## 1. Introduction

### 1.1. The mass transfer process with minimum irreversibility

In many processes, heat and mass transfer are distributed in time or space. The problem of thermodynamically perfect organization lies in the choice of such concentration and temperature change, in space or time, laws to minimize the entropy production

#### 1.1.1. Optimal organization of an irreversible mass transfer process

Consider the irreversible process of mass transfer, in which from one flow to another one substance is transmitted. The problem of minimal irreversibility of this process at a given average intensity of mass transfer takes the form:

under conditions

The minimum is searched by selecting the concentration *i*-th flow dependence on the concentration of the redistributed substance in it). Here

Minimal irreversibility conditions of mass transfer arise from the solution of (1) — (3). They can be described as follows [24]:* In the mass transfer process with minimum irreversibility the ratio of flow g and chemical potential μ2 derivatives with respect to the concentration c2 is proportional to the ratio of relative flow g square to the temperature, in any cross-section of l.*

Indeed, the entropy production after transition from

under the condition

The Lagrange function of problem (5), (6) takes the form

stationarity conditions with respect to

lead to the equation (4). The proportionality coefficient

For a specific task

we'll get the following equation from (4) :

At the constant temperature and pressure, this condition leads to the equation

and the constancy of flow

For the mass transfer law of the form

Derivatives are

After their substitution into (4), we obtain

During the mass transfer between phases the driving force of the process is expressed as the difference between the concentration of a redistributed component in one phase

#### 1.1.2. Example

Let optimality conditions of irreversible mass transfer have the form (11). From the view of flow

where M denotes the right side of (11). Substituting

Or

Assuming a linear dependence

where *a*, *b* — are some constants determined by processing experimental data of the equilibrium. Then, find the optimum profile

## 2. Irreversible work of separation and heat-driven separation

### 2.1. Introduction

The minimal amount of energy needed for separation a mixture with a given composition can be estimated using reversible thermodynamics. These estimates turn out to be very loose and unrealistic. They also do not take into account kinetic factors (laws and coefficients of heat and mass transfer, productivity of the system, etc.). In this paper we derive irreversible estimates of the work of separation that take into account all these factors.

The majority of separation systems are open systems that exchange mass and energy with the environment. If mass and heat transfer coefficients (determined by the size and construction of the apparatus) are finite and if the productivity of the system is finite then the processes in such systems are reversible. The energy flows, the compositions of the mass flows, and the productivity of the system are linked via the balance equations of energy, mass, and entropy. The latter also includes entropy production in the system. Minimal energy used for separation corresponds to minimal entropy production in the system subject to various constraints. This allows us to estimate this minimal energy.

There is a qualitative as well as a quantitative difference between the reversible and irreversible estimates obtained in this paper. For example, the irreversible estimate of the work of separation for poor mixtures (where the concentration of one of the components is close to one) tends to a finite nonzero limit, which depends on the kinetics factors. The reversible work of separation for such mixtures tends to zero. The reversible estimate differs from the amount of energy needed in practice for separation of poor mixtures by a factor of 10^{5}.

For heat-driven separation processes the novel results obtained in this paper include the estimate of the minimal heat consumption as a function of kinetic factors and the thermodynamic limit on the productivity of a heat-driven separation.

### 2.2. Thermodynamic balances of Separation Processes and the Link between Energy Consumption and Entropy Production

Consider the system, shown in Figure 1, where the flow of mixture with rate

In centrifuging, membrane separation, and adsorption–desorption cycles that are driven by pressure variations, no heat is supplied/removed and only mechanical work is spent. In absorption–desorption cycles, distillation, and so forth, no mechanical work is spent, only heat is consumed (heat-driven separation). In some cases the number of input and output flows can be larger. As a rule one can still represent the system as an assembly of separate blocks, whose structure is shown in Figure 1.

