Maximum work configuration of finite potential source endoreversible non-isothermal chemical engines

Lingen Chen; Shaojun Xia

doi:10.1515/jnet-2022-0045

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Maximum work configuration of finite potential source endoreversible non-isothermal chemical engines

Lingen Chen and Shaojun Xia

Published/Copyright: November 18, 2022

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 48 Issue 1

Abstract

Chemical engine is an abstract model of some devices, such as solid state, photochemical, and electrochemical devices, photovoltaic cell, and mass exchangers. Finite chemical-potential source is one of its features. Finite time thermodynamics provides effective theoretical tool for determining performance limits for given thermal systems, and determining optimal process paths of thermal systems for given performance objectives. Endoreversible model is its basic model. A model of endoreversible non-isothermal chemical engines operating between a finite chemical-potential source and an infinite chemical-potential sink with mass resistance and heat resistance is established. Mass transfer processes between chemical potential reservoir and working fluid of the model are assumed to obey Onsager equations in linear irreversible thermodynamics. With a fixed cycle period, optimal cycle configuration for the maximum work output of the model is derived by applying optimal control theory. The results obtained include optimal performance and optimal path results in many previous literatures, and can provide some theoretical guidelines for optimal designs of practical chemical plants.

Keywords: endoreversible non-isothermal chemical engine; finite potential source; finite time thermodynamics; linear irreversible thermodynamics; maximum work output; optimal control

1 Introduction

Finite time thermodynamics (FTT) provides effective theoretical tool for one to determine performance limits for given thermal systems, and to determine optimal process paths of thermal systems for given performance objectives [1–25]. Ondrechen et al. [26] investigated optimal cycle configuration (OCC) for maximum work output (MWO) from a finite heat source (FHS) by sequential reversible Carnot engine cycles. Ondrechen et al. [27] further investigated OCC for the MWO of a FHS endoreversible heat engine (EHE) with Newtonian heat transfer law (HTL) [q ∝ Δ(T)] (where T is temperature). Yan and Chen [28] investigated the same problem with linear phenomenological HTL [q ∝ Δ(T ⁻¹)]. Other researchers had investigated the similar problems with different HTLs, including generalized radiative [q ∝ Δ(T ⁿ)], generalized convective [q ∝ (ΔT)^m], mixed heat resistances and a generalized [ q ∝ ( Δ ( T n ) ) m ] HTLs. Amelkin et al. [29, 30] investigated the OCC for the MWO of a FHS multi-reservoir EHE.

FTT optimization for EHEs has been extended to the studies of generalized thermal processes and devices. Chemical engine is an abstract model of some devices, such as solid state, photochemical, and electrochemical devices, photovoltaic cell, and mass exchangers. De Vos [31–34] did extension studies for EHE idea, and developed an idea of endoreversible non-isothermal chemical engine (NICE) by extending heat reservoir to heat and mass reservoir and extending heat exchanger with only heat resistance to heat and mass exchanger with both heat resistance and mass resistance, and investigated the optimal performances of the NICEs, including solar energy cells, chemical reactions, etc. Gordon [35] found OCC for MWO from a finite chemical potential source (FCPS) by reversible sequential isothermal chemical engines (ICEs). Gordon and Orlov [36] further found OCC for the MWO of an endoreversible ICE, and showed that the OCC consisted of four processes, that is, two constant-chemical-potential processes and two instantaneous constant-mass-flux processes, which was analogy to Curzon-Ahlborn EHE [37]. Chen et al. [38, 39] obtained optimal power-efficiency relations of single [22] and combined [23] endoreversible ICEs. Lin et al. [40] obtained optimal performance of an irreversible ICE with mass resistance, internal irreversibility effect and mass leakage. Tsirlin et al. [41, 42] obtained optimal performances of endoreversible ICEs with objectives of power output [41] and entropy generation [42]. Chen et al. [43] analyzed effect of mass transfer laws (MTLs) on endoreversible ICE. Xia et al. [44] obtained optimal power-efficiency relation of irreversible ICE with diffusive MTL [g ∝ Δ(μ/(kT))] (where μ is the chemical potential).

FCPS is one of chemical engine’s features. Xia et al. [45, 46] studied the OCC for the MWO of an endoreversible ICE with FCPS and with linear MTL [g ∝ Δμ] [45], and further studied effects of MTLs [46]. Xia et al. [47] studied the OCC for the MPO of a FCPS multi-reservoir endoreversible ICE.

