Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis

Article Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis

Rafał Laskowski *, Adam Smyk, Artur Rusowicz and Andrzej Grzebielec

Institute of Heat Engineering, Warsaw University of Technology, 21/25 Nowowiejska Str., 00-665 Warsaw, Poland; [email protected] (A.S.); [email protected] (A.R.); [email protected] (A.G.) * Correspondence: [email protected]; Tel.: +48-22-234-5297

Academic Editor: Yan Jin Received: 1 October 2016; Accepted: 6 December 2016; Published: 10 December 2016

Abstract: The diameter and configuration of tubes are important design parameters of power condensers. If a proper tube diameter is applied during the design of a power unit, a high energy efficiency of the condenser itself can be achieved and the performance of the whole power generation unit can be improved. If a tube assembly is to be replaced, one should verify whether the chosen condenser tube diameter is correct. Using a diameter that is too large increases the heat transfer area, leading to over-dimensioning and higher costs of building the condenser. On the other hand, if the diameter is too small, water flows faster through the tubes, which results in larger flow resistance and larger pumping power of the cooling-water pump. Both simple and complex methods can be applied to determine the condenser tube diameter. The paper proposes a method of technical and economic optimisation taking into account the performance of a condenser, the low-pressure (LP) part of a turbine, and a cooling-water pump as well as the profit from electric power generation and costs of building the condenser and pumping cooling water. The results obtained by this method were compared with those provided by the following simpler methods: minimization of the entropy generation rate per unit length of a condenser tube (considering entropy generation due to heat transfer and resistance of cooling-water flow), minimization of the total entropy generation rate (considering entropy generation for the system comprising the LP part of the turbine, the condenser, and the cooling-water pump), and maximization of the power unit’s output. The proposed methods were used to verify diameters of tubes in power condensers in a200-MW and a 500-MW power units.

Keywords: power plant condenser; minimization of entropy generation rate; technical and economic optimisation of condenser tube diameter

1. Introduction Condensers are used in steam power plants to close the thermal cycle and transfer the heat of condensation to the environment. Condensers are normally shell-and-tube heat exchangers where steam fed from the low-pressure (LP) part of the turbine condenses. For steam condensation, water is used and it can be drawn from a large water reservoir or a watercourse, such as a river, sea or lake (an open cooling cycle) or from a basin located under a cooling tower (a closed cooling cycle) [1]. Parameters (temperature and the mass flow rate) of cooling water at the condenser inlet affect the pressure of condensing steam, which in turn impacts on the power generated in the LP part of the turbine. The higher the temperature of cooling water at the condenser inlet is, the higher the pressure of the condensing steam becomes [2–5]. As the cooling-water mass flow rate rises, steam pressure in the condenser decreases [6,7], while the resistance to flow and power supplied to the cooling-water pump increase. Thus, choosing appropriate parameters of the condenser’s operation and geometry

Entropy 2016, 18, 444; doi:10.3390/e18120444 www.mdpi.com/journal/entropy Entropy 2016, 18, 444 2 of 20

(tube diameter, heat transfer area, condenser length, and the number of tube) for a given power unit is an issue involving multiple criteria. The condenser is located under the LP part of the turbine; therefore the condenser length is determined by the length of the LP part. When steam power units are retrofitted, the condenser may be subject to an upgrade; during such an upgrade it is possible to install tubes of a corrected diameter that provides better condenser performance. Choosing condenser tubes of a diameter that is too large (over-dimensioning) increases the heat transfer area and costs of building the condenser. Furthermore, if, for a constant cooling-water mass flow rate, the tube diameter is too large, water flows through the tubes at a lower speed and heat transfer conditions deteriorate, e.g., the heat transfer coefficient on the water side is lower. On the other hand, if the diameter is too small, water flows faster through the tubes, which improves heat transfer conditions, but also results in larger flow resistance and larger pumping power (more electric power is used by the cooling-water pump). When determining the appropriate diameter of condenser tubes, one should consider multiple issues, and this paper focuses on the issue of choosing the right diameter. The tube diameter was examined for a condenser in a 200-MW and a 500-MW power units, since units of this capacity are currently in the process of modernization in Poland. In addition, the units differ in capacity and efficiency, and their condensers have tubes of different inner diameters. To this end, an economic method was applied, taking into account the profit from electric power generation and costs of pumping cooling-water and building the condenser. The results obtained by this method were compared with those provided by the following simpler methods: minimization of the entropy generation rate per unit length of a condenser tube (considering entropy generation due to heat transfer and resistance of cooling-water flow), minimization of the total entropy generation rate (considering entropy generation in the LP part of the turbine and the cooling-water pump), and maximization of the power unit’s output. Two fundamental physical phenomena occur in heat exchangers: transfer of heat from the hot medium to the cold one, and resistance of the media to flow. These two processes can be compared and their combined effect can be calculated based on entropy generation. According to the second law of thermodynamics, the heat flow and flow resistance, being irreversible processes, are accompanied by entropy generation. The entropy generation represents irreversible processes, involving losses, which is why designers of heat exchangers should bear in mind that the irreversible processes should be kept as minimum as possible. In a per-unit model of the entropy generation rate for a condenser, the entropy generation resulting from the heat flow and the resistance to flow on the cooling-water side per unit length of a condenser tube are taken into account [8–10]. The cooling-water mass flow rate affects the condenser performance and parameters, and the power required by the cooling-water pump. A change in steam pressure in the condenser is followed by a change in the power of the LP part of the turbine. Hence, in addition to entropy generation in the condenser due to heat flow and cooling-water flow resistance, the model of the total entropy generation rate also considers entropy generation in the LP part of the turbine and in the cooling-water pump. Since the main purpose of a steam power unit is generating electric power, the model involving the unit’s maximum output serves to examine the difference between the power generated by the turbine and the power required to drive the cooling-water pump. Assuming a constant rate of heat flow provided in fuel fed to a boiler, the maximum output of the power unit matches the output at the unit’s maximum efficiency. Papers analysing the efficiency and output of power units include [11–15]. The economic approach to choosing the shell-and-tube heat exchanger geometry commonly uses a cost-based method taking into account costs of building the heat exchanger and forcing out media with pumps on an annual basis [16–21]. Algorithms used to determine optimum heat exchanger geometry with the cost-based objective function include genetic algorithms and particle swarm algorithms [22–25], the Artificial Bee Colony (ABC) algorithm [26], the imperialist competitive algorithm (ICA) [27], the biogeography-based (BBO) algorithm [28], the gravitational search algorithm [29], and the electromagnetism-like algorithm (EM) [30]. Optimization functions involving the total cost of condenser construction and exergy losses can also be found in the literature [31–33]. The parameters of a condenser in a steam power unit affect the power generated by the LP part of the turbine. Therefore, in this paper, to analyze the choice of Entropy 2016, 18, 444 3 of 21 losses can also be found in the literature [31–33]. The parameters of a condenser in a steam power Entropy 2016, 18, 444 3 of 20 unit affect the power generated by the LP part of the turbine. Therefore, in this paper, to analyze the choice of the optimum inner diameter of condenser tubes from the economic point of view, a formulathe optimum was used inner taking diameter into ofaccount condenser the profit tubes fromfrom theelectric economic power point generation of view, by a formula the LP waspart usedof thetaking turbine into less account the cost the of profit pumping from electriccooling power water generationand of building by the the LP condenser part of the on turbine a one-year less the basis.cost ofThe pumping method cooling of the waterminimum and ofentropy building rate the generation condenser onwas a one-yearused as complementary basis. The method to ofthe the economicminimum method, entropy since rate the generation former wasreflects used the as co complementaryntribution of design to the economicparameters method, of equipment, since the includingformer reflects the steam the contribution condenser, ofto design losses parameters occurring ofduring equipment, operation including of the the equipment. steam condenser, To determineto losses the occurring optimum during condenser operation tube of diameter, the equipment. in addition To determine to the appropriate the optimum objective condenser function, tube onediameter, should also in addition consider to constraints, the appropriate such as objective minimum function, and maximum one should flow also velocities consider of constraints, water in thesuch tubes, as minimummaximum and pressure maximum drop flow and velocitiesgeometrical of waterconstraints in the [34 tubes,–37]. maximum The issue pressureof choosing drop the and rightgeometrical inner diameter constraints relates [34– 37not]. Theonly issue to ofheat choosing exchangers the right but inner also diameterto district relates heating not onlypipelines to heat [38–41].exchangers but also to district heating pipelines [38–41]. MathematicalMathematical modelling modelling and and heat heat transfer numericalnumerical simulations simulations of of power power condensers condensers have have been beencarried carried out for out many for years many by ayears number by of scientists.a number Zero-dimensional of scientists. [Zero-dimensional42–50], one-dimensional [42–50], [49 ], one-dimensionaltwo-dimensional [49], [50 –57two-dimensional], quasi-three-dimensional [50–57], quasi-three-dimensional [58–61], and even three-dimensional [58–61], and [even62,63 ] three-dimensionalmodels have been [62,63] developed. models have Due been to considerable developed. Due dimensions to considerable of power dimensions condensers of power (as the condenserslargest shell-and-tube (as the largest heat shell-and-tube exchangers), heat two-dimensional exchangers), two-dimensional models are most models often used are frommost amongoften usedmulti-dimensional from among multi-dimensional ones. In the paper, ones. a zero-dimensional In the paper, a zero-dimensional model of the condenser model (anof the initial condenser analysis) (anwas initial chosen analysis) to determine was chosen the optimalto determine condenser the op tubetimal diameter. condenser If, tube in the diameter. process If, of in retrofitting the process the ofunits retrofitting considered, the units further considered, analysis of further this issue analys is required,is of this tube issue diameters is required, are plannedtube diameters to be verified are plannedusing a to two-dimensional be verified using model a two-dimensional of the condenser. model of the condenser.

