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Optimum performance for energy transfer in a chemical reaction system
Physica A 446 (2016) 224–233
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Physica A journal homepage: www.elsevier.com/locate/physa
Optimum performance for energy transfer in a chemical reaction system N. Sánchez-Salas a,∗ , J.C. Chimal-Eguía b , M.A. Ramírez-Moreno a a
Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edif. 9 UP Zacatenco, 07738, México D.F, Mexico
b
Centro de Investigación en Computación del IPN., Av. Juan de Dios Bátiz s/n U.P. Zacatenco, CP 07738, México D.F, Mexico
highlights • Optimal performance of an isothermal engine driven by the chemical energy transfer is presented. • Efficient power and Omega criterion are applied as objective functions. • Optimal efficiency shows a similarity for the isothermal engine analyzed.
article
info
Article history: Received 21 August 2015 Received in revised form 4 November 2015 Available online 9 December 2015 Keywords: Optimization Kinetics Biochemistry Isothermal motor
abstract This paper presents an optimal performance analysis for an isothermal model which performs work at a steady state via the energy transfer from a chemical reaction, using the methodology of the so-called Finite Time Thermodynamics (FTT). Furthermore, three optimal operation modes namely, Maximum Power, Maximum Omega function and Efficient Power are presented for the model. Results show analogies in the optimal performance between thermal and isothermal engine models. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Molecular motors are machines that operate in cycles and convert chemical energy into useful work, mainly from the hydrolysis of the ATP molecule, to perform a variety of cellular functions such as DNA replication, repair and transcription of RNA, protein synthesis and transport between cell and other elements [1]. Such machines operate in a constant temperature environment unlike heat engines which are limited by the Carnot (ηC ) efficiency. The limit value of efficiency for molecular motors is one and can be reached only in the limit when the output power is zero. Biochemical processes of living cells occur at ambient or near-ambient temperatures [2,3] at which the rates of spontaneous leak processes are limited, but, cells selectively improve the kinetics of their energy-coupling reactions with enzymes, which effectively create alternative reaction paths with lower activation energies. Enzymes lower activation energies by 30 to 100 kJ/mol, which, at T = 298.15 K, corresponds to increases in reaction rates of 5 to 17 orders of magnitude [4]. Indeed, the survival of living systems often depends heavily on their reproduction rate and, thereby, on the rate of energy conversion in the cells. On the other hand, thermodynamic efficiency is an important factor in the yield of reproduction processes, so that, energy conversion in living cells is expected to occur where a trade-off between both performance aspects is considered.
∗
Corresponding author. E-mail address: [email protected] (N. Sánchez-Salas).
http://dx.doi.org/10.1016/j.physa.2015.11.030 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Scheme of a (bio)chemical reaction model, sketched by Lems [21].
In 1975 Curzon and Ahlborn [5] presented an expression for Efficiency at Maximum Power (EMP) modeling a thermal machine which executes a Carnot-type cycle, a new branch of thermodynamics emerged in the last decades, known as Finite Time Thermodynamics (FTT). The model proposed by Curzon and Ahlborn operates between two heat sources √ with high and low temperatures, Th and Tc (< Th ), respectively, and the expression for the EMP obtained was ηCA = 1 − Tc /Th , a result that in principle is independent of the model parameters and only depends on the temperatures of the heat reservoirs; analogously this is what happens with the efficiency for a reversible Carnot cycle (ηC = 1 − Tc /Th ). Although to obtain the efficiency of Curzon and Ahlborn, it is necessary to consider that the heat transfer involved obeys a linear law. Notice that previously other authors [6,7] have found the same expression for similar models. An important advantage observed in the results obtained in FTT is that optimal efficiency values are closer to those observed in real machines [8]. Within the context of the FTT several objective functions to be optimized under rich criteria have been introduced and, many models of heat engines have been proposed and analyzed which perform as motors, heat pumps or refrigerators [9,10]. It has recently been reported [11–13] that thermal engines show some kind of universality in the behavior of the efficiency at maximum power [13], although the analyzed models are different in nature and scale, for instance, macroscopic, stochastic or quantum [14–16]. Traditionally most of energy-transfer devices discussed by the FTT were heat engines although, some optimal performance analyses were also conducted where biochemical reaction models were analyzed [17–19]. In recent years, isothermal engines with kinetic [20–22] and mesoscopic [23,24] description have been published, as those which convert non-thermal energy (mainly chemistry) into useful work. The importance of these models is that the energy production processes that are observed in the molecular biological level machines obey similar principles. In this paper, an optimal performance of a system that produces work via chemical energy transfer by the coupling of two chemical reactions using the methodology of FTT is presented. The performed analysis takes into account three operating regimes namely: Maximum Power, Omega Function and Efficient Power criterion, respectively. Two special cases for the (bio)chemical device efficiency are presented: (i) The general numerical case and (ii) A limit case where analytic results are found. 2. Model for chemical energy transfer The model that would be analyzed was performed by Lems et al. in 2007 [21]. In his model, Lems studied a system of biochemical reactions shown in Fig. 1, the chemical energy is transferred at 298.15 K, in a coupled chemical reaction in which the favorable reaction C → D is used to manage the unfavorable reaction X → Y , a well known procedure in bioenergetics where the biological tasks are managed by the energetic coupling of the reactions [4]. The system considers two leak reactions, which dissipate chemical energy without contributing to production of work output. To describe the system operation, initially the potential of each individual reaction to deliver work is considered. The ability to perform work of a chemical reaction is determined by its affinity (A) of the reaction [21], representing the degree of non-equilibrium between reactants and products, and is determined by the chemical potential µi , and the stoichiometric coefficients νi . From the work proposed by Donder [25] it is known that the affinity can be written as the negative of the change in Gibbs free energy of reaction (−∆r G), A = −∆r G = −
νi µi .