#### 2.2.1. Heat-driven separation

Consider a heat-driven separation *j*-th substance in the *i*-th flow. The thermodynamic balance equations of mass, energy, and entropy here take the following form

where

Here,

Elimination of

and the flow of used heat for heat-driven separation is

The first term in the square brackets depends only on the parameters of the input and output flows and represents the reversible work of separation per unit of time (reversible power of separation). The second term there represents the process kinetics and corresponding energy dissipation.

For mixtures that are close to ideal gases and ideal solutions, molar enthalpies and entropies

where *R* is the universal gas constant. The reversible energy consumption here is

We denote here the Carnot efficiency of the ideal cycle of the heat engine as

Condition (25) can be rewritten as

Here,

where it is assumed that there is constant contact of the working body with the heat reservoirs and

It is easy to show that if

Substitution of

where

Conditions (29-31) single out the area of thermodynamically feasible heat-driven separation systems.

Expressions (27) and (28) and eq (25) can be further specified by assuming the constancy of heat capacities, that the mixture is binary, and so forth.

#### 2.2.2. Mechanical separation

Consider a separation system that uses mechanical work with rate *p*. Assume that no heat is supplied/removed *T* and the same pressure. Multiplication of eq (24) by *T* and subtraction of the result from the energy balance eq (23), where *p*, yields

here

After taking into account eq (27) that the enthalpy increment

The first term in this expression represents the minimal power for separation that corresponds to the reversible process

Here

is the reversible power of separation of the *i*-th flow into pure substances.

### 2.3. Minimal work of separation in irreversible processes

#### 2.3.1. Assumptions and problem formulation

Assume that the components of the input mixture are close to ideal gases or ideal solutions. The chemical potential of the *i*-th component can then be written in the following form

where *i*-th component.

First we consider a system that includes three elements, a reservoir with the time independent temperature *T*, pressure *P*, and vector of concentrations

We do not consider here how to implement the derived optimal dependence of the chemical potential of the working body because of two reasons. First, our main objective is to derive a lower bound on the work of separation. However, imposing constraints on feasible variations of chemical potential would lead to an increase in energy consumption. Second, we will demonstrate that for the majority of mass transfer laws the optimal mass transfer flow is time independent, and its implementation is straightforward.

The work of separation in an isothermal process for an adiabatically insulated system can be found using the Stodola formula in terms of the reversible work

The reversible work is equal to the increment of the system’s internal energy. Since as a result of the process

and it is independent of *N*, *A* corresponds to the minimum of the entropy increment

Because the working body’s parameters have the same values at the beginning and at the end of a cycle

#### 2.3.2. Optimal solution

The problem of minimization of

Assume

then the problems (39) and (40) can be decomposed into 2k problems

where

Problems eq (41) are averaged nonlinear programming problems. Their optimal solutions

or switches between two so-called basic values on the interval

If the function *g* (we omit subscripts for simplicity). If it is positive then the constancy of the rate in the optimal process is guaranteed.

The first term in this expression is always positive because the chemical potentials’ difference is the driving force of mass transfer and monotonically depends on the flow. For the majority of laws of mass transfer the inequality eq (43) holds. In particular, it holds if the flow of mass transfer is proportional to the difference of chemical potentials in any positive degree.

Consider mass transfer flow that depends linearly on the chemical potential difference for all *i*, *j.* Then

It is clear that the conditions eq (43) hold and the optimal rates of flows obey equalities (42).

Equalities (42) hold for any nonswitching solution. The minimal increment of the entropy production for such solution is

and the minimal work of separation is

The optimal rates are determined by the initial and final states which allows us to specify the estimate eq (46).

Near equilibrium the flows obey Onsanger’s kinetics eq (44), and from eq (46) it follows that

is the equivalent mass transfer coefficient on the i-th component and the minimal entropy production is

The lower bound for the average power of separation is

If

then expressions (47) and (50) take the form

Where

Note that the irreversible estimate of the work of separation eq (51) does not tend to zero for poor mixtures when the concentration of one of the components tends to one (Figure 3).