2. 2.Description Description of ofCondensers Condensers under under Consideration Consideration FigureFigure 11 showsshows the the part part of of the the thermal thermal system system including including the the subsystem subsystem considered considered of of the the 500-MW500-MW power power unit: unit: the the LP LP part part of of the the turbine, the condenser,condenser, andand the the cooling-water cooling-water pump, pump, with with the thenotation notation used. used.

FigureFigure 1. 1. TheThe subsystemsubsystemunder under consideration, consideration, being bein a partg a ofpart the 500-MWof the 500-MW power unit: power the low-pressureunit: the

low-pressurepart of the part turbine of (LP)the turbine with extractions (LP) with extextractions1 to ext3; ext the1 condenserto ext3; the (C); condenser and the (C); cooling-water and the cooling-waterpump (CWP). pump (CWP).

TheThe connections connections between between the the LP partpart ofof the the turbine, turbine, the the condenser, condenser, and and the cooling-waterthe cooling-water pump pumpare similar are similar in the casein the of thecase 200-MW of the unit,200-MW the difference unit, the beingdifference that the being LP part that in the the LP 200-MW part in unit the has 200-MWonly one unit extraction has only ( extone1). extraction (ext1).

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EntropyEntropy 2016 2016, 18, 18, 444, 444 4 of4 of 21 21 A diagram of a typical two-pass condenser in a steam power unit is shown in Figure2. The condensersAA diagram diagram of theof of twoa atypical typical power two-pass two-pass units are condenser alsocondenser two-pass in in a designs,asteam steam power butpower their unit unit tube is is shown bundlesshown in in areFigure Figure signiﬁcantly 2. 2. The The condensersdifferentcondensers (Figure ofof thethe3). twotwo powerpower unitsunits areare alsoalso twtwo-passo-pass designs,designs, butbut ththeireir tubetube bundlesbundles areare significantlysignificantly different different (Figure (Figure 3). 3).

Figure 2. A diagram of a typical two-pass condenser in a steam power unit. FigureFigure 2 .2 A. A diagram diagram of of a atypical typical two-pass two-pass condenser condenser in in a asteam steam power power unit. unit.

(a()a ) (b(b) )

FigureFigure 3 .3 Tube. Tube bundle bundle layout layout in in a acondenser condenser of of the the 500-MW 500-MW (a ()a and) and the the 200-MW 200-MW (b ()b units.) units. Figure 3. Tube bundle layout in a condenser of the 500-MW (a) and the 200-MW (b) units.

BasicBasic specifications specifications of of condensers condensers in in the the 200- 200-MWMW and and the the 500-MW 500-MW power power units units are are listed listed in in TablesTablesBasic 1 1and and specifications 2, 2, respectively. respectively. of condensers in the 200-MW and the 500-MW power units are listed in Tables1 and2, respectively. TableTable 1. 1. Basic Basic specifications specifications of of the the condenser condenser in in the the 200-MW 200-MW power power unit unit [42,43]. [42,43]. Table 1. Basic specifications of the condenser in the 200-MW power unit [42,43]. ItemItem SymbolSymbol Unit Unit Value Value Heat transfer area A m2 2 × 5710 = 11,420 HeatItem transfer area SymbolA Unitm2 2 × 5710 = Value 11,420 Number of tubes n - 2 × 6878 = 13,756 Number of tubes n 2 - 2 × 6878× = 13,756 Heat transfer area A  m 2 5710 = 11,420 Cooling-waterNumber of tubes mass flow rate n m2 -kg/s 2 2× ×40006878 = =8000 13,756 Cooling-water mass flow rate m2 kg/s 2 × 4000 = 8000 . × Inlet/outletCooling-water cooling mass water flow temperature, rate norm.m2 t2i/t2o kg/s°C 2 17/25.74000 = 8000 Inlet/outlet cooling water temperature, norm. t2i/t2o ◦ °C 17/25.7 Inlet/outlet cooling water temperature, norm. t /t2o C 17/25.7 2i .  Rated condensed-steam mass flow rate m1 kg/s 129 RatedRated condensed-steam condensed-steam mass mass flow rate flow rate m1 m1 kg/skg/s 129 129 TubeTube outer diameterouter diameter d2o d2o mmmm 30 30 Tube outer diameter d2o mm 30 Tube inner diameter d2i mm 28 Tube inner diameter d2i mm 28 TubeTube inner length diameter L d2i mmmm 28 9000 Tube length L mm 9000 Mean waterTube pressure length p2 L bar (abs)mm 9000 3 NumberMean of water passes pressure -p2 -Bar (abs) 3 2 Mean water pressure p2 Bar (abs) 3 NumberNumber of of passes passes - - - - 2 2

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Table 2. Basic speciﬁcations of the condenser in the 500-MW power unit.

Item Symbol Unit Value Heat transfer area A m2 2 × 9500 = 19,000 Number of tubes n - 2 × 16,000 = 32,000 . Cooling-water mass ﬂow rate m2 kg/s 2 × 6700 = 13,400 o Inlet/outlet cooling water temperature, norm. t /t2o C 24/32 2i . Rated condensed-steam mass ﬂow rate m1 kg/s 207.5 Tube outer diameter d2o mm 24 Tube inner diameter d2i mm 22.6 Tube length L mm 8000 Mean water pressure p2 bar (abs) 3 Number of passes - - 2

3. Mathematical Models for Determining the Optimum Condenser Tube Diameter To find the correct condenser tube diameter, one should consider quantities reflecting heat transfer and hydraulic resistance. The flow rate of heat in a condenser is described by the Péclet equation [44,45]:

. Q = U · A · ∆tln (1)

The logarithmic mean temperature difference has the form:

(ts − t2o) − (ts − t2i) ∆tln = (2) ln ts−t2o ts−t2i

The overall heat transfer coefﬁcient (U) was taken as for a cylindrical wall according to the equation: 1 = U δ (3) d2o + d2o · ln d2o + f + 1 α2·d2i 2·λm d2i λ f α1

The coefﬁcient of heat transfer (α2) from cooling water to tube walls was determined from the Dittus–Boelter equation [44,45]: 0.8 0.4 λ2 α2 = 0.023 · Re2 · Pr2 · (4) d2i

The equation for heat transfer on the side of condensing steam α1 was evaluated from the equation proposed by Szkłowier [47,48]:

z 0.33 α = C · Π0.1 · Nu −0.5 · 1 + s .0.15 · ε −0.04 · α (5) 1 m s n 2 f o p

The Nusselt number (Nun), the heat transfer coefﬁcient for steam condensation (αp) on a single clean horizontal tube, and the geometric parameter of steam inﬂow per bank (s f ) are determined from the following equations, respectively: αp · d2o Nun = (6) λ1

3 !0.25 ρk · λk · r · g αp = 0.728 · (7) νk · ∆tp · d2o f S · L s f = = (8) A πd2onL Entropy 2016, 18, 444 6 of 20

Szkłowier determined the similarity number Πs for a bank of tubes in a condenser as:

2 w1 · ρ1 Πs = (9) ρk · d2o · g

The velocity of steam at the bank inlet, essential for heat transfer conditions in the steam condenser, can be determined from the equation:

. . m1 m1 w1 = = (10) ρ1 · f ρ1 · s f · A

The heat transfer area is: A = π · d2o · L · n (11) On considering Equations (3) and (4), the heat ﬂow rate can be given including thermal resistance per unit length as: . 1 Q = · L · n · ∆tln (12) rl The thermal resistance per unit length is: 1 1 1 d2o 1 rl = + · ln + (13) π α2 · d2i 2 · λm d2i α1 · d2o

The minimum of the thermal resistance per unit length is where heat transfer conditions are the most favourable. To determine the condenser tube diameter, one should also take the resistance to flow into account. Hydraulic losses due to water flow through condenser tubes were obtained from the equation: 2 ρ2w2 L ∆p = λ f r (14) 2 d2i where the flow resistance coefficient is a function of the Reynolds number and roughness according to the equation: 0.25 = λ f r 2 (15) h k 5.74 i log + 0.9 3.7d2i Re2 The simultaneous effect of heat transfer and flow resistance can be considered according to the second law of thermodynamics, since these phenomena are accompanied by entropy generation resulting from irreversible processes.

