Page 1 Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 31-37 Thermo-economic analysis of regenerative heat engines Santanu Bandyopadhyay Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India e-mail: santanub@iitb.ac.in Received 19 May 2003; accepted 8 September 2003 Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are derived without assuming any particular equation of state associated with the working fluid. [Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines; Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine] 1 Introduction Carnot efficiency (? C = 1 – T min /T max ) is the maximum possible efficiency of a heat engine with which low-grade thermal energy may be reversibly transformed into high-grade mechanical energy. Ideal regenerative heat engines (such as Stirling and Ericsson heat engines), with perfect regeneration, also operate with the Carnot efficiency. To achieve Carnot efficiency, thermal exchanges between the reservoirs and the working fluid of the engine have to occur through reversible isothermal processes. These processes demand infinite heat exchanger surface area. A heat engine with finite heat exchanger area, result in zero power production. On the other hand, the efficiency of an internally reversible Carnot engine, deliver maximum power, given by ? MP = 1 – (T min /T max ) 1/2 (Ref. 1). The global need for fuel-efficient and environmentally viable power production, with thermodynamic reliability and economy, demands moderation of the traditional energy conversion processes with new approaches. Bera and Bandyopadhyay 2 have analyzed the effect of combustion on the thermoeconomic performances of Carnot, Otto and Joule-Brayton engines. Classical reversible heat engines are never realizable in practice, but the aim is to reach the highest limit of power production within the constraints of finite resources. With this end in view, regenerative heat engine cycles have been studied and their design philosophies have flourished. Regenerative heat engines have other benefits also. Exhaust emissions of a regenerative heat engine are low and may be easily controlled as the combustion is isolated from cyclic pressure and temperature changes experienced by the working fluid. Continuous complete combustion with 20-80 % excess air replaces intermittent combustion occurring in other piston engines. This is because quenching of the flame does not take place at the ‘cold’ metal surface. This leads to remarkably low noise levels 3 . Regenerative engines are so thermally efficient that they are prime contender for alternative power unit. The mean effective pressure and the mechanical efficiency of a regenerative engine are also quite high 4 . Hence, generations of physicists and engineers of past, focused on these types of engines. In this paper, internally reversible regenerative heat engines with imperfect regeneration are discussed and detailed understanding for optimal design of such engines are provided. Page 2 Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 31-37 Thermo-economic analysis of regenerative heat engines Santanu Bandyopadhyay Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India e-mail: santanub@iitb.ac.in Received 19 May 2003; accepted 8 September 2003 Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are derived without assuming any particular equation of state associated with the working fluid. [Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines; Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine] 1 Introduction Carnot efficiency (? C = 1 – T min /T max ) is the maximum possible efficiency of a heat engine with which low-grade thermal energy may be reversibly transformed into high-grade mechanical energy. Ideal regenerative heat engines (such as Stirling and Ericsson heat engines), with perfect regeneration, also operate with the Carnot efficiency. To achieve Carnot efficiency, thermal exchanges between the reservoirs and the working fluid of the engine have to occur through reversible isothermal processes. These processes demand infinite heat exchanger surface area. A heat engine with finite heat exchanger area, result in zero power production. On the other hand, the efficiency of an internally reversible Carnot engine, deliver maximum power, given by ? MP = 1 – (T min /T max ) 1/2 (Ref. 1). The global need for fuel-efficient and environmentally viable power production, with thermodynamic reliability and economy, demands moderation of the traditional energy conversion processes with new approaches. Bera and Bandyopadhyay 2 have analyzed the effect of combustion on the thermoeconomic performances of Carnot, Otto and Joule-Brayton engines. Classical reversible heat engines are never realizable in practice, but the aim is to reach the highest limit of power production within the constraints of finite resources. With this end in view, regenerative heat engine cycles have been studied and their design philosophies have flourished. Regenerative heat engines have other benefits also. Exhaust emissions of a regenerative heat engine are low and may be easily controlled as the combustion is isolated from cyclic pressure and temperature changes experienced by the working fluid. Continuous complete combustion with 20-80 % excess air replaces intermittent combustion occurring in other piston engines. This is because quenching of the flame does not take place at the ‘cold’ metal surface. This leads to remarkably low noise levels 3 . Regenerative engines are so thermally efficient that they are prime contender for alternative power unit. The mean effective pressure and the mechanical efficiency of a regenerative engine are also quite high 4 . Hence, generations of physicists and engineers of past, focused on these types of engines. In this paper, internally reversible regenerative heat engines with imperfect regeneration are discussed and detailed understanding for optimal design of such engines are provided. INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 32 The power-efficiency characteristics of a real engine help a designer to identify the operating region for optimal design of the heat engine and to realize the upper bounds on power production and its attainable efficiency. The power-efficiency characteristics for irreversible Carnot cycle, irreversible Joule-Brayton cycle, and Rankine cycle are equivalent to each other 5 . The power-efficiency characteristics of a regenerative engine are expected to be a strong function of regeneration efficiency. In this paper, the power-efficiency characteristics of regenerative heat engines are studied and the operating regions are identified. Knowing the governing equation of state related with any particular working fluid, different design parameters may easily be calculated. The results are derived without assuming any particular equation of state associated with the working fluid. However, for brevity, thermal capacity rates are assumed to be independent of temperature. 