If system includes not one but a number of output subsystems then it is clear that the estimate for the minimal work of separation is equal to the sum of the estimates for each subsystem.

The superscript j here denotes the subsystems.

#### 2.3.3. Separation of a System with finite capacity into m subsystems

Consider a system that is shown in Figure 4. Its initial state is described by the vector of concentrations

The work in the reversible separation process here is

The reversible work of separation is equal to the difference of the reversible work of separation of the initial mixture into pure components and the reversible work of separation for mixtures in each of the subsystems.

We again assume that flows

Here,

The minimal work of separation for the mixture with concentrations

Here,

The first term here coincides with the reversible work of separation

#### 2.3.4. Example

Consider separation of the binary mixture into pure components in time

The estimate eq (61) was derived in ref [1] by solving the problem of optimal separation of the binary mixture in the given time

Here,

#### 2.3.5. Example

Assume *g* and *P*:

Therefore,

### 2.4. Potential application of obtained estimates

We will illustrate the possibilities of the application of the derived estimates.

#### 2.4.1. Estimate of the power of separation in a continuous separation system

Consider a continuous separation system with the input flow *m* output flows

Equation (59) allows us to estimate the minimal power required for continuous separation in such system

Where

Mass balance equations yield

The number of conditions eq (66) is

If the number of flows

here

L is the concave function on

We have *k* linear equations for

#### 2.4.2. Example

Assume

From eq (65) we obtain

Equations (69) and (70) for

We obtain

#### 2.4.3. The selection of the separation sequence for a multicomponent mixture

In practice, separation of multicomponent mixtures is often realized via a sequence of binary separations. So, a three-component mixture is first separated into two flows, one of which does not contain one of the components. The second flow is then separated into two unicomponent flows. The reversible work of separation (that corresponds to the power

Consider a three-component mixture with concentration

The first component is first separated, then the second and the third are separated.

The second component is separated, and then the first and the third are separated.

We assume that the separation at each stage is complete. We get up to the constant multiplier

The first two terms in this sum represent the loss of irreversibility during the first stage of separation. For

Consider the first stage of case a for

When the second flow is separated into two flows their rates are

and the irreversible power is

The combined irreversible power is

Similarly in case b we get

The differential between these two values is

If

Note that it is not possible to formulate the general rule to choose the optimal separation sequence for a multicomponent mixture, in particular, on the basis of the reversible work of separation. It is necessary here to compare irreversible losses for each sequence.

#### 2.4.4. Example

Assume that the composition of the input three component mixture is ^{2}/(J s), ^{2}/(J s). From (eq 73) we find that the difference in power between sequences *a* and *b* is

The comparison of the combined minimal irreversible power for the same initial data shows that the power for separation of a mixture using sequence *b* is higher than the power used for sequence *a*, that is,

Thus, sequence *a* is preferable, and it is better to perform the complete separation by separating the first component.

### 2.5. Limiting productivity and minimal heat consumption for a heat-driven separation

In many separation processes a heat engine is used to create the differential of the chemical potential between the working body and the reservoirs (the driving force of mass transfer). Here, the working body is heated during contact with one reservoir and is cooled during contact with the other reservoir. One can represent the heat-driven separation system as a transformer of heat into the work of separation that generates power p, consumes heat flow from hot reservoir

It was shown in refs [12] and [6] that the potential of the direct transformation of heat to work is limited and the maximal generated power for the working body with the distributed parameters is

In this expression

The maximal power determines the heat flow consumed from the hot reservoir. Further increase of heat consumption for given values of heat transfer coefficients requires an increase of the temperature differential between the reservoirs and the working body and reduces the power.