(1)
i
Each of the reactions involved in the system gives (see Table 1). When A > 0, the products of the reaction are at a lower energy level than the reactants, the reaction is possible until thermodynamic equilibrium is reached, when A = 0 (see Fig. 3). At equilibrium, there are no reaction forces that handle the reaction and therefore the production no longer has the potential to generate work.
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C +D → D+Y C →D Y →X
Acoupled ACD AYX
Under steady state operation the affinities of the three chemical reactions involved are interrelated as follows, ACD = AYX + Acoupled .
(2)
Eq. (2) indicates the affinity ACD available between C and D, is the necessary energy to manage the unfavorable reaction YX with affinity AYX and also the energy for the reaction coupled with Acoupled was carried out spontaneously. As Lems [21] et al. pointed out, the value of ACD is important in the model, since biochemical reactions are carried out at room temperature at constant pressure. In the hydrolysis reaction of ATP, the main energy source for most cellular functions is estimated to have an affinity reaction between 48 and 55 kJ/mol [4,26], depending on given intracellular conditions. So, for this model a fixed value 50 kJ/mol to ACD will be considered. Another important aspect in the model’s description is to indicate the reaction rate in terms of the affinity of each of the reactions involved. These are described by the transition state theory developed by Eyring [27] where the net reaction rate v is the result of the forward and backward reaction rate: →
←
v= v−v.
(3)
Fig. 3 shows a processing scheme for reactive products where there is an intermediate state known as activation state. A positive value of the affinity of the reaction (A) has a larger energy barrier (∆Gact + A) for the backward reaction, while the energy barrier for the forward reaction remains unchanged. As a consequence, the forward reaction rate exceeds the rate of backward and forward net reaction, in other words, there is conversion of reactants to products. It is possible to find the relationship between the forward and backward rates in terms of the reaction affinity (A) as, →
←
v / v = eA/RT .
(4)
Combining (3) and (4), following the relationship for the net reaction rate based on the value of the affinity function finds: →
v = v (1 − e−A/RT ).
(5)
This expression force-flow is not linear in contrast to some cases of transfer heat energy as observed in the models of heat engines, where there are still linear relationships to large ranges of temperature differences; in this case the linear region is small (A < 1). For the model there are three rates for each of the reactions involved in the shape expressed in (5), these are: → → → vCD , vYX and vcoupled in turn related to the quantities − v CD , − v YX , − v coupled and ACD , AYX and Acoupled , respectively. − → − → − → In particular, for this system the value of v CD is taken with fixed value 1 mol/s and values v YX , v coupled are determined − → as follows: the forward reaction rates v for each of the reactions are given by the Eyring equation, expressed as follows, →
v = k0 e−1G
act /RT
.
(6)
For simplicity, it is considered that k0 (called frequency factor) has the same value for different reactions and this means that the composition of the substances involved in the reactions has a similar nature and equivalent conditions. It is taken − → − → that v CD and v YX are equal, therefore the activation barriers for the leakage reactions are also equal. Then, it is possible to find the following relations,
− → − → 1Gact −1Gact leak coupled v coupled v coupled RT = − =e . (7) − → → v CD v YX Wherein 1Gact leak is the activation energy of both leakage reactions CD and YX . On the other hand, a λ parameter is defined as:
λ=
act 1Gact leak − 1Gcoupled
. (8) RT The factor λ states the amount the coupled reaction is favorable to the effect on the leak reaction. With λ > 0 the activation − → barrier of the reaction leakage exceeds the activation barrier of the coupled reaction, and therefore, the value of v coupled − → surpasses the value v leak ; with λ = 0, the coupled reaction and leakage reaction are equally favorable and, in λ < 0, value − → → v leak exceeds that of − v coupled . As is known, catalysis is the process by which a chemical reaction rate is increased and is achieved with a different transition state that has lower activation energy. In this case λ can be considered as a coupled reaction catalysis parameter, that is, expresses how far the coupled reaction is catalyzed on the leakage reaction. Thus, the important variables of the model are the affinity of the coupled reaction (Acoupled ) that is the force that will cause unfavorable XY reaction to carry out, and the kinetic parameter λ, expressing how much influence leak reactions has in the process of converting chemical energy into work of the system. As will be analyzed, the behavior of the model is based on these two variables showing work output and efficiency behavior.
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Fig. 2. Graphical presentation of the energy barrier between the reactants and products of a chemical reaction [27,28].
