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New possibilities for energy production from renewable low-potential sources
Applied Energy 74 (2003) 203–209 www.elsevier.com/locate/apenergy
New possibilities for energy production from renewable low-potential sources Igor I. Samkhan* Yaroslavl State Technical University, Yaroslavl 150023, Russia
Abstract We examine an opportunity for increasing cyclic process efficiency at the expense of taking into account speeds and times of running processes. We consider dynamic equilibrium systems, in which the equations of state include both parameters of the system and environment. At low flow speeds, conventional quasi-static methods apply. At high speeds, new regenerative cycles become possible. This requires the introduction of an extended interpretation of the second law of thermodynamics. This involves specifying the performance of a cycle by the relative difference of temperatures of the working fluid (but not of thermal sources) at the upper and lower temperature levels of the cycle. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Renewable energy; Thermodynamics; Quasi-static and dynamic method; Second law; Efficiency; Power
1. Introduction The fundamental thermodynamics being widely used nowadays employs a method of quasi-static equilibrium to analyze processes. Finite-time thermodynamics, which has been actively developed for the last 20 years, discusses thermodynamic processes as non-equilibrium ones with respect to static parameters of state. Thus it is possible to analyze thermodynamic processes by using a method of dynamic equilibrium under which conditions of non-equilibrium (i.e. difference of potentials, time, rate, etc.) are taken into account as components of the dynamic equations of state for thermodynamic systems. Essentially, the proposed method of thermodynamic analysis is similar to the D’Alembert–Lagrange principle, which determines the ratio * Fax: +7-0852-217843. E-mail address: [email protected]; [email protected] (I.I. Samkhan). 0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(02)00147-2
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Nomenclature T P V Q L H W R Cp ; Cv F a M mg ; ms ; mf Cpg ; Cps ; Cpf Tav Qext Lext Qfr Lfr k; n S Cn ; C q Wf Rg , Rs P k !
Temperature Pressure Volume Heat Work Enthalpy Potential or kinetic energy Universal gas constant Thermal (heat) capacity P ¼ const and V ¼ const Flow cross-section area Sonic speed Mach number Mass flows of gas, steam and fluid Thermal capacities for gas, steam and fluid Average temperature heating Heat from external source Work to external mechanical receiver Heat of irreversible process Work of friction forces Index of quasi-static and dynamic adiabats Entropy Apparent thermal capacity dynamic adiabatic and quasi-static polytrope Heat of evaporation Evaporation rate Gas and steam constants Pressure difference Second law efficiency Efficiency of dynamic cycle Velocity Flow density
between static and dynamics in theoretical mechanics. Application of the quasi-static equilibrium description of non-equilibrium gas-dynamic power cycles in papers by Leont’ev and Shmidt [1] can be considered as a formalistic manifestation of such an approach. The method is discussed after an example investigating the gas flows [2,3]. Here the method is also used for a more precise definition of two-phase flow behaviour and as an introduction to a more comprehensive interpretation of the second law of thermodynamics.
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2. Equations of state for gas flows and steam-gas two-phase fluids According to [2], the calorific equations of state for an ideal gas flow can be presented as: Cp T ¼ C T þ PV þ !2 =2 ¼ C T þ PV þ PV
ðk 1ÞM 2 þ 1 dT dP Q L dF M2 ¼ 2 2 þ M2 k1 T P a a F
ð1Þ
ð2Þ
which correspond to the equation of energy balance Q=dH+L+!2/2 and which can be transformed via the equation PV=RT for quasi-equilibrium systems: T, P, V, Cp and Cv are temperature; pressure, volume, and thermal capacity of the fluid under P=const and V=constant respectively, R–gas constant, a–sonic speed; !– flow rate; M–Mach number; P–differential pressure of the static systems and surroundings; k–index of the quasi-static adiabatic; F–cross-section of the flow; Q, L, H–flows of heat, technical work and enthalpy, respectively. One of the specific features of a steam-gas two-phase flow is its dependence on a relative rate of evaporation (condensation) of fluid (steam), namely Wf ¼ @mf =@T. Under this condition and at low rates, the equation of state for a two-phase flow is Q þ mg þ ms VdP ¼ mg Cpg dT þ ms Cps dT þ mf Cpf dT þ qWf dT; ð3Þ where mg, ms, mf–mass flows of gas, steam and fluid repectively; Ri and Cp–gas constants and thermal capacities for gas, steam and fluid; q–heat of evaporation. Application of the equations permits a consideration of the non-equilibrium processes from the corresponding equilibrium ones.
