Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs

Abstract

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.

1. Introduction

The efficiencies of heat-engine operation employing various numbers (≥ 2) of heat reservoirs are investigated. In Section 2, we discuss heat-engine operation with the work output of the heat engines sequestered. In Section 3, we discuss heat-engine operation with the work output of the heat engines being totally frictionally dissipated. We consider mainly heat engines whose efficiencies depend on ratios of a higher and lower temperature or on simple functions of such ratios. Examples include heat engines operating not only via the Carnot cycle [1, 2, 3, 4, 5, 6, 7, 8, 9] but also via the Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [2, 3, 4, 5, 6, 7, 8, 9], and endoreversible heat engines operating at maximum power output assuming Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. [4], Section 4-9). But we also provide brief comments concerning more general cases. Endoreversible heat-engine operation assumes irreversible heat flows directly proportional to temperature differences but otherwise reversible operation [10, 11, 12]. Although we do not employ them in this chapter, we note that generalizations of the Curzon-Ahlborn efficiency, and also various related efficiencies, have also been investigated [13, 14, 15, 16, 17, 18, 19, 20, 21]. In particular, we note that alternative results [21] to the Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. [4], Section 4-9) have been derived [21]. But for definiteness and for simplicity, in this chapter, we employ the standard Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. [4], Section 4-9) for cyclic heat engines operating at maximum power output.

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 fill in details and correct a few mistakes in an earlier, briefer, consideration of the efficiencies of heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22]. We note that heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22] should not be confused with recycling heat engines’ frictionally dissipated work outputs into the hottest available reservoir [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], which is a different process that has been thoroughly investigated and discussed previously [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], and which we further investigate in another chapter [38] in this book.

We consider only cyclic heat engines. Noncyclic (necessarily one-time, single-use) 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.

2. Multiple-reservoir heat-engine efficiencies with work output sequestered

We designate the temperatures of the heat reservoirs via subscripts, with T₁ being the temperature of the initial, hottest, reservoir, T₂ the temperature of the second-hottest reservoir, T₃ 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 Q₁ from a hot reservoir at temperature T₁ and rejecting waste heat to a cold reservoir at temperature T₂. If its efficiency is ${\mathrm{ϵ}}_{1\to 2}$, its work output is

$$W_{1\to 2}=Q_1{\mathrm{ϵ}}_{1\to 2}.$$E1

It rejects waste heat $Q_1-W_{1\to 2}=Q_1\left(1-{\mathrm{ϵ}}_{1\to 2}\right)$ to the reservoir at temperature T₂. If there is a third reservoir at temperature T₃ and $W_{1\to 2}$ is sequestered, that is, not frictionally dissipated, and if the efficiency of heat-engine operation between the second and third reservoirs is ${\mathrm{ϵ}}_{2\to 3}$, a heat engine can then perform additional work

$$W_{2\to 3}=Q_1\left(1-{\mathrm{ϵ}}_{1\to 2}\right){\mathrm{ϵ}}_{2\to 3}$$E2

by employing the reservoir at temperature T₂ as a hot reservoir and the reservoir at temperature T₃ as a cold reservoir. All told it can do work:

$$\begin{array}{cc}{W}_{1\to 2}+W_{2\to 3}& =Q_1{\mathrm{ϵ}}_{1\to 2}+Q_1\left(1-{\mathrm{ϵ}}_{1\to 2}\right){\mathrm{ϵ}}_{2\to 3}\\ & =Q_1\left({\mathrm{ϵ}}_{1\to 2}+{\mathrm{ϵ}}_{2\to 3}-{\mathrm{ϵ}}_{1\to 2}{\mathrm{ϵ}}_{2\to 3}\right).\end{array}$$E3

By contrast, if the heat engine operates in a single step at efficiency ${\mathrm{ϵ}}_{1\to 3}$, employing the reservoir at temperature T₁ as a hot reservoir and the reservoir at temperature T₃ as a cold reservoir, it can do work

$$W_{1\to 3}=Q_1{\mathrm{ϵ}}_{1\to 3}.$$E4

Anticipating that we will eventually deal with n heat reservoirs, let us consider efficiencies of the form

$${\mathrm{ϵ}}_{i\to j}=1-{\left(\frac{T_i}{T_j}\right)}^x,$$E5

where i and j are positive integers in the respective ranges 1 ≤ i ≤ n − 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

$$\begin{array}{cc}{W}_{1\to 2}+W_{2\to 3}& =Q_1\left\{\begin{array}{l}\left[1-{\left(\frac{T_2}{T_1}\right)}^x\right]+\left[1-{\left(\frac{T_3}{T_2}\right)}^x\right]\\ -\left[1-{\left(\frac{T_2}{T_1}\right)}^x\right]\left[1-{\left(\frac{T_3}{T_2}\right)}^x\right]\end{array}\right\}\\ & =Q_1\left\{\begin{array}{c}2-{\left(\frac{T_2}{T_1}\right)}^x-{\left(\frac{T_3}{T_2}\right)}^x\\ -\left[1-{\left(\frac{T_2}{T_1}\right)}^x-{\left(\frac{T_3}{T_2}\right)}^x+{\left(\frac{T_2}{T_1}\right)}^x{\left(\frac{T_3}{T_2}\right)}^x\right]\end{array}\right\}\\ & =Q_1\left[1-{\left(\frac{T_2}{T_1}\right)}^x{\left(\frac{T_3}{T_2}\right)}^x\right]\\ & =Q_1\left[1-{\left(\frac{T_3}{T_1}\right)}^x\right]=W_{1\to 3.}\end{array}$$E6