3. Energetic function characteristics This section describes the behavior of energy system functions, work output and efficiency. Additionally, two objective functions are also presented; these were proposed in the context of the FTT to optimize thermal machines. In this case this methodology will be used to optimize the chemical reaction engine as described previously. 3.1. Work-output rate and efficiency Work output per unit time of the system is the product of the unfavorable chemical reaction affinity, that is, AYX and the net rate to the species Y is produced from the X , taking into account the leakage element, that is,
˙ out = (vcoupled − vYX )AYX . W
(9) A
A
CD The explicit expression (9) is given by (A.1) where the following changes of variable x = coupled and y = RT have been RT used. Eq. (9) is a function of x, y and λ, where y is related to the available input energy in the system, x is used for coupling the favorable reaction and the unfavorable one and λ is defined by Eq. (8). From the same nature of x and y both are functionality connected, see Eq. (2) and Fig. 2, where x and y are associated to the affinity A for each involved reaction. Whereas λ is related with 1Gact , so it is reasonable to consider it as independent of x and y in the net reaction rate described by the transition state theory [28]. The behavior of Eq. (9) can be seen in Fig. 3(a) as a function of x for a defined y and some fixed values of λ are shown in
˙ out for a given value of λ. Fig. 3(c) shows the the same graphic. Performance infers that there is a value x that maximizes W ˙ out grows monotonically when y work output as a function of y for x = 5 and different values of λ. It can be observed that W increases; then, there is no y-value that maximizes the work. From 3(d) it follows that the efficiency increases when y grows and also there is no y-value that maximizes this function. It is important to note that y is related with available energy, so it is convenient and reasonable to consider y equals a constant between 48 and 55 kJ/mol [4], as was previously mentioned. On the other hand, the efficiency for coupled chemical reaction device is the relationship between chemical energy transferred from YX for the energy obtained from the source reaction CD, that is, η=
˙ out W ˙ in W
=
(vcoupled − vXY )AXY . (vCD + vcoupled )ACD
(10)
Also expression in terms of x and y variables is presented in the Appendix (A.7). The behavior of the efficiency as a function of x shown in Fig. 3(b) for different values of λ is indicated. As can be seen, the efficiency increases when the λ parameter is bigger, because the effect of leakage reactions is negligible, having a large activation energy, while the coupled reaction is favored. In the limit λ → ∞, η = 1 − yx , theoretically the value 1 could be obtained if x = 0 (with y ̸= 0). When λ decreases it is observed that the value of x that maximizes the efficiency increases. 3.2. Omega function and efficient power Now the behavior for the Omega function is presented. This feature was proposed based on a unified optimization criterion [29] which represents the best compromise between the benefits and losses of energy to any device and has the advantage that it does not explicitly require the entropy expression to be applied, or any other environmental parameter,
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Fig. 3. (a) Work output performance and (b) system efficiency as a function of x, with y = 20.2. (c) Work output performance and (d) efficiency as a function of y, with x = 5, indicating values of λ.
Fig. 4. (a) Performance of Omega Function (Ω ) and (b) efficient Power (PE ) as a function of x and for different values of λ.
such as the case where the Exergy or the Ecological criteria are used as objective functions [30,31]. For this system the maximum efficiency is 1 and the minimum is 0. In this case the expression for Omega function [29] is given by:
Ω=
2η − 1
η
˙ out . W
(11)
Eq. (11) is depicted in Fig. 4(a) for the present system. It can be observed that the value of x that optimizes Omega varies very little to different values of λ. As was previously mentioned, the behavior of the device under maximum efficient power regime will also be discussed. This approach was recently proposed by Yilmaz [32] in the context of heat thermal engines, now will be applicable to a
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Fig. 5. Work output, Efficiency, Omega Function and Efficient Power, all functions are normalized in order to get an easier comparison with efficiency. For each function, the dashed line indicates the critical x-value where the maximum of each one is reached for λ = 5.
chemical reaction system and is defined as the product of the power output and efficiency, which states:
˙ out . PE = η W
(12)
Maximization of this function provides a compromise between power and efficiency, where the designed parameters at maximum efficient power conditions lead to more efficient engines than those at the maximum power conditions [32]. This function has an equivalent performance to the Omega function. It is worth mentioning that λ-values necessary to have Ω positive are greater than those necessary to have W positive. In fact, for λ = 0.5, Ω is negative and W positive (see Fig. 3), this happens because for this value the losses of energy are greater than benefits in the device. Subsequently, Fig. 5 shows the functions presented in this section, with λ = 5, highlighting x values that maximizes each function, which will be analyzed in more detail in the following section. 4. Optimal performance This section gives the results of the optimal model of a system that transfers energy (bio)chemical using the methodology of the FTT having traditionally been applied to heat engines operating between two reservoirs of temperature and, has a device that performs work under isothermal conditions through the transfer of chemical energy. The performance of the model is studied under three optimal regimes namely: (a) Maximum Work Output, (b) Unified criterion called Omega function and (c) Maximum Efficient Power. For the analysis x is taken as the independent variable, and what must be solved are the following equations:
˙ out ∂W ≡ 0, ∂x
(13)
∂Ω ≡ 0, ∂x
(14)
∂ PE ≡ 0. ∂x
(15)
Explicit expressions for the above equations appear in the Appendix. In particular, Eqs. (13)–(15) will be solved in two different cases of λ values, which are λ → ∞ and 0 < λ < ∞. 4.1. Case 0 < λ < ∞ Fig. 6 depicts the numerical values of x which solve numerically Eqs. (13)–(15), where the behavior mentioned in the previous section can be observed in more detail. It is also worthwhile to point out that the x-values which maximize each function have similar mathematical behavior. For λ values close to zero x-values are higher, whereas for λ values increase, x-values have a slow decline except for the efficiency where x-values decrease faster. Fig. 7 presents optimized efficiency under the three criteria as a function of λ, also maximum efficiency is shown. The figure shows that values of efficiency operating under maximum Omega criterion are greater with respect to those observed at Maximum Power Output and Maximum Efficient Power, respectively. The efficiency of the device increases as λ grows in any of the different regimes of operation; furthermore, the work output presents this same behavior, so the performance of the device in the limit as λ → ∞ is discussed.
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Fig. 6. Values of x which optimizes the efficiency for each criteria.
