Minimal dissipation conditions of heat exchange systems

International Journal of Heat and Mass Transfer 110 (2017) 539–544

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Minimal dissipation conditions of heat exchange systems Stanislaw Sieniutycz a, Anatoliy M. Tsirlin b, Andrei A. Akhremenkov b,⇑ a b

Faculty of Chemical & Process Engineering, Warsaw University of Technology, Warszawa, Poland Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalesskiy, Russia

a r t i c l e

i n f o

Article history: Received 13 January 2017 Received in revised form 5 March 2017 Accepted 10 March 2017

Keywords: Finite-time thermodynamics Heat exchange systems Entropy production

a b s t r a c t The study estimates minimal entropy production, the corresponding distribution of heat exchange surfaces and contact temperatures for heat exchange systems with fixed total heat load and constant heat transfer coefficients for the whole system. Mathematical conditions to reach this minimal entropy production are found. For a typical heat exchange system entropy production can be expressed through flow’s temperatures. Comparison of the actual entropy production with its minimal value leads to estimates of thermodynamic feasibility and efficiency of the system. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Classical estimates for limiting performance of technological systems (heat and refrigerating machines, separation systems, chemical reactors, etc.) are based on efficiency limits of reversible processes (Carnot efficiency, reversible work of separation, standard chemical affinity, etc.). These estimates are very important but cannot be reached by real systems as the reversible models do not take into account effects of flow intensity, contact surfaces and some other factors related to a fixed productivity and finite dimensions of the equipment. In some cases reversible estimations lose any sense, as in the case of stationary non-equilibrium systems with several reservoirs or systems with substance and energy inflows. The heat exchanger is an example of such a system. Therefore, thermodynamic estimation of the heat exchanger efficiency must take into account the effect of a restricted contact surface (integral coefficient of heat exchange) and of the finite heat load (that is, the finite amount of heat transmitted from hot to cold flows per unit of time). For the efficiency estimation of such systems we use the exergy method (see [1,2], etc.), that is the method of rating the system’s exergy loss. This exergy loss is proportional to the entropy production and to the ambient temperature T 0 . The minimum exergy loss at given temperatures of hot flows and for a given heat load corresponds to the maximum average temperature of cold flows at the outlet of the heat exchanger.

⇑ Corresponding author. E-mail addresses: [email protected] (S. Sieniutycz), [email protected]. ru (A.M. Tsirlin), [email protected] (A.A. Akhremenkov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.031 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

In this study we have solved the problem of the minimal entropy production (minimal exergy losses) in some typical multi-flow heat exchange systems. Such estimation permits to:
define the significance of such factors as temperatures and heat capacities flows, heat loads, integral coefficients of the heat exchange, etc., in a selected heat exchange system;
utilize the optimal heat exchange conditions to make a designed system’s configuration closer to the ideal when designing new systems;
evaluate thermodynamic efficiencies of the designed system by means of comparison of its actual entropy production with the least possible entropy production under the same conditions. The entropy production in a heat exchange system can be calculated for known fluxes of heat capacity rate (water equivalent), input and output temperatures of these flows. Flow’s water equivalent is product of flow rate g [kg/s] and specific heat C [J/kg K] (W ¼ gC). Alternatively, one can also calculate the sum of products of heat loads q and corresponding driving forces, say X, for each heat exchanger.

XðT 1 ; T 2 Þ ¼

1 1 T1  T2 q  ¼ ¼ ; T2 T1 T1T 2 aT 1 T 2

ð1Þ

where T 1 ; T 2 – the temperature of hot and cold flows, q ¼ aðT 1  T 2 Þ. Driving force usually it proportional to the q. In the latter case the entropy production is proportional to q2 and inversely proportional to the product of heat coefficient a and contact area, S. For a constant system’s exchange area, S, this entropy production could be arbitrarily small if the heat load q is arbitrarily small and the

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List of symbols C g j; i S T; u q W

specific heat (J/K kg) flow rate (kg/s) index entropy (J/K) temperature (K) heat load (J/s) water equivalent (J/K s)