2 Regenerative Heat Engines The ideal thermodynamic cycle corresponding to regenerative heat engine consists of two isothermal and two polytropic (of index n) processes. The temperature-entropy diagram of a typical regenerative cycle is shown in Fig. 1. Depending on the nature of the polytropic process, regenerative cycle reduces to Carnot (for adiabatic process), Ericsson (for isobaric process) or Stirling (for isometric process) cycles. The isothermal compression occurs between states 1 and 2. In the isothermal compression, heat is rejected by the working fluid to the cold reservoir. The isothermal expansion process, where injection of heat to the working fluid from the external hot reservoir takes place, occurs between states 3 and 4. The temperatures of the hot and the cold reservoirs are denoted by T max and T min , respectively. Processes from states 2 to 3 and 4 to 1 are polytropic (of index n). Portion of the heat rejected from the polytropic process 4 to 1 is supplied partly to the process 2 to 3 through a regenerator. Condition of the working fluid after the regeneration process is denoted by the states 2R and 4R (Fig. 1). Regenerator in the process, 2 to 2R supply heat, and external heat is provided by the hot reservoir from 2R to 4. Similarly, heat is rejected from 4R to 2. Therefore, total heat supplied to and rejected with are given as Q in = Q 2R3 + Q 34 … (1) Q out = Q 4R1 + Q 12 … (2) Assuming that the cycle is internally reversible and all the irreversibility is associated with the finite driving force of the heat transfer process (that is, reversibility of the heat engine), the entropy balance for the regenerative engine may be satisfied. Q 34 /T h + C n ln(T h /T c ) = Q 12 /T c + C n ln(T h /T c ) … (3) where T h (= T 3 = T 4 ) and T c (= T 1 = T 2 ) are the highest and the lowest temperatures attained by the working fluid. In Eq. (3), C n denotes the thermal capacity rate of the polytropic process. Knowing the equation of state that governs the working fluid and the polytropic index, n, the polytropic thermal capacity rate C n , can be determined. For working fluid obeying ideal gas laws, polytropic thermal capacity rate may be calculated in terms of thermal capacity rate at constant volume as C n = C v (n - ?)/(n - 1). For brevity, C n is assumed to be independent of temperature. The temperatures of the working fluid after the regeneration (Q R ) may be written from the energy balance of the regenerator. T 4R = T h – Q R /C n … (4) T 2R = T c + Q R /C n … (5) With the help of Eqs (4) and (5), energy exchange equations are rewritten as follows: Q in = C n (T h – T c ) – Q R + Q 34 … (6) Q out = C n (T h – T c ) – Q R + Q 12 … (7) Fig. 1 ? Temperature-entropy diagram of a regenerative heat engine Page 3 Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 31-37 Thermo-economic analysis of regenerative heat engines Santanu Bandyopadhyay Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India e-mail: santanub@iitb.ac.in Received 19 May 2003; accepted 8 September 2003 Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are derived without assuming any particular equation of state associated with the working fluid. [Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines; Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine] 1 Introduction Carnot efficiency (? C = 1 – T min /T max ) is the maximum possible efficiency of a heat engine with which low-grade thermal energy may be reversibly transformed into high-grade mechanical energy. Ideal regenerative heat engines (such as Stirling and Ericsson heat engines), with perfect regeneration, also operate with the Carnot efficiency. To achieve Carnot efficiency, thermal exchanges between the reservoirs and the working fluid of the engine have to occur through reversible isothermal processes. These processes demand infinite heat exchanger surface area. A heat engine with finite heat exchanger area, result in zero power production. On the other hand, the efficiency of an internally reversible Carnot engine, deliver maximum power, given by ? MP = 1 – (T min /T max ) 1/2 (Ref. 1). The global need for fuel-efficient and environmentally viable power production, with thermodynamic reliability and economy, demands moderation of the traditional energy conversion processes with new approaches. Bera and Bandyopadhyay 2 have analyzed the effect of combustion on the thermoeconomic performances of Carnot, Otto and Joule-Brayton engines. Classical reversible heat engines are never realizable in practice, but the aim is to reach the highest limit of power production within the constraints of finite resources. With this end in view, regenerative heat engine cycles have been studied and their design philosophies have flourished. Regenerative heat engines have other benefits also. Exhaust emissions of a regenerative heat engine are low and may be easily controlled as the combustion is isolated from cyclic pressure and temperature changes experienced by the working fluid. Continuous complete combustion with 20-80 % excess air replaces intermittent combustion occurring in other piston engines. This is because quenching of the flame does not take place at the ‘cold’ metal surface. This leads to remarkably low noise levels 3 . Regenerative engines are so thermally efficient that they are prime contender for alternative power unit. The mean effective pressure and the mechanical efficiency of a regenerative engine are also quite high 4 . Hence, generations of physicists and engineers of past, focused on these types of engines. In this paper, internally reversible regenerative heat engines with imperfect regeneration are discussed and detailed understanding for