Abstract

It is known that the maximum efficiency of conversion of thermal energy into mechanical work or separation work is achieved in reversible processes. If the intensity of the target flux is set, the processes in the thermodynamic system are irreversible. In this case, the role of reversible processes is played by the processes of minimal dissipation. The review presents the derivation of conditions for minimum dissipation in general form and their specification for heat and mass transfer processes with arbitrary dynamics. It is shown how these conditions follow the solution of problems on the optimal organization of two-flux and multiflux heat exchange. The algorithm for the synthesis of heat exchange systems with given water equivalents and the phase state of the flows is described. The form of the region of realizability of systems using thermal energy and the problem of choosing the order of separation of multicomponent mixtures with the minimum specific heat consumption are considered. It is shown that the efficiency of the rectification processes in the marginal productivity mode monotonously depends on the reversible efficiency, which makes it possible to ignore irreversible factors for choosing the order of separation in this mode.

Keywords: entropy production, conditions of minimal dissipation, optimal heat transfer, multithreaded heat exchange system, rectification, separation of multicomponent mixtures, boundary of the realizability of thermal machines

### 1. Problems and methodology of finite-time thermodynamics

Applied thermodynamics originates from the work of Sadi Carnot in 1824 [1]. One of the problems of thermodynamics is the study of problems on the limiting possibilities of thermodynamic systems. For a long time, these tasks boiled down to finding the maximum efficiency of heat and refrigeration machines, separation systems, and various chemical processes. The solution of these problems led to the fact that the maximum efficiency value was determined in the case when the process under study was reversible. Reversibility will include processes in which the coefficients of heat and mass transfer are arbitrarily large or the fluxes of energy and matter in the system under study are arbitrarily small. With the development of nuclear energy, a new task was set—to obtain such a cycle of a heat engine that would correspond to its maximum power with certain fixed exchange ratios with sources. This task is due to the fact that the capital expenditures for the construction of nuclear power facilities are high with a relatively low cost of fuel spent. Variants of solving the problem of optimization thermodynamics were proposed in [2, 3].

Further development of finite-time thermodynamics was stimulated by a great deal of work of very many investigators. Here, we list names of just a few first researchers: R.S. Berry, B. Andresen, K.H. Hoffmann, P. Salamon, L.I. Rozonoer, and some others (see [1–32]).

Typical problems of optimization thermodynamics include the following: processes with minimal irreversibility; determination of the limiting possibilities of heat engines, cold cycles, and heat pumps (maximum power, maximum efficiency, many realizable modes); and analysis of the processes of separation of mixtures.

The general approach to solving problems is as follows. It is assumed that the whole system is divided into subsystems. In each subsystem, at any time moment, the deviations of the intensive variables from their average values over the volume are negligible. Consequently, the change of these variables (temperatures, pressures, etc.) occurs only at the boundaries of the subsystems, which means that the system as a whole is in a nonequilibrium state. This assumption makes it possible to use the equation of state in the description of individual subsystems, which are valid under equilibrium conditions, and ordinary differential equations can be used to describe the dynamics of the subsystems. The solution of extremal problems in this case is performed by methods of the optimal control theory for lumped parameter systems.

To study the limiting possibilities of thermodynamic systems, it is first necessary to make balance relations for matter, energy, and entropy. Moreover, the balance ratio for entropy includes dissipation σ, that is, the production of entropy. It characterizes the irreversibility of processes in the system. If all processes are reversible, then the dissipation is zero. If the processes are irreversible, then dissipation takes positive values. Dissipation depends on the dynamics of the processes. The set of realizability of the system in the parameter space of input and output streams is determined by the nonnegativity of dissipation. Reversible processes lie on the boundary of this set.

When a minimum possible dissipation is found as a function of flux intensities, then the inequality σ ≥ σmin holds in an arbitrary real system; this contracts the region of realization. In this formulation, the set obtained incorporates the effects of process dynamics, the magnitude of fluxes, and the system's extent (due to the presence of heat and mass transfer coefficients).

In any real system, it is possible to narrow the realizability region if we find the minimum possible dissipation value as a function of the flux intensity (σ ≥ σmin). This will take into account the dynamics, the flux intensity, and even the size of the installation through the coefficients of heat and mass transfer.