3.1. Minimization of the Entropy Generation Rate per Unit Length of a Condenser Tube: SL = f(d2i) → min Entropy generation in the condenser results from heat flow and resistance to flow. The entropy generation rate per unit length of a condenser tube, including the entropy generation due to heat flow . . (SLq) and flow resistance of cooling water (SLp) can be expressed as [8–10]:

. 2 . 3 q 8m λ f r . . = + = + SL 2 2 2 5 SLq SLp (16) πλ2T2 Nu2 π ρ2 T2 d2i

The optimum inner diameter of condenser tubes is obtained by minimization of the entropy generation rate per unit length (Equation (16)). Entropy generation resulting from heat flow and resistance to flow is specified in the proposed equation per unit length of a condenser tube. In the above equation, entropy generation on the condensing steam side is not taken into consideration. In order to include entropy generation on the condensing steam side, entropy balance for the whole condenser is required, which is described by equations in Section 3.2. Entropy 2016, 18, 444 7 of 20

. 3.2. Minimization of the Total Entropy Generation Rate: S = f(d2i) → min A change in the inner diameter of condenser tubes affects the performance of the condenser itself, the LP part of the turbine, and the cooling-water pump. Hence, it is reasonable to analyze the subsystem comprising these components. For the subsystem under consideration (the LP part of the turbine, the condenser, and the cooling-water pump), the model of minimization of the entropy generation rate takes into account five entropy generation components: due to heat flow from condensing steam, heat flow to cooling water, and the resistance of cooling water flow through the condenser tubes, in the cooling-water pump, and in the LP part of the turbine. The entropy generation rate due to heat transfer on the steam side is:

. . Q S1 = − (17) Ts

The entropy generation rate due to heat transfer to cooling water is:

. . T2o S . = m c ln (18) 2,Q 2 2 T2i

The entropy generation rate due to the resistance of cooling water to ﬂow through condenser tubes [7] is: . 3 . λ f rm2 L S = (19) 2,p 2 2 d 2T2ρ2F2 2i The entropy generation rate for the pump equals:

. . Sp = m2 spo − spi (20)

The entropy generation rate for the LP part of the 500-MW turbine can be given as (Figure1):

. . . . . . . . St = mt(sext1 − sti) + mt − mext1 (sext2 − sext1) + mt − mext1 − mext2 (sext3 − sext2) + m1(sto − sext3) (21)

The entropy generation rate for the LP part of the 200-MW turbine can be written as:

. . . St = mt(sext1 − sti) + m1(sto − sext1) (22)

The total entropy generation rate for the system under consideration is the sum of the ﬁve components: ...... S = S + S . + S + S + S (23) 1 2,Q 2,p p t The optimum inner diameter of condenser tubes is obtained by minimization of the total entropy generation rate (23). The smallest increase in entropy due to irreversible processes matches the largest power output of the system.

3.3. Maximization of the Power Unit’s Output: P = f(p1(d2i)) → max A change in the inner diameter of condenser tubes affects the power generated by the LP part of the turbine and the power supplied to the cooling-water pump. The power generated by the LP part of the 500-MW turbine is (Figure1):

. . . . . . . Pt = mt(hti − hext1) + mt − mext1 (hext1 − hext2) + mt − mext1 − mext2 (hext2 − hext3) + m1(hext3 − hto) (24)

The power generated by the LP part of the 200-MW turbine is:

. . Pt = mt(hti − hext1) + m1(hext1 − hto) (25) Entropy 2016, 18, 444 8 of 20

The power supplied to the cooling-water pump equals:

. m2∆pp . Pp = = m2 hpo − hpi (26) ρ2ηp

The optimum inner diameter of condenser tubes is obtained by maximization of the power generated by the LP part of the turbine less the power used to drive the cooling-water pump:

P = Pt − Pp (27)

With this criterion the highest power output of the unit can be determined. In order to consider the income from electricity generation and costs of cooling-water pumping and condenser construction, an economic criterion was used; see Section 3.4.

3.4. The Economic Method—Profit Maximization: Z = f(d2i) → max The economic analysis was based on the unit’s ratings prior to the upgrade of its thermal system, including the condenser. A surplus profit/loss over N years of planned power plant lifetime was defined as the income from selling surplus electricity less the cost of building the condenser:

N ZN = ∑ Pt − Pp − Ptr − Ppr · ce · τ · at − cA · (A − Ar) (28) t=1

If, for the sake of simplicity, a constant annual electricity generation (the same in each year of operation) is assumed, Equation (28) can be transformed into:

N ZN = Pt − Pp − Ptr − Ppr · ce · τ · ∑ at − cA · (A − Ar) (29) t=1

Equation (29) can be rewritten with regard to annual effects Za:

1 Z = P − P − P − P · c · τ − c · (A − A ) (30) a t p tr pr e N A r ∑ at t=1 where ﬁxed cost ratio is equal to:

1 d(1 + d)N fc = = (31) N (1 + d)N − 1 ∑ ·at t=1

Finally, the economic criterion for choosing the condenser tube diameter can be written as follows: Za = Pt − Pp − Ptr − Ppr · ce · τ − fc · cA · (A − Ar) (32)

The optimum inner diameter of condenser tubes is obtained by maximization of proﬁt according to Equation (32). A change in the condenser tube diameter affects power of the turbine and the cooling-water pump, as well as heat transfer surface in the condenser. Equation (32) considers changes both in the net income from electricity sales and the cost of constructing the condenser.

4. Calculation Results Condenser parameters, including steam condensation pressure, were calculated using a condenser simulator for steady states. The mathematical model of the condenser used balance and heat ﬂow Entropy 2016, 18, 444 9 of 20

equations. For the given condenser geometry, the simulator's input variables were: temperature (t2i), pressure (p2i), and the mass flow rate of cooling water at the condenser inlet, and the mass flow rate of . condensing steam (m1); the output variables were: cooling-water temperature at the condenser outlet (t2o), and pressure of steam condensing in the condenser (ps). A detailed description of the condenser model can be found in [7,47,48]. The model takes into account the effects of inert gases and deposits and the fact that a change in the inner diameter of condenser tubes is followed by a change in their thickness. Calculations were performed for a constant heat flow rate equal to the heat flow rate at rated operating conditions. The rated heat flow rate for the condenser of the 200-MW (500-MW) unit is . . Q = 293 MW (Q = 453 MW, respectively). The number of tubes (n) and their length (L) are constant and equal to design data (Tables1 and2). The heat transfer area is given by Equation (11). A constant length of the condenser was assumed, since it is defined by the length of the LP part of the turbine. A constant number of condenser tubes was taken, since for slight alteration of the condenser tube diameter tube sheets can be used. If the alteration of the diameter is considerable, the tube sheets should be replaced. To evaluate the power output of the LP part of the turbine in the 200-MW unit, the following parameters at the inlet of the LP part of the turbine were used: steam pressure pti = 0.124 MPa, ◦ steam temperature tti = 178 C, and steam enthalpy hti = 2831 kJ/kg. Steam parameters at ◦ the extraction (ext1) are the following: pressure pext1 = 0.0299 MPa, temperature text1 = 69 C, . enthalpy hext1 = 2615.7 kJ/kg, and the steam mass flow rate from the extraction mext1 = 6.795 kg/s. The efficiency of the group of stages between the inlet of the LP part and the extraction was 0.86. For the LP part of the turbine in the 500-MW unit, the following parameters at the turbine inlet ◦ were taken: steam pressure pti = 0.554 MPa, steam temperature tti = 266 C, and steam enthalpy at the inlet hti = 2994 kJ/kg. Steam parameters at the first extraction were the following: pressure ◦ pext1 = 0.2283 MPa, temperature text1 = 177 C, enthalpy hext1 = 2824 kJ/kg, and the steam mass . flow rate from the extraction mext1 = 16.156 kg/s. Steam parameters at the second extraction were ◦ the following: pressure pext2 = 0.0717 MPa, temperature text2 = 90 C, enthalpy hext2 = 2635 kJ/kg, . and the steam mass flow rate from the extraction mext2 = 9.368 kg/s. Steam parameters at the third ◦ extraction were the following: pressure pext3 = 0.0275 MPa, temperature text3 = 67 C, enthalpy . hext3 = 2525 kJ/kg, and the steam mass flow rate from the extraction mext3 = 12.1 kg/s. The efficiency of the group of stages between the inlet of the LP part and the first extraction was 0.867, between the first and second extractions 0.90, and between the second and third extractions 0.735. For both the turbines, a constant efficiency of the group of stages between the last extraction and the outlet of the LP part of the turbine, equal to 0.8, was used. A change in the power of the LP part of the turbine resulted from a change in steam pressure in the condenser; the steam pressure in the condenser changes if the condenser tube diameter, and therefore the heat transfer area, is modified. Since pressure in the condenser varied relatively slightly, the parameters at the inlet of the LP part of the turbine and at the extractions were assumed to be constant. A constant efficiency of the cooling-water pump, equal to ηp = 0.72, was taken [42] for a 200-MW unit’s pump; for a cooling-water pump of a 500-MW unit, a constant efficiency was taken, equal to ηp = 0.86. The following values were used in the economic analysis: the electricity price ce = 180 zł/MWh (1zł = 4.4 euro), the annual power plant operation time τ = 7000 h, the price of one square metre of the 2 heat transfer area cA = 700 zł/m , and the fixed cost ratio fc = 0.12.

4.1. Calculation Results for the Condenser in the 200-MW Power Unit Based on data obtained from the condenser model, Figure4 shows velocity of cooling water in tubes and pressure drop on the cooling-water side. Entropy 2016, 18, 444 10 of 21 Entropy 2016, 18, 444 10 of 21 = The following values were used in the economic analysis: the electricity price ce = 180 zł/MWh The following values were used in the economic analysis: the electricity price ce 180 zł/MWh (1zł = 4.4 euro), the annual power plant operation time τ = 7000 h , the price of one square metre of (1zł = 4.4 euro), the annual power plant operation time τ = 7000 h , the price of one square metre of = 2 = the heat transfer area cA = 700 zł/m2 , and the fixed cost ratio fc = 0.12 . the heat transfer area cA 700 zł/m , and the fixed cost ratio fc 0.12 .