3. Specific features of adiabatic processes If Q ¼ 0, then Eqs. (1) and (2) indicate adiabatic processes with an apparent adiabatic index n which depends on the rates and accelerations of the processes. According to Eq. (1), adiabatic flows of an ideal gas can be shown as polytropes of quasi-static processes (4) Cn dT ¼ dH þ dL
ð4Þ
where Cn ¼ @W=@T–apparent thermal capacity of the dynamic adiabatic process similar to the apparent thermal capacity of a quasi-static polytrope C ¼ Q=@T: Besides, according to Eq. (2), such adiabatic processes, under low rates when M=(!/a) ! 0, k=const and dL=VdP are shown through the fundamental Poisson equation (PV n=const) where n=k. Under flow rates close to or higher than sonic speed (M > 1), Eq. (2) permits to decribe adiabatic processes in the form of
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isotherms (T=const) and isoenthalpies (H=const) for which the apparent index n=1. In this case, Eq. (2) is expressed as (5) dP=P ¼ kM 2 = kM 2 1 ðdF=FÞ ð5Þ where ‘‘’’ (‘‘+’’) in the denominator corresponds to compression (expansion) respectively of the flow. Equations for the adiabatic compression of two-phase flows can be shown as (6) lnðP2 =P1 Þ ¼ ½n=ðn 1Þ lnðT2 =T1 Þ
ð6Þ
which differs from the quasi-static adiabatic process of an ideal gas only by the dependence n on the fluid’s evaporation rate Wf. Thus, under a low evaporation rate (Wf !0), " # " # f Cgp mf Cpf k mf Cp 1þ n=ðn 1Þ ¼ þ ¼ ð7Þ k1 Rg mg Rg mg Cpg If the fluid evaporates completely during compression at a constant rate Wf ¼ mf =ðP2 P1 Þ ¼ const and ms ¼ mf ¼ const; then h i s f g 2m C þ m C þ C þ q= ð T T Þ g p f 2 1 p p n ¼ ð8Þ n1 2mg Rg þ mf Rs where Tl, T2 and P1, P2 are flow temperature and pressure at the beginning and end of compression. In the other case if mf ¼ 0, expression (6) coincides with the ‘‘adiabatic’’ equation of Poisson with the replacement of n by an adiabatic index k. Trajectories of the dynamic adiabatic process in a 2D system of static coordinates, for example PV or TS, are shown as intercrossing curves with adiabatic indices nl > n2 > n3 (Figs. 1 and 2). In this case, the trajectories (projections) of the dynamic adiabatic process can coincide with the trajectories of intercrossing adiabatic process or polytropes of processes of an ideal gas with isentropic indexes kl > k2 > k3 if ni=ki. While plotting some intercrossing adiabatic processes in TS coordinates, difficulties arise because of the dependence of the scale of coordinates on the substance properties arise due to ln (T2/T1)=(Cp/ S), and a loss of evident direct relations of entropy S=f (P, T) with heat-exchange processes takes place.
4. Peculiarities of using flows in cycling processes Intercrossing of adiabatic processes of working fluids with variable compositions, structures or rates of processes indicates the availability of regeneration cycles with an equivalent transformation of heat from an external source to work. One such cycle, with a gas-dynamic regeneration, is given in the TS diagram, Fig. 3. The cycle consists of the following processes: 1-2—isobaric heating by heat Ql=(H2–H1); 2-3–adiabatic expansion with production of work L1=Q1=H2H1; 3-4–adiabatic expansion with a rate growth to M > 1; 4-5–gas compression through mechanical
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Fig. 1. Adiabatic compression of the same gas flow with different adiabatic indexes n1 >n2 >n3 in a mechanical–geometrical nozzle and adiabatic quasi-static processes of three ideal gases with adiabatic indexes k=n.
