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Applied Thermal Engineering 68 (2014) 45e58

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Thermal analysis of closed feedwater heaters in nuclear power plants

Mario Álvarez-Fernández a,c,*, Luis del Portillo-Valdés b, Cristina Alonso-Tristán c

aDepartment of Mechanical Maintenance, Santa María de Garoña Nuclear Power Plant, 09212 Burgos, Spainb E.T.S. de Ingeniería, University of the Basque Country, Alameda de Urquijo, 48013 Bilbao, Spainc Escuela Politécnica Superior, University of Burgos, Avda. Cantabria, 09006 Burgos, Spain

h i g h l i g h t s

� The formulation for the calculation of feedwater heaters in NPP is introduced.� The model has been validated in the Santa María de Garoña NPP.� The model has been tested looking for mal-function in feedwater heaters.

a r t i c l e i n f o

Article history:Received 15 March 2013Accepted 4 April 2014Available online 16 April 2014

Keywords:Closed feedwater heaterNuclear power plantOverall heat transfer coefficientThermodynamic properties of wet-steam

* Corresponding author. Department of MechanicalGaroña Nuclear Power Plant, 09212 Burgos, Spain.

E-mail addresses: mario.alvarez@nuclenor.es,(M. Álvarez-Fernández).

http://dx.doi.org/10.1016/j.applthermaleng.2014.04.001359-4311/� 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

A model is introduced for the thermal analysis of closed feedwater heaters in which wet steam isextracted from the steam turbine (and piped into the heater). Where the fluid flowing through the steamturbine is wet steam, application of this model is of relevance to nuclear power plant diagnostics. Atpresent, all feedwater heater calculations are performed by setting the stage moisture removal effec-tiveness, on the assumption that the mass flows and/or the enthalpy rates of the steam bled from theturbine are known values. Our model enables detection of multiple malfunctions, even where inaccuratereadings may have been registered in the control room at an earlier point in time. Furthermore, there isno immediate need for on-site inspection, which is an important advantage in Boiling Water ReactorPower Plants where feedwater heaters are placed in high radiation dose areas.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Feedwater heaters increase the Rankine cycle efficiency byraising the temperature of the feedwater that is piped into thesteam generator, in such a way that the necessary thermal energytransferred to the steam to increase the enthalpy of the systemwillbe lower. Furthermore, certain stages of the turbines installed innuclear power plants have a special geometry on the inactive sideof their buckets, the purpose of which is to reduce the amount ofmoisture present in the steam. Steam moisture is sent to thefeedwater heaters (returning to the condenser and, therefore, to thecycle). This is one of the main objectives of feedwater heatersinstalled in nuclear power plants and it defines the minimumnumber that are needed, so that the steam quality can be kept overcertain values to ensure the integrity of the static and dynamiccomponents of the steam turbine.

Maintenance, Santa María de

marioalvarezfdez@gmail.com

6

Hence, the obvious importance of ensuring that any degradationof the feedwater heaters and the steam turbines is kept to a min-imum when the facility is in operation. If the water leaving theheaters is not of a specified temperature, or if the mass flow orspecific enthalpy going to the heater from the extractions points ishigher than the specified level, more power will be wasted. Effec-tive monitorization of the process can be a very useful tool fordiagnostic purposes.

Improvements to computer calculations applied to thermalengineering, have led to the development of a great number ofmodels that focus on the design of heat exchangers. They includethe complex BelleDelaware method [1] or the so-called particleswarm optimization technique [2] for optimal design, which has acomplete library of configurations, in order to select the one withthe minimum drop in pressure [3]. Computational Fluid Dynamics(CFD) have also been successfully applied to heat exchangerdesign [4].

Interesting papers can be found in the field of power plant di-agnostics, although they are in general applicable to three-zonefeedwater heaters that are typical in conventional power plants[5]. However, the most widely accepted procedure for calculating

Nomenclature

A heat transfer surface area, m2 (ft2)Cp heat capacity at constant pressure, kJ/kg�C (Btu/lbm�F)do outside tube diameter, _mm (in)F correction factor for the log mean temperature

difference calculationGTTD greater terminal temperature difference, �C (�F)h specific enthalpy or convection heat transfer

coefficient, kJ/kg (Btu/lbm) or kW/m2 �C (Btu/h ft2 �F)k thermal conductivity, W/m �C (Btu/h ft �F)LTTD lesser terminal temperature difference, �C (�F)_m mass flow, kg/s (lbm/h)N number of shell-side passesNu Nusselt numberp pressure (absolute), bar (psia)Pr Prandtl number_Q heat transferred, kW (Btu/h)R relationship between the temperature difference of

the hot and cold fluidsRT overall thermal resistance, m2 �C/W (h ft2 �F/Btu)Re Reynolds numberrfs shell-side fouling resistance, m2 �C/W (h ft2 �F/Btu)

rft feedwater fouling resistance, m2 �C/W (h ft2 �F/Btu)rm tube material resistance, m2 �C/W (h ft2 �F/Btu)rs shell-side film resistance, m2 �C/W (h ft2 �F/Btu)rt feedwater film resistance, m2 �C/W (h ft2 �F/Btu)t temperature, �C (�F)T tube thickness, _mm (in)U overall heat transfer coefficient, W/m2 �C (Btu/h ft2 �F)X qualityDp pressure drop, bar (psi)Dtm log mean temperature difference, �C (�F)3s stage moisture removal effectiveness

Subscriptsc cold fluid (feedwater)d fluid from the previous heaters or other processesh hot fluid (steam)i inlet sectiono outlet section2 intermediate section (c2 or h2)(1)or 1 drain cooling zone(2) or 2 condensing zone;d On-design value

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5846

the performance of closed feedwater heaters in the power industry,which was developed by the American Society of Mechanical Engi-neers, is called ASME PTC 12.1 [6]. The theoretical data used in thedevelopment of this standard can be found in the specializedliterature [7]. However, if used in nuclear power plants, in whichwet steam is extracted from the steam turbine, this procedurewould require data on steam quality and mass flow taken fromdirect measurements or from the thermal balance of the steamturbine. There is plenty of published literature on this subject [8],including techniques to estimate steam quality by either radioac-tive [9] or chemical [10] tracers. Recently, the Electric PowerResearch Institute (EPRI) published a paper [11] introducing asystem for in-situ measurement of steam wetness. The use oftracers is a good solution for performance assessment, but not forreal-time evaluation, where the aforementioned procedure couldbe applied in the future. However, these kinds of systems are notactually installed in modern facilities.

In the present paper, a new procedure is introduced, in order toestimate the enthalpy and the mass flow of the steam bled from theturbine that passes into the feedwater heater, that may be appliedto the diagnostics of these components in nuclear power plants.

