thermal con of ss 304

7/28/2019 Thermal Con of Ss 304

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Determination

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of Thermal

Pro cee di ngs o f NHT C 034th National Heat Transfer Conferenc

PMsburgh, Pennsylvania, August 20-22,200

Conductivity of 304 Stainless Steel Using ParameterEstimation Techniques

Bennie F. Blackwei12, Walter Gi113,Kevin J. Dowding4, and Thomas E. Voth4Sandia National LaboratoriesEngineering Sciences Center

MS 0828Albuquerque, NM 87185

ABSTRACTSensitivity coefficients were analyzed in order to guide the

design of an experiment to estimate the thermal conductivity of304 stainless steel. The uncertainty on the temperaturemeasurements was estimated by several means and its impact onthe estimated conductivity is discussed. The estimated thermalconductivity of 304 stainless steel is consistent with results fromother sources.NOMENCLATUREckNpiv.NfsTTkTmTmintxifij

specific heat, J/Kg-Kthermal conductivity, W/m-Kno. of parametersno. of sensorsno. of measurement timessum of squares function, see Eq. (6)temperature, Cscaled sensitivity coefficient, C, see Eq. (1)maximum temperature, Cminimum temperature, Ctime, ssensitivity matrixposition vectortemperature measurement for sensor i at timej, C

1.

2.3.4.

Sandia is a mult iprogram laboratory operated by Sandia Corporat ion , aLockheed Martin Company, for the United States Department of Energyunder Contract DE-AC04-94AL85000.Dktinguished Member of Technical Staff , Fellow ASME.Principaf Member of Technical Staff , Member ASME.Senior Member of Technicat Staff , Member ASME.

GreekA D-optimality condition, see Eq. (4)A+ dimensionless D-optimality condition, see Eq. (5)AT = Tma - TminAt data sample rateU thermal diffusivity, m2/sP density, Kg/m3.P estimated standard deviation in parameter p, units of pINTRODUCTIONThermal systems often incorporate a number of mechanical

joints between individual components. These joints allowassembly/disassembly, fitrnish mechanical support, and providpathways for redistribution of thermal energy. To predict thbehavior of a thermal system, it is necessary to incorporatethermal contact phenomena into computational tools. Aaccepted modeling approach is to use correlations that providecontact conductance based on joint characteristics, such amaterial pair, surface finishes, surface harnesses, and contacpressure. These parameters are important because the surfacirregularities deform upon contact and control the actuainterface area.An experimental apparatus has been designed to develop

contact conductance correlations for metal-to-metal interfacesAs part of the data reduction procedure, it is necessary to knowthe thermal conductivity of the metals on each side of thcontact interface. This thermal conductivity information is ofteobtained from either handbooks or separate experimentsdesigned to measure those values. An unpublished uncertaintyanalysis for the steady-state contact conductance experimen


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DISCLAIMERThis repo~ was prepared as an account of work sponsoredbyanagency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor impiy its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or ref iect those of the United StatesGovernment or any agency thereof.


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has indicated that the largest contributor is because of theuncertainty in the thermal conductivity of the metal. Also, arelative uncertainty of 10% on a handbook conductivitytranslates to a larger relative uncertainty in the estimated contactconductance. Therefore, it is important to reduce the uncertaintyin the conductivity in order to reduce the uncertainty in contactconductance. In this paper we take an alternative approach inthat we use the same basic hardware used in the contactconductance experiment to also measure the thermalconductivity. Since the thermal conductivity is measured onhardware that is prototypical for the contact conductanceexperiment, it is felt that the experimental uncertainty in thethermal conductivity measurements will be reduced over thatobtained from a separate thermal conductivity experiment.Description of Contact Conductance ExperimentA cutaway view of the contact conductance experiment is

shown in Fig. 1. The contact interface of interest is that betweenthe two hollow cylinders, each with an outside diameter of 8.89cm (3.5 in), wall thickness of 0.508 cm (0.2 in), and length of6.985 cm (2.75 in). (For the results reported here, a singlecontinuous cylinder of 13.97 cm (5.5 in) length was used