Greek symbols heat transfer coefficient (J/K s) entropy production (J/K s) k Lagrange coefficient

a r

heat coefficient a is fixed or if the coefficient a is arbitrarily large and the load q is fixed. The above statements are true for every heat exchanger and for the heat exchange system as a whole. However, since in the process reality neither of these limiting conditions (for a and q) could be implemented, we are allowed to assume that q and a are fixed and bounded quantities. In fact, this assumption is equivalent to the acceptance of some constraints which bound the entropy production within an admissible region of the process space. The present study estimates the minimal entropy production for such constrained systems. Many results stemming from the present theory have previously been known for the special case of two-flow heat exchangers [3–5]. Therefore, in what follows, we shall focus on multi-flow heat exchange systems. The minimal entropy production condition bounds the feasibility of heat exchange systems in the plane with coordinates q and a. Relation between entropy production and input/output parameters of flowing flows. The total differential of molar entropy expressed in terms of heat capacity and differentials of temperature and pressure has the form [2]


 cp @v ds ¼ dT  dp: T @T p

ð2Þ

where T - absolute temperature, cp - molar heat capacity at a constant pressure p, and v - molar volume. Integrating this equation with respect to temperature and pressure between initial and final values of T and p we shall get the entropy production per one mole. For a known expenditure of a molar flow, a partial entropy production can be attributed to this flow value. By summing these partial values with respect to flows we can get the entropy production in whole system. For fluids of a constant heat capacity and under constant pressure


 T# ; s  s0 ¼ cp ln T0

ð3Þ

T 0 and T # - the temperature at the inlet and outlet [K]. The entropy production due to the change of a state of the ith flow is equal to the product of its water equivalent and the logarithm of temperature ratio involving input and output temperatures

ri ¼ W i ln


 T #i ; T 0i

i ¼ 1; 2; . . . ;

production in cold flow. Heat is supplied to them at a higher temperature and output temperature is increased. To link entropy production of flows output temperature of cold flow should be found explicitly. Let us consider a double-flow heat exchanger, Fig. 1, characterized by heat load q and water equivalents W 1 and W 2 . We shall write down an equation linking the entropy production with system’s input temperatures, ðT 01 ; T 02 Þ, and output temperatures ðT #1 ; T #2 Þ

r ¼ r1 þ r2 ¼ W 1 ln



 T 01  q=W 1 T #2 þ W 2 ln : T 01 T #2  q=W 2

Now we assume that the values of hot flows and heat load are fixed, so that r1 is also fixed. From Eq. (5) a relation follows which links the output temperature of cold flow with the entropy production r

T #2 ¼

q r1 : Þ W 2 ð1  exp½ rW 2

ð6Þ

The output temperature of the cold flow increases monotonously with r. If r is fixed, a or q increase, to increase the output temperature of cold flow. For a double-flow heat exchange with the given heat load these holds the following Proposition is valid: Proposition 1. The entropy production of the system with the hot flow input temperature T 01 and heat capacity rate W 1 cannot be less than

r ¼ a

ð1  mÞ2 ; m

ð7Þ

(Fig. 2), where

m¼1

W1

a

ln

T 01 : T 01  q=W 1

ð8Þ

In the case when the cold flow’s input temperature T 02 and the heat capacity rate W 2 are fixed, the minimal entropy production is

r ¼ a

ðn  1Þ2 ; n

ð9Þ

ð4Þ

where W i ¼ g i C i - product of flow rate g i and specific heat C i (water equivalent). Total entropy production is the sum of entropy productions attributed to individual flows. 2. Double-flow heat exchange If heat load and input temperature for hot flow are fixed to decrease system’s entropy production we should reduce entropy

ð5Þ

Fig. 1. Double-flow heat exchanger.

S. Sieniutycz et al. / International Journal of Heat and Mass Transfer 110 (2017) 539–544

541

3. Multiflow heat exchange A calculation of the complex heat exchange system with several hot and cold flows implies the need to determine flow contact temperatures, the distribution of heat exchange surfaces and heat loads. To solve this problem heuristic algorithms are usually used or the problem is solved only for the given system’s structure, [10–14] (see Fig. 3). In the papers [15,16] the heat exchange system with three flows and a given structure is considered. It is shown that the entropy production with in a countercurrent flow is lower than in a concurrent flow. Below it is proved that for the system with an arbitrary structure the entropy production is bounded from below and gets conditions to reach this bound. The entropy production in the multiflow heat exchange system with a linear heat transport law is equal to a sum of entropy components:

r¼

Fig. 2. Attainability boundary for the double-flow heat exchanger.