The dependence of the used power on the productivity of irreversible separation processes is monotonic eq (63). Therefore, the limiting productivity of heat-driven separation processes corresponds to the maximal possible power produced by transformation of heat into work. Further increase of heat consumption

For the Newton (linear) law of mass transfer and heat–work transformer the dependence of the power on the heat used is

Here,

The minimal heat consumption

Substitution of the right-hand side of eq (74) instead of p in eq (63) yields the maximal possible productivity of the system (where

We obtain

and the limiting productivity is

Formulas (76) and (77) allow us to estimate the limiting productivity of a heat-driven separation process for Newton’s laws of heat transfer between the working body and reservoirs and mass transfer proportional to the differentials in chemical potentials (mass transfer is close to isothermal with the temperature T).

#### 2.5.1. Example

Consider heat-driven monoethanamide gas cleansing. One of the components is absorbed by the cold solution from the input gas mixture. This solution is then heated and this component is vaporized. The input mixture’s parameters are ^{2}/(kg s), ^{2}/(kg s).

Because the solution circulates and is heated and cooled in turns, the limiting power for transformation of heat into work is given by the expression (74) with the corresponding

The power for separation is given by eq (63).

We have

The minimal work required for a system with Onsanger’s equations are (see eq (63))

Thus,

Let us estimate the minimal heat consumption. From eq (75) we get

If the temperatures of the input and output flows are not the same then the minimal energy required for separation can be estimated using the thermodynamic balance equations (31) and (32) and the expression for

### 2.6. Conclusion

New irreversible estimates of the in-principle limiting possibilities of separation processes are derived in this paper. They take into account the unavoidable irreversibility caused by the finite rate of flows and heat and mass transfer coefficients. They also allow us to estimate the limiting productivity of a heat-driven separation and to find the most energy efficient separation sequence/regime of separation for a multicomponent mixture.

## 3. Optimization of membrane separations

### 3.1. Introduction

As the properties of membranes improve, the membrane separation of liquids and gases is more widely used in chemical engineering [8,10,11,20]. Since the mathematical modeling of membrane separations is simpler than that for most of the other separation processes, they could be controlled by varying the pressure, contact surface area, and the like during the separation process.

The minimal work needed to separate mixtures into pure components or into mixtures of given compositions can be minorized using well-known relationships of reversible thermodynamics [15]. However, this estimate is not accurate because it ignores the mass transfer laws and the properties of membranes, process productivity, possible intermediate processes of mixing, and so on. The estimates based on reversible thermodynamics are not suitable for determining the optimal sequence of operations in the separation of multicomponent systems, because they depend only on the compositions of feeds and end products and do not reflect the sequence of operations in which the end product was obtained. The work needed for separation consists of its reversible work and irreversible energy losses. The losses are equal to

### 3.2. Batch membrane separation

We will first consider a batch separation of a mixture in a system consisting of two chambers separated by a membrane permeable to only one active (to be separated) component of the mixture (Fig. 5). Let and *I*, respectively. These parameters can be varied during the process. At the initial moment of time *1* and the concentration *3* at a mass transfer rate *g*, which depends on its chemical potentials on both membrane sides, *4*. The process is isothermal, and the temperature *T* is specified and remains unchanged.

The intensive variables in the second chamber are the pressure *G* of the component that passed through the membrane in time

and, hence, the reversible work of separation, which is equal to the increment of the free energy of the system:

Consequently, the minimum of the produced work corresponds to the minimum of the irreversible losses of energy, which is proportional to

The increment of entropy in the system, the minimum of which should be determined for a separation process of duration

The amount of the active component that passed through the membrane is written as

The process duration

The variation of

It follows from Eq. (81) that

The solution of Eq. (81) determines the dependence of the mixture amount in the first chamber on the active component concentration

After expression (82) is substituted into Eq. (81), the latter takes the form

First, we will find such time variation of, chemical potential

We will write the Lagrangian function F for the problem given by Eqs. (79) and (80) in view of the fact that the constant factor 1/T does not affect the optimality condition:

The mass transfer rate g is equal to zero when

To cancel out λ, we integrate the both sides of this equality from zero to

Consequently, to determine

If the flux is proportional to the difference of chemical potentials,

it follows from optimality condition (84) that

The variation of

For mixtures close in properties to ideal gases, the chemical potential (molar Gibbs energy) of the active component of the mixture is written as

where

The variation of *g*. After

For the flux defined by Eq. (85) and defined by Eq. (86), Eq. (83) takes the form:

The solution to this equation is written as

Substituting the latter into Eq. (88) gives the time variation of the pressure:

After the optimal variation of

The optimal variation of the pressure and mole fraction of oxygen in the first chamber is shown in Fig. 6. It corresponds to the separation of a gas mixture composed of carbon dioxide, 120 moles of CO_{2}, and oxygen, 180 moles of O_{2} (active component), when ^{2}/(s J),

The produced work is

Although the chemical potential for ideal solutions is written like Eq. (87), the function

we obtain

For the flux defined by Eq. (85) and

For illustration, we considered the separation of water with a high salt concentration. Like ocean water, it contained 36 g/l of salt (inert component). The other process parameters were ^{2}/(s J),

### 3.3. Membrane separation process distributed along the filter

The parameters of the system can vary with length rather than with time, as in the previous system. The flow diagram of this system is shown in Fig. 8. The mixture to be separated, which is characterized by a molar flux *L*. As the mixture travels over the length *l*, the active component passes across the membrane into the second chamber. The concentration of the active component in the mixture to be separated at the outlet of the first chamber is *s(L)*. In irreversible continuous separation, the power p expended for separation is the sum of the reversible component

which is determined at the given conditions, and the irreversible losses *p*.

The flux of the component to be distributed at section *l* is equal to

The production of entropy is determined by the expression

Assume that

If the operating regime in the first chamber is close to plug flow, the material balance equations for section *l* give equations analogous to Eqs. (81).

The above equation can be used to obtain a relationship analogous to Eq. (83):

where

Equations (92), (93), and (95) represent an optimal control problem in which

In view of this replacement, the problem given by Eqs. (92), (93), and (95) can be written as

with the constraints

The concentration

and, hence,

The same follows from constraint (97) with

In distinction to batch membrane processes, the control action in a continuous membrane separation can be additionally represented by the coefficient of heat transfer

where

In constraint (98), the mass transfer rate can be written as Eq. (100), and equality (101) can be added to the constraints of the problem. The resulting problem, given by Eqs. (96), (98), and (101), is an isoperimetric variation problem. The necessary condition for the optimality of its solution is the requirement that the Lagrangian function should be stationary with respect to

where the multipliers *F* with respect to the desired variables are written as

The above equations give the process optimality conditions:

From constraints (98) and (103) we obtain

It follows from (101) and (102) that

After expressions (105) and (104) are substituted into conditions (102) and (103), respectively, we can use the known function

Let us write the above relationships specifically for the function g written as a linear function of the difference of chemical potentials, Eq. (85), and chosen functions

Constraints (102)–(105) lead to the equations

For brevity, we will introduce the notation

and we obtain

The concentration of the active component in the first chamber declines with increasing l. Therefore, under optimal operating conditions,

To find

The evaluation of the integral gives us the desired formula:

where

Equation (108) in view of Eq. (109) yields the optimal dependence of the difference of chemical potentials on the concentration

Consequently, Eq. (95) takes the form:

Integrating this equation with specified initial conditions, we can find the variation of the concentration of the active component over the length of the first chamber under optimal operating conditions:

Substituting this expression into Eqs. (110) and (111) yields the variation of the desired variables over the length:

The minimal value of the production of entropy corresponding to the above solution is written as

We will introduce

where *s(L)* is the total contact surface area.