optimal design of such engines are provided. INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 32 The power-efficiency characteristics of a real engine help a designer to identify the operating region for optimal design of the heat engine and to realize the upper bounds on power production and its attainable efficiency. The power-efficiency characteristics for irreversible Carnot cycle, irreversible Joule-Brayton cycle, and Rankine cycle are equivalent to each other 5 . The power-efficiency characteristics of a regenerative engine are expected to be a strong function of regeneration efficiency. In this paper, the power-efficiency characteristics of regenerative heat engines are studied and the operating regions are identified. Knowing the governing equation of state related with any particular working fluid, different design parameters may easily be calculated. The results are derived without assuming any particular equation of state associated with the working fluid. However, for brevity, thermal capacity rates are assumed to be independent of temperature. 2 Regenerative Heat Engines The ideal thermodynamic cycle corresponding to regenerative heat engine consists of two isothermal and two polytropic (of index n) processes. The temperature-entropy diagram of a typical regenerative cycle is shown in Fig. 1. Depending on the nature of the polytropic process, regenerative cycle reduces to Carnot (for adiabatic process), Ericsson (for isobaric process) or Stirling (for isometric process) cycles. The isothermal compression occurs between states 1 and 2. In the isothermal compression, heat is rejected by the working fluid to the cold reservoir. The isothermal expansion process, where injection of heat to the working fluid from the external hot reservoir takes place, occurs between states 3 and 4. The temperatures of the hot and the cold reservoirs are denoted by T max and T min , respectively. Processes from states 2 to 3 and 4 to 1 are polytropic (of index n). Portion of the heat rejected from the polytropic process 4 to 1 is supplied partly to the process 2 to 3 through a regenerator. Condition of the working fluid after the regeneration process is denoted by the states 2R and 4R (Fig. 1). Regenerator in the process, 2 to 2R supply heat, and external heat is provided by the hot reservoir from 2R to 4. Similarly, heat is rejected from 4R to 2. Therefore, total heat supplied to and rejected with are given as Q in = Q 2R3 + Q 34 … (1) Q out = Q 4R1 + Q 12 … (2) Assuming that the cycle is internally reversible and all the irreversibility is associated with the finite driving force of the heat transfer process (that is, reversibility of the heat engine), the entropy balance for the regenerative engine may be satisfied. Q 34 /T h + C n ln(T h /T c ) = Q 12 /T c + C n ln(T h /T c ) … (3) where T h (= T 3 = T 4 ) and T c (= T 1 = T 2 ) are the highest and the lowest temperatures attained by the working fluid. In Eq. (3), C n denotes the thermal capacity rate of the polytropic process. Knowing the equation of state that governs the working fluid and the polytropic index, n, the polytropic thermal capacity rate C n , can be determined. For working fluid obeying ideal gas laws, polytropic thermal capacity rate may be calculated in terms of thermal capacity rate at constant volume as C n = C v (n - ?)/(n - 1). For brevity, C n is assumed to be independent of temperature. The temperatures of the working fluid after the regeneration (Q R ) may be written from the energy balance of the regenerator. T 4R = T h – Q R /C n … (4) T 2R = T c + Q R /C n … (5) With the help of Eqs (4) and (5), energy exchange equations are rewritten as follows: Q in = C n (T h – T c ) – Q R + Q 34 … (6) Q out = C n (T h – T c ) – Q R + Q 12 … (7) Fig. 1 ? Temperature-entropy diagram of a regenerative heat engine BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 33 Heat exchangers are assumed to be counter- current. Total thermal conductance in the hot side of the engine is given as K h = C n ln((T max – T c – Q R /C n )/(T max - T h )) + Q 34 /(T max – T h ) … (8) Similarly, for cold side of the heat engine one can get K c = C n ln((T h – T min – Q R /C n )/(T c – T min )) + Q 12 /(T c – T min ) … (9) The maximum possible regeneration is C n (T h – T c ). The regeneration process may be modeled with an efficiency of (1 - e). Hence Q R = (1 - e) C n (T h – T c ) … (10) Defining the non-dimensional quantities such as t h = T h /T max , t c = T c /T max , t C = T min /T max , k = K h /(K h + K c ), s = C n /(K h + K c ), q = Q/(T max (K h + K c )), and w = W/(T max (K h + K c )), above equations may be written in dimensionless form. Note that, 1 = t h = t c = t C . Combining these equations the energy input to and rejected with are given by q in = es(t h – t c ) + (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))] ˜ es(t h – t c ) + (1 – t h )[k – es (t h – t c )/(1 – t h ))] = (1 – t h ) k … (11) and similarly q out = es(t h – t c ) + (t c – t C )[(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))] ˜ (t c – t C ) (1 - k) … (12) The reversibility [Eq. (3)] of the heat engine translates to (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))]/t h = (t c – t C ) [(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))]/t c … (13) Neglecting higher terms of the expanded logarithm function, this leads to (1 – t h )k/t h – se(t h – t c )/t h = (t c – t C ) (1 – k)/t c – se(t h – t c )/t c … (14) Denoting t = t c /t h , the power generated by the internally reversible engine and its efficiency can be expressed as w ˜ (k(1 – k)(1 - t)( t - t C ) - se(1 – t) 2 (k + (1 – k) t C ))/( t - se(1 – t) 2 ) … (15) and ? ˜ 1 - (k(1 – k) t( t - t C ) + (1 – k)se t C (1 – t) 2 )/(k (1 – k) ( t - t C ) – kse(1 – t) 2 ) … (16) The power output [Eq. (15)] or the efficiency [Eq. (16)] of the engine may be maximized to optimize the performance of the regenerative engine. Note that these approximations are reasonable, except for very low working temperature ratio ( t ? t C ). For a very low working temperature ratio original equations have to be solved numerically. Simultaneous solution of these equations will ensure the concurrent employment of the first and the second laws. 3 Power-Efficiency Characteristics Internally reversible Carnot engine operates between the limits of thermal ‘short-circuit’ and thermal ‘open-circuit’ conditions 5 . No power is produced either when the engine operates at thermal short-circuit condition with zero efficiency or the engine operates at thermal open-circuit condition with maximum possible efficiency (Carnot efficiency, ? C ). The maximum power corresponds to Curzon- Ahlborn 1 efficiency for internally reversible Carnot engine. Unlike this, the power-efficiency characteristic of a real heat engine corresponds to zero power and zero efficiency at both the limits of thermal short-circuit and open-circuit conditions 5 . The power- efficiency characteristics of most of the real heat engines are akin to loop-like behaviour. Real heat engines, with finite resources, exhibit possibility of operation at maximum power or at maximum efficiency. The power-efficiency characteristics of an internally reversible regenerative engine are shown in Fig. 2 for different regeneration efficiencies. From Fig. 2, it may be noted that the efficiency and the power output both exhibit a maximum. These Page 4 Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 31-37 Thermo-economic analysis of regenerative heat engines Santanu Bandyopadhyay Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India e-mail: santanub@iitb.ac.in Received 19 May 2003; accepted 8 September 2003 Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are derived without assuming any particular equation of state associated with the working fluid. [Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines; Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine] 1 Introduction Carnot efficiency (? C = 1 – T min /T max ) is the maximum possible efficiency of a heat engine with which low-grade thermal energy may be reversibly transformed into high-grade mechanical energy. Ideal regenerative heat engines (such as Stirling and Ericsson heat engines), with perfect regeneration, also operate with the Carnot efficiency. To achieve Carnot efficiency, thermal exchanges between the reservoirs and the working fluid of the engine have to occur through reversible isothermal processes. These processes demand infinite heat exchanger surface area. A heat engine with finite heat exchanger area, result in zero power production. On the other hand, the efficiency of an internally reversible Carnot engine, deliver maximum power, given by ? MP = 1 – (T min /T max ) 1/2 (Ref. 1). The global need for fuel-efficient and environmentally viable power production, with thermodynamic reliability and economy, demands moderation of the traditional energy conversion processes with new approaches. Bera and Bandyopadhyay 2 have analyzed the effect of combustion on the thermoeconomic performances of Carnot, Otto and Joule-Brayton engines. Classical reversible heat engines are never realizable in practice, but the aim is to reach the highest limit of power production within the constraints of finite resources. With this end in view, regenerative heat engine cycles have been studied and their design philosophies have flourished. Regenerative heat engines have other benefits also. Exhaust emissions of a regenerative heat engine are low and may be easily controlled as the combustion is isolated from cyclic pressure and temperature changes experienced by the working fluid. Continuous complete combustion with 20-80 % excess air replaces intermittent combustion occurring in other piston engines. This is because quenching of the flame does not take place at the ‘cold’ metal surface. This leads to remarkably low noise levels 3 . Regenerative engines are so thermally efficient that they are prime contender for alternative power unit. The mean effective pressure and the mechanical efficiency of a regenerative engine are also quite high 4 . Hence, generations of physicists and engineers of past, focused on these types of engines. In this paper, internally reversible regenerative heat engines with imperfect regeneration are discussed and detailed understanding for optimal design of such engines are provided. INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 32 The power-efficiency characteristics of a real engine help a designer to identify the operating region for optimal design of the heat engine and to realize the upper bounds on power production and its attainable efficiency. The power-efficiency characteristics for irreversible Carnot cycle, irreversible Joule-Brayton cycle, and Rankine cycle are equivalent to each other 5 . The power-efficiency characteristics of a regenerative engine are expected to be a strong function of regeneration efficiency. In this paper, the power-efficiency characteristics of regenerative heat engines are studied and the operating regions are identified. Knowing the governing equation of state related with any particular working fluid, different design parameters may easily be calculated. The results are derived without assuming any particular equation of state associated with the working fluid. However, for brevity, thermal capacity rates are assumed to be independent of temperature. 2 Regenerative Heat Engines The ideal thermodynamic cycle corresponding to regenerative heat engine consists of two isothermal and two polytropic (of index n) processes. The temperature-entropy diagram of a typical regenerative cycle is shown in Fig. 1. Depending on the nature of the polytropic process, regenerative cycle reduces to Carnot (for adiabatic process), Ericsson (for isobaric process) or Stirling (for isometric process) cycles. The isothermal compression occurs between states 1 and 2. In the isothermal compression, heat is rejected by the working fluid to the cold reservoir. The isothermal expansion process, where injection of heat to the working fluid from the external hot reservoir takes place, occurs between states 3 and 4. The temperatures of the hot and the cold reservoirs are denoted by T max and T min , respectively. Processes from states 2 to 3 and 4 to 1 are polytropic (of index n). Portion of the heat rejected from the polytropic process 4 to 1 is supplied partly to the process 2 to 3 through a regenerator. Condition of the working fluid after the regeneration process is denoted by the states 2R and 4R (Fig. 1). Regenerator in the process, 2 to 2R supply heat, and external heat is provided by the hot reservoir from 2R to 4. Similarly, heat is rejected