A Derivation of Eqs. (11), (12), and (14)

Efficiency at maximum power output for an engine with a passive piston

Abstract

Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is theoretically studied. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the side wall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call stochastic mean field model.

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1 Introduction

Equilibrium thermodynamics reveals the relation between work and heat, and the upper bound for extracted work from an arbitrarily heat cycle [1, 2]. The milestone of equilibrium thermodynamics is that thermodynamic efficiency for any heat cycle between two reservoirs characterized by the temperatures and is bounded by the Carnot efficiency: achieved by quasi-static operation [3]. There are many studies on the efficiency of engines including both external and internal combustion engines. The steam engines and steam turbines belong to the former category whose ideal cycles are the Carnot cycle, the Stirling cycle and so on [3, 4]. The diesel and free-piston engines are examples of the latter, and their ideal cycles are the Otto cycle, the Brayton cycle and so on [5, 6]. It is also known that the maximum efficiency for the ideal external combustion engines is , while that for the ideal internal ones is usually smaller than . For a practical point of view, an engine with is useless, because its power is zero.

The extension of thermodynamics toward finite-time operations, so-called finite time thermodynamics, has been investigated by many authors [10, 11, 12, 14, 17, 15, 20, 21, 19, 16, 18, 22, 23, 24, 25, 26, 27, 13, 31, 28, 29, 8, 9, 7, 30, 32]. Chambadal and Novikov independently proposed, and later Curzon and Ahlborn rediscovered that the efficiency at maximum power output (MP) is given by the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency: [10, 11, 9, 12, 8, 7]. Recently it is found that Reitlinger originally proposed in 1929 [8, 7]. The validity of the CNCA efficiency near equilibrium has been justified through the linear irreversible thermodynamics [14], molecular kinetics [15, 16] or low-dissipation assumption [17]. It is believed that the CNCA efficiency is, in general, only the efficiency at MP near equilibrium situations. Indeed, there are many situations to exceed the CNCA efficiency in idealized setups [17, 19, 15]. Although there are several studies for finite time thermodynamics including external and internal combustion engines or fluctuating heat engines [31, 28, 29, 30, 32], they are mostly interested in force-controlled engines [15, 20, 21, 16, 18, 24, 25, 26, 27, 19, 31, 28, 29, 32], where a piston or a partitioning potential is controlled by an external agent. On the other hand, the efficiency at MP for an engine partitioned by a passive piston without any external force control, has not been well-studied so far.

Figure 1: (Color online). A schematic picture of our setup, where identical hard core particles are enclosed in a container partitioned by an adiabatic piston of mass at . The density and the temperature for the outside gas are kept to be constants. The temperature of the thermal wall at is controlled by an external agent, while thermodynamic quantities such as the density and the temperature fluctuate in time.

The aim of this paper is to clarify the efficiency at MP for the engine with a passive piston, which is an idealized model of internal combustion engines without mechanical controls. We consider a hard core gas confined by a massive piston in a chamber, where the piston freely moves in one-direction by the pressure difference (see Fig. 1). We use the molecular dynamics (MD) simulation of hard core gases to examine a theoretically derived efficiency at MP on the basis of an effective model, which we call stochastic mean field model (SMF).

Because the engine we consider is an internal combustion engine, the maximum efficiency is smaller than the Carnot efficiency. Our study is relevant from the following two reasons. Firstly, we can find many situations, where the direct mechanical control of a piston is difficult. For example, the structure of internal combustion engines is usually too complicated to control inside mechanically [6]. Therefore, we need to clarify the effect of the uncontrollable motion of a piston on the efficiency. Secondly, the study of engines having passive pistons is important even for finite time thermodynamics. In the absence of mechanical control of a piston or a partitioning wall, heat flow when we attach a thermal wall is inevitable. Because heat flow from a reservoir is not usually taken into account in conventional finite time thermodynamics, it is important to verify whether the existing theoretical results are unchanged under the existence of such heat flow [10, 11, 12, 14, 17, 15, 20, 21, 19, 16, 18, 22, 23, 24, 25, 26, 27, 13, 28, 29]. Indeed, we will show that conventional results are only valid for our system when the heat flow is negligible as in dilute gases. Thus, we believe that our study for the simplest engine with a passive piston from a thermodynamic point of view is important.