Then, from the balance equations, it is necessary to derive the connection between the system performance indicators and dissipation σ. Performance indicators usually monotonically deteriorate with the increase of σ. The best values of efficiency indicators are achieved in a reversible process, which allows using them similarly to the Carnot efficiency indicators.

Next, it is necessary to solve the problem of the organization of processes in such a way that, with the given constraints, the dissipation as a function of the flux intensities is minimal. This is the most difficult step in analyzing the capabilities of thermodynamic systems.

Consider the process of studying the limiting possibilities in more detail, and begin with thermodynamic balances. Thermodynamic balances show the relationship between the fluxes (matter, energy, and entropy) that the system exchanges with the environment and the changes in these values in the system [19]. Let us summarize all the fluxes, considering incoming fluxes as positive and outcoming

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

fluxes as negative. Fluxes can be convective and diffusive. Convective fluxes are forced into the system and removed from it. The diffusive flux depends on the differences between the intensive variables of the system at the point where it enters and the intensive variables of the environment.

The energy balance shows the rate of change in the energy of a system, which is determined by the flux of energy that enters or is removed along with the convective fluxes of matter, the change in energy due to the diffusional exchange of matter, the currents of conductively transmitted heat, and the power of the work done. Material balance shows the change in the number of moles of substances in the system. Entropy balance shows the change in the entropy of the system, which occurs due to the influx of entropy together with the incoming substances, the influx or removal of heat, and the production of entropy due to the irreversibility of exchange processes.

If the system operates cyclically, the balances can be recorded on average for the equipment working cycle. In this case, the total change in energy, amount of matter, and entropy per cycle is zero, since the state of the system at the start and the end of the cycle is the same. Balances are transformed into a system of relations of averages over cycle-averaged components.

The equations of thermodynamic balances show the relationship between process efficiency indicators, external fluxes, and the structure of the system. The increase in entropy σ causes an increase in the entropy of output fluxes. At the same time, either the temperature of the fluxes at the outlet decreases, or the outlet flux of heat increases at a constant temperature. This leads to a reduction in the work of separation, the mechanical work produced by the system.

Consider the operation of a thermal machine that converts the heat received from a hot source with temperature T<sup>þ</sup> into work. The working fluid gives a part of the energy to a cold source with a temperature T�. The working fluid changes its state cyclically. As an indicator of efficiency, we will consider the thermal efficiency (<sup>η</sup> <sup>¼</sup> <sup>p</sup>=qþ)—the ratio of the work produced to the amount of heat collected from a hot source.

Let us denote the average intensity of the heat flux taken from the hot source q<sup>þ</sup> and that given to the cold source, q�. For the generated power p, we write the equations of energy balances:

$$q\_{+} - q\_{-} - p = 0,\tag{1}$$

and

$$\frac{q\_{+}}{T\_{+}} - \frac{q\_{-}}{T\_{-}} + \sigma = 0.\tag{2}$$

Since the state of the working fluid either does not change in time (for steam and gas turbines) or changes cyclically (for steam engines), then there are zeros in the right parts of the equations.

Thermal efficiency <sup>η</sup> <sup>¼</sup> <sup>p</sup>=q<sup>þ</sup> follows from Eq. (1) in the form

$$
\eta = \frac{p}{q\_+} = \mathbf{1} - \frac{q\_-}{q\_+}.\tag{3}
$$

Taking into account the fact that the Eq. (2) implies

$$\frac{q\_{-}}{q\_{+}} = \frac{T\_{-}}{T\_{+}} + \sigma \frac{T\_{-}}{q\_{+}} \,. \tag{4}$$

Heat and Mass Transfer - Advances in Science and Technology Applications

Therefore,

$$\eta = \left(\mathbf{1} - \frac{T\_{-}}{T\_{+}}\right) - \sigma \frac{T\_{-}}{q\_{+}} = \frac{\mathbf{1} - T\_{-}/T\_{+}}{\mathbf{1} + \sigma T\_{-}/p}.\tag{5}$$

Thermal efficiency η is equal to Carnot efficiency when any irreversible phenomena are absent in the system.

