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

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

$$W\_{1 \to 2} = Q\_1 \epsilon\_{1 \to 2}.\tag{8}$$

*W<sup>D</sup>*

*W<sup>D</sup>* <sup>1</sup>!<sup>3</sup> is

Note that

work *<sup>W</sup>D,*extra

**133**

<sup>1</sup>!<sup>3</sup>*,*max :

<sup>1</sup>!<sup>3</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

We now maximize *W<sup>D</sup>*

<sup>1</sup>!<sup>2</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

*T*1 � �*<sup>x</sup>*

� � � �*<sup>x</sup>*

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

¼ 0 )

) *d dT*<sup>2</sup>

) 1 *T*1 � *<sup>T</sup>*<sup>3</sup> *T*2 2 ¼ 0

Thus, the optimum value *T*2,opt of *T*2, which maximizes *W<sup>D</sup>*

<sup>1</sup>!<sup>3</sup>*,*max <sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>2</sup> � ð Þ *<sup>T</sup>*1*T*<sup>3</sup> <sup>1</sup>*=*<sup>2</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>2</sup> � *<sup>T</sup>*<sup>3</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>2</sup> � <sup>2</sup> *<sup>T</sup>*<sup>3</sup>

<sup>¼</sup> <sup>2</sup>*Q*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

<sup>1</sup>!3*,*max <sup>&</sup>gt; *<sup>Q</sup>*<sup>1</sup> if *<sup>T</sup>*<sup>3</sup>

mean of *T*<sup>1</sup> and *T*3. Applying Eqs. (11) and (12), the maximum value *W<sup>D</sup>*

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>2</sup> � *<sup>T</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

*dW<sup>D</sup>* 1!3 *dT*<sup>2</sup>

*W<sup>D</sup>*

*W<sup>D</sup>*

*<sup>W</sup>D,*extra

<sup>1</sup>!<sup>3</sup>*,*max <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

<sup>2</sup>!<sup>3</sup> <sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>2</sup>

� *<sup>T</sup>*<sup>3</sup> *T*2

<sup>1</sup>!<sup>3</sup> with respect to *T*2:

*d dT*<sup>2</sup>

*T*1 � � � �*<sup>x</sup>*

*:*

<sup>2</sup> � *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup>*

*T*2 *T*1 þ *T*3 *T*2 � �

) *<sup>T</sup>*2*,*opt <sup>¼</sup> ð Þ *<sup>T</sup>*1*T*<sup>3</sup> <sup>1</sup>*=*<sup>2</sup>

*T*1 " #*<sup>x</sup>*

� �*<sup>x</sup>=*<sup>2</sup> " #

*T*1 � �*<sup>x</sup>=*<sup>2</sup>

*T*1 � �*<sup>x</sup>=*<sup>2</sup> " #

*T*1 � �*<sup>x</sup>=*<sup>2</sup> " #*:*

> *T*1 � �*<sup>x</sup>=*<sup>2</sup>

This obtains if *T*3*=T*<sup>1</sup> <1*=*4 for *x* ¼ 1 and if *T*3*=T*<sup>1</sup> < 1*=*16 for *x* ¼ 1*=*2. Also, comparing the last line of Eq. (6) with Eq. (13), we find for the maximum extra

<sup>1</sup>!<sup>3</sup>*,*max � *W*<sup>1</sup>!<sup>3</sup>

*T*1 � �*<sup>x</sup>=*<sup>2</sup> " #

> *T*1 � �*<sup>x</sup>=*<sup>2</sup>

*T*3 *T*1 � �*<sup>x</sup>*

*T*1 � �*<sup>x</sup>=*<sup>2</sup> " #

<sup>¼</sup> <sup>2</sup>*Q*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> 2 1 � *<sup>T</sup>*<sup>3</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>2</sup> � <sup>2</sup> *<sup>T</sup>*<sup>3</sup>

¼ *Q*<sup>1</sup> 1 þ

<sup>þ</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

� � � � � �*<sup>x</sup>*

� *<sup>T</sup>*<sup>3</sup> *T*2

¼ 0

*:*

( ) " #*<sup>x</sup>*

� *<sup>T</sup>*<sup>3</sup> *T*1

> < 1 <sup>2</sup> <sup>⇔</sup> *T*3 *T*1 < 1

� *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

� <sup>1</sup> � *<sup>T</sup>*<sup>3</sup> *T*1

> *T*3 *T*1

� � � �*<sup>x</sup>* ( )

� 1 þ

� �*<sup>x</sup>* " #

� 2 *T*3 *T*1 � �*<sup>x</sup>=*<sup>2</sup> " #≥0*:*

*T*1 � � � �*<sup>x</sup>*

� *<sup>T</sup>*<sup>3</sup> ð Þ *<sup>T</sup>*1*T*<sup>3</sup> <sup>1</sup>*=*<sup>2</sup>

� � � �*<sup>x</sup>*

*T*2

¼ 0

(11)

(12)

<sup>1</sup>!3, is the geometric

<sup>2</sup>2*=<sup>x</sup> :* (14)

<sup>1</sup>!<sup>3</sup>*,*max of

(13)

(15)

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 a reservoir at temperature *T*2. But now, in addition, we let the work output *W<sup>D</sup>* <sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> be totally frictionally dissipated and rejected into the reservoir at temperature *T*<sup>2</sup> (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 *<sup>Q</sup>*<sup>1</sup> � *<sup>W</sup><sup>D</sup>* <sup>1</sup>!<sup>2</sup> ¼ *Q*1ð Þ 1 � ϵ<sup>1</sup>!<sup>2</sup> has been rejected and the work output *W<sup>D</sup>* <sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> 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<sup>2</sup>!<sup>3</sup> can then perform additional work

$$W\_{2\to 3} = Q\_1 \epsilon\_{2\to 3} \tag{9}$$

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. (*W<sup>D</sup>* <sup>2</sup>!<sup>3</sup> may or may not be frictionally dissipated, so it only optionally carries the superscript *D*.) All told the total work output is

$$\begin{split} \boldsymbol{W}\_{1\rightarrow 3}^{D} &= \boldsymbol{W}\_{1\rightarrow 2}^{D} + \boldsymbol{W}\_{2\rightarrow 3}^{D} = \boldsymbol{Q}\_{1}\boldsymbol{\epsilon}\_{1\rightarrow 2} + \boldsymbol{Q}\_{1}\boldsymbol{\epsilon}\_{2\rightarrow 3} \\ &= \boldsymbol{Q}\_{1}(\boldsymbol{\epsilon}\_{1\rightarrow 2} + \boldsymbol{\epsilon}\_{2\rightarrow 3}). \end{split} \tag{10}$$

If ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj <sup>x</sup>* , 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. (5) and (10), we have:

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

$$\begin{split} W\_{1 \to 3}^{D} &= W\_{1 \to 2}^{D} + W\_{2 \to 3}^{D} = Q\_{1} \left\{ \left[ \mathbf{1} - \left( \frac{T\_{2}}{T\_{1}} \right)^{x} \right] + \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{2}} \right)^{x} \right] \right\} \\ &= Q\_{1} \left[ 2 - \left( \frac{T\_{2}}{T\_{1}} \right)^{x} - \left( \frac{T\_{3}}{T\_{2}} \right)^{x} \right]. \end{split} \tag{11}$$

We now maximize *W<sup>D</sup>* <sup>1</sup>!<sup>3</sup> with respect to *T*2:

$$\begin{split} \frac{d\boldsymbol{W}\_{1\rightarrow 3}^{D}}{dT\_{2}} &= \mathbf{0} \Rightarrow \frac{d}{dT\_{2}} \left[ 2 - \left(\frac{T\_{2}}{T\_{1}}\right)^{x} - \left(\frac{T\_{3}}{T\_{2}}\right)^{x} \right] = \mathbf{0} \\ &\Rightarrow \frac{d}{dT\_{2}} \left(\frac{T\_{2}}{T\_{1}} + \frac{T\_{3}}{T\_{2}}\right) = \mathbf{0} \\ &\Rightarrow \frac{\mathbf{1}}{T\_{1}} - \frac{T\_{3}}{T\_{2}^{2}} = \mathbf{0} \\ &\Rightarrow T\_{2,\text{opt}} = (T\_{1}T\_{3})^{1/2}. \end{split} \tag{12}$$

Thus, the optimum value *T*2,opt of *T*2, which maximizes *W<sup>D</sup>* <sup>1</sup>!3, is the geometric mean of *T*<sup>1</sup> and *T*3. Applying Eqs. (11) and (12), the maximum value *W<sup>D</sup>* <sup>1</sup>!<sup>3</sup>*,*max of *W<sup>D</sup>* <sup>1</sup>!<sup>3</sup> is

$$\begin{split} W\_{1 \rightarrow 3, \text{max}}^{D} &= Q\_1 \left\{ 2 - \left[ \frac{(T\_1 T\_3)^{1/2}}{T\_1} \right]^x - \left[ \frac{T\_3}{(T\_1 T\_3)^{1/2}} \right]^x \right\} \\ &= Q\_1 \left[ 2 - \left( \frac{T\_3}{T\_1} \right)^{\ge 1} - \left( \frac{T\_3}{T\_1} \right)^{\ge 1/2} \right] \\ &= Q\_1 \left[ 2 - 2 \left( \frac{T\_3}{T\_1} \right)^{\ge 1} \right] \\ &= 2Q\_1 \left[ 1 - \left( \frac{T\_3}{T\_1} \right)^{\ge 1/2} \right] . \end{split} \tag{13}$$

Note that

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*),

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

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 a reservoir at temperature *T*2.

dissipated and rejected into the reservoir at temperature *T*<sup>2</sup> (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

*W*<sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup>*:* (8)

<sup>1</sup>!<sup>2</sup> ¼ *Q*1ð Þ 1 � ϵ<sup>1</sup>!<sup>2</sup> has been rejected and the

*W*<sup>2</sup>!<sup>3</sup> ¼ *Q*1ϵ<sup>2</sup>!<sup>3</sup> (9)

<sup>2</sup>!<sup>3</sup> may or may not be friction-

<sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> has been totally frictionally dissipated into the reser-

<sup>2</sup>!<sup>3</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> þ *Q*1ϵ<sup>2</sup>!<sup>3</sup> <sup>¼</sup> *<sup>Q</sup>*1ð Þ <sup>ϵ</sup><sup>1</sup>!<sup>2</sup> <sup>þ</sup> <sup>ϵ</sup><sup>2</sup>!<sup>3</sup> *:* (10)

, where *i* and *j* are positive integers in the respective ranges

<sup>1</sup>!<sup>2</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> be totally frictionally

or *W*1!*n*/*Q*1, cannot exceed the Carnot limit.

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

But now, in addition, we let the work output *W<sup>D</sup>*

If both the waste heat *<sup>Q</sup>*<sup>1</sup> � *<sup>W</sup><sup>D</sup>*

work output *W<sup>D</sup>*

work output is

**132**

If ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj*

decelerates, or when the factory or appliance engine is turned off.

reservoir at temperature *T*<sup>3</sup> as a cold reservoir. (*W<sup>D</sup>*

<sup>1</sup>!<sup>3</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

*W<sup>D</sup>*

*<sup>x</sup>*

0<*x*≤ 1, applying Eqs. (5) and (10), we have:

voir at temperature *T*2, and there is a third reservoir at temperature *T*3, a heat

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

ally dissipated, so it only optionally carries the superscript *D*.) All told the total

<sup>1</sup>!<sup>2</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

1≤ *i*≤ *n* � 1 and *i*<*j*≤ *n*, and where *x* is a positive real number in the range

engine operating at efficiency ϵ<sup>2</sup>!<sup>3</sup> can then perform additional work

**frictionally dissipated**

*Thermodynamics and Energy Engineering*

$$\mathbf{W}\_{1\rightarrow 3\text{-max}}^{D} > \mathbf{Q}\_1 \text{ if } \left(\frac{T\_3}{T\_1}\right)^{\mathbf{x}/2} < \frac{1}{2} \Leftrightarrow \frac{T\_3}{T\_1} < \frac{1}{2^{2/\mathbf{x}}}.\tag{14}$$

This obtains if *T*3*=T*<sup>1</sup> <1*=*4 for *x* ¼ 1 and if *T*3*=T*<sup>1</sup> < 1*=*16 for *x* ¼ 1*=*2. Also, comparing the last line of Eq. (6) with Eq. (13), we find for the maximum extra work *<sup>W</sup>D,*extra <sup>1</sup>!<sup>3</sup>*,*max :

$$\begin{split} \boldsymbol{W}\_{1\rightarrow 3\text{max}}^{\text{Dexfra}} &= \boldsymbol{W}\_{1\rightarrow 3\text{max}}^{\text{Dexfra}} - \boldsymbol{W}\_{1\rightarrow 3} \\ &= 2Q\_{1} \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x/2}} \right] - Q\_{1} \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x}} \right] \\ &= Q\_{1} \left\{ 2 \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x/2}} \right] - \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x}} \right] \right\} \\ &= Q\_{1} \left[ 2 - 2 \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x/2}} - \mathbf{1} + \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x}} \right] \\ &= Q\_{1} \left[ \mathbf{1} + \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x}} - 2 \left( \frac{T\_{3}}{T\_{1}} \right)^{\text{x}/2} \right] \ge \mathbf{0}. \end{split} \tag{15}$$