The above literatures mainly focused on optimal performances and OCCs of EHEs and endoreversible ICEs, which include only the pure heat or mass transfer process, and more generalized models of heat engines and chemical engines should include simultaneous heat and mass transfer. De Vos [32] provided an endoreversible NICE model with two infinite potential mass reservoirs. Sieniutycz and Kubiak [48] and Sieniutycz [49–52] studied optimal performances of infinite-potential-reservoir endoreversible NICE based on Lewis similar criterion and linear irreversible thermodynamics (LIT) [48, 49], and studied effect of internal dissipation of working fluid qualitatively [50–52]. Cai et al. [53] investigated the MPO of an endoreversible NICE, in which HMT processes obeyed Newtonian HTL and linear MTL [g ∝ Δ(μ)], respectively. Guo et al. [54] investigated the MPO of an irreversible NICE using the weak-dissipation assumption.

Recently, Chen and Xia [55–58] focused on OCCs of single- [55] and multi- [56] stage irreversible ICEs with FCPS as well as the optimal performances of single- [57] and multi- [58] stage endoreversible NICEs with infinite chemical potential source. In Refs. [57, 58], an endoreversible NICE model, in which heat and mass transfer processes were assumed to obey Onsager equations in LIT [59], was established.

One of the goals of finite time thermodynamics is to pursue universal research results. There are no work concerns the OCC study for NICEs in the open literatures. Based on Refs. [57, 58] this paper will establish a FCPS endoreversible NICE model, in which heat and mass transfer processes are assumed to obey Onsager equations in LIT [39], and derive OCC for the MWO of the endoreversible NICE by applying optimal control theory.

2 Chemical engine model

Figure 1 shows an endoreversible NICE model operating between a FCPS and an infinite chemical-potential sink (ICPS). The key difference between the model shown in Figure 1 and that in Refs. [57, 58] is that the FCPS is introduced in the model of this paper. Total mass of FCPS at time t is denoted as M ₁(t), and temperature and specific heat capacity of mixture in it are denoted as T ₁(t) and c _p1, respectively. One has M ₁(0) = M ₁₀ and T ₁(0) = T ₁₀ at the initial time t = 0. Since FCPS is finite in size, both its temperature and chemical potential (concentration) of key component B ₁ in it can change with heat and mass absorbed by working fluid of endoreversible NICE. Let chemical potential and concentration (expressed by mole fraction) of key component B ₁ in it be μ ₁(c ₁) and c ₁(t), respectively. Chemical reaction in endoreversible NICE is assumed to be a single reversible isomerization reaction B ₁ ⇔ B ₂ [50–52, 60]. Concentration of key component B ₁ in working fluid at FCPS side is c _1′(t), and that of key component B ₂ in working fluid at ICPS side is c _2′(t). The corresponding chemical potentials are μ 1 ′ c 1 ′ and μ 2 ′ c 2 ′ , respectively, and the corresponding temperatures are T 1 ′ ( t ) and T 2 ′ ( t ) , respectively. The ICPS is infinite in size, and temperature, chemical potential and concentration of key component B ₂ in it keep constants, which are denoted as T ₂, μ ₂(c ₂) and c ₂, respectively. Let total energy and total mass absorbed by endoreversible NICE from the FCPS over a whole cycle be E ₁ and G ₁, respectively, and those released by endoreversible NICE to ICPS be E ₂ and G ₂, respectively. Cycle period is τ.

Figure 1:

Model of a FCPS endoreversible NICE.

Consider that heat and mass transfer process between chemical potential reservoir and working fluid of endoreversible NICE obey Onsager equations in LIT [59]. Therefore, for the FCPS side of endoreversible NICE, one has

(1) E 1 = ∫ 0 τ E ̇ 1 ( t ) d t = ∫ 0 τ α 1 ( t ) 1 T 1 ′ − 1 T 1 + γ 1 ( t ) μ 1 T 1 − μ 1 ′ T 1 ′ d t

(2) G 1 = ∫ 0 τ g 1 ( t ) d t = ∫ 0 τ γ 1 ( t ) 1 T 1 ′ − 1 T 1 + h 1 ( t ) μ 1 T 1 − μ 1 ′ T 1 ′ d t

where E ̇ 1 ( t ) and g ₁(t) are energy and mass transfer rates, respectively; α ₁(t) and h ₁(t) are phenomenological heat and mass transfer coefficients, respectively; and γ ₁(t) is cross heat and mass transfer coefficient.

