Thermal Conductivity of Cryogenic Deuterium

Transcription

1 hermal onductvty of ryogenc euterum Alan he Pttsford Mendon Hgh chool Pttsford, Y Advsor: r. R.. raxton Laboratory for Laser nergetcs Unversty of Rochester Rochester, Y ovember 007

2 Alan he Abstract Accurate values of the thermal conductvty and specfc heat of deuterum are necessary to make hgh qualty cryogenc laser fuson targets. ue to the dffculty of measurng these values at cryogenc temperatures, the 3ω method s used. hs nvolves passng an oscllatng current through a platnum wre embedded n deuterum and measurng the thrd-harmonc voltage that results from the temperature dependence of the resstance. A two dmensonal model has been bult that calculates heat flow n both the axal and radal drectons.e., along and perpendcular to the wre ncludng the frequency-dependent skn depth. he model has been ft to four sets of expermental data the wre n a vacuum and n gaseous, lqud, and sold deuterum by adjustng the parameters of thermal conductvty and specfc heat. Generally, the model and data agree well. he model has enabled the thermal conductvty to be accurately determned at cryogenc temperatures for gaseous, lqud, and sold deuterum.. Introducton he thermal conductvty of deuterum s used n the modelng of cryogenc laser fuson targets. In ths process, a small fuel pellet, consstng of a mxture of deuterum and trtum, s heated and compressed by hgh-energy laser beams. hs creates condtons smlar to those of the un, allowng the nuclear fuson of deuterum and trtum to occur through the reacton 4 He n. - nergy s released n the form of the knetc energy of the resultng neutron. he fuel pellet s ntally kept at cryogenc temperatures to maxmze the amount of fuel n the small volume. Knowledge of the thermal conductvty of deuterum s necessary for modelng the heat flow wthn the target. Knowng the conductvty to better precson mproves target qualty. he conductvty s also used n modelng the behavor of the target when exposed to the envronment.

3 Alan he 3 Intally, the target s covered by a shroud to protect t from the envronment. Immedately before the laser shot, the shroud s quckly removed. It s of nterest to model what happens n the nterval between the removal of the shroud and the shot, such as the rate of ncrease of pressure n the center of the target due to the conducton of heat resultng from external radaton. In 954, early measurements of the thermal conductvty of deuterum were made by Powers. he setup was of the parallel plate type, where heat was deposted at a known rate at one end of a cylnder contanng deuterum and exted through a heat bath at the other end. he temperature gradent across the cylnder was measured by a multple-juncton-dfference thermocouple. he experment was prone to errors ncludng heat flow nto the walls requrng correcton, convecton due to the geometry of the setup, and lack of symmetry from the ntruson of measurement devces. Frst ntroduced by ahll 3 n 986 and subsequently used by varous authors, 4-8 the 3ω method measures thermal conductvty n a geometry where the avenues of heat flow are smpler to accommodate than n prevous methods. here s also less susceptblty to convecton. he method uses a wre or thn flm as both a heater and a thermometer to produce and measure temperature oscllatons n a chosen medum. Plastc euterum Platnum re opper V Fgure. agram of the experment. An oscllatng current s passed through the wre, and the voltage across a porton of the wre s measured. he plastc and copper are heat baths.

4 Alan he 4 In recent experments, 9 the 3ω method was used to determne the thermal conductvty of cryogenc deuterum. As ndcated n Fgure., a platnum wre s embedded n deuterum. A snusodal current through the wre deposts heat nto the system. Heat exts the system by the heat baths, whch are set at the desred temperature. he tme-averaged thermal equlbrum of the system steady state s establshed after a bref nterval, after whch the voltage across a porton of the wre s measured. he temperature dependence of the resstance of the wre results n a small thrd-harmonc voltage, whch has both n-phase and out-of-phase components wth respect to the phase of the power. It s possble to determne the oscllatory temperature from ths voltage. he temperature s out of phase wth the oscllatng power at hgh drve frequences and n phase n the lmt as the frequency approaches zero. A computer program was bult to model ths experment. he model accounts for the twodmensonal geometry of the experment there s thermal conducton parallel and perpendcular to the wre. It has been used to determne the sgnfcance of the effects and to provde mproved accuracy over the one-dmensonal numercal solutons, whch gnore heat flow along the wre, that were prevously used to model experments. 0 y matchng the predcted temperature ampltudes of the program to the expermental data, the thermal conductvtes of sold, lqud, and gaseous deuterum have been determned.. Analyss of the 3ω Method hs secton shows how the ampltude of the temperature oscllatons n the wre can be determned from the expermentally obtaned voltages. he resstance of the wre changes wth the temperature : R R α, where R 0 s the resstance at temperature 0 and α s a constant α dr / d.