2. Governing equations

Heat transfer in a feedwater heater can be formulated as:

_Q ¼ UAFDtm (1)

Each feedwater heater zonewill have an individual heat transfervalue, _Q , the sum of which will equal total heat transferred by thefeedwater heater. It will also have an overall heat transfer coeffi-cient, U, and a heat transfer surface area, A. The value of the loga-rithmic mean temperature difference (LMTD), Dtm, may becalculated for pure parallel-flow or counter-flow heat exchangerswith the following equation [12]:

Dtm ¼ GTTD� LTTDlnðGTTD=LTTDÞ (2)

where, the greater terminal temperature difference, GTTD, and thelesser terminal temperature difference, LTTD, are calculated [13] inparallel flow heaters by:

GTTD ¼ thi � tci (3a)

LTTD ¼ tho � tco (3b)

and in counterflow heaters by:

GTTD ¼ thi � tco (4a)

LTTD ¼ tho � tci (4b)

In Eq. (1), the correct factor, F, depends upon the particular flowarragment in shell-and-tubes heat exchangers [14].

In the case of nuclear power plants, the hot fluid inlet temper-ature, thi, is the saturation temperature at heater pressure; as thispressure is known by a pressure transmitter usually installed in theheater, the temperature can be calculated by using the IAPWS (In-ternational Association for the Properties of Water and Steam)formulation [15].

If Eq. (1) is resolved for a single zone (condensing) feedwaterheater, all temperatures in the preceding equations will be known,and the value of the logarithmic log mean temperature difference,Dtm, may be easily calculated. However, the single zone design isnot the normal design for power plant heaters with two or morezones (desuperheating, condensing and drain cooling). The inter-mediate temperatures of the cold fluid (feedwater) should there-fore be calculated, as will be discussed later, by using an indirectprocedure.

In Eq. (1), the heat transfer surface area, A, can be calculatedfrom the geometry of the tubes, which are specifications that willbe supplied by the heater manufacturer. This value should be cor-rected, where tubes may have become plugged.

Finally, the overall heat transfer coefficient, U, is usually indi-cated by the manufacturer, but it should be noted that the value isgreatly influenced by specific operating conditions. For

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 47

convenience, the overall heat transfer coefficient calculations arepresented in full in Appendix A. Note that this calculation requiresdata on thermal conductivity, heat capacity at constant pressureand steam viscosity, in order to calculate the shell-side film resis-tance, rs, by using Eqs. (A.2a) and (A.5). The IAPWS formulation isnot valid for the calculation of these properties under wet steamconditions, which is the case of nuclear power plants. Thus, somesimplification will be required, but with very little influence on thefinal result: as explained further on, the calculations in this paperconsider that the shell-side film resistance is only influenced bychanges in the mass flow in the drain-cooling section.

A further coefficient was introduced some time ago for moreaccurate adjustments inwet steam calculations: the stage moistureremoval effectiveness, 3s. In these facilities, the blades have someirregularities on their inactive surfaces (typically groves) where anywater still present in the wet steam is centrifuged and removed,increasing the quality of the steam that will be used by the nextstage. A wet steam turbine blade with radial groves is shown inFig. 1.

Effectiveness is the relationship between the percent of totalflow to that stage that is removed as water and the percent of shellmoisture before moisture removal. Thus, moisture removal in theturbine stage will be percentage effectiveness multiplied by themoisture at the stage before moisture removal multiplied by thetotal flow entering the following stage [16]. According to the labelsin Fig. 2:

_mhi ¼ 3s$ _mðt2Þ$1� Xðt2Þ1� Xhi

(5)

The effectiveness of moisture reduction, as the flow of steamenters the turbine section, is:

3s >_mhi_mðt2Þ

(6)

For convenience, Eq. (6) is calculated in full in Appendix C.

Fig. 1. Blade removed from a low pressure turbine in a nuclear power plant.

Fig. 2. Turbine extraction to a feedwater heater.

2.1. Balance of closed feedwater heaters in nuclear power plants inon-design condition

The heaters in nuclear power plants are usually two-zonecounterflow feedwater heaters with a condensing zone and adrain cooling zone (a separate structure, integrated inside the shell,and completely separate from other components in the steam flow)as shown in Fig. 3. At the bottom of the heater, a liquid seal islocated at the position of the heater drain valve, so that the highestlevel is always below the tube bundle (so that cooling of thecondensed steam cannot take place) and the lowest level is alwaysabove the inlet to the cooling zone (so that only saturated liquid canpass into this section). Conventionally, the level is held by a PIcontroller, but recent models can be found in the literature tocontrol the water level in the heater by optimizing the efficiency[17].

When the steam extracted from the steam turbine goes into thefeedwater heater, it is condensed and then subcooled. So, thefeedwater is heated in two zones: first, by the drain cooled steam,and then, by the condensed steam. As the overall heat transfercoefficients and heat transfer surface areas are different, thecalculation should be divided into two steps; the feedwater tem-perature when leaving the drain cooled zone (entering thecondensed zone) will be unknown. Fig. 4 shows a schematicdrawing of a heater and plots the different parameters that areneeded to develop the thermal analysis and the temperature vari-ations of the hot and cold fluid.

The following parameters will be known: mass flow and ther-modynamic properties of the steam coming from the previousheater ( _md and hd); mass flow and thermodynamic properties ofthe feedwater ( _mc, tci, tco, pci and pco), heater pressure (ph) andthermodynamic properties of the steam leaving the heater (tho andpho). The pressure pco will be calculated from the drop in pressure

Fig. 3. Sectional drawing of a feedwater heater typical of a nuclear power plant [HL: high level; WL: working level; LL: low level].

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5848

through the tubes: pco¼ pci�Dpc, and pho from the drop in pressurethrough the shell: pho ¼ ph � Dph.

Thus, the following variables will be unknown: mass flow andthermodynamic properties of the steam extracted from the turbineand, in some cases, from other processes such as valves/turbineseals, turbine shell drains, moisture removal provisions, and excesssteam from steam seal regulator ( _mhi and hhi). The objective of thisstudy is to demonstrate how these variables may be calculated.

The mass flow leaving the heater remains unknown, but re-quires no more than an easy mass balance calculation:_mho ¼ _md þ _mhi.