from 4R to 2. Therefore, total heat supplied to and rejected with are given as Q in = Q 2R3 + Q 34 … (1) Q out = Q 4R1 + Q 12 … (2) Assuming that the cycle is internally reversible and all the irreversibility is associated with the finite driving force of the heat transfer process (that is, reversibility of the heat engine), the entropy balance for the regenerative engine may be satisfied. Q 34 /T h + C n ln(T h /T c ) = Q 12 /T c + C n ln(T h /T c ) … (3) where T h (= T 3 = T 4 ) and T c (= T 1 = T 2 ) are the highest and the lowest temperatures attained by the working fluid. In Eq. (3), C n denotes the thermal capacity rate of the polytropic process. Knowing the equation of state that governs the working fluid and the polytropic index, n, the polytropic thermal capacity rate C n , can be determined. For working fluid obeying ideal gas laws, polytropic thermal capacity rate may be calculated in terms of thermal capacity rate at constant volume as C n = C v (n - ?)/(n - 1). For brevity, C n is assumed to be independent of temperature. The temperatures of the working fluid after the regeneration (Q R ) may be written from the energy balance of the regenerator. T 4R = T h – Q R /C n … (4) T 2R = T c + Q R /C n … (5) With the help of Eqs (4) and (5), energy exchange equations are rewritten as follows: Q in = C n (T h – T c ) – Q R + Q 34 … (6) Q out = C n (T h – T c ) – Q R + Q 12 … (7) Fig. 1 ? Temperature-entropy diagram of a regenerative heat engine BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 33 Heat exchangers are assumed to be counter- current. Total thermal conductance in the hot side of the engine is given as K h = C n ln((T max – T c – Q R /C n )/(T max - T h )) + Q 34 /(T max – T h ) … (8) Similarly, for cold side of the heat engine one can get K c = C n ln((T h – T min – Q R /C n )/(T c – T min )) + Q 12 /(T c – T min ) … (9) The maximum possible regeneration is C n (T h – T c ). The regeneration process may be modeled with an efficiency of (1 - e). Hence Q R = (1 - e) C n (T h – T c ) … (10) Defining the non-dimensional quantities such as t h = T h /T max , t c = T c /T max , t C = T min /T max , k = K h /(K h + K c ), s = C n /(K h + K c ), q = Q/(T max (K h + K c )), and w = W/(T max (K h + K c )), above equations may be written in dimensionless form. Note that, 1 = t h = t c = t C . Combining these equations the energy input to and rejected with are given by q in = es(t h – t c ) + (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))] ˜ es(t h – t c ) + (1 – t h )[k – es (t h – t c )/(1 – t h ))] = (1 – t h ) k … (11) and similarly q out = es(t h – t c ) + (t c – t C )[(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))] ˜ (t c – t C ) (1 - k) … (12) The reversibility [Eq. (3)] of the heat engine translates to (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))]/t h = (t c – t C ) [(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))]/t c … (13) Neglecting higher terms of the expanded logarithm function, this leads to (1 – t h )k/t h – se(t h – t c )/t h = (t c – t C ) (1 – k)/t c – se(t h – t c )/t c … (14) Denoting t = t c /t h , the power generated by the internally reversible engine and its efficiency can be expressed as w ˜ (k(1 – k)(1 - t)( t - t C ) - se(1 – t) 2 (k + (1 – k) t C ))/( t - se(1 – t) 2 ) … (15) and ? ˜ 1 - (k(1 – k) t( t - t C ) + (1 – k)se t C (1 – t) 2 )/(k (1 – k) ( t - t C ) – kse(1 – t) 2 ) … (16) The power output [Eq. (15)] or the efficiency [Eq. (16)] of the engine may be maximized to optimize the performance of the regenerative engine. Note that these approximations are reasonable, except for very low working temperature ratio ( t ? t C ). For a very low working temperature ratio original equations have to be solved numerically. Simultaneous solution of these equations will ensure the concurrent employment of the first and the second laws. 3 Power-Efficiency Characteristics Internally reversible Carnot engine operates between the limits of thermal ‘short-circuit’ and thermal ‘open-circuit’ conditions 5 . No power is produced either when the engine operates at thermal short-circuit condition with zero efficiency or the engine operates at thermal open-circuit condition with maximum possible efficiency (Carnot efficiency, ? C ). The maximum power corresponds to Curzon- Ahlborn 1 efficiency for internally reversible Carnot engine. Unlike this, the power-efficiency characteristic of a real heat engine corresponds to zero power and zero efficiency at both the limits of thermal short-circuit and open-circuit conditions 5 . The power- efficiency characteristics of most of the real heat engines are akin to loop-like behaviour. Real heat engines, with finite resources, exhibit possibility of operation at maximum power or at maximum efficiency. The power-efficiency characteristics of an internally reversible regenerative engine are shown in Fig. 2 for different regeneration efficiencies. From Fig. 2, it may be noted that the efficiency and the power output both exhibit a maximum. These INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 34 characteristics are similar to those of Carnot heat engine with heat leak. It may be viewed from Eqs (6) and (7) where the first term corresponds to equivalent external heat leak. However, the power-efficiency characteristic curve with perfect regeneration is equivalent to that of an internally reversible Carnot engine without thermal leakage (Fig. 2). The characteristic curve passes through a maximum power point (w max ) and a maximum efficiency point (? max ). The operating region is the portion of the power-efficiency curve lying in between maximum power and maximum efficiency point. Beyond this range, the power production and the operating efficiency, both deteriorate. Within the operating range, as the operating efficiency of the heat engine decreases from the maximum attainable limit, the power production increases but the fuel utilization decreases. These counteractive activities bring the efficiency of the real engine to lie in between the maximum power and maximum operating efficiency points 2 . Therefore, the operating region may be defined as the region that simultaneously satisfy both the inequalities ? max = ? = ? MP and w max = w = w ME . These can be summarized as a single criterion 1 = t MP = t = t ME = t C … (17) This is the