If the specific mass transfer coefficient of the membrane material *s(L)* are known, we can find the optimal distribution of the membrane surface area over the length of the filter:

For near-ideal gas mixtures, we can write

where *T* are assumed to be specified, and

When

The optimal curves for the pressure and mass transfer coefficient are plotted in Fig. 9, in which the data refer to the separation of a gas mixture composed of carbon dioxide CO_{2} and oxygen O_{2} (active component) when^{2}/(s J), ^{2}/(s J),

At the filter outlet,

The consumed power is

For ideal solutions, the calculation is almost the same except for the form in which the chemical potentials are written. For the first chamber,

where *v* is the molar volume of the active component.

For the second chamber,

The dependence of the solution pressure in the second chamber on the concentration is written as

For illustration, we considered the separation of water with a high salt concentration. Like ocean water, it contained 36 g/l of salt (inert component). The other process parameters were ^{2}/(s J), ^{2}/(s J),

The consumed power is

### 3.4. Conclusion

The minimal losses of energy for irreversible membrane separations with specified production rates are estimated. The variation of the driving force (difference of chemical potentials) and the distribution of the membrane surface area over the filter length corresponding to the process with minimal energy losses are found.

The obtained estimates can be used for assessing the deviation of the actual membrane separation from the optimal process and for comparing the thermodynamic efficiency of membrane separation processes with different flow diagrams, as well as for formulating and solving problems regarding the optimal sequence of operations in the separation of multicomponent mixtures.

## 4. Optimization of diffusion systems

### 4.1. Introduction

The problem of deriving work from a irreversible thermodynamic system and the inverse problem of maintaining its irreversible state by consuming energy are central in thermodynamics. For systems that are not in equilibrium with respect to temperature, the first (direct) of the above problems is solved using heat engines and the second one (inverse) is solved using heat pumps. For systems that are not in equilibrium with respect to composition, the second problem is solved using separation systems and the first one is solved using diffusion engines. As a rule, separation systems and diffusion engines are based on membranes.

There is a lot of studies of membrane separation systems and diffusion engines in the literature [5,7]. In the present paper, these systems will be considered using the theory of finite-time thermodynamics. The finite-time thermodynamics, which evolved in the past years, studies the limiting performance of irreversible thermodynamic systems when the duration of the processes is finite and the average rate of the streams is specified [14, 17]. For example, some problems for heat engines, such as maximizing the power at given heat transfer coefficients and maximizing the efficiency at given power for different conditions of contact between the working body and surroundings, are already solved. In this case, the irreversible processes of the interaction of subsystems each of which is in internal equilibrium are considered.

For systems that are not uniform in concentration, it is most important to study the limiting performance of separation systems. In this case, however, the inverse problem of studying the performance of diffusion engines is of definite interest as well. The simplest variant of this problem was first formulated by Rozonoer [17]. The review of the literature shows that this problem was discussed rather superficially.

In the present paper, we will study the limiting performance of membrane systems in the separation processes with fixed rates, focusing on the following problems:

Minimizing the amount of energy necessary for the separation of a feed mixture with a given composition into separation products with given compositions at a given average production rate.

Maximizing the power and efficiency of diffusion engines.

The solution of these problems depends strongly on whether the feed mixture used by the engine is gaseous or liquid because this determines the form of the chemical potentials of components and, hence, the driving forces of the process. For near-ideal gas mixtures, the chemical potential of component *I* of the mixture takes the form [15]:

where *I* and

we can rewrite the expression for the chemical potential in the form:

where

Although the chemical potential for liquids has the same form as Eq. (118), the form of the function

we obtain

It is assumed that the processes are isothermal and the temperatures of all subsystems are equal to T. The problems listed above will be considered for gaseous mixtures and then for liquid solutions.

### 4.2. Limiting performance of diffusion systems for gaseous mixtures

#### 4.2.1. Maximum work in a membrane process

Consider a system consisting of a thermodynamic reservoir, the intensive variables of which are fixed and are independent of mass transfer fluxes, and a working body, the intensive variables of which can be varied with time by one or another way. The system can consume external energy or generate work. In the first case, the work will be negative; in the second, positive.

