02 GTP-14-1384

Gabriel MarinescuAlstom Power,

Baden 5401, Switzerland

e-mail: [email protected]

Peter SteinAlstom Power,



Michael SellAlstom Power,



Natural Cooling and Startupof Steam Turbines: Validityof the Over-ConductivityFunctionThe temperature drop during natural cooling and the way in which the steam turbinerestarts have a major impact on the cyclic lifetime of critical parts and on the cyclic lifeof the whole machine. In order to ensure the fastest startup without reducing the lifetimeof the turbine critical parts, the natural cooling must be captured accurately in calcula-tion and the startup procedure optimized. During the cool down and restart, all turbinecomponents interact both thermally and mechanically. For this reason, the thermal ana-lyst has to include, in his numerical model, all turbine significant parts—rotor, casingstogether with their internal fluid cavities, valves, and pipes. This condition connectedwith the real phenomenon lead-time—more than 100 hours for natural cooling—makesthe analysis time-consuming and not applicable for routine projects. During the pastyears, a concept called “over-conductivity” was introduced by Marinescu et al. (2013,“Experimental Investigation Into Thermal Behavior of Steam Turbine Components—Temperature Measurements With Optical Probes and Natural Cooling Analysis,” ASMEJ. Eng. Gas Turbines Power, 136(2), p. 021602) and Marinescu and Ehrsam (2012,“Experimental Investigation on Thermal Behavior of Steam Turbine Components: Part2—Natural Cooling of Steam Turbines and the Impact on LCF Life,” ASME Paper No.GT2012-68759). According to this concept, the effect of the fluid convectivity and radia-tion is replaced by a scalar function K(T) called over-conductivity, which has the sameheat transfer effect as the real convection and radiation. K(T) is calibrated against themeasured temperature on a Alstom KA26-1 steam turbine (Ruffino and Mohr, 2012,“Experimental Investigation on Thermal Behavior of Steam Turbine Components: Part1—Temperature Measurements With Optical Probes,” ASME Paper No. GT2012-68703). This concept allows a significant reduction of the calculation time, which makesthe method applicable for routine transient analyses. The paper below shows the theoreti-cal background of the over-conductivity concept and proves that when applied on othermachines than KA26-1, the accuracy of the calculated temperatures remains within15–18 �C versus measured data. A detailed analysis of the link between the over-conductivity and the energy equation is presented as well. [DOI: 10.1115/1.4030411]

Introduction

In order to design reliably a steam turbine for fast starting andflexible operation, it is essential to understand the thermal behav-ior of the turbine components during natural cooling and startup.The thermal stress arising during startup is in close connectionwith the temperature gradient, when the machine cools down fromnominal condition to standstill condition. The stress calculationsproved a significant impact of the natural cooling on the cycliclife, especially on the turbine rotor.

On the other side, the transient analysis of steam turbines facesa big challenge regarding the calculation time. Typically, the cool-ing time of a steam turbine is 100 hours or more, which makes anaccurate calculation time consuming, presently not applicable forroutine projects. A concept called over-conductivity that reducessignificantly the calculation time was introduced in Refs. [1] and[2]. The main idea of this concept is a scalar function K(T) thatmultiplies the fluid conductivity of each finite element within theturbine cavities. K(T) is calibrated in such a way that it renders thesame heat transfer effect as the real convection and radiation. AsK(T) was calibrated on a Alstom KA26-1 steam turbine (seeRef. [2]), a verification of the results’ accuracy on other machines

is mandatory. The paper below shows the robustness of the over-conductivity function when applied on three different machines,other than the KA26-1 turbine used for calibration. The resultsshow that the deviation of the calculated temperatures versusmeasurements—including the rotor critical locations—remainswithin a bandwidth of 15–18 �C. Additionally, the paper showsthe close link between the over-conductivity function and theenergy equation.

The Natural Cooling Process

For a steam turbine (see Fig. 1), the natural cooling starts oncethe control valve closes. The natural cooling is a time dependentprocess that can last more than 100 hours. During this process, theheat is released gradually from the hot components to the coldcomponents and from there to environment, until the whole tur-bine reaches the ambient temperature. Typically, the followingthree phases define the natural cooling as a physical process:

• Phase 1: The control valve starts to close, the turbine cavityis evacuated and remains for several hours at condenser pres-sure, below the ambient pressure.