The growth of σ leads to a growth entropy of output streams; under other equal conditions, this growth reduces the flux temperature at the outlet or at a fixed temperature increases the waste heat flux. And in this and in another case, this leads to a decrease in the mechanical work produced by the system or the work of separation. Energy efficiency of thermodynamic system, characterized by the relation of useful work, produced in it, to the energy costs, reaches a maximum in the invertible processes, when σ ¼ 0.

#### 2. Processes with a minimal dissipation

It is known that the maximum efficiency of conversion of thermal energy into mechanical work or separation work is achieved in reversible processes. If the intensity of the target flux is set, the processes in the thermodynamic system are irreversible. In this case, the role of reversible processes is played by the processes of minimal dissipation, so it is necessary to determine conditions under which thermodynamic processes exhibit minimal dissipation for a prescribed average intensity (prescribed averaged value of driving forces).

#### 2.1 The minimal dissipation's conditions

Consider two systems interacting with each other. Intensive variables for the ith system will be denoted by ui and extensive variables by xi. In general, these are vector variables. When systems are in contact, the difference between u1 and u<sup>2</sup> leads to the appearance of flux J uð <sup>1</sup>; u2Þ. Function J is continuous, is differentiable, and has the following properties:

$$\frac{\partial l\_j}{\partial u\_{1\circ}} > 0, \qquad \qquad \frac{\partial l\_j}{\partial u\_{2\circ}} < 0,\tag{6}$$

$$J(u\_1, u\_2) = 0, \quad \text{at} \qquad u\_2 = u\_1$$

for scalar u1 and u2.

The difference between vectors u1 and u2 (of the same sign as flux Jj) leads to appearance of driving forces Xj. Each force defines exclusively u1<sup>j</sup> and u2<sup>j</sup>, satisfying conditions analogous to those in Eq. (6). Entropy production σ, which characterizes the process irreversibility, is equal to the scalar product of the flux vector and the driving force vector, and average value of entropy is described by the formula

$$\overline{\sigma} = \frac{1}{L} \int\_0^L \sum\_{j=1}^m J\_j(u\_1, u\_2) X\_j(u\_{1j}, u\_{2j}) dl,\tag{7}$$

where the independent variable l has the interpretation of a space-time or a contacting area measure. The integrand of this functional is defined non-negatively. Minimal Dissipation Processes in Irreversible Thermodynamics and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.84703

� � We shall assume that in our algorithm (at least) one intensive variable appears, by definition u2ð Þl , which may assume its values from within a certain manifold V. Yet, because of the variability of extensive variables of the first subsystem dY1<sup>j</sup> ¼ �Jjðu1; <sup>u</sup>2<sup>Þ</sup> , the second variable changes in accordance with the formula dl

$$\frac{du\_{1j}}{dl} = \rho\_j(u\_1, u\_2), \quad u\_1(0) = u\_{10}, \quad j = 1, \ldots, m. \tag{8}$$

Average values of all or some selected fluxes are prescribed:

$$\frac{1}{L} \int\_{0}^{L} J\_{j}(u\_{1}, u\_{2}) dl = \overline{J\_{j}}, \quad j = 1, \dots, k\_{1}, \quad k\_{1} \le m. \tag{9}$$

Further on, we consider only the case of a scalar flux. The problem for vector fluxes and its solution is considered with details in [32, 33].

The scalar flux problem involves minimizing of the integral

$$\overline{\sigma} = \frac{1}{L} \int\_0^L f(u\_1, u\_2) X(u\_1, u\_2) dl \to \min\_{u\_2 \in V} \tag{10}$$

subject to constraining conditions:

$$\frac{du\_1}{dl} = \rho(u\_1, u\_2), \quad u\_1(0) = u\_{10},\tag{11}$$

$$\frac{1}{L} \int\_{0}^{L} f(u\_1, u\_2) dl = \overline{\mathcal{J}}.\tag{12}$$

The problem (10)–(12) simplifies in an important case when the rate of change of variable u1 is proportional to the flux:

$$
\rho(u\_1, u\_2) = c(u\_1) f(u\_1, u\_2). \tag{13}
$$

In this case the condition of minimal dissipation assumes the form

$$J^2(u\_1, u\_2) = \lambda\_2 \left(\frac{\partial \mathcal{J}(u\_1, u\_2)}{\partial u\_2} : \frac{\partial \mathcal{X}(u\_1, u\_2)}{\partial u\_2}\right),\tag{14}$$

whereas the condition of prescribed flux intensity can be written as

$$\int\_{u\_{10}}^{u\_{1L}} \frac{du\_1}{c(u\_1)} = \overline{f} \cdot L. \tag{15}$$

