Conditions of minimal dissipation in thermodynamic processes.

## 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 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

When a minimum possible dissipation is found as a function of flux intensities, then the inequality

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 (

Then, from the balance equations, it is necessary to derive the connection between the system performance indicators and dissipation

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 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

Consider the operation of a thermal machine that converts the heat received from a hot source with temperature

Let us denote the average intensity of the heat flux taken from the hot source

and

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

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

Therefore,

Thermal efficiency

The growth of

## 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

for scalar

The difference between vectors

where the independent variable

We shall assume that in our algorithm (at least) one intensive variable appears, by definition

Average values of all or some selected fluxes are prescribed:

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

subject to constraining conditions:

The problem (10)–(12) simplifies in an important case when the rate of change of variable

In this case the condition of minimal dissipation assumes the form

whereas the condition of prescribed flux intensity can be written as

The value of

If the flux is proportional to the driving force with constant coefficient

### 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

whereas the heat flux is

where

If the process takes place in time, then the parameter

In agreement with conditions (14), (15) describing the minimum dissipation subject a prescribed average intensity of heat flux

The first of these conditions determines

For the Newtonian law of heat transfer

with a constant heat capacity (water equivalent)

Therefore, for an arbitrary

As it follows from Eq. (20),

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

minimal entropy production is obtained in the form

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

Process | Conditions of minimal dissipation and entropy production |
---|---|

Heat transfer | |

Vector flux, linearly depending on driving forces | |

One-sided isothermal mass transfer | |

Two-sided isothermal equimolar mass transfer |

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

The conditions under which the minimum possible production of the entropy of the *(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 which the inlet temperature is less than the calculated temperature* *, do not participate in heat exchange*.

Computational relations for Newtonian heat transfer are

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

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

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

—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:

—Hot fluxes with initial temperatures less than

If a part of the hot fluxes condenses in the process of heat transfer, then in the expression for

Here, it is taken into account that the temperature

Thus, the expression for

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

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

For hot (cooled) fluxes, except for water equivalents, their temperatures at the inlet to the heat exchanger

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:

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:

The first of these terms is negative, the second is positive, and their sum is always greater than

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

The formal statement will take the form:

The Lagrangian of this problem

The conditions of its stationarity in

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

—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

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:

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

In this case, we assume that when the hot flux with the highest input temperature (first) is cooled to a temperature of

Cold fluxes are ordered by their outlet temperature, so that

Its temperature will drop to the set temperature at the output of the second stream.

Its temperature will drop to its initial temperature.

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

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

where

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

where

The curves of the current contact temperatures decrease monotonically with the growth of

The interval *the homogeneity interval*.

For each such interval of

Both equivalent fluxes change their phase states.

The hot equivalent flux is cooled and the cold is heated.

One of the fluxes changes its phase state, and the other is cooled or heated.

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

—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

Depending on which of these contact combinations is implemented, it is possible to select the type of cell hydrodynamics and find

## 4. The region of realizability of systems

An irreversible factor affecting machine power or pump performance is finite heat transfer coefficients

As

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,

The maximum difference between

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

In this case,

As

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.

## 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:

The ratio of target mass flux

Using material balances of Eq. (46), we shall express

Here,

We transform Eq. (48) to the form

Here,

The entropy of mixing per one mole of mixture is:

Note that the ratio T-/F depends on reversible factors only. In the reversible process, the entropy production

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

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

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

Qualitative expressions linking characteristic coefficients

Here,

The coefficients

**The sufficiency condition for independence of** **of**

*If the partial derivative* *depends continuously on some scalar function* *and the ratio* *is a function of* *, then the thermal efficiency in the maximum productivity mode is defined exclusively in terms of variables* *, 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

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

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

We arrange the substances according to the property

Let

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

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 non-negative. 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.

## 6. Conclusions

This chapter discusses the problems of optimization of thermodynamics and methods of analysis of systems and describes the types of thermodynamic balances, the relationship between the performance of the process, and the production of entropy. Also, it is shown that in the absence of irreversibility, the thermal efficiency is equal to the Carnot efficiency.

The conditions are found under which the thermodynamic processes at a given average intensity have minimal dissipation, expressions for determining the minimum dissipation and entropy with the Newtonian heat transfer law are obtained, and expressions for the cases of vector flux, one-sided isothermal, and two-sided equimolar mass transfer are given.

The synthesis algorithm makes it possible to build heat exchange systems with minimal irreversibility, in which restrictions on water equivalents, temperatures, and phase states of the flows are fulfilled, which imply combining the fluxes into two equivalent ones. The nature of the set of realizable modes of heat engines and pumps is described. It is shown that the efficiency corresponding to the maximum power mode does not depend on heat transfer coefficients, but is only a function of the Carnot efficiency.

Separation processes are considered, and estimates of the thermal efficiency of the separation process and the shape of the realizable area boundary are obtained for them. It is shown that the efficiency in the mode of maximum performance depends only on the reversible efficiency. The rule of temperature multipliers is described, which allows to determine the separation order in multistage systems.

## Notes

- †Ad memoriam Anatoly V. Zaev