Theoretical Evaluation of the Maximum Work of Free-Piston Engine Generators

Shinji Kojima

doi:10.1515/jnet-2015-0089

Article Publicly Available

Theoretical Evaluation of the Maximum Work of Free-Piston Engine Generators

Shinji Kojima

Published/Copyright: May 13, 2016

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 42 Issue 1

Abstract

Utilizing the adjoint equations that originate from the calculus of variations, we have calculated the maximum thermal efficiency that is theoretically attainable by free-piston engine generators considering the work loss due to friction and Joule heat. Based on the adjoint equations with seven dimensionless parameters, the trajectory of the piston, the histories of the electric current, the work done, and the two kinds of losses have been derived in analytic forms. Using these we have conducted parametric studies for the optimized Otto and Brayton cycles. The smallness of the pressure ratio of the Brayton cycle makes the net work done negative even when the duration of heat addition is optimized to give the maximum amount of heat addition. For the Otto cycle, the net work done is positive, and both types of losses relative to the gross work done become smaller with the larger compression ratio. Another remarkable feature of the optimized Brayton cycle is that the piston trajectory of the heat addition/disposal process is expressed by the same equation as that of an adiabatic process. The maximum thermal efficiency of any combination of isochoric and isobaric heat addition/disposal processes, such as the Sabathe cycle, may be deduced by applying the methods described here.

Keywords: free piston; engine; thermal efficiency; Joule heat; friction; optimization; adjoint equations; Otto cycle; Brayton cycle

1 Introduction

A free-piston engine is a type of heat engines, in which the piston motion is not controlled by mechanical linkage such as a crank in internal combustion engine but determined by the interaction of forces from a working fluid, a rebound device such as a gas-spring chamber, and a load device such as a gas compressor or a linear alternator [1, 2]. Modern applications of the free-piston engine concept include free-piston internal combustion engine generators that eliminate the crank mechanism (which is heavy and produces large frictional losses due to side thrust) by replacing it with electrical coils and permanent magnets in the piston and cylinder walls [1, 2]. These applications are being investigated by many research groups for uses such as in hybrid electric vehicles as range extenders [2–9]. Although one of the potential advantages of the free-piston concept is the operational flexibility through the variable compression ratio that leads to higher efficiency [1, 2], optimal cycle operation has seldom been considered and to the authors knowledge no investigation exists with respect to Joule heat produced by electricity. For example, Lin et al. [10] discussed the optimal motion trajectory of a four-stroke free-piston Miller cycle engine using optimal control theory. However, the result is limited to the Miller cycle and no comparison among various types of cycles was given. Furthermore, the Joule heat is not investigated. Thus, the level of efficiency that can be achieved by the free-piston engine generators is yet unknown.

With regard to the optimal cycle operation, there exists a field of thermodynamics, the so-called finite-time thermodynamics (FTT) [11] or endoreversible thermodynamics (ERT) [12], which accounts the irreversibility occurring between a system and its environment or the key dissipative effects: The former name represents an aspect of the finite-time interaction between the system and the environment or the dissipative effects occurring in finite time, while the latter represents another important aspect of the assumption/approximation that no internal irreversibility (self-increase in entropy) occurs in the system or its core subsystems. Both aspects are the essence of the field, which has brought the first landmark, i. e. a more realistic limit to the performance of heat engines than the Carnot efficiency [11–13]. Although the birth of the field could be recognized more than 30 years ago [11, 12], the attempts to find more realistic limits to the performance of various processes still continue to be done actively. For examples, by referring the recent papers in this journal, Açıkkalp and Yamık [14] have obtained the maximum available work of irreversible spark ignition, compression ignition, and Brayton cycles with temperature-dependent specific heat, and given a comprehensive list of papers about the optimized work or exergy of irreversible systems including internal combustion engines and Brayton cycles; Wagner and Hoffmann [15] have modeled a proton exchange membrane (PEM) fuel cell and captured the functional dependencies of the cell characteristics quite well; and the maximum work associated with some quantum systems has been discussed [16, 17].

In the present study, utilizing the adjoint equations that originate from the calculus of variations the limit of the maximum thermal efficiency of the Otto and the Brayton cycles is estimated with consideration of the loss of the work due to friction and Joule heat. Although the working fluid is ideally assumed as being a thermally and calorically perfect gas, our formulation enables analytic solutions and transparent interpretation of the results, which is expected to facilitate the analysis for more practical working fluids.

The present assumption of the perfect gas implicitly means the endoreversibility, i. e. no internally self-increase in entropy of the working fluid, while taking friction and the Joule heat into account requires the finite-time analysis. Therefore, the present work can be categorized as one of FTT and ERT.

With respect to the methods of optimization, FTT and ERT often require optimal paths [18] such as the optimization of the piston movement as in the present study, and usually utilize the Euler–Lagrange formalism [19, 20] or the canonical form for the adjoint variables [21, 22], both of which derive from the calculus of variation. In the present study, however, the adjoint equations are directly derived without going through the canonical form.

In Section 2, we describe our variational problem to maximize the work done by the engine generators and derive the corresponding adjoint equations. In Section 3, the solution of the adjoint equations for the adiabatic process of each of the two cycles is derived, while in Section 4 the solution for the heat addition/disposal process is derived. In Section 5, we connect the two processes (the adiabatic and the heat addition/disposal), for which we have found appropriate boundary/connecting conditions to solve the adjoint equations. In Section 6, we discuss the two kinds of losses (Joule heat and friction) for each cycle. In Section 7, we show the results of sample calculations for the generator with an off-the-shelf linear motor and a calorific value of methane–air combustion.

2 Problem formulation

In this paper, we consider two kinds of losses: one is the Joule heat produced by the electric current flowing through the linear motor generator and the other is the heat produced by viscous dissipation due to piston motion. However, we neglect the electromagnetic force that is exerted on the piston since we do not use the equation of piston motion similar to the usual treatment of thermodynamics. Note that if the equation of motion is considered, state variables such as pressure, volume and temperature of the working fluid are determined by their initial values, and therefore they cannot be the variables to maximize the work done of the cycles.

We maximize the following function J:

(2.1)max⇐J≡∫t0tfmPv˙−μmv˙A2−RI2−∑i=12λigidt,

where

(2.2)g1≡mcvT˙+Pv˙−q˙=0,

(2.3)g2≡E−LI˙−RI−Kemv˙A=0

(2.4) mPv−RgT=0.

Here, the working fluid is assumed to be thermally and calorically perfect gas; m denotes the mass of the fluid; μ represents the viscous coefficient of the moving piston; A indicates the cross-sectional area of the piston; R denotes the internal resistance of the motor; parameter λ represents the Lagrange multipliers; cv denotes the specific heat at constant volume of the fluid; T, P, and v represent the temperature, pressure, and specific volume of the fluid, respectively; q indicates the amount of heat added to the fluid; E represents the voltage applied to the motor; L and I indicate the inductance and the electric current of the motor, respectively; Ke denotes the counter electromagnetic force constant of the motor; and Rg denotes the gas constant of the fluid.

Equation (2.3) indicates that the sign of the current is negative when the piston moves in the direction of the expansion of the working fluid. Equation (2.1) represents the maximization of the work done while minimizing the loss. If the work done is maximized without minimizing the loss, the maximization is just that of the area enclosed by the cycle trajectory on the P−v diagram and is not related to the lapse of time. However, the two kinds of losses are functions of the piston velocity and the current that may change with time. Therefore, we must maximize the area by minimizing the current and the piston velocity for a specified duration of one cycle.

In the following, we analyze J of the expansion stage in which the working fluid is not compressed with the assumption that q˙≠0 for the duration from time t0 to time t1 and q˙=0 for time t1 to time tf. J of the compression stage in which the working fluid does not expand is also analyzed similarly to that of the expansion stage. The following argument can be read as if we analyze the expansion stage only, but it also holds for the compression stage (except in the part where specific arguments about expansion and compression stages are given). In general, the word “heat addition” can be interpreted as “heat disposal” if q˙<0. The work done of a whole cycle is just the sum of the work done of each expansion and compression stage.