Fig. 7. Optimized efficiency for (bio)chemical reaction model analyzed for each criterion.
4.2. Case λ → ∞ This limit case indicates the ideal situation in which the reaction YX leakage is negligible with respect to the coupled reaction. For this case the following equations must be solved,
˙ out ∂W ≡ 0, ∂x λ→∞ ∂ Ω ≡ 0, ∂ x λ→∞ ∂ P E ≡ 0. ∂x
(16)
(17)
(18)
λ→∞
The resulting equations for this particular case are shown in the Appendix, each one can be solved analytically for x yielding the following results xmaxW = −2 +
√
2 2 + y,
(19)
4 + y,
(20)
9 + 4y.
(21)
xmaxΩ = −2 +
xmaxPE = −3 +
Furthermore, the expression for the efficiency in this limit is x
η =1− . y
(22)
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Substituting Eqs. (19), (20), (21) to (22), analytic expressions for efficiencies in the various proposed regimens are found. Performing a series expansion of these efficiencies in terms of y value around 0 can be obtained. For instance, the expression for efficiency at maximum work output is,
ηW ≈
1 2
+
y 16
y2
−
64
+ ··· .
(23)
A similar expression was reported by Golubeva et al. [24] for a molecular machine model described as a Brownian particle diffusing in a periodic tilted potential, and by Van den Broeck [22] and his collaborators as an isothermal molecular machine operating at maximum power. The following expression corresponds to the efficiency of the device to operate in conditions of maximum Omega function.
ηΩ ≈
3 4
+
y 64
−
y2 512
+ ··· .
(24)
The series for the efficiency under maximum efficient power is presented,
ηPE ≈
2 3
+
y 27
−
2y2 243
+ ··· .
(25)
Equivalent expressions for heat engines operating between two reservoirs of temperature were found, independently of scales and nature of the proposed models [11–13]. 5. Concluding remarks In this paper the methodology of the so-called FTT is applied; although traditionally used for the study of heat engines, these were used to analyze the performance of an isothermal engine operating by transferring chemical energy. The behavior of the efficiency and work output for the engine as a function x and the kinetic parameter λ is shown. There is a characteristic value at x that maximizes these functions. Two objective functions were also presented, the Omega and the Efficient Power functions, which exhibit similar behavior, that is, show maximums for particular values at x for a given λ value. It is possible to find the numerical values of x that optimize each of the functions and then substituting these values into the efficiency in order to analyze the performance of the chemical engine; the results indicated that, operating under maximum Omega function conditions provides better values for efficiency than those obtained under the other two criteria. From the numerical results it follows that the kinetic parameter λ, defined as the difference between the activation energy for the leak, and the activation energy for the coupled reaction, modulates the operation of the chemical engine; moreover, for larger values of this parameter, work output and efficiency increase. For a real situation an increase of λ is equivalent to favor the coupled reaction by means of catalysis. The values that optimize the listed function have a range between 0 and 4, such that for values of x near to 0 obtains high efficiency, but work output is low. To higher values of x the efficiency decreased a little but a better performance is achieved for work output production. Finally, the theoretical limit case was analyzed when λ → ∞ that is, when the leak reaction is neglected with respect to the coupled reaction. In this case, it is possible to find analytic expressions for each x that maximize the objective functions; at the same time expressions for optimized efficiencies in each of the schemes are found. In order to compare efficiencies, Taylor series expansions were performed for each of the results with respect to y around 0. It is interesting to note that the series show analogies as in the case of other thermal and isothermal engine models, suggesting a similarity in the performance for energy converters. Acknowledgments Authors thank CONACYT, EDI-IPN and COFAA-IPN for supporting this work. N. Sánchez-Salas is grateful for financial support from MINECO (Spain) under Grant. ENE2013-40644-R. Authors would like also to thank A. Calvo-Hernández and F. Angulo-Brown for many stimulating discussions. Appendix
˙ out = (vcoupled − vYX )(ACD − Acoupled ) W
(A.1)
where
→ vcoupled = − v coupled (1 − eACD /RT ) → v =− v (1 − eAYX /RT ) YX
YX
(A.2) (A.3)
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with x =
Acoupled RT
and y =
ACD , RT
rewritten as,
→ vcoupled = − v coupled (1 − e−x ) → v =− v (1 − ex−y ). YX
(A.4) (A.5)
YX
From the former the work output in terms of x, y and λ is
˙ out = W
˙ out W − →
RT v
= [eλ (1 − e−x ) − (1 − ex−y )](y − x).
(A.6)
YX
From (10) and the above, efficiency is expressed as,
η=
eλ (1 − e−x ) − (1 − ex−y )
(1 − e−y ) + eλ (1 − e−x )
1−
x y
.
(A.7)
Case 0 < λ < ∞
˙ out ∂W = eλ [e−x (1 − x + y) − 1] + 1 + ex−y (y − x − 1) ∂x ∂Ω = e−x−y 2ex+y − 2ex+y+λ − 2e−2x (1 + x − y) + ey+λ (2 − 2x + y) . ∂x
(A.8) (A.9)
The following expression becomes cumbersome to manage so it is left indicated. Although, it is easy to obtain,
∂PE ∂ ˙ = ηW out . ∂x ∂x Case λ → ∞ ˙ out ∂W = e−x (1 − x + y) − 1 ∂x λ→∞ ∂ Ω = e−x (2 − 2x + y) − 2 ∂ x λ→∞ x −x ∂ P E = 1− e (−2 + x − y) + 2 . ∂x y
(A.10)
(A.11)
(A.12)
(A.13)
λ→∞
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Physica A journal homepage: www.elsevier.com/locate/physa
Optimum performance for energy transfer in a chemical reaction system N. Sánchez-Salas a,∗ , J.C. Chimal-Eguía b , M.A. Ramírez-Moreno a a
Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edif. 9 UP Zacatenco, 07738, México D.F, Mexico
b
Centro de Investigación en Computación del IPN., Av. Juan de Dios Bátiz s/n U.P. Zacatenco, CP 07738, México D.F, Mexico
highlights • Optimal performance of an isothermal engine driven by the chemical energy transfer is presented. • Efficient power and Omega criterion are applied as objective functions. • Optimal efficiency shows a similarity for the isothermal engine analyzed.
