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

We designate the temperatures of the heat reservoirs via subscripts, with *T*<sup>1</sup> being the temperature of the initial, hottest, reservoir, *T*<sup>2</sup> the temperature of the second-hottest reservoir,*T*<sup>3</sup> the temperature of the third-hottest reservoir, etc., and *Tn* the temperature of the *n*th, coldest, reservoir.

Let a heat engine operate between two reservoirs, extracting heat *Q*<sup>1</sup> from a hot reservoir at temperature *T*<sup>1</sup> and rejecting waste heat to a cold reservoir at temperature *T*2. If its efficiency is ϵ<sup>1</sup>!2, its work output is

$$W\_{1 \to 2} = Q\_{\mathbf{1}} \epsilon\_{\mathbf{1} \to \mathbf{2}}.\tag{1}$$

It rejects waste heat *Q*<sup>1</sup> � *W*<sup>1</sup>!<sup>2</sup> ¼ *Q*1ð Þ 1 � ϵ<sup>1</sup>!<sup>2</sup> to the reservoir at temperature *T*2. If there is a third reservoir at temperature *T*<sup>3</sup> and *W*<sup>1</sup>!<sup>2</sup> is sequestered, that is, not frictionally dissipated, and if the efficiency of heat-engine operation between the second and third reservoirs is ϵ<sup>2</sup>!3, a heat engine can then perform additional work

$$W\_{2\to 3} = Q\_1(1 - e\_{1\to 2})e\_{2\to 3} \tag{2}$$

by employing the reservoir at temperature *T*<sup>2</sup> as a hot reservoir and the reservoir at temperature *T*<sup>3</sup> as a cold reservoir. All told it can do work:

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs DOI: http://dx.doi.org/10.5772/intechopen.89047*

$$\begin{split} W\_{1\to 2} + W\_{2\to 3} &= Q\_1 \mathfrak{e}\_{1\to 2} + Q\_1 (\mathfrak{1} - \mathfrak{e}\_{1\to 2}) \mathfrak{e}\_{2\to 3} \\ &= Q\_1 (\mathfrak{e}\_{1\to 2} + \mathfrak{e}\_{2\to 3} - \mathfrak{e}\_{1\to 2} \mathfrak{e}\_{2\to 3}). \end{split} \tag{3}$$

By contrast, if the heat engine operates in a single step at efficiency ϵ1!3, employing the reservoir at temperature *T*<sup>1</sup> as a hot reservoir and the reservoir at temperature *T*<sup>3</sup> as a cold reservoir, it can do work

$$W\_{1\to 3} = Q\_1 \epsilon\_{1\to 3}.\tag{4}$$

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

$$\mathbf{e}\_{i \to j} = \mathbf{1} - \left(\frac{T\_i}{T\_j}\right)^{\mathbf{x}},\tag{5}$$

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!<sup>3</sup> = *W*1!<sup>2</sup> + *W*2!3, as we will now show. We have

$$\begin{split} W\_{1\to 2} + W\_{2\to 3} &= Q\_1 \left\{ \begin{bmatrix} \mathbf{1} - \left(\frac{T\_2}{T\_1}\right)^X \end{bmatrix} + \left[\mathbf{1} - \left(\frac{T\_3}{T\_2}\right)^X\right] \\\ -\left[\mathbf{1} - \left(\frac{T\_2}{T\_1}\right)^X\right] \left[\mathbf{1} - \left(\frac{T\_3}{T\_2}\right)^X\right] \end{bmatrix} \\\ &= Q\_1 \left\{ \begin{array}{l} 2 - \left(\frac{T\_2}{T\_1}\right)^X - \left(\frac{T\_3}{T\_2}\right)^X \\\ -\left[\mathbf{1} - \left(\frac{T\_2}{T\_1}\right)^X - \left(\frac{T\_3}{T\_2}\right)^X + \left(\frac{T\_2}{T\_1}\right)^X \left(\frac{T\_3}{T\_2}\right)^X\right] \end{array} \right\} \\\ &= Q\_1 \left[ \mathbf{1} - \left(\frac{T\_2}{T\_1}\right)^X \left(\frac{T\_3}{T\_2}\right)^X \right] \\\ &= Q\_1 \left[ \mathbf{1} - \left(\frac{T\_3}{T\_1}\right)^X \right] = W\_{1 \to 3} \end{split} \tag{6}$$

We note that *x* = 1 for the Carnot, Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [1–9] and *x* = 1/2 for endoreversible heat engines operating at Curzon-Ahlborn efficiency [10–12] (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 [1–12]. 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 [1–12]. 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) [2–9] 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!<sup>3</sup> = *W*1!<sup>2</sup> + *W*2!3, this additivity of *W* obtains for any number of steps, that is, we have

$$W\_{1\to n} = W\_{1\to 2} + W\_{2\to 3} + \dots + W\_{n-1\to n} = \sum\_{j=1}^{n-1} W\_{j\to j+1} \tag{7}$$

we employ the standard Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9) for cyclic heat engines operating at maximum power output.

and Second Laws of Thermodynamics.

*Thermodynamics and Energy Engineering*

with respect to noncyclic heat engines.

**sequestered**

**130**

Concluding remarks are provided in Section 5.

*Tn* the temperature of the *n*th, coldest, reservoir.

ture *T*2. If its efficiency is ϵ<sup>1</sup>!2, its work output is

at temperature *T*<sup>3</sup> as a cold reservoir. All told it can do work:

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

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 [22–37], which is a *different* process that has been thoroughly investigated and discussed previously [22–37], and which we further investigate in another chapter [38] in this book.

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

General remarks, especially concerning entropy, are provided in Section 4.

We designate the temperatures of the heat reservoirs via subscripts, with *T*<sup>1</sup> being the temperature of the initial, hottest, reservoir, *T*<sup>2</sup> the temperature of the second-hottest reservoir,*T*<sup>3</sup> the temperature of the third-hottest reservoir, etc., and

Let a heat engine operate between two reservoirs, extracting heat *Q*<sup>1</sup> from a hot reservoir at temperature *T*<sup>1</sup> and rejecting waste heat to a cold reservoir at tempera-

It rejects waste heat *Q*<sup>1</sup> � *W*<sup>1</sup>!<sup>2</sup> ¼ *Q*1ð Þ 1 � ϵ<sup>1</sup>!<sup>2</sup> to the reservoir at temperature *T*2. If there is a third reservoir at temperature *T*<sup>3</sup> and *W*<sup>1</sup>!<sup>2</sup> is sequestered, that is, not frictionally dissipated, and if the efficiency of heat-engine operation between the second and third reservoirs is ϵ<sup>2</sup>!3, a heat engine can then perform additional work

by employing the reservoir at temperature *T*<sup>2</sup> as a hot reservoir and the reservoir

*W*<sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup>*:* (1)

*W*<sup>2</sup>!<sup>3</sup> ¼ *Q*1ð Þ 1 � ϵ<sup>1</sup>!<sup>2</sup> ϵ<sup>2</sup>!<sup>3</sup> (2)

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

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 [3–9, 13–19]. 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 *Wj*!*<sup>j</sup>*+1/*Q <sup>j</sup>* (1 ≤ *j* ≤ *n* � 1), *Wj*!*j*+*k/Q <sup>j</sup>* (1 ≤ *j* ≤ *n* � 1 and 1 ≤ *k* ≤ *n* � *j*), or *W*1!*n*/*Q*1, cannot exceed the Carnot limit.