In our problem, the state variables are P or T and I, and the control variables are q and v˙. From eq. (2.1) the control Hamiltonian can be defined as follows:

(2.5)H≡mPv˙−μ(mv˙A)2−RI2+λ1m(−Pv˙+q˙)+λ2(−E+RI+Kemv˙A)={(1−λ1)P+λ2KeA}mv˙+λ1mq˙−μ(mv˙A)2−RI2+λ2(RI−E)

where λ1/mcv and −λ2/L are interpreted as the adjoint variables of the state variables T and I, respectively.

As described in Section 3, we can derive the following inequalities based on the solution of the adjoint equations:

(2.6)1−λ1>0,

and λ1>0 for the expansion stage and λ1<0 for the compression stage. Therefore, the Pontryagin’s maximum principle requires that Pv˙ is as large as possible and that q˙≥0 for the expansion stage and q˙≤0 for the compression stage.

From eqs. (2.1), (2.2), and (2.3), we obtain the following variation of J:

(2.7)δJ≡∫t0tfmPδv˙+mv˙δP−2μmv˙Amδv˙A−2RIδI−λ1mcvδT˙+Pδv˙+v˙δP−δq˙−λ2δE−LδI˙−RδI−Kemδv˙Adt,=mP−2μmv˙A1A−λ1P+λ2Ke1Aδv−λ1mcvδT+λ1mδq+λ2LδIt=tft=t0 +∫t0tfm−P˙+2μmv¨A1A+λ˙1P+λ1P˙−λ˙2Ke1Aδv+mv˙−λ1v˙δP+mλ˙1cvδT−λ˙1mδq−λ2δE−2RI+λ˙2L−λ2RδIdt.

Using eq. (2.4), δT can be expressed by δP and δv, and then eq. (2.7) becomes the following:

(2.8)δJ=m1−γγ−1λ1P−2μmv˙A1A+λ2Ke1Aδv−λ1mvδPγ−1+λ1mδq+λ2LδIt=tft=t0 +∫t0tfmλ1−1P˙+2μmv¨A1A+γγ−1λ˙1P−λ˙2Ke1Aδv+m1−λ1v˙+λ˙1vγ−1δP−λ˙1mδq−λ2δE−2RI+λ˙2L−λ2RδIdt,

where γ denotes the specific heat ratio of the working fluid.

From eq. (2.8) we obtain the following stationary conditions of J (i. e. the adjoint equations and the terminal condition of each Lagrange multiplier):

(2.9) 1mδJδPt=1−λ1v˙+1γ−1λ˙1v=0,

(2.10)1mδJδvt=λ1−1P˙+2μmv¨A1A+γγ−1λ˙1P−λ˙2Ke1A=0,

(2.11) δJδIt=−2RI+λ˙2L−λ2R=0,

(2.12)1mδJδqt=−λ˙1t=0,

(2.13)δJδEt=−λ2t=0,

(2.14) λ1tf=0,

(2.15)λ2tf=0.

Here, the first thing we notice is that eqs. (2.11) and (2.13) give the following solution if we do not specify Et:

2RI=0→It=0.

However, this solution is trivial since the Joule heat is absent. Thus, we assume that Et is specified, and consequently eq. (2.13) need not be considered hereafter.

For the ease of analysis, we can use the following dimensionless form of eqs. (2.2) to (2.4) and (2.9) to (2.15) if desired:

(2.16)c_vT˙_+γ−1a6P_ν˙_−a7q˙_=0,

(2.17) a4E_−a1L_I˙_−R_I_−a3K_em_ν˙_A_=0,

(2.18) a6P_ν_=Rg_T_,

(2.19)1−λ_1ν˙_+1γ−1λ˙_1ν_=0,

(2.20)a5λ_1−1P˙_A_+2μ_m_ν¨_A_+γγ−1a5λ˙_1P_A_−a2λ˙_2K_e=0,

(2.21)2R_I_+a1λ˙_2L_−λ_2R_=0,

(2.22) λ˙_1t=0,

(2.23)λ_1tf=0,

(2.24)λ_2tf=0,

where

(2.25)a1≡LrRrtr,

(2.26)a2≡IrKertrμrmrvrAr=KerμrmrvrIrArtr,

(2.27)a3≡KerRrIrmrvrArtr=KerRrmrvrIrArtr,

(2.28)a4≡ErRrIr,

(2.29)a5≡λ1rPrArμrmrArvrtr,

(2.30)a6≡PrvrRgrTr=1,

(2.31)a7≡qrcvrTr,

where the suffix r represents the reference values for normalization.

It should be noted that a6=1 and Rg_=1 since we set Prvr=RgrTr. Further, we set λ2r=Ir since the units of λ2 and I are the same, which is recognized by eq. (2.11).

In addition, we choose the values of cvr，Lr，Rr，Ker，μr，mr，and Ar such that the dimensionless properties c_v, L_, R_, K_e, μ_, m_, and A_ become 1. Then eqs. (2.16) to (2.18) and (2.19) to (2.24) become the following:

(2.32)T˙_+γ−1a6P_ν˙_−a7q˙_=0,

(2.33)a4E_−a1I˙_−I_−a3ν˙_=0,

(2.34)a6P_ν_=Rg_T_,

(2.35)1−λ_1ν˙_+1γ−1λ˙_1ν_=0,

(2.36)a5λ_1−1P˙_+2ν¨_+γγ−1a5λ˙_1P_−a2λ˙_2=0,

(2.37)2I_+a1λ˙_2−λ_2=0,

(2.38)λ˙_1t=0.

(2.39)λ_1(tf)=0

(2.40) λ_2tf=0,

In addition, we obtain the following relation from eqs. (2.26) and (2.29):

a5a2=λ1rPrArIrKer,

which indicates that λ1r is originally dimensionless since the units of PrAr and IrKer are identical. Thus, we set λ1r=1.

3 Adiabatic process

First, we consider the case with q˙=0, for which eqs. (2.2) and (2.4) give the following well-known relation of an adiabatic process:

(3.1)P=C1v−γ,

(3.2)T=C2v1−γ,

where C1 and C2 are constants.

Since P and v exist in the remaining eqs. (2.3), (2.9), and (2.10), we need eq. (3.1) to express P in v and then to find v using eqs. (2.3), (2.9), and (2.10). Then, T and P can be determined by eqs. (3.1) and (3.2).

Substituting eq. (3.1) into eq. (2.10), we obtain

(3.3)C1γ1−λ1v˙+C1γγ−1λ˙1v+2μmν¨A1A−λ˙2Ke1Avγ+1=0.

Using eq. (2.9), eq. (3.3) becomes

(3.4)2μmν¨A=λ˙2Ke.

Solving eqs. (3.4), (2.3), (2.11), and (2.15) simultaneously, the variables I, v, and λ2 can be determined.

On the other hand, λ1 can be determined by eq. (2.9) into which v is substituted. Equation (2.9) gives

(3.5)ddtln1−λ1=γ−1ddtlnv.

Integrating eq. (3.5) backward in time from the terminal condition given in eq. (2.14), it becomes apparent in the expansion stage that λ1 increases from 0 and approaches 1 while 1−λ1 becomes small as long as the time is traced back because dv−dt<0 and γ>1; while in the compression stage λ1 becomes negative and smaller than 0 since dv−dt>0 and consequently 1−λ1 becomes large; that is, 1−λ1>0 always holds irrespective of the expansion and compression stages.

The analytic solution of the simultaneous eqs. (2.3), (3.4), and (2.11) can be obtained as follows.

First, the dimensionless forms of these equations are

(3.6)a4E_−a1I˙_−I_−a3v˙_=0,

(3.7) 2ν¨_=a2λ˙_2,

(3.8)2I_+a1λ˙_2−λ_2=0,

where eqs. (3.6) and (3.8) are the same as eqs. (2.33) and (2.37), respectively. The solution of eqs. (3.6) to (3.8) can be obtained if four boundary conditions, λ2tf=0, vt1, vtf, and one of the three variables λ2t1, It1, and Itf are given. Each variable at time t1 must coincide with that of the solution for the heat addition/disposal process described in Section 4.