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], Section 4-9). 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

$$W_{1\to n}=W_{1\to 2}+W_{2\to 3}+\dots +W_{n-1\to n}=\sum _{j=1}^{n-1}{W}_{j\to j+1.}$$E7

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 − 1), W_j→j+k/Q_j (1 ≤ j ≤ n − 1 and 1 ≤ k ≤ n − j), or W_1→n/Q₁, cannot exceed the Carnot limit.

3. Multiple-reservoir heat-engine efficiencies with work output totally frictionally dissipated

Let a heat engine operate between two reservoirs, extracting heat Q₁ from a hot reservoir at temperature T₁ and rejecting waste heat to a cold reservoir at temperature T₂. If its efficiency is ${\mathrm{ϵ}}_{1\to 2}$, its work output is

$$W_{1\to 2}=Q_1{\mathrm{ϵ}}_{1\to 2}.$$E8

It rejects waste heat $Q_1-W_{1\to 2}=Q_1\left(1-{\mathrm{ϵ}}_{1\to 2}\right)$ to a reservoir at temperature T₂. But now, in addition, we let the work output $W_{1\to 2}^D=Q_1{\mathrm{ϵ}}_{1\to 2}$ be totally frictionally dissipated and rejected into the reservoir at temperature T₂ (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 $Q_1-W_{1\to 2}^D=Q_1\left(1-{\mathrm{ϵ}}_{1\to 2}\right)$ has been rejected and the work output $W_{1\to 2}^D=Q_1{\mathrm{ϵ}}_{1\to 2}$ has been totally frictionally dissipated into the reservoir at temperature T₂, and there is a third reservoir at temperature T₃, a heat engine operating at efficiency ${\mathrm{ϵ}}_{2\to 3}$ can then perform additional work

$$W_{2\to 3}=Q_1{\mathrm{ϵ}}_{2\to 3}$$E9

by employing the reservoir at temperature T₂ as a hot reservoir and the reservoir at temperature T₃ as a cold reservoir. ($W_{2\to 3}^D$ may or may not be frictionally dissipated, so it only optionally carries the superscript D.) All told the total work output is

$$\begin{array}{cc}{W}_{1\to 3}^D& =W_{1\to 2}^D+W_{2\to 3}^D=Q_1{\mathrm{ϵ}}_{1\to 2}+Q_1{\mathrm{ϵ}}_{2\to 3}\\ & =Q_1\left({\mathrm{ϵ}}_{1\to 2}+{\mathrm{ϵ}}_{2\to 3}\right).\end{array}$$E10

If ${\mathrm{ϵ}}_{i\to j}=1-{\left(T_i/T_j\right)}^x$, where i and j are positive integers in the respective ranges $1\le i\le n-1$ and $i<j\le n$, and where x is a positive real number in the range $0<x\le 1$, applying Eqs. (5) and (10), we have:

$$\begin{array}{cc}{W}_{1\to 3}^D& =W_{1\to 2}^D+W_{2\to 3}^D=Q_1\left\{\left[1-{\left(\frac{T_2}{T_1}\right)}^x\right]+\left[1-{\left(\frac{T_3}{T_2}\right)}^x\right]\right\}\\ & =Q_1\left[2-{\left(\frac{T_2}{T_1}\right)}^x-{\left(\frac{T_3}{T_2}\right)}^x\right].\end{array}$$E11

We now maximize $W_{1\to 3}^D$ with respect to T₂:

$$\begin{array}{cc}\frac{{\mathit{dW}}_{1\to 3}^D}{{\mathit{dT}}_2}=0& \Rightarrow \frac{d}{{\mathit{dT}}_2}\left[2-{\left(\frac{T_2}{T_1}\right)}^x-{\left(\frac{T_3}{T_2}\right)}^x\right]=0\\ & \Rightarrow \frac{d}{{\mathit{dT}}_2}\left(\frac{T_2}{T_1}+\frac{T_3}{T_2}\right)=0\\ & \Rightarrow \frac{1}{T_1}-\frac{T_3}{T_2^2}=0\\ & \Rightarrow T_{2,\mathrm{opt}}={\left(T_1{T}_3\right)}^{1/2}.\end{array}$$E12