5 Alan he 5 he snusodal current appled to the wre can be expressed n complex form: ωt * ωt [ Ie I e ] I t I 0s sn ωt I 0c cos ωt, - where I s complex and * I ndcates ts complex conjugate. Generally, I 0 and I s 0 s real. he temperature has a steady tme-averaged component,, and real and magnary components that are n phase and out of phase wth the power, respectvely: t ωt * ωt [ e e ] t t. -3 he temperature oscllates at the second harmonc, snce the power, P I R, brngs n t e ω terms from -. Usng - to -3, the voltage across the wre s ωt * ωt ωt * ωt [ I e I e ] R t [ e e ] V t I t R t 0 α 0, -4 whch can be wrtten as ωt * ωt 3ωt * 3ωt V t [ Vωe Vω e V3 ωe V3 ωe ], -5 where Vω I R0 and V 3ω αi. -6 he ampltude of the frst harmonc voltage s the product of the current and the tme-averaged resstance, and the ampltude of the thrd harmonc voltage s proportonal to the product of the frst harmonc current and the second harmonc temperature. vdng the two parts of -6 to elmnate the current shows that the ampltude of the temperature oscllaton can be determned from the rato of the thrd harmonc voltage to the frst harmonc voltage: 0, V3ω R0. -7 V α ω

6 Alan he 6 3. omputer Model A computer model was bult to model the experment. It was constructed n R, Z geometry: perpendcular and parallel to the wre, respectvely see Fgure.. he model ncludes the platnum wre, deuterum, and the surroundng heat baths. he radus of the wre 7.5 µm s small when compared to the radal wdth of the deuterum medum to 3 mm, and hence a varable-sze grd was mplemented n the deuterum, wheren the radal wdths of the cells ncrease geometrcally wth ther dstance from the wre. he cell wdths n Z are kept unform for all meda. he average of the platnum and deuterum conductvtes s used as the conductvty at the platnum-deuterum boundary. here s no heat flux across the lower boundary of the platnum wre the Z axs. a R all Heat ath b euterum de Heat ath Platnum re he equaton solved for each cell s where the heat flux Q s gven by Z Fgure. a A depcton of the computer model ncludng the cells used to solve the heat flow equaton. de heat baths and the wall heat bath are named dfferently for dentfcaton but functon dentcally. b A 3 depcton of the model, where each cell s a rng. Q, 3- t Q κ. 3-

7 Alan he 7 Here s the specfc heat of the cell, s the temperature, κ s the thermal conductvty and s the heat deposted per unt volume. hs can be solved as an ntal value problem to gve the full soluton of the form -3, or t can be solved for the complex n the steady state, n whch case / t s replaced wth ωt ωt ω[ e * e ]. he ntal value soluton s gven by [ θ κ θ κ ], 3-3 t where t s the tme step, ' s the temperature after the tme step, and θ s a numercal parameter between 0 and. hen θ 0, ' at each cell s gven explctly n terms of known quanttes. However, the small t necessary for stablty requres many tme steps to reach the steady state. For θ 0. 5, stablty s acheved however large t s. he choce θ s typcally made. hs gves a backward dfference equaton n whch the terms couple the unknown at each cell center to the at the four adjacent cells. hether the ntal value soluton wth θ > 0 or the steady state soluton s requred, a set of smultaneous equatons result that can be solved va matrx nverson see Appendx. he program constructs a qundagonal coeffcent matrx that corresponds to a grd of cells as llustrated n the smplfed grd of Fgure 3.. R euterum Fgure 3. Z A smplfed 3 4 grd. Platnum For ths grd, wth the bottom row as the platnum wre and the top two rows as the deuterum, there s a correspondng qundagonal matrx equaton:

8 Alan he A A F A F A F F A F A F F A F A F 3-4 where each box corresponds to a column n R and ndcates or. he center dagonal addresses the relevant cell and, A,, and F address heat flow to the four adjacent cells. he power suppled by the heat source s ncluded n, and s nonzero only for platnum cells. he full equatons for the matrx are gven n the Appendx. he matrx equaton 3-4 s solved teratvely usng the Incomplete holesky onjugate Gradent 3 method. hs method was orgnally envsaged for all real quanttes, but was adapted here to handle all complex numbers to accommodate the ω term n and produce a complex whose real and magnary parts correspond to n-phase and out-of-phase. 4. olutons 4. Platnum re n Vacuum A set of expermental data for a platnum wre n vacuum was obtaned to determne the thermal conductvty and specfc heat of the wre. Heat flow s neglgble n R and the problem can be consdered as n Z. he program was used to calculate solutons n Z of the ntal value problem usng 3-3 to show the tme evoluton of the temperature of the system.

9 Alan he 9 Fgure 4. shows the temperature at the center of the wre as a functon of tme. It takes about 30 ms to reach the steady state, whch s when the measurements are taken n the experment. Also, t can be seen that the temperature oscllatons are out of phase wth the power. Fgure 4. Graphs of the temperature n the center of the wre and the power aganst tme for a platnum wre n vacuum wth the current oscllatng at 00 Hz. he spatal dependence of the steady-state solutons along Z s shown n Fgure 4. for four frequences from Hz to 000 Hz. he temperature ampltude s zero at each end due to the sde heat baths. At dfferent frequences the temperature ampltudes along Z have dstnctve shapes, whch can be understood n terms of the penetraton depth 0 d of the temperature ampltudes. he penetraton depth s a measure of how far the temperature oscllatons penetrate before falloff and s a functon of frequency, thermal conductvty, and specfc heat: d κ ω. 4- he penetraton depth shortens as the frequency ncreases because there s nsuffcent tme for the changes n temperature to be thoroughly transmtted. At low frequences of 0 Hz and less, the temperature profles Fgures 4.a and 4.b mantan the arch shape, showng the relatvely complete penetraton of the temperature changes. he quadratc shape results from the maxmum temperature beng at the mddle of the wre.

10 Alan he 0 a Hz. d 3.3 mm b 0 Hz. d 4. mm In phase Out of phase c 00 Hz. d.33 mm d 000 Hz. d 0.4 mm Fgure 4. Graph of n-phase and out-of-phase temperature ampltudes aganst dstance z along the platnum wre at a Hz, b 0 Hz, c 00 Hz, and d 000 Hz. he penetraton depths d are also gven. At Hz, the power s oscllatng at a suffcently low frequency that there s enough tme for the temperature to change n phase wth the power and the n-phase component domnates. At the upper frequency range 00 Hz and greater, there s nsuffcent tme for the temperature gradent begnnng at the heat baths to extend far nto the center of the wre. hs s llustrated by the relatve peaks n temperature ampltude near the heat baths at hgher frequences Fgures 4.c and 4.d. mlar behavor was found n Ref. 8. It s also at these frequences that the magnary component domnates the real component, and thus the temperature becomes out of phase wth the power as seen n Fgure 4.. mlar steady-state solutons were obtaned over a broad range of frequences, and the

11 Alan he average temperature ampltude at each frequency was obtaned by averagng over the platnum cells between the voltage leads see Fgure.. he results are plotted n Fgure 4.3 wth the conductvty and specfc heat chosen to best match the expermental data. a ±0% n platnum conductvty b ±0% n platnum specfc heat xpermental data In phase 0% Out of phase 0% Fgure 4.3 he n-phase and out-of-phase temperature ampltudes as a functon of frequency. he model predctons are shown by the sold curves. he dashed curves show ±0% varatons n a the conductvty and b the specfc heat of the platnum. In the low frequency lmt, the calculated temperature ampltudes are domnated by the n-phase component, whle the out-of-phase component falls to zero. In the hgh frequency lmt, the out-of-phase component domnates, and the n-phase component falls to zero. he data s evdently flawed at less than 0 Hz, snce the n-phase component should approach a constant and the out-of-phase component should fall to zero. he dstnctve shapes of the real and magnary curves and the close agreement between smulaton and experment above 0 Hz allowed for the smultaneous determnaton of both thermal conductvty and specfc heat. he thermal conductvty and specfc heat of the platnum wre, assumed to be 7.5 µm n radus, were determned to be 530 /m K and J/m 3 K, respectvely, wth an uncertanty of about ±0% snce the curves n Fgure 4.3 wth ±0% varatons provde poorer fts. hese values dffer from the lterature values of 430 /m K and J/m 3 K. hanges of ±0% n the platnum conductvty Fgure 4.3a shft the temperature