article
info
Article history: Received 21 August 2015 Received in revised form 4 November 2015 Available online 9 December 2015 Keywords: Optimization Kinetics Biochemistry Isothermal motor
abstract This paper presents an optimal performance analysis for an isothermal model which performs work at a steady state via the energy transfer from a chemical reaction, using the methodology of the so-called Finite Time Thermodynamics (FTT). Furthermore, three optimal operation modes namely, Maximum Power, Maximum Omega function and Efficient Power are presented for the model. Results show analogies in the optimal performance between thermal and isothermal engine models. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Molecular motors are machines that operate in cycles and convert chemical energy into useful work, mainly from the hydrolysis of the ATP molecule, to perform a variety of cellular functions such as DNA replication, repair and transcription of RNA, protein synthesis and transport between cell and other elements [1]. Such machines operate in a constant temperature environment unlike heat engines which are limited by the Carnot (ηC ) efficiency. The limit value of efficiency for molecular motors is one and can be reached only in the limit when the output power is zero. Biochemical processes of living cells occur at ambient or near-ambient temperatures [2,3] at which the rates of spontaneous leak processes are limited, but, cells selectively improve the kinetics of their energy-coupling reactions with enzymes, which effectively create alternative reaction paths with lower activation energies. Enzymes lower activation energies by 30 to 100 kJ/mol, which, at T = 298.15 K, corresponds to increases in reaction rates of 5 to 17 orders of magnitude [4]. Indeed, the survival of living systems often depends heavily on their reproduction rate and, thereby, on the rate of energy conversion in the cells. On the other hand, thermodynamic efficiency is an important factor in the yield of reproduction processes, so that, energy conversion in living cells is expected to occur where a trade-off between both performance aspects is considered.
∗
Corresponding author. E-mail address: [email protected] (N. Sánchez-Salas).
http://dx.doi.org/10.1016/j.physa.2015.11.030 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Scheme of a (bio)chemical reaction model, sketched by Lems [21].
In 1975 Curzon and Ahlborn [5] presented an expression for Efficiency at Maximum Power (EMP) modeling a thermal machine which executes a Carnot-type cycle, a new branch of thermodynamics emerged in the last decades, known as Finite Time Thermodynamics (FTT). The model proposed by Curzon and Ahlborn operates between two heat sources √ with high and low temperatures, Th and Tc (< Th ), respectively, and the expression for the EMP obtained was ηCA = 1 − Tc /Th , a result that in principle is independent of the model parameters and only depends on the temperatures of the heat reservoirs; analogously this is what happens with the efficiency for a reversible Carnot cycle (ηC = 1 − Tc /Th ). Although to obtain the efficiency of Curzon and Ahlborn, it is necessary to consider that the heat transfer involved obeys a linear law. Notice that previously other authors [6,7] have found the same expression for similar models. An important advantage observed in the results obtained in FTT is that optimal efficiency values are closer to those observed in real machines [8]. Within the context of the FTT several objective functions to be optimized under rich criteria have been introduced and, many models of heat engines have been proposed and analyzed which perform as motors, heat pumps or refrigerators [9,10]. It has recently been reported [11–13] that thermal engines show some kind of universality in the behavior of the efficiency at maximum power [13], although the analyzed models are different in nature and scale, for instance, macroscopic, stochastic or quantum [14–16]. Traditionally most of energy-transfer devices discussed by the FTT were heat engines although, some optimal performance analyses were also conducted where biochemical reaction models were analyzed [17–19]. In recent years, isothermal engines with kinetic [20–22] and mesoscopic [23,24] description have been published, as those which convert non-thermal energy (mainly chemistry) into useful work. The importance of these models is that the energy production processes that are observed in the molecular biological level machines obey similar principles. In this paper, an optimal performance of a system that produces work via chemical energy transfer by the coupling of two chemical reactions using the methodology of FTT is presented. The performed analysis takes into account three operating regimes namely: Maximum Power, Omega Function and Efficient Power criterion, respectively. Two special cases for the (bio)chemical device efficiency are presented: (i) The general numerical case and (ii) A limit case where analytic results are found. 2. Model for chemical energy transfer The model that would be analyzed was performed by Lems et al. in 2007 [21]. In his model, Lems studied a system of biochemical reactions shown in Fig. 1, the chemical energy is transferred at 298.15 K, in a coupled chemical reaction in which the favorable reaction C → D is used to manage the unfavorable reaction X → Y , a well known procedure in bioenergetics where the biological tasks are managed by the energetic coupling of the reactions [4]. The system considers two leak reactions, which dissipate chemical energy without contributing to production of work output. To describe the system operation, initially the potential of each individual reaction to deliver work is considered. The ability to perform work of a chemical reaction is determined by its affinity (A) of the reaction [21], representing the degree of non-equilibrium between reactants and products, and is determined by the chemical potential µi , and the stoichiometric coefficients νi . From the work proposed by Donder [25] it is known that the affinity can be written as the negative of the change in Gibbs free energy of reaction (−∆r G), A = −∆r G = −
νi µi .