For ICPS side of endoreversible NICE, one has

(3) E 2 = ∫ 0 τ E ̇ 2 ( t ) d t = ∫ 0 τ α 2 ( t ) 1 T 2 − 1 T 2 ′ + γ 2 ( t ) μ 2 ′ T 2 ′ − μ 2 T 2 d t

(4) G 2 = ∫ 0 τ g 2 ( t ) d t = ∫ 0 τ γ 2 ( t ) 1 T 2 − 1 T 2 ′ + h 2 ( t ) μ 2 ′ T 2 ′ − μ 2 T 2 d t

where g ₂(t) and E ̇ 2 ( t ) are mass and energy transfer rates, respectively; h ₂(t) and α ₂(t) are phenomenological mass and heat transfer coefficients, respectively; and γ ₂(t) is cross heat and mass transfer coefficient.

Assume that working fluid of endoreversible NICE starts to absorb heat and mass from FCPS at time t = 0, it turns to release heat and mass to low-chemical-potential sink at time t = t ₁ (0 < t ₁ < τ), and there is an instantaneous constant-entropy and constant-mass-flux conversion process of working fluid between FCPS and ICPS. Thus one obtains

(5) α 1 ( t ) = α 1 , 0 ≤ t ≤ t 1 0 , t 1 ≤ t ≤ τ ; γ 1 ( t ) = γ 1 , 0 ≤ t ≤ t 1 0 , t 1 ≤ t ≤ τ ; h 1 ( t ) = h 1 , 0 ≤ t ≤ t 1 0 , t 1 ≤ t ≤ τ

(6) α 2 ( t ) = 0 , 0 ≤ t ≤ t 1 α 2 , t 1 ≤ t ≤ τ ; γ 2 ( t ) = 0 , 0 ≤ t ≤ t 1 γ 2 , t 1 ≤ t ≤ τ ; h 2 ( t ) = 0 , 0 ≤ t ≤ t 1 h 2 , t 1 ≤ t ≤ τ

where α ₁, γ ₁, h ₁, α ₂, γ ₂ and h ₂ are constants.

Cycle work output (W) of endoreversible NICE is:

(7) W = ∫ 0 τ P ( t ) d t = ∫ 0 τ [ E ̇ 1 ( t ) − E ̇ 2 ( t ) ] d t = ∫ 0 τ α 1 ( t ) 1 T 1 ′ − 1 T 1 + γ 1 ( t ) μ 1 T 1 − μ 1 ′ T 1 ′ − α 2 ( t ) 1 T 2 − 1 T 2 ′ − γ 2 ( t ) μ 2 ′ T 2 ′ − μ 2 T 2 d t

where P(t) is instantaneous power of endoreversible NICE. Since NICE is endoreversible, entropy change of working fluid of NICE through a whole cycle should be zero. One obtains

(8) ∫ 0 τ E ̇ 1 ( t ) − g 1 ( t ) μ 1 ′ T 1 ′ − E ̇ 2 ( t ) − g 2 ( t ) μ 2 ′ T 2 ′ d t = 0

The law of mass conservations (that is, G ₁ = G ₂) further gives

(9) ∫ 0 τ γ 1 ( t ) 1 T 1 ′ − 1 T 1 + h 1 ( t ) μ 1 T 1 − μ 1 ′ T 1 ′ − γ 2 ( t ) 1 T 2 − 1 T 2 ′ − h 2 ( t ) μ 2 ′ T 2 ′ − μ 2 T 2 d t = 0

Because there is only key component B ₁ participating in mass transfer process at FCPS side, one obtains

(10) d M 1 / d t = d ( M 1 c 1 ) / d t = − g 1 , M 1 ( 0 ) = M 10 , c 1 ( 0 ) = c 10

(11) d ( M 1 T 1 ) / d t = − E ̇ 1 / c p 1 , T 1 ( 0 ) = T 10

From Eqs. (10) and (11), one further obtains

(12) d c 1 / d t = − g 1 ( 1 − c 1 ) / M 1

(13) d T 1 / d t = ( g 1 T 1 − E ̇ 1 / c p 1 ) / M 1

Mass ( m ̃ 1 ) of inert component in FCPS keeps constant during process of finite-rate mass-transfer, and one has the following relation

(14) M 1 ( 1 − c 1 ) = m ̃ 1

Combining Eq. (14) with Eqs. (12) and (13), respectively, gives

(15) d c 1 / d t = − g 1 ( 1 − c 1 ) 2 / m ̃ 1

(16) d T 1 / d t = ( g 1 T 1 − E ̇ 1 / c p 1 ) ( 1 − c 1 ) / m ̃ 1

3 Solving procedure

The problem is to determine MWO of endoreversible NICE for given cycle period τ, that is, to determine maximum W in Eq. (7) and corresponding optimal time paths of the concentrations c _i(t) and the temperature T _i(t) (i = 1, 1′, 2′) with the constraints of Eqs. (8), (9), (12) and (13). Correspondingly, a modified Lagrange function L is given by