In two-zone feedwater heaters, Eq. (1) should be divided into:

_Q ð1Þ ¼ Uð1ÞAð1ÞFð1ÞDtmð1Þ (7a)

_Q ð2Þ ¼ Uð2ÞAð2ÞFð2ÞDtmð2Þ (7b)

where, subscript (1) refers to its properties in the drain coolingzone and subscript (2) refers to its properties in the condensingzone. The heater manufacturer provides the specifications for the

Fig. 4. Feedwater heater, subsystems under consideration

overall transfer coefficient, U, and the heat transfer surface area, A.If the first value is unknown, it can be determined by using theequations in Appendix A. The correction factor F(1) for the log meantemperature difference calculation in the drain-cooling zone, whenit is applicable, depends on the flow arrangement in this particularsection [14] and it is usually supplied by themanufacturer, or can becalculated according to the Bowman formulation [18]. The value ofthe correction factorwill be one in condensers and in 1-1 tube-and-shell heat exchangers, so in the condensing zone F(2) ¼ 1.

In counterflow feedwater heaters, Eq. (2) should be divided into:

Dtm1 ¼ ðth2 � tc2Þ � ðtho � tciÞln th2�tc2

tho�tci

(8a)

Dtm2 ¼ ðthi � tcoÞ � ðth2 � tc2Þln thi�tco

th2�tc2

(8b)

Obviously, the temperatures of the hot fluid at the inlet and atthe intermediate section (between the condensing and the drain

and variation of steam and feedwater temperatures.

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 49

cooling zones) will be equal and can be calculated as the heaterpressure is known: thi ¼ th2 ¼ tsat(p ¼ ph).

The following equation may be used to calculate the heattransfer values, _Q:

_Q ¼ _Q1 þ _Q2 ¼ _mcðhco � hciÞ (9)

where, the specific enthalpy of the cold fluid at the inlet,hci ¼ enthalpy(t ¼ tci,p ¼ pci), and the specific enthalpy of the coldfluid at the outlet, hco ¼ enthalpy(t ¼ tco,p ¼ pco), can be calculatedby using the IAPWS formulation.

Combining Eqs. (7a), (7b), (8a), (8b) and (9), in the case of on-design conditions:

U1dA1F1

�th2;d � tc2;d

�� �tho;d � tci;d

�ln th2;d�tc2;d

tho;d�tci;d

þ U2dA2

�thi;d � tco;d

�� �th2;d � tc2;d

�ln thi;d�tco;d

th2;d�tc2;d

¼ mc;d�hco;d � hci;d

�(10)

The value of tc2;d can be calculated by iteration of Eq. (10), takinginto account that tco;d > tc2;d > tci;d. So, the values of the heat that istransferred in the condensing zone, _Q1;d, and the heat transferredin the drain cooling zone, _Q2;d, can be solved by using Eqs. (7a) and(7b).

The mass flow extracted from the turbine can be calculatedfrom:

_mhi;d ¼_Q1;d

hh2;d � hho;d�X

_md;d (11)

And the specific enthalpy of the steam entering the heater fromthe steam turbine:

hhi;d ¼_Q2;d þ

�_mhi;d þ

P_md;d

�hh2;d �

P_md;dhd;d

_mhi;d(12)

Once the enthalpy hhi;d has been calculated, the quality of thesteam can be solved by applying the IAPWS formulation:Xhi;d ¼ quality(p ¼ ph; h ¼ hhi;d). Thus, Eq. (5) determines theeffectiveness of the moisture removal stage:

3s;d ¼_mhi;d$

�1� Xhi;d

�_mðt2;dÞ$

�1� Xðt2;dÞ

� (13)

2.2. Balance of closed feedwater heaters in nuclear power plants inoff-design conditions

Obviously, when the heater manufacturer specifies the value ofthe overall heat transfer coefficient, it refers to the design condition.In off-design conditions, this coefficient should be corrected withthe new feedwater thermodynamic properties and the mass flow(affecting feedwater film resistance, rt), and the new thermody-namic properties and mass flow of the steam (affecting the value ofthe shell-side film resistance, rs). The other coefficients of theoverall heat transfer coefficients are the fouling resistance values(the effects of which increase with age, but are considered constantin this paper) and the tubematerial resistance values. The last valueshould be corrected for loss of thickness or the build up of crud thatmay influence the tube walls; for the purposes of this paper, thegeometry of the tubes will be considered constant.

First of all, the values of these coefficients (see Appendix A) haveto be calculated for on-design conditions. The thermodynamic and

transport properties of the feedwater in the drain cooling zone, (1),should be calculated from the pressure and temperature of thewater at the inlet section of the heater: pci;d and tci;d. The samepressure may be used in the condensing zone, (2), and the tem-perature can be taken from the average temperature between theintermediate temperature, tc2;d and the final temperature, tco;d.

The overall heat transfer coefficient should be corrected for theoff-design condition, by recalculating the values of the resistancecoefficients, in cases where these coefficients are affected bychanges in the properties of the fluid or the mass flow. The resis-tance coefficients used in off-design calculations will be those thataffect the drain cooling zone, as will be explained in the followingsection, where the influence of changes in the thermodynamic andtransport properties of the steam and feedwater can be neglected,because the (hot and cold) fluid is compressed-water that takes thethermodynamic state in on-design operation as its reference. Thus,the overall resistance coefficient of drain cooling zone will beRT1 ¼ rfs(1) þ rm(1) þ rft(1) þ rt(1) þ rs(1):

RT1 ¼ 3:522$10�5�

dodo � 2T

�þ do2000k

ln�

dodo � 2T

�

þ 5:2833$10�5 þ rtð1Þ;d

�_mc:d_mc

�0:835

þ rsð1Þ;d

�_mho;d_mho

�0:55

(14)

The actual value of the overall heat transfer coefficient for eachzone could now be calculated, although a further variable has yet tobe introduced: the mass flow of the steam ð _mho ¼ _mhi þ

P_mdÞ.

Eqs. (7a), (8a) and (11) for the drain cooling zone and Eqs. (7b),(8b) and (12) for the condensing zone allow us to define twoequations with five unknowns (mass flow, _mhi, enthalpy of thesteam extracted, hhi, intermediate temperature, tc2, and overall heattransfer coefficients, U1 and U2). But according to Eq. (14), theoverall heat transfer coefficient is a function of the mass flow of thesteam, which means that, in reality, two equations with three un-knowns are achieved:

_mhihhi þX

ð _mdhdÞ ��_mhi þ

X_md

�hh2

¼ U2A2ðthi � tcoÞ � ðth2 � tc2Þ

ln thi�tcoth2�tc2

(15)

�_mhi þ

X_md

�ðhh2 � hhoÞ ¼ U1A1F1

ðth2 � tc2Þ � ðtho � tciÞln th2�tc2

tho�tci

(16)

2.2.1. Drain cooling zoneAccording to Appendix A, the shell-and-tube fouling resistances,

rfs and rft, and tube material resistance, rm, will be constants for aparticular feedwater heater. Furthermore, the feedwater filmresistance, rt, can be calculated, because the instrumentation of theheater will supply the actual mass flow. All those resistances maybe calculated with Eq. (14).