X

W iþ ln

i

where

n¼1þ

W2

a

ln

T 02 þ q=W 2 : T 02

ð10Þ

Fig. 2 shows the minimum entropy production as a function of heat exchanger heat load q [W] and integral heat transfer coefficient a [W/K]. Values of the temperature of hot flow and its water equivalent are fixed. These estimations are valid for the linear heat transport law

q ¼ aðT 1  T 2 Þ

ð11Þ

and for every section of the heat exchanger. a – the heat transfer coefficient [J/s K m2]. The minimal entropy production, Eqs. (7) and (9), can be achieved in a countercurrent tube heat exchanger with a constant lengthwise exchange coefficient a, provided that the flow temperature ratio changes inversely to the heat capacity ratio. In this case, the temperature ratio for every section of the exchanger is

T 1 ðlÞ 1 ¼n¼ : T 2 ðlÞ m

ð12Þ

T i#þ X T j# þ W j ln : T i0þ T j0 j

Here the summation is over heat exchanges, each characterized by the index i for hot flows, and the index j for cold flows. Let us obtain the lower estimate of the entropy production in the multi-flow heat exchange system and corresponding to this low estimate the temperature distributions, as well as distributions of heat exchange coefficients and heat loads in heat exchangers. This estimation permits to find a thermodynamic efficiency

q¼

r 61 r

ð15Þ

for a real working system. When designing this system, one can use our calculation to obtain the system’s efficiency sufficiently close to that following from our calculations which utilize distributions of temperatures and heat-exchange contact area. The case when temperatures of hot flows are fixed. Let us consider the input temperature of heating flow T 0iþ and the heat capacity rate WðT 0iþ Þ ¼ W i þ as fixed quantities. We will denote the heat load for a flow with temperature T 0iþ as qi , and the corresponding heat exchange coefficient as ai . The distribution of heat exchange surfaces is equal to the distribution of the heat exchange coefficients, therefore the following coefficient

a¼

X

ai ;

ð16Þ

i

The proof of Proposition 1 as a consequence of minimal dissipation conditions was given in [8]. The result was specialized for the linear heat transport law [4,6,7]. This result will now be used for the multi-flow heat exchanger. Let us remark that the condition of constant intensity of sources and constant temperatures of working medium in the heat engine constitutes an optimal condition for the heat engine with finite capacity sources [9]. The ratio of the minimal entropy production r and the real entropy production defines the thermodynamic efficiency of a heat exchanger. As shown in [3], for an arbitrary heat transport law qðT 1 ; T 2 Þ, the entropy production is minimal if at each point of hot and cold flows’ contact the following conditions hold for q2 ðT 1 ; T 2 Þ

q2 ðT 1 ; T 2 Þ ¼ k

@q 2 T ; @T 2 2

ð13Þ

where k is a constant, which depends on a contact surface and the heat load.

ð14Þ

Fig. 3. Structure of the multi-flow heat exchanger.

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will be assumed as a constant quantity. We shall also use the sum of heat loads

@ ri m2  1 ¼  2i ¼ k2 ; @qi mi T #iþ

q¼

or

X qi :

ð17Þ

i

When T iþ ; W iþ and q are fixed, then the average enthalpy of hot flows in the outlet flow of the system is fixed as well. Outlet temperatures of hot flows depend on inlet temperatures and the heat load as follows

T #iþ ¼ T 0iþ  qi =W iþ

i ¼ 1; . . . ; n:

ð18Þ

The problem of entropy production minimization has the form

r¼

X

ri !

i

min

uðT iþ ;T 0iþ Þ;ai ;qi

;

ð19Þ

where uðT iþ ; T 0iþ Þ the current temperature of the cold flow which exchanges the energy with the hot flow of the input temperature T 0iþ and the current temperature T iþ . Derivation of computational relations. We shall solve the problem (16)–(19) in two stages. At the first stage, we assume that qi and ai are fixed for all i. Keeping these conditions in mind, we define the correlations of current temperatures for the heated flows ui , and the heating flows T iþ corresponding to the minimal entropy production ri (denoted in Eq. (9) by r ) for the heating flow with initial temperature T 0iþ . At the second stage, we define the contact surface distribution ai and the heat load distribution qi , both capable to minimize r subject to the constraints (16) and (17). The first problem is solved in Section 2, Eqs. (9) and (10). For every inlet temperature of a hot flow

mi ¼

uðT i ; T 0i Þ W iþ T 0iþ ¼1 ln ; T iþ ai T 0iþ  Wqiþi

ri ¼ ai

ð1  mi Þ2 ; mi

i ¼ 1; 2; . . . n:

i ¼ 1; 2; . . . n:

ð20Þ

ð21Þ

The second stage results in a and q distribution problem subject to the constraints (16), (17) and the condition

r¼

X

ri ½T 0iþ ; ai ; W iþ ; qi  ! min :

