Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs

The efficiencies of heat-engine operation employing various numbers ( ≥ 2) of heat reservoirs are investigated. Operation with the work output of the heat engines sequestered, as well as with it being totally frictionally dissipated, is discussed. We consider mainly heat engines whose efficiencies depend on ratios of a higher and lower temperature or on simple functions of such ratios but also provide brief comments concerning more general cases. We show that, if a hot reservoir supplies a heat engine whose waste heat is discharged and whose work output is totally frictionally dissipated into a cooler reservoir, which in turn supplies heat-engine operation that discharges waste heat into a still cooler reservoir, the total work output can exceed the heat input from the initial hot reservoir. This extra work output increases with increasing numbers ( ≥ 3) of reservoirs. We also show that this obtains within the restrictions of the First and Second Laws of Thermodynamics.

We show that, if a hot reservoir supplies a heat engine whose waste heat is discharged and whose work output is totally frictionally dissipated into a cooler reservoir, which in turn supplies heat-engine operation that discharges waste heat into a still cooler reservoir, the total work output can exceed the heat input from the initial hot reservoir. This extra work output increases with increasing numbers (≥ 3) of reservoirs. We also show that this obtains within the restrictions of the First and Second Laws of Thermodynamics.
We consider only cyclic heat engines. Noncyclic (necessarily one-time, singleuse) heat engines are not limited by the Carnot bound and can in principle operate at unit (100%) efficiency. A simple example is the one-time expansion of a gas pushing a piston. Other examples include rockets: the piston (payload) is launched into space by a one-time power stroke (but typically most of the work output accelerates the exhaust gases, not the payload) and firearms: the piston (bullet) is accelerated by a one-time power stroke and then discarded (but some, typically less than with rockets, of the work output accelerates the exhaust gases resulting from combustion of the propellant). Even if the work output of a noncyclic engine could be frictionally dissipated and the resulting heat returned to the system, there would be, at best, restoration of temperature to its initial value but not restoration of the piston to its initial position. Hence the method investigated in this chapter is useless with respect to noncyclic heat engines.
General remarks, especially concerning entropy, are provided in Section 4. Concluding remarks are provided in Section 5.

Multiple-reservoir heat-engine efficiencies with work output sequestered
We designate the temperatures of the heat reservoirs via subscripts, with T 1 being the temperature of the initial, hottest, reservoir, T 2 the temperature of the second-hottest reservoir, T 3 the temperature of the third-hottest reservoir, etc., and T n the temperature of the nth, coldest, reservoir.
Let a heat engine operate between two reservoirs, extracting heat Q 1 from a hot reservoir at temperature T 1 and rejecting waste heat to a cold reservoir at temperature T 2 . If its efficiency is ϵ 1!2 , its work output is It rejects waste heat Þto the reservoir at temperature T 2 . If there is a third reservoir at temperature T 3 and W 1!2 is sequestered, that is, not frictionally dissipated, and if the efficiency of heat-engine operation between the second and third reservoirs is ϵ 2!3 , a heat engine can then perform additional work by employing the reservoir at temperature T 2 as a hot reservoir and the reservoir at temperature T 3 as a cold reservoir. All told it can do work: By contrast, if the heat engine operates in a single step at efficiency ϵ 1!3 , employing the reservoir at temperature T 1 as a hot reservoir and the reservoir at temperature T 3 as a cold reservoir, it can do work Anticipating that we will eventually deal with n heat reservoirs, let