j! ss cylinder?;

instead of the split cylinders. This cylinder was machined fromthe same billet as the split cylinders.) A flange of 12.7 cm (5.in) diameter by 0.635 cm (0.25 in) thickness is present on thupper and lower cylinder ends. The copper block is composedof two halves: the (solid) contact plate is 12.7 cm (5.0 indiameter by 1.905 cm (0.75 in) thick; the body is 12.7 cm (5.in) diameter by 3.848 cm (1.515 in) and has serpentine channelsmachined in it to enhance the heat exchange effectiveness fothe fluid circulated through it. The contact plate and body arbrazed together and are referred to collectively as the coppeblock.Time-dependent temperature measurements are provided b

thermocouples mounted in the top and bottom copper blockidentified in Fig. 1. A single 30-gauge thermocouple (0.25mm, 0.010 in diameter, Type K) measures the temperature oeach of the copper blocks; this thermocouple is located 4.483cm (1.765 in) from the copper/stainless steel flange interface (ithe body) and at the bottom of a 6.35 cm (2.25 in) deep radiahole. Two separate temperature-controlled baths supply fluid tthe top/bottom (OFHC) copper blocks. A simplified crossection of the heating/cooling blocks and stainless steel cylindeis shown in Fig. 2; for the experimental results presented here,

A rtationdu

sta [on

CL I

CL ;Figure 1. Cutaway view of contact conduc-tance experiment Figure 2. Cross section of experimental apparatus


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continuous cylinder was used as opposed to the two piececylinder shown in Fig. 2.Thermocouples are mounted in the stainless steel cylinder

walls at 14 axial stations with a uniform spacing of 0.953 cm(0.375 in). Station 1 is near the top of the cylinder and is located0.795 cm (0.313 in) from the inside face of the flange. Station14 is located at a mirror image position near the bottom flange.Stations 7 and 8 are located at M.475 cm (M.187 in) from the x= O position (axial center line), respectively. At each axialstation along the stainless steel cylinder, there arethermocouples at four angular stations, each 90 degrees apartfor a total of 56 thermocouples mounted in the cylinder wall.The thermocouples are arranged in four columns: two columnsof Type K and two columns of Type T, both of 30-gage wire(0.254 mm or 0.010 in diameter).The thermocouples were installed by drilling 1.17 mm

(0.0461 in) diameter by 2.54 mm (O.1 in) deep holes in thecylinder walls. The thermocouples were formed by strippinginsulation from single-strand wire, welding the junction, anddipping the junction in a eutectic mix of indiumhin. Theeutectic was also placed in the holes, and it remains liquid andensures good contact between the thermoelectric elements andthe stainless steel. The leads were wrapped around the outsidediameter of the cylinder for one turn and held down withKaptonTM tape so as to minimize conduction losses along thelead wires.The vacuum system consists of a bell jar vacuum chamber, a

base plate with feed-through ports, and a complete vacuumpumping system with controls and gaging. The nominaldimensions of the glass bell jar vacuum chamber are 45.7 cm by76.2 cm by 0.826 cm (18 in by 30 in by 0.325 in) wall thickness.The system is composed of a high-speed roughing pump and afour-stage diffusion pump capable of maintaining 10-7Torr.The data was acquired using the PC-based LabViewm with a

16 bit A-D system; the sample interval was one second.EXPERIMENT DESIGN ISSUESThe temperature of the top and bottom copper blocks can be

independently controlled through their own individual fluidbaths; thk allows considerable flexibility in the temperaturehistory that can be imposed. Blackwell and Dowding (1999)explored optimum experiment design issues for the estimationof thermal conductivity from temperature data of this type. Theyconsidered two different boundary condition scenarios, termedsymmetric and anti-symmetric because of the boundsyconditions imposed on the copper blocks. The symmetricboundary condition increased the temperature of both the topand bottom copper block by an amount AZ as rapidly as thetemperature baths would allow. The anti-symmetric boundarycondition increased the top Cu block temperature by ATD,whereas the lower block temperature was decreased by ATfl.The temperature profile for the symmetric scenario is

symmetrical about the midplane. Even if a midplane contactingsurface was present, it would not impact the thermalconductivity estimation because of the presence of thisymmetry or adiabatic surface. Furthermore, it was found thathe symmetric scenario will have a smaller variance in thestimated conductivity than the anti-symmetric scenario. Asuch all results presented here will be for the symmetricboundary conditions.The thermocouple results for a run at a pressure of roughly