ð22Þ

aP0; qP0

i

Here the Lagrange function takes the following form

L¼

X

ri ðT 0iþ ; ai ; W iþ ; qi Þ  k1

i

X

a i  k2

i

X qi :

ð23Þ

@ ri ¼ k1 ; @ ai

@ ri ¼ k2 @qi

8i:

ð24Þ

@mi W i T 0iþ 1  mi ¼ 2 ln ¼ ; @ ai ai ai T #iþ

ð26Þ

ð27Þ

From (25) it follows that for the optimal heat exchange organization, m is independent of T 0iþ and the output temperatures for all flows should be equal to each other:

T #iþ ¼ T #þ : T #þ is unambigiously determined by Eq. (17) since

q¼

X W i ðT 0iþ  T #þ Þ:

ð28Þ

i

After defining

Wþ ¼

X W iþ ;

ð29Þ

i

X T 0iþ W iþ ;

T 0þ W ¼

ð30Þ

i

we obtain

T 0þ W  q : Wþ

T #þ ¼

ð31Þ

In order to express m by initial data rewrite Eq. (20) as follows

ai ¼

W iþ ðln T 0iþ  ln T #þ Þ : 1m

ð32Þ

As ai is non-negative then T 0iþ P T #þ . Therefore in a heat exchange system only hot flows with a temperature higher than T 0iþ > T #þ should be used. If T 0iþ 6 T #þ then all contact surface areas for this flow should be zero. This result corresponds to the fact that, in optimal cycles of the heat engine with several sources, it is not advantageous to contact with hot sources having temperatures lower then T þmin and cold sources with temperature greater then T max . After summing the left and the right hand sides of (32), where the integral heat exchange coefficient is fixed, we find m in the form

m¼1

In order to calculate derivatives in (24), let us extract the following derivatives

1X W iþ ðln T 0iþ  ln T #þ Þ:

a

ð33Þ

i

Thus the optimal distribution of heat exchange coefficient is

W iþ ðln T 0iþ  ln T #þ Þ

ai ¼ a X

ðln T 0iþ  ln T #þ Þ

;

ð34Þ

i

the heat load distribution is

qi ¼ W iþ ðT 0iþ  T #þ Þ;

@mi 1 ¼ ; @qi ai T #iþ

ð35Þ

and the least possible entropy production is

r ¼ a

@ ri m2  1 ¼ ai i 2 : @mi mi Hence Eq. (24) takes the following form


2 @ ri 1  mi ¼ ¼ k1 ; @ ai mi

1  m2i : m2i k2

T #iþ ¼

i

where k1 and k2 are some constants independent of i. The stationarity conditions for L lead to equations

i ¼ 1; 2; . . . n

ð25Þ

ð1  mÞ2 : m

ð36Þ

After comparing Eq. (36) with the entropy production r of a real heat exchange system characterized by total heat exchange coefficient a, hot flow inlet temperatures T 0iþ , heat capacity rates W iþ and the outlet enthalpies of heating flows W iþ ; T #iþ we can esti mate the system’s efficiency as q ¼ rr .

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In order to make the system characteristics closer to ideal, we should distribute the heating flows and the heat exchange surfaces according to Eqs. (35) and (34) and choose contact temperatures according to Eq. (33) i.e. the condition of temperature constancy in terms of m. This can be done by reducing the heat exchange surface in heat exchangers with characterized by temperature ratio for hot and cold flows higher than the system’s average value. For heat exchangers with the temperature ratio lower then the average value, the heat exchange surface should be enlarged. Similarly, the heat intake should be increased for heating flows with output temperatures higher than the average outlet temperature of heating flows. The findings can be summarized in Proposition 2a. Proposition 2a. For heat exchange systems with given number of hot flows, inlet temperatures, heat capacity rates, total heat load and heat exchange surface in a linear heat transport law, the entropy production satisfies inequality r P rþ . The final formulas describing the optimal choice for output temperatures of heated flow, the heat load, the heat exchange coefficient, the temperature ratio for contact flows and the least possible dissipation take the following form: k X

T #þ ¼

T i0þ W iþ q

;

i¼1

k X W iþ i¼1

q0

¼ W iþ ðT i0þ  T #þ Þ;

a0 ¼

aW iþ ðln T i0þ ln T #þ Þ

k X

;