Fig. 2. Adiabatic compression of the same gas flow as in Fig. 1 with different adiabatic indexes n1 >n2 > n3 in a mechanical–geometrical nozzle and adiabatic quasi-static processes of three ideal gases with adiabatic indexes k=n.
and geometric effects with a consumption of work L2=RT3 ln (P1/P3) and with a rise of gas braking temperature; 5-1–adiabatic braking of the flow; 1-6–irreversible regeneration of a residual kinetic energy of the flow to its thermal energy Qfr=L2 through friction forces; 6-2–heating of the flow from an external heat source Qext=QlL2. The amount of mechanic energy Lext which is delivered to an external consumer is usually determined through an energy balance or Lext=Ql+L2=Qext. Closing of this cycle corresponds to a quotient form of energy balance equation dH=L+W=0, which supplements previously indicated similar variants dH=Q+L=0 in Carnot’s cycle and dH=Q+W in the working cycle [1]. Similar cycles, with a full regeneration of heat, can be produced for steam-gas two-phase flows.
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Fig. 3. A closed power gas cycle with a combination of quasi-static and dynamic processes.
5. Supplementary (more comprehensive) interpretation of the second law of thermodynamics The final results make it necessary to introduce a new (more comprehensive) interpretation of the second law of thermodynamics. Contrary to the classical treatment of this law, according to which the efficiency of cycles depends only on a relative temperature difference between hot T1 and cool T2 reservoirs, the new interpretation defines the potential efficiency of cycles by temperatures of the working fluid at the upper T1 and at the lower T2 of temperature levels of the cycle. The new interpretation has larger fields of application by comparison with the traditional one. In particular, in a quasi-static approximation, both methods are identical. Moreover the known forms of temperature limitations of efficiency of Carnot’s cycle k also occur in dynamic cycles with maximum regeneration. For example, though the efficiency of a dynamic cycle (Fig. 3) which is determined by a ratio of useful external work Lext to the total flow (with regeneration) of thermal energy Ql at an average heating temperature Tav: ¼ Lext =Q1 ¼ Qext =Q1 ¼ 1 T3 =Tav coincides externally with definition of the value k , but it still has a different physical sense, namely productivity (a measure of capacity for doing work) of the cycle, transferring the given temperature dependence not only to a generated mechanical energy, as it was thought previously, but also to the thermal energy, which is fed in from external sources. Besides, the efficiency of the cycle in this case can be much higher, than for Carnot’s cycle because of the absence of discharging a heat waste from the cycle to an environment.
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The same result which complies with a definition of entropy as one of the form of thermal capacity [4] can be obtained from a comprehensive analysis of quasi-static ideal-gas Carnot’s cycle as indicated in Ref. [5].
6. Conclusion We examined static non-equilibrium systems as well as equilibrium ones by using dynamic equations of state and energy, which together with the system parameters include a potential difference between the system and environment. Application of such dynamic equations of state brings about additional possibilities to control the properties of a working fluid and to develop new thermodynamic processes, which have not been taken into account before. In particular, it has become possible to adjust the adiabatic exponents of working fluids and to combine their trajectories with the trajectories of intersecting adiabatic processes and polytropes of quasi-static processes. The new positive effects require the introduction of a more comprehensive interpretation of the second law of thermodynamics according to which the maximum cycle efficiency is determined by a relative temperature difference of a working fluid at the upper and at the lower levels of the cycle. In quasi-static approximations, the given interpretation formally coincides with the classical one and, in general, it can express a measure of capacity for work of a cycle, making it possible to further increase the level of thermomechanical transformations. This case admits availability of some new ideal regenerative cycles, where the heat from an external source, is converted to work supplied to an external user in an equivalent manner.
References [1] Leont’ev A, Shmidt K. The ideal cycle of a closed gas-turbine plant without a compressor. News letters of the Russian Academy of Science, Energetica 1997;3:132–41. [2] Samkhan I. Specific features of thermodynamics of gas flow. In: Proceedings of the 8th International Energy Forum, Las Vegas, 23-28 July 2000. p. 865–870. [3] Samkhan I. Thermodynamic processes from positions of a method of dynamic balance. In: Russian National Symposium on power engineering. Kazan, Russia, 10–14 September 2001. p. 262–265. [4] Samkhan I. On the relationship between the specific heat and entropy of thermodynamic systems. Physics-Doklady 1996;41(1):327–30. [5] Samkhan I. Thermodynamics of cycles from the position of equivalence principle. In: Proceedings of the 2nd International Symposium, Kazan, Russia, 1998. p. 315–316.