4.1. Calculation Results for the Condenser in the 200-MW Power Unit 4.1. Calculation Results for the Condenser in the 200-MW Power Unit EntropyBased2016 18on data obtained from the condenser model, Figure 4 shows velocity of cooling water in Based, on, 444data obtained from the condenser model, Figure 4 shows velocity of cooling water10 of in 20 tubes and pressure drop on the cooling-water side. tubes and pressure drop on the cooling-water side. 4 4.0 4 4.0 3.5 3.3 3.5 3.3 3 2.7 3 2.7

2.5 2.0 bar , , m/s 2 2 2.5 2.0 bar , , m/s 2 w 2 dp w 2 1.3 dp 2 1.3 1.5 0.7 1.5 0.7 1 0.0 1 0.0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 the tube inner diameter d2i, mm the tube inner diameter d2i, mm w2 - velocity of cooling water dp2 - pressure drop w2 - velocity of cooling water dp2 - pressure drop

Figure 4.4. VelocityVelocity of cooling water in tubes and pre pressuressure drop on the cooling-water side in the Figure 4. Velocity of cooling water in tubes and pressure drop on the cooling-water side in the condenser as functions of the tube inner diameter.diameter. condenser as functions of the tube inner diameter. For a constant cooling-water mass flow rate, the larger the condenser tube diameter, the lower For a constant cooling-water mass mass flow flow rate, th thee larger the condenser tube diameter, the lower velocity in tubes, which follows from the continuity equation. For a lower velocity of cooling water velocity in tubes, which follows from the continuity equation. For For a lower velocity of cooling water in tubes, pressure drop on the cooling-water side is smaller—Equation (14). If velocity of water in inin tubes, pressure drop on the cooling-watercooling-water side is smaller—Equationsmaller—Equation (14). If If velocity velocity of of water in condenser tubes is lower, the heat transfer coefficient on the water and steam sides is smaller, and, condenser tubes tubes is is lower, lower, the heat transfer coeffici coefficientent on on the the water water and and steam steam sides sides is is smaller, smaller, and, and, consequently, the overall heat transfer coefficient becomes smaller (Figure 5). With a constant heat consequently, the overall heat transfertransfer coefficient coefficient becomes smaller (Figure 55).). WithWith aa constantconstant heatheat flow rate, decreasing the overall heat transfer coefficient despite the increase in the heat transfer flowflow rate, decreasingdecreasing thethe overall overall heat heat transfer transfer coefficient coefficient despite despite the the increase increase in the in heatthe heat transfer transfer area area results in deterioration of heat transfer conditions, and, as a consequence, to a drop in the UA arearesults results in deterioration in deterioration of heat of transfer heat transfer conditions, conditions, and, as and, a consequence, as a consequence, to a drop to ina drop the UA in productthe UA product (Figure 6). Hence, the rise in condensing steam pressure occurs, which can be seen in product(Figure6 ).(Figure Hence, 6). the Hence, rise in the condensing rise in condensing steam pressure steam occurs, pressure which occurs, can be which seen in can Figure be seen5. The in Figure 5. The drop in condensing steam pressure with increasing heat transfer area, which was Figuredrop in 5. condensing The drop steamin condensing pressure withsteam increasing pressure heatwith transfer increasing area, heat which transfer was reported area, which in [64 was,65], reported in [64,65], relates to a constant tube diameter and variable length or number of tubes. In reportedrelates to in a constant[64,65], relates tube diameter to a constant and variable tube diam lengtheter or and number variable of tubes.length In or the number case considered of tubes. In in the case considered in the paper, the number and length of tubes are constant, while the tube inner the case paper, considered the number in andthe paper, length ofthe tubes number are constant,and length while of tubes the tube are innerconstant, diameter while varies, the tube which inner is diameter varies, which is why a slight pressure increase can be observed. diameterwhy a slight varies, pressure which increase is why cana slight be observed. pressure increase can be observed. 5 0.0449 5 0.0449 4.5 0.0443 4.5 0.0443 4 0.0437 K)

2 4 0.0437 K) 2 3.5 0.0431 3.5 0.0431 p1, bar p1, p1, bar p1,

U, kW/(m 3 0.0426

U, kW/(m 3 0.0426 2.5 0.0420 2.5 0.0420 2 0.0414 2 0.0414 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 the tube inner diameter d2i, mm the tube inner diameter d2i, mm U - overall heat transfer coefficient p1 - steam pressure U - overall heat transfer coefficient p1 - steam pressure

Figure 5. The overall heat transfer coefficient and steam pressure in the condenser as functions of Figure 5. The overalloverall heatheattransfer transfer coefﬁcient coefficient and and steam steam pressure pressure in in the the condenser condenser as functionsas functions of theof the tube inner diameter. thetube tube inner inner diameter. diameter.

Following the decrease in the product of the overall heat transfer coefﬁcient and the heat transfer area, the thermal resistance per unit length (13) rises (Figure6). The effect of the tube inner diameter on power generated by the LP part of the turbine (25) and on power required to drive the cooling-water pump (26) is shown in Figure7. Entropy 2016, 18, 444 11 of 21

Following the decrease in the product of the overall heat transfer coefficient and the heat Entropytransfer 2016 area,, 18, 444the thermal resistance per unit length (13) rises (Figure 6). 11 of 21

EntropyFollowing2016, 18, 444 the decrease0.0036 in the product of the overall heat transfer 40000coefficient and the11 heat of 20 transfer area, the thermal resistance per unit length (13) rises (Figure 6). 0.0034 38750

0.0036 40000 0.0032 37500

0.0034 38750 0.003 36250

0.0032 37500 UA, kW/K

length, mK/W 0.0028 35000

0.003 36250 0.0026 33750 thermal resistance per unit unit per resistance thermal UA, kW/K

length, mK/W 0.0028 35000 0.0024 32500 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0.0026 33750 thermal resistance per unit unit per resistance thermal the tube inner diameter d2i, mm 0.0024 32500 thermal resistance per unit length UA 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

the tube inner diameter d2i, mm Figure 6. The thermal resistance per unit length and the product of the overall heat transfer thermal resistance per unit length UA coefficient and the heat transfer area as functions of the tube inner diameter. Figure 6. The thermal resistance per unit length and the product of the overall heat transfer coefﬁcient FigureThe effect 6. The of thethermal tube innerresistance diameter per unit on powerlength geanneratedd the product by the ofLP the part overall of the heat turbine transfer (25) and and the heat transfer area as functions of the tube inner diameter. on powercoefficient required and the to heatdrive transfer the cooling-water area as functions pump of the (26) tube is showninner diameter. in Figure 7.

The effect of 58000the tube inner diameter on power generated by the LP part of5000 the turbine (25) and on power required57900 to drive the cooling-water pump (26) is shown in Figure 7. 4550 57800 4100 5800057700 50003650 5790057600 45503200 5780057500 41002750 Pt, kW Pt, 5770057400 36502300 Pp, kW 5760057300 32001850 5750057200 27501400 Pt, kW Pt, 5740057100 2300950 Pp, kW 5730057000 1850500 57200 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 1400 57100 950 the tube inner diameter d2i, mm 57000 500 Pt - power of LP part of the turbine Pp - pump power 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 the tube inner diameter d , mm Figure 7.7.Power Power generated generated by theby LPthe part LP ofpart the turbineof the andturbine power2i and required power to required drive the cooling-waterto drive the pumpcooling-water as functions pump of as the Ptfunction tube- power inners of of diameter.the LP tubepart innerof the diameter. turbine Pp - pump power Figure 7. Power generated by the LP part of the turbine and power required to drive the An increase in the tube inner diameter is followed by a slight increase in steam pressure, which cooling-water pump as functions of the tube inner diameter. results in the drop in power of the LP part of thethe turbine, since steam expands to a higher pressure. pressure. The lower pumping power is a consequence of a lower velocity of cooling water in condenser tubes The lowerAn increase pumping in the power tube isinner a consequence diameter is of followed a lower by velocity a slight of increase cooling waterin steam in condenserpressure, which tubes due to their larger inner diameter.diameter. resultsThe in powerthe drop unit’s in power net output, of the i.e., LP thepart difference of the turbine, between since the steam power expands generated to a byhigher the LP pressure. part of The lowerThe power pumping unit’s power net output, is a consequence i.e., the difference of a lowe betweenr velocity the of power cooling generated water in bycondenser the LP part tubes of the turbineturbine andand that that used used by by the the cooling-water cooling-water pump, pump, and and the entropythe entropy generation generation rate per rate unit per length unit duelength to theirof a condenser larger inner tube diameter. (16 with its two components are plotted in Figure 8). of a condenserThe power tube unit’s (16 net with output, its two i.e., components the difference are plottedbetween in the Figure power8). generated by the LP part of the turbineThe minimum and that entropyused by generationthe cooling-water rate per pump, unit lengthand the of entropy a condenser generation tube rate according per unit to lengthEquation of a (16) condenserwas obtained tube (16 for with the its tube two inner components diameter are of plotted 27 mm. in The Figure larger 8). the condenser tube diameter, the lower the speed of cooling-water in tubes, which results in a drop in entropy generation per unit length of a condenser tube due to resistance to ﬂow. As the speed of cooling-water in tubes becomes lower, the Nusselt number on the water side is smaller, and entropy generation increases due to heat ﬂow per unit length of a condenser tube according to Equation (16). The maximum difference between the power generated by the LP part of the turbine and that required to drive the cooling-water pump (27) was obtained for the tube inner diameter of 31 mm.