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > i¼1 > > > > > > k X > >  > W iþ ðln T i0þ  ln T #þ Þ; > m ¼ 1  1a > > > i¼1 > > > > > > ð1mÞ2  > > rþ ¼ a m ; > > > >  a0 ¼ q0 ¼ W i ¼ 0; T i0þ 6 T #þ ; ; W iþ ðln T i0þ ln T #þ Þ

k X

T i W i þq

T # ¼

i¼1

k X

;

W i i¼1

qi ¼ W i ðT #  T i Þ;

ai ¼

aW i ðln T # ln T i Þ

k X

;

W i ðln T # ln T i Þ

i¼1 1 m

¼ 1 þ a1 

k X W i ðln T #  ln T i Þ; i¼1



r ¼ a m1 þ m  2 ; a ðT i Þ ¼ q ðT i Þ ¼ W i ¼ 0; T i P T # :

9 > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > ;

ð38Þ

The last requirement states that in the system with the minimal entropy production all input temperatures of cold flows should neither exceed nor be equal to T # .

r P r :

ð39Þ

This inequality is a thermodynamic feasibility condition satisfied in multiflow heat exchange systems. Heat exchangers for which this condition is violated cannot be implemented. In a real system parameters of cold and hot flows are usually given. To evaluate the thermodynamic efficiency of such system we should take use the maximal value of r or r , to get more detailed estimate of efficiency.

ð37Þ

4. Example Fig. 4 shows the heat exchange system with three hot and cold flows. Hot flows index is þ, whereas cold flows index is . Temperatures, in Kelvins, are shown on the figure next to the arrows. Heat exchange coefficients [kJ/sK] are given in circles, and flows of water equivalents for each inlet flow are W iþ and W i . For these conditions we get heat loads qij for each heat exchanger (Table 1). In accordance with Eq. (5), the entropy production in

where k is the number of hot flows, which it is advisable to use (T i0þ > T #þ ). Since input temperatures and water equivalents for hot flows are given it’s obtain input flows of heat and entropy. For given heat load output heat flows also given. Average temperature for output temperatures the higher, the less its entropy flow. So minimization of entropy production in system is equal to maximization of average temperatures of output flows. Temperatures of cold flows are fixed. To estimate r we assume that characteristics of hot flows are given and the task is to select cold flows parameters in such a way that the entropy production is minimal. But it may also happen that characteristics of cold flows are given. We shall denote variables of cold flows (temperatures, heat capacity, rates, etc.) with the index ”—”. Using previous arguments, we can find the value of minimal entropy production for a system with fixed temperatures of cold flows, T 1 P T i P T 2 and heat capacity rates, W i . Proposition 2b. For heat exchange systems with given number of cold flows, inlet temperatures, heat capacity rates, total heat load and heat exchange surface, the entropy production satisfies inequality r P r which means that. Below we present the computation relations derived for this case:

Fig. 4. The heat exchange system.

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Table 1 Heat loads qij .

5. Conclusion

jni

1

2

3

1 2 3

885 591.9 1233.5

416 375.3 983.7

271.5 452 549.2

this system, calculated as the sum of entropy production increases for each flow, is described by the equation. The entropy production in this system (14) calculated as sum of entropy production increases for each flow.

r¼

3 3 X T i#þ X T j# W iþ ln þ W i ln : T T j0 i0þ i¼1 j¼1

ð40Þ

The numerical value of the entropy production is equal to

r ¼ 5574 [kJ/sK].

We use the set of formulas in Eq. (37) to perform calculations of the optimal thermodynamic system with total heat load q ¼ 5851 [kJ/s] and heat exchange coefficients [in conductance units] a ¼ 48 [kJ/sK]. We will get the optimal output temperature for hot flows T #þ ¼ 453:4 K. Comparing this temperature with inlet temperatures for hot flows, we conclude that the third flow with the temperature 450 K should be excluded from the system. Recalculation of T #þ for two hot flows for the same values of q and a gives T #þ ¼ 457:2. Optimal values of heat loads for first and second flows are respectively equal to qðT 01þ Þ ¼ 3712 [kJ/s] and qðT 02þ Þ ¼ 2140 [kJ/s]; the optimal contact surface distribution for these two flows, Eq. (37), corresponds with aðT 01þ Þ ¼ 29:9 [kJ/ sK], aðT 02þ Þ ¼ 18:1 [kJ/sK]. The ratios of effective temperatures should be equal for cold and hot flows, and are equal to m = 0.752. The minimal entropy production in this system is equal to r ¼ 3:93 [kJ/sK]. For this system the thermodynamic efficiency is equal to

q¼

r ¼ 0; 705: r

Combining results for real and optimal systems we can formulate the following recommendations: 1. The flow with the input temperature 450 K should be removed from the system. Heat exchange surfaces for other flows grow for the first flow from 19 to 30, for the second one from 14 to 18 [kJ/sK]. 2. Heat exchange surfaces for each flows should be distributed in such a way that the ratio of effective temperatures is close to 0.75. Note that in the real system this value changed from 0.63 to 0.88. 3. The output temperatures of hot flows should be close to 457.2 K.