12 Alan he ampltude curves both vertcally and horzontally. Increasng the conductvty allows heat to be more easly conducted out to the heat baths, shftng the curves down. he results do not depend on conductvty n the hgh frequency lmt. hanges of ±0% n the specfc heat Fgure 4.3b shft the curves horzontally. Increases n the specfc heat shft the curves to the left. he hgh and low frequency lmts reman unaffected. he results do not depend on specfc heat n the low frequency lmt. here s, however, another uncertanty n these values due to the uncertanty of the wre radus, whch was measured by a lght mcroscope as 7.5 µm. rrors assocated wth dffracton at the wre s edge are on the order of the wavelength of the lght, whch s approxmately µm and sgnfcant n comparson to the radus. An dentcal ft can be obtaned usng the standard values of thermal conductvty and specfc heat f the radus s adjusted from 7.5 µm to 8.4 µm a % ncrease. In the rest of ths work, the lterature values wll be used for platnum under the assumpton that the wre radus was 8.4 µm. 4. Gaseous euterum he model s essental when deuterum s ncluded, snce heat can now flow n the R drecton. he effects are llustrated n Fgure 4.4, whch shows sgnfcant nonunformty along the wre at 0 Hz. he wre can barely be seen, as t s just the nner 8.4 µm. he effects on the temperature ampltude due to the sde heat baths n the model are qualtatvely smlar to those found from the model. he temperature ampltudes along Z have the arch shape as n the 0 Hz case of Fgure 4.b. he maxmum [n-phase, out-of-phase] temperature ampltudes have been reduced from [00 mk, 30 mk] to [5 mk, 9 mk], because the heat deposted n the platnum wre s conducted nto the gas.

13 Alan he 3 a In phase mk b Out of phase mk Fgure 4.4 emperature ampltudes for a platnum wre n gaseous deuterum at 0 Hz splt nto a n-phase and b out-of-phase components. In phase Out of phase Fgure 4.5 In-phase and out-ofphase temperature ampltudes as a functon of drve frequency for a platnum wre n.0 torr of deuterum gas at 0.9 K. he sold lne ndcates that the wall heat bath s at 3 mm and the dashed lne ndcates mm. From Fgure 4.5 smlar to Fgure 4.3 but for deuterum gas, the thermal conductvty of deuterum gas was determned to be /m K, whch s 37% greater than the value of 0.00 /m K quoted n Ref. 5. hs ft was obtaned wth a specfc heat of 360 J/m 3 K, whch s 7.5% greater than the orgnal of 335 J/m 3 K. 5 espte beng flawed at the low frequency range below Hz, where the magnary component goes negatve, the data s ft well by the model. he radal dstance of the platnum wre to the wall heat bath n the experment ranged