(1)
i
Each of the reactions involved in the system gives (see Table 1). When A > 0, the products of the reaction are at a lower energy level than the reactants, the reaction is possible until thermodynamic equilibrium is reached, when A = 0 (see Fig. 3). At equilibrium, there are no reaction forces that handle the reaction and therefore the production no longer has the potential to generate work.
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C +D → D+Y C →D Y →X
Acoupled ACD AYX
Under steady state operation the affinities of the three chemical reactions involved are interrelated as follows, ACD = AYX + Acoupled .
(2)
Eq. (2) indicates the affinity ACD available between C and D, is the necessary energy to manage the unfavorable reaction YX with affinity AYX and also the energy for the reaction coupled with Acoupled was carried out spontaneously. As Lems [21] et al. pointed out, the value of ACD is important in the model, since biochemical reactions are carried out at room temperature at constant pressure. In the hydrolysis reaction of ATP, the main energy source for most cellular functions is estimated to have an affinity reaction between 48 and 55 kJ/mol [4,26], depending on given intracellular conditions. So, for this model a fixed value 50 kJ/mol to ACD will be considered. Another important aspect in the model’s description is to indicate the reaction rate in terms of the affinity of each of the reactions involved. These are described by the transition state theory developed by Eyring [27] where the net reaction rate v is the result of the forward and backward reaction rate: →
←
v= v−v.
(3)
Fig. 3 shows a processing scheme for reactive products where there is an intermediate state known as activation state. A positive value of the affinity of the reaction (A) has a larger energy barrier (∆Gact + A) for the backward reaction, while the energy barrier for the forward reaction remains unchanged. As a consequence, the forward reaction rate exceeds the rate of backward and forward net reaction, in other words, there is conversion of reactants to products. It is possible to find the relationship between the forward and backward rates in terms of the reaction affinity (A) as, →
←
v / v = eA/RT .
(4)
Combining (3) and (4), following the relationship for the net reaction rate based on the value of the affinity function finds: →
v = v (1 − e−A/RT ).
(5)
This expression force-flow is not linear in contrast to some cases of transfer heat energy as observed in the models of heat engines, where there are still linear relationships to large ranges of temperature differences; in this case the linear region is small (A < 1). For the model there are three rates for each of the reactions involved in the shape expressed in (5), these are: → → → vCD , vYX and vcoupled in turn related to the quantities − v CD , − v YX , − v coupled and ACD , AYX and Acoupled , respectively. − → − → − → In particular, for this system the value of v CD is taken with fixed value 1 mol/s and values v YX , v coupled are determined − → as follows: the forward reaction rates v for each of the reactions are given by the Eyring equation, expressed as follows, →
v = k0 e−1G
act /RT
.
(6)
For simplicity, it is considered that k0 (called frequency factor) has the same value for different reactions and this means that the composition of the substances involved in the reactions has a similar nature and equivalent conditions. It is taken − → − → that v CD and v YX are equal, therefore the activation barriers for the leakage reactions are also equal. Then, it is possible to find the following relations,
− → − → 1Gact −1Gact leak coupled v coupled v coupled RT = − =e . (7) − → → v CD v YX Wherein 1Gact leak is the activation energy of both leakage reactions CD and YX . On the other hand, a λ parameter is defined as:
λ=
act 1Gact leak − 1Gcoupled
. (8) RT The factor λ states the amount the coupled reaction is favorable to the effect on the leak reaction. With λ > 0 the activation − → barrier of the reaction leakage exceeds the activation barrier of the coupled reaction, and therefore, the value of v coupled − → surpasses the value v leak ; with λ = 0, the coupled reaction and leakage reaction are equally favorable and, in λ < 0, value − → → v leak exceeds that of − v coupled . As is known, catalysis is the process by which a chemical reaction rate is increased and is achieved with a different transition state that has lower activation energy. In this case λ can be considered as a coupled reaction catalysis parameter, that is, expresses how far the coupled reaction is catalyzed on the leakage reaction. Thus, the important variables of the model are the affinity of the coupled reaction (Acoupled ) that is the force that will cause unfavorable XY reaction to carry out, and the kinetic parameter λ, expressing how much influence leak reactions has in the process of converting chemical energy into work of the system. As will be analyzed, the behavior of the model is based on these two variables showing work output and efficiency behavior.
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Fig. 2. Graphical presentation of the energy barrier between the reactants and products of a chemical reaction [27,28].