If E_ is a function of time t, it is difficult to obtain the analytic solution. Thus, E_ is assumed to be a constant hereafter.

Since eqs. (3.6) to (3.8) are a set of linear second-order differential equations, the analytic solution is obtained by rewriting it into a set of first-order differential equations to determine the eigenvalues and eigenvectors of its characteristic equation as follows:

By defining a new variable z_ as

(3.9)z_≡v˙_,

eqs. (3.6) to (3.8) can be rewritten as the following matrix form that includes eq. (3.9):

(3.10)I˙_λ˙_2ν˙_z˙_=a4a1E_−1a1I_−a3a1ν˙_1a1λ2_−1a12I_z_a221a1λ2_−1a12I_=−1a100−a3a1−2a11a1000001−a2a1a22a100I_λ2_ν_z_+a4a1000E_.

The four eigenvalues of the matrix in the most right-hand side of eq. (3.10) are α, −α, and two 0’s, where

(3.11)α≡1+a2a3a1,

and the general solution of eq. (3.10) is as follows:

(3.12)[I_λ2_v_z_]=C1[1−a1αa22a21α1]exp(αt_)+C2[1+a1αa22a2−1α1]exp(−αt_)+C3[−a3−2a3t_1]+C4[0010]+[a4E_2a4E_00]

where the four constants C1, C2, C3, and C4 depend on the boundary values and the applied voltage E_.

4 Heat addition/disposal process

Next, we consider the case with q˙≠0, for which eq. (2.12) (λ˙1t=0) is effective unless q˙t is specified. Then eqs. (2.9) and (2.10) become the following, respectively:

(4.1)1mδJδPt=1−λ1v˙=0,

(4.2)1mδJδv(t)=(λ1−1)P˙+2μ(mv¨A)(1A)−λ˙2Ke1A=0

4.1 Isochoric heat addition/disposal

Equation (4.1) leads to v˙=0 (heat addition/disposal at a constant volume) due to the inequality 1−λ1>0 (eq. (2.6)). Then eqs. (2.2) and (2.3) become the following, respectively:

(4.3)cvT˙−q˙=0,

(4.4)E−LI˙−RI=0,

and eq. (4.2) becomes

(4.5)1mδJδv=λ1−1P˙−λ˙2Ke1A=0.

Meanwhile, eq. (2.11) does not change.

However, when the isochoric condition (4.1) forcibly holds in the case that the maximum and the minimum of v are specified, eqs. (4.2) and (4.5) need not be satisfied. Instead, the following inequalities are required for the isochoric heat addition/disposal since J must be large as possible to prevent further expansion in the expansion stage:

(4.6)0>1mδJδvt=λ1−1P˙−λ˙2Ke1A=λ1−1γ−1q˙v−λ˙2Ke1A

and

(4.7)0<1mδJδvt=λ1−1γ−1q˙v−λ˙2Ke1A

to prevent further compression in the compression stage, where the following relation obtained from eqs. (2.4) and (4.3) has been substituted into the middle of eq. (4.5):

(4.8)P˙=Rgvq˙cv=γ−1vq˙.

Since λ1−1<0 due to eq. (2.6), the first term of the right-hand side of eqs. (4.6) and (4.7) is negative for the expansion stage with q˙>0 and positive for the compression stage with q˙<0. Thus, eqs. (4.6) and (4.7) are satisfied when λ˙2>0 for the expansion stage and λ˙2<0 for the compression stage as of the samples described in Section 7.2.

Using eq. (4.3), T can be determined when q˙ is given. I is determined by eq. (4.4) with the boundary value It0 or It1. λ2 is determined by substituting I into eq. (2.11) and must satisfy the relation λ2t0=0 (eq. (A.9) in Appendix A).

First, we obtain I from eq. (4.4) assuming that the value of I_t0 is known. The dimensionless form of eq. (4.4) is as follows:

I˙_=1a1a4−I_,

whose solution is

(4.9)I_t_=I_t_0−a4exp−t_−t_0a1+a4.

On the other hand, eq. (3.8) gives

(4.10)λ˙_2=1a1λ_2−2I_.

Substituting eq. (4.9) into eq. (4.10), the following is obtained:

(4.11)λ˙_2t_=−1a1exp−t_−t_0a1I_t_0−a4+expt_−t_0a1I_t_0+a4−λ_2t_0.

Using the relation λ2t0=0 (eq. (A.9)), λ2_t_ is completely determined as follows:

(4.12)λ2_t_=−2a4cosht_−t_0a1−1+I(t0)sinht_−t_0a1.

4.2 Isobaric heat addition/disposal

Next, we consider the case that the maximum and minimum pressures of the working fluid are specified. In this case the following inequalities, instead of eq. (4.1), inevitably hold for the heat addition/disposal process unless v˙=0 since 1−λ1>0 (eq. (2.6)):

(4.13)1mδJδPt=1−λ1v˙>0,

for the expansion stage, and

(4.14)1mδJδPt=1−λ1v˙<0,

for the compression stage. Although eqs. (4.13) and (4.14) require a further increase and decrease in pressure for the expansion and compression stages, respectively, the condition P˙=0 is forcibly ensured by controlling q˙ in accordance with eqs. (4.21) and (4.22) that follow.

Assuming P˙=0, the following can be obtained from eq. (4.2):

(4.15)1mδJδvt=2μmν¨A1A−λ˙2Ke1A=0→2μmν¨A=λ˙2Ke.

The following equations must also be satisfied at the same time:

(4.16)mcvT˙+Pv˙−q˙=0,

(4.17)E−LI˙−RI−Kemv˙A=0,

(4.18)mPv−RgT=0,

(4.19)δJδIt=−2RI+λ˙2L−λ2R=0,

(4.20)1mδJδqt=−λ˙1t=0.

Equations (4.16) to (4.20) are the same as eqs. (2.2) to (2.4), (2.11), and (2.12), respectively. Since P˙=0, eq. (4.18) leads to

(4.21)v˙=RgT˙P.

Substituting eq. (4.21) into eq. (4.16), we obtain

(4.22)T˙=q˙cP.

Two variables T and λ1 are determined by eqs. (4.22) and (4.20), respectively. The other three variables I, v, and λ2 must be simultaneously solved using eqs. (4.15), (4.17), and (4.19). However, the dimensionless forms of the latter three equations are identical to eqs. (3.6) to (3.8) that are derived for the adiabatic process. While eqs. (4.17) and (4.19) are originally common to both the adiabatic and the heat addition/disposal processes, the coincidence of eq. (4.15) with eq. (3.4) is surprising. Equation (4.15) is obtained by setting P˙=0 and λ˙1=0 in eq. (2.10), whereas eq. (3.4) is obtained by the cancellation of the first term by the third term in the middle of eq. (2.10) after substituting eq. (3.1) into eq. (2.10). In brief, the origins of the two equations, eqs. (4.15) and (3.4), are different. However, the common eqs. (3.6) to (3.8) give the solutions I, v, and λ2 of both the processes and facilitates the setting of the boundary condition as described in the following section.

5 Connecting two processes

We first connect the isobaric heat addition/disposal process with the adiabatic process before connecting the isochoric process since the former connection is easier and requires only determining the unknown constants in eq. (3.12).

5.1 The Brayton cycle

When connecting the isobaric heat addition/disposal process with the adiabatic process, three variables I, v, and λ2 can be determined by eqs. (3.6) to (3.8), which are common to the two processes, consequently removing the boundary value matching at time t1 between the two. The difference between the heat addition/disposal and the adiabatic processes appears only in the equations that determine T and P, eqs. (4.16) and (4.18); while λ1 is determined by eq. (2.9) in both processes.