criterion for optimal design of a real heat engine and selection of optimal operating stages for combined-cycle power generation 6 . In some cases, full theoretical power-efficiency curve cannot be measured due to mechanical or material constraints. Therefore, the operating region may not be fully accessible. In these cases, designer is expected to select the best possible performance subject to such constraints. In practice, neither maximum power nor maximum efficiency can be the sole objective of an energy conversion device. However, better understandings of these limiting cases are essential for the multiple-objective design approach. Maximizing power production of the regenerative engine one can get t MP = (b + (b 2 – (a + b + d) (b – d – a t C )) 1/2 )/(a + b + d) … (18) where a = k(1 - k), b = sea(1 – t C ), and d = se(k + (1 – k) t C ). Maximum efficiency of the regenerative engine corresponds to 2 t ME (a + f + d) = [2a t C + d(1 + t C ) + f - h] + ((2a t C + d(1 + t C ) + f – h) 2 – 4(a + f + d)( a t C 2 + d t C - h)) 1/2 … (19) where f = sek(1 – t C ) and h = se(1 – k) t C (1 – t C ). In Fig. 2, loci of maximum power and maximum efficiency for different regeneration efficiencies are also shown. From Fig. 2, it may be observed that the operating region reduces and hence, the flexibility of the engineer, as the regeneration efficiency deteriorates. Variations of maximum power, maximum efficiency, power at maximum efficiency, and efficiency at maximum power with the variations in regeneration inefficiencies are shown in Fig. 3. The performance of the heat engine deteriorates with decreasing regeneration performances. Below a certain regenerator efficiency ( e = 0.2 as shown in Fig. 3), the operating region collapses for all practical purpose. It is interesting to note that unlike maximum power, maximum efficiency, and efficiency at maximum power, power at maximum efficiency does not increase monotonically with increasing regeneration efficiency. This implies that for any practical regenerator with very high regeneration efficiency (95% efficiency for the case shown in Fig. 3), significant power may generate at maximum efficiency of the engine. Therefore, for most of the Fig. 2 ? Power-efficiency characteristics of a regenerative heat engine for different regeneration efficiencies Page 5 Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 31-37 Thermo-economic analysis of regenerative heat engines Santanu Bandyopadhyay Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India e-mail: santanub@iitb.ac.in Received 19 May 2003; accepted 8 September 2003 Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are derived without assuming any particular equation of state associated with the working fluid. [Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines; Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine] 1 Introduction Carnot efficiency (? C = 1 – T min /T max ) is the maximum possible efficiency of a heat engine with which low-grade thermal energy may be reversibly transformed into high-grade mechanical energy. Ideal regenerative heat engines (such as Stirling and Ericsson heat engines), with perfect regeneration, also operate with the Carnot efficiency. To achieve Carnot efficiency, thermal exchanges between the reservoirs and the working fluid of the engine have to occur through reversible isothermal processes. These processes demand infinite heat exchanger surface area. A heat engine with finite heat exchanger area, result in zero power production. On the other hand, the efficiency of an internally reversible Carnot engine, deliver maximum power, given by ? MP = 1 – (T min /T max ) 1/2 (Ref. 1). The global need for fuel-efficient and environmentally viable power production, with thermodynamic reliability and economy, demands moderation of the traditional energy conversion processes with new approaches. Bera and Bandyopadhyay 2 have analyzed the effect of combustion on the thermoeconomic performances of Carnot, Otto and Joule-Brayton engines. Classical reversible heat engines are never realizable in practice, but the aim is to reach the highest limit of power production within the constraints of finite resources. With this end in view, regenerative heat engine cycles have been studied and their design philosophies have flourished. Regenerative heat engines have other benefits also. Exhaust emissions of a regenerative heat engine are low and may be easily controlled as the combustion is isolated from cyclic pressure and temperature changes experienced by the working fluid. Continuous complete combustion with 20-80 % excess air replaces intermittent combustion occurring in other piston engines. This is because quenching of the flame does not take place at the ‘cold’ metal surface. This leads to remarkably low noise levels 3 . Regenerative engines are so thermally efficient that they are prime contender for alternative power unit. The mean effective pressure and the mechanical efficiency of a regenerative engine are also quite high 4 . Hence, generations of physicists and engineers of past, focused on these types of engines. In this paper, internally reversible regenerative heat engines with imperfect regeneration are discussed and detailed understanding for optimal design of such engines are provided. INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 32 The power-efficiency characteristics of a real engine help a designer to identify the operating region for optimal design of the heat engine and to realize the upper bounds on power production and its attainable efficiency. The power-efficiency characteristics for irreversible Carnot cycle, irreversible Joule-Brayton cycle, and Rankine cycle are equivalent to each other 5 . The power-efficiency characteristics of a regenerative engine are expected to be a strong function of regeneration efficiency. In this paper, the power-efficiency characteristics of regenerative heat engines are studied and the operating regions are identified. Knowing the governing equation of state related with any particular working fluid, different design parameters may easily be calculated. The results are derived without assuming any particular equation of state associated with the working fluid. However, for brevity, thermal capacity rates are assumed to be independent of temperature. 