Assuming that Ro1 ¼ rfs(1) þ rm(1) þ rft(1) þ rt(1), by substitutingthe value of rs(1), from Eq. (14), in Eq. (A.1), we have:

_mho_mho;d

¼�

rs1;d1=U1 � Ro1

�20=11

(17)

If the heat capacity at constant pressure is considered constant,which is a normal approximation for subcooled water, the rela-tionship _mho= _mho;d can be rewritten as:

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5850

_mho_mho;d

¼ _mhocph1ðth2 � thoÞ_mho;dcph1;d

�th2;d � tho;d

�$�th2;d � tho;d

�ðth2 � thoÞ

(18)

Considering the heat transfer rate of the cold fluid instead of thehot fluid:

_mho_mho;d

¼ _mccpc1ðtc2 � tciÞ_mc;dcpc1;d

�tc2;d � tci;d

�$�th2;d � tho;d

�ðth2 � thoÞ

(19)

and, assuming that Y ¼ 1_mc;d

$thi;d�tho;dtc2;d�tci;d

, and R1 ¼ th2 � tho/tc2 � tci, thenEq. (16) can be rewritten as:

�rs1;d

1=U1 � Ro1

�20=11

¼ YR1

_mc (20)

Finally, isolating the value of the overall heat transfer coefficient,U1:

U1 ¼"rs1;d

�R1Y

_mc

�11=20þ Ro1

#�1

(21)

The overall heat transfer coefficient, U1, and the intermediatefeedwater temperature, tc2, can be calculated with Eq. (21) and Eq.(B.5a) or Eq. (B.5b) together with Eq. (B.6) by iteration. So, the massflow extracted from the steam turbine can be calculated by Eq. (17),recalling that _mhi ¼ _mho �

P_md.

2.2.2. Condensing zoneThe values of the intermediate feedwater temperature, tc2, and

extracted mass flow, _mhi, were calculated using the drain coolingzone formulation. Now, from Appendix B:

U2A2 ¼ �Cc2$ln�1� tco � tc2

thi � tc2

�(22)

where Cc2 ¼ _mc$Cpcð2Þ;d.By isolating the steam extracted enthalpy, hhi, in Eq. (14), and

substituting the value U2A2 for the one obtained in Eq. (22):

hhi ¼1_mhi

8<:� _mc$Cpcð2Þ;d$

ln�1� tco�tc2

thi�tc2

�ln�thi�tcoth2�tc2

� ½ðthi � tcoÞ � ðth2 � tc2Þ�

þ�_mhi þ

X_md

�hh2 �

Xð _mdhdÞ

9=;

(23)

In the condensing zone, th2 and thi can be considered equal,which means that the steam pressure is also considered constantduring the condensation process. This simplification is usuallyadopted. Furthermore, taking into account that ln(1 � tco � tc2/thi � tc2) ¼ ln(thi � tco/th2 � tc2), Eq. (23) can be rewritten as:

hhi ¼1_mhi

n_mc$Cpcð2Þ;dðtco�tc2Þþ

�_mhiþ

X_md

�hh2�

Xð _mdhdÞ

o(24)

Eq. (24) shows the thermal balance of the condensing section ofthe feedwater heater.

The schematic calculation described in these sections, for on-design and off-design conditions, is represented in Fig. 5.

3. Experimental analysis

Data supplied from a nuclear power plant were used in ourvalidation of the equations andmethods proposed in this study: theSanta María de Garoña Nuclear Power Plant (SMGNPP), situated inthe north of Spain. Its two trains of feedwater heaters are located inparallel and are schematically represented in Fig. 6.

The steam supplied from the reactor (boiling water reactor type,BWR) is piped to the high-pressure (HP) turbine along four pipes,each with control and stop valves. After the expansion in the HPturbine (10 stages), the steam is passed along four pipes, two ofwhich run into low-pressure (LP) turbine “A” and the other two intoLP turbine “B”.

Between the HP turbine and the LP turbines, the pipes that runinto LP turbine “A” are connected to heater 12A and the pipes thatrun into LP turbine “B” are connected to heater 12B.

After these connections, the pipes run into four Moisture Sep-arators (MS), where the steam quality is improved by moistureremoval, and then on to heater 11A (from one of the MS in train “A”and one of the MS in train “B”) and heater 11B (in the same way).The seal steam used in the turbine valves and in the high pressureside of the HP turbine goes to heater 11A.

Then, the steam is expanded in LP turbines “A” and “B”. Each onehas two opposite flows going to the condenser (10 stages per flow).The following extractions are in place during the steam expansionthrough the LP turbines: in stage 12 to heaters 11A/B, in stage 14 toheaters 9A/B, in stage 16 to heaters 8A/B and in stage 18 to heaters7A/B (located in the condenser). Several drains are placed in stages15, 17 and 19, in order to improve steam quality. The steam that ispiped to Heaters 7A/B has not been through the steam-seal regu-lator (SSR), which is designed to be supplied by 10,000 lb/h(1.26 kg/s) of seal steam from the valves and the HP turbine. Fig. 6also represents the different flows used for turbine and valve seals.

Finally, and not represented in Fig. 6, the steam drained fromheaters 12A/B goes to heaters 11A/B, etc. The steam drained fromheaters 7A/B goes directly to the condenser.

3.1. Heaters 12A/B analysis

As explained above, heaters 12A/B are supplied with steam fromthe HP turbine discharge. Very little degradation was reported forthis particular turbine after each preventive maintenance inspec-tion (every eight years), so its efficiency can be considered equal tothe design specification. The enthalpy of the extracted steam istherefore comparable to the design specifications.

The model introduced in this paper was used to calculate thedifferent parameters of these heaters using actual parameterstaken from operational processes over a 4-year period (2008e2011). The model was implemented with these parameters and theEngineering Equation Solver (EES) [19] software programme. Ad-justments for the nine tubes plugged into heater 12B were intro-duced. The registered values are presented in Table 1, alongside thevalues for the overall heat transfer coefficients (U1 and U2), and thequality (Xhi) and mass flow ð _mhiÞ of steam extracted to the heaters.In cases where the registered parameters were wrong due toinstrumentation failure, the value is shown in italics alongside thenew value that appears between square brackets.

The value of ph in heater 12A had been fluctuating from Aug-09,when it was 11.24 bars, until the 2010 refueling outage. After theoutage and following calibration of the pressure transmitter, thepressure readings returned to normal and registered stable values.Hence, the selected value for the calculations was changed to 11.25bars in Jan-10.