(17) L = E ̇ 1 − E ̇ 2 + λ 1 E ̇ 1 − g 1 μ 1 ′ T 1 ′ − E ̇ 2 − g 2 μ 2 ′ T 2 ′ + λ 2 ( g 1 − g 2 ) + u 1 ( t ) d c 1 d t + g 1 ( 1 − c 1 ) 2 m ̃ 1 + u 2 ( t ) d T 1 d t − ( g 1 T 1 − E ̇ 1 / c p 1 ) ( 1 − c 1 ) m ̃ 1

where λ ₁ and λ ₂ are Lagrange constants to be determined, and u ₁(t) and u ₂(t) are time-dependent functions. Eq. (17) further gives

(18) L = E ̇ 1 1 + λ 1 T 1 ′ − E ̇ 2 1 + λ 1 T 2 ′ + g 1 λ 2 − λ 1 μ 1 ′ T 1 ′ − g 2 λ 2 − λ 1 μ 2 ′ T 2 ′ + u 1 ( t ) d c 1 d t + g 1 ( 1 − c 1 ) 2 m ̃ 1 + u 2 ( t ) d T 1 d t − ( g 1 T 1 − E ̇ 1 / c p 1 ) ( 1 − c 1 ) m ̃ 1

The necessary conditions to determine the extreme value of Eq. (18) are Euler–Lagrange equations, as followings:

(19) ∂ L ∂ c 1 − d d t ∂ L ∂ ( d c 1 / d t ) = 0 , ∂ L ∂ T 1 − d d t ∂ L ∂ ( d T 1 / d t ) = 0

(20) ∂ L ∂ c 1 ′ − d d t ∂ L ∂ d c 1 ′ / d t = 0 , ∂ L ∂ T 1 ′ − d d t ∂ L ∂ d T 1 ′ / d t = 0

(21) ∂ L ∂ c 2 ′ − d d t ∂ L ∂ d c 2 ′ / d t = 0 , ∂ L ∂ T 2 ′ − d d t ∂ L ∂ d T 2 ′ / d t = 0

Assume that the effect of the temperature T on the chemical potential μ is neglected, and the chemical potential μ of the component only depends on its concentration c. Substituting Eq. (18) into Eqs. (19)–(21) yields:

When 0 ≤ t ≤ t ₁, one has
(22) 1 + λ 1 T 1 ′ + u 2 ( 1 − c 1 ) m ̃ 1 c p 1 ∂ E ̇ 1 ∂ c 1 + λ 2 − λ 1 μ 1 ′ T 1 ′ + u 1 ( 1 − c 1 ) 2 − u 2 ( 1 − c 1 ) T 1 m ̃ 1 ∂ g 1 ∂ c 1 + 2 u 1 g 1 ( c 1 − 1 ) + u 2 ( g 1 T 1 − E ̇ 1 / c p 1 ) m ̃ 1 − d u 1 d t = 0

(23) 1 + λ 1 T 1 ′ ∂ E ̇ 1 ∂ T 1 + λ 2 − λ 1 μ 1 ′ T 1 ′ + u 1 ( 1 − c 1 ) 2 m ̃ 1 ∂ g 1 ∂ T 1 + u 2 ( c 1 − 1 ) m ̃ 1 g 1 − 1 c p 1 ∂ E ̇ 1 ∂ T 1 − d u 2 d t = 0

(24) 1 + λ 1 T 1 ′ − u 2 ( c 1 − 1 ) m ̃ 1 c p 1 ∂ E ̇ 1 ∂ c 1 ′ + λ 2 − λ 1 μ 1 ′ T 1 ′ + u 1 ( 1 − c 1 ) 2 + u 2 ( c 1 − 1 ) T 1 m ̃ 1 ∂ g 1 ∂ c 1 ′ − λ 1 g 1 T 1 ′ ∂ μ 1 ′ ∂ c 1 ′ = 0