<sup>∗</sup> The value of u1<sup>L</sup> is determined regardless of the optimal solution u uð Þ<sup>1</sup> . <sup>2</sup>

If the flux is proportional to the driving force with constant coefficient α, then the minimum entropy production equals

$$
\overline{\sigma} = \frac{\overline{f}^2}{a}.\tag{16}
$$

#### 2.2 Minimal dissipation's conditions of selected processes

Consider the conditions for the minimum dissipation of heat exchange. Let us take the temperature of the body being heated as the controlling intense variable. The driving force in the minimum dissipation problem is

$$X(T\_1, T\_2) = \left(\frac{1}{T\_2} - \frac{1}{T\_1}\right),\tag{17}$$

whereas the heat flux is q Tð <sup>1</sup>; T2Þ. In the majority of cases, we may assume the energy balance in the form

$$\frac{dT\_1}{dl} = -\frac{1}{c\_1(T\_1)}q(T\_1, T\_2), \quad T\_1(0) = T\_{10},\tag{18}$$

where c1 ð Þ T<sup>1</sup> is the heat capacity of the hot source.

If the process takes place in time, then the parameter l has the meaning of time, and the parameter L—the duration of the process. If a pipe heat exchanger is considered, in which the hot flux temperature changes from section to section, the value of c is the water equivalent of the flux, and L is the length of the heat exchanger.

In agreement with conditions (14), (15) describing the minimum dissipation subject a prescribed average intensity of heat flux q, we can obtain a condition of minimum dissipation for an arbitrary law of heat transfer:

$$q^2(T\_1, T\_2) : \frac{\partial q}{\partial T\_2} T\_2^2 = -\lambda\_2 = \text{const},\tag{19}$$

$$\int\_{T\_{1L}}^{T\_{10}} c\_1(T\_1)dT\_1 = \overline{q} \cdot L, \qquad \int\_{T\_{1L}}^{T\_{10}} \frac{c\_1(T\_1)dT\_1}{q(T\_1, T\_2)} = L. \tag{20}$$

The first of these conditions determines T <sup>∗</sup> <sup>2</sup>ðT1; λ2Þ, second—T1<sup>L</sup>, and third constant λ2.

For the Newtonian law of heat transfer

$$q = a(T\_1 - T\_2) \tag{21}$$

with a constant heat capacity (water equivalent) c, we obtain from conditions (18)–(20)

$$a^2(T\_1 - T\_2)^2 = -\lambda\_2(-a)T\_2^2 \Rightarrow a\left(\frac{T\_1}{T\_2} - \mathbf{1}\right)^2 = \lambda\_2.\tag{22}$$

Therefore, for an arbitrary l of an optimal process, the ratio <sup>T</sup> T 2 1 should be constant. This constant equals

$$\frac{T\_1}{T\_2} = \mathbf{1} + \sqrt{\frac{\lambda\_2}{a}}.\tag{23}$$

As it follows from Eq. (20), T1<sup>L</sup> ¼ T10 � qL=c. Finally, the condition (20) leads to the following equality:

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

$$\frac{\sqrt{\frac{\lambda\_2}{a}}}{1 + \sqrt{\frac{\lambda\_2}{a}}} = -\frac{c}{aL} \ln\left(1 - \frac{\overline{q}L}{cT\_{10}}\right). \tag{24}$$

Substituting Eqs. (23) and (24) into the expression

$$
\sigma = \frac{c}{L} \int\_{T\_{\rm L}}^{T\_{\rm 10}} \left( \frac{1}{T\_2(T\_1)} - \frac{1}{T\_1} \right) dT\_{\rm 1} \tag{25}
$$

minimal entropy production is obtained in the form

$$\sigma\_{\rm min} = \frac{c^2 \ln^2 \left( \mathbf{1} - \frac{\overline{q}L}{cT\_{10}} \right)}{\left[ aL + c \ln \left( \mathbf{1} - \frac{\overline{q}L}{cT\_{10}} \right) \right] L} \,\tag{26}$$

Table 1 presents analogous conditions of minimal dissipation for some wellknown processes and corresponding expressions for minimal entropy production.