Thus, the optimum value T₂,opt of T₂, which maximizes $W_{1\to 3}^D$, is the geometric mean of $T_1$ and $T_3$. Applying Eqs. (11) and (12), the maximum value $W_{1\to 3,max}^D$ of $W_{1\to 3}^D$ is

$$\begin{array}{cc}{W}_{1\to 3,max}^D& =Q_1\left\{2-{\left[\frac{{\left(T_1{T}_3\right)}^{1/2}}{T_1}\right]}^x-{\left[\frac{T_3}{{\left(T_1{T}_3\right)}^{1/2}}\right]}^x\right\}\\ & =Q_1\left[2-{\left(\frac{T_3}{T_1}\right)}^{x/2}-{\left(\frac{T_3}{T_1}\right)}^{x/2}\right]\\ & =Q_1\left[2-2{\left(\frac{T_3}{T_1}\right)}^{x/2}\right]\\ & =2Q_1\left[1-{\left(\frac{T_3}{T_1}\right)}^{x/2}\right].\end{array}$$E13

Note that

$$W_{1\to 3,max}^D>Q_1\text{if}{\left(\frac{T_3}{T_1}\right)}^{x/2}<\frac{1}{2}\iff \frac{T_3}{T_1}<\frac{1}{2^{2/x}}.$$E14

This obtains if $T_3/T_1<1/4$ for $x=1$ and if $T_3/T_1<1/16$ for $x=1/2$. Also, comparing the last line of Eq. (6) with Eq. (13), we find for the maximum extra work $W_{1\to 3,max}^{D,\text{extra}}$:

$$\begin{array}{cc}{W}_{1\to 3,max}^{D,\text{extra}}& =W_{1\to 3,max}^D-W_{1\to 3}\\ & =2Q_1\left[1-{\left(\frac{T_3}{T_1}\right)}^{x/2}\right]-Q_1\left[1-{\left(\frac{T_3}{T_1}\right)}^x\right]\\ & =Q_1\left\{2\left[1-{\left(\frac{T_3}{T_1}\right)}^{x/2}\right]-\left[1-{\left(\frac{T_3}{T_1}\right)}^x\right]\right\}\\ & =Q_1\left[2-2{\left(\frac{T_3}{T_1}\right)}^{x/2}-1+{\left(\frac{T_3}{T_1}\right)}^x\right]\\ & =Q_1\left[1+{\left(\frac{T_3}{T_1}\right)}^x-2{\left(\frac{T_3}{T_1}\right)}^{x/2}\right]\ge 0.\end{array}$$E15

It is easily shown that $W_{1\to 3,max}^{D,\text{extra}}\ge 0$, with the equality obtaining if and only if $\frac{T_3}{T_1}=1\Rightarrow W_{1\to 3,max}^D=W_{1\to 3}=0\Rightarrow W_{1\to 3,max}^D-W_{1\to 3}=W_{1\to 3,max}^{D,\text{extra}}=0$. For, denoting the ratio ${\left(\frac{T_3}{T_1}\right)}^{x/2}$ as r and setting ${\mathit{dW}}_{1\to 3,max}^{D,\text{extra}}/\mathit{dr}=0$ yields

$$\begin{array}{cc}\frac{{\mathit{dW}}_{1\to 3,max}^{D,\text{extra}}}{\mathit{dr}}=0& \Rightarrow \frac{d}{\mathit{dr}}\left(r^2-2r\right)=0\\ & \Rightarrow 2r-2=0\\ & \Rightarrow r=1.\end{array}$$E16

Thus $W_{1\to 3,max}^D$ is minimized at 0 if $r={\left(\frac{T_3}{T_1}\right)}^{x/2}=1\Rightarrow \frac{T_3}{T_1}=1$. For all $\frac{T_3}{T_1}<1,W_{1\to 3,\text{extra}}^D>0$. Moreover, applying Eqs. (5), (13), and (15), note that

$$\begin{array}{ll}& \underset{T_3/T_1\to 0}{lim}{W}_{1\to 3,max}^D=2Q_1=2\underset{T_3/T_1\to 0}{lim}{W}_{1\to 3}\\ \Rightarrow & \underset{T_3/T_1\to 0}{lim}{W}_{1\to 3,max}^{D,\text{extra}}=2Q_1-Q_1=Q_1=\underset{T_3/T_1\to 0}{lim}{W}_{1\to 3}.\end{array}$$E17

Now consider heat-engine operation employing four heat reservoirs, with all work totally frictionally dissipated (except possibly at the last step; thus, $W_{3\to 4}^D$ only optionally carries the superscript D). Thus we have

$$\begin{array}{cc}{W}_{1\to 4}^D& =W_{1\to 2}^D+W_{2\to 3}^D+W_{3\to 4}^D=Q_1{\mathrm{ϵ}}_{1\to 2}+Q_1{\mathrm{ϵ}}_{2\to 3}+Q_1{\mathrm{ϵ}}_{3\to 4}\\ & =Q_1\left({\mathrm{ϵ}}_{1\to 2}+{\mathrm{ϵ}}_{2\to 3}+{\mathrm{ϵ}}_{3\to 4}\right).\end{array}$$E18