**133**

It is easily shown that *<sup>W</sup>D,*extra <sup>1</sup>!3*,*max ≥ 0, with the equality obtaining if and only if *<sup>T</sup>*<sup>3</sup> *<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!3*,*max <sup>¼</sup> *<sup>W</sup>*1!<sup>3</sup> <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!3*,*max � *<sup>W</sup>*1!<sup>3</sup> <sup>¼</sup> *<sup>W</sup>D,*extra <sup>1</sup>!3*,*max ¼ 0. For, denoting the ratio *<sup>T</sup>*<sup>3</sup> *T*1 *<sup>x</sup>=*<sup>2</sup> as *<sup>r</sup>* and setting *dWD,*extra <sup>1</sup>!3*,*max *<sup>=</sup>dr* <sup>¼</sup> 0 yields

$$\frac{dW\_{1-3,\text{max}}^{D,\text{exra}}}{dr} = 0 \Rightarrow \frac{d}{dr}(r^2 - 2r) = 0$$

$$\Rightarrow 2r - 2 = 0$$

$$\Rightarrow r = 1.$$

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

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

and

*T*2*,*opt *T*1

*T*<sup>4</sup> *T*3*,*opt

*T*3*,*opt *T*2*,*opt

Applying Eqs. (20)–(23), we obtain

Applying Eqs. (22)–(24), we obtain

*T*<sup>4</sup> *T*1 <sup>¼</sup> *<sup>T</sup>*<sup>2</sup> *T*1 *T*3 *T*2 *T*<sup>4</sup> *T*3

<sup>¼</sup> *<sup>T</sup>*2*,*opt *T*1

<sup>¼</sup> *<sup>T</sup>*2*,*opt *T*1 � �<sup>3</sup>

> *T*2*,*opt *T*1

)

*W<sup>D</sup>*

<sup>1</sup>!<sup>4</sup>*,*max <sup>&</sup>gt; *<sup>Q</sup>*<sup>1</sup> if *<sup>T</sup>*<sup>4</sup>

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

*W<sup>D</sup>*

for *x* ¼ 1*=*2. Also, applying Eqs. (5) and (26),

We obtain

**135**

<sup>¼</sup> *<sup>T</sup>*1*T*3*,*opt � �1*=*<sup>2</sup> *T*1

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

<sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *T*2*,*opt*T*<sup>4</sup>

<sup>¼</sup> *<sup>T</sup>*3*,*opt

<sup>¼</sup> *<sup>T</sup>*2*,*opt*T*<sup>4</sup> � �<sup>1</sup>*=*<sup>2</sup> *T*2*,*opt

> *T*2*,*opt *T*1

*T*3*,*opt *T*2*,*opt

> <sup>¼</sup> *<sup>T</sup>*3*,*opt *T*2*,*opt

<sup>1</sup>!<sup>4</sup>*,*max ¼ *Q*<sup>1</sup> 3 � 3

*T*3*,*opt *T*1 � �<sup>1</sup>*=*<sup>2</sup>

)

)

� �1*=*<sup>2</sup> <sup>¼</sup> *<sup>T</sup>*<sup>4</sup>

*<sup>T</sup>*1*T*3*,*opt � �1*=*<sup>2</sup> <sup>¼</sup> *<sup>T</sup>*3*,*opt

<sup>¼</sup> *<sup>T</sup>*3*,*opt *T*2*,*opt

in general

*T*4 *T*3*,*opt

<sup>¼</sup> *<sup>T</sup>*3*,*opt *<sup>T</sup>*2*,*opt � �<sup>3</sup>

> <sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *T*3*,*opt

<sup>¼</sup> <sup>3</sup>*Q*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>4</sup>

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

This obtains if *<sup>T</sup>*4*=T*<sup>1</sup> <sup>&</sup>lt;ð Þ <sup>2</sup>*=*<sup>3</sup> <sup>3</sup> <sup>¼</sup> <sup>8</sup>*=*27 for *<sup>x</sup>* <sup>¼</sup> 1 and if *<sup>T</sup>*4*=T*<sup>1</sup> <sup>&</sup>lt; ð Þ <sup>2</sup>*=*<sup>3</sup> <sup>6</sup> <sup>¼</sup> <sup>64</sup>*=*<sup>729</sup>

< 2 <sup>3</sup> <sup>⇔</sup> *T*4 *T*1 < 2 3 � �<sup>3</sup>*=<sup>x</sup>*

*T*1 � �*<sup>x</sup>=*<sup>3</sup> <sup>¼</sup> *<sup>T</sup>*3*,*opt *T*1 � �1*=*<sup>2</sup>

*<sup>T</sup>*2*,*opt � �1*=*<sup>2</sup>

*T*1 � �1*=*<sup>2</sup>

<sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *<sup>T</sup>*2*,*opt � �<sup>1</sup>*=*<sup>2</sup>

<sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *<sup>T</sup>*2*,*opt � �<sup>1</sup>*=*<sup>2</sup>

> <sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *T*3*,*opt *:*

in particular

<sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *<sup>T</sup>*3*,*opt � �<sup>3</sup>

<sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *T*1 � �<sup>1</sup>*=*<sup>3</sup>

*T*4 *T*1 � �*<sup>x</sup>=*<sup>3</sup> " #

*T*1 � �*<sup>x</sup>=*<sup>3</sup> " #*:* *:*

(22)

(24)

(25)

(26)

*:* (27)

*:* (23)

Thus *W<sup>D</sup>* <sup>1</sup>!3*,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *<sup>T</sup>*<sup>3</sup> *T*1 *<sup>x</sup>=*<sup>2</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>T</sup>*<sup>3</sup> *<sup>T</sup>*<sup>1</sup> ¼ 1. For all *T*3 *<sup>T</sup>*<sup>1</sup> <sup>&</sup>lt;1*, W<sup>D</sup>* <sup>1</sup>!<sup>3</sup>*,*extra <sup>&</sup>gt; 0. Moreover, applying Eqs. (5), (13), and (15), note that

$$\begin{aligned} \lim\_{T\_3/T\_1 \to 0} \mathcal{W}\_{1 \to 3, \text{max}}^D = 2Q\_1 = 2 \lim\_{T\_3/T\_1 \to 0} \mathcal{W}\_{1 \to 3} \\ \Rightarrow \lim\_{T\_3/T\_1 \to 0} \mathcal{W}\_{1 \to 3, \text{max}}^{D\_3 \text{extra}} = 2Q\_1 - Q\_1 = Q\_1 = \lim\_{T\_3/T\_1 \to 0} \mathcal{W}\_{1 \to 3} \end{aligned} \tag{17}$$