Four boundary values are necessary to solve eqs. (3.6) to (3.8) and it is appropriate to choose λ2t0=0, λ2tf=0, vt0, and vtf as these values. The validity of the condition λ2t0=0 is described in Appendix A, while the condition λ2tf=0 is simply eq. (2.15).

Applying these four boundary values to eq. (3.12), we obtain the following:

(5.1)2a2expαt_02a2exp−αt_0−2a302a2expαt_fexp−αt_f−2a301αexpαt_0−1αexp−αt_0t_011αexpαt_f−1αexp−αt_ft_f1C1C2C3C4=−2a4E_−2a4E_ν_t_0ν_t_f,

where the values ν_t_0 and ν_tf are originally unknown when we give only the maximum and the minimum of P for the Brayton cycle. However, if we give the value ν_t_0 besides the maximum and the minimum of P, the values ν_t1 and ν_t_f are determined by eqs. (5.1), (3.12), and (3.1).

The solution of eq. (5.1) can be obtained as follows:

(5.2)C1=a2αb4−t_0+t_f+a3ν_t_0−ν_tfb1,

(5.3)C2=C1expαt_0+t_f,

(5.4)C3=2b2b4+b3ν_t_0−ν_t_fb1,

(5.5)C4=b3−t_fν_t_0+t_0ν_t_f−b2b4t_0+t_f−a3ν_t_0+ν_t_fb1,

where

(5.6)b1≡2a3b2+b3t_0−t_f,

(5.7)b2≡a2expαt_0−expαt_f,

(5.8)b3≡αexpαt_0+expαt_f,

(5.9)b4≡a4E_.

Equation (3.12) with the four constants given in eqs. (5.2) to (5.5) determines I_ as follows:

(5.10)I_=C11−a1αa2expαt_+C1expαt_0+t_f1+a1αa2exp−αt_+2a2expαt_0−expαt_fa4E_+αexpαt_0+expαt_fν_t_0−ν_t_fb1−a3+a4E_

(5.11)I_(t_0)=−I_(t_f)=−a1α2b1(exp(αt_0)−exp(αt_f))[[(−t_0+t_f)a4E_+a3(v_(t_0)−v_(t_f))]]

Equation (5.11) is the same as eq. (A.14) and is consistent with eq. (A.9), λ2t0=0.

Here, it is necessary to note that we cannot choose arbitrary amount of heat addition in the expansion stage because of the following reasons. First, the pressure ratio εP of the Brayton cycle is determined by the trajectory of the piston:

(5.12)εP≡PmaxPmin=ν_t_fν_t_1γ.

Second, the trajectory and the amount of heat addition qH in the expansion stage determine Pmax since dq=cPdT in the isobaric heat addition process:

(5.13)Pmax=γ−1γqHν_t_1−ν_t_0.

Combining eqs. (5.12) and (5.13), the following is obtained:

(5.14)qH=γγ−1ν_t_1−ν_t_0ν_t_fν_t_1γPmin,

i. e. qH is uniquely determined by our optimum trajectory and Pmin.

5.2 The Otto cycle

Assuming that eqs. (4.6) and (4.7) hold, we have connected the solution of isochoric heat addition/disposal to that of the adiabatic process as follows.

The solution for isochoric heat addition/disposal eq. (4.9) gives the value of current I_ at time t_1 (the end of heat addition/disposal and the start of the adiabatic process) as

(5.15)I_t_1=I_t_0−a4exp−t_1−t_0a1+a4.

Using this value as one of the boundary values of the adiabatic process, we have determined the four constants C1, C2, C3, and C4 in eq. (3.12). The remaining three boundary values are λ2tf=0, vt0, and vtf, with the latter two assumed to be given. Thus, the four boundary conditions of the adiabatic process are as follows:

(5.16)I_t_1=C11−a1αa2expαt_1+C21+a1αa2exp−αt_1+C3−a3+a4E_,

(5.17)0=λ2_t_f=C12a2expαt_f+C22a2exp−αt_f+C3−2a3+2a4E_,

(5.18)ν_t_0=C11αexpαt_1+C2−1αexp−αt_1+C3t_1+C4,

(5.19)ν_t_f=C11αexpαt_f+C2−1αexp−αt_f+C3t_f+C4,

where the isochoric condition vt0 = vt1 is considered in eq. (5.18).

Solving eqs. (5.16) to (5.19), we obtain

(5.20)C1=−a2[[a2a3g2+{[I_(t_1)exp(αt_1)−b4(exp(αt_1)− g3exp(αt_f))](t_1−t_f)+a3(exp(αt_1)− g3exp(αt_f))(v_(t_0)−v_(t_f))}]]αg1

(5.21)C2=a2expαt_1+t_fa2a3g2−I_t_1expαt_f+b4expαt_f+g4expαt_1t_1−t_f−a3expαt_f+g4expαt_1ν_t_0−ν_t_fαg1,

(5.22)C3=a2g0g2+b4g6+g7ν_t_0−ν_t_fαa3g1,

(5.23) C4=1αg1a22a3g5−g7t_fν_t_0−t_1ν_t_fα2−a2αI_t_1−2t_1expαt_1+t_f+t_fexp2αt_1+exp2αt_f+b4g0t_1g4−t_fexpαt_1+t_1g3+t_fexpαt_f−a3g0ν_t_0g4−ν_t_fexpαt_1+ν_t_0g3+ν_t_fexpαt_f,

where

(5.24)g0≡expαt_1−expαt_f,

(5.25)g1≡a2a3g0g6+g7t_1−t_fα,

(5.26)g2≡g0I_t_1,

(5.27)g3≡1+a1α,

(5.28)g4≡−1+a1α,

(5.29)g5≡exp2αt_1−exp2αt_fI_t_1,

(5.30)g6≡−2+a1αexpαt_1+2+a1αexpαt_f,

(5.31)g7≡g4exp2αt_1+g3exp2αt_f,

(5.32)b4≡a4E_.

Equation (5.15) indicates that the boundary value It_1 is determined by the current It_0 that is unknown at this stage of the analysis. Substituting eq. (5.15) into eqs. (5.26) and (5.29), eq. (3.12) with eqs. (5.20) to (5.32) gives the solution for each variable including I_(t_f) as the first-order function of It_0. For example,

(5.33)I_t_f=C11−a1αa2expαt_f+C21+a1αa2exp−αt_f+C3−a3+a4E_,

where the three constants C1, C2, and C3 are the functions of It_0. Therefore, It_0 and the four constants including C4 can be determined if the relation between I_t_f and I_t0 is known.

As described in Section 5.1, eq. (A.14) holds for the Brayton cycle. Due to the small magnitude of the value of a4 compared with the value I_t0 in eq. (5.15), we have assumed that the first equality in eq. (5.11) holds for the Otto cycle and successfully obtained the solution for each variable. The solution I_t0 is as follows:

(5.34)〚a1a2a3a4exp[βt_fa1](exp[t_0a1]−exp[t_1a1]) (exp[2βt_1a1] – exp[2βt_fa1])+β{−a3exp[t_1+βt_fa1](exp[βt_1a1] – exp[βt_fa1])(v_(t_f0)−v_(t_f)){(−1+β)exp[βt_1a1]+(1+β) exp[βt_fa1]}+a4(t_1−t_f)[2exp[β(t_1+2t_f)a1](exp[t_0a1]−exp[t_1a1])+E_exp[t_1+βt_fa1](exp[βt_1a1] – exp[βt_fa1]){(−1+β)exp[βt_1a1]+(1+β) exp[βt_fa1]}]}〛〚a1a2a3β{4exp[t_1+βt_1+2βt_fa1]−(2−β)exp[t_1+2βt_1+βt_fa1]−βexp[t_0+3βt_fa1]+βexp[t_0+β(2t_1+t_f)a1]−(2+β)exp[t_1+3βt_fa1]}+(t_1−t_f){2exp[t_0+βt_1+2βt_fa1]+(−1+β)exp[t_1+2βt_1+βt_fa1]+(1+β)exp[t_1+3βt_fa1]}〛

where

(5.35)β≡1+a2a3.