If ${\mathrm{ϵ}}_{i\to j}=1-{\left(T_i/T_j\right)}^x$, where i and j are positive integers in the respective ranges $1\le i\le n-1$ and $i<j\le n$, and where $x$ is a positive real number in the range $0<x\le 1$, applying Eqs. (5) and (18), we have:

$$\begin{array}{cc}{W}_{1\to 4}^D& =W_{1\to 2}^D+W_{2\to 3}^D+W_{3\to 4}^D\\ & =Q_1\left\{\left[1-{\left(\frac{T_2}{T_1}\right)}^x\right]+\left[1-{\left(\frac{T_3}{T_2}\right)}^x\right]+\left[1-{\left(\frac{T_4}{T_3}\right)}^x\right]\right\}\\ & =Q_1\left[3-{\left(\frac{T_2}{T_1}\right)}^x-{\left(\frac{T_3}{T_2}\right)}^x-{\left(\frac{T_4}{T_3}\right)}^x\right].\end{array}$$E19

We wish to maximize $W_{1\to 4}^D$. Based on Eq. (12) and the associated discussions, the optimum value $T_{j,\mathrm{opt}}$ of $T_j$ of reservoir $j \left(1<j<n\iff 2\le j\le n-1\right)$, which maximizes $W_{j-1\to j+1}^D$, is the geometric mean of $T_{j-1}$ and $T_{j+1}$. Thus we have

$$T_{2,\mathrm{opt}}={\left(T_1{T}_{3,\mathrm{opt}}\right)}^{1/2}$$E20

and

$$T_{3,\mathrm{opt}}={\left(T_{2,\mathrm{opt}}{T}_4\right)}^{1/2}.$$E21

Applying Eqs. (20) and (21), we obtain

$$\frac{T_{2,\mathrm{opt}}}{T_1}=\frac{{\left(T_1{T}_{3,\mathrm{opt}}\right)}^{1/2}}{T_1}={\left(\frac{T_{3,\mathrm{opt}}}{T_1}\right)}^{1/2}$$E22

and

$$\frac{T_4}{T_{3,\mathrm{opt}}}=\frac{T_4}{{\left(T_{2,\mathrm{opt}}{T}_4\right)}^{1/2}}={\left(\frac{T_4}{T_{2,\mathrm{opt}}}\right)}^{1/2}.$$E23

Applying Eqs. (20)–(23), we obtain

$$\begin{array}{cc}\frac{T_{3,\mathrm{opt}}}{T_{2,\mathrm{opt}}}& =\frac{T_{3,\mathrm{opt}}}{{\left(T_1{T}_{3,\mathrm{opt}}\right)}^{1/2}}={\left(\frac{T_{3,\mathrm{opt}}}{T_1}\right)}^{1/2}\\ & =\frac{{\left(T_{2,\mathrm{opt}}{T}_4\right)}^{1/2}}{T_{2,\mathrm{opt}}}={\left(\frac{T_4}{T_{2,\mathrm{opt}}}\right)}^{1/2}\\ & \Rightarrow {\left(\frac{T_{3,\mathrm{opt}}}{T_1}\right)}^{1/2}={\left(\frac{T_4}{T_{2,\mathrm{opt}}}\right)}^{1/2}\\ & \Rightarrow \frac{T_{2,\mathrm{opt}}}{T_1}=\frac{T_{3,\mathrm{opt}}}{T_{2,\mathrm{opt}}}=\frac{T_4}{T_{3,\mathrm{opt}}}.\end{array}$$E24

Applying Eqs. (22)–(24), we obtain

$$\begin{array}{cc}\frac{T_4}{T_1}& =\frac{T_2}{T_1}\frac{T_3}{T_2}\frac{T_4}{T_3}\text{in general}\\ & =\frac{T_{2,\mathrm{opt}}}{T_1}\frac{T_{3,\mathrm{opt}}}{T_{2,\mathrm{opt}}}\frac{T_4}{T_{3,\mathrm{opt}}}\text{in particular}\\ & ={\left(\frac{T_{2,\mathrm{opt}}}{T_1}\right)}^3={\left(\frac{T_{3,\mathrm{opt}}}{T_{2,\mathrm{opt}}}\right)}^3={\left(\frac{T_4}{T_{3,\mathrm{opt}}}\right)}^3\\ & \Rightarrow \frac{T_{2,\mathrm{opt}}}{T_1}=\frac{T_{3,\mathrm{opt}}}{T_{2,\mathrm{opt}}}=\frac{T_4}{T_{3,\mathrm{opt}}}={\left(\frac{T_4}{T_1}\right)}^{1/3}.\end{array}$$E25

Applying Eqs. (19) and (25), we obtain

$$\begin{array}{cc}{W}_{1\to 4,max}^D& =Q_1\left[3-3{\left(\frac{T_4}{T_1}\right)}^{x/3}\right]\\ & =3Q_1\left[1-{\left(\frac{T_4}{T_1}\right)}^{x/3}\right].\end{array}$$E26

