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Ind. Eng. Chem. Res. 1999, 38, 3046-3050
The Driving Force Distribution for Minimum Lost Work in Chemical Reactors Close to and Far from Equilibrium. 1. Theory S. Kjelstrup,*,| E. Sauar,‡ D. Bedeaux,† and H. van der Kooi§ Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, Laboratory of Applied Thermodynamics and Phase Equilibria, Delft University of Technology, 2628 BL Delft, The Netherlands, Department of Physical Chemistry, Norwegian University of Science and Technology, 7034 Trondheim, Norway
We present a mathematical procedure for the determination of the driving force distribution in a chemical reactor that has minimum lost work for a given production rate. It is shown how the path of minimum lost work is determined from knowlegde of reaction kinetics, using the reaction A f B as an example. The normal chemical reaction has a nonlinear relation between the rate, r, and the driving force, -A/T, where A is the affinity and T is the absolute temperature. Minimum lost work is obtained when A/T + r d(A/T)/dr is constant. This criterion is converted into a more practical criterion for the operating temperature along the reactor. The inverse operating temperature should be parallel to the inverse equilibrium temperature, when the enthalpy of reaction is constant. 1. Introduction
the requirement of a constant production rate, J (in mol/ h):
Chemical reactors are normally designed to operate at the maximum reaction rate, see, e.g., ref 6. In this article, we shall study another reactor path, namely, the path that looses the minimum of useful work. The lost work per unit time (from now on called only the lost work) in a chemical reactor is given by the GouyStodola theorem (see, e.g., ref 1):
∫
Wlost ) T0Ω σ dx
(1)
where σ is the entropy production rate per unit volume. The reaction takes place at temperature T, but the lost work is referred to at the temperature of the surroundings, T0. The lost work is obtained by integrating the entropy production rate per unit volume over the reactor volume. For simplicity, we used a constant crosssectional area of the reactor, Ω. The transport takes place only along the length coordinate axis, x. Possible reductions in lost work can give economic gains. The relation between reductions in lost work and economic gains is not unique; however, it varies, for instance, with energy prices. We consider it valuable to do energy optimization studies also separate from economic analyses. Minimization of Wlost means minimization of the integral of σ, so we shall proceed with the integral only. We shall refer to minimization of the integral of σ as minimization of the lost work. The minimum value of the integral shall be called the minimum total entropy production rate. The minimization is carried out with * To whom correspondence should be addressed. Telephone/ fax: 47-73594179/1676. E-mail:
[email protected]. † Leiden University. ‡ Carnegie Mellon University. § Delft University of Technology. | Norwegian University of Science and Technology.
J)Ω
∫r dx
(2)
where r is the reaction rate. Minimization without this constraint gives the meaningless result of σ ) 0. It is normal in engineering contexts to optimize under constraints, and it is then common practice to use the Euler-Lagrange method.2,14 The question that we shall ask by the minimization procedure is the following: Given J, how can we operate the reactor so that we obtain minimum lost work? The answer to the problem is an ideal answer in the way that it results in operating conditions or designs which may be unattainable. It is not the aim here to discuss in detail the practical construction or operation of the ideal reactor. We shall present a method that can be used to improve actual operations. Minimum lost work can mean the highest possible temperature of the outlet cooling fluid, and in a subsequent article we shall demonstrate that this is so. The present work can be seen as a continuation of our effort to find the reduced lost work in process equipments. The path of minimum lost work was obtained by us for distillation columns,8,11 but has long been the target of optimization.7 The study of chemical reactors in this context is newer.12,13,15 We have previously used a linear relation between the fluxes and the forces of transport in the cases we have studied. In this work we study situations where the rate, r, is not necessarily a linear function of the driving force (the reaction affinity or Gibbs energy of reaction). A short version of the article has been presented.5 We shall use a plug flow reactor as our model case, as evident already from eqs 1 and 2. The results for batch reactors will be the same. In the batch reactor one integrates over time, while in the plug flow reactor, one integrates over space. The developments in time or
10.1021/ie980744+ CCC: $18.00 © 1999 American Chemical Society Published on Web 07/15/1999
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3047
space are in this respect equivalent, so we need only give equations for one type of reactor (see also ref 12).