4X10-5 Torr are shown in Fig. 3. This pressure is sufficiently low

Time, sFigure 3. Experimental temperature results for Run042999.to eliminate convection, and the temperatures are sufficientlylow that radiation is not significant. While the top and bottomcopper block temperatures are from single thermocouplemeasurements, the Station 1-14 results are each the average ofour thermocouples at the same axial station and equally spacearound the circumference of the cylinder. Statistics on thestemperature measurements will be discussed in a subsequensection. Because of the finite capacity of the temperature bathsthe change in temperature of the top and bottom copper blockonly approximates a step change boundary condition. Also, notthat even though the top and bottom blocks were fed from thsame temperature bath, their temperatures were not the samduring the rapid transient. At early times the top bloctemperature may be as much as 4C above the bottom bloctemperature. It is speculated that the reason for this discrepancyis that the piping lengths supplying the top and bottom Cblocks are different. Had the top and bottom block temperaturehistories been the same, then pairs of thermocouples (Stationand Station 14, etc.) should also read the same. The farther ththermocouples are from the heated ends, the more closely ththermocouple pairs agree.During the cool down phase, the thermocouple pairs on th


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stainless steel cylinder cross over. The data reduction procedureto be discussed below will account for the fact that the top andbottom boundary conditions were different; longitudinalsymmetry was not assumed.The optimal design of experiments studies sensitivity

coefficients in order to make decisions on quantities such asexperiment duration, sensor locations, sample rate, etc. In thisinstance sensitivity coefficients are defined as partial derivativesof field variables (temperature in this case) with respect toparameters of interest (conductivity in this case). We have foundit useful to utilize scaled sensitivity coefficients. For ourexperiments the scaled thermal conductivity sensitivitycoefficient of interest is defined as

T/,(?,t ,k) = kg. (1)Note that the scaled sensitivity coefficient is a field variable justlike temperature, and it has the units of temperature. The scalingof the sensitivity coefficient is important in that it allowssensitivity coefficients to be directly compared to acharacteristic temperature rise of the experiment. For thisexperiment the characteristic temperature rise is the rise from itsinitial value. We utilized software designed specifically tocompute both temperature and sensitivity coefficient fields.Details of this methodology can be found in Blackwell et al.(1999) and Dowding et al. (1999). This software utilizes aControl Volume Finite Element Method to solve the energyequation and also the derived sensitivity partial differentialequations. It is felt that the sensitivity equation method is moreaccurate than finite difference determination of sensitivitycoefficients and requires less user intervention to determineappropriate finite difference step sizes.The computation of the temperature and sensitivity coefficient

fields requires a computational mesh be developed for theexperiment. If the copper blocks are included in thecomputational model, then the contact conductance between thetop/bottom copper blocks and the stainless steel flanges must beincluded in this model. Two-dimensional simulations of theexperiment were performed in which a range of copper/stainlesssteel contact conductance was assumed. From thesecalculations it was determined that the temperature field throughthe wall thickness at Stations 1 and 14 was uniform. This allowsthe geometrical extent of the model to be reduced to includeonly that portion of the cylinder between Stations 1 and 14 andeliminates the need for contact conductance at the copper/stainless steel interface. In fact, this says that the model is reallyone-dimensional as the boundary conditions are uniform at theStations 1 and 14 cross sections. However, the data from thisexperiment was reduced using a two-dimensional axisyrnmetriccomputational model, since the software for computing thesensitivity coefficients dld not have a 1-D model.Sensitivity coefficients have been calculated for the 14