14 Alan he 4 from mm to 3 mm. euterum gas shows a sgnfcantly better ft wth the wall heat bath at 3 mm Fgure 4.5. he effects of the wall heat bath can be solated by examnng the soluton n R, dsallowng heat flow to the sde heat baths. Heat s deposted n the platnum, flows across the platnum-deuterum boundary nto the deuterum, and fnally flows nto the wall heat bath. In-phase Out-of-phase Fgure 4.6 emperature ampltudes aganst radus R n deuterum gas at 0.9 K. he blue and red ndcate the wall heat bath at and 3 mm, respectvely. old lnes show 0 Hz and dashed lnes show 0. Hz. Inphase ampltudes are postve and outof-phase ampltudes are negatve. In Fgure 4.6, the sold red and blue curves show 0 Hz temperature ampltudes along R wth the wall heat bath at 3 mm and mm, respectvely. nce the penetraton depth s much less than mm, there s only a small dfference n the temperature profles between the two cases. he dashed curves are the same except that the frequency s 0. Hz. he large penetraton depth at the lower frequency s dsrupted by the wall heat bath, causng the maxmum n-phase temperature n the mm case [69 mk] to be sgnfcantly lower than that of the 3 mm case [84 mk]. he decrease n temperature ampltude that results from decreasng the radal dstance of the wall heat bath s also seen n Fgure 4.5. Only the parts of the curves less than a certan frequency threshold 0 Hz are affected, snce the smaller penetraton depths at frequences above the threshold do not confront the wall heat bath. hus the wall heat bath s suffcently far from the wre to not affect the value of thermal conductvty obtaned from Fgure 4.5.

15 Alan he Lqud euterum Out of phase In phase Fgure 4.7 In-phase and out-ofphase temperature ampltudes as a functon of drve frequency for a platnum wre n lqud deuterum at 4.3 K. he sold lnes ndcate that the wall heat bath s at mm and the dashed lnes ndcate 0. mm. A comparson of predctons and expermental data for lqud deuterum s shown n Fgure 4.7. he agreement s generally good, but the out-of-phase data s suspect of premature falloff at 0 Hz and less. In an attempt to explan ths behavor, the wall heat bath was moved closer n the smulaton, to 0. mm dashed curves. However, ths fals to provde a convncng ft, and the closest heat bath n the experment was at.0 mm. he dstance of the wall heat bath only affects the low frequency range. he structure seen between 00 Hz and 000 Hz s also questonable and may be due to a thn layer of frozen ar between the platnum and the deuterum. he range of good data between 0 Hz and 00 Hz s nsuffcent to determne both thermal conductvty and specfc heat. Yet a good ft was obtaned usng the standard specfc heat, J/m 3 K, and the standard thermal conductvty, 0.35 /m K old euterum From Fgure 4.8, the thermal conductvty of sold deuterum was determned to be 0.40 /m K, whch s 46% greater than the orgnal 0.74 /m K. 5 hs ft was obtaned wth the standard specfc heat of J/m 3 K. 5 he data s very good n the lmted range, but ths

16 Alan he 6 also contrbutes to uncertanty. thout a dstngushng shape such as seen n Fgures 4.3 and 4.5, the conductvty and specfc heat cannot both be absolutely determned, as dfferent combnatons of conductvty and specfc heat can produce the same ft. Out of phase In phase Fgure 4.8 In-phase and out-ofphase temperature ampltudes as a functon of drve frequency for a platnum wre n deuterum sold at 8.6 K. a In phase mk b Out of phase mk Fgure 4.9 emperature ampltudes of sold deuterum at 0 Hz splt nto a n-phase and b out-of-phase components, showng the relatve unformty of temperature ampltudes n Z. In sold deuterum, temperature ampltudes are more unform n Z Fgure 4.9 compared wth gaseous deuterum Fgure 4.4 because the relatvely hgh specfc heat of sold deuterum results n smaller temperature ampltudes; thus the heat flow along Z proportonal to / z s much smaller.

17 Alan he 7 5. ummary Improved values of thermal conductvty and specfc heat of cryogenc deuterum have been obtaned by a new computer program and are summarzed n able 5.. he program s able to ft data sets for gaseous, lqud, and sold deuterum. he program has confrmed the publshed values of thermal conductvty and specfc heat for platnum. he model provdes a more accurate representaton than conventonal solutons, and has demonstrated that the 3ω method can be appled to cryogenc deuterum. he adaptaton of the Incomplete holesky onjugate Gradent 3 method to use complex coeffcents s beleved to be new. able 5.. etermned values of thermal conductvty and specfc heat of deuterum, where values n parentheses are prevous best values obtaned from Ref. 5. hermal onductvty /m K pecfc Heat J/m 3 K Gas 0.9 K Lqud 4.3 K old 8.6 K Future work should nclude the acquston and analyss of mproved expermental data. here are some obvous problems wth the data: n the case of gaseous deuterum, there s a drop to negatve out-of-phase temperature ampltudes below Hz, and n the case of lqud deuterum, there s an unpredcted falloff below 0 Hz and structure between 00 Hz and 000 Hz. Possble expermental mprovements may be to ncrease the spectral range of the data and mprove the measurement of the wre dameter. In spte of the problems wth some of the data, most of the data was found to be of good qualty allowng mproved values of the thermal conductvty of deuterum to be obtaned. hese values wll be used to model cryogenc target behavor and to mprove target qualty.