• Phase 2 called steam ingestion phase: The glands system ismaintained in service feeding the turbine cavity with steam.

• Phase 3: Once the ingestion phase ends, the condenser pumpshutdowns, the ambient air enters the turbine cavity, and thepressure increases to ambient pressure.

Contributed by the Turbomachinery Committee of ASME for publication in theJOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 14,2014; final manuscript received April 18, 2015; published online May 12, 2015.Editor: David Wisler.

Journal of Engineering for Gas Turbines and Power NOVEMBER 2015, Vol. 137 / 112601-1Copyright VC 2015 by ASME

Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 05/12/2015 Terms of Use: http://asme.org/terms

During steam ingestion, the steam flows at critical conditionthrough each gland and expands into the turbine cavity. There, itis driven by the kinetic energy and the temperature gradient. Afterthe ingestion phase ends, only the temperature gradient drives thesteam flow. During the whole natural cooling period, the rotorruns at low speed (barring gear mode).

Research Status on Natural Cooling

The main difficulty of the turbine cooling calculation is how todefine the thermal boundary conditions at the fluid–metal inter-face, especially during the natural cooling. The almost negligiblepressure gradient in the turbine cavity and the very long integra-tion time (more than 100 hours) prevents to efficiently use the tra-ditional tools of computational fluid dynamics. Solutions toovertake these difficulties are presented below.

In 2011, Spelling et al. [3] presented a calculation methodbased on a node-centered finite-volume technique on each con-duction domain. The method was applied extensively for differenttransient regimes, including natural cooling, but no details aboutthe thermal boundary conditions on the fluid–metal interface aregiven.

In 2014, Mukhopadhyay et al. presented a very interestingpaper [4] about the transient conjugate heat transfer analysis of a3D steam turbine casing. The comparison versus measured datashows a good accuracy for the turbine cooling phase. Being a con-jugate heat transfer approach, no heat transfer coefficients (HTCs)were required on the fluid–metal interface. The calculation did notinclude the natural cooling.

Regarding the numerical results validation, Mohr and Ruffinopresented, in 2012, an experimental method to measure the rotortemperature with optical probes during natural cooling [5]. Thethermal survey was conducted on an Alstom KA26-1 IP (interme-diate pressure) steam turbine with optical probes for the rotor tem-perature and with standard thermocouples for inner and outercasing temperatures (see Fig. 2).

Starting from these measurements, Marinescu and Ehrsam [2]modeled the KA26-1 natural cooling process introducing a func-tion K(T) called over-conductivity,

k�ðTÞ ¼ KðTÞ � kairðTÞ (1)

This function allowed modeling the transient heat transfer processof the whole machine with 12–15 �C accuracy along 96 physicalhours.

The over-conductivity function was calibrated in such a waythat it replaces the effect of the transient convection within theturbine cavities by an equivalent higher conductivity, rendering

on each finite element the same heat transfer effect. The methodhas the advantage to be fast enough and consequently applicablefor routine projects, but raised a question on the validity whenapplied on other machines.

Natural Cooling and the Over-Conductivity Function

The over-conductivity function K(T) was presented in detail inRef. [2]. This approach allows the heat transfer calculation withinthe turbine cavities, where the pressure gradient is negligible andthe temperature gradient drives the fluid flow. K(T) is defined as asecond-degree polynomial function of the local temperature

KðTÞ ¼ a1T2 þ a2T þ a3 (2)

The constants a1, a2, and a3 are calculated based on the measuredtemperatures on a specific steam turbine unit. Applied for eachthermocouple, presented in Fig. 2, the calculation algorithm pre-sented in Ref. [2] gave a corresponding Kj(T) function, where j isthe thermocouple index (see Fig. 3). The averaged K(T) is theover-conductivity function. Obviously, always K(T)> 1.

At this point the following two remarks must be noted:

• The scatter of the Kj(T) functions at high temperatures (tem-perature/live steam temperature above 0.55) is an indicationthat at high temperatures, the pressure gradient is not fullynegligible. This scatter defines the method accuracy.