thermocouple stations. The results for Stations 1-7 are presentedin Fig. 4 along with the temperature history at Station 1. Note

a L 1 1 I ~o 250 500 750 10(Time, sFigure 4. Scaled thermal conductivity sensitivitycoefficients (left scale) and Station 1 temperaturehistory (right scale).that the sensitivity coefficient at Station 1 is zero, this is becausethe temperature at a specified temperature boundary conditionindependent of the thermal conductivity. Station 7 has thlargest sensitivity coefficient, indicating that it contributes thmost information about the thermal conductivity. Since ththermocouple locations near the midplane of the experimenthave the largest sensitivity coefficients, one might ask why ththermocouples were not located closer to the rnidplane. Thoriginal intent of the experiment was to measure the contacconductance using the steady state temperature profile anconductivity to estimate the heat flux. For this situationuniform placement of thermocouples was thought to be betterThe thermocouple locations were already specified before thconductivity estimation experiment was conceived. Thsensitivity coefficient results for Stations 8-14 are very similato those for Stations 1-7 and so are not shown.The duration of the heating phase and cooling phase of Ru

042999 (Fig. 3) was chosen based on engineering judgementand a visual inspection of the sensitivity coefficients. A morformal approach would involve optimal experiment designtechniques. The D-optimality condition discussed in Beck anArnold (1977) was used in subsequent analysis of thexperiment design. This condition involves maximizing thdeterminant of the matrix

A = Det(XTX), (2where X is the N~. Nt by NP sensitivity matrix, the number osensors is N~, the number of times is N@ and the number o


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parameters is NP. In this particular application, we are onlyestimating a single parameter, thermal conductivity of thestainless steel; hence NP = 1. There are 14 sensors on thestainless steel cylinder, but since Stations 1 and 14 will be usedas boundary conditions, there are only 12 axial stationscontributing parameter estimation information; the sensitivitycoefficients are zero at the specified temperature boundaryconditions. The number of time measurements will be treated asvariable through the relationship

t = NrAt , (3)

where t is the experiment duration, and At is the data samplerate, for all the results presented here, At =1 s. For this simplecase of a single parameter, the optimality condition reduces to(1)s dT 2A = ?lj., m i j (4)One could increase A by adding sensors or taking moremeasurements as long as i)T/~k is nonzero. Furthermore,because 13T/~k is typically related to the temperature range, alarger temperature variation causes A to increase. To eliminatethese dependencies, a normalized version of the optimalitycondition was used in this study and is defined to be

N, N, aT 2A+ = 1

[ )_ZZkW ~(5)N$Nl(Tmal- Tmjn)2i=lj= I L j

where (Tmax,Tmin) is the (maximum, minimum) temperatureover time and sensor location. The quantity A+ can be viewedas an information content per data sample, and we want tochoose the heating and cooling durations that maximize thisquantity. The time dependence of A+ comes through theimplicit dependence of Nl on time, Eq. (3).Note the presence of the scaled thermal conductivity

sensitivity coefficients in Eq. (5). These scaled sensitivitycoefficients are precisely those shown in Fig. 4. When thecooling phase is initiated (by means of a temperature boundarycondition), there is no sudden increase or decrease in sensitivitycoefficients as there is when experiments are driven with a heatflux boundary condition, and the heat flux is turned off.Consequently, the cooling phase does not add much additionalinformation in the situation being studied here.The D-optimality condition defined in Eq. (5) for Run 042999

is presented in Fig. 5. The heating duration for this experimentwas 644 s; beginning at this time cooling fluid was circulatedthrough the top and bottom copper blocks. The cooling durationcan be treated as a variable to be selected based on the time atwhich A+ in Fig. 5 is a maximum; this time is 756 s. Even