Entropy 2016, 18, 444 12 of 21

0.18 56800 0.16 56400 0.14 56000 0.12 55600 0.1 55200 0.08 54800 Entropy 2016, 18, 444 12 of 20 Entropy 2016, 18, 444 kW Pt-Pp, 12 of 21 0.06 54400 0.04 54000 S/L, SLq, SLp, W/(mK) SLq, S/L, 0.18 56800 0.02 53600 0.16 56400 0 53200 0.14 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 56000

0.12 the tube inner diameter d2i, mm 55600

0.1 S/L - the entropy generation rate per unit length SLq SLp 55200Pt-Pp 0.08 54800 Figure 8. The entropy0.06 generation rate per unit length of a condenser tube with54400 its two components, kW Pt-Pp, and the difference0.04 between the power generated by the LP part of the turbine 54000and that used by the S/L, SLq, SLp, W/(mK) SLq, S/L, cooling-water pump0.02 as functions of the tube inner diameter. 53600 0 53200 The minimum entropy generation rate per unit length of a condenser tube according to 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Equation (16) was obtained for the tube inner diameter of 27 mm. The larger the condenser tube the tube inner diameter d , mm diameter, the lower the speed of cooling-water in tubes,2i which results in a drop in entropy S/L - the entropy generation rate per unit length SLq SLp Pt-Pp generation per unit length of a condenser tube due to resistance to flow. As the speed of cooling-water in tubes becomes lower, the Nusselt number on the water side is smaller, and entropyFigure generation 8. The entropy increases generation due torate rate heat per per unit flow length per unitof of a condenser length of tube tubea condenser with with its its two tube components, according to Equationand the (16). difference The maximum between differencethe power generatedbetween bythe by th thepowere LP part generated of the turbine by the and LP that part used of the by the turbine and thatcooling-water required pump to drive as function functionsthe cooling-waters of the tube pumpinner diameter. (27) was obtained for the tube inner diameter of 31 mm. FigureThe minimum 99 displaysdisplays entropy thethe totaltotal generation entropyentropy generationgenerationrate per unit raterate length forfor thethe of systemsystem a condenser underunder considerationconsiderationtube according (23),(23), to comprisingEquation (16) the the was LP LP partobtained of the for turbine, the tube the inner cond condenser, enser,diameter and of the 27 cooling-watermm. The larger pump, the condenserand the annual tube surplusprofitdiameter,surplusproﬁt the according lower the to speed Equation of (32).cooling-water in tubes, which results in a drop in entropy generation per unit length of a condenser tube due to resistance to flow. As the speed of cooling-water in tubes82 becomes lower, the Nusselt number on the water0.5 side is smaller, and entropy generation increases80 due to heat flow per unit length of a condenser0.0 tube according to Equation (16). The maximum78 difference between the power generated by the-0.5 LP part of the turbine and that required to drive76 the cooling-water pump (27) was obtained for the-1.0 tube inner diameter of /year 31 mm. ł 74 -1.5 Figure 9 displays the total entropy generation rate for the system under consideration (23), S, W/K comprising the LP part72 of the turbine, the condenser, and the cooling-water-2.0 pump, and the annual Za, mln z mln Za, surplusprofit according70 to Equation (32). -2.5 68 -3.0 82 0.5 66 -3.5 80 0.0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 78 -0.5 the tube inner diameter d2i, mm 76 S - the total entropy generation rate Za - the annual profit -1.0 /year ł 74 -1.5

Figure 9. The totalS, W/K entropy generation rate for the system under consideration, comprising the LP part Figure 9. The total72 entropy generation rate for the system under consideration,-2.0 comprising the LP

partof the of turbine, the turbine, the condenser, the condenser, and the and cooling-water the cooling-water pump, andpump, the annualand the surplusprofit annual surplusprofit z mln Za, as functions as 70 -2.5 functionsof the tube of inner the tube diameter. inner diameter. 68 -3.0 The minimum total 66 entropy generation rate for the system under consideration -3.5 occurs for the 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 diameter of 31 mm, i.e., the one for which the ma maximumximum power is generated (Figure 88).). TheThe highesthighest the tube inner diameter d , mm annual surplus profit, being the essential criterion for choosing2i the condenser tube diameter, was achieved at the diameter ofS 29- the mm. total entropy For condenser generation tuberate diametersZa - the annual between profit 27 mm and 33 mm, the graph of the annual surplus profit is rather flat but has a maximum. In this case, a possibly small changeFigure in the 9. The diameter total entropy is recommended, generation rate for thefor the condenser system under in question consideration, to a diameter comprising of 29 the or 30LP mm. part of the turbine, the condenser, and the cooling-water pump, and the annual surplusprofit as functions of the tube inner diameter.

The minimum total entropy generation rate for the system under consideration occurs for the diameter of 31 mm, i.e., the one for which the maximum power is generated (Figure 8). The highest

Entropy 2016, 18, 444 13 of 21 Entropy 2016, 18, 444 13 of 21 annual surplus profit, being the essential criterion for choosing the condenser tube diameter, was achievedannual surplus at the diameterprofit, being of 29 the mm. essential For conden criterserion tubefor choosing diameters the between condenser 27 mm tube and diameter, 33 mm, wasthe graphachieved of theat the annual diameter surplus of 29 profit mm. is For rather conden flatser but tube has diametersa maximum. between In this 27 case, mm aand possibly 33 mm, small the graph of the annual surplus profit is rather flat but has a maximum. In this case, a possibly small changeEntropy 2016 in the, 18, diameter 444 is recommended, for the condenser in question to a diameter of 29 or 30 mm.13 of 20 change in the diameter is recommended, for the condenser in question to a diameter of 29 or 30 mm. 4.2. Calculation Results for the Condenser in the 500-MW Power Unit 4.2. Calculation Results for the Co Condenserndenser in the 500-MW Power Unit Similarly to the 200-MW unit condenser above, Figures 10–13 depict changes in heat flow parametersSimilarly of theto the condenser 200-MW in theunit 500-MW condenser unit. above, Figure Figures Figures 10 shows 10–1310 velocity–13 depictdepict of coolingchanges changes water in in heat heat in tubesflow flow andparameters pressure of drop the condenser on the cooling-wa in the 500-MW ter side. unit. Figure Fi Figuregure 11 displays 10 shows the velocity overall of heat cooling transfer water coefficient in tubes andand steampressure pressure drop onin thethe cooling-watercooling-wacondenser aster functions side. Figure of the 11 displaystube inner the diameter. overall heat The transferthermal coefficient coefficientresistance perand unit steam length pressure (13) andin the the condenser product of as the functions overall of he theat transfer tube inner coefficient diameter. and The the thermal heat transfer resistance area asper functions unit length of (13)(13) the and andtube thethe inner productproduct diameter of of the the overall overallare plo heat hettedat transfer transferin Figure coefficient coefficient 12. The and andeffect the the heatof heat the transfer transfer tube areainner area as diameterasfunctions functions on of thepowerof tubethe generated tube inner inner diameter by diameterthe areLP plottedpart are of plo inthe Figuretted turbine in 12 Figure .(24) The and effect12. on The ofpower the effect tube required of inner the diametertotube drive inner the on cooling-waterdiameterpower generated on power pump by generated the(26) LP is shown part by of the in the FigureLP turbine part 13. of (24) the and turbine on power (24) and required on power to drive required the cooling-water to drive the cooling-waterpump (26) is shown pump in(26) Figure is shown 13. in Figure 13. 3 1.6 3 1.6 2.5 1.3 2.5 1.3 2 1.1 2 1.1

1.5 0.8 , bar , m/s 2 2 , bar w 1.5 0.8 dp , m/s 2 2

w 1

0.5 dp 1 0.5 0.5 0.3 0.5 0.3 0 0.0 020 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0.0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 the tube inner diameter d2i, mm

the tube inner diameter d2i, mm w2 - velocity of cooling water dp2 - pressure drop w2 - velocity of cooling water dp2 - pressure drop Figure 10. Velocity of cooling water in tubes and pressure drop on the cooling-water side in the Figure 10. Velocity of cooling water in tubes and pressure drop on the cooling-water side in the condenserFigure 10. asVelocity functions of ofcooling the tube water inner in diameter.tubes and pressure drop on the cooling-water side in the condenser as functions of the tube inner diameter. condenser as functions of the tube inner diameter.

4 0.0624 4 0.0624 3.5 0.0616 3.5 0.0616 K)

2 3 0.0608 K)

2 3 0.0608

2.5 0.0600 bar p1, U, kW/(m 2.5 0.0600 bar p1, U, kW/(m 2 0.0592 2 0.0592 1.5 0.0584 1.520 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0.0584 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 the tube inner diameter d2i, mm

the tube inner diameter d2i, mm U - overall heat transfer coefficient p1 - steam pressure U - overall heat transfer coefficient p1 - steam pressure FigureFigure 11 11.. The overall heatheat transfertransfercoefﬁcient coefficient and and steam steam pressure pressure in in the the condenser condenser as functionsas functions of theof theFiguretube tube inner 11 inner. diameter.The diameter. overall heat transfer coefficient and steam pressure in the condenser as functions of the tube inner diameter.