New possibilities for energy production from renewable low-potential sources Igor I. Samkhan* Yaroslavl State Technical University, Yaroslavl 150023, Russia
Abstract We examine an opportunity for increasing cyclic process efficiency at the expense of taking into account speeds and times of running processes. We consider dynamic equilibrium systems, in which the equations of state include both parameters of the system and environment. At low flow speeds, conventional quasi-static methods apply. At high speeds, new regenerative cycles become possible. This requires the introduction of an extended interpretation of the second law of thermodynamics. This involves specifying the performance of a cycle by the relative difference of temperatures of the working fluid (but not of thermal sources) at the upper and lower temperature levels of the cycle. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Renewable energy; Thermodynamics; Quasi-static and dynamic method; Second law; Efficiency; Power
1. Introduction The fundamental thermodynamics being widely used nowadays employs a method of quasi-static equilibrium to analyze processes. Finite-time thermodynamics, which has been actively developed for the last 20 years, discusses thermodynamic processes as non-equilibrium ones with respect to static parameters of state. Thus it is possible to analyze thermodynamic processes by using a method of dynamic equilibrium under which conditions of non-equilibrium (i.e. difference of potentials, time, rate, etc.) are taken into account as components of the dynamic equations of state for thermodynamic systems. Essentially, the proposed method of thermodynamic analysis is similar to the D’Alembert–Lagrange principle, which determines the ratio * Fax: +7-0852-217843. E-mail address: [email protected]; [email protected] (I.I. Samkhan). 0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(02)00147-2
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Nomenclature T P V Q L H W R Cp ; Cv F a M mg ; ms ; mf Cpg ; Cps ; Cpf Tav Qext Lext Qfr Lfr k; n S Cn ; C q Wf Rg , Rs P k !
Temperature Pressure Volume Heat Work Enthalpy Potential or kinetic energy Universal gas constant Thermal (heat) capacity P ¼ const and V ¼ const Flow cross-section area Sonic speed Mach number Mass flows of gas, steam and fluid Thermal capacities for gas, steam and fluid Average temperature heating Heat from external source Work to external mechanical receiver Heat of irreversible process Work of friction forces Index of quasi-static and dynamic adiabats Entropy Apparent thermal capacity dynamic adiabatic and quasi-static polytrope Heat of evaporation Evaporation rate Gas and steam constants Pressure difference Second law efficiency Efficiency of dynamic cycle Velocity Flow density
between static and dynamics in theoretical mechanics. Application of the quasi-static equilibrium description of non-equilibrium gas-dynamic power cycles in papers by Leont’ev and Shmidt [1] can be considered as a formalistic manifestation of such an approach. The method is discussed after an example investigating the gas flows [2,3]. Here the method is also used for a more precise definition of two-phase flow behaviour and as an introduction to a more comprehensive interpretation of the second law of thermodynamics.
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2. Equations of state for gas flows and steam-gas two-phase fluids According to [2], the calorific equations of state for an ideal gas flow can be presented as: Cp T ¼ C T þ PV þ !2 =2 ¼ C T þ PV þ PV
ðk 1ÞM 2 þ 1 dT dP Q L dF M2 ¼ 2 2 þ M2 k1 T P a a F
ð1Þ
ð2Þ
which correspond to the equation of energy balance Q=dH+L+!2/2 and which can be transformed via the equation PV=RT for quasi-equilibrium systems: T, P, V, Cp and Cv are temperature; pressure, volume, and thermal capacity of the fluid under P=const and V=constant respectively, R–gas constant, a–sonic speed; !– flow rate; M–Mach number; P–differential pressure of the static systems and surroundings; k–index of the quasi-static adiabatic; F–cross-section of the flow; Q, L, H–flows of heat, technical work and enthalpy, respectively. One of the specific features of a steam-gas two-phase flow is its dependence on a relative rate of evaporation (condensation) of fluid (steam), namely Wf ¼ @mf =@T. Under this condition and at low rates, the equation of state for a two-phase flow is Q þ mg þ ms VdP ¼ mg Cpg dT þ ms Cps dT þ mf Cpf dT þ qWf dT; ð3Þ where mg, ms, mf–mass flows of gas, steam and fluid repectively; Ri and Cp–gas constants and thermal capacities for gas, steam and fluid; q–heat of evaporation. Application of the equations permits a consideration of the non-equilibrium processes from the corresponding equilibrium ones.