As shown in [34], the conditions of minimal dissipation make it significantly easier to estimate the limiting possibilities of thermodynamic systems. In a system with multithreaded heat exchange [35], the total heat load q and the total heat transfer coefficient α are fixed. At the input of the system, k heating fluxes with temperatures Ti0 and water equivalents Wi come in. It is necessary to choose the parameters of heat fluxes, the structure of the system, and the distribution of heat transfer coefficients.

The conditions under which the minimum possible production of the entropy of the σ <sup>∗</sup> trait is reached are also defined in [34]: (1) At each point of contact of the heating and heated streams, the minimum dissipation conditions must be satisfied. (2) Temperatures of heating fluxes at the outlet of the system should be equal to each other, as well as the temperature of the heated fluxes at the outlet. (3) Heating fluxes, in


Table 1. Conditions of minimal dissipation in thermodynamic processes. which the inlet temperature is less than the calculated temperature T, do not participate in heat exchange.

Computational relations for Newtonian heat transfer are

$$\begin{aligned} \overline{T} &= \frac{\sum\_{i=1}^{k} T\_{i0} W\_i - \overline{q}}{\sum\_{i=1}^{k} W\_i} \\ q^\* \left( T\_{i0} \right) &= W\_i (T\_{i0} - \overline{T}), \\ a^\* \left( T\_{i0} \right) &= \frac{\overline{a} W\_i (\ln T\_{i0} - \ln \overline{T})}{\sum\_{i=1}^{k} W\_i (\ln T\_{i0} - \ln \overline{T})}, \\ m &= 1 - \frac{1}{\overline{a}} \sum\_{i=1}^{k} W\_i (\ln T\_{i0} - \ln \overline{T}), \\ \overline{\sigma}^\* &= \overline{a} \frac{\left(1 - m\right)^2}{m}, \\ a^\* \left( T\_{i0} \right) &= q^\* \left( T\_{i0} \right) = W\_i = \mathbf{0}, \quad T\_{i0} \le \overline{T}. \end{aligned} \tag{27}$$

The system in which the entropy production calculated with parameters of all fluxes

$$
\sigma = \sum\_{\nu} W\_{\nu} \ln \frac{T\_{\nu}^{out}}{T\_{\nu}^{in}} \tag{28}
$$

is lower than a certain value cannot exist in reality.

Analogous relations can easily be obtained in the case when the inlet parameters of heated fluxes are prescribed.

#### 3. Synthesis of heat exchange systems

In [36] the problem of the limiting possibilities of the heat exchange system ("ideal" heat exchange) was considered. The minimum possible entropy production σ <sup>∗</sup> was found in the system with the given values of water equivalents and input temperatures of hot or cold fluxes and given the total heat load and the total heat transfer coefficient. It is shown that for the case when the heat flux is proportional to the temperature difference (Newtonian dynamics), this irreversibility limit can be reached if at each point of contact the ratio of the absolute temperatures of the fluxes is the same, and their temperatures at the outlet of the system are the same for all fluxes whose input temperatures are fixed (hot or cold).

Conditions of ideal heat exchange impose very strict requirements on the characteristics of the system:

—Each double-flux cell must be a counter-flux heat exchanger.

—The ratio of the water equivalents of the hot and cold flux in it should be equal to the ratio in degrees Kelvin of the temperature of the cold flux at the outlet of the heat exchange cell to the temperature of the hot flux at its inlet—conditions of thermodynamic consistency.

—This ratio and its corresponding minimum possible entropy production at fixed temperatures and water equivalents of hot fluxes are related to their inlet temperatures T<sup>0</sup> <sup>i</sup> , the water equivalents Wi, and the total heat transfer coefficient K as:

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

$$m = 1 - \frac{1}{K} \sum\_{i=1}^{n} \mathcal{W}\_i \left( \text{l/} \, T\_i^0 - \ln \overline{T}\_+ \right), \begin{pmatrix} \\ \\ \\ \end{pmatrix} \begin{pmatrix} & & & \\ & & & \\ & & & \\ & & & \\ \end{pmatrix} \tag{29}$$

—The temperature of the hot streams at the outlet should be the same and, as it follows from the conditions of the energy balance, is equal to:

$$\overline{T}\_{+} = \frac{\sum\_{i=1}^{k} T\_{i0} W\_{i} - \overline{q}}{\sum\_{i=1}^{k} W\_{i}},\tag{30}$$

—Hot fluxes with initial temperatures less than T<sup>þ</sup> do not participate in the heat exchange system.