(25) 1 + λ 1 T 1 ′ − u 2 ( c 1 − 1 ) m ̃ 1 c p 1 ∂ E ̇ 1 ∂ T 1 ′ − λ 1 E ̇ 1 T 1 ′ 2 + λ 2 − λ 1 μ 1 ′ T 1 ′ + u 1 ( 1 − c 1 ) 2 m ̃ 1 + u 2 ( c 1 − 1 ) T 1 m ̃ 1 ∂ g 1 ∂ T 1 ′ + λ 1 g 1 μ 1 ′ T 1 ′ 2 = 0
When t ₁ ≤ t ≤ τ, one has
(26) − ∂ E ̇ 2 ∂ c 2 ′ 1 + λ 1 T 2 ′ − λ 2 − λ 1 μ 2 ′ T 2 ′ ∂ g 2 ∂ c 2 ′ + λ 1 g 2 T 2 ′ ∂ μ 2 ′ ∂ c 2 ′ = 0

(27) − ∂ E ̇ 2 ∂ T 2 ′ 1 + λ 1 T 2 ′ + λ 1 E ̇ 2 T 2 ′ 2 − λ 2 − λ 1 μ 2 ′ T 2 ′ ∂ g 2 ∂ T 2 ′ − λ 1 g 2 μ 2 ′ T 2 ′ 2 = 0

Eqs. (22)–(27) determine OCC for MWO of FCPS endoreversible NICE. The problem should be solved numerically.

4 Analyses for special cases and discussions

One of the goals of finite time thermodynamics is to pursue universal research results. Sections 4.1–4.5 will illustrate that the research results of the FCPS endoreversible NICE model in this paper have strong universality, and the existing research results are special cases of this paper. The following special case analysis and the corresponding research results also fully illustrate the reliability of the model and methodology in this paper.

4.1 Infinite heat source endoreversible heat engine

If M ₁₀ → ∞, γ ₁ = γ ₂ = h ₁ = h ₂ = 0, E ₁ = Q ₁ and E ₂ = Q ₂, i.e. the previous FCPS endoreversible NICE turns to be an infinite heat source EHE, and the OCC is with two isothermal processes and two instantaneous adiabatic processes [37]. From Eqs. (1), (3), (7) and (8), one obtains optimal power-efficiency relation:

(28) P = W τ = α 1 ( 1 − η T ) / T 2 − 1 / T 1 [ 1 + α 1 / α 2 ( 1 − η T ) ] 2

where η T = 1 − T 2 ′ / T 1 ′ is thermal efficiency. Eq. (28) is the results for the infinite heat reservoir EHE with linear phenomenological HTL in Refs. [37, 61, 62].

4.2 Finite heat source endoreversible heat engine

When heat and mass transfer process becomes pure heat transfer process, one has γ ₁ = γ ₂ = h ₁ = h ₂ = 0, E ₂ = Q ₂, E ₁ = Q ₁, E ̇ 2 = q 2 , E ̇ 1 = q 1 , where Q ₂ and Q ₁ are the amounts of heat transferred corresponding to low- and high-temperature sides of EHE, respectively, and q ₁ and q ₂ are the corresponding heat transfer rates, respectively. The previous FCPS endoreversible NICE becomes the FHS EHE. Eqs. (23), (25) and (27), respectively, become

(29) ∂ q 1 ∂ T 1 1 + λ 1 T 1 ′ + u 2 M 1 c p 1 ∂ q 1 ∂ T 1 − d u 2 d t = 0

(30) ∂ q 1 ∂ T 1 ′ 1 + λ 1 T 1 ′ − λ 1 q 1 T 1 ′ 2 + u 2 M 1 c p 1 ∂ q 1 ∂ T 1 ′ = 0

(31) − ∂ q 2 ∂ T 2 ′ 1 + λ 1 T 2 ′ + λ 1 q 2 T 2 ′ 2 = 0

From Eqs. (29)–(31), one further obtains

(32) 1 T 1 ′ − 1 T 1 = Q 1 α 1 t 1 = c o n s t

(33) 1 T 2 ′ − 1 T 2 = − Q 2 α 2 ( τ − t 1 ) = c o n s t

Eqs. (32) and (33) are results for MWO of the FHS EHE with linear phenomenological HTL in Refs. [38, 63–65].

4.3 Infinite potential source endoreversible isothermal chemical engine

If M ₁₀ → ∞, α ₁ = α ₂ = 0 and T 1 = T 1 ′ = T 2 ′ = T 2 = T , the previous FCPS endoreversible NICE becomes infinite potential reservoir endoreversible ICE. The OCC is with two constant-chemical-potential mass-transfer processes and two instantaneous constant-mass-flux processes. The optimal power-efficiency relation of endoreversible ICE is [38, 40]

(34) P ̄ = h 1 η ( 1 − η ) ( μ 1 − μ 2 ) 2 / 1 + ( h 1 / h 2 ) 0.5 2

where η = μ 1 ′ − μ 2 ′ / ( μ 1 − μ 2 ) . The optimal power-efficiency relation of endoreversible ICE with two infinite mass reservoirs is parabolic-type curve.