Now consider heat-engine operation employing four heat reservoirs, with all work totally frictionally dissipated (except possibly at the last step; thus, *W<sup>D</sup>* <sup>3</sup>!<sup>4</sup> only optionally carries the superscript *D*). Thus we have

$$\begin{aligned} W\_{1\to 4}^D &= W\_{1\to 2}^D + W\_{2\to 3}^D + W\_{3\to 4}^D = Q\_1 \mathbf{e}\_{1\to 2} + Q\_1 \mathbf{e}\_{2\to 3} + Q\_1 \mathbf{e}\_{3\to 4} \\ &= Q\_1 (\mathbf{e}\_{1\to 2} + \mathbf{e}\_{2\to 3} + \mathbf{e}\_{3\to 4}). \end{aligned} \tag{18}$$

If ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj <sup>x</sup>* , 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. (5) and (18), we have:

$$\begin{split} W\_{1 \to 4}^{D} &= W\_{1 \to 2}^{D} + W\_{2 \to 3}^{D} + W\_{3 \to 4}^{D} \\ &= Q\_{1} \left\{ \left[ \mathbf{1} - \left( \frac{T\_{2}}{T\_{1}} \right)^{x} \right] + \left[ \mathbf{1} - \left( \frac{T\_{3}}{T\_{2}} \right)^{x} \right] + \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{3}} \right)^{x} \right] \right\} \\ &= Q\_{1} \left[ \mathbf{3} - \left( \frac{T\_{2}}{T\_{1}} \right)^{x} - \left( \frac{T\_{3}}{T\_{2}} \right)^{x} - \left( \frac{T\_{4}}{T\_{3}} \right)^{x} \right]. \end{split} \tag{19}$$

We wish to maximize *W<sup>D</sup>* <sup>1</sup>!4. Based on Eq. (12) and the associated discussions, the optimum value *Tj,*opt of *Tj* of reservoir *j* ð Þ 1< *j*<*n* ⇔ 2≤ *j*≤ *n* � 1 , which maximizes *W<sup>D</sup> <sup>j</sup>*�1!*j*þ1, is the geometric mean of *Tj*�<sup>1</sup> and *Tj*þ1. Thus we have

$$T\_{\text{2,opt}} = \left(T\_1 T\_{\text{3,opt}}\right)^{1/2} \tag{20}$$

and

$$T\_{\text{3opt}} = \left(T\_{\text{2opt}}T\_{\text{4}}\right)^{1/2}.\tag{21}$$

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

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

$$\frac{T\_{\text{2,opt}}}{T\_1} = \frac{\left(T\_1 T\_{\text{3,opt}}\right)^{1/2}}{T\_1} = \left(\frac{T\_{\text{3,opt}}}{T\_1}\right)^{1/2} \tag{22}$$

and

It is easily shown that *<sup>W</sup>D,*extra

*Thermodynamics and Energy Engineering*

*T*1 *<sup>x</sup>=*<sup>2</sup>

lim *T*3*=T*1!0

) lim *T*3*=T*1!0

<sup>1</sup>!3*,*max <sup>¼</sup> *<sup>W</sup>*1!<sup>3</sup> <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>*

*dWD,*extra <sup>1</sup>!3*,*max *dr* <sup>¼</sup> <sup>0</sup> )

<sup>1</sup>!3*,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *<sup>T</sup>*<sup>3</sup>

*W<sup>D</sup>*

optionally carries the superscript *D*). Thus we have

¼ *Q*1ð Þ ϵ<sup>1</sup>!<sup>2</sup> þ ϵ<sup>2</sup>!<sup>3</sup> þ ϵ<sup>3</sup>!<sup>4</sup> *:*

<sup>1</sup>!<sup>2</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

0<*x*≤ 1, applying Eqs. (5) and (18), we have:

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>3</sup> � *<sup>T</sup>*<sup>2</sup>

<sup>1</sup>!<sup>2</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

*<sup>x</sup>*

*<sup>W</sup>D,*extra

<sup>2</sup>!<sup>3</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

<sup>2</sup>!<sup>3</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

� *<sup>T</sup>*<sup>3</sup> *T*2 *<sup>x</sup>*

*<sup>x</sup>*

the optimum value *Tj,*opt of *Tj* of reservoir *j* ð Þ 1< *j*<*n* ⇔ 2≤ *j*≤ *n* � 1 , which

*T*2*,*opt ¼ *T*1*T*3*,*opt

*T*3*,*opt ¼ *T*2*,*opt*T*<sup>4</sup>

*T*1 *<sup>x</sup>*

> *T*1 *<sup>x</sup>*

1≤ *i*≤ *n* � 1 and *i*<*j*≤ *n*, and where *x* is a positive real number in the range

3!4

<sup>þ</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

*T*2 *<sup>x</sup>*

> � *<sup>T</sup>*<sup>4</sup> *T*3

*<sup>j</sup>*�1!*j*þ1, is the geometric mean of *Tj*�<sup>1</sup> and *Tj*þ1. Thus we have

<sup>1</sup>*=*<sup>2</sup>

*<sup>x</sup>*

as *<sup>r</sup>* and setting *dWD,*extra

*d dr <sup>r</sup>*

) *r* ¼ 1*:*

*T*1 *<sup>x</sup>=*<sup>2</sup>

*T*3*=T*1!0

<sup>3</sup>!<sup>4</sup> ¼ *Q*1ϵ<sup>1</sup>!<sup>2</sup> þ *Q*1ϵ<sup>2</sup>!<sup>3</sup> þ *Q*1ϵ<sup>3</sup>!<sup>4</sup>

, where *i* and *j* are positive integers in the respective ranges

<sup>þ</sup> <sup>1</sup> � *<sup>T</sup>*<sup>4</sup>

*:*

<sup>1</sup>!4. Based on Eq. (12) and the associated discussions,

*T*3

<sup>1</sup>*=*<sup>2</sup> (20)

*:* (21)

<sup>1</sup>!<sup>3</sup>*,*extra <sup>&</sup>gt; 0. Moreover, applying Eqs. (5), (13), and (15), note that

<sup>1</sup>!<sup>3</sup>*,*max ¼ 2*Q*<sup>1</sup> � *Q*<sup>1</sup> ¼ *Q*<sup>1</sup> ¼ lim

Now consider heat-engine operation employing four heat reservoirs, with all work totally frictionally dissipated (except possibly at the last step; thus, *W<sup>D</sup>*

<sup>1</sup>!<sup>3</sup>*,*max ¼ 2*Q*<sup>1</sup> ¼ 2 lim

) 2*r* � 2 ¼ 0

if *<sup>T</sup>*<sup>3</sup>

*<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>*

Thus *W<sup>D</sup>*

*W<sup>D</sup>*

<sup>1</sup>!<sup>4</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

If ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj*

<sup>1</sup>!<sup>4</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

We wish to maximize *W<sup>D</sup>*

*W<sup>D</sup>*

maximizes *W<sup>D</sup>*

and

**134**

*<sup>T</sup>*<sup>1</sup> <sup>&</sup>lt;1*, W<sup>D</sup>*

*T*3

denoting the ratio *<sup>T</sup>*<sup>3</sup>

<sup>1</sup>!3*,*max ≥ 0, with the equality obtaining if and only

<sup>1</sup>!3*,*max � *<sup>W</sup>*1!<sup>3</sup> <sup>¼</sup> *<sup>W</sup>D,*extra