In this paper we define the thermodynamically optimal organization of the heat exchange in order to achieve the least possible entropy production for a system with the fixed heat load and the fixed total heat transfer coefficient. We also define the appropriate heat load distribution and the heat exchange coefficient distribution for the input flows. Our study permits to estimate the thermodynamic efficiency for a arbitrary heat exchange system and to refine it, as well as to analyze the system’s dependence on such factors as the temperature variations of flows or changes in the heat exchange surfaces. Optimal conditions for a heat exchanger cannot be implemented for some systems for any distribution of heat exchange surfaces. It means that the structure parameters of these systems should be changed. For example, in heat exchange system with one cold flow and several hot flows optimal conditions could not be reached. But for these system given results allow to estimate it efficiency. Analogous results can be obtained for other heat transport laws. In this case the condition of the least possible dissipation should be used for a given but different kinetics. References [1] V.M. Brodjanskiy, V. Fratsher, K. Mikhalek, Exergy methods and its applications, Energoatomizdat, Moscow, 1988. [2] F. Bosnjakovic, Technical Thermodynamics, Holt R&W, New York, 1965. [3] A.M. Tsirlin, Optimal control of heat- and mass-exchange process, Izv. AN SSSR. Technicheskaya Kibernetika 2 (1991) 81–86. [4] B. Andresen, J.M. Gordon, Optimal heating and cooling strategies for heat exchangers design, J. Appl. Phys. 1 (1992) 71–78. [5] R.S. Berry, V.A. Kasakov, S. Sieniutycz, Z. Szwast, A.M. Tsirlin, Themodynamic Optimization of Finite Time Processes, Wiley, Chichester, 1999. [6] S.A. Amelkin, B. Andresen, J.M. Burzler, K.H. Hoffmann, A.M. Tsirlin, Thermomechanical systems with several heat reservoirs: maximum power processes, J. Non-Equilib. Thermodyn. 30 (1) (2005) 67–80. [7] S.A. Amelkin, B. Andresen, J.M. Burzler, K.H. Hoffmann, A.M. Tsirlin, Maximum power processes for multi-source endoreversible heat engines, J. Phys. D: Appl. Phys. 37 (9) (2004) 1400–1404. [8] A.M. Tsirlin, Optimal control of the irreversible processes of heat and mass transfer, Soviet J. Comput. Syst. Sci. 2 (1991) 171–179. [9] P. Salamon, A. Nitzan, B. Andresen, R.S. Berry, Minimum entropy production and the optimization of heat engines, Phys. Rev. A. 21 (1980) 2115. [10] V.V. Kafarov, V.P. Meshalkin, V.L. Perov, Mathematical foundations of computer-aided design of chemical plants, Ximiya, 1979. [11] S. Sieniutycz, J. Jezowski, Energy Optimization in Process Systems, Elsevier LTD, UK, 2009. [12] K. Hartmann, I. Hacker, L. Rockstroh, Modelierung und Optimierung Verfahrenstechnischer Systeme, Akademie Verlag, Berlin, 1978. [13] A. Tedder, D.F. Rudd, Parametric studies in industrial distillation: Part II. Heuristic optimization, AIChE J. 24 (2) (1978) 316–323. [14] A.M. Tsirlin, A.A. Akhremenkov, I.N. Grigorevsky, Minimal irreversibility and optimal distributions of heat transfer surface area and heat load in heat transfer systems, Theor. Found. Chem. Eng. 42 (2) (2008) 2037. [15] Ruan Deng-Fang, Yuan Xiao-Feng, You-Rong Li, Shuang-Ying Wu, Entropy generation analysis of parallel and counter-flow three-fluid heat exchangers with three thermal communications, J. Non-Equilib. Thermodynam. 36 (2) (2011) 141154. [16] S.A. Amelkin, B. Andresen, P. Salamon, A.M. Tsirlin, V.N. Umaguzhina, Limits of mechanical systems with multiple sources, Izvestiya Akademii Nauk. Energetika. (1) (1999) 152–158.