The equations that determine T, P, and λ1 are the same as for the Brayton cycle.

5.3 Other cycles

In Section 4 we have not excluded the possibility of the case where both P and v are not constant during heat addition/disposal. In such a case, it is very difficult to analytically solve the simultaneous eqs. (4.1), (4.2), (2.2) to (2.4), (2.11), and (2.12) since the equations are nonlinear and inhomogeneous due to a source q˙. However, the combination of isochoric and isobaric heat addition/disposal such as the Sabathe cycle can be constructed by applying the methods described in Sections 5.1 and 5.2, if an appropriate boundary condition is found to connect the two stages. Note that eq. (A.14) may also hold for such cycles, but the proof is beyond the scope of this work.

6 Loss of Maximized Work

In the foregoing sections, the solutions that maximize J in eq. (2.1) have been obtained for the Brayton and the Otto cycles. Hence, now we calculate the loss corresponding to the maximized J based on each solution.

First, eq. (2.1) can be rewritten as follows:

(6.1)max⇐J≡J1−J2,

where

(6.2)J1≡∫t0tfmPv˙−λ1g1dt,

(6.3)J2≡∫t0tfμmv˙A2+RI2+λ2g2dt,

J1 represents the work done of a conventional heat engine to be maximized with the constraint g1=0, whereas J2 represents the function to be minimized, i. e. the loss due to the Joule heat and friction with the constraint g2=0 of a linear motor generator. Here we define Jmax as the maximized J,

(6.4)Jmax≡J1,Jmax−J2,Jmax,

where

(6.5)J1.Jmax≡∫t0tfmPv˙dt=m∫t0tfPv˙dt=mrPrvr∫t_0t_fP_ν˙_dt_,

(6.6)J2,Jmax≡∫t0tfμmv˙A2+RI2dt=trRrIr2∫t_0t_fa3a2ν˙_2+I_2dt_,

and from eqs. (2.26) to (2.27)

(6.7)a3a2=μrRrmrvrIrArtr2.

Each state variable in the integration of eqs. (6.5) and (6.6) must have the value at which J is maximized, and J2,Jmax is the loss corresponding to the gross work done J1,Jmax.

The dimensionless terms J_1,Jmax and J_2,Jmax satisfy the following:

(6.8)Jmax=mrPrvrJ_1,Jmax−J_2,Jmax,

where

(6.9)J_1,Jmax≡∫t_0t_fP_ν˙_dt_,

(6.10)J_2,Jmax=J_2,Jmaxfriction+J_2,JmaxJoule,

(6.11)J_2,Jmaxfriction≡a8a3a2∫t_0t_fν˙_2dt_,

(6.12)J_2,JmaxJoule≡a8∫t_0t_fI_2dt_

(6.13)a8≡trRrIr2mrPrvr=trRrIr2mrRgrTr.

According to eqs. (6.11) and (6.12), the relative magnitude between the two kinds of losses (friction and the Joule heat) depends on the nondimensional value of a3/a2. Meanwhile, the sum of the two kinds of losses relative to the work done depends on the nondimensional value of a8.

6.1 Loss of the Brayton cycle

As clarified in Section 4.2, eqs. (3.6) to (3.8) describe the process of isobaric heat addition/disposal as well as the adiabatic process. From the solution given in eq. (3.12), we obtain

(6.14)I_=C11−a1αa2expαt_+C21+a1αa2exp−αt_+C3−a3+a4E_,

(6.15)ν_=1αC1expαt_−C2exp−αt_+C3t_+C4.

Substituting eqs. (6.14) and (6.15) into eqs. (6.11) and (6.12), we obtain

(6.16)J_2,Jmaxfriction=a8a3a2α4C1C3h1+C12h2+h3+αC32h4,

(6.17)J_2,JmaxJoule=a8C1a22α4a2h0h1+C11+h5h2+1−h5h3+h02h4,

where

(6.18)h0≡a4E_−a3C3,

(6.19)h1≡expαt_f−expαt_0,

(6.20)h2≡exp2αt_f−exp2αt_0,

(6.21)h3≡2αt_f−t_0expαt_0+t_f

(6.22)h4≡t_f−t_0,

(6.23)h5≡a12α2,

and the constants C1, C2, C3, and C4 are given in eqs. (5.2) to (5.5).

As described in Appendix B, the solution ν_t_ for the compression stage is the time reversal of the solution for the expansion stage. Further, in Appendix C it is shown that each of the solutions λ2_t_ and I_t_ for the compression stage is the sign reversal of that for the expansion stage. Therefore, the loss during the compression stage is the same as that during the expansion stage, and consequently the loss due to the piston and the motor in the whole cycle is 2J2,Jmax.

When the loss is absent, the work done of the Brayton cycle is well known as the following w (per mass of working fluid):

(6.24)w=1−1εPγ−1/γqH=1−εP1γ−1qH,

where

(6.25)εP≡PmaxPmin,

and qH is the amount of heat added to a unit mass of working fluid during an isobaric heat addition process. Therefore, the net thermal efficiency ηnet that accounts for the loss 2J2 is as follows:

(6.26)ηnet=w−2J2m/w−2J2mqHeqH=η−2J2mqH=η−2PrvrJ_2qH,

where

(6.27)η≡w/qH=1−εP1γ−1.

For the Brayton cycle, J2,Jmax is a function of the dimensionless parameters a1, a2, a3, a4, a8, and E_ as given in eqs. (6.16) to (6.23). Thus, we can determine the sensitivities of J2,Jmax with respect to these parameters. For example, we have obtained the following sensitivity with respect to E_:

(6.28)∂J_2,Jmaxfriction∂E_=a82a3a2αa4b1C1h2+h3a2αh4+4b2h1,

(6.29)∂J_2,JmaxJoule∂E_=a8a422a2h0h1+C11+h5h2+1−h5h3a2−t_0+t_fb1+22C1a2αh1+h0h41−a32b2b1.

Substituting the specification data of a commercial linear motor (given in Table 1 of Section 7), we obtain the above two sensitivities as –0.16 % and –0.25 % of J_2,Jmaxfriction and J_2,JmaxJoule, respectively, for the baseline sample case in Section 7.1 (no. 1 in Table 2). This leads to the conclusion that the influence of the variation in applied voltage E_ is negligibly small.

Table 1:

Generator specification for the sample calculation.

Linear motor^a	Inductance (H)	2.7 × 10⁻³	Lr
	Resistance (Ω)	2.34	Rr
	Back EMF constant [V/(m/s)]	90	Ker
	Thrust constant (N/A)	90	–
	Nominal current (A)	1	Ir
	Nominal voltage (V)	1	Er
Moving parts including piston	Viscous coefficient^b [N/(m/s)]	20.8	μr
Mechanical part	Stroke of piston (mm)	90	–
	Cross-sectional area of piston (cm²)	64	Ar
	Compression ratio (–)	10, 20	ε
	One stroke duration (ms)	5, 10	tr
Flammable mixture (working fluid, assumed as thermally and calorically perfect gas)	Pressure at BDC (Pa)	10⁵	Pr
	Temperature at BDC (K)	350	Tr
	Gas constant^c (J/kg/K)	0.2869 × 10³	Rgr
	Specific heat ratio^d (–)	1.336	γ
	Mass (g)	0. 637355	mr
	Specific chemical heat release^e (J/kg)	2.76 × 10⁶	qH for the Otto cycle^f

Note: ^aProperties of a commercially available linear motor. ^bThe measured value for the linear motor, which is comparable to the frictional force on the cylinder liner of a single-cylinder diesel engine at 1,200 rpm given in Ref. [23]. ^cFor air [24]. ^dFor air at 1 atm and 1,000 K [24]. ^eBased on the lower heating value of methane and assuming its near-stoichiometric mixture with air [25]. ^fFor the Brayton cycle, qH is determined by eq. (5.14).