<sup>2</sup> � <sup>2</sup>*<sup>r</sup>* <sup>¼</sup> <sup>0</sup>

<sup>1</sup>!3*,*max *<sup>=</sup>dr* <sup>¼</sup> 0 yields

<sup>¼</sup> <sup>1</sup> ) *<sup>T</sup>*<sup>3</sup>

*W*<sup>1</sup>!<sup>3</sup>

*T*3*=T*1!0

*<sup>T</sup>*<sup>1</sup> ¼ 1. For all

*W*<sup>1</sup>!<sup>3</sup>*:*

<sup>1</sup>!3*,*max ¼ 0. For,

(16)

(17)

<sup>3</sup>!<sup>4</sup> only

(18)

(19)

$$\frac{T\_4}{T\_{\text{3\text{opt}}}} = \frac{T\_4}{\left(T\_{\text{2\text{opt}}}T\_4\right)^{1/2}} = \left(\frac{T\_4}{T\_{\text{2\text{opt}}}}\right)^{1/2}.\tag{23}$$

Applying Eqs. (20)–(23), we obtain

$$\begin{split} \frac{T\_{\text{3\text{opt}}}}{T\_{2\text{opt}}} &= \frac{T\_{\text{3\text{opt}}}}{\left(T\_{1}T\_{\text{3\text{opt}}}\right)^{1/2}} = \left(\frac{T\_{\text{3\text{opt}}}}{T\_{1}}\right)^{1/2} \\ &= \frac{\left(T\_{\text{2\text{opt}}}T\_{4}\right)^{1/2}}{T\_{2\text{opt}}} = \left(\frac{T\_{4}}{T\_{2\text{opt}}}\right)^{1/2} \\ &\Rightarrow \left(\frac{T\_{\text{3\text{opt}}}}{T\_{1}}\right)^{1/2} = \left(\frac{T\_{4}}{T\_{2\text{opt}}}\right)^{1/2} \\ &\Rightarrow \frac{T\_{2\text{opt}}}{T\_{1}} = \frac{T\_{3\text{opt}}}{T\_{2\text{opt}}} = \frac{T\_{4}}{T\_{3\text{opt}}}. \end{split} \tag{24}$$

Applying Eqs. (22)–(24), we obtain

$$\begin{aligned} \frac{T\_4}{T\_1} &= \frac{T\_2}{T\_1} \frac{T\_3}{T\_2} \frac{T\_4}{T\_3} \text{ in general} \\ &= \frac{T\_{2,\text{opt}}}{T\_1} \frac{T\_{3,\text{opt}}}{T\_{2,\text{opt}}} \frac{T\_4}{T\_{3,\text{opt}}} \text{ in particular} \\ &= \left(\frac{T\_{2,\text{opt}}}{T\_1}\right)^3 = \left(\frac{T\_{3,\text{opt}}}{T\_{2,\text{opt}}}\right)^3 = \left(\frac{T\_4}{T\_{3,\text{opt}}}\right)^3 \\ &\Rightarrow \frac{T\_{2,\text{opt}}}{T\_1} = \frac{T\_{3,\text{opt}}}{T\_{2,\text{opt}}} = \frac{T\_4}{T\_{3,\text{opt}}} = \left(\frac{T\_4}{T\_1}\right)^{1/3}. \end{aligned} \tag{25}$$

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

$$\begin{split} \mathbf{W}\_{1 \rightarrow 4, \text{max}}^{D} &= \mathbf{Q}\_{1} \left[ \mathbf{3} - \mathbf{3} \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} \right] \\ &= \mathbf{3Q}\_{1} \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} \right]. \end{split} \tag{26}$$

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

$$\mathcal{W}\_{1\to 4,\text{max}}^{D} > Q\_1 \text{ if } \left(\frac{T\_4}{T\_1}\right)^{\ge 3} < \frac{2}{3} \Leftrightarrow \frac{T\_4}{T\_1} < \left(\frac{2}{3}\right)^{3/\ge}. \tag{27}$$

This obtains if *<sup>T</sup>*4*=T*<sup>1</sup> <sup>&</sup>lt;ð Þ <sup>2</sup>*=*<sup>3</sup> <sup>3</sup> <sup>¼</sup> <sup>8</sup>*=*27 for *<sup>x</sup>* <sup>¼</sup> 1 and if *<sup>T</sup>*4*=T*<sup>1</sup> <sup>&</sup>lt; ð Þ <sup>2</sup>*=*<sup>3</sup> <sup>6</sup> <sup>¼</sup> <sup>64</sup>*=*<sup>729</sup> for *x* ¼ 1*=*2. Also, applying Eqs. (5) and (26),

$$\begin{split} W\_{1 \rightarrow 4, \text{max}}^{\text{Jexfram}} &= W\_{1 \rightarrow 4, \text{max}}^{D} - W\_{1 \rightarrow 4} \\ &= 3Q\_{1} \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} \right] - Q\_{4} \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}} \right] \\ &= Q\_{1} \left\{ 3 \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} \right] - \left[ \mathbf{1} - \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}} \right] \right\} \\ &= Q\_{1} \left[ 3 - 3 \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} - \mathbf{1} + \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}} \right] \\ &= Q\_{1} \left[ 2 + \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}} - 3 \left( \frac{T\_{4}}{T\_{1}} \right)^{\mathbf{x}/3} \right] \ge \mathbf{0}. \end{split} \tag{28}$$

fixed. The temperatures *T*<sup>2</sup> through *Tn*�<sup>1</sup> 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

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

<sup>¼</sup> *TjTj*þ<sup>2</sup> � �1*=*<sup>2</sup> *Tj*

<sup>¼</sup> *Tj*þ<sup>2</sup> *TjTj*þ<sup>2</sup>

� �1*=*<sup>2</sup> <sup>¼</sup> *Tj*þ<sup>2</sup>

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* � 2,

> <sup>¼</sup> *Tj*þ<sup>1</sup> *Tj* � �<sup>2</sup>

⇔ *Tj*þ<sup>1</sup> *Tj*

� �*<sup>x</sup>*

<sup>1</sup>!*n,*max <sup>¼</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *Tn*

< *n* � 2 *n* � 1

respective ranges 1≤ *i* ≤ *n* � 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