Entropy 2016, 18, 444 14 of 21 EntropyEntropy2016 2016, 18, ,18 444, 444 14 of14 21 of 20 Entropy 2016, 18, 444 14 of 21

0.0044 65000 0.0044 65000 0.00430.0044 6388965000 0.0043 63889 0.00420.0043 6277863889 0.0042 62778 0.00410.0042 6166762778 0.0041 61667 0.00410.004 6055661667 0.004 60556 0.00390.004 5944460556 0.0039 59444 UA, kW/K UA, UA, kW/K UA, length, mK/W length, 0.00380.0039 5833359444

length, mK/W length, 0.0038 58333 0.0037 57222 kW/K UA, length, mK/W length, 0.0038 58333 0.0037 57222 thermal resistancethermal per unit 0.0036 56111

thermal resistancethermal per unit 0.0037 57222 0.0036 56111 thermal resistancethermal per unit 0.00350.0036 5500056111 0.0035 55000 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0.0035 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3555000 20 21 22 23the 24 tube 25 inner 26 27 diameter 28 29 30d2i, 31mm 32 33 34 35 the tube inner diameter d2i, mm thermalthe tube resistance inner diameter per unit dlength2i, mm UA thermal resistance per unit length UA thermal resistance per unit length UA Figure 12. The thermal resistance per unit length and the product of the overall heat transfer FigureFigure 12. 12The. The thermal thermal resistance resistance per unitper lengthunit length and theand product the product of the overallof the heatoverall transfer heat coefﬁcienttransfer coefficientFigure 12. andThe the thermal heat transfer resistance area peras functions unit length of the an tubed the inner product diameter. of the overall heat transfer andcoefficient the heat and transfer the heat area transfer as functions area as of functions the tube of inner the diameter.tube inner diameter. coefficient and the heat transfer area as functions of the tube inner diameter. 149000 3296 149000 3296 148800149000 28683296 148800 2868 148600148800 24402868 148600 2440 148400148600 20122440 148400 2012 148400 2012

Pt, kW 148200 1584 Pp, kW Pp,

Pt, kW 148200 1584 148000 1156 kW Pp, 148000 1156 147800148000 7281156 147800 728 147600147800 300728 147600 300 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 147600 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35300 20 21 22 23the 24 tube 25 inner 26 27 diameter 28 29 d 302i, mm 31 32 33 34 35 the tube inner diameter d2i, mm Pt - power ofthe LP tube part innerof the diameter turbine d2i,Pp mm - pump power Pt - power of LP part of the turbine Pp - pump power Pt - power of LP part of the turbine Pp - pump power Figure 13. Power generated by the LP part of the turbine and power required to drive the FigureFigure 13. 13.Power Power generated generated bythe byLP the part LP of part the turbineof the andturbine power and required power torequired drive the to cooling-water drive the cooling-waterFigure 13. Power pump generated as function bys ofthe the LP tube part inner of diameter.the turbine and power required to drive the pumpcooling-water as functions pump of the as function tube inners of diameter. the tube inner diameter. cooling-water pump as functions of the tube inner diameter. The entropy generation rate per unit length of a condenser tube according to Equation (16) The entropy generation rate per unit length of a condenser tube according to Equation (16) withThe Theits entropy two entropy components generation generation and rate ratethe per difference per unit unit length lengthbetween of of a condenserpowera condenser generated tube tube according by according the LP topart Equationto ofEquation the turbine (16) (16) with with its two components and the difference between power generated by the LP part of the turbine itsandwith two power its components two required components and to drive the and difference the the cooling-water difference between between pump power power(27) generated are generated shown by in thebyFigure the LP LP part14. part of theof the turbine turbine and and power required to drive the cooling-water pump (27) are shown in Figure 14. powerand power required required to drive to drive the cooling-water the cooling-water pump pump (27) are(27) shownare shown in Figure in Figure 14. 14. 0.1 147800 0.1 147800 0.090.1 147660147800 0.09 147660 0.080.09 147520147660 0.08 147520 0.070.08 147380147520 0.07 147380 0.060.07 147240147380 0.06 147240 0.050.06 147100147240 0.05 147100 0.040.05 146960147100 0.04 146960 kW Pt-Pp, 0.030.04 146820146960 kW Pt-Pp, 0.03 146820 kW Pt-Pp,

S/L, SLq, SLp, W/(mK) SLp, SLq, S/L, 0.020.03 146680146820

S/L, SLq, SLp, W/(mK) SLp, SLq, S/L, 0.02 146680

S/L, SLq, SLp, W/(mK) SLp, SLq, S/L, 0.010.02 146540146680 0.01 146540 0.010 146400146540 0 146400 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35146400 20 21 22 23the 24 tube 25 inner 26 27 diameter 28 29 d 302i, mm 31 32 33 34 35 the tube inner diameter d2i, mm S/L - the entropythe generation tube inner rate diameterper unit length d2i, mmSLq SLp Pt-Pp S/L - the entropy generation rate per unit length SLq SLp Pt-Pp S/L - the entropy generation rate per unit length SLq SLp Pt-Pp Figure 14. The entropy generation rate per unit length of a condenser tube with its two components Figure 14. The entropy generation rate per unit length of a condenser tube with its two components FigureandFigure the 14. 14 differenceThe. The entropy entropy between generation generation the power raterate generatedperper unit length by th eof of LP a a condenser condenserpart of the tube turbine tube with with and its its twothat two componentsused components by the and the difference between the power generated by the LP part of the turbine and that used by the andcooling-waterand the the difference difference pump between between as functionthe the powerspower of the generatedgenerated tube inner by bydiameter. th thee LP LP part part of of the the turbine turbine and and that that used used by bythe the cooling-water pump as functions of the tube inner diameter. cooling-watercooling-water pump pump as as functions functions of of thethe tubetube innerinner diameter.

Entropy 2016, 18, 444 15 of 20 Entropy 2016, 18, 444 15 of 21

The minimumminimum entropyentropy generation generation rate rate per per unit unit length length of a condenser of a condenser tube according tube according to Equation to Equation(16) was obtained(16) was for obtained the tube innerfor the diameter tube inner of 24 mm.diameter The maximumof 24 mm. difference The maximum between difference the power betweengenerated the by thepower LP partgenerated of the turbineby the andLP thatpart required of the toturbine drive theand cooling-water that required pump to drive (27) wasthe cooling-waterobtained for the pump tube (27) inner was diameter obtained of fo 28r mm.the tube inner diameter of 28 mm. Figure 1515 displays displays the the total total entropy entropy generation generation rate for rate the systemfor the under system consideration, under consideration, comprising comprisingthe LP part ofthe the LP turbine, part of thethe condenser,turbine, the and cond theenser, cooling-water and the cooling-water pump, and the pump, annual and surplus the annual proﬁt surplusaccording profit to Equation according (32). to Equation (32).

126.5 0.3 126 0.2 125.5 0.0 125 -0.2 /year 124.5 -0.3 ł 124 -0.5 S, W/K 123.5 -0.6 Za, mln z mln Za, 123 -0.8 122.5 -0.9 122 -1.1 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

the tube inner diameter d2i, mm S - the total entropy generation rate Za - the annual profit

Figure 15.15. The totaltotal entropyentropy generation generation rate rate for for the the system system under under consideration, consideration, comprising comprising the LPthe part LP partof the of turbine, the turbine, the condenser, the condenser, and the and cooling-water the cooling-water pump, andpump, the and annual the surplus annual proﬁtsurplus as functionsprofit as functionsof the tube of inner the tube diameter. inner diameter.

The minimum total entropy generation rate for the system under consideration occurs for the diameter ofof 2828 mm, mm, i.e., i.e., the the one one for for which which the maximumthe maximum power power is generated is generated (Figure (Figure 14). The 14). highest The highestannual surplusannual surplus proﬁt was profit achieved was ac athieved the diameter at the diameter of 25 mm. of 25 mm.

5. Conclusions Conclusions The paper presents a methodology for for choosing choosing the the diameter diameter of of tubes tubes in in a a power power condenser, condenser, involvinginvolving thermodynamic thermodynamic methods methods and and an an economic economic one. one. This This issue issue is is relevant relevant to to both both the design of new condensers and the replacement of usedused tubetube assembliesassemblies inin upgradedupgraded condensers.condensers. Four methods werewere presented:presented: minimization minimization of theof entropythe entropy generation generation rate per rate unit per length unit of length a condenser of a condensertube (considering tube (considering entropy generation entropy generation due to heat due transferto heat transfer and resistance and resistance of cooling-water of cooling-water flow); flow);minimization minimization of the totalof the entropy total entropy generation generation rate for ra thete systemfor the system comprising comprising the LP partthe LP of thepart turbine, of the turbine,the condenser, the condenser, and the and cooling-water the cooling-water pump; pump maximization; maximization of the of power the power unit’s unit’s output; output; and and the theeconomic economic method method (where (where the criterionthe criterion of selection of selection is the isannual the annual profit profit calculated calculated as the as income the income from fromelectric electric power generationpower generation less the costsless ofthe pumping costs of cooling pumping water cooling and building water the and condenser). building the condenser).The methodology was applied to assess the appropriateness of tube diameters for a condenser in a 200-MWThe methodology and a 500-MW was applied power units.to assess The the choice appr ofopriateness the diameter of tube was diameters made for equal for a numbercondenser of intubes a 200-MW and their and lengths a 500-MW (per unitpower length). units. The choice of the diameter was made for equal number of tubesFor and the their 200-MW lengthsunit (per condenser,unit length). the minimum entropy generation rate per unit length of a condenserFor the tube 200-MW according unit condenser, to Equation the (16) minimum was obtained entropy for generation the tube innerrate per diameter unit lengthof 27 of mm. a condenserThe maximum tube netaccording power to output Equati ofon the (16) unit was was obtained obtained for forthe thetube tube inner inner diameter diameter of 27 of mm. 31 mm.The maximumThe minimum net powertotal entropy output generationof the unit ratewas forobtain theed system for the comprising tube inner the diameter LP part of of 31 the mm. turbine, The minimumthe condenser, total andentropy the generation pump occurred rate for for the the syst sameem comprising diameter. the The LP optimum part of the inner turbine, diameter the condenser,corresponding and to the the pump maximum occurred annual for profit the (Zsaa)me is 29diameter. mm. With The most optimum condensers inner of diameter 200-MW correspondingpower units, the to tubethe maximum inner diameter annual is 28profit mm, ( whileZa) is the29 mm. outer With one ismost 30 mm. condensers For condenser of 200-MW tube power units, the tube inner diameter is 28 mm, while the outer one is 30 mm. For condenser tube diameters between 27 mm and 33 mm, the graph of the annual surplus profit is rather flat, but has a

Entropy 2016, 18, 444 16 of 20 diameters between 27 mm and 33 mm, the graph of the annual surplus profit is rather flat, but has a maximum. In this case, a possibly small change in the diameter is recommended, for the condenser in question to a diameter of 29 or 30 mm. For the 500-MW unit condenser, the minimum entropy generation rate per unit length of a condenser tube according to Equation (16) was obtained for the tube inner diameter of 24 mm. The tube inner diameter for which the minimum total entropy generation rate for the system comprising the LP part of the turbine, the condenser, and the pump was achieved matches the one for which the maximum power generation was obtained, and equals 28 mm. The highest annual profit was achieved at the optimum inner diameter of 25 mm. With most condensers of a 500-MW power unit, the tube inner diameter is 22.6 mm, while the outer one 24 mm. The methods of entropy generation minimization (considering both heat transfer and flow resistance) and the method of the unit’s power output maximization are approximate methods for determining the optimum condenser tube diameter. The results shown indicate that the diameter for which the minimum total entropy generation occurs matches the one for which the maximum power is achieved. The diameter obtained with both these methods is slightly larger than the one calculated with the economic (profit-based) method which comprehensively takes into account the power generated by the LP part of the turbine, the power required to drive the pump, the cost of building the condenser, and, indirectly, the entropy generation in the system under consideration. In this paper, the optimum diameter of condenser tubes was found for constant number and length of the tubes. However, this is not the only approach that can be taken; a constant heat transfer area or a constant pressure in the condenser can also be assumed (while changing the number of condenser tubes accordingly).