• K(T) was established based on the natural cooling tempera-tures measured on an Alstom KA26-1 IP steam turbine. Thatmeans this function includes the effect of the internal radia-tion specific to this machine. But the internal radiation is notthe same on other machines. This is the second limitation ofthe over-conductivity function.

Fig. 2 Alstom KA26-1 IP steam turbine instrumentation

Fig. 3 The over-conductivity function (source: Ref. [1])

Fig. 1 Alstom KA26-1 IP steam turbine during instrumentation(source: Ref. [1])

J_ID: GTP DOI: 10.1115/1.4030411 Date: 12-May-15 Stage: Page: 2 Total Pages: 9

112601-2 / Vol. 137, NOVEMBER 2015 Transactions of the ASME


Over-Conductivity and the Energy Equation

The over-conductivity function presented in Eq. (1) can supplyimportant qualitative data regarding the flow pattern within theturbine cavity. As long as, during natural cooling, the pressuregradient of the fluid is almost negligible, the temperature gradientis the flow driver. Consequently, we can assume an equivalent ve-locity V proportional with the fluid temperature gradient rT,

V ¼ f ðT; pÞ � rT (3)

Equation (3) is applicable only within the fluid domain, and doesnot include the tangency condition on the fluid–metal interface.For this reason, V in the paper below is called “equivalent veloc-ity” and could be interpreted as velocity of the heat wave travelingin the turbine cavity.

The parameter f(T, p) is a scalar function of pressure and localfluid temperature. Equation (3) relies on the fluid behavior duringnatural cooling—as long as the fluid temperature is constant, thevelocity is zero. Starting from Eqs. (1) and (3), we will show howthe f(T, p) analytical equation can be found. In this order, let uswrite the energy equation for axisymmetric domains in two ways.

(a) The standard form

qcp

@T

@tþ u

@T

@xþ v

@T

@r

� �¼ k

@2T

@x2þ @

2T

@r2

� �þ U (4)

where

U ¼ 2l@v

@r

� �2

þ v

r

� �2

þ @u

@x

� �2

� 1

3� r � Vð Þ2

" #

þ l@v

@xþ @u

@r

� �2

(5)

is the dissipation function. U captures the friction effect,fully negligible for the natural cooling, as we will see atthe end of this section. If we neglect U and separate thevelocity and the temperature gradient, then Eq. (4)becomes

qcp

@T

@tþ qcpV � rT ¼ k � DT (6)

(b) The form corresponding to the finite element assump-tion, which means V¼ 0 and the convection and radia-tion effect wrapped up within an equivalent fluidconductivity k�

qcp

@T

@t¼ k� � DT (7)

But a link between k� and k was already defined in Eq. (1).Then comparing Eqs. (6), (7), and (1) we get

k� ¼ 1� f ðT; pÞ � qcp

k� rT � rT

DT

� �� k (8)

which gives the final equation of f(T, p)

f ðT; pÞ ¼ kqcp

� 1� KðTÞð Þ � DT

rT � rT(9)

As long as K(T)> 1, the temperature Laplacian DT gives thef(T, p) sign. That means for the finite elements that cool downDT< 0 and then f(T, p)> 0. Conversely, for the finite elements

that heat up DT> 0 and then f(T, p)< 0. Because K(T) and thetemperature gradient rT are time- and space-dependent, the func-tion f(T, p) is time- and space-dependent as well.

The 2D transient analysis was conducted using a finite elementmodel, where the mesh includes both metal parts and fluid parts(see Fig. 4).

On the metal parts, the conductivity was declared to be metalconductivity; meanwhile on the fluid parts, the conductivity wasdeclared according to Eq. (1). All boundary conditions both forinitial state and transient regime were kept the same as presentedin Ref. [1]. The finite element model was run as follows:

• a first run dedicated to natural cooling, for which the resultsare presented below

• a second run dedicated to startup, for which the results arepresented in the section entitled Startup and the Over-Conductivity Function

The natural cooling run supplied the time variation of the nodaltemperature and nodal temperature gradient based on which wesucceeded to visualize the behavior of the over-conductivity func-tion K(T) and the fluid equivalent velocity V at several locationswithin the turbine cavity (see points A, B C, D, and E on Fig. 5).