<sup>¼</sup> *Tj*þ<sup>2</sup> *Tj* � �1*=*<sup>2</sup>

> *Tj* � �1*=*<sup>2</sup>

⇔ *Tj*þ<sup>1</sup> *Tj*

� �<sup>1</sup>*=*ð Þ *<sup>n</sup>*�<sup>1</sup>

*T*1

⇔ *Tn T*1 <

<sup>¼</sup> *Tn T*1

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* � 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* � 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*

<sup>¼</sup> *Tj*þ<sup>2</sup> *Tj* � �<sup>1</sup>*=*<sup>2</sup>

*:*

, where *i* and *j* are positive integers in the

� �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup> " #*,* (36)

� �ð Þ *<sup>n</sup>*�<sup>1</sup> *<sup>=</sup><sup>x</sup>*

*,* (37)

*n* � 2 *n* � 1 (33)

(35)

*:* (34)

*Tj*þ<sup>1</sup> *Tj*

*Tj*þ<sup>2</sup> *Tj*þ<sup>1</sup>

<sup>¼</sup> *Tj*þ<sup>1</sup> *Tj*

<sup>¼</sup> *Tj*þ<sup>1</sup> *Tj*

<sup>¼</sup> *Tj*þ<sup>1</sup> *Tj* � �*<sup>n</sup>*�<sup>1</sup>

*Tj*þ<sup>2</sup> *Tj*þ<sup>1</sup>

*Tj*þ<sup>2</sup> *Tj*þ<sup>1</sup>

) *Tj*þ<sup>2</sup> *Tj*

⇔ *Tn T*1

If, as per Eq. (5), ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj*

*W<sup>D</sup>*

*T*1

� �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup>

<sup>1</sup>!*n,*max <sup>&</sup>gt; *<sup>Q</sup>*<sup>1</sup> if *Tn*

*n*-reservoir system. We obtain:

Eqs. (31) and (32), we obtain:

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

and

we obtain:

values of *j*.

*W<sup>D</sup>*

and

**137**

It is easily shown that *<sup>W</sup>D,*extra <sup>1</sup>!<sup>4</sup>*,*max ≥ 0, with the equality obtaining if and only if *T*<sup>4</sup> *<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!<sup>4</sup>*,*max <sup>¼</sup> *<sup>W</sup>*<sup>1</sup>!<sup>4</sup> <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!<sup>4</sup>*,*max � *<sup>W</sup>*<sup>1</sup>!<sup>4</sup> <sup>¼</sup> *<sup>W</sup>D,*extra <sup>1</sup>!<sup>4</sup>*,*max ¼ 0. For, denoting the ratio *<sup>T</sup>*<sup>4</sup> *T*1 � �*<sup>x</sup>=*<sup>3</sup> as *<sup>r</sup>* and setting *dWD,*extra <sup>1</sup>!<sup>4</sup>*,*max *=dr* ¼ 0 yields

$$\frac{d W^{D, \text{extra}}}{dr} = 0 \Rightarrow \frac{d}{dr} \left(r^3 - 3r\right) = 0$$

$$\Rightarrow 3r^2 - 3 = 0$$

$$\Rightarrow r^2 = 1$$

$$\Rightarrow r = 1.$$

Thus *<sup>W</sup>D,*extra <sup>1</sup>!<sup>4</sup>*,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *<sup>T</sup>*<sup>4</sup> *T*1 � �*<sup>x</sup>=*<sup>3</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>T</sup>*<sup>4</sup> *<sup>T</sup>*<sup>1</sup> ¼ 1. For all *T*<sup>4</sup> *<sup>T</sup>*<sup>1</sup> <sup>&</sup>lt; <sup>1</sup>*, WD,*extra <sup>1</sup>!<sup>4</sup>*,*max <sup>&</sup>gt;0. Moreover, applying Eqs. (5), (26), and (28), note that

$$\begin{aligned} \lim\_{T\_4/T\_1 \to 0} W^D\_{1 \to 4, \text{max}} &= \mathfrak{Z} \mathcal{Q}\_1 = \mathfrak{Z} \lim\_{T\_4/T\_1 \to 0} W\_{1 \to 4} \\ \Rightarrow \lim\_{T\_4/T\_1 \to 0} W^{D\_{\text{extra}}}\_{1 \to 4, \text{max}} &= \mathfrak{Z} \mathcal{Q}\_1 - \mathcal{Q}\_1 = \mathfrak{Q} \mathcal{Q}\_1 = \mathfrak{Z} \lim\_{T\_4/T\_1 \to 0} W\_{1 \to 4} .\end{aligned} \tag{30}$$

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:

$$T\_{j+1} = \left(T\_j T\_{j+2}\right)^{1/2},\tag{31}$$

where *j* is any positive integer in the range 1≤ *j*≤ *n* � 2 and

$$T\_{j+2} = \left(T\_{j+1}T\_{j+3}\right)^{1/2},\tag{32}$$

where *j* is any positive integer in the range 1≤ *j*≤ *n* � 3. The respective temperatures *T*<sup>1</sup> and *Tn* of the extreme (hottest and coldest) reservoirs are assumed to be

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

fixed. The temperatures *T*<sup>2</sup> through *Tn*�<sup>1</sup> 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:

$$\frac{T\_{j+1}}{T\_j} = \frac{\left(T\_j T\_{j+2}\right)^{1/2}}{T\_j} = \left(\frac{T\_{j+2}}{T\_j}\right)^{1/2} \tag{33}$$

and

*<sup>W</sup>D,*extra

*Thermodynamics and Energy Engineering*

It is easily shown that *<sup>W</sup>D,*extra

*T*1 � �*<sup>x</sup>=*<sup>3</sup>

*T*<sup>4</sup>

*T*<sup>4</sup>

**136**

*<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>*

denoting the ratio *<sup>T</sup>*<sup>4</sup>

Thus *<sup>W</sup>D,*extra

lim *T*4*=T*1!0

point: 8*=*27 >1*=*4 and 64*=*729> 1*=*16).