4.4 Finite potential source endoreversible isothermal chemical engine

When heat and mass transfer process becomes isothermal mass transfer process, one has α ₁ = α ₂ = 0 and T 1 = T 1 ′ = T 2 ′ = T 2 = T . The previous FCPS endoreversible NICE becomes FCPS endoreversible ICE. Eqs. (22), (24) and (26), respectively, become

(35) ∂ g 1 ∂ c 1 μ 1 ′ + λ 2 + u 1 ( 1 − c 1 ) 2 m ̃ 1 ∂ g 1 ∂ c 1 + u 1 2 g 1 ( c 1 − 1 ) m ̃ 1 − d u 1 d t = 0

(36) ∂ g 1 ′ ∂ c 1 ′ μ 1 ′ + λ 2 + g 1 ∂ μ 1 ′ ∂ c 1 ′ = 0

(37) ∂ g 2 ∂ c 2 ′ μ 2 ′ + λ 2 + g 2 ∂ μ 2 ′ ∂ c 2 ′ = 0

From Eqs. (35)–(37), one further obtains

(38) μ 1 − μ 1 ′ = G 1 T / ( h 1 t 1 ) = c o n s t

(39) μ 2 ′ − μ 2 = G 1 T / [ ( τ − t 1 ) h 2 ] = c o n s t

Eqs. (38) and (39) are results for MWO of FCPS endoreversible ICE with linear MTL in Refs. [45, 46]. According to Refs. [45, 46], there is no analytical solution for the maximum power output and the corresponding efficiency, which can only be calculated by numerical method. The equivalent chemical potentials of the key components in the working fluid at the high potential side of the chemical engine and the high potential reservoir are defined as μ ̄ 1 = ∫ 0 t 1 μ 1 ( t ) d t / t 1 and μ ̄ 1 ′ = ∫ 0 t 1 μ 1 ′ ( t ) d t / t 1 . At this time, the optimal power-efficiency relation of the chemical engine is the same as Eq. (34), which is also parabolic-type curve.

4.5 Infinite potential source endoreversible non-isothermal chemical engine

If M ₁₀ → ∞, both temperature T ₁ and chemical potential μ ₁ of FCPS are constants, and the OCC consists of two heat and mass transfer processes with constant-temperature and constant-chemical-potential, and two instantaneous processes with constant-entropy and constant-mass-flux. Eqs. (1)–(4), respectively, become

(40) E 1 = α 1 1 T 1 ′ − 1 T 1 + γ 1 μ 1 T 1 − μ 1 ′ T 1 ′ t 1

(41) G 1 = γ 1 1 T 1 ′ − 1 T 1 + h 1 μ 1 T 1 − μ 1 ′ T 1 ′ t 1

(42) E 2 = α 2 1 T 2 − 1 T 2 ′ + γ 2 μ 2 ′ T 2 ′ − μ 2 T 2 ( τ − t 1 )

(43) G 2 = γ 2 1 T 2 − 1 T 2 ′ + h 2 μ 2 ′ T 2 ′ − μ 2 T 2 ( τ − t 1 )

From Eq. (8), one obtains [31–33]

(44) E 1 − G 1 μ 1 ′ T 1 ′ − E 2 − G 2 μ 2 ′ T 2 ′ = 0

From G ₁ = G ₂, and Eqs. (7) and (44), one further obtains [31–33]

(45) W = E 1 1 − T 2 ′ T 1 ′ + T 2 ′ μ 1 ′ T 1 ′ − μ 2 ′ T 2 ′ G 1

Let W = E ₁ η _T + G ₁ η _μ, and there is a vector efficiency η = (η _T, η _μ) in Eq. (45), where η _T and η _μ are, respectively, given by [31–33]

(46) η T = 1 − T 2 ′ T 1 ′ , η μ = T 2 ′ T 1 ′ μ 1 ′ − μ 2 ′

If the cycle is reversible, one has T 1 ′ = T 1 , μ 1 ′ = μ 1 , T 2 ′ = T 2 and μ 2 ′ = μ 2 . The reversible efficiency of the cycle is η _rev = (η _T,rev, η _μ,rev), where η _T,rev and η _μ,rev are, respectively, given by [31–33]