us consider efficiencies of the form where i and j are positive integers in the respective ranges 1 ≤ i ≤ n À 1 and i < j ≤ n and where x is a positive real number in the range 0 < x ≤ 1. Applying Eqs. (3) and (5), W 1!3 = W 1!2 + W 2!3 , as we will now show. We have We note that x = 1 for the Carnot, Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [1][2][3][4][5][6][7][8][9] and x = 1/2 for endoreversible heat engines operating at Curzon-Ahlborn efficiency [10][11][12] (see also Ref. [4], . For all of these cycles, the temperature in the numerator is that of the coldest available reservoir for a given cycle [1][2][3][4][5][6][7][8][9][10][11][12]. For the Carnot, Ericsson, and Stirling cycles, and for endoreversible heat engines operating at Curzon-Ahlborn efficiency, the temperature in the denominator is that of the hottest available reservoir for a given cycle [1][2][3][4][5][6][7][8][9][10][11][12]. For the air-standard Otto and air-standard Brayton cycles, the temperature in the denominator is that at the end of the adiabatic-compression process but before the addition of heat from the hottest available reservoir (substituting, in air-standard cycles, for combustion of fuel) [2][3][4][5][6][7][8][9] in a given cycle. The Second Law of Thermodynamics forbids x > 1 if the temperature in the numerator is that of the coldest available reservoir for a given cycle and the temperature in the denominator is that of the hottest available reservoir for a given cycle, because then the Carnot efficiency would be exceeded. Since for the aforementioned heat engines, and indeed for any heat engine for which Eq. (5) is applicable, W 1!3 = W 1!2 + W 2!3 , this additivity of W obtains for any number of steps, that is, we have For more complex efficiencies than those of Eq. (5), for example, those of the Diesel and dual cycles, which are functions of more than two temperatures, and also for some more complex efficiencies that are functions of two temperatures, the equality of Eq. (7) may not always obtain [3][4][5][6][7][8][9][13][14][15][16][17][18][19]. But whether or not the equality of Eq. (7) obtains, the Second Law of Thermodynamics requires that, whichever reservoirs are employed, the efficiency with all work outputs sequestered, whether W j!j+1 /Q j (1 ≤ j ≤ n À 1), W j!j+k /Q j (1 ≤ j ≤ n À 1 and 1 ≤ k ≤ n À j), or W 1!n /Q 1 , cannot exceed the Carnot limit.

Multiple-reservoir heat-engine efficiencies with work output totally frictionally dissipated
Let a heat engine operate between two reservoirs, extracting heat Q 1 from a hot reservoir at temperature T 1 and rejecting waste heat to a cold reservoir at temperature T 2 . If its efficiency is ϵ 1!2 , its work output is It rejects waste heat Þto a reservoir at temperature T 2 . But now, in addition, we let the work output W D 1!2 ¼ Q 1 ϵ 1!2 be totally frictionally dissipated and rejected into the reservoir at temperature T 2 (indicated via a superscript D). This is in fact by far the most common mode of heat-engine operation. With rare exceptions (e.g., a heat engine's work output being sequestered for a long time interval as gravitational potential energy in the construction of a building, or essentially permanently in the launching of a spacecraft), heat engines' work outputs are typically totally frictionally dissipated immediately or on short time scales (see Ref. [6], Chapter VI (especially Sections 54, 60, and 61); and Ref. [7], Sections 6.9-6.14 and 16.8). Indeed, this is true of almost all engines, heat engines or otherwise. The work outputs of all engines of vehicles (automobiles, trains, ships, submarines, aircraft, etc.) operating at constant speed, and of all factory and appliance engines operating at constant speed, are immediately and continually frictionally dissipated. The work output temporarily sequestered as kinetic energy when a vehicle accelerates, or when a factory or appliance engine is turned on, is frictionally dissipated a short time later when the vehicle decelerates, or when the factory or appliance engine is turned off.