) lim *T*4*=T*1!0

integer ≥ 4), we obtain:

*W<sup>D</sup>*

*<sup>W</sup>D,*extra

*<sup>T</sup>*<sup>1</sup> <sup>&</sup>lt; <sup>1</sup>*, WD,*extra

<sup>1</sup>!4*,*max <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

<sup>1</sup>!4*,*max � *W*1!<sup>4</sup>

*T*1 � �*x=*<sup>3</sup> " #

> *T*1 � �*x=*<sup>3</sup>

*T*4 *T*1 � �*<sup>x</sup>*

as *<sup>r</sup>* and setting *dWD,*extra

*T*1 � �*x=*<sup>3</sup> " #

� *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>4</sup>

� <sup>1</sup> � *<sup>T</sup>*<sup>4</sup>

*T*4 *T*1

<sup>1</sup>!<sup>4</sup>*,*max ≥ 0, with the equality obtaining if and only if

<sup>1</sup>!<sup>4</sup>*,*max *=dr* ¼ 0 yields

<sup>¼</sup> <sup>1</sup> ) *<sup>T</sup>*<sup>4</sup>

*T*4*=T*1!0

*W*<sup>1</sup>!<sup>4</sup>

*<sup>T</sup>*<sup>1</sup> ¼ 1. For all

*,* (31)

*,* (32)

*<sup>W</sup>*<sup>1</sup>!<sup>4</sup>*:* (30)

<sup>1</sup>!<sup>4</sup>*,*max � *<sup>W</sup>*<sup>1</sup>!<sup>4</sup> <sup>¼</sup> *<sup>W</sup>D,*extra

<sup>3</sup> � <sup>3</sup>*<sup>r</sup>* � � <sup>¼</sup> <sup>0</sup>

<sup>2</sup> � <sup>3</sup> <sup>¼</sup> <sup>0</sup>

� � � �*<sup>x</sup>* ( )

� 1 þ

*T*4 *T*1

� �*<sup>x</sup>* " #

� 3

� �*x=*<sup>3</sup> " #

*d dr <sup>r</sup>*

) 3*r*

) *r* <sup>2</sup> <sup>¼</sup> <sup>1</sup>

) *r* ¼ 1*:*

*T*1 � �*<sup>x</sup>=*<sup>3</sup>

*T*4*=T*1!0

<sup>1</sup>!<sup>4</sup>*,*max ¼ 3*Q*<sup>1</sup> � *Q*<sup>1</sup> ¼ 2*Q*<sup>1</sup> ¼ 2 lim

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

Generalizing Eqs. (20)–(30) for an *n*-reservoir system (*n* = any positive

*Tj*þ<sup>1</sup> ¼ *TjTj*þ<sup>2</sup>

*Tj*þ<sup>2</sup> ¼ *Tj*þ<sup>1</sup>*Tj*þ<sup>3</sup>

where *j* is any positive integer in the range 1≤ *j*≤ *n* � 2 and

� �<sup>1</sup>*=*<sup>2</sup>

� �<sup>1</sup>*=*<sup>2</sup>

where *j* is any positive integer in the range 1≤ *j*≤ *n* � 3. The respective temperatures *T*<sup>1</sup> and *Tn* of the extreme (hottest and coldest) reservoirs are assumed to be

<sup>1</sup>!<sup>4</sup>*,*max <sup>&</sup>gt;0. Moreover, applying Eqs. (5), (26), and (28), note that

*T*1 � � � �*<sup>x</sup>*

*T*1

(28)

(29)

≥0*:*

<sup>1</sup>!<sup>4</sup>*,*max ¼ 0. For,

<sup>¼</sup> <sup>3</sup>*Q*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>4</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> 3 1 � *<sup>T</sup>*<sup>4</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>3</sup> � <sup>3</sup> *<sup>T</sup>*<sup>4</sup>

¼ *Q*<sup>1</sup> 2 þ

<sup>1</sup>!<sup>4</sup>*,*max <sup>¼</sup> *<sup>W</sup>*<sup>1</sup>!<sup>4</sup> <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>*

*dWD,*extra <sup>1</sup>!<sup>4</sup>*,*max *dr* <sup>¼</sup> <sup>0</sup> )

<sup>1</sup>!<sup>4</sup>*,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *<sup>T</sup>*<sup>4</sup>

<sup>1</sup>!<sup>4</sup>*,*max ¼ 3*Q*<sup>1</sup> ¼ 3 lim

$$\frac{T\_{j+2}}{T\_{j+1}} = \frac{T\_{j+2}}{\left(T\_j T\_{j+2}\right)^{1/2}} = \left(\frac{T\_{j+2}}{T\_j}\right)^{1/2}.\tag{34}$$

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* � 2, we obtain:

$$\begin{aligned} \frac{T\_{j+2}}{T\_{j+1}} &= \frac{T\_{j+1}}{T\_j} \\ \Rightarrow \frac{T\_{j+2}}{T\_j} &= \frac{T\_{j+1}}{T\_j} \frac{T\_{j+2}}{T\_{j+1}} = \left(\frac{T\_{j+1}}{T\_j}\right)^2 \Leftrightarrow \frac{T\_{j+1}}{T\_j} = \left(\frac{T\_{j+2}}{T\_j}\right)^{1/2} \\ \Leftrightarrow \frac{T\_n}{T\_1} &= \left(\frac{T\_{j+1}}{T\_j}\right)^{n-1} \Leftrightarrow \frac{T\_{j+1}}{T\_j} = \left(\frac{T\_n}{T\_1}\right)^{1/(n-1)} .\end{aligned} \tag{35}$$

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* � 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* � 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), ϵ*<sup>i</sup>*!*<sup>j</sup>* ¼ 1 � *Ti=Tj* � �*<sup>x</sup>* , 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, 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:

$$\boldsymbol{W}\_{1\rightarrow n\text{-max}}^{D} = (n-1)\,\boldsymbol{Q}\_{1}\left[\mathbf{1} - \left(\frac{T\_{n}}{T\_{1}}\right)^{\times/(n-1)}\right],\tag{36}$$

$$W\_{1 \to \eta, \max}^{D} > Q\_1 \text{ if } \left(\frac{T\_n}{T\_1}\right)^{\mathbf{x}/(n-1)} < \frac{n-2}{n-1} \Leftrightarrow \frac{T\_n}{T\_1} < \left(\frac{n-2}{n-1}\right)^{(n-1)/\mathbf{x}},\tag{37}$$

and

$$\begin{split} W\_{1\rightarrow\eta,\max}^{D\text{-}\text{-}\text{-}\text{-}\text{-}\text{-}\text{-}W\_{1\rightarrow\eta} \\ &= (n-1)Q\_{1}\left[1-\left(\frac{T\_{n}}{T\_{1}}\right)^{\times/(n-1)}\right] - Q\_{1}\left[1-\left(\frac{T\_{n}}{T\_{1}}\right)^{\times}\right] \\ &= Q\_{1}\left\{(n-1)\left[1-\left(\frac{T\_{n}}{T\_{1}}\right)^{\times/(n-1)}\right] - \left[1-\left(\frac{T\_{n}}{T\_{1}}\right)^{\times}\right]\right\} \\ &= Q\_{1}\left[n-1-(n-1)\left(\frac{T\_{n}}{T\_{1}}\right)^{\times/(n-1)}-1+\left(\frac{T\_{n}}{T\_{1}}\right)^{\times}\right] \\ &= Q\_{1}\left[n-2+\left(\frac{T\_{n}}{T\_{1}}\right)^{\times}-(n-1)\left(\frac{T\_{n}}{T\_{1}}\right)^{\times/(n-1)}\right] \ge 0. \end{split} \tag{38}$$

Note the *linear* divergence of *W<sup>D</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

temperature ratio of *T*1*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�<sup>3</sup>

efficiency with *T*1*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�<sup>3</sup>

**4. General remarks, especially concerning entropy**

*logarithmic* divergence of *W<sup>D</sup>*

*W<sup>D</sup>*

*W<sup>D</sup>*

available.