The outlet feedwater temperature in heater 12A from Nov-08 toMar-09 was an erroneous reading. As the temperature element was

Fig. 5. Schematic diagram of the model, including on-design and off-design calculations.

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 51

fixed in the 2009 refueling outage, the temperature readings forJan-2009 (�270 �C) were changed to a value of 148.9 �C in thecalculations.

Finally, the pressure signal in heater 12B between 2008 and2009 was lost, until the pressure transmitter was fixed in the 2009

Fig. 6. Balance of Plant e Santa María de Garoña [HP: high-pressure turbine; LP: low-preejectors; SPE: steam-packing exhauster; #: Stage; MU: make-up water; CST: condensed sto

refueling outage. The pressure for the calculations was changed to11.28 bars in Jul-08 and Jan-09.

In Fig. 7, the calculated values of the overall heat transfer co-efficients U1 (drain cooling zone) and U2 (condensing zone) areplotted, as are the values of the coefficients in the design condition

ssure turbine; HTR: feedwater heater; SSR: steam-seal regulator; SJAE: steam-jet airrage tank].

Table 1Parameters used for the calculations ( _mc , tci, tco, ph and tho) and results obtained (U1, U2, Xhi and _mhi). Column one shows the values specified by themanufacturer for on-designoperation.

Design Jan-08 Jul-08 Jan-09 Jul-09 Jan-10 Jul-10 Jan-11 Jul-11

12A _mc (kg/s) 342.8 342.6 344.3 343.3 345.1 344.2 345.5 345.5 346.3tci (�C) 143.7 142.1 142.3 141.9 142.5 142.0 142.3 142.1 142.1tco (�C) 182.9 175.8 176.8 180.9 181.3 180.9 180.4 180.5 180.6ph (bars) 11.41 11.33 11.42 11.39 11.21 10.7 [11.25] 11.24 11.30 11.30tho (�C) 149.3 148.9 149.1 �270 [148.9] 146.6 145.6 147.5 148.1 148.1U1 (W/m2 �C) 2509.7 2386.9 2389.6 2373.2 2715.2 2806.3 2564.4 2473.5 2473.4U2 (W/m2 �C) 3929.1 2155.4 2270.5 3230.6 3564.5 3351.4 3223.8 3191.6 3245.6Xhi (�) 0.87 0.71 0.73 0.83 0.92 0.94 0.86 0.84 0.85_mhi (kg/s) 30.9 32.3 32.3 32.5 28.6 27.7 30.2 31.3 31.3

12B _mc (kg/s) 342.8 342.7 343.5 342.8 344.5 344.8 345.2 344.4 342.6tci (�C) 143.7 141.5 141.7 141.5 141.7 141.5 130.9 136.2 137.8tco (�C) 182.9 180.0 179.8 179.4 179.5 179.4 179.2 179.1 180.3ph (bars) 11.41 11.27 0.95 [11.28] 0.93 [11.28] 11.30 11.36 11.31 11.29 11.34tho (�C) 149.3 146.5 147.9 147.4 147.4 147.1 147.4 146.9 146.7U1 (W/m2 �C) 2509.6 2615.9 2466.3 2504.1 2526.0 2545.0 1867.2 2146.1 2267.3U2 (W/m2 �C) 3929.1 3089.3 3038.9 2928.0 2942.2 2873.0 3256.8 3064.9 3264.1Xhi (�) 0.87 0.88 0.83 0.83 0.84 0.85 0.87 0.85 0.87_mhi (kg/s) 30.9 29.7 31.4 30.9 30.7 30.5 39.7 35.5 33.8

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5852

(taken from the heater specifications). A vertical straight line rep-resents the dates onwhich the heaters were cleaned and inspected:train “A” in 2009, and train “B”, in 2011.

The abnormally low values in January and in July, 2008, of theoverall heat transfer coefficient in the condensing zone of heater12A should also be noted in comparisonwith the design coefficient.When the equations introduced in the present paper were applied,in order to calculate the quality of the steam extracted from theturbine, a lower than expected value was also obtained. This valueand those of the overall heat transfer coefficients and themass flowof the extracted steam were obtained on the assumption of con-stant transfer surface areas, A1 and A2; in other words, that the levelof the heater is constant and equal to the level in the on-designcondition.

Figs. 8 and 9 plot the values that represent the variations insteam quality, Xhi, and the overall heat transfer coefficients, U1 andU2, as calculated by the model, versus the transfer surface area inthe condensing zone. The effect of the transfer areas in the overallheat transfer coefficients and steam quality is clearly visible.

On the assumption that steam quality undergoes no significantvariations, which is the normal situation in this heater, if the steamdischarged from the HP turbine remains approximately constant,then the operating data, which refers to data from January and July,2008, leads us to the conclusion that the level in the heater waslower than the design point. If we consider the steam quality ofXhi ¼ 0.87 registered in January, 2008, the resulting heat transfersurface area in Fig. 9 would be A2 ¼ 1,034.4 m2: meaning that the

Fig. 7. Overall heat transfer coefficients calculated by using the present method. Thedashed line shows the on-design operation values (U1 ¼ 2509.7 W/m2 �C,U2 ¼ 3929.1 W/m2 �C).

heat transfer surface area in the condensing zone would be around42 m2 higher than the designed one (on-design condition,A2 ¼ 992,2 m2). Considering the dimensional characteristics andsectional drawing of the heater, shown in Fig. 10 and Fig. 3, theimplication is that the first two sub-sections of the cooling sectionwere not effective, causing a decrease in the feedwater tempera-ture, as shown in Table 1.

After this particular issue was identified, possible heater ab-normalities were investigated. In August of 2008, an inspection ofthe level control valve of heater 12A noted that the position of thevalve did not coincide with the position recorded in the controlroom. Later on, a valve closure failure was recorded, in October,when the heater level dropped to zero. It was possible to maintainthe level by positioning the valve in manual and controlling thelevel with the emergency drain valve. In November, the positionwas changed, but was inaccurately calibrated, perhaps due to theconstraints of working in an area with high levels of radiationdoses. The levels were adjusted in the 2009 refueling outage(March), in the 2010 refueling outage (March) and in the 2011outage (May). If the other parameters were properly adjusted (theinfluence of the different parameters will be discussed in the nextsection), the readings suggest that the level was too highly adjustedin 2009 andwas set too low (but closer to the design point) in 2011;the calibration appears to be correct in the 2010 refueling outage.

All these issues could be easily detected using the modeldescribed in this paper, even when inaccurate readings may have

Fig. 8. Overall heat transfer coefficients modeled at different heater levels.