2 Regenerative Heat Engines The ideal thermodynamic cycle corresponding to regenerative heat engine consists of two isothermal and two polytropic (of index n) processes. The temperature-entropy diagram of a typical regenerative cycle is shown in Fig. 1. Depending on the nature of the polytropic process, regenerative cycle reduces to Carnot (for adiabatic process), Ericsson (for isobaric process) or Stirling (for isometric process) cycles. The isothermal compression occurs between states 1 and 2. In the isothermal compression, heat is rejected by the working fluid to the cold reservoir. The isothermal expansion process, where injection of heat to the working fluid from the external hot reservoir takes place, occurs between states 3 and 4. The temperatures of the hot and the cold reservoirs are denoted by T max and T min , respectively. Processes from states 2 to 3 and 4 to 1 are polytropic (of index n). Portion of the heat rejected from the polytropic process 4 to 1 is supplied partly to the process 2 to 3 through a regenerator. Condition of the working fluid after the regeneration process is denoted by the states 2R and 4R (Fig. 1). Regenerator in the process, 2 to 2R supply heat, and external heat is provided by the hot reservoir from 2R to 4. Similarly, heat is rejected from 4R to 2. Therefore, total heat supplied to and rejected with are given as Q in = Q 2R3 + Q 34 … (1) Q out = Q 4R1 + Q 12 … (2) Assuming that the cycle is internally reversible and all the irreversibility is associated with the finite driving force of the heat transfer process (that is, reversibility of the heat engine), the entropy balance for the regenerative engine may be satisfied. Q 34 /T h + C n ln(T h /T c ) = Q 12 /T c + C n ln(T h /T c ) … (3) where T h (= T 3 = T 4 ) and T c (= T 1 = T 2 ) are the highest and the lowest temperatures attained by the working fluid. In Eq. (3), C n denotes the thermal capacity rate of the polytropic process. Knowing the equation of state that governs the working fluid and the polytropic index, n, the polytropic thermal capacity rate C n , can be determined. For working fluid obeying ideal gas laws, polytropic thermal capacity rate may be calculated in terms of thermal capacity rate at constant volume as C n = C v (n - ?)/(n - 1). For brevity, C n is assumed to be independent of temperature. The temperatures of the working fluid after the regeneration (Q R ) may be written from the energy balance of the regenerator. T 4R = T h – Q R /C n … (4) T 2R = T c + Q R /C n … (5) With the help of Eqs (4) and (5), energy exchange equations are rewritten as follows: Q in = C n (T h – T c ) – Q R + Q 34 … (6) Q out = C n (T h – T c ) – Q R + Q 12 … (7) Fig. 1 ? Temperature-entropy diagram of a regenerative heat engine BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 33 Heat exchangers are assumed to be counter- current. Total thermal conductance in the hot side of the engine is given as K h = C n ln((T max – T c – Q R /C n )/(T max - T h )) + Q 34 /(T max – T h ) … (8) Similarly, for cold side of the heat engine one can get K c = C n ln((T h – T min – Q R /C n )/(T c – T min )) + Q 12 /(T c – T min ) … (9) The maximum possible regeneration is C n (T h – T c ). The regeneration process may be modeled with an efficiency of (1 - e). Hence Q R = (1 - e) C n (T h – T c ) … (10) Defining the non-dimensional quantities such as t h = T h /T max , t c = T c /T max , t C = T min /T max , k = K h /(K h + K c ), s = C n /(K h + K c ), q = Q/(T max (K h + K c )), and w = W/(T max (K h + K c )), above equations may be written in dimensionless form. Note that, 1 = t h = t c = t C . Combining these equations the energy input to and rejected with are given by q in = es(t h – t c ) + (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))] ˜ es(t h – t c ) + (1 – t h )[k – es (t h – t c )/(1 – t h ))] = (1 – t h ) k … (11) and similarly q out = es(t h – t c ) + (t c – t C )[(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))] ˜ (t c – t C ) (1 - k) … (12) The reversibility [Eq. (3)] of the heat engine translates to (1 – t h )[k – s ln(1 + e(t h – t c )/(1 – t h ))]/t h = (t c – t C ) [(1 – k) – s ln(1 + e(t h – t c )/( t c – t C ))]/t c … (13) Neglecting higher terms of the expanded logarithm function, this leads to (1 – t h )k/t h – se(t h – t c )/t h = (t c – t C ) (1 – k)/t c – se(t h – t c )/t c … (14) Denoting t = t c /t h , the power generated by the internally reversible engine and its efficiency can be expressed as w ˜ (k(1 – k)(1 - t)( t - t C ) - se(1 – t) 2 (k + (1 – k) t C ))/( t - se(1 – t) 2 ) … (15) and ? ˜ 1 - (k(1 – k) t( t - t C ) + (1 – k)se t C (1 – t) 2 )/(k (1 – k) ( t - t C ) – kse(1 – t) 2 ) … (16) The power output [Eq. (15)] or the efficiency [Eq. (16)] of the engine may be maximized to optimize the performance of the regenerative engine. Note that these approximations are reasonable, except for very low working temperature ratio ( t ? t C ). For a very low working temperature ratio original equations have to be solved numerically. Simultaneous solution of these equations will ensure the concurrent employment of the first and the second laws. 3 Power-Efficiency Characteristics Internally reversible Carnot engine operates between the limits of thermal ‘short-circuit’ and thermal ‘open-circuit’ conditions 5 . No power is produced either when the engine operates at thermal short-circuit condition with zero efficiency or the engine operates at thermal open-circuit condition with maximum possible efficiency (Carnot efficiency, ? C ). The maximum power corresponds to Curzon- Ahlborn 1 efficiency for internally reversible Carnot engine. Unlike this, the power-efficiency characteristic of a real heat engine corresponds to zero power and zero efficiency at both the limits of thermal short-circuit and open-circuit conditions 5 . The power- efficiency characteristics of most of the real heat engines are akin to loop-like behaviour. Real heat engines, with finite resources, exhibit possibility of operation at maximum power or at maximum efficiency. The power-efficiency characteristics of an internally reversible regenerative engine are shown in Fig. 2 for different regeneration efficiencies. From Fig. 2, it may be noted that the efficiency and the power output both exhibit a maximum. These INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 34 