Figure 6 shows the time variation of the temperature gradientmodule calculated at A, B, C, D, and E. As expected, on a largetime scale, all gradients diminish and converge to zero. The con-vergence rate is not uniform, which means the temperature of the

Fig. 4 The mesh of the finite element model

Fig. 5 Position of points A, B, C, D, and E

Journal of Engineering for Gas Turbines and Power NOVEMBER 2015, Vol. 137 / 112601-3


metal parts does not diminish at the same rate. The order of mag-nitude ranges within 0–0.20 �C/mm.

Figure 7 shows the time variation of the temperature LaplacianDT calculated at the points mentioned above. It is interesting tosee that at point E, the Laplacian changes the sign at approx.14 hours after natural cooling start. The negative Laplacian is inconnection with the vacuum in the turbine cavity during the steamingestion phase. Vacuum means the fluid expands and the fluidtemperature drops below the metal temperature. Once the steamingestion ends, the steam is replaced gradually by ambient air andthe radiation from the hot parts (rotor and inner casing) prevails.

The Laplacian increases up to a time (approx. 42 hours afternatural cooling start) when the heat lost to ambient becomes moreimportant and the temperature drops down to ambient—both theLaplacian and the gradient tend to zero.

Figure 8 shows the time variation of the function f(p,T) definedin Eq. (9) and calculated at A, B, C, D, and E. Being a proportion-ality factor between the equivalent velocity and the temperaturegradient on the same direction, it is not mandatory to diminish tozero. The divergent variation of the function f at points A, B, andE confirms that within the turbine cavity, the temperature gradientdrops down faster than the equivalent velocity.

Based on these results, we can find the time variation of theequivalent velocity in the turbine cavity and explain the flow pat-tern. Figure 9 shows the variation of the equivalent velocity andits components at point A.

The negative radial velocity within the first 14 hours is an indi-cation that during the first hours, after natural cooling start, therotor and inner casing cool down faster than the valve. At almost40 hours after natural cooling start, the radial component of the

velocity exceeds the axial component mainly because the heat lostradially through outer casing is bigger than the heat lost axiallythrough bearings. This tendency remains valid for more than100 hours. The equivalent velocity ranges within 0.02 m/s–0.04 m/s. Finally, we have to note that as long as the numerical modeldoes not include the mass and the impulse equations, the equiva-lent velocity is not applicable on the fluid–metal interface.

Startup and the Over-Conductivity Function

The turbine startup includes a phase called “steam qualitycheck” (see Ref. [6] and Fig. 10), when the control valve is closedand there is no active flow in the turbine cavity. In this case, thestandard tools for heat transfer analysis are not applicable. Manytimes, after steam quality check, a prewarming phase follows toheat the turbine rotor before ramping up the turbine. The pre-warming can last from few tens of minutes up to few tens ofhours. The impact of the prewarming temperature and lead timeon the turbine rotor cyclic life was analyzed in Ref. [6].

As long as, during the steam quality check phase, the turbinecavity is closed, the gradient of the fluid pressure is negligible.Only the temperature gradient drives the fluid flow. That meansthe over-conductivity function is applicable and can be used tocalculate the transient temperature map of the whole machine.At cold start, the condensation condition is fulfilled on theinternal faces of the machine. In order to get accurate calculatedtemperatures, it is mandatory to take the steam condensation intoaccount.

Fig. 6 Variation of the temperature gradient module $Tj j

Fig. 7 Time variation of the Laplacian function DT

Fig. 8 Time variation of the function f(p,T)

Fig. 9 Time variation of the velocity module and its compo-nents at point A



The same Alstom KA26-1 IP turbine was used to model thetransient heat transfer during startup. The machine was stabilizedat cold condition before startup, all thermocouples indicating17–20 �C. The startup consisted of the following (see Fig. 10):

• t1: gas-turbine ignition.• t2: the glands system was switched ON, the control valve

being closed. On this machine, only the steam quality wasverified, prewarming phase not applied.

