4. The region of realizability of systems

An irreversible factor affecting machine power or pump performance is finite heat transfer coefficients α<sup>i</sup> between heat sources and the working fluid. We substitute the minimum possible entropy production into Eq. (5) and obtain the condition determining the maximum power:

$$p = \eta\_k q\_+ - \sigma\_{\min}(q\_+) T\_- \,. \tag{41}$$

� � As σmin <sup>q</sup><sup>þ</sup> increases faster than qþ, there exist a maximum of power in the region of realizability.

To minimize the production of entropy, it is necessary that with each contact of the working medium with the sources the conditions of minimum dissipation, which depend on the dynamics of heat transfer, are met. For a source of infinite capacity and the temperature of the working fluid in contact with, it should be constant. For Newtonian dynamics, the ratio of working fluid temperature and sources should have been constant. So, if the temperature of the source changes due to the final capacity, then the temperature of the working fluid should change, remaining proportional to the temperature of the source.

For sources of infinite capacity, the optimal cycle of a heat machine with maximum power for any heat transfer dynamics should consist of two isotherms and two adiabats, and it turned out that the efficiency corresponding to the maximum power (it is called the Novikov-Curzon-Ahlborn, ηnca) is only a function of the Carnot efficiency:

$$
\eta\_{m\alpha} = \mathbf{1} - \sqrt{\mathbf{1} - \eta\_{\mathbf{C}}} \,\tag{42}
$$

The maximum difference between η<sup>C</sup> and ηnca is achieved when the ratio of absolute temperature of the hot and cold sources is 0.25, when ηC=0.75.

For power that is less than the maximum possible, the maximum efficiency of the heat machine is equal to

$$\eta\_{\text{max}}(\underline{p}) = 1 - \frac{1}{2T\_{+}} \left[ T\_{+} + T\_{-} - \frac{\underline{p}}{a} - \sqrt{(T\_{+} - T\_{-})^{2} + \left(\frac{\underline{p}}{a}\right)^{2} - 2\frac{\underline{p}}{a}(T\_{+} + T\_{-})} \right]. \tag{43}$$

In this case, α is expressed as

$$a = \frac{a\_+ a\_-}{a\_+ + a\_-}.\tag{44}$$

As p ! 0, the efficiency ηmax tends to Carnot efficiency and as

$$p \to p\_{\text{max}} = a \left( \sqrt{T\_+} - \sqrt{T\_-} \right)^2. \tag{45}$$

Corresponding thermal efficiency approaches the efficiency value obtained by Novikov, Curzon, and Ahlborn (42).

The nature of the set of realization modes is shown in Figure 1.

Similar results can be obtained for the heat pumps. Since the flux of costs is mechanical energy, the set of realizable modes has the form of a convex upward and unbounded parabola.

Figure 1. Power of thermal machine as a function of driving heat flux.

#### 5. Rectification processes

In the separation process, energy is spending on getting the work of separation. The work of separation can be obtained as an increase in the free energy of the streams leaving the system compared to the energy of the mixture flux at the system inlet. The energy expended can be thermal or mechanical. In systems of separation with thermal energy, the set of realizable modes coincides in the form with heat engines. In this case, the rectification processes will be the most important and energy-intensive. In the section below, the process of thermal separation of a two-component mixture is considered, and considerations which allow one to proceed to the determination of the order of separation of multicomponent mixtures are obtained.

Let the following parameters be defined for a mixture of two components: qi , Ti, si, pi , hi, xi, μi—molar consumption, temperature, molar entropy, pressure, enthalpy, concentration of key component, and its chemical potential in ith stream. Assume the index i ¼ 0 for the separated stream, index i ¼ 1 for the stream of the enriched key component (for which x1>x0), and index i ¼ 2 for the stream cleared of the key component ðx<sup>2</sup> <sup>&</sup>lt;x0Þ. Heat flux <sup>q</sup> +, brought (coming) from a source of <sup>þ</sup> temperature Tþ, is supplied to the separation system, whereas heat flux q� is rejected to a source of temperature T�. Equations of thermodynamic balances are of the form:

$$\begin{aligned} \mathbf{g}\_0 &= \mathbf{g}\_1 + \mathbf{g}\_2, & \mathbf{g}\_0 \mathbf{x}\_0 - \mathbf{g}\_1 \mathbf{x}\_1 - \mathbf{g}\_2 \mathbf{x}\_2 = \mathbf{0}, \\\ q\_+ &- q\_- + \mathbf{g}\_0 h\_0 - \mathbf{g}\_1 h\_1 - \mathbf{g}\_2 h\_2 = \mathbf{0}, \\\ \frac{q\_+}{T\_+} &- \frac{q\_-}{T\_-} + \mathbf{g}\_0 s\_0 - \mathbf{g}\_1 s\_1 - \mathbf{g}\_2 s\_2 + \sigma = \mathbf{0}. \end{aligned} \tag{46}$$