If a part of the hot fluxes condenses in the process of heat transfer, then in the expression for m (Eq. (24)), the water equivalent of the corresponding term tends to infinity. Assign the index k to the condensing fluxes and find the limit.

� � � � �� <sup>q</sup> W <sup>k</sup> <sup>k</sup> ln T<sup>0</sup> <sup>k</sup> � ln T<sup>þ</sup> <sup>¼</sup> Wk ln T<sup>0</sup> <sup>k</sup> � ln Tk <sup>0</sup>� <sup>W</sup> when Wk tends to infinity. <sup>k</sup> Using L'Hospital's rule to disclose the uncertainties, we find that

$$\lim\_{W\_k \to \infty} W\_k \left( \ln T\_k^0 - \ln \left( \sfprod\_{k}^{0} - \frac{q\_k}{W\_k} \right) \right) \left( \frac{q\_k}{T\_k^0} = \frac{\mathcal{g}\_k r\_k}{T\_{bk}}.\right. \tag{31}$$

Here, it is taken into account that the temperature T<sup>0</sup> <sup>k</sup> is equal to the condensation temperature and the thermal load is the product of the flux rate of the latent heat of vaporization.

Thus, the expression for m in the presence of condensing fluxes will be rewritten in the form:

$$m = 1 - \frac{1}{K} \quad \sum\_{i=k} W\_i \left( \ln T\_i^0 - \ln \overline{T}\_+ \right) \left( - \sum\_k \frac{\mathbf{g}\_k \eta\_k^0}{T\_k b\_k} \right) \tag{32}$$

In a multithreaded system integrated with the technological process, the values of water equivalents of both hot and cold fluxes are set, and often their outlet temperatures are set. Therefore, the performance of the ideal heat exchange system cannot be achieved. It is natural to set the task of synthesis of the heat exchange system of the minimum irreversibility at more rigid restrictions on characteristics of streams. The conditions of ideal heat transfer can only serve as a "guiding star" like Carnot's efficiency for thermal machines, and the value of the ratio σ <sup>∗</sup> to the real production of entropy in the designed system is an indicator of its thermodynamic perfection.

Next, we propose the calculated relations for the bottom estimate of the minimum dissipation in the system with the above restrictions and the synthesis of a hypothetical system in which such an estimate is implemented.

Consider a multithreaded heat exchange system containing a set of hot (index i) and cold fluxes (index j), with given water equivalents Wi,Wj. For each of the cold (heated) fluxes, its inlet and outlet temperatures are set to T0 and Tj>T<sup>0</sup> <sup>j</sup> . <sup>j</sup>

For hot (cooled) fluxes, except for water equivalents, their temperatures at the inlet to the heat exchanger T0 are set. If some flux in the system changes its phase <sup>i</sup>

state, then for it except water equivalents, the flux rate gi , gj and heat of vaporization (condensation) ri, rj are fixed. The ambient temperature will be denoted as T0.

Under these conditions, the thermal load of the system is equal to the total energy required for heating all cold fluxes and is determined by the equality:

$$\overline{q} = \sum\_{j} q\_{j} = \sum\_{j} \mathcal{W}\_{j} \left( \overline{T}\_{j} - T\_{j}^{0} \right). \tag{33}$$

The difference in the conditions imposed on the hot and cold fluxes is due to the fact that for cold fluxes leaving the system with a temperature less than a predetermined one, heating is required, i.e., additional energy costs, and for hot ones, if their outlet temperature is greater than a predetermined one, cooling is required, which is much easier.