Table 2:

Variation in the parametric study.

Case no.	Stage	Cycle frequency (Hz)	t_1^a while t_0=0 and t_f=1	ε with P_min=1	Stroke (m)
1	Expansion	50	0.1	10	0.09
1c	Compression		0.961879 for the Brayton cycle
1c	Compression		0.1 for the Otto cycle
2	Expansion		0.1	20	0.095
3			0.05	10	0.09
4		100	0.1	10	0.09
5				20	0.095
6			0.05	10	0.09

Note: ^aFor the Brayton cycle, another value of t_1 is also used as described in Section 7.1.

We note that J2,Jmax has no sensitivity with respect to t1, which is obvious since J2 depends only on v and I, the latter of which also depends only on v by the constraint eq. (2.3), i. e. J2 depends only on v or the trajectory of the piston that does not depend on t1.

6.2 Loss of the Otto cycle

For the Otto cycle, the current during isochoric heat addition/disposal is given in eq. (4.9), which differs from that in the adiabatic process. According to eqs. (6.11) and (6.12) the loss is expressed as follows:

(6.30)J_2,Jmaxfriction≡a8a3a2∫t_0t_1ν˙_2dt_+∫t_1t_fν˙_2dt_=a8a3a2∫t_1t_fν˙_2dt_,

(6.31)J_2,JmaxJoule≡a8∫t_0t_1I_2dt_+∫t_1t_fI_2dt_,

where v_˙=0 during isochoric heat addition/disposal is accounted for.

The most right-hand side of eq. (6.30) is calculated by eq. (6.16) with the replacement of t_0 by t_1 in eqs. (6.18) to (6.23). Similarly, the second integration in the right-hand side of eq. (6.31) is calculated by eq. (6.17) with the replacement of t_0 by t_1 in eqs. (6.18) to (6.23). The four constants C1, C2, C3, and C4 in eqs. (6.16) to (6.18) are determined by eqs. (5.20) to (5.23).

The first integration in the right-hand side of eq. (6.31) is obtained by the substitution of eq. (4.9) into it as follows:

(6.32)∫t_0t_1I_2dt_=a1I_t_0−a41−exp−t_1−t_0a1 ×12I_t_0−a4exp−t_1−t_0a1+1+2a4+a42t_1−t_0.

Due to isochoric heat addition/disposal, the solution (I_, λ2_, ν_) for the compression stage of the Otto cycle is not the time reversal of that for the expansion stage, in contrast to the Brayton cycle. Accordingly, in the Otto cycle the loss due to the piston and the motor is not2J2,Jmax but

(6.33)J_2,Jmaxcycle=J_2,Jmaxexpansion+J_2,Jmaxcompression,

where

(6.34)J_2,Jmaxexpansion≠J_2,Jmaxcompression.

When the loss is absent, the work done of the Otto cycle is well known as the following w (per mass of working fluid):

(6.35)w=1−1εγ−1qH=1−ε1−γqH,

where

(6.36)ε≡vmaxvmin

represents the compression ratio, and qH denotes the amount of heat added to a unit mass of the working fluid during the isochoric heat addition process. Therefore, the net thermal efficiency ηnet that accounts for the loss J_2,Jmaxcycle is as follows:

(6.37)ηnet=w−J2cyclem/qH=η−J2cyclemqH=η−PrvrJ_2cycleqH,

where

(6.38)η≡w/qH=1−ε1−γ.

No friction occurs during isochoric heat addition and eq. (6.32) indicates that ∫t_0t_1I_2dt_ is independent to E_. The two sensitivities of J_2,Jmaxfriction and J_2,JmaxJoule with respect to E_ can be numerically obtained by the difference between the two calculations with small difference in E_. Using the specification data of a commercial linear motor (given in Table 1 of Section 7), we obtain the above two sensitivities as –0.22 % and –0.13 % of J_2,Jmaxfriction and J_2,JmaxJoule, respectively, for the baseline sample case in Section 7.2 (no. 1 in Table 2). This leads to the same conclusion as for the Brayton cycle, i. e. the influence of the variation in applied voltage E_ is negligibly small.

In contrast to the Brayton cycle, J2,Jmax is sensitive with respect to t1, which is obvious since v does not change from t0 to t1 and consequently the values of J_2,JmaxfrictionandJ_2,JmaxJoule depend on t1.

7 Sample calculation

In this section, we show the results of calculations based on the values shown in Table 1 for an off-the-shelf linear motor.

Table 1 includes the reference values for the dimensionless equations described in Section 2. The start time of the heat addition/disposal process and the end time of the adiabatic process are t_0=0 and t_f=1, respectively, in the calculation.

We have calculated six expansion stages and one compression stage for each of the Brayton and the Otto cycles with the variation of heat addition/disposal duration t_1−t_0, compression ratio ε, and cycle frequency, as shown in Table 2. The calculated compression stage (case 1c in Table 2) makes up a cycle with the corresponding expansion stage (case 1 in Table 2). The heat addition qH in the expansion stage for the Otto cycle is the specific chemical heat release given in Table 1, whereas that for the Brayton cycle is determined by eq. (5.14) and is much smaller than that for the Otto cycle. Meanwhile, the heat disposal qC in the compression stage for both cycles is determined by the following equation:

(7.1)qC=1−ηqH,

where η is determined by eq. (6.27) or (6.38) for the Brayton or the Otto cycle, respectively. The heat release rate q˙ is assumed as a constant q˙=qHt1−t0 for the expansion stage and −qCt1−t0 for the compression stage.

7.1 Brayton cycle

Figure 1 shows the P_−ν_ diagram of the Brayton cycle made up of cases 1 and 1c in Table 2. Figures 2–4 show the time histories of P_, ν_, and I_ for each stage of the same cycle. Figures 3 and 4 clearly indicate the time-reversal symmetry of ν_ (proved in Appendix B) and the sign-reversal symmetry of I_ (proved in Appendix C), respectively. Since vtf (vmax for the expansion stage and vmin for the compression stage) and pressure ratio εP determines vt1 by the adiabatic relation eq. (3.1), and vt1 determines t1 by the solution eq. (6.15), the duration t1−t0 generally differs between the expansion and compression stages as shown in Figure 2. If the duration is the same, εP becomes different between the two stages and the cycle cannot be realized. The duration t1−t0 for the compression stage that gives the same εP as of the expansion stage is easily obtained by utilizing the time-reversal symmetry of the solution vt between the two stages.

$Figure 1: P_−ν_$$\underline P - \underline \nu $$ diagram of the Brayton cycle made up of cases 1 and 1c in Table 2.$

Figure 1:

P_−ν_ diagram of the Brayton cycle made up of cases 1 and 1c in Table 2.

Figure 2:

The pressure histories of the Brayton cycle shown in Figure 1.

Figure 3:

The specific volume histories of the Brayton cycle shown in Figure 1.

Figure 4:

The current histories of the Brayton cycle shown in Figure 1.

The histories of current, specific volume, and its time derivative in the expansion stage with t_1−t_0=0.1 (cases 1, 2, 4, and 5) are shown in Figures 5–7, respectively, which reveal the influence of the compression ratio and cycle frequency (engine speed). The influence of the compression ratio is not as great as that of the cycle frequency.

$Figure 5: Histories of current in the expansion stage with a heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$ of 0.1. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.$

Figure 5:

Histories of current in the expansion stage with a heat addition duration t_1−t_0 of 0.1. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

Figure 6:

Histories of specific volume. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

Figure 7:

Histories of time derivative of specific volume. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

The influence of compression ratio, cycle frequency (engine speed), and the duration of heat addition t_1−t_0 on the pressure in the expansion stage is shown in Figures 8 and 9. Both figures indicate that the maximum pressure (the pressure in the heat addition process) increases with a larger cycle frequency (a shorter heating period) and with a larger compression ratio due to the adiabatic relation in eq. (3.1). Further, the maximum pressure is nearly the same when the length of the heating period t1−t0 and the compression ratio are the same irrespective of cycle frequency due to nearly identical volume histories (Figure 6) in the heating period t1−t0. Dimensionless time t_ is inversely proportional to cycle frequency if the original time t is the same.