Author Contributions: Rafał Laskowski, Adam Smyk, Artur Rusowicz and Andrzej Grzebielec performed the calculations analyzed the data and wrote the paper. All authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

A heat transfer area, m2 at discount factor, - c2 specific heat of water, J/(kg·K) ce electricity price, zł/MWh 2 cA price of one square meter of the heat transfer area, zł/m Cm coefficient of heat transfer intensity for steam and water mixture in a bank of tubes d discount rate, - d2i tube inner diameter, m d2o tube outer diameter, m surface area of steam flow across the section between tubes of a bank in the condenser f within its outer circumference, m2 fc fixed cost ratio, 1/year 2 F2 cross-section of the cooling water flow, m g gravitational acceleration, m/s2 h specific enthalpy, J/kg k roughness, mm L tube length, m n number of tubes, - N years of power plant operation, year Nu Nusselt number, - Nun Nusselt number for steam condensing on a single tube, - . m rate of mass flow through one tube, kg/s . m steam mass flow rate to the condenser, kg/s . 1 m mass flow rate of cooling water, kg/s . 2 mt steam mass flow rate at the inlet of the low-pressure part of the turbine, kg/s p pressure, Pa (abs) ∆pp pressure rise across the pump, Pa Pp pump power, W Entropy 2016, 18, 444 17 of 20

Pt power generated in the low-pressure part, W P difference in power between the low-pressure part and the pump, W Pr Prandtl number, - . Q flow rate of the heat transferred, kW . q heat flow rate per unit tube length, W/m 0 r phase transition heat of condensing steam, r = h”(ps) − h (ps), kJ/kg rl thermal resistance per unit length, mK/W Re Reynolds number, - s specific entropy, J/(kg·K) S circumference of steam inflow, measured across a section between tubes outside the bank, m . S sum of entropy rates, W/K s f ratio of steam inflow in the area between tubes on the outer circumference of a bank, - SL entropy generation rate per unit length of a condenser tube, W/(mK) . S2,p entropy generation rate due to the resistance of cooling-water flow, W/K . S . entropy generation rate due to heat transfer to water, W/K . 2,Q S entropy generation rate on the steam side, W/K . 1 S entropy generation rate in the low-pressure part, W/K . t Sp entropy generation rate in the pump, W/K . SLp entropy generation due to flow resistance of cooling water, W/(mK) . SLq entropy generation due to heat flow, W/(mK) T2i cooling water temperature at the condenser inlet, K T2o cooling water temperature at the condenser outlet, K T2 average cooling water temperature, (T2o + T2i)/2, K Ts steam saturation temperature, K ◦ ∆tln logarithmic temperature difference across the steam condenser, C ◦ ∆tp mean temperature difference between steam and the external surface of tubes, C U overall heat transfer coefficient, kW/(m2·K) w velocity, m/s z number of tube runs in a condenser, - Za annual surplusprofit, zł/year Zn profit over N years of operation, zł α coefficient of heat transfer, kW/(m2·K) 2 αp heat transfer coefficient for steam condensation on a single horizontal tube, kW/(m ·K) δf thickness of the fouling layer, m εo ratio of air content in the condenser to steam flow, - ηp pump efficiency, - λ1 thermal conductivity of steam, kW/(mK) λ2 thermal conductivity of cooling water, kW/(mK) λ f thermal conductivity of the fouling layer, kW/(mK) λ f r flow resistance coefficient, - λk thermal conductivity of condensate, kW/(mK) λm thermal conductivity of tube material, kW/(mK) 2 νk kinematic viscosity of condensate, m /s Πs similarity number for a bank of tubes, - ρ density, kg/m3 3 ρk density of condensate, kg/m τ annual power plant operation time, h

Subscripts 1 relates to steam 2 relates to cooling water ext relates to parameters at the extraction in steam turbine f fouling m tube material ti relates to parameters at the inlet of the low-pressure part to relates to parameters at the outlet of the low-pressure part pi relates to parameters upstream the pump po relates to parameters downstream the pump r relates to reference (nominal) parameters Entropy 2016, 18, 444 18 of 20

References

1. Nag, P.K. Power Plant Engineering; Tata McGraw-Hill Education: New York, NY, USA, 2002. 2. Haseli, Y.; Dincer, I.; Naterer, G.F. Optimum temperatures in a shell and tube condenser with respect to exergy. Int. J. Heat Mass Transf. 2008, 51, 2462–2470. [CrossRef] 3. Laskowski, R.; Smyk, A. Analysis of the working conditions of a steam condenser using measurements and an approximation model. Rynek Energii 2014, 1, 110–115. 4. Laskowski, R.; Smyk, A.; Lewandowski, J.; Rusowicz, A. Cooperation of a Steam Condenser with a Low-pressure Part of a Steam Turbine in Off-Design Conditions. Am. J. Energy Res. 2015, 3, 13–18. 5. Atria, S.I. The influence of condenser cooling water temperature on the thermal efficiency of a nuclear power plant. Ann. Nucl. Energy 2015, 80, 371–378. 6. Anozie, A.N.; Odejobi, O.J. The search for optimum condenser cooling water flow rate in a thermal power plant. Appl. Therm. Eng. 2011, 31, 4083–4090. [CrossRef] 7. Laskowski, R.; Smyk, A.; Lewandowski, J.; Rusowicz, A.; Grzebielec, A. Selecting the cooling water mass flow rate for a power plant under variable load with entropy generation rate minimization. Energy 2016, 107, 725–733. [CrossRef] 8. Bejan, A. Entropy generation minimization: The new thermodynamics of finite size devices and finite time processes. J. Appl. Phys. 1996, 79, 1191–1218. [CrossRef] 9. Laskowski, R.; Rusowicz, A.; Smyk, A. Verification of the condenser tubes diameter based on the minimization of entropy generation. Rynek Energii 2015, 1, 71–75. 10. Laskowski, R.; Rusowicz, A.; Grzebielec, A. Estimation of a tube diameter in a ‘church window’ condenser based on entropy generation minimization. Arch. Thermodyn. 2015, 36, 49–59. 11. Erdem, H.H.; Akkaya, A.V.; Cetin, B.; Dagdas, A.; Sevilgen, S.H.; Sahin, B.; Teke, I.; Gungor, C.; Atas, S. Comparative energetic and exergetic performance analyses for coal-fired thermal power plants in Turkey. Int. J. Therm. Sci. 2009, 48, 2179–2186. [CrossRef] 12. Aljundi, I.H. Energy and exergy analysis of a steam power plant in Jordan. Appl. Therm. Eng. 2009, 29, 324–328. [CrossRef] 13. Datta, A.; Sengupta, S.; Duttagupta, S. Exergy analysis of a coal-based 210 MW thermal power plant. Int. J. Energy Res. 2007, 31, 14–28. 14. Wołowicz, M.; Milewski, J.; Badyda, K. Feedwater repowering of 800 MW supercritical steam power plant. J. Power Technol. 2012, 92, 127–134. 15. Milewski, J.; Bujalski, W.; Wołowicz, M.; Futyma, K.; Kucowski, J. Off-design operation of an 900 MW-class power plant with utilization of low temperature heat of flue gases. J. Power Technol. 2015, 95, 221–227. 16. Soltan, B.K.; Saffar-Avval, M.; Damangir, E. Minimizing capital and operating costs of shell and tube condensers using optimum baffle spacing. Appl. Therm. Eng. 2004, 24, 2801–2810. [CrossRef] 17. Fertaka, S.; Thibault, J.; Gupta, Y. Design of shell-and-tube heat exchangers using multiobjective optimization. Int. J. Heat Mass Transf. 2013, 60, 343–354. [CrossRef] 18. Allen, B.; Gosselin, L. Optimal geometry and flow arrangement for minimizing the cost of shell-and-tube condensers. Int. J. Energy Res. 2008, 32, 958–969. [CrossRef] 19. Wildi-Tremblay, P.; Gosselin, L. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. Int. J. Energy Res. 2007, 31, 867–885. [CrossRef] 20. Caputo, A.C.; Pelagagge, P.M.; Salini, P. Heat exchanger design based on economic optimization. Appl. Therm. Eng. 2008, 28, 1151–1159. [CrossRef] 21. Azad, A.V.; Amidpour, M. Economic optimization of shell and tube heat exchanger based on constructal theory. Energy 2011, 36, 1087–1096. [CrossRef] 22. Hajabdollahi, H.; Ahmadi, P.; Dincer, I. Thermoeconomic optimization of a shell and tube condenser using both genetic algorithm and particle swarm. Int. J. Refrig. 2011, 34, 1066–1076. [CrossRef] 23. Sadeghzadeh, H.; Ehyaei, M.A.; Rosen, M.A. Techno-economic optimization of a shell and tube heat exchanger by genetic and particle swarm algorithms. Energy Convers. Manag. 2015, 93, 84–91. [CrossRef] 24. Selbas, R.; Kızılkan, O.; Reppich, M. A new design approach for shell-and-tube heat exchangers using genetic algorithms from economic point of view. Chem. Eng. Process. 2006, 45, 268–275. [CrossRef] 25. Patel, V.K.; Rao, R.V. Design optimization of shell-and-tube heat exchanger using particle swarm optimization technique. Appl. Therm. Eng. 2010, 30, 1417–1425. [CrossRef] Entropy 2016, 18, 444 19 of 20