(47) η T , r e v = 1 − T 2 T 1 , η μ , r e v = T 2 T 1 μ 1 − μ 2

From Eqs. (40)–(45), if γ ₁ = γ ₂ = h ₁ = h ₂ = 0, E ₁ = Q ₁ and E ₂ = Q ₂ further, the results for EHE with linear phenomenological HTL [q ∝ Δ(T ⁻¹)] are also obtained; if α ₁ = α ₂ = 0 and T 1 = T 1 ′ = T 2 ′ = T 2 = T further, the results for endoreversible ICE with linear MTL [g ∝ Δμ] are also obtained. From Eqs. (40) and (41), one obtains

(48) 1 T 1 ′ = 1 T 1 + h 1 E 1 − γ 1 G 1 t 1 α 1 h 1 − γ 1 2

(49) μ 1 ′ T 1 ′ = μ 1 T 1 − α 1 G 1 − γ 1 E 1 t 1 α 1 h 1 − γ 1 2

From Eqs. (42) and (43), one further obtains

(50) 1 T 2 ′ = 1 T 2 − h 2 E 2 − γ 2 G 1 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 )

(51) μ 2 ′ T 2 ′ = μ 2 T 2 + α 2 G 1 − γ 2 E 2 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 )

Substituting Eqs. (48)–(51) into Eq. (44) yields

(52) E 1 − μ 1 G 1 T 1 + h 1 E 1 2 − 2 γ 1 G 1 E 1 + α 1 G 1 2 t 1 α 1 h 1 − γ 1 2 − E 2 − μ 2 G 1 T 2 + h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 ) = 0

From Eq. (52), one can obtains E ₂(E ₁, G ₁, t ₁) as a function of E ₁, G ₁ and t ₁. Since total work output of endoreversible ICE is W = E ₁ − E ₂, the problem is to determine the maximum W with the constraint of Eq. (52). Correspondingly, a modified Lagrange function is given by

(53) L = E 1 − E 2 + λ E 1 − μ 1 G 1 T 1 + h 1 E 1 2 − 2 γ 1 G 1 E 1 + α 1 G 1 2 t 1 α 1 h 1 − γ 1 2 − E 2 − μ 2 G 1 T 2 + h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 )

where λ is Lagrange multiplier, which is a constant should be determined. By using ∂L/∂E ₁ = 0, ∂L/∂E ₂ = 0, ∂L/∂t ₁ = 0 and ∂L/∂G ₁ = 0, one obtains

(54) 1 + λ 1 T 1 + 2 h 1 E 1 − 2 γ 1 G 1 t 1 α 1 h 1 − γ 1 2 = 0

(55) − 1 + λ − 1 T 2 + 2 h 2 E 2 − 2 γ 2 G 1 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 ) = 0

(56) h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( τ − t 1 ) 2 α 2 h 2 − γ 2 2 − h 1 E 1 2 − 2 γ 1 N 1 E 1 + α 1 G 1 2 t 1 2 α 1 h 1 − γ 1 2 = 0

(57) − μ 1 T 1 + 2 α 1 G 1 − 2 γ 1 E 1 t 1 α 1 h 1 − γ 1 2 + μ 2 T 2 + 2 h 2 E 2 − 2 γ 2 G 1 ( τ − t 1 ) ( α 2 h 2 − γ 2 2 ) = 0

From Eq. (56), one further obtains the time t ₁, which is given by

(58) t 1 = τ / 1 + h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( α 1 h 1 − γ 1 2 ) h 1 E 1 2 − 2 γ 1 G 1 E 1 + α 1 G 1 2 ( α 2 h 2 − γ 2 2 )

Combining Eqs. (54) and (55) with Eq. (58) yields

(59) 1 T 1 − 1 T 2 + 1 + h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( α 1 h 1 − γ 1 2 ) h 1 E 1 2 − 2 γ 1 G 1 E 1 + α 1 G 1 2 ( α 2 h 2 − γ 2 2 ) × ( 2 h 1 E 1 − 2 γ 1 G 1 ) τ α 1 h 1 − γ 1 2 + ( 2 h 2 E 2 − 2 γ 2 G 1 ) α 2 h 2 − γ 2 2 h 1 E 1 2 − 2 γ 1 G 1 E 1 + α 1 G 1 2 ( α 2 h 2 − γ 2 2 ) h 2 E 2 2 − 2 γ 2 G 1 E 2 + α 2 G 1 2 ( α 1 h 1 − γ 1 2 ) = 0