If both the waste heat Þhas been rejected and the work output W D 1!2 ¼ Q 1 ϵ 1!2 has been totally frictionally dissipated into the reservoir at temperature T 2 , and there is a third reservoir at temperature T 3 , a heat engine operating at efficiency ϵ 2!3 can then perform additional work by employing the reservoir at temperature T 2 as a hot reservoir and the reservoir at temperature T 3 as a cold reservoir. (W D 2!3 may or may not be frictionally dissipated, so it only optionally carries the superscript D.) All told the total work output is where i and j are positive integers in the respective ranges 1 ≤ i ≤ n À 1 and i < j ≤ n, and where x is a positive real number in the range 0 < x ≤ 1, applying Eqs. (5) and (10), we have: We now maximize W D 1!3 with respect to T 2 : Thus, the optimum value T 2 ,opt of T 2 , which maximizes W D 1!3 , is the geometric mean of T 1 and T 3 . Applying Eqs. (11) and (12), the maximum value Note that This It is easily shown that W D,extra 1!3,max ≥ 0, with the equality obtaining if and only if T 3 as r and setting dW D,extra Thus Moreover, applying Eqs. (5), (13), and (15), note that Now consider heat-engine operation employing four heat reservoirs, with all work totally frictionally dissipated (except possibly at the last step; thus, W D 3!4 only optionally carries the superscript D). Thus we have where i and j are positive integers in the respective ranges 1 ≤ i ≤ n À 1 and i < j ≤ n, and where x is a positive real number in the range 0 < x ≤ 1, applying Eqs. (5) and (18), we have: We wish to maximize W D 1!4 . Based on Eq. (12) and the associated discussions, the optimum value T j,opt of T j of reservoir j 1 < j < n ⇔ 2 ≤ j ≤ n À 1 ð Þ , which maximizes W D jÀ1!jþ1 , is the geometric mean of T jÀ1 and T jþ1 . Thus we have and Applying Eqs. (20) and (21), we obtain and Applying Eqs. (20)- (23), we obtain Applying Eqs. (22)- (24), we obtain Applying Eqs. (19) and (25), we obtain We now slightly modify Eqs. (14)- (17) to apply for our four-reservoir system. We obtain This obtains if (5) and (26), It is easily shown that W D,extra 1!4,max ≥ 0, with the equality obtaining if and only if Thus W D,extra 1!4,max is minimized at 0 if r ¼ T 4 Comparing Eqs. Generalizing Eqs. (20)-(30) for an n-reservoir system (n = any positive integer ≥ 4), we obtain: where j is any positive integer in the range 1 ≤ j ≤ n À 2 and where j is any positive integer in the range 1 ≤ j ≤ n À 3. The respective temperatures T 1 and T n of the extreme (hottest and coldest) reservoirs are assumed to be fixed. The temperatures T 2 through T nÀ1 of all intermediate reservoirs are all assumed to be optimized in accordance with Eqs. (31) and (32). With that understood, for brevity and to avoid using different subscripts for the extreme and intermediate reservoirs, the subscript "opt" is omitted in Eqs. (31)-(35). Applying Eqs. (31) and (32), we obtain: and Applying Eqs. (33) and (34), and recognizing that Eqs. (33) and (34) obtain for all values of j such that j is any positive integer in the range 1 ≤ j ≤ n À 2, we obtain: The first two lines of Eq. (35) obtain for all values of j such that j is any positive integer in the range 1 ≤ j ≤ n À 2, and the third line of Eq. (35) obtain for all values of j such that j is any positive integer in the range 1 ≤ j ≤ n À 1. The first two lines of Eq. (35) pertain to any three adjacent heat reservoirs, and hence 2 appears in the exponents of the second line thereof; the third line of Eq. (35) pertains to all n heat reservoirs, and hence n À 1 appears in the exponents thereof. The second and third lines of Eq. (35) mutually justify each other: the third line of Eq. (35) must obtain because the second line thereof obtains for all values of j; and, conversely, given that the third line of Eq. (35) obtains, the second line thereof must obtain for all values of j.