with the rate
r ) k1cA - k2cB
(8)
2. Chemical Reactions Far from Equilibrium 2.1. The General Problem. The entropy production rate per unit volume is determined from irreversible thermodynamics. The entropy production rate is always the sum of products of fluxes and conjugate forces.9 When a chemical reaction is the only process of importance,
[ AT]
σ)r-
(3)
where r is the reaction rate and -A/T is its thermodynamic driving force. Here, A is the affinity (the Gibbs energy of reaction) and T is the temperature. (We have used the opposite sign convention for A as that originally introduced by deDonder). The reaction rate is, furthermore,
(AT)[AT]
r ) -l
(4)
where l is a phenomenological coefficient. The minimization problem can be formulated as an Euler-Lagrange optimization problem:14
δ A δ (x) T
∫[-rAT + λr] dx′ ) 0
( )
(5)
where λ is the Lagrange multiplier, to be obtained in terms of J. Before we consider the solution, let us get a better physical understanding of the situation. The argument for finding the minimum entropy production resembles the calculation of limit costs in economic theory. Consider within the bounds of a constant output that the extra yield, dr, at a location x, results in the extra entropy production, dσ. If we can have the same yield at another location, without paying as much for it in terms of dσ, it is beneficial to move the production to that location. This means that we want in the end to achieve a situation where dσ/dr is constant. In this situation, there is no benefit to move production anywhere else. From the condition of constant dσ/dr, and σ ) -r(A/T), we find
A d T A )C - -r T dr
()
c ) cA ) 1 - cB
∫
J ) Ωt [k1c - k2(1 - c)] dx
(7)
(10)
The driving force of the reaction is
1-c A 1 - ln Keq - ) - (µB - µA) ) -R ln T T c
[ (
]
)
(11)
where Keq ) k2/k1 is the equilibrium constant, R is the gas constant, and the mixture is assumed to be ideal. The rate, eq 8, is related to the force, eq 11, in a nonlinear way. The entropy production of the total system is obtained by introducing the local value of the rate and the corresponding driving force into eq 3. Since A/T has a local relation to c, we can differentiate with respect to c instead of with respect to A/T. According to functional theory, we can also replace the functional derivative with the local derivative, and obtain from eq 5,
A ∂ -r + λr ) 0 ∂c T
[
]
(12)
with the result
[ (1 -c c) - ln K ] - k Rr+ k (1 -1 c + 1c) ) λ
R ln
eq
1
(13)
2
The first term to the right is the thermodynamic force. We see that the force is not necessarily constant when we have minimum lost work. It is the sum of the force plus the reaction rate times [R/(k1 + k2)](1/(1 - c) + 1/c) that is constant. The force that gives the minimum total entropy production rate is
opt
(6)
(9)
The output of B per unit of time is given (compare eq 2):
[-AT]
where C is a constant. The solution to this formulation of the problem must coincide with the solution to eq 5, as in fact it does with C ) -λ. 2.2. Nonlinear Chemical Reactions. We are now ready to study chemical reactions far from equilibrium. The problem in eq 4 is that the phenomenological coefficient is not known. The reaction rate is normally given by rate constants and concentrations of reactants and products in reaction kinetics. Consider therefore as an example the reaction
AhB
and with first-order reaction rate constants k1 and k2. The concentration of A is cA and of B it is cB. Because of mass conservation, we can introduce normalized concentrations, giving
1 1 Rr + -λ )k1 + k2 1 - c c
(
)
(14)
The term containing (1/(1 - c) + 1/c) determines the deviation of the force from the constant λ. The reaction rate is larger in the beginning of the process than near the end. To find the value of λ, we solve c(x) from eq 13, substituting the result in (10). We then obtain λ as a function of J and the rate constants of the reaction. This relation can next be used with eq 14 to determine the value of the optimal force along the reactor. Equations like (10)-(14) can be used in this manner to determine the force that gives minimum lost work in the reactor. More details on the solution method have been developed.4 If we now return to the formulation from irreversible thermodynamics, we find the optimum solution for the