The ratio of target mass flux g1 and heat flux q<sup>þ</sup> may be accepted as the thermal efficiency of the separation process:

$$
\eta = \frac{\mathbf{g\_1}}{\mathbf{q\_+}}.\tag{47}
$$

Using material balances of Eq. (46), we shall express g2 in terms of g1 and introduce coefficient a ¼ ðx1 � x0Þ=ðx0 � x2Þ. Then, the second flux satisfies g2 ¼ ag1. Eq. (46) assumes the forms

$$q\_{+} - q\_{-} + \mathbf{g}\_{1}(\Delta h\_{01} + a\Delta h\_{02}) = \mathbf{0},\tag{48}$$

$$\frac{q\_{+}}{T\_{+}} - \frac{q\_{-}}{T\_{-}} + \mathfrak{g}\_{1}(\Delta s\_{01} + a\Delta s\_{02}) + \sigma = \mathbf{0}.\tag{49}$$

Minimal Dissipation Processes in Irreversible Thermodynamics and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.84703

Here, Δs01 ¼ s0 � s1, Δs02 ¼ s0 � s2 are the entropy increases, whereas Δh01 ¼ h0 � h1, Δh02 ¼ h0 � h2 are enthalpy increases in corresponding streams. It is mandatory that the concentrations of the key component in streams are prescribed.

We transform Eq. (48) to the form q� <sup>¼</sup> <sup>q</sup><sup>þ</sup> <sup>þ</sup> <sup>g</sup>1ðΔh01 <sup>þ</sup> <sup>a</sup>Δh02<sup>Þ</sup> and substitute the expression obtained into Eq. (49). The frontier of the realization set is characterized by the following equation:

$$g\_1 = \frac{1}{F} \left( 1 - \frac{T\_-}{T\_+} \right) q\_+ - \sigma \frac{T\_-}{F}. \tag{50}$$

Here, F ¼ T�ðΔs01 þ aΔs02Þ � Δh01 � ah02. The increases of enthalpy and entropy contained in F have the forms

$$
\Delta h\_{0i} = C\_p (T\_0 - T\_i), \quad i = 1, 2,\tag{51}
$$

$$
\Delta s\_{0i} = \mathbf{C}\_{p0} \ln T\_0 - \mathbf{C}\_{pi} \ln T\_i - R \ln \frac{P\_0}{P\_i} + \Delta s\_{\text{mix}0} - \Delta s\_{\text{mix}i}, \quad i = 1, 2.
$$

The entropy of mixing per one mole of mixture is:

$$
\Delta \mathfrak{s}\_{\text{mix}i} = R[(\mathfrak{1} - \mathfrak{x}\_i) \ln \left( \mathfrak{1} - \mathfrak{x}\_i \right) + \mathfrak{x}\_i \ln \mathfrak{x}\_i], \quad i = \overline{\mathfrak{0}, \mathfrak{2}}.\tag{52}
$$

Note that the ratio T-/F depends on reversible factors only. In the reversible process, the entropy production σ is zero, and the thermal efficiency reaches a maximum equal to the multiplier at the heat flux in Eq. (42).

As a productivity you can take any of the streams, even the stream of a separated mixture, because with given compositions of the streams they are proportional.

A reversible estimate of the thermal efficiency of the separation process and the shape of the border of the realizability region can be clarified by finding the minimum possible for a given productivity and dynamics of heat and mass transfer value σ and its dependence on the coefficients of dynamics and heat flux.

If the dynamics of heat transfer can be approximated by the Fourier law and the mass transfer flux is proportional to the difference of chemical potentials, then the minimum dissipation is proportional to the square of the cost of heat. The boundary of the set of realizable modes in this case has a parabolic form

$$
\mathfrak{g} = b\mathfrak{q} - a\mathfrak{q}^2.\tag{53}
$$

Then, the efficiency of a separation column in the maximum productivity mode is equal to one half of the reversible efficiency:

$$
\eta^\* = 0, \mathfrak{S}\eta^0 = 0, \mathfrak{S}b.\tag{54}
$$

Qualitative expressions linking characteristic coefficients a and b with parameters of the separation column were obtained [38]:

$$b = \frac{T\_B - T\_D}{T\_B A\_G} = \frac{\eta\_k}{A\_G}, \quad a = \left[\frac{1}{\beta\_B T\_B T\_+} + \frac{1}{\beta\_D T\_D T\_-} + \frac{2(\varkappa\_D - \varkappa\_B)}{k r^2}\right] \frac{T\_D}{A\_G}.\tag{55}$$

Here, AG is the molar reversible work of mixture separation, equal to the difference between molar free energy of streams leaving the column and the free energy of raw stream, TD, TB are the temperatures in the condenser and the kettle of the column, r is the molar evaporation heat, βD, β<sup>B</sup> are the coefficients of heat exchange in the condenser and the kettle, and k is the effective coefficient of mass transfer for column height.