Entropy production is the difference between the total entropy of outgoing fluxes and the total entropy of incoming fluxes. Initially, we assume that all fluxes enter and leave the system in the same phase state, the pressure change in the system is small, and the heat capacity is constant. Then, the change in the entropy of each flux is the product of its water equivalent by the logarithm of the ratio of its inlet and outlet temperatures in degrees Kelvin [37]. So, it follows from the conditions of the thermodynamic entropy balance that:

$$
\sigma = \sigma\_+ + \sigma\_- = \sum\_i \mathcal{W}\_i \left( \ln \overline{T}\_i - \ln T\_i^0 \right) + \sum\_j \mathcal{W}\_j \left( \ln \overline{T}\_j - \ln T\_j^0 \right). \tag{34}
$$

The first of these terms is negative, the second is positive, and their sum is always greater than σ <sup>∗</sup> >0.

Note that all variables determining the value of the entropy growth of cold fluxes are given by the conditions of the problem, so that the minimum entropy production corresponds to the minimum at a given thermal load of the first summand by temperatures Ti.

The formal statement will take the form:

$$\sigma\_{+} = \sum\_{i} \mathcal{W}\_{i} \left( \ln \overline{T}\_{i} - \ln T\_{i}^{0} \right) \to \min \Big/ \sum\_{i} \mathcal{W}\_{i} \left( T\_{i}^{0} - \overline{T}\_{i} \right) = \overline{q} = \sum\_{j} \mathcal{W}\_{j} \left( \overline{T}\_{j} - T\_{j}^{0} \right). \tag{35}$$

The Lagrangian of this problem

$$L = \sum\_{i} \mathcal{W}\_{i} \left( \ln \overline{T}\_{i} - \ln T\_{i}^{0} \right) - \lambda \sum\_{i} \mathcal{W}\_{i} \left( \overline{T}\_{i} - T\_{i}^{0} \right). \tag{36}$$

The conditions of its stationarity in Ti lead to equality:

$$
\overline{T}\_i = \frac{1}{\lambda}.\tag{37}
$$

Thus, in any water equivalents and input temperatures of hot streams, minimum dissipation corresponds to such an organization of heat transfer for which the temperatures of hot streams at the exit are the same.

In general, coolant fluxes at the system inlet can have different phase states: vapor, liquid, or vapor-liquid mixture. The same states can be at the output of the stream.

—If the flux does not change its phase state, but changes only the temperature, then we assume that its temperature at the input to the cell T<sup>0</sup> <sup>k</sup> , the water equivalent Minimal Dissipation Processes in Irreversible Thermodynamics and Their Applications DOI: http://dx.doi.org/10.5772/intechopen.84703

Wk, and for cold fluxes the temperature at the output Tk are known. The temperature of the hot streams at the output of the T<sup>þ</sup> system is selectable (see Eq. (30)).

—If the cold flux changes its phase state so that at the inlet it is a liquid at boiling point and at the outlet it is saturated with steam (let us define it as "evaporating"), the weight flux rate gj , the boiling point Tbj, and the heat of vaporization rj are given. The same is true for hot "condensing" streams. They have a saturated steam state at the inlet and a liquid state at the boiling point at the outlet.

Thus, the first step in the synthesis algorithm of heat exchange systems is the preparation of initial data, in which actual fluxes and their characteristics are converted into calculated fluxes. They can be of two types: those that do not change their phase state (heated and cooled) and those that change it at the boiling point (evaporating and condensing). End-to-end fluxes are not included in the calculation. To calculate the total heat load production, use the following expression:

$$
\sum\_{jh} \mathcal{W}\_{jh} \left( \overline{T}\_{jh} - T^0\_{jh} \right) + \sum\_{bj} \mathcal{g}\_{bj} r\_{b\epsilon} = \overline{q}. \tag{38}
$$

Minimum dissipation implies fulfillment of the "counterflow principle": the cold streams with higher temperatures must be in contact with the hot flux with a higher temperature. The latter requirement, as well as the equality of temperatures of hot streams at the outlet, corresponds to the conditions of the ideal heat transfer [36].

As the hot fluxes move from one contact cell to the next, their temperature changes due to the recoil of the heat flux. At the output of the system, the heat flux given by them is q, and the temperature is Tþ. Let us denote by q the given heat load in some intermediate state of hot fluxes. As the hot streams cool down, it changes from zero to q.

