along the space variables the entropy production is brought to a minimum. The necessary condition for a minimum is satisfied at every point of the function; in fact, d2F/dTx? = 2 / T 2
>0
which is the Legendre condition. Literature Cited Aguirre, P i 0 Shtesis de Sistemas de Generacidn y Demanda de Energia en Plantas Quimicas. Ph.D. Dissertation, Universidad Nacioncl del Litoral, Santa Fe, Argentina, 1987. Aguirre, Pio; Scenna, Nicolis Optimal thermodynamic synthesis of dual purpose desalination plants. Chem. Eng. Sci. 1988, in press. Bolza, Oskar Lectures on the Calculus of Variations; Chelsea: New York, 1973. Gyarmarti, I. Non-Equilibrium Thermodynamics, Field Theory and Variational Principles; Springer-Verlag: Berlin, 1970. Irazoqui, Horacio Optimal thermodynamic Synthesis of Thermal Energy Recovery Systems. Chem. Eng. Sci. 1986, 41(5), 1243-1255. Linnhoff, B.; Flower, J. Synthesis of Heat Exchange Networks. AIChE J. 1978,24,4. Nishitani, H.; Kunugita, E. On the vector optimization of Heat exchange. J. Chem. Eng. Jpn. 1982, 15, 6. Onsager, L. Phys. Rev. 1931a, 37, 405. Onsager, L. Phys. Rev. 1931b, 38, 2265. Scenna, Nicol6s Synthesis of Thermal Desalination Processes, Part 11, Multieffect evaporation. Desalination 1987, 64, 123-135. Slattery, J. Momentum, Energy, and Mass Transfer in Continua; McGraw Hill: New York, 1972. Tondeur, D.; Kvaalen, Equipartition of Entropy Production. An Optimality Criterion for Transfer and Separation Processes. Ind. Eng. Chem. Res. 1987,26, 56-65. Umeda, T.; Itoh, J.; Shinoko, K. Heat Exchange System Synthesis. Chem. Eng. Prog. 1978, 74, 70. Wisniewski, S.; Staniszewski, B.; Szymanik, R. Thermodynamics of Nonequilibrium Processes; PWN-Warszana: Poland, 1976.
* To whom correspondence should be addressed. 'Fellow of the National Research Council of Argentina (CONICET). Instituto de Desarrollo y Disefio. Instituto de Desarrollo Tecnoldgico para la Industria Quimica. f
or more concisely, which is Laplace's equation, V2(ln T ) = O (44 For heat-exchange devices, it is possible to identify a characteristic direction in which the temperature distribution becomes outstanding. This direction, n x ) , is normal to the surface corresponding to the heat-exchange area. Then, from this assumption,
and, consequently,
Nicolis J. Scenna**' INGAR~ Avellaneda 3657 3000 S a n t a Fe, Argentina
Pi0 A. Aguirret INTECS Guemes 3450 3000 S a n t a Fe, Argentina Received for review June 15, 1988 Revised manuscript received January 25, 1989 Accepted February 17, 1989