Figure 5. D-optimality condition as a function oftime.though the experiment was run in excess of 1000s, the majorityof the information about thermal conductivity is containedthe first 756 s of the experiment. The data was reduced usingrun duration of 1000s.THERMAL CONDUCTIVITY ESTIMATIONThe thermal conductivity is estimated from the experimental

temperatures in Fig. 3 such that the least square error betweenthe computational model of the experiment and thexperimental temperatures is minimized. This sum of squarefunction is given by

The iterativeaccomplished

N. N,s = E ~ (Tij-y~)2 (6

i=lj=l

solution to this minimization problem wathrough the DAKOTA (1999) software, whic

allows one to connect stand-alone thermal analysis softwarwith stand-alone optimization software. The communicationbetween the various software is through external data files. Thiapproach allows the thermal anal ysis software and optimizationsoftware to develop independent of each other. Additionadetails on this approach have been presented in Blackwell anEldred (1997) and Dowding and Blackwell (1998).The computational model consisted of the walls of th

stainless steel cylinder. The end boundary conditions were thexperimentally measured temperatures at Stations 1 and 14. Thboundary conditions on the side walls of the cylinder weradiabatic. The estimated thermal conductivity was 14.34 W/mK; this result compares favorably with other measurements


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given in Table 1; the other measurements were linearlyTable 1: Comparison of thermal conductivity of 304 Stain-less Steel with other results. Units are W/m-K and are validfor 31 C; uncertainty bound is HoThis Work Taylor, et al. (1997) Incropera and De Witt (1990)14.34~0.58 14.74+ 1.5 14.97

interpolated at a temperature of 31C, the average temperatureof Run 042999. The uncertainty statement contained in Taylor,et al. (1997) stated that the conductivity was accurate to +5%. Itis not known if this uncertainty bound was la or 2cT; it wasassumed to be 16. The details of the uncertainty analysis for thiswork are presented below.Since time-dependent temperature data is used in the

parameter estimation process, the computational model requiresboth thermal conductivity and volumetric heat capacity. The useof temperature boundary conditions to drive the modelprecluded the possibility of estimating both conductivity andvolumetric heat capacity from the temperature measurementspresented here. The conductivity estimation process wasperformed by assuming a value for the volumetric heat capacityand then minimizing the least square error. If the volumetricheat capacity is arbitrarily changed, then the resultingconductivity estimate will rdso change such that the ratio(thermal diffusivity) is a constant. Consequently, the parameterestimation process really estimates the thermal diffusivity. Thedensity and heat capacity values used in the data reductionprocess were p = 7916*18 Kg/m3 (this work, 1(s) and c=497*1O J/Kg-K (Taylor et al., 1997; given as 2% accuracy,assumed to be la).Beck and Arnold (1977) indicate that for additive,

uncorrelated errors with zero mean, the standard deviation in theestimated thermal diffusivity is related to the standard deviationin the temperature measurements through

(7)

This relationship points out why maximizing A is important interms of minimizing the errors in the estimated conductivity.The dimensional and dimensionless A are related through Eq.(4) and Eq. (5) and can be written as

N$N,AT2A= . A+.a (8)

The variance and standard deviation in the estimatedconductivity can be written as

eaz 1 (3Tz i5a() () i3T- z N,NtA+ AT r ~ = & (9)+AT

The issue we are faced with now is how to estimate 6~. Thiwill be explored in more detail in the following section.ESTIMATES OF STANDARD DEVIATION IN TEMPERA-TURE MEASUREMENTSIt is important to study the temperature residuals

corresponding to the estimated thermal diffusivit y. Thesresiduals are an indicator of the standard deviationtemperature and are defined as the difference between thexperiment and the model temperatures. Residuals for Ru042999 are presented in Fig. 6. The standard deviation in th

tiu$ 1Sta8LA Sta9??TITl -1stall

~o 1000Time, sFigure 6. Temperature residuals for Run 042999.