If, as per Eq.
where i and j are positive integers in the respective ranges 1 ≤ i ≤ n À 1 and i < j ≤ n, and where x is a positive real number in the range 0 < x ≤ 1, then, applying Eqs. (5) and (31)-(35), we now generalize Eqs. (13)- (17) and (26)- (30), as well as the associated discussions, to apply for our n-reservoir system. We obtain: It is easily shown that W D,extra 1!n,max ≥ 0, with the equality obtaining if and only if Thus W D,extra 1!n,max is minimized at 0 if r ¼ T n Note that the values in Eqs. (36), (38), and (40) increase monotonically with increasing n and that the fulfillment of the inequality in Eq. (37) becomes monotonically easier with increasing n. Equation (40) is valid not only for Carnot efficiency (x ¼ 1) but even for Curzon-Ahlborn efficiency (x ¼ 1=2), indeed for any x finitely greater than 0 in the range 0 < x ≤ 1, because T n ð Þ ! 1 in the limit T n =T 1 ! 0, albeit ever more slowly with decreasing x.
By contrast, even granting Carnot efficiency (x ¼ 1) [22]: Note the linear divergence of W D 1!n,max in the limit T n =T 1 ! 0 with n fixed as per Eq. (40) even not assuming Carnot efficiency, as contrasted with the paltry logarithmic divergence of W D 1!n,max in the limit n ! ∞ with T n =T 1 fixed even granting Carnot efficiency as per the derivation [22] of Eq. (41).
But we note that the temperature of the cosmic background radiation is only 2:7 K, while the most refractory materials remain solid at temperatures slightly exceeding 2700 K. This provides a temperature ratio of T 1 =T n ≈ 10 3 ⇔ T n =T 1 ≈ 10 À3 . Could even larger values of T 1 =T n be possible, at least in principle? Perhaps, maybe, if frictional dissipation of work into heat might somehow be possible into a gaseous hot reservoir at temperatures exceeding the melting point or even the critical temperature (the maximum boiling point at any pressure) of even the most refractory material. Yet even with the paltry logarithmic divergence of W D 1!n,max in the limit n ! ∞ with T 1 =T n fixed as per Eq. (41) and even with a temperature ratio of T 1 =T n ≈ 10 3 ⇔ T n =T 1 ≈ 10 À3 , assuming Carnot efficiency by Eq. (41) W D 1!n,max =Q 1 ≈ ln 10 3 ≈ 7. Hence by Eq. (41) an advanced civilization employing 7 concentric Dyson spheres [39,40] can procure 7 times as much work output (to the nearest whole number) as its host star's total energy output. Actually the limit n ! ∞ with T 1 =T n fixed is not sufficiently closely approached to apply Eq. (41): we should instead apply Eq. (36). Applying Eq. (36) and assuming Carnot efficiency with T 1 =T n ≈ 10 3 ⇔ T n =T 1 ≈ 10 À3 , W D 1!n,max =Q 1 ≈ 4. Hence by Eq. (36) an advanced civilization employing 4 concentric Dyson spheres [39,40] can procure 4 times as much work output (to the nearest whole number) as its host star's total energy output.

General remarks, especially concerning entropy
It is important to emphasize that the super-unity cyclic-heat-engine efficiencies W D 1!n,max =Q 1 that can obtain with work output totally frictionally dissipated if n ≥ 3 ð Þare consistent with both the First and Second Laws of Thermodynamics. The two laws are not violated because, if the work output of a heat engine is frictionally dissipated as heat into a cooler reservoir, both laws allow this heat to be partially converted to work again if another, still cooler, reservoir is available.
In this Section 4 we do not restrict heat-engine efficiencies to the form given by Equation (5), nor necessarily assume efficiencies of the same form at each step j ! j þ 1 or j ! j þ k (1 ≤ k ≤ n À j). The validity of this Section 4 requires only that the efficiency with all work sequestered, or at any one given step j ! j þ 1 whether work is sequestered or not, be within the Carnot limit, in accordance with the Second Law.