• t3: the control valve started to open. The acceleration phasestarted. The rotor speed increased to grid synchronization.

• t4: idle regime. The rotor speed reached the synchronization.The machine was ready for ramp-up.

• t5: the machine reached 10% power regime.

The same over-conductivity function K(T) was used to modelthe heat transfer in the turbine cavities. It must be noted that dur-ing natural cooling the operating points, corresponding to eachthermocouple, travel from high to low temperatures; meanwhileduring startup, the same operating points travel from low to hightemperatures, all of them remaining on the same K(T) curve (seeFig. 11). For example, the cold start test started on Dec. 11, 2010at 16:33. The position of the operating points can be seen succes-sively at 16:33, 16:58, and 17:06. Conversely, the natural cooling

test started on Dec. 13, 2010. The position of the operating pointscan be seen successively at 18:44, 19:48, 00:58, and 10:15.

The turbine glands were modeled following the physics of thesteam flow including the condensation phenomenon. Duringstartup from time t1–t5, the pressure in suction cavity is alwaysbelow the ambient pressure, meanwhile in the pressure cavity it isabove the pressure in the turbine cavity and suction cavity. Conse-quently, there is steam ingestion in the turbine cavity, as presentedin Fig. 12. From time t5 to base load, the steam pressure in the tur-bine cavity is higher than the pressure in the pressure cavity, andhence the primary flow changes the direction.

The thermal boundary conditions were calculated at each inte-gration timestep using the average equation of mass and energyconservation. For example, on passages 11–12 (see Fig. 12), theenergy equation is

M12h�12 ¼ M11h�12 þ HTC11 � ðTm� TÞ þWindage (10)

At station 30, the mass and energy conservation equations are

M30 ¼ M12 þM22

M30h�30 ¼ M12h�12 þM22h�22

(11)

Figure 13 shows the temperature map at 30 min after the glandssystem is switched ON, when the turbine cavity is closed. Aslightly higher temperature at the hot gland side versus the coldgland side can be seen because the room available on the hot sideis smaller than the cold side.

Figure 14 shows a comparison between the measured and cal-culated temperature at thermocouple T24.1. The maximum devia-tion ranges within 0–18 �C and occurred mainly due tocondensation, which distributes the heat asymmetrically aroundthe inner casing; meanwhile, the finite element modes assumed anaxisymmetric inner casing. We note as well the “bump” of tem-perature variation between t3 and t4 that captured the effect ofhigher steam mass flow required during the rotor acceleration.

Validity of the Over-Conductivity Function

The over-conductivity function K(T) was calibrated based on aAlstom KA26-1 IP turbine data (see Ref. [2]). In order to verifyhow much this function is dependent upon the turbine configura-tion, the following machines were selected for verification [7]:

(a) Alstom 460 MW HP (high pressure) turbine. This is a HP50 Hz, single-flow class machine (see Fig. 15).The outer casing was instrumented with thermocouples(Th22, Th32, Th42, and Th52). The valves were

Fig. 10 Temperature variation at thermocouples T11.1, T24.1,Tm33, and Tm42 (source: Ref. [6])

Fig. 11 Startup and natural cooling on over-conductivitydiagram Fig. 12 Gland steam flow from time t1 to t5



instrumented as well with thermocouples (ThV04, ThV07,and ThV09).

(b) Alstom 1100 MW HP turbine. This is a HP 50 Hz, single-flow class machine (see Fig. 15).On this machine, the turbomax indication was comparedagainst the calculated temperature at the same location.Turbomax is a temperature sensor located on the inlet spiral(see the indication “Turbomax” in Fig. 16).

(c) Alstom 1100 MW IP turbine. This is a 50 Hz, IP double-flow class machine (see Fig. 17).

The turbomax indication was also used to compare the calcu-lated temperature versus measurements. Due to the double-flowconfiguration of this machine, the exhaust columns and the controlvalve were separated from the outer casing and a condition ofequal heat flux from the axisymmetric domains to plain domainswas imposed.

All thermal models used the K(T) function (2) and followed thesame calculation process.