The organization of this paper is as follows. We explain our setup and operation protocol for the temperature of the thermal wall in Sec. 2. We introduce SMF in Sec. 3 to analyze the power and efficiency. We examine the validity of SMF in Sec. 4 comparing the time evolution of MD simulation and that of SMF. In Sec. 5, we obtain the efficiency at MP for our engines containing dilute hard core gases theoretically, which is close to the CNCA efficiency in the massive piston limit. We also find that the efficiency at MP for moderately dense gases is smaller than the CNCA efficiency even in the linear non-equilibrium regime. In Sec. 6, to clarify the efficiency in the linear non-equilibrium regime, we explicitly derive the Onsager matrix. We clarify the finite density effect for the efficiency and stress the importance of the heat flux when we attach a bath at on the efficiency at MP in this section. We discuss the difference between our results and previous results in Sec. 7 and conclude the paper with some remarks in Sec. 8. In App. A, we show part of the derivation of SMF. In App. B, we discuss the time evolution of the temperature profile after we attach a hot reservoir. In App. C, the definition of the work and heat for our system is discussed. In App. D, the effects of piston mass and inelasticity of the piston are studied and we discuss the effect of the sidewall-friction on the piston in App. E. Throughout this paper, variables with “” denote stochastic variables.

2 Setup

In our system, hard core particles of each mass and diameter are enclosed in a three-dimensional container partitioned by an adiabatic piston of mass and the area on the right side of -direction, a diathermal wall attached with a thermal bath on the left side of -direction and four adiabatic walls on the other directions (Fig. 1). There exists a constant pressure satisfying from out side of the piston (right side of the piston). The density and the temperature for the outside gas are kept to be constants. We assume that adhesion between particles and the walls of the container as well as the one between particles can be ignored. The piston is assumed to move in one dimension without any sidewall friction. Post-collision velocity and pre-collision velocity in -direction for a colliding particle and the piston are related as:

(1)
(2)

where the contribution from the horizontal motion of particles to the wall is canceled as a result of statistical average. Here, represents the momentum change of the piston because of the collision for the particle of velocity , where is the restitution coefficient between the particles and the piston. The reason why we introduce the restitution coefficient is that the wall consists of a macroscopic number of particles and part of impulses of each collision can be absorbed into the wall as the excitation of internal oscillation.

We adopt the Maxwell reflection rule for a collision between a particle and a diathermal wall attached with the bath at . The post-collisional velocity toward the wall at is chosen as a random variable obeying the distribution

(3)

whose domain is given by and .

Figure 2: (Color online) A set of schematic figures of the operation protocol. We attach a heat bath at on the diathermal wall at . (a) For , is kept to be , and at , is switched to be that at simultaneously, and (b) is kept to be this state until . Then, we again replace the bath by that at simultaneously. After repeating the switching of , the heat cycle reaches a steady cycle.

Let us consider a heat cycle for heating and cooling processes (Fig. 2). Initially, the enclosed gas and the gas outside are in a mechanical equilibrium state, which satisfies and . At , we attach a heat bath at on the diathermal wall. For , is kept to be (Fig. 2 (a)), and at , is switched to be simultaneously, and is kept to be this state until . Then, we again replace the bath at by the one at simultaneously (Fig. 2 (b)). After repeating the switching and attaching of the baths, the heat cycle reaches a steady cycle. It should be noted that the enclosed gas is no longer thermal equilibrium during the cycle. During the operation, we ignore the time necessary for the switching the heat bath. The finite switching time only lowers the power but does not affect the efficiency of the cycle and what the maximum-power-output process is.

In this paragraph, we explain some additional remarks for the MD simulation. We assume that particles are colliding elastically each other and with side walls. The collision rule between the piston and a particle is given by Eqs. (1) and (2). We introduce typical length and time scale as and for later convenience. The number of particle is fixed through our simulation. The collisional force from outside the piston is modeled by as will be defined in Eq. (6).

3 Stochastic mean field model

Let us introduce the stochastic mean field (SMF) model to describe the dynamics of the piston and the energy balance of our system by using two independent stochastic variables: fluctuating density and fluctuating temperature . The reason why we call our model the SMF is that the piston moves in a stochastic manner because of impulses of the hard-core particles and we average out the spatial inhomogeneity of the gas. Here, and satisfy the stochastic equations:

(4)
(5)

where the stochastic force is introduced as

(6)

Here, and denote Poissonian noises of the unit amplitude whose event probabilities are respectively given by

(7)
(8)

where we have introduced the radial distribution function at contact [33]. The symbol in Eq. (6) represents Itô type stochastic product [34, 35, 36]. is Heaviside function satisfying for and for . The density and temperature for the gas outside are kept to be constants in time, i.e., and . We introduced the velocity distribution function (VDF) for the gas as . It should be noted that a set of Eqs. (5) and (6) is an extension of our previous study toward a finite density hard core gas when the density and the temperature change in time and this is the reason why we adopt Itô product in Eq. (6) [36]. We adopt the equation of state for hard core gases of volume fraction is given by [37]

(9)

Next, we propose the time evolution for . The differential of the internal energy for the gas is given by

(10)
(11)
(12)
(13)
(14)

Here, denotes the total heat flow from the thermal bath at . denote the heat flux from the internal thermal conduction and represents the remaining heat flow [15, 16]. denotes the kinetic energy transfer from the piston to the gas. In summary, main part of our SMF model consists of two coupled equations: the equation of motion for the piston (5) and the energy equation for the enclosed gas (10). In App. A, we derive Eqs. (11), (12), and (14).

