NATIONAL RADIO ASTRONOMY OBSERVATORY CHARLOTTESVILLE) VIRGINIA. ELECTRONICS DIVISION INTERNAL REPORT No. 200

Transcription

1 NATONAL RADO ASTRONOMY OBSERVATORY CHARLOTTESVLLE) VRGNA ELECTRONCS DVSON NTERNAL REPORT No 00 HEAT FLOW N A COOLED COAXAL TRANSMSSON LNE DAN L FENSTERMACHER* AUGUST 979 NUMBER OF COPES: 50 * SUMMER STUDENT

2 HEAT FLOW N A COOLED COAXAL TRANSMSSON LNE ABSTRACT Presened here are he resuls of a heorecal sudy of he hea flow and emperaure gradens whn a coaxal ransmsson lne The ype of cable suded s frequenly used n mcrowave research where cryogenc emperaures are requred o manan a mnmal nose level n he crcus Two refrgerang saons were aken no accoun ncludng her placemen and effcency The soluon o he lnear dfferenal model s presened n s enrey along wh graphcal dsplays of he effecs of geomery and maerals on emperaures he equvalen conducvy of he nner conducor and hea flow

3 TABLE OF CONTENTS nroducon The Dfferenal Model and s Soluon 3 Boundary Condons 5 Equvalen Thermal Ressance of he nner Conducor 6 The General Soluon for he Consans Calculaons 7 Conclusons Page LST OF FGURES Skech of he Coaxal Transmsson Lne Model of a Dfferenal Elemen dz 3 Temperaure Dsrbuon Along nner Conducor End Temperaure vs Toal Lengh of Lne 5 End Temperaure for Varous Parameer Values 6 Equvalen Conducvy vs Dsance Beween Refrgeraors Rao of Hea Removal ( 0) vs Toal Lengh of Lne 0 8 Toal Hea Removed vs Lengh of Lne

4 nroducon The problem under consderaon here s he flaw of hea hrough a coaxal cable connecng an ousde source o a mcrowave devce cooled o 0 K The geomery of he cable s aken o be cylndrcally symmerc over s 5 cm lengh wh refrgerang saons n hermal conac wh he ouer conducor (See Fgure ) n seng up a dfferenal model of hea flaw found necessary o make he followng assumpons: The lenghwse flow of hea hrough he elecrcally nsulang delecrc s neglgble when compared o ha n he wo surroundng meal conducors The varaon n hermal conducvy wh emperaure can be approxmaed by a smple average whou addng unreasonable uncerany o he calculaons 3 The refrgeraors are n deal hermal conac wh he ouer conducor Along wh hese assumpons he knowledge ha emperaure drop n a gven maeral s proporonal o hea flow allowed he consrucon of a lnear model wh a famlar soluon

5 T(a) / /r/ / / / (/ 7 / 7 / J A z / / / / / 7 7 / / / / / T + // / / QW Q(z) Q(A) go0 La FGURE : Skech of he coaxal ransmsson lne hvel CONDL/CTu (B) dt; E C end( C TO e D ec r c T) C/E FGURE : The model of a dfferenal elemen dz

6 T he D ff erenal Model and s Soluon The model have skeched n Fgure represens he hea currens and emperaures of neres n he coaxal lne Longudnal flow oward he cold end s aken as he +z drecon and radal flow s pcured as he perpendcular pah beween he nner and ouer conducors The varables have he followng connoaons: hermal ressvy of he nner and ouer conducors respecvely degrees (uns ) wa cm z hermal conducvy of he eflon cylnder around he nner conducor was v degree cmz Q( z) seady sae hea flaw (was) T(z) seady sae emperaure (degrees K) The hermal consans of he maerals hemselves are relaed o R and G by geomercal facors of he lne n he nfnesmal lengh dz here s an equal nflux and ouflux of hea wh some emperaure drop n he drecon of flow The four zdependen varables we are seekng are T T Q l and Q From he model he followng four equaons evdenly characerze he process: () dt = (R dz) Q () dt = (R dz) Q (3) dq = (G dz)(t Tl) () dq = (G dz)(t T) Each dfferenal change n he four varables s n balance wh he hea flaw or emperaure graden exsng n he lengh d z

7 The coupled se of lnear dfferenal equaons shown above can be solved by proposng a soluon of he form e Xz such ha he operaor dz can be replaced by he consan X By dvdng equaons o by dz and makng hs subsuon we arrve a a se of lnear algebrac equaons n he four unknowns n marx form hese are wren: () X 0 R 0 () 0 X 0 R (3) G G X 0 () G G 0 T T Ql Q r Knowng ha he deermnan of he marx mus now be dencally zero yelds he characersc equaon n egenvalue X: A (GR + GR )X = 0 Roos: X = 0 0 ± J GR The presence of wo zeroroos allows a consan and lnear erm o ener no he soluon whch hen becomes: (5) T (z) = A + Bz + Ce Xz + DeXz By separaely applyng 3 of he equaons we can deermne he form of he oher varables (6) Apply (): Q (z) = C Az D Az + Xe R (7) Apply (3) T (z) = A + Bz R Az Ne Az B C Az D Az (8) Apply (): Q (z) = + R R Fnally wh he hermal properes of each maeral known he only remanng ask s o apply approprae boundary condons from he physcal makeup of he problem n order o deermne he consans A B C and D