$Figure 8: Pressure histories in the expansion stage. The red and cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1; meanwhile, the blue lines correspond to a heat addition duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.05 and a compression ratio of 10.$

Figure 8:

Pressure histories in the expansion stage. The red and cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration of t_1−t_0 = 0.1; meanwhile, the blue lines correspond to a heat addition duration of t_1−t_0 = 0.05 and a compression ratio of 10.

$Figure 9: Trajectories in the expansion stage on a P – v diagram. The red and cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$=0.1; meanwhile, the blue lines correspond to a heat addition duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.05 and a compression ratio of 10.$

Figure 9:

Trajectories in the expansion stage on a P – v diagram. The red and cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration t_1−t_0=0.1; meanwhile, the blue lines correspond to a heat addition duration of t_1−t_0 = 0.05 and a compression ratio of 10.

Pressure ratio εP and thermal efficiency η when neglecting the loss J2 are shown along compression ratio in Figures 10 and 11, respectively. Since the thermal efficiency η is determined by εP and εP depends on maximum pressure, εP and η (Figures 10 and 11) as well as maximum pressure (Figures 8 and 9) for the same duration of heating period t1−t0 vary little even when cycle frequency changes. However, the smallness of εP, η and the resultant small qH given in eq. (5.14) makes the specific work done J1/m=w=ηqH of the Brayton cycle smaller than the loss J2/m as follows.

$Figure 10: Pressure ratio: Solid lines and dotted lines represent heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1 and 0.05, respectively.$

Figure 10:

Pressure ratio: Solid lines and dotted lines represent heat addition duration t_1−t_0 = 0.1 and 0.05, respectively.

$Figure 11: Gross thermal efficiency. Solid lines and dotted lines represent a heat addition duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1 and 0.05, respectively.$

Figure 11:

Gross thermal efficiency. Solid lines and dotted lines represent a heat addition duration of t_1−t_0 = 0.1 and 0.05, respectively.

The t_1 that maximizes qH is obtained through the following equation that is derived from eq. (5.14):

(7.2)ν_t_1=γγ−1ν_t_0.

The numerical value of t_1 that satisfies eq. (7.2) for the optimized trajectory ν_t_ can be obtained by a simple iterative procedure. Using this value of t_1 we have estimated thermal efficiency after subtracting the loss J2 from the work J1 (net thermal efficiency ηnet) and found that ηnet is negative in the practical range of compression ratio (Figure 12) because each kind of loss is greater than 50 % of the gross work done w (Figure 13). The corresponding εP and η are also shown in Figures 14 and 15.

Figure 12:

Net thermal efficiency in the case of the maximized amount of heat addition.

Figure 13:

Ratio of the two losses to the gross work done in the case of the maximized amount of heat addition: Green lines represent the Joule heat and blue lines represent the friction loss.

Figure 14:

Pressure ratio in the case of the maximized amount of heat addition: No dependence on cycle frequency.

Figure 15:

Gross thermal efficiency in the case of the maximized amount of heat addition: No dependence on cycle frequency.

7.2 Otto cycle

Figure 16 shows the P_−ν_ diagram of the Otto cycle made up of cases 1 and 1c in Table 2. Figures 17–19 show the time histories of P_, ν_, and I_ for each stage of the same cycle. Figure 18 indicates no time-reversal symmetry of ν_ in contrast to the Brayton cycle (Figure 3). Although Figure 19 appears to suggest the sign-reversal symmetry of I_, eq. (4.9) denies this due to the shift of I_ by the term a4, although this shift is small. The duration t1−t0 is the same between expansion and compression stages, which is also in contrast to the Brayton cycle.

$Figure 16: P_−ν_$$\underline {{P}} - \underline \nu $$ diagram of the Otto cycle made up of cases 1 and 1c in Table 2.$

Figure 16:

P_−ν_ diagram of the Otto cycle made up of cases 1 and 1c in Table 2.

Figure 17:

The pressure histories of the Otto cycle shown in Figure 16.

Figure 18:

The specific volume histories of the Otto cycle shown in Figure 16.

Figure 19:

The current histories of the Otto cycle shown in Figure 16.

The histories of current, specific volume, and its time derivative in the expansion stage with t_1−t_0=0.1 (cases 1, 2, 4, and 5) are shown in Figures 20–22, respectively, which reveal the influence of compression ratio and cycle frequency (engine speed): The influence of compression ratio is not as great as that of cycle frequency.

$Figure 20: Histories of current in expansion stage with heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$ of 0.1. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.$

Figure 20:

Histories of current in expansion stage with heat addition duration t_1−t_0 of 0.1. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

Figure 21:

Histories of specific volume. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

Figure 22:

Histories of time derivative of specific volume. Red lines represent a compression ratio of 10 and the cyan lines represent a compression ratio of 20.

The influence of compression ratio, cycle frequency (engine speed), and the duration of heat addition t_1−t_0 on the pressure in the expansion stage are shown in Figures 23 and 24. Both figures indicate that maximum pressure does not depend on cycle frequency in contrast to the Brayton cycle whose pressure of heat addition process becomes larger with a larger cycle frequency. Also, maximum pressure becomes larger with a larger compression ratio and is the same when the compression ratio is the same irrespective of cycle frequency because of the identical volume in the heating period (Figures 21 or 24).

$Figure 23: Pressure histories in the expansion stage. Red lines and the cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$=0.1; and blue lines represent t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.05 with a compression ratio of 10.$

Figure 23:

Pressure histories in the expansion stage. Red lines and the cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration t_1−t_0=0.1; and blue lines represent t_1−t_0 = 0.05 with a compression ratio of 10.

$Figure 24: Trajectories in expansion stage on P – v diagram. Red lines and the cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1; and blue lines represent t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.05 with a compression ratio of 10.$

Figure 24:

Trajectories in expansion stage on P – v diagram. Red lines and the cyan lines represent a compression ratio of 10 and 20, respectively, with a heat addition duration of t_1−t_0 = 0.1; and blue lines represent t_1−t_0 = 0.05 with a compression ratio of 10.

Thermal efficiency η when neglecting the loss J2 and thermal efficiency after subtracting the loss J2 from the work J1 (net thermal efficiency ηnet) are shown along with compression ratio in Figures 25 and 26, respectively.

Since the gross thermal efficiency η is determined by compression ratio ε, η does not depend on the duration of heating period and cycle frequency (Figure 25). Accordingly, ηnet with the same duration for the whole cycle varies little (Figure 26) even when cycle frequency changes, because J2,Jmax varies little compared to the gross work done.

$Figure 25: Gross thermal efficiency: Solid lines and dotted lines (which are hidden by the former) represent heat addition/disposal duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1 and 0.05, respectively. No dependence on cycle frequency and the duration t_1−t_0$${\underline t _1} - {\underline t _0}$$.$

Figure 25:

Gross thermal efficiency: Solid lines and dotted lines (which are hidden by the former) represent heat addition/disposal duration of t_1−t_0 = 0.1 and 0.05, respectively. No dependence on cycle frequency and the duration t_1−t_0.

$Figure 26: Net thermal efficiency: Solid lines and dotted lines represent heat addition/disposal duration of t_1−t_0$${\underline t _1} - {\underline t _0}$$ = 0.1 and 0.05, respectively.$

Figure 26:

Net thermal efficiency: Solid lines and dotted lines represent heat addition/disposal duration of t_1−t_0 = 0.1 and 0.05, respectively.

On the other hand, ηnet with the same heat addition duration grows larger with larger cycle frequency (Figure 26). Also, the loss due to the Joule heat relative to the work done becomes smaller with larger cycle frequency, whereas the loss due to friction becomes large (Figure 27). The reason for this is that the duration of no friction due to the constant volume of heat addition/disposal becomes shorter with larger cycle frequency. Meanwhile, both losses relative to the work done become smaller with a larger compression ratio (Figure 27).