Figure 3: (Color online) Comparison between SMF and the SMF without heat conduction (w/o cond.) for and . The initial volume fraction is calculated to be . The solid line and the cross points represent the dynamics of temperature for SMF and the dilute version of SMF, respectively. The inset represents the detailed time evolution for .

The heat flux is estimated from the solution of the heat diffusion equation for the temperature profile :

(15)

under the situation that the thermal conductivity and density are constants in space and time, where the piston position is fixed at . Imposing the boundary conditions , and on Eq. (15), the solution of Eq. (15) is given by

(16)

Assuming that and change in time adiabatically, i.e. [37, 38] and , we obtain the approximate heat flux as

(17)

where we have adopted expressions in Refs. [37, 38] for density and temperature dependence of the thermal conductivity .

Because the heat conduction relaxes fast to a steady state for the dilute gas, we can simplify Eq. (10) as

(18)

though heat conduction exists. We numerically confirm that the gradient of the temperature for the dilute gas relaxes faster than that for the dense gas in App. B. Indeed, we compare the dynamics of temperature in Fig. 3 for SMF and the SMF without heat conduction using Eq. (18), the difference between two methods is negligible. Here we have adopted the initial volume fraction as . We choose which is long enough for the relaxation of the system. We will also show that does not affect the efficiency at MP for the dilute gas later. Thus, we use Eq. (18) for the dilute gas instead of Eq. (10).

In this paragraph, let us explain the numerical details of SMF. The numerical integration is performed through Adams-Bashforth method, with and . Calculating , and are respectively replaced by and , where , and . Because Eq. (10) turns out to be unstable if the heat conduction in Eq. (14) is larger than that of Eq. (12), we impose the condition if through the numerical stability of our simulation. The simulation data are averaged in steady cycles, where the averaged quantity is represented by .

4 Time evolution

To verify the validity of the SMF model, we compare the time evolution of the MD simulation and SMF. We examine the dilute and moderately dense gases in Sec. 4.1 and 4.2, respectively.

4.1 Dilute case

We consider a dilute gas of the diameter which corresponds to at . Time evolutions of the volume (the position of the piston) for are drawn in Fig. 4 (a) for , and (b) for , . We have confirmed that this for each is larger than the relaxation time to the corresponding steady state. The simulation data are averaged from 11th cycle to 20th cycle, where the solid and dashed lines, respectively, represent the data for MD simulation and those for simulation of our SMF model. Similarly, Figs. 4 (c) and (d) are the time evolutions for the temperature of the gas, and Figs. 4 (e) and (f) are the time evolutions for the piston velocity. Dot-dashed lines represent the operation protocol of . It is remarkable that our SMF model correctly predicts the time evolution of MD.

Let us explain the behavior of the system shown in Fig. 4. When the heating process starts, the enclosed gas starts expanding, to find a new mechanical equilibrium density determined by the condition , because the pressure for the enclosed gas becomes larger than that for the outside after the heating. Similarly, the gas is compressed when the cooling process starts. It should be stressed that the heating (cooling) and expansion (compression) processes take place simultaneously.

Figure 4: (Color online) The time evolutions of steady cycles for . They are categorized into two types: damped oscillating type for (left) and over-damped type for (right). The time evolutions for the piston position ((a) and (b)), the temperature ((c) and (d)), and the piston velocity ((e) and (f)) are plotted. Time evolutions for the corresponding physical quantities for MD simulation (solid line) agree with those for the SMF model (dashed line).
Figure 5: (Color online) The time evolution of temperature for MD, the SMF, and the dilute approximation of SMF are compared. For heating regime , the dilute SMF overestimates the heat gain, while the SMF works better than the dilute version, in particular, for small . The inset represents the detailed time evolution for , where SMF captures the MD simulation results.