The reservoir and the working body interact through a membrane that is permeable only to one (active) component of the mixture. The mass transfer rate *g* depends on the chemical potentials of the active component in the reservoir

where

When the process duration

The variation of the system entropy will be caused by the decrease in the reservoir entropy, the increase in the entropy of the working body, and the production of entropy due to the irreversible mass transfer

the variation of the entropies of the reservoir and working body with time

In this case, the function

Let us find the quantitative relationship between the work *A*, which can be extracted (consumed) in this process, and the value of *G* and the concentration

As the amount of the second component is maintained constant, we obtain

It follows from (123) and (124) that

The equations for the material, energy, and entropy balances around the system take the form:

where *h*, *s* are the molar enthalpies and entropies of the mixture in the working body and reservoir, respectively. They are related by the equation [15]:

The pressure in the working body can vary with time, provided that

The second term in the right-hand side of this equality can be calculated using

The maximum of the produced (minimum of the spent) work corresponds to the minimum of entropy production in the mass transfer process.

The problem of finding the minimum of

is convex with respect to *L* with respect to

The multiplier *L* with respect to

Consequently, the chemical potential of the active component of the working body for any rate satisfying (133) should be controlled so that the mass transfer rate should be constant.

The law of variation of the control variable, such as the working-body pressure, corresponding to this solution will not be constant in time because the mixture composition is varied during the process according to Eq. (125), in which the flux is determined by Eq. (134).

For mass transfer law (120), the minimal entropy produced is

In the case where the system contains a source of a finite capacity at constant temperature and pressure instead of the reservoir (source of an infinite capacity), the fraction of the active component varies according to an equation similar to (125). As a result, the chemical potential

Instead of the calendar time, the problem can be studied using the time of contact, when the working body moves and its parameters at every point of the loop remain constant. This can be used to determine the optimal laws of pressure variation for the zones of contact between the working body and source.

#### 4.2.2. Diffusion-mechanical cycle for maximum power

Let us consider the direct cycle of work extraction in a system consisting of a working body and two reservoirs with different chemical potentials. In the first reservoir, the chemical potential of the key element is equal to

**Alternating contact with reservoirs**. Consider the case where the working body alternately contacts the first and second reservoirs and its parameters are cyclically varied with time. Let

with the constraints placed on the increment in the amount of the working-body:

To calculate the basic values of

The number of basic values of *L* that is strictly convex with respect to

or

The roots for this equation for *L* is maximal at the basic points, we can write

which determines the value of

Let us specify the obtained relations for

It follows from (137) that

Substituting

The maximum of L with respect to

The fractions of time

The maximal work in time

where

**Constant contact with sources**. In heat engines, there can be either alternate or constant contact between the working body and sources. In the latter case, the parameters of the working body are distributed and the process in it can be regarded close to reversible if the distribution of the parameters is caused by the conductive flux. Likewise, a constant contact with sources is possible in systems that are not homogeneous in concentration, such as separation systems and diffusion engines.

In this case, the maximal power takes the form of a nonlinear programming problem:

with the constraint

The optimality constraint for this problem leads to the relation:

which together with equality (140) determines the desired variables.

LetEquality (141) can be written in the form:

The constraint

The solution to Eqs. (142) and (143) can be written as

The value of maximal power corresponding to this choice is

where the equivalent mass transfer coefficient is defined as

### 4.3. Limiting performance of diffusion systems for liquid mixtures

The result obtained above for the membrane systems consisting of a working body and a source of finite or infinite capacity using gaseous mixtures can be translated in the same form to liquid solutions with allowance for the different form of the chemical potential. Diffusion engines are most often designed for the treatment of saline water. Let us consider two flow-sheets of liquid diffusion engines.