3048 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
Lagrange multiplier λ as follows. Equation 5 gives
(AT) +
( ( )) ∂ ln r A ∂ T
-1
)λ
(AT)
opt
∂ ln l λ 1 A )- + -λ 2 2 T A ∂ ln T
[( ) ]
(AT)
-
(15)
By substituting eq 4 into eq 15, we obtain:
-
and the constant force is
()
(16)
∫
(AT) dx A tΩ∫[l + ( )l′] dx T
λ)
-1
(17)
To complete the list of expressions for reactions close to equilibrium, we also give the local entropy production rate per volume:
(AT)
∂l A ∂ T
()
(18)
The condition that l is independent of A/T means that l can only depend on the state variables in a limited way. The coefficient can depend on the temperature, pressure, and some of the concentrations, but we must be able to vary A/T independent of l. 3. Reactions Close to Equilibrium We have reported earlier that the optimal force was always constant when the phenomenological coefficient was independent of the force (the principle of equipartition of forces).11 We shall see that the above solutions reduce to the constant force solution. Close to equilibrium, l is always independent of the force. Even if l is independent of the force, the coefficient may vary along the path because composition, temperature, and pressure vary along the reactor. By introducing l′ ) 0 in the equations above, we obtain from eq 4
( ( ))
-1
(24)
2
σmin(x) ) l(x)
∂ ln r A ∂ T
-1
(AT)
l′ , l
with
l′ )
(23)
The force distribution that gives minimum lost work is uniform over the length of the reactor (in a steady-state flow reactor) or over the time (in a batch reactor). Since A < 0 for a spontaneous reaction, λ is a negative constant. The force can be found from the integral of l(x), J, t, and Ω. The force is close to constant according to eq 15 when
The solution for λ is therefore equal to
-2J + tΩ l′
(tΩJ )[∫l(x) dx]
)
EoF
A )T
(19)
2 EoF
(tΩJ ) [∫l(x′) dx′]
) l(x)
2
-2
(25)
It is only when the phenomenological coefficient is constant that σ is constant through the reactor. In contrast, ref15 defined the optimum-reactor path as a path with constant entropy production. The minimum total entropy production rate becomes
∑min) Ω∫σmin dx J 2 ) ( ) [∫l(x) dx]-1 tΩ
(26) (27)
Equation 27 shows that ∑min becomes smaller, the larger l(x) is for the given reactor. (Equation 27 gives zero as the limit value for ∑min when ∫l(x) dx f ∞.) The total resistance to the reaction can be defined R ) [∫l(x) dx]-1. Equation 27 can hence be used to define the minimum lost work for a process. To see also that eq 14 reduces to a constant force when the reaction is close to equilibrium, consider the perturbation δc on ceq:
c ) ceq + δc
(28)
The concentration at equilibrium, ceq, is obtained for r ) 0:
ceq )
k2 k1 + k2
(29)
and from eq 16 Close to equilibrium, we therefore have
A λ )- (x) T EoF 2
[ ]
(20)
The force is constant, independent of the value of l(x). The subscript EoF means equipartition of the force. From eq 17 the Lagrange multiplier is
∫
2J λ ) - [ l(x) dx]-1 tΩ
(21)
so that the given production rate in terms of λ becomes
λ J ) - tΩ l(x) dx 2
∫
(22)
r ) (k1 + k2)δc
(30)
(k1 + k2)2 A - ) Rδc T k1k2
(31)
and
(
)
1 Rr A 1 >0 - ) + T k1 + k2 (1 - ceq) ceq
(32)
The first of these equations gives a constant force. The next equation says that even if A/T(x) is constant, r(x) need not be so since ceq varies along the path. The path
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3049