The time evolutions of the physical quantities can be categorized into two types: (a) damped-oscillating type and (b) over-damped type depending on the mass ratio . Taking the average of Eq. (18) and assuming that the piston is heavy , the time evolution of the averaged temperature is written as

(19)
(20)

Assuming that the displacement of the piston is small , the average of Eq. (5) is written as

(21)

where we have introduced the viscous friction coefficients and . The right-hand side of Eq. (21) is equivalent to the force acting on a harmonic oscillator in a viscous medium. If the viscous drag is sufficiently small, i.e. , the motion of the piston is the damped-oscillating type (Fig. 4(a)), while the motion turns out to be the over-damped type, if is not small (Fig. 4(b)).

4.2 Moderately dense case

Let us examine the validity of SMF for a moderately dense gas. We adopt which corresponds to at . In Fig. 5, simulation results for MD, SMF, and the SMF without heat conduction are plotted. It is obvious that the heat conduction plays an important role for the moderately dense gas in contrast to the dilute case (See the inset of Fig. 5). Although the time evolution of MD for small is well predicted by SMF (See the inset of Fig. 5), the agreement is relatively poor for . The agreement for is also not good, though the difference is not large. Note that the discrepancy for is not relevant for the efficiency at MP, because we need only . The improvement of SMF for is left as a future work.

Figure 6: (Color online) The average power is plotted against . Apparently, there exists for the maximum power operation, which corresponds to the necessary time for gas to expand toward the mechanical equilibrium. The dotted curve drawn as the guide line proportional to .
Figure 7: (Color online) Efficiencies at maximum power operations for dilute gases for . We plot the result of SMF (open piles). The open squares and open triangles are simulation data for the SMF without heat conduction, while filled ones are the data for the corresponding MD simulation. The observed efficiencies are close to (dashed line) and Eq. (32)(solid line).

5 Existence of Maximum Power and its Efficiency

In this section, we discuss the efficiency of the engine at MP. We show that the efficiency at MP for the dilute gas corresponds to the CNCA efficiency if the piston is sufficiently massive and elastic in Sec. 5.1, while that for the moderately dense gas is smaller than the CNCA efficiency as will be presented in Sec. 5.2.

5.1 Dilute case

Let us illustrate that the MP exists for our engine. We define the work and the heat spent per a cycle as

(22)
(23)

where and represent the integral over a single cycle and the integral for the bath at , respectively, with the definition in Eq. (12). It should be noted that Eq. (23) is consistent with previous works [15, 16] and the validity for the definition of work Eq. (22) is discussed in App. C. The efficiency for a single operation protocol [39] is defined as

(24)

We also introduce the conventional efficiency, which is defined as

(25)

In this section, we average the data from 11th cycle to 110th cycle.

The contact time dependence of the power , for the under-damped type (squares) and the over-damped type (circle) are shown in Fig. 6, where and are fixed and . Apparently, the MP is achieved at time , which corresponds to the necessary time for the gas to expand toward the mechanical equilibrium. We note that the long time heating or cooling ruins the power, because the extracted work is, at most, . Thus, the power decreases as a function of : for , which is drawn as a dashed line in Fig. 6.

We, here, explain that the obtained work is balanced with the work done by the viscous friction for gases. Multiplying onto Eq. (21) and integrating over the cycle, we obtain , because the integral of the left hand side of Eq. (21) is zero. Thus, the obtained work is balanced with the work done by the viscous friction for gases.

We present the results for the efficiency at MP (Fig. 7) for massive elastic piston and . We discuss the effect of piston mass and its inelasticity in App. D. The open squares and triangles are the simulation data for the SMF without heat conduction characterized by Eq. (18), while filled ones are the data for the corresponding MD simulation. Although and are different quantities, they agree with each other. As a comparison with previous studies, we plot the CNCA efficiency (dotted lines). Our SMF model correctly predicts the efficiency at MP for MD simulations for . We note that the efficiency for our model are close to the CNCA efficiency.

Here, we derive the semi-analytical expression on on the basis of SMF in the limit . In this limit, rapidly relaxes to bath temperature, right after is switched. The average of the work Eq. (22) can be approximated by

(26)

where we have introduced the volume change of the gas through the cycle

(27)

and choose . Integrating the equation of the energy conservation (18), we obtain

(28)

where we have introduced and . Averaging Eq. (28) and expanding in terms of , we obtain

(29)

where we have ignored the heat leak due to the fluctuation of the piston . Therefore, the efficiency is given by

(30)

Assuming that depends on the power of with a power index :

(31)

we obtain the analytical expression on for MP:

(32)

which is shown in Fig. 7 by solid lines. The exponent is estimated from the simulation of SMF, where for (Fig. 8).The physical meaning of would be explained in Sec. 6. As is shown in Fig. 7, Eq. (32) agrees with the results of MD for . We expect that the exponent is reduced to in the limit and , as follows. Although there exists the tiny heat leak during the expansion process, we may approximately ignore the leak because the heating process is almost isochoric, as will be discussed in Sec. 7. Recalling Poisson’s relation for an adiabatic process of ideal monoatomic gases between state and : , where and () respectively represent the position of the piston and temperature for the state , the exponent agrees with the simulation result. In Sec. 6, we will prove that corresponds to the tight coupling condition for the Onsager matrix in linearly irreversible thermodynamics. Substituting the obtained for into Eq. (32), we obtain

(33)

We note that Eq. (33) is identical to the expansion of up to :

(34)

We can here conclude that the efficiency at MP for an engine with an elastic passive piston whose mass is sufficiently massive confining dilute gases is the CNCA efficiency.

Figure 8: (Color online) The volume change of the enclosed gas at MP is plotted against for .

5.2 Moderately dense case

We have analyzed the efficiency for dilute gases in the previous subsection. Here, we discuss the efficiency at MP for a moderately dense hard core gas. The efficiency at MP is plotted in the main figure of Fig. 9, where SMF model almost correctly predicts the results of our MD simulation. The data for SMF at are averaged over cycles after cycles for initial relaxation to improve their numerical accuracy. The other data are averaged from 11th cycle to 110th cycle. We find that the efficiency for moderately dense hard core gases is smaller than that for dilute ones to compensate the heat flux as will be shown in the next section.

Figure 9: (Color online) The main figure represents the efficiency at MP for moderately dense hard core gases. SMF almost correctly predicts the efficiency for MD simulation. We note that the efficiency is much smaller than the CNCA one, which is caused by the inevitable heat flux . The inset represents the expansion ratio defined in Eq. (59) for moderately dense gases. The exponent is estimated to be .

6 Linearly irreversible thermodynamics

In the previous section, we have suggested that the efficiency at MP output for the dilute gas can be described by the CNCA efficiency in the limit and , while that for the moderately dense gas is smaller than the CNCA efficiency. In this section, we show that results in linear non-equilibrium situation can be understood by the relations between the currents and the thermodynamic forces on the basis of the Curie-Prigogine symmetry principle [40]:

(35)
(36)

where the Onsager matrix satisfies and . In the following, we assume that the piston is elastic and massive limit , and we abbreviate the average of an arbitrary stochastic quantity as . We examine the dilute gas in Sec. 6.1 and clarify the finite density effect in Sec. 6.2.

6.1 Dilute case

Let us derive the Onsager matrix in our setup for the dilute gas following Refs. [16, 18]. We consider the linear non-equilibrium situation as , where and are the mid-temperature and the temperature difference , respectively, satisfying . Here, the total entropy production per a unit cycle is rewritten as

(37)

where we have used and . On the basis of the relation

(38)

and are respectively given by

(39)
(40)

Let us derive and by taking . is written as

(41)

The first term on the right-hand side of Eq. (41) vanishes in the limit . Then, from Eqs. (35), (39), and (40) we obtain

(42)
(43)

Here, we have introduced as the inevitable dissipation due to the finite velocity of the piston. Now the heat is given by

(44)

which can be rewritten as

(45)
(46)

in the leading order of and the limit . From Eq. (20), we have used in the limit . Next, let us determine and . can be determined from the condition , i.e., the work-consuming state:

(47)

Then, we obtain the reciprocal relation

(48)

Taking terms depending only on in Eq. (44), we obtain

(49)
(50)

where we have ignored the higher order term including . Equations (42), (46), (48) and (50) are the explicit expressions of the Onsager matrix.

Here, we show that corresponds to the tight coupling limit of the Onsager matrix, where flux is proportional to . Because the determinant is readily calculated as

(51)

where we have used Eq. (47) with Eq. (40), i.e. and under the nearly equilibrium condition . The tight coupling limit corresponds to , which is equal to the value obtained in Sec. 5. The CNCA efficiency is derived on the basis of Eqs. (35) and (36) in the tight coupling limit, following the similar procedure in Ref. [14]. It should be noted that the control parameter for our engine is not but , in contrast to Ref. [14].

6.2 Moderately dense case

We stress that the efficiency at MP of the engine for the moderately dense gas is much smaller than the CNCA efficiency even in linear non-equilibrium regime , which is the result of the inevitable loose coupling of the Onsager matrix as follows. Solving the average of Eq. (10) in terms of , we obtain

(52)
(53)

where we have introduced the scaled flux . See also Eq. (20) for the comparison with the dilute case. Because the additional heat flux exists, Eqs. (37) and (44) are, respectively, replaced by

(54)

where we have introduced