temperature can be estimated from(lo

and was found to be 0.049C. Visually the band of k26~ in Fig6 appears to capture 95% of the residuals. Ideally the residualsshould be randomly distributed around zero. Clearly thresiduals in Fig. 6 are not random, suggesting that there may binconsistencies between the measurements and thcomputational model. Most likely this pattern is because of thmodel not including all the physics present in the experiment.However, note the small magnitude of the residuals, indicatinggood agreement between the model and data.At steady state, uncertainty in the temperature measurement

is because of random noise from the data acquisition system anfixed bias from the data acquisition amplifiers or variationsthermocouple alloy composition. To assess the noise level ansimultaneously remove the bias, the instrumented cylinder wasubjected to long-term soaking at constant temperature levelsa temperature-controlled oven. A thermistor with a calibration


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traceable to NIST was installed on the cylinder, and itstemperature was used to develop a thermocouple-by-thermocouple correction curve for bias errors. Because thethermocouples were installed on the parts and connected to thedata acquisition system as in the actual experiment, thecorrection represents an end-to-end calibration of thetemperature measurement system. Bias corrections were on theorder of 0.5C and varied with thermocouple. The remainingerror (assumed to be random noise) amounted to roughly6T =M. 10C over all the thermocouples.For dynamic temperature measurements, such as reported

here, an additional source of uncertainty is the installation of thethermocouple in the cylinder wall. First, the thermocouple,being of finite size and installed in a metal part with atemperature gradient, leads to ambiguities in interpreting itsreading. Further, nonperfect contact between the thermocoupleand the well in which it was installed coupled with thedifference in thermocouple wire material properties leads to atime lag in the reading. Thus the uncertainty in the dynamicsituation is dependant on installation details, the local time rateof change of temperature, and the local thermal gradient in thecylinder. To quantify the contribution of these effects touncertainty, the time response of the four circumferentialthermocouples at each axial station were compared bycalculating the standard deviation about the station mean at eachdata time. These results are shown in Figure 7 for axial Stations

0.5M]Sta1

Run042999

time, s

Figure 7. Standard deviation in four circumferentialstations for axial stations 1-71-7; similar results were obtained for axial Stations 8-14. It wasfound that for axial stations closest to the boundary (l-3), whichunderwent relatively high rates of change in high-gradientconditions, the spatial standard deviation was as high as 0.5C(Station 1) to about 0.2C (Station 3). For the other stations thespatial standard deviation was at or below O.10C, when steady

state was approached. Note that the sensors with the largeserrors (Stations 1-3 and 12- 14) have the smallest thermaconductivity sensitivity coefficient (see Fig. 4).There is a strong correlation between temperature rise rat

temperature gradient, and circumferential standard deviationThe temperature rise rate and the temperature gradient fStation 1 were computed using the experimental data and finidifferences, and these results are given in Fig. 8. The peaspatial temperature gradient lags the peak temperature rise ratThe peak in the standard deviation curve (taken from Fig. 7 fStation 1) occurs at a time between when the maximum timand spatial temperature gradients occur. The maximumtemperature gradient at Station 1 is approximately 0.28 C/mmFor a thermocouple well of 1.17 mm diameter, the positionthe 0.254 mm (0.010 in) diameter wire pair could vary bymuch as *0.44 mm (N.017 in); this assumes that the two wireare side by side in a direction perpendicular to the axis of thcylinder. Using the above (maximum) temperature gradient0.28C/mm and the positional uncertainty of the center linethe thermocouple wire, the uncertainty in temperatureapproximately IHI.12C; this result is consistent with the nesteady-state results presented in Figure 7. This result does nconsider that the presence of the thermocouple alters thtemperature one is @ying to measure.In summary, the three temperature uncertainty estimates var

by an order of magnitude and lie in the approximate rang

0.5 0.10.45 0.080.40.35 0.06

~ 0.3L 0.04b o.25z 0.2 0.02 @z o~ 0.15 0:$ 0.1 ~g 0.05 -0.02 ~>0 -0.04G -0.05 -0.06

-0.1-0.15 -0.08-0,20 250 500 750 1000Time, s

Figure 8. Temperature gradient, temperature riserate, and circumferential standard deviation foraxial Station 1.


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thermal con of ss 304

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