18 Alan he 8 Acknowledgements I would especally lke to thank r. raxton for hs tme and nspratonal gudance throughout ths project, and for provdng me wth the opportunty to work at the Laboratory for Laser nergetcs. I would also lke to thank Mr. Roger Gram for makng ths project possble. Fnally, I thank everyone n the 007 Hgh chool ummer Program for beng an excellent frend. References. R. Hardng et al., Phys. Plasmas 3, R.. Powers et al., J. Am. hem. oc. 76, G. ahll and R. O. Pohl, Phys. Rev. 35, O. rge and. R. agel, Rev. c. Instrum. 58, L. Lu,. Y, and. L. Zhang, Rev. c. Instrum. 7, A. Jacquot et al., J. Appl. Phys. 9, F. hen, et al., Rev. c. Instrum. 75, ames and G. hen, Rev. c. Instrum. 76, R. Gram, unpublshed results G. ahll, Rev. c. Instrum. 6, G. ahll, Rev. c. Instrum. 73, H. Press et al., umercal Recpes: he Art of centfc omputng, ambrdge Unversty Press, ambrdge, 986, p Kershaw, J. omput. Phys. 6, V. J. Johnson ed., Propertes of Materals at Low emperature Phase A ompendum, ew York, Pergamon Press, 96 p P.. ouers, Hydrogen Propertes for Fuson nergy, Unversty of alforna Press, erkeley, 986, pp. 67 and 69.

19 Alan he 9 Appendx hs secton detals the numercal steady state soluton of 3- and the calculaton of the qundagonal matrx coeffcents of 3-4. Integratng the heat equaton 3- over the volume of cell see Fgure A. gves [ ] m V Q Q Q Q t V A- where m s the specfc heat of cell, are the cell nterface areas, and V s the cell volume. a Interor pont b oundary pont Fgure A. a A typcal nteror cell wth the heat flow Q nto ts four adjacent cells. b A depcton of a pont on the wall heat bath. R s the number of cells n the R drecton. Usng t e z r t z r ω,,, whch dfferentates to t e t ω ω A-3 r Q κ, etc. for nteror ponts A-4 0 r Q κ, etc. for boundary ponts, A-5 one fnds [ ] [ ] R R m V V β β β β ω A-6 Q Q Q Q r r -R R - Q Q Q Q z z r -R R -

20 Alan he 0 where r κ, r κ, z κ, z κ A-7 r κ β, 0 β, / z κ β, / z κ β A-8 Here s the surface area of the nterface, κ s the thermal conductvty, 0 0, and r and z are the dstances between cell centers n R and Z, respectvely. he terms represent the heat flow across nterfaces between typcal nteror cells whle β represents the heat flow across boundares. he terms and β for each nterface are exclusvely chosen,.e., one must be zero. here s no heat flow across the lower boundary, snce t s the exact center of the symmetrc cylnder of the 3 depcton of the model Fgure.b. he fluxes across the east and west sde heat baths are represented smlarly to the wall heat bath, except that / z s the dstance n Z from the cell center to the sde heat bath. quaton A-6 can be expressed as R R F A, A-9 where A-9. A A-9. m V β β β β ω A-9.3 A-9.4 F A-9.5 V A-9.6 he system of smultaneous equatons forms a qundagonal matrx equaton 3-4 that

21 Alan he can be solved by Kershaw s teratve IG 3 method, adapted to provde complex solutons for. In cases, a par of dagonals becomes zero, and the matrx equaton smplfes to a trdagonal matrx that can be solved easly. th mnor modfcatons to A-9. to A-9.6, the ntal value equaton 3-3 can also be solved.