Fig. 9. Steam quality at the extraction point modeled at different heater levels.

Fig. 11. Simulation of heater temperatures with different levels (tci is supposed con-stant) [TTD ¼ Terminal Temperature Difference; DCA ¼ Drain Cooler Approach].

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 53

been registered in the control room at an earlier point in time andwith no on-site inspection.

Furthermore, the study has identified the importance of intro-ducing the heater level and correcting the heat transfer surfaceareas. However, this parameter is not recorded on the computer (ithas to be checked on the control room instrumentation), and thepoint at which the steam cooling begins is not measurable: in casethe level drops, it is not possible to quantify the cooling andcondensing sections by using the instrumentation.

The feedwater outlet temperature and the steam outlet tem-perature were calculated with the model presented in this paper,considering that the steam extracted from the turbine had designvalues of mass flow and quality, even though the data recorded forheater 12A in January of 2008 were used. The results are plotted inFig. 11, which shows the effect of the level. When the condensingzone transfer surface area, A2, is adjusted to higher values, thefeedwater temperature increases slightly, increasing the Rankineefficiency if the feedwater inlet temperature remains constant.

Fig. 10. Sectional drawing of heater 12A/B (tu

However, the steam temperature drained from the heater, passingonto the next one, decreases, which leads to a drop in the feedwaterinlet temperature. This negative effect in the value of tho is verysignificative when high values of A2 are simulated. It means that, infact, the terminal temperature difference will decrease when theshell liquid level is reduced. The plot shows an experimentallydemonstrated effect. Furthermore, the upper value of A2 will be onethat ensures the mechanical integrity of the drain cooler.

3.2. Sensitivity analysis

The objective of the present model is to calculate the thermo-dynamic properties of wet steam extracted from turbines and pipedto feedwater heaters in nuclear power plants. In the previous sec-tion, taking into consideration the resulting values at different

be plate) according manufacturer’s data.

Table 3Estimated errors in steam quality calculation due to inherent errors in measurementapparatus.

_mc (kg/s) tci (�C) tco (�C) ph (bars) tho (�C) xhi (�) Error (%)

339.4 143.1 182.3 11.60 149.9 0.826 6.5346.1 144.3 183.5 11.66 148.7 0.918 7,5

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5854

times in heaters 12A and 12B, the importance has been highlightedof introducing the real level (the real heat transfer surface areas). Infact, the problemwith this particular issue is that the level was notproperly adjusted. If the real level were lower than the design value,the steam quality value in the model would be lower than theobserved value.

However, this parameter has to be considered when a thermalbalance develops between the feedwater heaters. The remainingparameters that are applied in the equations have a significantinfluence on the results. In this section, the influence of eachparameter will be discussed.

The parameters that applied to heater 12B in July of 2009,shown in Table 1, will be used as an example. If the calculated valuefor the on-design condition of steam quality, Xhi¼ 0.87 (rather thanthe value of Xhi ¼ 0.84 that was obtained), is taken as the rightvalue, the measured values that achieve the on-design quality canbe re-calculated. For comparative purposes, Table 2 shows themeasured value of each instrumented parameter (as well as theheat transfer surface area estimated from the manufacturer data,after nine plugged tubes in this particular heater at that time havebeen discounted), the value that satisfies the on-design quality andthe discrepancy between them.

In this particular power plant, the temperature is measuredwithtype “T” thermocouples with an error of�0.6�C(�1�F) or�0.4%, thepressure is measured with pressure transmitters with an error of�0.2%FS (Full Scale), and the flow is measured with flowmeterswith an error of �1%. All these error margins refer to inherentsensor error.

With regard to the heat transfer surface area (either in thecondensing zone or in the drain-cooling zone), its value should bethe design value, except in those abnormal situations, such as theone described in the previous section, or if a reduction in thetransfer capability is taken into consideration. But, in this lattercase, the expected value [2] in Table 2 should be lower than theestimated one.

The other parameters, considered individually, could onlysatisfy the design quality, if the measurement instruments recor-ded outside the calibrated values. However, the sum total ofinherent instrument error should be taken into account.

Taking this inherent error of the measurement apparatus intoconsideration, the limit values for quality in the design conditionare represented in Table 3.

In conclusion, when the quality reading is within �6% of thedesigned value, the discrepancy can be justified by the inherenterror of process measuring instruments. Otherwise, the resultsshould be investigated in the same way as in the previous section.

This observation clearly supports the high accuracy of thecalculation.

3.3. Thermal analysis of the feedwater heater train

As explained in Section 3, only heaters 12A and 12B are suppliedby the steam discharged from the HP turbine, which means that

Table 2Sensitivity analysis from data measured and calculated in heater 12B in July of 2009.[1]: Measured value according to data in Table 1; [2]: Value that satisfies the on-design quality; [3]: Error.

[1] [2] [3]

A2 (m2) 984.3 992.5 0.8%_mc (kg/s) 344.5 355.7 3.3%tci (�C) 141.7 144.8 3.1 �Ctco (�C) 179.5 180.7 1.2 �Ctho (�C) 147.4 146.5 �0.9 �Cph (bars) 11.3 13.8 22.1%

other components of the plant have no influence on the mass flowentering the heater. Heaters 11A and 11B are supplied by steamfrom the LP turbine, the drained cooled steam fromheaters 12A and12B and by the moisture removed from the MS. The other heatersare therefore influenced by the mass flow coming from the MS.

If individual flows into the heater are to be calculated, when thepresent model is applied to off-design operation, an additionalformulation is necessary, such as the Martin equation for turbine/valve seals. This formulation will always include enthalpy as avariable, which is affected by the efficiency of the turbine sectionwhere the extraction point is located. In the model, mass flow andenthalpy are considered in terms of the mixture of all the steamfrom the extraction points; in reality, the model can be appliedwithout the introduction of this formulation. However, only the on-design operation has been considered here, in order to compare theresults with the reference values. If the model is used in actualoperation, and the rest of the plant is not modeled, possible mal-functions may be due either to the heater or to the turbine. Thisarea falls outside the scope of the present paper.

The thermal balance of the feedwater heater train was calcu-lated with data from the manufacturer’s documentation for theheaters and steam turbine, as shown in Table 4, in order to validatethe model presented in this paper. The model of the completefeedwater train was implemented in EES software. The results areshown in Fig. 12, as well as the thermal balance supplied by themanufacturer. Table 5 shows the main values for comparison, ob-tained with the present model, and those supplied by the manu-facturer in the thermal kit included in the SMGNPP documentation.