26. Sahin, A.S.; Kilic, B.; Kilic, U. Design and economic optimization of shell and tube heat exchangers using Artificial Bee Colony (ABC) algorithm. Energy Convers. Manag. 2011, 52, 3356–3362. [CrossRef] 27. Hadidi, A.; Hadidi, M.; Nazari, A. A new design approach for shell-and-tube heat exchangers using imperialist competitive algorithm (ICA) from economic point of view. Energy Convers. Manag. 2013, 67, 66–74. [CrossRef] 28. Hadidi, A.; Nazari, A. Design and economic optimization of shell-and-tube heat exchangers using biogeography-based (BBO) algorithm. Appl. Therm. Eng. 2013, 51, 1263–1272. [CrossRef] 29. Mohanty, D.K. Gravitational search algorithm for economic optimization design of a shell and tube heat exchanger. Appl. Therm. Eng. 2016, 107, 184–193. [CrossRef] 30. Abed, A.M.; Abed, I.A.; Majdi, H.S.; Al-Shamani, A.N.; Sopian, K. A new optimization approach for shell and tube heat exchangers by using electromagnetism-like algorithm (EM). Heat Mass Transf. 2016, 52, 2621–2634. [CrossRef] 31. Ozcelik, Y. Exergetic optimization of shell and tube heat exchangers using agenetic based algorithm. Appl. Therm. Eng. 2007, 27, 1849–1856. [CrossRef] 32. Eryener, D. Thermoeconomic optimization of baffle spacing for shell and tube heat exchangers. Energy Convers. Manag. 2006, 47, 1478–1489. [CrossRef] 33. Can, A.; Buyruk, E.; Eryener, D. Exergoeconomic analysis of condenser type heat exchangers. Exergy Int. J. 2002, 2, 113–118. [CrossRef] 34. Sanaye, S.; Hajabdollahi, H. Multi-objective optimization of shell and tube heat exchangers. Appl. Therm. Eng. 2010, 30, 1937–1945. [CrossRef] 35. Ponce-Ortega, J.M.; Serna-González, M.; Jiménez-Gutiérrez, A. Use of genetic algorithms for the optimal design of shell-and-tube heat exchangers. Appl. Therm. Eng. 2009, 29, 203–209. [CrossRef] 36. Ayala, H.V.H.; Keller, P.; Morais, M.F.; Mariani, V.C.; Coelho, L.S.; Rao, R.V. Design of heat exchangers using a novel multiobjective free search differential evolution paradigm. Appl. Therm. Eng. 2016, 94, 170–177. [CrossRef] 37. Muralikrishna, K.; Shenoy, U.V. Heat exchanger design targets for minimum area and cost. Chem. Eng. Res. Des. 2000, 78, 161–167. [CrossRef] 38. Tol, H.I.; Svendsen, S. Improving the dimensioning of piping networks and network layouts in low-energy district heating systems connected to low energy buildings: A case study in Roskilde, Denmark. Energy 2012, 38, 276–290. [CrossRef] 39. Nussbaumer, T.; Thalmann, S. Influence of system design on heat distribution costs in district heating. Energy 2016, 101, 496–505. [CrossRef] 40. Smyk, A.; Pietrzyk, Z. Dobór ´srednicy ruroci ˛agóww sieci ciepłowniczej z uwzgl˛ednieniemoptymalnych pr˛edko´sciwody sieciowej. Rynek Energii 2011, 6, 98–105. (In Polish) 41. Murat, J.; Smyk, A. Dobór ´srednicy ruroci ˛agóww układzie rozgał˛e´zno-pier´scieniowymdla przykładowych struktur sieci ciepłowniczej. Rynek Energii 2015, 9, 13–19. (In Polish) 42. Salij, A.; St˛epień,J.C. Performance of Turbine Condensers in Power Units of Thermal Systems; Kaprint: Warsaw, Poland, 2013. (In Polish) 43. Rusowicz, A. Issues Concerning Mathematical Modelling of Power Condensers; Warsaw University of Technology: Warsaw, Poland, 2013. (In Polish) 44. Cengel, Y.A. Heat Transfer; McGraw-Hill: New York, NY, USA, 1998. 45. Holman, J.P. Heat Transfer; McGraw-Hill: New York, NY, USA, 2002. 46. Grzebielec, A.; Rusowicz, A. Thermal Resistance of Steam Condensation in Horizontal Tube Bundles. J. Power Technol. 2011, 91, 41–48. 47. Szkłowier, G.G.; Milman, O.O. Issledowanije I Rasczot Kondensacionnych Ustrojstw Parowych Turbin; Energoatomizdat: Moskwa, Russia, 1985. (In Russian) 48. Smyk, A. The Influence of Thermodynamic Parameters of a Heat Cogeneration System of the Nuclear Heat Power Plant on Fuel Saving in Energy System. Ph.D. Thesis, Warsaw University of Technology, Warsaw, Poland, 1999. (In Polish) 49. Chmielniak, T.; Trela, M. Diagnostics of New-Generation Thermal Power Plants; Wydawnictwo IMP PAN: Gdańsk,Poland, 2008. 50. Saari, J.; Kaikko, J.; Vakkilainen, E.; Savolainen, S. Comparison of power plant steam condenser heat transfer models for on-line condition monitoring. Appl. Therm. Eng. 2014, 62, 37–47. [CrossRef] Entropy 2016, 18, 444 20 of 20

51. Brodowicz, K.; Czaplicki, A. Condensing Vapour Flow Resistance Throught Tubes Bundle in the Presence of Condensate on Tubes. In Proceedings of the 8th International Heat Transfer Conference, San Francisco, CA, USA, 17–22 August 1986; Volume 4, pp. 1689–1694. 52. Carlucci, L. Computations of flow and heat transfer in power plant condenser. In Proceedings of the 8th International Heat Transfer Conference, San Francisco, CA, USA, 17–22 August 1986; Volume 5, pp. 2541–2546. 53. Fujii, T.; Honda, H.; Oda, K. Condensation of Steam on Horizontal Tube—the Influence of Oncoming Velocity and Thermal Conduction at the Tube Wall. In Proceedings of the 18th National Heat Transfer Conference, San Diego, CA, USA, 6–8 August 1979; pp. 35–43. 54. Fujii, T.; Uehara, H.; Hirata, K.; Oda, K. Heat transfer and flow resistance in condensation of low pressure steam flowing through tube banks. Int. J. Heat Mass Transf. 1972, 15, 247–259. [CrossRef] 55. Malin, M.R. Modelling flow in an experimental marine condenser. Int. Commun. Heat Transf. 1997, 24, 597–608. [CrossRef] 56. Roy, R.P.; Ratisher, M.; Gokhale, V.K. A Computational Model of a Power Plant Steam Condenser. J. Energy Resour. Technol. 2001, 123, 81–91. [CrossRef] 57. Zeng, H.; Meng, J.; Li, Z. Numerical study of a power plant condenser tube arrangement. Appl. Therm. Eng. 2012, 40, 294–303. [CrossRef] 58. Zhang, C. Numerical Modeling Using a Quasi-Three-Dimensional Procedure for Large Power Plant Condenser. J. Heat Transf. 1994, 116, 180–188. [CrossRef] 59. Zhang, C.; Bokil, A. A quasi-three-dimensional approach to simulate the two-phase fluid flow and heat transfer in condensers. Int. J. Heat Mass Transf. 1997, 40, 3537–3546. [CrossRef] 60. Zhang, C.; Sousa, A.C.M.; Venart, J.E.S. Numerical Simulation of Different Types of Steam Surface Condensers, Journal of Energy Resources Technology. Trans. ASME 1991, 113, 63–70. 61. Zhou, L.X.; Li, F.Y.; Li, W.H. Quasi-three-dimensional numerical study of shell side of condenser. Proc. CSEE 2008, 28, 25–30. 62. Nedelkovski, I.; Vilos, I.; Geramitoioski, T. Finite element solution of Navier-Stokes equations for steam flow and heat transfer. Trans. Eng. Comput. Technol. 2005, 5, 171–175. 63. Prieto, M.M.; Suarez, I.M.; Montanes, E. Analisys of the thermal performance of a church window steam condenser for different operational conditions using three models. Appl. Therm. Eng. 2003, 23, 163–178. [CrossRef] 64. Bekdemir, S.; Ozturk, R.; Yumurtac, Z. Condenser Optimization in Steam Power Plant. J. Therm. Sci. 2003, 12, 176–179. [CrossRef] 65. Ozdamar, G.; Ozturk, R. Optimization of a condenser in a thermal power plant. In Proceedings of the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Valletta, Malta, 16–18 July 2012.

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