The extra work that is made available via frictional dissipation into cooler reservoirs is paid for by an extra increase in entropy. Consider the work available via heat-engine operation between reservoir j at temperature T j and reservoir j þ 2 at temperature T jþ2 without versus with frictional dissipation into reservoir j þ 1 at temperature T jþ1 T j > T jþ1 > T jþ2 À Á . Without frictional dissipation a heat engine performs work (42) by employing the reservoir at temperature T j as a hot reservoir and the reservoir at temperature T jþ1 as a cold reservoir. It rejects waste heat Q j À W j!jþ1 ¼ Q j 1 À ϵ j!jþ1 À Á to the reservoir at temperature T jþ1 . If a third reservoir at temperature T jþ2 and W j!jþ1 is sequestered, that is, not frictionally dissipated, a heat engine can then perform additional work: by employing the reservoir at temperature T jþ1 as a hot reservoir and the reservoir at temperature T jþ2 as a cold reservoir. All told it can do work: With total frictional dissipation of W j!jþ1 into reservoir j þ 1 at temperature T jþ1 , we still have But now we let the work output W D j!j¼1 ¼ Q 1 ϵ jþ1!jþ2 be totally frictionally dissipated into the reservoir at temperature T jþ1 (indicated via a superscript D). If there is a third reservoir at temperature T jþ2 , a heat engine can then perform additional work: All told it can do work: The extra work is paid for by the extra increase in entropy owing to frictional dissipation into extra heat Q D extra of the work output as per Eqs. (42) and (45) into reservoir j þ 1 at temperature T jþ1 . This extra increase in entropy is [In the last four steps of Eq. (50), we applied Eqs. (42), (45), (48), and (49).] Thus In no case do we assume an efficiency with all work sequestered, or at any one given step j ! j þ 1 whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except that it is conserved-and we never violate conservation of energy.) We note that, while frictional dissipation of work into intermediate reservoirs can yield extra work W D extra in heat-engine operation (albeit at the expense of ΔS D extra ), it seems to be of no help in reverse, that is, refrigerator or heat pump, operation. For, in refrigerator or heat pump operation, with an intermediate reser-  [4], Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. [5], Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17; Ref. [6], Chapter XXI; Ref. [7], Sections 6.7, 6.8, 7.3, and 7.4); and Ref. [9], pp. 233-236 and Problems 1, 2, 4, 6, and 7 of Chapter 8. [Problem 2 of Chapter 8 in Ref. [9] considers absorption refrigeration, wherein the entire energy output is into an intermediate-temperature (most typically ambient-temperature) reservoir, and hence for which also there is no energy left over to be frictionally dissipated.]

Conclusion
We investigated the increased heat-engine efficiencies obtained via operation employing increasing numbers (≥ 3) of heat reservoirs and with work output totally frictionally dissipated into all reservoirs except the first, hottest, one at temperature T 1 and (possibly) also the last, coldest, one at temperature T n . We emphasize again that our results are consistent with both the First and Second Laws of Thermodynamics. The two laws are not violated because, if the work output of a heat engine is frictionally dissipated as heat into a cooler reservoir, both laws allow this heat to be partially converted to work again if another, still cooler, reservoir is available.
We do, however, challenge an overstatement of the Second Law that is sometimes made, namely, that energy can do work only once. Energy can indeed do work more than once, because the Second Law does not forbid recycling of energy, so long as total entropy does not decrease as a result. This criterion of non-decrease of total entropy is obeyed, as per Section 4. In no case do we assume an efficiency with all work sequestered, or at any one given step j ! j þ 1 whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except that it is conserved-and we never violate conservation of energy).
While in this chapter we do not challenge the First or Second Laws of Thermodynamics, we should note that there have been many challenges to the Second Law, especially in recent years [41][42][43][44][45][46]. By contrast, the First Law has been questioned only in cosmological contexts [47][48][49] and with respect to fleeting violations thereof associated with the energy-time uncertainty principle [50,51]. But there are contrasting viewpoints [50,51] concerning the latter issue.