In order to prepare a consistent set of input data, all threemachines were operated for natural cooling in the same way:

Fig. 13 Temperature distribution at 30 min after the glands system is switched ON

Fig. 14 Measured and calculated temperature at T24.1

Fig. 15 Alstom HP 460 MW turbine. Temperature distributionat base load regime.



(a) Each machine was stabilized more than 12 hours at baseload.

(b) At natural cooling start (time¼ 0), the control valve closes.The turbine cavity is evacuated and remains at the con-denser pressure.

(c) The glands system is activated. The steam ingestion in theturbine cavity starts and lasts 1–5 hours, according to eachturbine specification.

(d) Once the steam ingestion ends, the condenser valve closesand the ambient air enters the turbine cavity. The naturalcooling continues until the next event.

Figures 18 and 19 show the calculated temperature distributionon the Alstom 460 MW HP turbine at 8 hours and 60 hours afternatural cooling start. Figure 20 shows the comparison between thecalculated and measured temperature at thermocouple Th22. Forthis particular machine, the 15 �C temperature difference betweenmeasurement and calculation recorded during the first 3 hours (seeFig. 20) is an indication that the steam mass flow ingested throughthe hot gland was lower than the nominal value, but the similarslope of the measured and calculated temperature confirms thatthe over-conductivity function contributed correctly to the globalheat transfer process. Another good verification is presented inFig. 21.

The turbine valves are not axisymmetric parts, meanwhile theouter casing was assumed axisymmetric. The conversion of the3D nonaxisymmetric parts to 2D is presented in detail in Refs. [2]and [5]. The good matching presented in Fig. 21 is a confirmationthat the 3D–2D conversion works correctly.

Figures 22 and 23 show the calculated temperature distributionon the Alstom 1100 MW HP turbine at 8 hours and 60 hours afternatural cooling start. We retrieve the same behavior as KA26-1 IPturbine—the rotor and casings cool down slightly faster than thecontrol valves during the first hours than the control valves.

Fig. 17 Alstom IP 1100 MW turbine. Temperature distributionat base load regime.

Fig. 18 Alstom 460 MW HP turbine. Temperature distributionat 8 hr after natural cooling start.


Fig. 20 Alstom 460 MW HP turbine. Temperature variation atTh22. Calculated versus measured data.

Fig. 16 Alstom HP 1100 MW turbine. Temperature distributionat base load regime.



Figure 24 shows a comparison between the measured and cal-culated temperature at turbomax location. The deviation rangeswithin 10–12 �C. On Alstom machines, the turbomax sensor cav-ity is filled with MgO, which has a lower conductivity than the

inlet spiral metal. That means the turbomax temperature is slightlydelayed versus the sharp temperature drop within the first hoursafter natural cooling start. That explains the deviation marked in acircle on Fig. 24.

Figures 25 and 26 show the calculated temperature distributionon the Alstom 1100 MW IP turbine at 8 hours and 60 hours afternatural cooling start. Figure 27 shows a comparison between themeasured and calculated temperature at turbomax location. Beinga double-flow turbine, the outer face of the outer casing in the 2D

Fig. 21 Alstom KA26-1 HP turbine. Temperature variation atTh32. Calculated versus measured data.



Fig. 24 Alstom 1100 MW HP turbine. Temperature variation atTurbomax location. Calculated versus measured data.

Fig. 25 Alstom 1100 MW IP turbine. Temperature distributionat 8 hr after natural cooling start.

Fig. 26 Alstom 1100 MW IP turbine. Temperature distributionat 60 hr after natural cooling start.



model is almost fully covered by the exhaust columns and thecontrol valve. This feature required a special attention to the inter-face between the axisymmetric and the plain domains. In thisorder, the exhaust columns and the control valve were separatedfrom the outer casing and a condition of equal heat flux wasimposed on each interface. By this approach, the deviationbetween the measured and calculated temperatures remainedwithin 5–12 �C (see Fig. 27). The highest deviation occurs withinthe first hours after natural cooling start due to the slow turbomaxtime reaction, as noted above for the 1100 MW HP machine.