8 Boundary Condons Gven jus one refrgeraor we can se up boundary condons from he problem by nspecon The emperaure a he ho end for boh conducors s room emperaure (3000) emperaure a he sngle refrgeraor s known (0 0 ) and well nsulae he nner conducor a he cold end by keepng unconneced and n a vacuum (Q (z) goes o 0 a he cold end) Wh hs confguraon he emperaure a he cold end of he nner conducor s: where L = dsance from ho end o he refrgeraor (L) emperaure of he refrgeraor T (0) room emperaure a he ho end (he formula vald for XL > 3 so ha snh XL == cosh XL) The general case has more han a sngle refrgeraor and mus be reaed as separae secons of cable (perhaps each wh dfferen maerals) joned n a "connuous" way Each secon has s own X and consans B C D whch descrbe Thus each requres boundary condons The wo emperaures a he ho end and emperaure and hea flow a he cold end are he same gven any number of secons For each secon afer he frs more BCs are obaned as follows: The ouer conducor emperaure for boh secons s he refrgeraors emperaure A he nner conducor he hea flow ou of one secon s he same as he flow no he nex and here can be no emperaure jump here Smply saed connues plus refrgeraor gve he boundary condons The consans for all he secons can hen be solved smulaneously wh one large marx nverson ha wll be demonsraed laer

9 Equvalen Thermal Ressance of he nner Conducvy Up o now he end of he nner conducor has been aken as compleely nsulaed from hea flow and we have seen how o fnd he "open crcu" emperaure a ha pon When he cable s conneced o a devce however hea wll flow no ha devce based on s emperaure and he equvalen hermal ressance of he coax and he devce Because of he lnear couplng of hea flaw o emperaure and consan coeffcens n he dfferenal equaons he soluon s ndeed a lnear one such ha each boundary condon has a lnear effec on each of he varables of neres For example hs means ha he emperaure a he cold end of he nner conducor vares lnearly wh he boundary condon for he amoun of hea dranng from : The quany T o T (L) = R equv Q(L) + T o s he "open crcu" emperaure (whch exss when Q (L) = 0) and he consan of proporonaly s he equvalen ressance of he nnerconducor BeforewecandeermneR we mus solve he marx equv equaon for he consans A B C D The General Soluon for he Consans The mehod of generang boundary condons for each secon of he coax was descrbed above Here we wll examne he praccal case of wo refrgeraors a dsances L and L along he lne and show how he consans can be found The funconal forms of T(z) and Q(z) are known from equaons (5) (8) excep for he consans for each secon of he coax Le hese be reaed as varables and z be fxed for each parcular boundary condon Then 8 equaons are generaed as follows:

10 T (0) = 300 room emperaure ) = 0 second refrgeraor T (0) = 300 room emperaure T (L ) T(L ) = 0 connuy = 77 frs refrgeraor Q (L ) Q(L ) = 0 connuy = 77 frs refrgeraor l ) = 0 nsulaed end The prmed varables denoe he second secon of lne n marx form hese are wren: [m] C] = B] hus [] B] Az where [K) s he marx of z e Az erms evaluaed a fxed z C] s he vecor of coeffcens (A B D) B] s he vecor of boundary condons ( ) From he marx form of he soluon s agan evden ha each boundary condon has a lnear effec on he coeffcens By seng all B = 0 excep (L ) whch we wll denoe by Q o no he devce he rao of T(L o Q o can be deermned s n fac: T (L ) Q0 X L =R =m +m L+me +m8 equv where [m] = [} Along wh he open crcu emperaure hs equvalen ressance (n degrees/wa) complees he characerzaon of he hea flow hrough he nner conducor Calculaons The physcal dmensons of he coax are shown n Fgure The crossseconal areas of he nner and ouer conducors and delecrc are and 06 cm respecvely Takng averages of he hermal

11 8 conducves beween he refrgeraor emperaures we oban he followng numbers:* Conducves n Was/cm degree Beryllum Copper Sanless Seel Teflon (nner conducor) (ouer conducor) (delecrc) The values for R /(area g ) and R and R used n he model are relaed o he above by R = = /(area g ) where g g are he conducves To fnd he geomercal facor for G we negrae s conducvy radally over he hollow cylnder of delecrc and fnd G = g denoes he radus These values are shown below: efl 7n( ) where r r R R The dsance o he 77 refrgeraor s aken as L = 7 cm and he dsance o he 0 refrgeraor s L = 50 cm Ths s he "sandard" confguraon referred o n he graphs The frs assumpon made can now be checked The smalles rao of conducves (for longudnal hea flow) beween conducor and delecrc * s also possble o propose a change of varables a hs pon o accoun for some nonconsan funcon g(t) One negraes g(t) over he approprae emperaure range and hus obans a new varable n whch he dfferenal equaons are agan lnear The resrcon however s ha all 3 conducves mus have he dencal Tdependence for he new varable o be useful No praccal use was found for he parcular maerals here bu wh some compromses an applcaon for hs change of varable could be found