We now slightly modify Eqs. (14)–(17) to apply for our four-reservoir system. We obtain

$$W_{1\to 4,max}^D>Q_1\text{if}{\left(\frac{T_4}{T_1}\right)}^{x/3}<\frac{2}{3}\iff \frac{T_4}{T_1}<{\left(\frac{2}{3}\right)}^{3/x}.$$E27

This obtains if $T_4/T_1<{\left(2/3\right)}^3=8/27$ for $x=1$ and if $T_4/T_1<{\left(2/3\right)}^6=64/729$ for $x=1/2$. Also, applying Eqs. (5) and (26),

$$\begin{array}{cc}{W}_{1\to 4,max}^{D,\text{extra}}& =W_{1\to 4,max}^D-W_{1\to 4}\\ & =3Q_1\left[1-{\left(\frac{T_4}{T_1}\right)}^{x/3}\right]-Q_1\left[1-{\left(\frac{T_4}{T_1}\right)}^x\right]\\ & =Q_1\left\{3\left[1-{\left(\frac{T_4}{T_1}\right)}^{x/3}\right]-\left[1-{\left(\frac{T_4}{T_1}\right)}^x\right]\right\}\\ & =Q_1\left[3-3{\left(\frac{T_4}{T_1}\right)}^{x/3}-1+{\left(\frac{T_4}{T_1}\right)}^x\right]\\ & =Q_1\left[2+{\left(\frac{T_4}{T_1}\right)}^x-3{\left(\frac{T_4}{T_1}\right)}^{x/3}\right]\ge 0.\end{array}$$E28

It is easily shown that $W_{1\to 4,max}^{D,\text{extra}}\ge 0$, with the equality obtaining if and only if $\frac{T_4}{T_1}=1\Rightarrow W_{1\to 4,max}^D=W_{1\to 4}=0\Rightarrow W_{1\to 4,max}^D-W_{1\to 4}=W_{1\to 4,max}^{D,\text{extra}}=0$. For, denoting the ratio ${\left(\frac{T_4}{T_1}\right)}^{x/3}$ as r and setting ${\mathit{dW}}_{1\to 4,max}^{D,\text{extra}}/\mathit{dr}=0$ yields

$$\begin{array}{cc}\frac{{\mathit{dW}}_{1\to 4,max}^{D,\text{extra}}}{\mathit{dr}}=0& \Rightarrow \frac{d}{\mathit{dr}}\left(r^3-3r\right)=0\\ & \Rightarrow 3r^2-3=0\\ & \Rightarrow r^2=1\\ & \Rightarrow r=1.\end{array}$$E29

Thus $W_{1\to 4,max}^{D,\text{extra}}$ is minimized at 0 if $r={\left(\frac{T_4}{T_1}\right)}^{x/3}=1\Rightarrow \frac{T_4}{T_1}=1$. For all $\frac{T_4}{T_1}<1,W_{1\to 4,max}^{D,\text{extra}}>0$. Moreover, applying Eqs. (5), (26), and (28), note that

$$\begin{array}{ll}& \underset{T_4/T_1\to 0}{lim}{W}_{1\to 4,max}^D=3Q_1=3\underset{T_4/T_1\to 0}{lim}{W}_{1\to 4}\\ \Rightarrow & \underset{T_4/T_1\to 0}{lim}{W}_{1\to 4,max}^{D,\text{extra}}=3Q_1-Q_1=2Q_1=2\underset{T_4/T_1\to 0}{lim}{W}_{1\to 4}.\end{array}$$E30

Comparing Eqs. (13)–(17) with Eqs. (26)–(30), note the larger values in Eqs. (26), (28), and (30) than in Eqs. (13), (15), and (17), respectively, and the easier fulfillment of the inequality in Eq. (27) than in Eq. (14) (concerning the latter point: $8/27>1/4$ and $64/729>1/16$).

Generalizing Eqs. (20)–(30) for an n-reservoir system (n = any positive integer ≥ 4), we obtain:

$$T_{j+1}={\left(T_j{T}_{j+2}\right)}^{1/2},$$E31

where $j$ is any positive integer in the range $1\le j\le n-2$ and

$$T_{j+2}={\left(T_{j+1}{T}_{j+3}\right)}^{1/2},$$E32

where $j$ is any positive integer in the range $1\le j\le n-3$. 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 $T_{n-1}$ 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:

$$\frac{T_{j+1}}{T_j}=\frac{{\left(T_j{T}_{j+2}\right)}^{1/2}}{T_j}={\left(\frac{T_{j+2}}{T_j}\right)}^{1/2}$$E33

and

$$\frac{T_{j+2}}{T_{j+1}}=\frac{T_{j+2}}{{\left(T_j{T}_{j+2}\right)}^{1/2}}={\left(\frac{T_{j+2}}{T_j}\right)}^{1/2}.$$E34