3. Specific features of adiabatic processes If Q ¼ 0, then Eqs. (1) and (2) indicate adiabatic processes with an apparent adiabatic index n which depends on the rates and accelerations of the processes. According to Eq. (1), adiabatic flows of an ideal gas can be shown as polytropes of quasi-static processes (4) Cn dT ¼ dH þ dL
ð4Þ
where Cn ¼ @W=@T–apparent thermal capacity of the dynamic adiabatic process similar to the apparent thermal capacity of a quasi-static polytrope C ¼ Q=@T: Besides, according to Eq. (2), such adiabatic processes, under low rates when M=(!/a) ! 0, k=const and dL=VdP are shown through the fundamental Poisson equation (PV n=const) where n=k. Under flow rates close to or higher than sonic speed (M > 1), Eq. (2) permits to decribe adiabatic processes in the form of
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isotherms (T=const) and isoenthalpies (H=const) for which the apparent index n=1. In this case, Eq. (2) is expressed as (5) dP=P ¼ kM 2 = kM 2 1 ðdF=FÞ ð5Þ where ‘‘’’ (‘‘+’’) in the denominator corresponds to compression (expansion) respectively of the flow. Equations for the adiabatic compression of two-phase flows can be shown as (6) lnðP2 =P1 Þ ¼ ½n=ðn 1Þ lnðT2 =T1 Þ
ð6Þ
which differs from the quasi-static adiabatic process of an ideal gas only by the dependence n on the fluid’s evaporation rate Wf. Thus, under a low evaporation rate (Wf !0), " # " # f Cgp mf Cpf k mf Cp 1þ n=ðn 1Þ ¼ þ ¼ ð7Þ k1 Rg mg Rg mg Cpg If the fluid evaporates completely during compression at a constant rate Wf ¼ mf =ðP2 P1 Þ ¼ const and ms ¼ mf ¼ const; then h i s f g 2m C þ m C þ C þ q= ð T T Þ g p f 2 1 p p n ¼ ð8Þ n1 2mg Rg þ mf Rs where Tl, T2 and P1, P2 are flow temperature and pressure at the beginning and end of compression. In the other case if mf ¼ 0, expression (6) coincides with the ‘‘adiabatic’’ equation of Poisson with the replacement of n by an adiabatic index k. Trajectories of the dynamic adiabatic process in a 2D system of static coordinates, for example PV or TS, are shown as intercrossing curves with adiabatic indices nl > n2 > n3 (Figs. 1 and 2). In this case, the trajectories (projections) of the dynamic adiabatic process can coincide with the trajectories of intercrossing adiabatic process or polytropes of processes of an ideal gas with isentropic indexes kl > k2 > k3 if ni=ki. While plotting some intercrossing adiabatic processes in TS coordinates, difficulties arise because of the dependence of the scale of coordinates on the substance properties arise due to ln (T2/T1)=(Cp/ S), and a loss of evident direct relations of entropy S=f (P, T) with heat-exchange processes takes place.
4. Peculiarities of using flows in cycling processes Intercrossing of adiabatic processes of working fluids with variable compositions, structures or rates of processes indicates the availability of regeneration cycles with an equivalent transformation of heat from an external source to work. One such cycle, with a gas-dynamic regeneration, is given in the TS diagram, Fig. 3. The cycle consists of the following processes: 1-2—isobaric heating by heat Ql=(H2–H1); 2-3–adiabatic expansion with production of work L1=Q1=H2H1; 3-4–adiabatic expansion with a rate growth to M > 1; 4-5–gas compression through mechanical
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207
Fig. 1. Adiabatic compression of the same gas flow with different adiabatic indexes n1 >n2 >n3 in a mechanical–geometrical nozzle and adiabatic quasi-static processes of three ideal gases with adiabatic indexes k=n.