3. Energetic function characteristics This section describes the behavior of energy system functions, work output and efficiency. Additionally, two objective functions are also presented; these were proposed in the context of the FTT to optimize thermal machines. In this case this methodology will be used to optimize the chemical reaction engine as described previously. 3.1. Work-output rate and efficiency Work output per unit time of the system is the product of the unfavorable chemical reaction affinity, that is, AYX and the net rate to the species Y is produced from the X , taking into account the leakage element, that is,
˙ out = (vcoupled − vYX )AYX . W
(9) A
A
CD The explicit expression (9) is given by (A.1) where the following changes of variable x = coupled and y = RT have been RT used. Eq. (9) is a function of x, y and λ, where y is related to the available input energy in the system, x is used for coupling the favorable reaction and the unfavorable one and λ is defined by Eq. (8). From the same nature of x and y both are functionality connected, see Eq. (2) and Fig. 2, where x and y are associated to the affinity A for each involved reaction. Whereas λ is related with 1Gact , so it is reasonable to consider it as independent of x and y in the net reaction rate described by the transition state theory [28]. The behavior of Eq. (9) can be seen in Fig. 3(a) as a function of x for a defined y and some fixed values of λ are shown in
˙ out for a given value of λ. Fig. 3(c) shows the the same graphic. Performance infers that there is a value x that maximizes W ˙ out grows monotonically when y work output as a function of y for x = 5 and different values of λ. It can be observed that W increases; then, there is no y-value that maximizes the work. From 3(d) it follows that the efficiency increases when y grows and also there is no y-value that maximizes this function. It is important to note that y is related with available energy, so it is convenient and reasonable to consider y equals a constant between 48 and 55 kJ/mol [4], as was previously mentioned. On the other hand, the efficiency for coupled chemical reaction device is the relationship between chemical energy transferred from YX for the energy obtained from the source reaction CD, that is, η=
˙ out W ˙ in W
=
(vcoupled − vXY )AXY . (vCD + vcoupled )ACD
(10)
Also expression in terms of x and y variables is presented in the Appendix (A.7). The behavior of the efficiency as a function of x shown in Fig. 3(b) for different values of λ is indicated. As can be seen, the efficiency increases when the λ parameter is bigger, because the effect of leakage reactions is negligible, having a large activation energy, while the coupled reaction is favored. In the limit λ → ∞, η = 1 − yx , theoretically the value 1 could be obtained if x = 0 (with y ̸= 0). When λ decreases it is observed that the value of x that maximizes the efficiency increases. 3.2. Omega function and efficient power Now the behavior for the Omega function is presented. This feature was proposed based on a unified optimization criterion [29] which represents the best compromise between the benefits and losses of energy to any device and has the advantage that it does not explicitly require the entropy expression to be applied, or any other environmental parameter,
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Fig. 3. (a) Work output performance and (b) system efficiency as a function of x, with y = 20.2. (c) Work output performance and (d) efficiency as a function of y, with x = 5, indicating values of λ.
Fig. 4. (a) Performance of Omega Function (Ω ) and (b) efficient Power (PE ) as a function of x and for different values of λ.
such as the case where the Exergy or the Ecological criteria are used as objective functions [30,31]. For this system the maximum efficiency is 1 and the minimum is 0. In this case the expression for Omega function [29] is given by:
Ω=
2η − 1
η
˙ out . W
(11)
Eq. (11) is depicted in Fig. 4(a) for the present system. It can be observed that the value of x that optimizes Omega varies very little to different values of λ. As was previously mentioned, the behavior of the device under maximum efficient power regime will also be discussed. This approach was recently proposed by Yilmaz [32] in the context of heat thermal engines, now will be applicable to a
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Fig. 5. Work output, Efficiency, Omega Function and Efficient Power, all functions are normalized in order to get an easier comparison with efficiency. For each function, the dashed line indicates the critical x-value where the maximum of each one is reached for λ = 5.
chemical reaction system and is defined as the product of the power output and efficiency, which states:
˙ out . PE = η W
(12)
Maximization of this function provides a compromise between power and efficiency, where the designed parameters at maximum efficient power conditions lead to more efficient engines than those at the maximum power conditions [32]. This function has an equivalent performance to the Omega function. It is worth mentioning that λ-values necessary to have Ω positive are greater than those necessary to have W positive. In fact, for λ = 0.5, Ω is negative and W positive (see Fig. 3), this happens because for this value the losses of energy are greater than benefits in the device. Subsequently, Fig. 5 shows the functions presented in this section, with λ = 5, highlighting x values that maximizes each function, which will be analyzed in more detail in the following section. 4. Optimal performance This section gives the results of the optimal model of a system that transfers energy (bio)chemical using the methodology of the FTT having traditionally been applied to heat engines operating between two reservoirs of temperature and, has a device that performs work under isothermal conditions through the transfer of chemical energy. The performance of the model is studied under three optimal regimes namely: (a) Maximum Work Output, (b) Unified criterion called Omega function and (c) Maximum Efficient Power. For the analysis x is taken as the independent variable, and what must be solved are the following equations:
˙ out ∂W ≡ 0, ∂x
(13)
∂Ω ≡ 0, ∂x
(14)
∂ PE ≡ 0. ∂x
(15)
Explicit expressions for the above equations appear in the Appendix. In particular, Eqs. (13)–(15) will be solved in two different cases of λ values, which are λ → ∞ and 0 < λ < ∞. 4.1. Case 0 < λ < ∞ Fig. 6 depicts the numerical values of x which solve numerically Eqs. (13)–(15), where the behavior mentioned in the previous section can be observed in more detail. It is also worthwhile to point out that the x-values which maximize each function have similar mathematical behavior. For λ values close to zero x-values are higher, whereas for λ values increase, x-values have a slow decline except for the efficiency where x-values decrease faster. Fig. 7 presents optimized efficiency under the three criteria as a function of λ, also maximum efficiency is shown. The figure shows that values of efficiency operating under maximum Omega criterion are greater with respect to those observed at Maximum Power Output and Maximum Efficient Power, respectively. The efficiency of the device increases as λ grows in any of the different regimes of operation; furthermore, the work output presents this same behavior, so the performance of the device in the limit as λ → ∞ is discussed.
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Fig. 6. Values of x which optimizes the efficiency for each criteria.
Fig. 7. Optimized efficiency for (bio)chemical reaction model analyzed for each criterion.