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 $1\le j\le n-2$, we obtain:

$$\begin{array}{ll}& \frac{T_{j+2}}{T_{j+1}}=\frac{T_{j+1}}{T_j}\\ \Rightarrow & \frac{T_{j+2}}{T_j}=\frac{T_{j+1}}{T_j}\frac{T_{j+2}}{T_{j+1}}={\left(\frac{T_{j+1}}{T_j}\right)}^2\iff \frac{T_{j+1}}{T_j}={\left(\frac{T_{j+2}}{T_j}\right)}^{1/2}\\ \iff & \frac{T_n}{T_1}={\left(\frac{T_{j+1}}{T_j}\right)}^{n-1}\iff \frac{T_{j+1}}{T_j}={\left(\frac{T_n}{T_1}\right)}^{1/\left(n-1\right)}.\end{array}$$E35

The first two lines of Eq. (35) obtain for all values of j such that j is any positive integer in the range $1\le j\le n-2$, and the third line of Eq. (35) obtain for all values of $j$ such that $j$ is any positive integer in the range $1\le j\le n-1$. 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 $n-1$ 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. (5), ${\mathrm{ϵ}}_{i\to j}=1-{\left(T_i/T_j\right)}^x$, where $i$ and $j$ are positive integers in the respective ranges $1\le i\le n-1$ and $i<j\le n$, and where $x$ is a positive real number in the range $0<x\le 1$, 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:

$$W_{1\to n,max}^D=\left(n-1\right)Q_1\left[1-{\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\right],$$E36

$$W_{1\to n,max}^D>Q_1\text{if}{\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}<\frac{n-2}{n-1}\iff \frac{T_n}{T_1}<{\left(\frac{n-2}{n-1}\right)}^{\left(n-1\right)/x},$$E37

and

$$\begin{array}{cc}{W}_{1\to n,max}^{D,\text{extra}}& =W_{1\to n,max}^D-W_{1\to n}\\ & =\left(n-1\right)Q_1\left[1-{\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\right]-Q_1\left[1-{\left(\frac{T_n}{T_1}\right)}^x\right]\\ & =Q_1\left\{\left(n-1\right)\left[1-{\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\right]-\left[1-{\left(\frac{T_n}{T_1}\right)}^x\right]\right\}\\ & =Q_1\left[n-1-\left(n-1\right){\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}-1+{\left(\frac{T_n}{T_1}\right)}^x\right]\\ & =Q_1\left[n-2+{\left(\frac{T_n}{T_1}\right)}^x-\left(n-1\right){\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\right]\ge 0.\end{array}$$E38

It is easily shown that $W_{1\to n,max}^{D,\text{extra}}\ge 0$, with the equality obtaining if and only if $\frac{T_n}{T_1}=1\Rightarrow W_{1\to n,max}^D=W_{1\to n}=0\Rightarrow W_{1\to n,max}^D-W_{1\to n}=W_{1\to n,max}^{D,\text{extra}}=0$. For, denoting the ratio ${\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}$ as r and setting ${\mathit{dW}}_{1\to n,max}^{D,\text{extra}}/\mathit{dr}=0$ yields

$$\begin{array}{cc}\frac{{\mathit{dW}}_{1\to n,max}^{D,\text{extra}}}{\mathit{dr}}=0& \Rightarrow \frac{d}{\mathit{dr}}\left[r^{n-1}-\left(n-1\right)r\right]=0\\ & \Rightarrow \left(n-1\right)r^{n-2}-\left(n-1\right)=0\\ & \Rightarrow r^{n-2}=1\\ & \Rightarrow r=1.\end{array}$$E39

Thus $W_{1\to n,max}^{D,\text{extra}}$ is minimized at 0 if $r={\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}=1\Rightarrow \frac{T_n}{T_1}=1$. For all $\frac{T_n}{T_1}<1,W_{1\to n,max}^{D,\text{extra}}>0$. Moreover, applying Eqs. (5), (36), and (38), note that

$$\begin{array}{ll}& \underset{T_n/T_1\to 0,n\text{fixed}}{lim}{W}_{1\to n,max}^D=\left(n-1\right)Q_1=\left(n-1\right) \underset{T_n/T_1\to 0,n\text{fixed}}{lim}{W}_{1\to n}\\ \Rightarrow & \underset{T_n/T_1\to 0,n\text{fixed}}{lim} W_{1\to n,max}^{D,\text{extra}}=\underset{T_n/T_1\to 0,n\text{fixed}}{lim} \left(W_{1\to n,max}^D-W_{1\to n}\right)\\ =& \left(n-1\right)Q_1-Q_1=\left(n-2\right)Q_1=\left(n-2\right)\underset{T_n/T_1\to 0,n\text{fixed}}{lim}{W}_{1\to n}.\end{array}$$E40

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 ($x=1$) but even for Curzon-Ahlborn efficiency ($x=1/2$), indeed for any $x$ finitely greater than $0$ in the range $0<x\le 1$, because ${\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\to 0\iff 1-{\left(\frac{T_n}{T_1}\right)}^{x/\left(n-1\right)}\to 1$ in the limit $T_n/T_1\to 0$, albeit ever more slowly with decreasing $x$.