Fig. 2. Adiabatic compression of the same gas flow as in Fig. 1 with different adiabatic indexes n1 >n2 > n3 in a mechanical–geometrical nozzle and adiabatic quasi-static processes of three ideal gases with adiabatic indexes k=n.
and geometric effects with a consumption of work L2=RT3 ln (P1/P3) and with a rise of gas braking temperature; 5-1–adiabatic braking of the flow; 1-6–irreversible regeneration of a residual kinetic energy of the flow to its thermal energy Qfr=L2 through friction forces; 6-2–heating of the flow from an external heat source Qext=QlL2. The amount of mechanic energy Lext which is delivered to an external consumer is usually determined through an energy balance or Lext=Ql+L2=Qext. Closing of this cycle corresponds to a quotient form of energy balance equation dH=L+W=0, which supplements previously indicated similar variants dH=Q+L=0 in Carnot’s cycle and dH=Q+W in the working cycle [1]. Similar cycles, with a full regeneration of heat, can be produced for steam-gas two-phase flows.
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Fig. 3. A closed power gas cycle with a combination of quasi-static and dynamic processes.
5. Supplementary (more comprehensive) interpretation of the second law of thermodynamics The final results make it necessary to introduce a new (more comprehensive) interpretation of the second law of thermodynamics. Contrary to the classical treatment of this law, according to which the efficiency of cycles depends only on a relative temperature difference between hot T1 and cool T2 reservoirs, the new interpretation defines the potential efficiency of cycles by temperatures of the working fluid at the upper T1 and at the lower T2 of temperature levels of the cycle. The new interpretation has larger fields of application by comparison with the traditional one. In particular, in a quasi-static approximation, both methods are identical. Moreover the known forms of temperature limitations of efficiency of Carnot’s cycle k also occur in dynamic cycles with maximum regeneration. For example, though the efficiency of a dynamic cycle (Fig. 3) which is determined by a ratio of useful external work Lext to the total flow (with regeneration) of thermal energy Ql at an average heating temperature Tav: ¼ Lext =Q1 ¼ Qext =Q1 ¼ 1 T3 =Tav coincides externally with definition of the value k , but it still has a different physical sense, namely productivity (a measure of capacity for doing work) of the cycle, transferring the given temperature dependence not only to a generated mechanical energy, as it was thought previously, but also to the thermal energy, which is fed in from external sources. Besides, the efficiency of the cycle in this case can be much higher, than for Carnot’s cycle because of the absence of discharging a heat waste from the cycle to an environment.
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The same result which complies with a definition of entropy as one of the form of thermal capacity [4] can be obtained from a comprehensive analysis of quasi-static ideal-gas Carnot’s cycle as indicated in Ref. [5].
6. Conclusion We examined static non-equilibrium systems as well as equilibrium ones by using dynamic equations of state and energy, which together with the system parameters include a potential difference between the system and environment. Application of such dynamic equations of state brings about additional possibilities to control the properties of a working fluid and to develop new thermodynamic processes, which have not been taken into account before. In particular, it has become possible to adjust the adiabatic exponents of working fluids and to combine their trajectories with the trajectories of intersecting adiabatic processes and polytropes of quasi-static processes. The new positive effects require the introduction of a more comprehensive interpretation of the second law of thermodynamics according to which the maximum cycle efficiency is determined by a relative temperature difference of a working fluid at the upper and at the lower levels of the cycle. In quasi-static approximations, the given interpretation formally coincides with the classical one and, in general, it can express a measure of capacity for work of a cycle, making it possible to further increase the level of thermomechanical transformations. This case admits availability of some new ideal regenerative cycles, where the heat from an external source, is converted to work supplied to an external user in an equivalent manner.
References [1] Leont’ev A, Shmidt K. The ideal cycle of a closed gas-turbine plant without a compressor. News letters of the Russian Academy of Science, Energetica 1997;3:132–41. [2] Samkhan I. Specific features of thermodynamics of gas flow. In: Proceedings of the 8th International Energy Forum, Las Vegas, 23-28 July 2000. p. 865–870. [3] Samkhan I. Thermodynamic processes from positions of a method of dynamic balance. In: Russian National Symposium on power engineering. Kazan, Russia, 10–14 September 2001. p. 262–265. [4] Samkhan I. On the relationship between the specific heat and entropy of thermodynamic systems. Physics-Doklady 1996;41(1):327–30. [5] Samkhan I. Thermodynamics of cycles from the position of equivalence principle. In: Proceedings of the 2nd International Symposium, Kazan, Russia, 1998. p. 315–316.