Table 5 shows the values of mass flow, specific enthalpy andsteam quality piped into Heaters 8 and 7 that are contained in thethermal kit supplied by the manufacturer, which are split into twoparts: the steam extracted from the stage moisture removal pro-vision, and the steam extracted from the shell moisture removalprovision; the thermodynamic properties of the steam going to theheaters were calculated by a simple energy balance of the twoflows. The calculations with the present model give the values fortotal steam extracted from the turbine and piped directly into theheater. Furthermore, all the mass flows represented in Table 5 arethe sum of all flows entering or leaving heaters “A” and “B”.

In general, the values of the steam flowing through and leavingthe heater, _mhi and _mho, are very similar, specially in the latter case.The following consideration should be taken into account in theother parameters: the calculations introduced in the present paper

Table 4Feedwater heater data used in the calculations as supplied by the manufacturer.

HTR12 HTR11 HTR9 HTR8 HTR7

U1 (W/m2 �C) 2509.7 2464.4 2538.2 2413.3 1811.4U2 (W/m2 �C) 3929.1 3696.5 3855.5 3446.7 2612.0k (W/m �C) 27.69A1 (m2) 113.8 267.6 281.4 353.0 668.0A2 (m2) 992.2 873.3 771.1 845.4 1168.0do (mm) 15.88T (mm) 2.54Dph (bars) 0.34 0.32 0.34 0.31 0.14Dpc (bars) 0.86 0.86 1.07 1.11 0.48

Fig. 12. Thermal balance of feedwater heaters on the SMGNPP e 100% Load. (a) Supplied by the manufacturer. (b) Calculated with the method proposed in this paper.

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 55

were performed by using the IAPWS-97 formulation; the calcula-tions supplied in the thermal kit were developed in 1966.

It should be noted that the calculations in the thermal kit use Eq.(5) to estimate the quality of the steam extracted from the turbine,applying the assigned values of moisture removal effectiveness. Inthe model introduced in this paper, Eq. (23) is used to calculate thequality value directly from the enthalpy value. In the particular caseof heaters 12, 11 and 9, the effectiveness introduced in the thermalkit would be zero, which means that the quality will be equal to thequality of the steam flowing through the preceding stage in theSMGNPP, because moisture removal provisions are located in lowpressure turbine stages from stage 15. The values of steam qualityobtained by applying the present model are very similar to thevalues calculated in the SMGNPP thermal kit. Table 5 shows the %deviations between the values calculated from the figures in the

Table 5Thermal balance of the feedwater heater train e 100% Load.

SMGNPP thermal kit Calculated

HTR _mhi (kg/s) _mho (kg/s) hhi (kJ/kg) Xhi 3s (%) _mhi (kg/s)

12 61.25 61.25 2526.3 0.87 0.0 61.6811 94.34 155.59 1309.8 0.32 0.0 95.129 26.24 181.83 2482.5 0.90 0.0 25.628 31.91 213.74 1917.8 0.67 7.5 30.127 41.23 254.97 1610.1 0.57 16.0 44.30

SMGNPP documentation and the values calculated by using thepresent model.

4. Results and discussion

The main objective of the present paper is to present theformulation needed to calculate the thermodynamic properties andthe mass flow of the steam passing into the feedwater heaters thatare installed in nuclear power plants. As described in Sec. 2.1 andSec. 2.2, it is necessary to calculate these properties in on-designoperation (using data supplied by the manufacturer), as well asanother variables defined here, and use them as reference values inthe off-design calculation. The steps to be followed are summarizedin Fig. 5.

Absolute variation in %

_mho (kg/s) hhi (kJ/kg) Xhi _mhi _mho hhi Xhi

61.68 2518.2 0.87 0.7 0.7 0.3 0.0156.80 1304.7 0.32 0.8 0.8 0.4 0.0182.42 2535.3 0.93 2.4 0.3 2.1 3.3212.54 2016.7 0.72 5.6 0.6 5.2 7.5256.84 1518.3 0.53 7.5 0.7 5.7 7.0

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e5856

The model uses the well-known log mean temperature differ-ence and NTU-effectiveness methods. Furthermore, Eq. (21) makesit possible to calculate the feedwater temperature when the coldfluid leaves the drain-cooling zone (goes into the condensing zone),even if the mass flow and the enthalpy of the steam bled to theheater are unknown. Then, these properties can be easily calculatedby applying the mass and energy balances to each zone. Impor-tantly, the model easily adapts itself to the simplifications adoptedby these methods: the overall heat transfer coefficient is constantthroughout the heater, the flow rate of each fluid is constant, thespecific heat of each fluid is constant, there is no vapor condensa-tion or boiling liquid in part of the heater, heat losses are negligible,there is equal heat-transfer surface in each pass, and the temper-ature of the shell-side fluid in any shell-side pass is uniform overany cross section [18]. All these simplifications can be assumedtaking into account that these calculations are done in the drain-cooling zone (liquideliquid heat exchanger with a simplearrangement).

As a result, the method described here was applied to a realfacility, in order to verify whether this tool is useful for predictivemaintenance applications. The formulation and procedurescurrently used in the industry do not calculate the feedwaterheaters as individual components, so steam turbine performance inthe on-design condition has to be used (which is the biggestconstraint of these methods). The present method is able toperform these calculations individually in each heater. However,the results are non-measured variables, using the instrumentedparameters of the process.

For comparison and validation purposes, the on-design opera-tion described in Sec. 3.3 was used, in which the individual analysisof each heater is clear, instead of the traditional methodology, asshown in Fig. 12, where the flows going to the heater have to beintroduced. Both methods show similar results, as indicated inTable 5.

However, themain application of these calculations should be inoff-design operation, an issue that has been experimentallyexamined in Sec. 3.1. Once the steam properties and the mass flowhave been calculated, the rest of the parameters involving theheater performance can be easily defined. But the first are inthemselves very useful for diagnostic purposes, and are indepen-dent of any particular operational condition. In Fig. 7 the overallheat transfer coefficient of the heater in both zones and in bothheaters is lower than the designed one, because the instrumentedparameters were also different from the designed ones, as shown inTable 1.

A further application could be the calculation of the steamproperties and mass flow extracted from the steam turbines.However, the inherent error of themeasuring apparatus often leadsto non-coherent results, with operation of the steam turbines someway off the design specifications. This issue has been discussed inSec. 3.2 and as indicated, care should be exercised when applyingthe present method.

5. Conclusion

A model has been introduced for the thermal analysis of feed-water heaters in which wet steam is extracted from the steamturbine (and piped into the heater). The model has relevant ap-plications in nuclear power plants, where the fluid flowing throughthe steam turbine is wet steam.