#### 4.3.1. Diffusion engine with a constant contact between the working body and the sources

Let the system consist of two liquids with the same temperature separated by a semipermeable membrane. One of the liquids is a pure solvent and the other is a solution in which some substance of concentration *C* is dissolved. The membrane is permeable only to the solvent. The equilibrium in the system is reached as soon as the chemical potentials calculated by formula (119) become equal to each other:

Let the difference of pressure across the membrane be denoted as

Equation (144) is called the Van’t Hoff equation for osmotic pressure.

Consider the system shown in Fig. 13. The chamber to the left of the membrane contains a pure solvent at an environmental pressure equal to

where

Let

The additional flux across the membrane increases the volume of the solution, which drives a turbine and generates power

Consequently, the power r and efficiency

where the diffusion engine efficiency is the work extracted from 1 m^{3} of the concentrated solution. From here on, according to the accepted system of units, the units of power and efficiency referred to a unit membrane surface area are J/(m^{2} s) and J/m^{3}, respectively. If the relationship between

As

which is the upper bound for the maximal power.

The estimate produced by Eq. (146) can be refined if we take into consideration that *g*, *C* are related to each other by Eq. (145) and the equation of material balance on the dissolved component

Expressing *C* and *p* and

The points of maximum with respect to g for two concave functions (149) and (150) coincide. Consequently, to find the optimal value of

Equation (151) can be rewritten as

and its right-hand side can be denoted for brevity as M. Its solution will be denoted as

Numerical solution of Eq. (152) makes it possible to refine the value of the limiting power of the diffusion engine and find the corresponding operating conditions. Equation (151) determines

It should be noted that the ideal solution bounds the value of the concentration of the working solution:

The concentration should not be very high: otherwise, the molecules of the dissolved component will interact with each other and relation (144) is upset.

**Diffusion Engine with an Alternate Contact between the Working Body and Sources.** In the schematic diagram of the diffusion engine discussed in the preceding section, the working body was an open system working in constant contact with two sources under steady-state conditions. One of them supplied a concentrated solution and the other supplied a solvent.

Figure 14 shows the schematic diagram for a diffusion engine in which the working body alternately contacts each of the sources, receiving a solvent through one membrane and giving it up to a concentrated solution through another membrane. In this case, the pressure and flow rate of the working body are periodically varied: pressure increases for a lower flow rate (power

We will write the balance equations for this diagram and study its limiting performance, ignoring the energy losses for driving the flow of the concentrated solution through the bottom chamber and assuming that the concentration of the dissolved component in the

The engine power is

where

The efficiency will be defined as the ratio of power *p* to the flow rate

The rate of mass transfer is determined by the relations:

where

Figure 15 demonstrates the cycle of the working body of this diffusion engine. The power *abcd*.

The power of the diffusion engine will be determined when the relationship between the osmotic pressures in the chambers and the flow rates is ignored. To do it, we will solve the problem of constrained optimization:

with the constraints:

It follows from Eq. (154) that

Let us introduce the equivalent permeability:

and write the equation:

Then

The maximum of this expression, which is equal to

is reached at

Keeping in mind that the osmotic pressures in the chambers are related to the concentrations by Van’t Hoff equation (144) and the concentrations are related to the flow rates *g*, we obtain

In view of these relations, expression (155) for the engine power takes the form:

The expression for the efficiency is written as

The points of maximum with respect to g for the criteria (156) and (157) coincide. Therefore, we can use either of them in the conditions of optimality to find *p* with respect to *g* leads to an equation for the optimal flow rate:

The solution to Eq. (158) will be *p* take their maximal values. The values of flow rate *p* and

### 4.4. Conclusion

The estimates obtained in the present paper for the limiting performance of diffusion engines can be used to make their reversible-thermodynamics analysis more accurate and consider the influence of the kinetic factors (mass transfer relations, membrane permeabilities) and production flow rate. These estimates can also be used for the optimization of more complex membrane systems. The capacity of membrane systems increases in proportion to the membrane permeability. In this case, the performance of membranes is decreased by the nonuniformity of concentrations in the solution, polarization phenomena, and the other factors ignored in obtaining the above estimates.