By contrast, even granting Carnot efficiency ($x=1$) [22]:

$$\underset{n\to \infty ,T_n/T_1\text{fixed}}{lim}{W}_{1\to n,max}^D=Q_{1}ln\frac{T_1}{T_n}=\left(\underset{T_n/T_1\to 0,n\text{fixed}}{lim}{W}_{1\to n}\right)ln\frac{T_1}{T_n}.$$E41

Note the linear divergence of $W_{1\to n,max}^D$ in the limit $T_n/T_1\to 0$ with $n$ fixed as per Eq. (40) even not assuming Carnot efficiency, as contrasted with the paltry logarithmic divergence of $W_{1\to n,max}^D$ in the limit $n\to \infty$ 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 $2.7\mathrm{K}$, while the most refractory materials remain solid at temperatures slightly exceeding $2700\mathrm{K}$. This provides a temperature ratio of $T_1/T_n\approx {10}^3\iff T_n/T_1\approx {10}^{-3}$. 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 $W_{1\to n,max}^D$ in the limit $n\to \infty$ with $T_1/T_n$ fixed as per Eq. (41) and even with a temperature ratio of $T_1/T_n\approx {10}^3\iff T_n/T_1\approx {10}^{-3}$, assuming Carnot efficiency by Eq. (41)$W_{1\to n,max}^D/Q_1\approx ln{10}^3\approx 7$. 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 $n\to \infty$ 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 $T_1/T_n\approx {10}^3\iff T_n/T_1\approx {10}^{-3}$, $W_{1\to n,max}^D/Q_1\approx 4$. 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.

4. General remarks, especially concerning entropy

It is important to emphasize that the super-unity cyclic-heat-engine efficiencies $W_{1\to n,max}^D/Q_1$ that can obtain with work output totally frictionally dissipated $\left(\text{if}n\ge 3\right)$ 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.

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 $j\to j+1$ or $j\to j+k$ ($1\le k\le n-j$). The validity of this Section 4 requires only that the efficiency with all work sequestered, or at any one given step $j\to j+1$ 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 $j+2$ at temperature $T_{j+2}$ without versus with frictional dissipation into reservoir $j+1$ at temperature $T_{j+1}\left(T_j>T_{j+1}>T_{j+2}\right)$. Without frictional dissipation a heat engine performs work

$$W_{j\to j+1}=Q_j{\mathrm{ϵ}}_{j\to j+1}$$E42

by employing the reservoir at temperature $T_j$ as a hot reservoir and the reservoir at temperature $T_{j+1}$ as a cold reservoir. It rejects waste heat $Q_j-W_{j\to j+1}=Q_j\left(1-{\mathrm{ϵ}}_{j\to j+1}\right)$ to the reservoir at temperature $T_{j+1}$. If a third reservoir at temperature $T_{j+2}$ and $W_{j\to j+1}$ is sequestered, that is, not frictionally dissipated, a heat engine can then perform additional work:

$$W_{j+1\to j+2}=Q_j\left(1-{\mathrm{ϵ}}_{j\to j+1}\right){\mathrm{ϵ}}_{j+1\to j+2}$$E43

by employing the reservoir at temperature $T_{j+1}$ as a hot reservoir and the reservoir at temperature $T_{j+2}$ as a cold reservoir. All told it can do work:

$$\begin{array}{cc}{W}_{j\to j+2}& =W_{j\to j+1}+W_{j+1\to j+2}=Q_j{\mathrm{ϵ}}_{j\to j+1}+Q_j\left(1-{\mathrm{ϵ}}_{j\to j+1}\right){\mathrm{ϵ}}_{j+1\to j+2}\\ & =Q_j\left({\mathrm{ϵ}}_{j\to j+1}+{\mathrm{ϵ}}_{j+1\to j+2}-{\mathrm{ϵ}}_{j\to j+1}{\mathrm{ϵ}}_{j+1\to j+2}\right).\end{array}$$E44

With total frictional dissipation of $W_{j\to j+1}$ into reservoir $j+1$ at temperature $T_{j+1}$, we still have

$$W_{j\to j+1}^D=W_{j\to j+1}=Q_1{\mathrm{ϵ}}_{j\to j+1}.$$E45

But now we let the work output $W_{j\to j=1}^D=Q_1{\mathrm{ϵ}}_{j+1\to j+2}$ be totally frictionally dissipated into the reservoir at temperature $T_{j+1}$ (indicated via a superscript $D$). If there is a third reservoir at temperature $T_{j+2}$, a heat engine can then perform additional work:

$$W_{j+1\to j+2}^D=Q_1{\mathrm{ϵ}}_{j+1\to j+2}.$$E46

All told it can do work:

$$\begin{array}{cc}{W}_{j\to j+2}^D& =W_{j\to j+1}^D+W_{j+1\to j+2}^D=Q_j{\mathrm{ϵ}}_{j\to j+1}+Q_j{\mathrm{ϵ}}_{j+1\to j+2}\\ & =Q_j\left({\mathrm{ϵ}}_{j\to j+1}+{\mathrm{ϵ}}_{j+1\to j+2}\right).\end{array}$$E47