characteristics are similar to those of Carnot heat engine with heat leak. It may be viewed from Eqs (6) and (7) where the first term corresponds to equivalent external heat leak. However, the power-efficiency characteristic curve with perfect regeneration is equivalent to that of an internally reversible Carnot engine without thermal leakage (Fig. 2). The characteristic curve passes through a maximum power point (w max ) and a maximum efficiency point (? max ). The operating region is the portion of the power-efficiency curve lying in between maximum power and maximum efficiency point. Beyond this range, the power production and the operating efficiency, both deteriorate. Within the operating range, as the operating efficiency of the heat engine decreases from the maximum attainable limit, the power production increases but the fuel utilization decreases. These counteractive activities bring the efficiency of the real engine to lie in between the maximum power and maximum operating efficiency points 2 . Therefore, the operating region may be defined as the region that simultaneously satisfy both the inequalities ? max = ? = ? MP and w max = w = w ME . These can be summarized as a single criterion 1 = t MP = t = t ME = t C … (17) This is the criterion for optimal design of a real heat engine and selection of optimal operating stages for combined-cycle power generation 6 . In some cases, full theoretical power-efficiency curve cannot be measured due to mechanical or material constraints. Therefore, the operating region may not be fully accessible. In these cases, designer is expected to select the best possible performance subject to such constraints. In practice, neither maximum power nor maximum efficiency can be the sole objective of an energy conversion device. However, better understandings of these limiting cases are essential for the multiple-objective design approach. Maximizing power production of the regenerative engine one can get t MP = (b + (b 2 – (a + b + d) (b – d – a t C )) 1/2 )/(a + b + d) … (18) where a = k(1 - k), b = sea(1 – t C ), and d = se(k + (1 – k) t C ). Maximum efficiency of the regenerative engine corresponds to 2 t ME (a + f + d) = [2a t C + d(1 + t C ) + f - h] + ((2a t C + d(1 + t C ) + f – h) 2 – 4(a + f + d)( a t C 2 + d t C - h)) 1/2 … (19) where f = sek(1 – t C ) and h = se(1 – k) t C (1 – t C ). In Fig. 2, loci of maximum power and maximum efficiency for different regeneration efficiencies are also shown. From Fig. 2, it may be observed that the operating region reduces and hence, the flexibility of the engineer, as the regeneration efficiency deteriorates. Variations of maximum power, maximum efficiency, power at maximum efficiency, and efficiency at maximum power with the variations in regeneration inefficiencies are shown in Fig. 3. The performance of the heat engine deteriorates with decreasing regeneration performances. Below a certain regenerator efficiency ( e = 0.2 as shown in Fig. 3), the operating region collapses for all practical purpose. It is interesting to note that unlike maximum power, maximum efficiency, and efficiency at maximum power, power at maximum efficiency does not increase monotonically with increasing regeneration efficiency. This implies that for any practical regenerator with very high regeneration efficiency (95% efficiency for the case shown in Fig. 3), significant power may generate at maximum efficiency of the engine. Therefore, for most of the Fig. 2 ? Power-efficiency characteristics of a regenerative heat engine for different regeneration efficiencies BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 35 practical regenerator with very high regeneration efficiency, designer has the significant design flexibility without significant loss of efficiency and power production from the engine. Variations of maximum power, maximum efficiency, power at maximum efficiency and efficiency at maximum power with the thermal capacity rate of polytropic process are shown in Fig. 4. At the Carnot limit (C n ? 0) best performance is observed, whereas C n ? 8 indicates the isothermal process and the engine becomes impossible to operate. Again it may be observed that power at maximum efficiency shows a maximum against heat capacity rate of the polytropic processes. Therefore, depending upon the efficiency of the regenerator, heat capacity of the polytropic processes may be adjusted to obtained significant power from the engine operating at maximum efficiency. Perfect regeneration ( e = 0) and/or Carnot heat engine (s = 0) may be characterized by the relation s e = 0. In either of these cases Eqs (18) and (19) reduce to the following expressions. t MP (s e = 0) = ( t C ) 1/2 … (20) and t ME (s e = 0) = t C … (21) Therefore, isothermal heat engines with perfect regeneration operate with Curzon-Ahlborn 1 efficiency at maximum power point and with Carnot efficiency at maximum efficiency point. Note that, internally reversible Otto and Joule-Brayton engines also operate with the same efficiency at maximum power point 2 . 4 Distribution of Thermal Conductance Power production or efficiency of a regenerative heat engine can further be maximized subject to the distribution of heat exchanger thermal conductance. The optimum distribution for maximum power output comes out to be 2k MP = 1 - se(1 – t MP )(1 – t C )/( t MP – t C ) … (22) and for maximum efficiency (( t ME – t C )k ME + t C ) 2 = t C ( t ME – se(1 – t ME ) 2 ) … (23) Eqs (22) and (23) suggest that, 1 = 2k MP(or ME) or 1 - k MP(or ME) = k MP(or ME) . That is, in other words, K c(at MP or ME) = K h(at MP or ME) … (24) Therefore, the cold side exchanger requires more thermal conductance. For imperfect regeneration, the amount of heat rejection increases and a larger cold side exchanger reduce the external entropy generation by allowing the heat engine to reject energy at lower Fig. 3 ? Variation of maximum power, power at maximum efficiency, maximum efficiency and efficiency at maximum power for different regeneration efficiencies (k = 0.5, s = 1.0, and t C = 0.3) Fig. 4 ? Variation of maximum power, power at maximum efficiency, maximum efficiency and efficiency at maximum power for different thermal capacity rate of the polytropic process (k = 0.5, e = 0.01, and t C = 0.3)Read More

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