The coefficients a and b can be found not only by Eq. (55) but also by the results of measurements on the current column. This allows us to solve many problems associated with finding a set of realizability, including the problem of choosing the order of separation. It is important that the efficiency in maximum performance mode depends only on the reversible efficiency. The following condition, sufficient for to be independent of a, is valid.

The sufficiency condition for independence of η <sup>∗</sup> of a [34]:

ð Þ If the partial derivative <sup>∂</sup><sup>σ</sup> <sup>a</sup>; <sup>b</sup>; <sup>q</sup> depends continuously on some scalar function <sup>z</sup>ða; <sup>q</sup><sup>Þ</sup> <sup>∂</sup><sup>q</sup> ð Þ and the ratio <sup>σ</sup> <sup>a</sup>; <sup>b</sup>; <sup>q</sup> q is a function of z, then the thermal efficiency in the maximum productivity mode is defined exclusively in terms of variables b, characterizing the reversible process. The condition is satisfied for thermal machines and for binary rectification.

In Figure 2 shows an example of the boundaries of realizable sets in cases where σðða; b; qÞÞ does not depend on a.

With decreasing dynamic coefficients, the entropy production increases. The set of realizable modes is compressed, while the maximum performance points with a corresponding heat flux remain on a straight line with a slope of η <sup>∗</sup> :

#### 5.1 Order of separation: rule of temperature multipliers

We arrange the substances according to the property γ used for separation (boiling point in the rectification processes). We normalize γ so that it is in the range from 0 to 1. The order of separation of substances may be direct or reverse. In the case of a direct order, the stream with the components γ < γ 1 is first separated, and then, the stream with large values of γ is divided into two sub-streams so that the first one has γ 1 ≤ γ < γ 2 and the second one γ ≥ γ 2. In the case of reverse order, the stream with the components γ > γ 2 is first separated, and then, the stream with lower values of γ is divided into two sub-streams so that the first one has γ 1 ≤ γ < γ 2 and the second one γ ≤ γ 1.

Let A denote the work of separation of the mixture. The A includes the factor RT0, where R is the universal gas constant and T0 is the ambient temperature. Then, the entropy of the separated fluxes will decrease in proportion to the work of separation A. In a reversible case, a decrease in the entropy of the material flux should be compensated by an increase in the entropy of the heat flux. If T<sup>þ</sup> and T�

Figure 2. Nature of change of the realization frontier with the irreversibility increase.

#### Minimal Dissipation Processes in Irreversible Thermodynamics and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.84703

are the temperatures of heat supply and removal, then the increase in entropy of the heat flux can be expressed as δsq ¼ qð1=T� � 1=TþÞ. The total cost of heat depends on the order of separation, while the total work does not depend on the order of separation. At each separation stage, the heat flux is proportional to the separation <sup>¼</sup> work at this stage, with the proportionality multiplier TþT� KT . Temperature <sup>T</sup>þ�T� factors are determined by the choice of the separation boundary. Denote by KT1 and KT2 the temperature factors corresponding to the separation boundary γ 1 and γ 2, respectively, and by A21 and A22 the work of separation of the mixture in the second stage in the direct and inverse order of separation. The total cost of heat per mole of the input mixture in the direct and reverse order of separation can be written as:

$$q\_1 = K\_{T1}(A\_0 - A\_{21}) + K\_{T2}A\_{21}, \quad q\_2 = K\_{T2}(A\_0 - A\_{22}) + K\_{T1}A\_{22}.\tag{56}$$

To determine the separation order, it is necessary to calculate the difference:

$$
\Delta q = (K\_{T1} - K\_{T2})[A\_0 - A\_{21} - A\_{22}].\tag{57}
$$

If the result of the calculation Eq. (57) is negative, then it is reasonable to choose a direct separation order. If the result is positive—a reverse order.

In the case of a multistage system, this rule applies to each of two successive stages. It is easy to see that the expression in square brackets in Eq. (57) is nonnegative. From here follows the rule of temperature multipliers (see [39]): The separation boundaries must be chosen so that the temperature multipliers do not decrease from stage to stage. In the case when the separation efficiency in the maximum performance mode depends only on the reversible efficiency, the rule of temperature multipliers is also valid. It is important that the information that is needed to calculate temperature factors is much more accessible and accurate than the information on the dynamics of the processes in the column.