Combining Eq. (58) with Eq. (57) yields

(60) μ 2 T 2 − μ 1 T 1 + 1 + h 2 ε 2 2 − 2 γ 2 N 1 ε 2 + α 2 N 1 2 ( α 1 h 1 − γ 1 2 ) h 1 ε 1 2 − 2 γ 1 N 1 ε 1 + α 1 N 1 2 ( α 2 h 2 − γ 2 2 ) × 2 α 1 N 1 − 2 γ 1 ε 1 τ α 1 h 1 − γ 1 2 + 2 h 2 ε 2 − 2 γ 2 N 1 τ α 2 h 2 − γ 2 2 h 1 ε 1 2 − 2 γ 1 N 1 ε 1 + α 1 N 1 2 ( α 2 h 2 − γ 2 2 ) h 2 ε 2 2 − 2 γ 2 N 1 ε 2 + α 2 N 1 2 ( α 1 h 1 − γ 1 2 ) = 0

For the given parameters of T _i, μ _i, α _i, γ _i and h _i (i = 1, 2), on can solve Eqs. (52), (59) and (60) for E _{1, opt}, E _{2, opt} and G _1,opt numerically. The optimal time (t _{1, opt}) is further obtained by substituting E _{1, opt}, E _{2, opt} and G _1,opt into Eq. (58). From Eqs. (47)–(51), one further obtains the efficiency (η _max P = (η _T, max P, η _{μ, max P})) corresponding to the MWO, where η _T, max P and η _{μ, max P} are, respectively, given by [57]

(61) η T , max P = ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 ) ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 ) − ( h 2 E 2 , o p t − γ 2 G 1 , o p t ) T 2 × η T C − T 2 ( h 1 E 1 , o p t − γ 1 G 1 , o p t ) t 1 , o p t α 1 h 1 − γ 1 2 − T 2 ( h 2 E 2 , o p t − γ 2 G 1 , o p t ) ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 )

(62) η μ , max P = ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 ) ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 ) − ( h 2 E 2 , o p t − γ 2 G 1 , o p t ) T 2 × η μ C − T 2 ( α 1 G 1 , o p t − γ 1 E 1 , o p t ) t 1 , o p t α 1 h 1 − γ 1 2 − T 2 ( α 2 N 1 , o p t − γ 2 E 2 , o p t ) ( τ − t 1 , o p t ) ( α 2 h 2 − γ 2 2 )

5 Conclusions

Chemical engine is an abstract model of some devices, such as solid state, photochemical, and electrochemical devices, photovoltaic cell, and mass exchangers. FCPS is one of its features. FTT provides effective theoretical tool for determining performance limits for given thermal systems, and for determining optimal process paths of thermal systems for given performance objectives. Endoreversible model is its basic model. An endoreversible NICE model operating between a FCPS and an ICPS is established. The heat and mass transfer processes between chemical potential reservoir and working fluid of the endoreversible NICE are assumed to obey Onsager equations in LIT. For the given cycle period, the optimality condition for MWO of endoreversible NICE is obtained by applying optimal control theory. The object studied herein is rather general, and the obtained results include those for linear phenomenological HTL [q ∝ Δ(T ⁻¹)] EHEs with infinite and finite heat reservoirs, linear MTL [g ∝ Δμ] endoreversible ICEs with infinite and finite potential reservoirs, and endoreversible NICEs with infinite potential reservoirs. The calculating curves based on the proposed optimization model of the FCPS endoreversible NICE in this paper will be given in our following studies. The idea and solving method applied herein can be developed to studies on optimizing performances and paths of other chemical devices. The results could provide some guidelines for optimal designs of practical chemical devices, including solid state, photochemical, and electrochemical devices, mass exchangers, etc.

Corresponding author: Lingen Chen, Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China; and School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China, E-mail: lingenchen@hotmail.com

Funding source: The National Natural Science Foundation of China

Award Identifier / Grant number: 52171317

Award Identifier / Grant number: 51976235

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This paper is supported by The National Natural Science Foundation of China (Project Nos. 51976235 and 52171317). The authors wish to thank the reviewer for careful, unbiased, and constructive suggestions, which led to this revised manuscript.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2022-06-25

Accepted: 2022-10-18

Published Online: 2022-11-18

Published in Print: 2023-01-27

Articles in the same Issue

https://doi.org/10.1515/jnet-2022-0045

Keywords for this article

endoreversible non-isothermal chemical engine; finite potential source; finite time thermodynamics; linear irreversible thermodynamics; maximum work output; optimal control