Second Law.

performs work

**139**

temperature *Tj*þ<sup>1</sup> *Tj* >*Tj*þ<sup>1</sup> >*Tj*þ<sup>2</sup>

Eq. (41) *W<sup>D</sup>*

total energy output.

<sup>1</sup>!*n,*max in the limit *Tn=T*<sup>1</sup> ! 0 with *n* fixed as

, assuming Carnot efficiency by

<sup>1</sup>!*n,*max *=Q*<sup>1</sup> ≈4. Hence by Eq. (36)

<sup>1</sup>!*n,*max in the limit *n* ! ∞ with *Tn=T*<sup>1</sup> fixed even

per Eq. (40) even *not* assuming Carnot efficiency, as contrasted with the paltry

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

<sup>1</sup>!*n,*max in the limit *n* ! ∞ with *T*1*=Tn* fixed as per Eq. (41) and even with a

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*=Tn* fixed is not sufficiently closely approached to apply Eq. (41): we should instead apply Eq. (36). Applying Eq. (36) and assuming Carnot

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

It is important to emphasize that the super-unity cyclic-heat-engine efficiencies

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* � *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

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 *Tj* and reservoir *j* þ 2 at temperature *Tj*þ<sup>2</sup> without versus with frictional dissipation into reservoir *j* þ 1 at

*Wj*!*j*þ<sup>1</sup> ¼ *Qj*

. Without frictional dissipation a heat engine

ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> (42)

<sup>1</sup>!*n,*max *=Q*<sup>1</sup> that can obtain with work output totally frictionally dissipated ð Þ if *n*≥3 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

<sup>1</sup>!*n,*max *<sup>=</sup>Q*<sup>1</sup> <sup>≈</sup> ln 10<sup>3</sup> <sup>≈</sup> 7. Hence by Eq. (41) an advanced civilization

, *W<sup>D</sup>*

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*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�3. Could even larger values of *T*1*=Tn* be possible, at least in principle? Perhaps, maybe,

granting Carnot efficiency as per the derivation [22] of Eq. (41).

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

It is easily shown that *<sup>W</sup>D,*extra <sup>1</sup>!*n,*max ≥0, with the equality obtaining if and only if *Tn <sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!*n,*max <sup>¼</sup> *<sup>W</sup>*<sup>1</sup>!*<sup>n</sup>* <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>* <sup>1</sup>!*n,*max � *<sup>W</sup>*<sup>1</sup>!*<sup>n</sup>* <sup>¼</sup> *<sup>W</sup>D,*extra <sup>1</sup>!*n,*max ¼ 0. For, denoting the ratio *Tn T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup> as *<sup>r</sup>* and setting *dWD,*extra <sup>1</sup>!*n,*max *=dr* ¼ 0 yields

$$\frac{dW\_{1\to n,\text{max}}^{\text{Defextra}}}{dr} = 0 \Rightarrow \frac{d}{dr} \left[r^{n-1} - (n-1)r\right] = 0$$

$$\Rightarrow (n-1)r^{n-2} - (n-1) = 0$$

$$\Rightarrow r^{n-2} = 1$$

$$\Rightarrow r = 1.$$

Thus *<sup>W</sup>D,*extra <sup>1</sup>!*n,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *Tn T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *Tn <sup>T</sup>*<sup>1</sup> ¼ 1. For all *Tn <sup>T</sup>*<sup>1</sup> <sup>&</sup>lt; <sup>1</sup>*, WD,*extra <sup>1</sup>!*n,*max <sup>&</sup>gt;0. Moreover, applying Eqs. (5), (36), and (38), note that

$$\lim\_{T\_{\pi}/T\_{1}\to 0, \pi \text{ fixed}} W\_{1\to n, \text{max}}^{D} = (n-1)Q\_{1} = (n-1) \lim\_{T\_{\pi}/T\_{1}\to 0, \mu \text{ fixed}} W\_{1\to n}$$

$$\Rightarrow \lim\_{T\_{\pi}/T\_{1}\to 0, \mu \text{ fixed}} W\_{1\to n, \text{max}}^{D, \text{extra}} = \lim\_{T\_{\pi}/T\_{1}\to 0, \mu \text{ fixed}} \left(W\_{1\to n, \text{max}}^{D} - W\_{1\to n}\right) \tag{40}$$

$$= (n-1)Q\_{1} - Q\_{1} = (n-2)Q\_{1} = (n-2) \lim\_{T\_{\pi}/T\_{1}\to 0, \mu \text{ fixed}} W\_{1\to n}$$

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 *Tn T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup> ! 0 ⇔ 1 � *Tn T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup> ! 1 in the limit *Tn=T*<sup>1</sup> ! 0, albeit ever more slowly with decreasing *x*.

By contrast, even granting Carnot efficiency (*x* ¼ 1) [22]:

$$\lim\_{n \to \infty\_0} \lim\_{T\_n/T\_1 \text{ fixed}} W\_{1 \to n, \text{max}}^D = Q\_1 \ln \frac{T\_1}{T\_n} = \left( \lim\_{T\_n/T\_1 \to 0, n \text{ fixed}} W\_{1 \to n} \right) \ln \frac{T\_1}{T\_n}.\tag{41}$$

Note the *linear* divergence of *W<sup>D</sup>* <sup>1</sup>!*n,*max in the limit *Tn=T*<sup>1</sup> ! 0 with *n* fixed as per Eq. (40) even *not* assuming Carnot efficiency, as contrasted with the paltry *logarithmic* divergence of *W<sup>D</sup>* <sup>1</sup>!*n,*max in the limit *n* ! ∞ with *Tn=T*<sup>1</sup> 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*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�3. Could even larger values of *T*1*=Tn* 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<sup>D</sup>* <sup>1</sup>!*n,*max in the limit *n* ! ∞ with *T*1*=Tn* fixed as per Eq. (41) and even with a temperature ratio of *T*1*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�<sup>3</sup> , assuming Carnot efficiency by Eq. (41) *W<sup>D</sup>* <sup>1</sup>!*n,*max *<sup>=</sup>Q*<sup>1</sup> <sup>≈</sup> ln 10<sup>3</sup> <sup>≈</sup> 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*=Tn* 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*=Tn* ≈10<sup>3</sup> ⇔ *Tn=T*<sup>1</sup> ≈10�<sup>3</sup> , *W<sup>D</sup>* <sup>1</sup>!*n,*max *=Q*<sup>1</sup> ≈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.