of minimum lost work is therefore generally not a path of constant entropy production rate per volume. Chemical reactors are normally designed to give the maximum reaction rate of the chemical reaction. The mathematical scheme above can be used to find also the path of the maximum reaction rate with constant lost work. The problem of finding the driving force that gives the maximum reaction rate for a given total entropy production rate is mathematically equivalent to the problem of finding the driving force that gives minimum entropy production for a constant total production.11 This means that a trade-off between the production and the energy costs can be studied within the same framework.10 3.1. The Optimal Path in Practice. The criterion found for the optimal force in eqs 14 or 20 is not a practical criterion. We can transform the force criterion to a more practical criterion by integrating the GibbsHelmholz’ equation from an equilibrium state (eq) to the corresponding optimal (opt) state; see ref 13:
∫eqoptd(AT) ) -∫eqopt∆rH d(T1 )
(33)
Since (-A/T) ) 0 in equilibrium, the optimal force is
[AT]
∫eqopt∆rH d(T1 )
)opt
(34)
Under normal operation there is no chemical equilibrium between reactants and the product (the driving force is nonzero). The pressure, temperature, and composition vary along the reactor. The reaction may come to equilibrium, however, at any position for the given pressure and temperature. Or given the composition and pressure, the equilibrium temperature can be found. When the composition along the reactor is given, the condition (A/T)(x) ) 0 corresponds to an equilibrium temperature everywhere in the reactor. The sequence of states with (A/T)(x) ) 0 that defines the equilibrium path in the reaction coordinate space is therefore welldefined. The same is the optimum operating path. Both paths are theoretical, however. The equilibrium temperature for a given degree of conversion is most easily found by putting r ) 0 in eq 8. When we know the enthalpy of reaction as a function of temperature, we can carry out the integral. If the enthalpy of reaction is constant in the temperature interval, we have
() A T
opt
(
) -∆rH
)
1 1 Topt Teq
(35)
Since (A/T)opt is known from eq 14 or eq 20, we can find Topt. We see that the last equation gives a constant difference between 1/Topt and 1/Teq. The enthalpy of reaction is normally known as a function of temperature. This makes it easy to predict when deviations from the simple solution can be expected. An example of an inverse optimal temperature profile and inverse equilibrium temperature profile through a reactor is plotted in Figure 1. The two lines are parallel through the reactor when ∆H is constant. The reaction normally has its largest driving force in the beginning of the reactor. A deviation from parallell lines is therefore more likely in the beginning of the reactor than in the end.
Figure 1. Example of inverse optimal temperature as a function of the degree of conversion in a chemical reactor (upper curve). The inverse temperature for the mixture in equilibrium is also drawn (lower curve).
4. Concluding Remarks We have described above how one can design a chemical reactor with minimum lost work, also when the reaction is far away from equilibrium. Through the study of a reactor with a nonlinear relation between rate and affinity, we have given expressions for the optimal force and devised a method for its determination and control. We can expect minimum lost work by equipartition of forces when the reaction enthalpy varies little with composition, close to equilibrium, when the reaction rate is a linear function of the thermodynamic driving force, and when the derivative in eq 15 is constant. The principle of equipartition of forces can always be used for transport of heat, mass, and charge since the transport coefficients do not depend on the forces in these cases.3,8,11 The analysis was carried out for a steady-state plug flow reactor. An equivalent line of reasoning can be carried out for a batch reactor. The only difference is that one integrates over time instead of over space. The optimization that leads to the minimum total entropy production rate has two equivalent mathematical formulations. We have so far discussed the solution to the following question: Which force gives the minimum total entropy production rate with constant output of product? But this distribution of forces is also the answer to this question: Which force gives the maximum reaction rate with constant total entropy production. Constant total entropy