Conclusions

The over-conductivity function K(T) defined in Eq. (2) provedto be robust and reliable when applied on steam turbines with dif-ferent configurations. The over-conductivity function was verifiedon the following three machines:

• Alstom 460 MW HP 50 Hz turbine• Alstom 1100 MW HP 50 Hz turbine• Alstom 1100 MW IP 50 Hz turbine

The verifications showed that the deviation of the calculatedtemperatures ranges within 0–18 �C versus measurements alongthe whole 96 hr of natural cooling process. The deviation scatter isthe same both on the calibration machine and on the verificationmachines, which means the over-conductivity function K(T) isdependent on the local temperature gradient and not dependent onthe turbine configuration.

During natural cooling or startup process, when the controlvalve is closed, the temperature gradient on the fluid domains isthe main driver of the flow within the turbine cavity. The follow-ing equation captures this feature:

V ¼ f ðT; pÞ � rT

The proportionality function f(T, p) has an analytical equationdepending on the over-conductivity K(T),

f ðT; pÞ ¼ kqcp

� 1� KðTÞð Þ � DT

rT � rT

These results allowed the estimation of the equivalent velocity V inthe turbine cavity during the natural cooling process—0 m/s–0.05 m/s.

Nomenclature

a1, a2, a3 ¼ calibration parametersCp ¼ specific heat at constant pressure

f ¼ proportionality function between V and rTh* ¼ fluid total enthalpy

HP ¼ high pressureHTC ¼ heat transfer coefficient

IP ¼ intermediate pressureK(T) ¼ over-conductivity function

M ¼ fluid mass flowt ¼ time

T ¼ fluid temperatureTm ¼ metal temperature

V ¼ equivalent velocityx, r ¼ axial and radial coordinates

D ¼ Laplacian operatork ¼ fluid thermal conductivity

k* ¼ equivalent fluid thermal conductivityq ¼ densityr ¼ gradient operator

References[1] Marinescu, G., Mohr, W., Ehrsam, A., Ruffino, P., and Sell, M., 2013,

“Experimental Investigation Into Thermal Behavior of Steam TurbineComponents—Temperature Measurements With Optical Probes and NaturalCooling Analysis,” ASME J. Eng. Gas Turbines Power, 136(2), p. 021602.

[2] Marinescu, G., and Ehrsam, A., 2012, “Experimental Investigation on ThermalBehavior of Steam Turbine Components: Part 2—Natural Cooling of Steam Tur-bines and the Impact on LCF Life,” ASME Paper No. GT2012-68759.

[3] Spelling, J., Joecker, M., and Martin, A., 2011, “Thermal Modeling of a SolarSteam Turbine With a Focus on Start-Up Time Reduction,” ASME Paper No.GT2011-45686.

[4] Mukhopadhyay, D., Brilliant, H., M., and Zheng, X., 2014, “Development of aConjugate Heat Transfer Simulation Methodology for Prediction of Steam Tur-bine Cool-Down Phenomena and Shell Deflection,” ASME Paper No. GT2014-25874.

[5] Ruffino, P., and Mohr, W., 2012, “Experimental Investigation Into ThermalBehaviour of Steam Turbine Components: Part 1—Temperature MeasurementsWith Optical Probes,” ASME Paper No. GT2012-68703.

[6] Marinescu, G., Sell, M., Ehrsam, A., and Brunner, P., 2013, “ExperimentalInvestigation Into Thermal Behavior of Steam Turbine Components: Part 3—Startup and Impact on LCF Life,” ASME Paper No. GT2013-94356.

[7] Marinescu, G., Stein, P., and Sell, M., 2014, “Experimental Investigation IntoThermal Behavior of Steam Turbine Components: Part 4—Natural Cooling andRobustness of the Over-Conductivity Function,” ASME Paper No. GT2014-25247.

Fig. 27 Alstom 1100 MW IP turbine. Temperature variation atturbomax location. Calculated versus measured data.



http://dx.doi.org/10.1115/1.4025556

http://dx.doi.org/10.1115/GT2012-68759

http://dx.doi.org/10.1115/GT2011-45686

http://dx.doi.org/10.1115/GT2014-25874

http://dx.doi.org/10.1115/GT2012-68703

http://dx.doi.org/10.1115/GT2013-94356

http://dx.doi.org/10.1115/GT2014-25247

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02 GTP-14-1384