In this case, we assume that when the hot flux with the highest input temperature (first) is cooled to a temperature of T<sup>0</sup> <sup>2</sup> , the first and second fluxes are combined, so that their water equivalents are summed. A similar union occurs with the third flux, etc., until the temperature of the equivalent hot flux drops to the previously calculated formula (30) Tþ. If a condensing flux is at a certain temperature Tbi in the number of hot fluxes, the temperature of the equivalent flux is constant and equal to Tbi until the equivalent flux transfers the heat of condensation qbi ¼ gbiri. The dependence of the temperature of the hot flux equivalent on the given heat load Tþð Þq , we will call the contact temperature of the hot fluxes.

Cold fluxes are ordered by their outlet temperature, so that j ¼ 1 corresponds to the flux with the highest output temperature. For cold fluxes, the value of q is equal to the current required heat load, i.e., the heat they need to obtain to satisfy the conditions imposed on their temperature and output state. The greater the q, the lower the cold flux temperature corresponding to this value. In this case, we assume that when the temperature of the first flux decreases, two events are possible:


In the first case, the first cold flux is calculated combined with the second. In the second case, it is excluded from the system and transferred to the heating of the second stream. This procedure continues until an equivalent cold flux reaches the lowest cold flux temperature at the system inlet. The number of threads included in the equivalent cold flux is changed by adding fluxes with lower temperatures at the outlet and due to the exclusion from streams with the highest temperatures at the entrance. But each value of q corresponds to the value of T�ð Þq of the contact temperature of cold fluxes.

The dependencies of the current contact temperatures can be calculated from energy balance conditions similar to the expression (25). For equivalent hot flux:

$$T\_+(q) = \frac{\sum\_{i=1}^{S\_+(T\_+)} W\_i T\_i^0 - q}{\sum\_{i=1}^{S\_+(T\_+)} W\_i},\tag{39}$$

where SþðTþÞ is the set of indices of hot fluxes for which the inlet temperature is greater than the current contact temperature (T<sup>0</sup> <sup>i</sup> >Tþð Þq ).

Similarly, for the contact temperature of the equivalent cold flux, we have:

$$T\_{-}(q) = \frac{\sum\_{j \star S\_{-}(T\_{-})} W\_{j} T\_{j} - q}{\sum\_{j \star S\_{-}(T\_{-})} W\_{j}},\tag{40}$$

where S�ðT�Þ is the set of indices of cold fluxes for which the contact temperature T� satisfies the inequality Tj>T�ð Þ<sup>q</sup> <sup>&</sup>gt;T<sup>0</sup> j .

The curves of the current contact temperatures decrease monotonically with the growth of q, with Tþð Þq >T�ð Þq . On each of these curves, the points (nodes) are selected, in which either the composition of the fluxes entering the equivalent flux changes or the condensation/evaporation process takes place. In the latter case, horizontal sections appear on the curves. On the curve T�ð Þq , there can be vertical jumps if the flux temperature T<sup>0</sup> <sup>j</sup> >Tj�1.

The interval δq<sup>ν</sup> from one of the nodes on any of the contact curves to the nearest node on the same or another curve is characterized by the same composition and phase state of the contacting fluxes. We will call it the homogeneity interval.

For each such interval of δqν, three combinations of contacting fluxes are possible:


Contact temperature curves provide all the data necessary to calculate the heat transfer coefficient of the cell in which the contact is made:

—Water equivalents of Wþ,W� is equal to the sum of the equivalents of water fluxes which are part of the equivalent contacting fluxes.

—Temperatures of equivalent fluxes at the inlet and outlet of the interval of homogeneity is known.

—The thermal load of such a computational cell is δqν.

Depending on which of these contact combinations is implemented, it is possible to select the type of cell hydrodynamics and find Kν. Finding the heat transfer coefficient K<sup>ν</sup> for each ν interval of homogeneity and summing these coefficients over all intervals, we obtain the total coefficient K, which can be achieved by organizing the countercurrent heat exchange of equivalent fluxes. In turn, knowledge of the heat load q and the total heat transfer coefficient allows us to calculate the minimum possible entropy production σ <sup>∗</sup> by the formula (29) and estimate the degree of thermodynamic perfection of the constructed system as <sup>η</sup> <sup>¼</sup> <sup>σ</sup> σ0 ∗ , where σ<sup>0</sup> is total entropy production in the system.

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