12 s and occurs beween he sanless seel ouer conducor and he eflon a he cold end Thus here s a leas mes as much longudnal hea flow anywhere n eher of he conducors as n he delecrc Havng wo conducors hen makes he frs assumpon vald o whn a 5% correcon From he sandard confguraon of refrgeraors and for hese maerals we obaned he value of 37 for he open crcu emperaure and an equvalen hermal ressance of 3 degrees/mwa or conducance of 3 mwa/ degree Conclusons On he followng pages presen a large amoun of nformaon n graphcal form ha was exraced from he model The followng conclusons can be drawn regardng he coax lne he equvalen conducvy and refrgeraon: The emperaure graden s essenally lnear along he lne for he frs few cenmeers bu s hghly dependen on he placemen of refrgeraors The emperaure of each refrgeraor has a very srong effec on he coldend emperaure of he nner conducor and hus s crucal o opmze he effcency of he refrgeraors 3 Only he coldend conducves of he 3 maerals have a sgnfcan effec on he fnal emperaure wh he order of sgnfcance beng nner conducor delecrc and ouer conducor The value used for room emperaure has lle effec on he fnal emperaure of he lne 5 For a gven se of maerals he equvalen conducvy a he end of he nner conducor s almos enrely a funcon of he dsance beween refrgeraors

13 0 6 The opmum placemen of he 77 refrgeraor f has 3 mes he capacy of he 0 0 saon s 7/0 of he way o he 00 refrgeraor ndependen of he lengh of he lne All of hese nvesgaons were done by he HP 985 compuer n he elecroncs dvson The program remans on fle for furher sudy and o provde daa for comparson wh expermenal measuremens

14 3oo Sc3s S TFm5ERATVRE DSTK 6(T/a A/ LoNG N r oc)cro N \ SC a H H 0 M bh\n 6 Nplppopp : * L Ell j L sr/ ^ E l l L Ml f a a A " mummy ostavce (Cm) +Z DRE cr) ow z LA) :

15 0 60 so Yo 30 T E " V P EK o NO; c : :! : L ENO W PER ATuRE vs Torlc 67V q 77 of LME L f! f f / / A!! } f ; / /C L a H a PA ț El + f A / / / L : L P f mo M : ; j :!! ; G C M ( ; O 5 7 E/vGTH OF L me (cm) L l : H / / A e : / FT blllm llll Mlngu mullfl """"ANM; =lalhfmm / r f A! j ; : Ị / ; /! /

16 T l l 7V0 T&" P (RE Po Vf/ QVJ e mrter ALUEs MM A 0 A /5 0dEEN 0aMPrr ; 00Prll 000 mnlfl u ; D r7 L E q MN " L L 3ao ll Nel L H Lzr; b\ *"%% 5 c : w om po op orll mllar sr PP" PM Al :00 En rsadglpaggggllgllnn gg ME " MM; DE WO fl hnnn L n 0 a EME J L H J f H o A/ Ar Fs7 PARE A/ 76R ee L AT VE To Vrev 000 k f

17 6 % Y 3? 36 6 Qvl V 30 Z Ec)vLENT condl CTvT y VS ST NC E CET EW E E TOR S` R d z A / f l p l z r6 o ; ; */ V 0 ; A f V hal :770 7)( re 0**T L ; f! : fl$ f! ț z s ovstance 6e7A/EEN EFR6AT0AS (") j F V 0 n El ll MN : / : A l lsmm MMUMMNMNMM L A : ;! l : f " l f *

18 y /0 RAJo OF HEAT EMO A L PR! f A A 9 / T/ 0 E HA ± ro Y e lf e 7 L 7/ /yr : J l D VS 7 MMNUMOMM MN?? T E7/0 v 7OrAL L m ;/v P gll amna 0 L " wj l P 6BJ 0 mod 5mm7 llk rdrmlv o msnnnmllll f MN ONNMEMNMNUMN MNMNMMENEMMN J ON M 00! L ll r : n n OMEN MNMEBMMMNMMN 6 /0 TOTAL CEVGTH o / ME (CM) : : a «{ f L

19 0 ToT REA Rr r (WA f " v p s TOTAL HeAr RE/ o CO VS LENGT op P Fs HHf ll al L ll mnm ll A / : f a L / / : wl mm MA / " l mu [ J! : : L [ f 6 /0 TOTAL LFNGTH oa (em) : ; f : ;