production in this context means that the integral over the (varying) local entropy production is constant. The solution for the force at the maximum reaction rate may explain the results of Scho¨n and Andresen. These authors found results equivalent to a constant force in a reactor with the maximum reaction rate; see ref 12. We can thus draw two isoforce operating lines parallel to the equilibrium line in Figure 1, one according to eq 20 and one for the maximum reaction rate (not shown). Isoforce operating paths can be constructed for a series of unspecified, but constant, J’s, independent of the knowledge of reaction kinetics. Construction of a particular operating path for a reactor with specific output requires knowledge of reaction kinetics, however, whether or not the reaction is close to or far from equilibrium. Knowledge of the path that gives the minimum lost work is needed when one wants to make a trade-off between energy efficiency and production rate in a given reactor. With rising energy prices, it may be profitable to increase reactor investments and lower the energy input for the same product, or to slow down the
3050 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
production rate with given energy input. For instance, it has been suggested that a reduction in the lost work by as much as 30% can be obtained by reducing the output of a reactor by only 5%.10 With increasing prices on energy, such judgments may become more and more important. The present analysis may help to define the trade-off situations. Acknowledgment This work was done while S.K. was on sabbatical stay in Leiden University and in the Technical University of Delft. These institutions as well as NWO are therefore acknowledged for support. E.S. acknowledges a research grant from the Norwegian Research Council and Carnegie Mellon University for hosting a 3-month stay. Literature Cited (1) Bejan, A. Entropy Generation Through Heat and Fluid Flow; Wiley: New York, 1982. (2) Bejan, A. Entropy generation minimization: The new thermodynamics of finite-size devices and finite-time processes. J. Appl. Phys. 1996, 79, 1191-1218. (3) De Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover: London, 1985. (4) Kjelstrup, S.; Island, T. V.; Sauar, E.; Bedeaux, D. Energioptimal drift og design av en eller flere kjemiske reaktorer. Patent Appl. NO nr. 982798, 1998. (5) Kjelstrup, S.; Sauar, E.; van der Kooi, H.; Bedeaux, D. Reactor design by the principle of equipartition of forces and its extensions. In Second Workshop on Dissipation in Physical
Systems, Sept 1-3, 1977, Borkow, Poland; Radowicz, A., Ed.; Kielce: Poland, 1998; pp 133-145. (6) Levenspiel, O. Chemical Reactor Engineering; Wiley: New York, 1972. (7) Platonov, V.; Bergo, B. Multicomponent Mixtures Separation; Khimiya: Moscow, 1965. (8) Ratkje, S. K.; Sauar, E.; Hansen, E.; Lien, K. M.; Hafskjold, B. Analysis of entropy production rates for design of distillation columns. Ind. Eng. Chem. Res. 1995, 34, 3001-3007. (9) Ross, J.; Mazur, P. J. Chem. Phys. 1961, 35, 19. (10) Sauar, E. Energy Efficient Process Design by Equipartition of Forces: With Applications to Distillation and Chemical Reaction, Ph.D. Thesis, Department of Physical Chemistry, Norwegian University of Science and Technology, Trondheim, Norway, 1998. (11) Sauar, E.; Kjelstrup, S.; Lien, K. M. Equipartition of forces. A new principle for process design and operation. Ind. Eng. Chem. Res. 1996, 35, 4147-4153. (12) Sauar, E.; Kjelstrup, S.; Lien, K. M. Equipartition of forcessextensions to chemical reactors. Comput. Chem. Eng. 1997, 529-534. (13) Sauar, E.; Rivero, R.; Kjelstrup, S.; Lien, K. M. Diabatic column optimization compared to isoforce columns. Energy Convers. Manage. 1997, 38, 1777-1783. (14) Schechter, R. S. The variational method in engineering; McGraw-Hill: New York, 1967. (15) Scho¨n, C.; Andresen, B. Finite-time optimization of chemical reactions: nA ) mB. J. Phys. Chem. 1996, 100, 8843-8853.
Received for review November 23, 1998 Revised manuscript received April 15, 1999 Accepted April 17, 1999 IE980744+