The extra work

$$\begin{array}{cc}{W}_{\text{extra}}^D& =W_{j+1\to j+2}^D\\ & =Q_j{\mathrm{ϵ}}_{j\to j+1}{\mathrm{ϵ}}_{j+1\to j+2}\\ & =W_{j\to j+1}{\mathrm{ϵ}}_{j+1\to j+2}\\ & =W_{j\to j+1}^D{\mathrm{ϵ}}_{j+1\to j+2}\end{array}$$E48

is paid for by the extra increase in entropy owing to frictional dissipation into extra heat $Q_{\mathrm{extra}}^D$ of the work output as per Eqs. (42) and (45)

$$Q_{\mathrm{\text{extra}}}^D=W_{j\to j+1}=W_{j\to j+1}^D=Q_j{\mathrm{ϵ}}_{j\to j+1}$$E49

into reservoir $j+1$ at temperature $T_{j+1}$. This extra increase in entropy is

$$\Delta S_{\mathrm{\text{extra}}}^D=\frac{Q_{\mathrm{\text{extra}}}^D}{T_{j+1}}=\frac{Q_j{\mathrm{ϵ}}_{j\to j+1}}{T_{j+1}}=\frac{W_{j\to j+1}}{T_{j+1}}=\frac{W_{j\to j+1}^D}{T_{j+1}}=\frac{W_{\mathrm{\text{extra}}}^D}{{\mathrm{ϵ}}_{j+1\to j+2}{T}_{j+1}}.$$E50

[In the last four steps of Eq. (50), we applied Eqs. (42), (45), (48), and (49).] Thus

$$W_{\mathrm{\text{extra}}}^D=T_{j+1}\Delta S_{\mathrm{\text{extra}}}^D{\mathrm{ϵ}}_{j+1\to j+2}.$$E51

In no case do we assume an efficiency with all work sequestered, or at any one given step j → j + 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 $W_{\mathrm{\text{extra}}}^D$ in heat-engine operation (albeit at the expense of $\Delta S_{\mathrm{\text{extra}}}^D$), 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 reservoir $j+1$ at temperature $T_{j+1},Q_{j+2}+W_{j+2\to j+1}=Q_{j+1},Q_{j+1}+W_{j+1\to j}=Q_j$, hence $Q_{j+2}+W_{j+2\to j+1}+W_{j+1\to j}=Q_{j+2}+W_{j+2\to j}=Q_j$. Without an intermediate reservoir $j+1$ at temperature $T_{j+1},Q_{j+2}+W_{j+2\to j}=Q_j$. The bottom line $Q_{j+2}+W_{j+2\to j}=Q_j$ is identical with or without an intermediate reservoir $j+1$ at temperature $T_{j+1}$. With or without the intermediate reservoir $j+1$ at temperature $T_{j+1}$, all of the energy must end up as $Q_j$; thus, there is none left over to be frictionally dissipated. Hence the presence or absence of this intermediate reservoir makes no difference with respect to reverse, that is, refrigerator or heat pump, operation: See Ref. [1], Section 20-3; Ref. [2], Section 5.12 and Problem 5.22; Ref. [3], Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. [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.]

5. 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 T₁ 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 + 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.

Acknowledgments

I am very grateful to Dr. Donald H. Kobe, Dr. Paolo Grigolini, Dr. Daniel P. Sheehan, Dr. Bruce N. Miller, and Dr. Marlan O. Scully and for many very helpful and thoughtful insights, as well as for very perceptive and valuable discussions and communications, which greatly helped my understanding of thermodynamics and statistical mechanics. Also, I am indebted to them, as well as to Dr. Bright Lowry, Dr. John Banewicz, Dr. Bruno J. Zwolinski, Dr. Roland E. Allen, Dr. Abraham Clearfield, Dr. Russell Larsen, Dr. James H. Cooke, Dr. Wolfgang Rindler, Dr. Richard McFee, Dr. Nolan Massey, and Dr. Stan Czamanski for lectures, discussions, and/or communications from which I learned very much concerning thermodynamics and statistical mechanics. I thank Dr. Stan Czamanski and Dr. S. Mort Zimmerman for the very interesting general scientific discussions over many years. I also thank Dan Zimmerman, Dr. Kurt W. Hess, and Robert H. Shelton for the very interesting general scientific discussions at times. Additionally, I thank Robert H. Shelton for very helpful advice concerning diction.

Conflict of interest

The author declares no conflict of interest.