Up until now, the thermodynamic properties of the steamextracted from the heater were solved by applying the stagemoisture removal effectiveness. However, the values of this effec-tiveness, when available from the manufacturer in the thermal kits,are often unrelated to the load. Furthermore, use of this parameter

calculated at the design point of the unit may omit any possibledegradation of the stage moisture removal provisions.

In the present model, the thermodynamic properties of thesteam entering the heater were unknowns. When these values arecalculated they become an important tool for the analysis of plantefficiency. The model has been tested using the data available fromthe SMGNPP thermal kit and actual operating data. In all cases, theimportance of accurate instrumentation of the process has beenemphasized and examples have been given of how to detectinstrumentation error.

As a next step, subsequent research aims to complete themodeling of the Rankine cycle of a nuclear power plant, simulatingthe expansions in all sections of the steam turbine [20] and cor-recting the mass flow and the moisture content in each one.

Acknowledgements

The authors wish to thank NUCLENOR for supplying operationaldata recorded at the Santa María de Garoña Nuclear Power Plant,which were essential for validation of the equations and methodsdescribed in this paper.

Appendix A. The overall heat transfer coefficient in closedfeed-water heaters

Heat transference in the heater, referred to as overall heattransfer, U, appears in Eq. (1). Its value can be calculated from thefollowing equation:

U ¼ 1rs þ rfs þ rm þ rft þ rt

(A.1)

where, rs is the “shell side film resistance”, rfs is the “shell sidefouling resistance”, rm is the “tube material resistance”, rft is the“feedwater fouling resistance” and rt is the “feedwater filmresistance”.

The value of the “film resistances” can be calculated in tube-shell heaters by:

rs ¼ 1hoEf

(A.2a)

rt ¼ 1hi

�dodi

�(A.2b)

where, subscript i indicates the tube side fluid, subscript o indicatesthe shell side tube, h is the convection heat transfer coefficient, d isthe tube diameter and Ef is the fin efficiency [21].

In Eqs. (A.2a) and (A.2b), the convection heat transfer coefficientcan be calculated inside the tubes with the Rabas improved equa-tion [22]:

Nui ¼ 0:015Re0:835i Pr0:462i (A.3)

and by the Kern equation outside the tubes [23]:

Nuo ¼ 0:36Re0:55o Pr0:33o (A.4)

In Eqs. (A.3) and (A.4), Nu is the Nusselt number, Re is theReynolds number and Pr is the Prandtl number. Both equations areused for the (single phase) cooling zone in the present paper.

The value of the “tube material resistance” can be calculated intubular heaters by:

M. Álvarez-Fernández et al. / Applied Thermal Engineering 68 (2014) 45e58 57

rm ¼ do2000k

ln�

dodo � 2T

�(A.5)

where, do is the outside diameter of the tube and T is the tubethickness.

Finally, the value of the “shell/feedwater fouling resistances”can be calculated by:

rfs ¼ roEf

(A.6a)

rft ¼ riAo

Ai(A.6b)

The values of ro and ri are usually fixed value functions of thetype of fluid. Values for the feedwater (ri) and the steam (ro) may befound in the technical literature [24]. The impact of fouling resis-tance on the feedwater heater performance over time has beenanalyzed in Ref. [25].

In the case of heaters in steam power plants, both values areusually ro ¼ 5,283$10�5 m2 C/W (0,0003 hft2F/Btu) in the draingcooling zone, ro ¼ 0 m2 C/W in the condensing zone andri ¼ 3.522$10�5 m2 C/W (0.0002 hft2F/Btu) in both zones.

Appendix B. The NTU or heater effectiveness method appliedto off-design calculations

In the heater drain cooling zone, the feedwater heat capacity atconstant pressure can be calculated, in on-design condition, by:

Cpcð1Þ;d ¼ Q1;d

mc;d�tc2;d � tci;d

� (B.1)

The value of the product C ¼ _mCp in operating mode, referredto as the “heat capacity flow rate”, if no variations are considered inthe value of the heat capacity at constant pressure, will be:

Cc1 ¼_mcQ1;d

mc;d�tc2;d � tci;d

� (B.2)

In the case of steam flowing through the shell side:

Ch1 ¼ ð _mhi þP

_mdÞQ1;d�_mhi;d þ

P_md;d

��th2;d � tho;d

� (B.3)

The value of the number of transfer units (NTU) [14] in the draincooling zone will be:

NTU1 ¼ U1A1

Cc1(B.4)

And heater effectiveness, using the relationship for shell-and-tube heat exchangers [14]:

31 ¼ 1� exp½NTU1ðC1 � 1Þ�1� C1exp½NTU1ðC1 � 1Þ� (B.5a)

31 ¼ 2

26641þ C1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ C2

1

q$

1þ exp�� NTU1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ C2

1

q �

1� exp�� NTU1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ C2

1

q �3775�1

(B.5b)

where, C1 ¼ Cc1/Ch1. Eq. (B.5a) will be applicable to 1-1 shell-and-tube heat exchangers (pure counter-flow heat exchangers) andEq. (B.5b) to 1-2 shell-and-tube heat exchangers.

Finally, the intermediate feedwater temperature will be:

tc2 ¼ 31ðthi � tciÞ þ tci (B.6)

In the condensing zone, the value of the heat capacity flow ratewill be zero for the steam, Ch2 ¼ 0 (th2 ¼ thi). Thus, the equations inthis particular case will be:

Cc2 ¼_mcQ2;d

mc;d�tco;d � tc2;d

� (B.7)

NTU2 ¼ U2A2

Cc2(B.8)

32 ¼ 1� expð�NTU2Þ (B.9)

tco ¼ 32ðthi � tc2Þ þ tc2 (B.10)

Appendix C. The stage moisture removal effectiveness andthe reduction of moisture

If a mass balance is applied to the turbine extraction in Fig. 2,considering only the liquid phase:

_mt3ð1� Xt3Þ ¼ _mt2ð1� Xt2Þ � _mhið1� XhiÞ (C.1)

Dividing by _mt2ð1� Xt2Þ, and taking the value of the effective-ness into account, 3s, in Eq. (5):

_mt3ð1� Xt3Þ_mt2ð1� Xt2Þ

¼ 1� 3s (C.2)

The condition under which (1 � Xt3) < (1 � Xt2) will be:

_mt2ð1� 3sÞ < _mt3 (C.3)

Finally, substituting _mt3 ¼ _mt2 � _mhi into Eq. (C.3):

3s >_mhi_mt2

(C.4)

Incorporating this condition into Eq. (5), when the moisture inthe next stage is reduced, then:

Xhi < Xt2 (C.5)

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