$Figure 27: Ratio of the two losses to the work done for a heat addition duration t_1−t_0$${\underline t _1} - {\underline t _0}$$ of 0.05: Green lines represent the Joule heat and blue lines represent the friction loss.$

Figure 27:

Ratio of the two losses to the work done for a heat addition duration t_1−t_0 of 0.05: Green lines represent the Joule heat and blue lines represent the friction loss.

8 Summary and conclusion

The adjoint equations and their solutions can be formulated with seven dimensionless parameters.
The remarkable feature of the optimized Brayton cycle is that the piston trajectory of the heat addition/disposal process is expressed by the same equation as that of adiabatic process.
With typical specification data of linear motor generators and exothermicity of a flammable mixture, we have calculated the maximum limit of the thermal efficiencies and the behaviors of the optimized Otto cycle and the optimized Brayton cycle. However, the amount of heat addition for the Brayton cycle depends on the duration of the heat addition, and consequently the smallness of the pressure ratio of the Brayton cycle makes the net work done negative even when the duration of heat addition is optimized to give the maximum amount of heat addition.
For the Otto cycle, the net work done is positive, and both kinds of losses relative to the gross work done become smaller with the larger compression ratio.

The maximum thermal efficiency of any combination of isochoric and isobaric heat addition/disposal such as the Sabathe cycle may be deduced by applying the methods described here.

Acknowledgement

The author appreciates the help of Prof. Oomichi and his students with their experimental data of linear motor.

Notation

Any underlined symbol means dimensionless.

a: dimensionless parameters
A: cross-sectional area of piston
cP: specific heat at constant pressure of working fluid
cv: specific heat at constant volume of working fluid
C: constants in the solution of adiabatic process
E: voltage applied to generator
g: defined by eq. (2.2) or (2.3)
I: electric current of generator
J: function to be maximized defined by eq. (2.1)
Jmax: maximized J
J1,Jmax: maximized gross work done
J2,Jmax: loss corresponding to J1,Jmax
Ke: counter-electromagnetic force constant of generator
L: inductance of generator
m: mass of working fluid
P: pressure of working fluid
q: heat entered into working fluid in heat addition/disposal process
qH: heat entered into working fluid in heat addition process
qC: heat deprived from working fluid in heat disposal process
R: resistance of generator
Rg: gas constant of working fluid
t: time
t0: the start time of heat addition/disposal process
t1: the end time of heat addition/disposal process and the start time of adiabatic process
tf: the end time of adiabatic process
T: temperature of working fluid
v: specific volume of working fluid
w: work done per mass of working fluid of a cycle
Greek symbols
β: defined by eq. (5.35)
γ: specific heat ratio
ε: compression ratio
εP: pressure ratio
η: thermal efficiency of the cycles
ηnet: thermal efficiency accounting for the loss
λ: Lagrange multiplier
Suffix
friction: due to friction
Joule: due to the Joule heat
max: maximum
min: minimum
r: reference value for normalization

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Appendix A

Boundary condition of the current

The boundary conditions to determine the four constants in the solution of eq. (3.12) of an adiabatic process can be chosen arbitrarily unless they contradict each other. However, the following conditions must be satisfied because the self-inductive current prevents the discrete jump of the current:

(A.1)I_expanst_0expans=I_compt_fcomp,

(A.2)I_expanst_fexpans=I_compt_0comp,

where the suffix “expans” represents the variables in the expansion stage while the suffix “comp” represents the variables in the compression stage.

Although eqs. (A.1) and (A.2) enable the repetitive calculation of the sequence of expansion and compression stages possibly to give some steady-state solution, it does not give any analytic expression of initial value such as I_t_0. Thus, we have determined the three constants C1, C2, C4 as functions of the constant C3 based on the boundary values vt0, vtf that are assumed to be known, and the terminal condition λ2tf=0 (eq. (2.15)). Then we have obtained the following:

(A.3)I_t_0=λ2_t_02−a1αΩaexpαt_f+Ωbexpαt_0a2expαt_0−expαt_f2,

(A.4) I_t_f=λ2_t_f2−a1αΩaexpαt_0+Ωbexpαt_fa2expαt_0−expαt_f2,

(A.5)λ2_t_0=2Λ−Ωaexpαt_f−Ωbexpαt_0a2expαt_0−expαt_f2,

(A.6)λ2_t_f=2Λ−Ωaexpαt_0−Ωbexpαt_fa2expαt_0−expαt_f2,

where

(A.7)Λ≡a4E_−a3C3,

and Ωa and Ωb denote the constants that depend on a1, a2, a3, a4, E_, C3, α, t_0, t_f, ν_t_0, and ν_t_f. Also, Ωabecomes−ΩbandΩbbecomes−Ωaift_0andt_fare exchanged.

If t_0 and t_f are exchanged in eqs. (A.5) and (A.6), λ2_t_0 and λ2_t_f are exchanged in the left-hand side of each equation while the right-hand sides do not change. Then we obtain

(A.8)λ2_t_f=λ2_t_0.

Since the terminal condition eq. (2.15) holds, eq. (A.8) means

(A.9)λ2_t_0=0,

which is the boundary value that we have sought and use in eq. (5.1).

Furthermore, substituting eqs. (A.9) and (2.15) into eqs. (A.3) and (A.4), the following are obtained:

(A.10)I_t_0=−a1αΩaexpαt_f+Ωbexpαt_0a2expαt_0−expαt_f2,

(A.11)I_t_f=−a1αΩaexpαt_0+Ωbexpαt_fa2expαt_0−expαt_f2,

Since Ωa and Ωb only depend on t_0, t_f, ν_t_0, and ν_t_f besides the prescribed parameters and ν_t_ is time reversible for the Brayton cycle (Appendix B), the following equations that are obtained by exchanging t_0 and t_f in eqs. (A.10) and (A.11) truly hold for the Brayton cycle:

(A.12)I_t_f=a1αΩaexpαt_f+Ωbexpαt_0a2expαt_0−expαt_f2,

(A.13)I_t_0=a1αΩaexpαt_0+Ωbexpαt_fa2expαt_0−expαt_f2.

Equations (A.10) to (A.13) mean the following:

(A.14)I_t_f=−I_t_0,

which is equivalent to eq. (A.9) and satisfies eqs. (A.1) and (A.2).

Appendix B

Time-reversibility of volume

Since the solutions (I_, λ2_, ν_) for both the expansion and compression stages of the Brayton cycle have the same form (eq. (3.12)) and the boundary conditions (eq. (5.1)) do not change when ν_t_0 and ν_t_f are exchanged together with t_0 and t_f, respectively, the solution ν_t_ for the compression stage is the time reversal of the solution for the expansion stage, which we express as time reversible.

Appendix C

Sign-reversibility of current

For the Brayton cycle, it is shown that the current I_t_ and λ2_t_ in the compression stage have reversed signs from those in the expansion stage, which we express as sign reversible as follows:

Equation (3.8) indicates that I_t_ is sign reversible if λ2_t_ is sign reversible.
Equation (3.7) indicates that λ˙2_t_ is sign reversible because ν_t_ is time reversible (this means that ν_t_ in the compression stage is the time reversal of ν_t_ in the expansion stage) and hence its time derivatives v˙_t_ and v¨_t_ are sign reversible.
Since λ2_t_f=λ2_t_0=0 (eqs. (A.8) and (A.9)), point 2 means λ2_t_ is also sign reversible.

Thus, points 1 and 3 conclude that I_t_ is sign reversible.

Received: 2015-12-15

Revised: 2016-2-5

Accepted: 2016-2-26

Published Online: 2016-5-13

Published in Print: 2017-1-1

Articles in the same Issue

https://doi.org/10.1515/jnet-2015-0089

Keywords for this article

free piston; engine; thermal efficiency; Joule heat; friction; optimization; adjoint equations; Otto cycle; Brayton cycle