4.2. Case λ → ∞ This limit case indicates the ideal situation in which the reaction YX leakage is negligible with respect to the coupled reaction. For this case the following equations must be solved,
˙ out ∂W ≡ 0, ∂x λ→∞ ∂ Ω ≡ 0, ∂ x λ→∞ ∂ P E ≡ 0. ∂x
(16)
(17)
(18)
λ→∞
The resulting equations for this particular case are shown in the Appendix, each one can be solved analytically for x yielding the following results xmaxW = −2 +
√
2 2 + y,
(19)
4 + y,
(20)
9 + 4y.
(21)
xmaxΩ = −2 +
xmaxPE = −3 +
Furthermore, the expression for the efficiency in this limit is x
η =1− . y
(22)
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Substituting Eqs. (19), (20), (21) to (22), analytic expressions for efficiencies in the various proposed regimens are found. Performing a series expansion of these efficiencies in terms of y value around 0 can be obtained. For instance, the expression for efficiency at maximum work output is,
ηW ≈
1 2
+
y 16
y2
−
64
+ ··· .
(23)
A similar expression was reported by Golubeva et al. [24] for a molecular machine model described as a Brownian particle diffusing in a periodic tilted potential, and by Van den Broeck [22] and his collaborators as an isothermal molecular machine operating at maximum power. The following expression corresponds to the efficiency of the device to operate in conditions of maximum Omega function.
ηΩ ≈
3 4
+
y 64
−
y2 512
+ ··· .
(24)
The series for the efficiency under maximum efficient power is presented,
ηPE ≈
2 3
+
y 27
−
2y2 243
+ ··· .
(25)
Equivalent expressions for heat engines operating between two reservoirs of temperature were found, independently of scales and nature of the proposed models [11–13]. 5. Concluding remarks In this paper the methodology of the so-called FTT is applied; although traditionally used for the study of heat engines, these were used to analyze the performance of an isothermal engine operating by transferring chemical energy. The behavior of the efficiency and work output for the engine as a function x and the kinetic parameter λ is shown. There is a characteristic value at x that maximizes these functions. Two objective functions were also presented, the Omega and the Efficient Power functions, which exhibit similar behavior, that is, show maximums for particular values at x for a given λ value. It is possible to find the numerical values of x that optimize each of the functions and then substituting these values into the efficiency in order to analyze the performance of the chemical engine; the results indicated that, operating under maximum Omega function conditions provides better values for efficiency than those obtained under the other two criteria. From the numerical results it follows that the kinetic parameter λ, defined as the difference between the activation energy for the leak, and the activation energy for the coupled reaction, modulates the operation of the chemical engine; moreover, for larger values of this parameter, work output and efficiency increase. For a real situation an increase of λ is equivalent to favor the coupled reaction by means of catalysis. The values that optimize the listed function have a range between 0 and 4, such that for values of x near to 0 obtains high efficiency, but work output is low. To higher values of x the efficiency decreased a little but a better performance is achieved for work output production. Finally, the theoretical limit case was analyzed when λ → ∞ that is, when the leak reaction is neglected with respect to the coupled reaction. In this case, it is possible to find analytic expressions for each x that maximize the objective functions; at the same time expressions for optimized efficiencies in each of the schemes are found. In order to compare efficiencies, Taylor series expansions were performed for each of the results with respect to y around 0. It is interesting to note that the series show analogies as in the case of other thermal and isothermal engine models, suggesting a similarity in the performance for energy converters. Acknowledgments Authors thank CONACYT, EDI-IPN and COFAA-IPN for supporting this work. N. Sánchez-Salas is grateful for financial support from MINECO (Spain) under Grant. ENE2013-40644-R. Authors would like also to thank A. Calvo-Hernández and F. Angulo-Brown for many stimulating discussions. Appendix
˙ out = (vcoupled − vYX )(ACD − Acoupled ) W
(A.1)
where
→ vcoupled = − v coupled (1 − eACD /RT ) → v =− v (1 − eAYX /RT ) YX
YX
(A.2) (A.3)
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with x =
Acoupled RT
and y =
ACD , RT
rewritten as,
→ vcoupled = − v coupled (1 − e−x ) → v =− v (1 − ex−y ). YX
(A.4) (A.5)
YX
From the former the work output in terms of x, y and λ is
˙ out = W
˙ out W − →
RT v
= [eλ (1 − e−x ) − (1 − ex−y )](y − x).
(A.6)
YX
From (10) and the above, efficiency is expressed as,
η=
eλ (1 − e−x ) − (1 − ex−y )
(1 − e−y ) + eλ (1 − e−x )
1−
x y
.
(A.7)
Case 0 < λ < ∞
˙ out ∂W = eλ [e−x (1 − x + y) − 1] + 1 + ex−y (y − x − 1) ∂x ∂Ω = e−x−y 2ex+y − 2ex+y+λ − 2e−2x (1 + x − y) + ey+λ (2 − 2x + y) . ∂x
(A.8) (A.9)
The following expression becomes cumbersome to manage so it is left indicated. Although, it is easy to obtain,
∂PE ∂ ˙ = ηW out . ∂x ∂x Case λ → ∞ ˙ out ∂W = e−x (1 − x + y) − 1 ∂x λ→∞ ∂ Ω = e−x (2 − 2x + y) − 2 ∂ x λ→∞ x −x ∂ P E = 1− e (−2 + x − y) + 2 . ∂x y
(A.10)
(A.11)
(A.12)
(A.13)
λ→∞
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