HEAT-BALANCE INTEGRAL METHOD FOR HEAT TRANSFER IN SUPERFLUID HELIUM. Bertrand BAUDOUY

Transcription

1 THERMAL SCIENCE: Vol. 1 (9), No., pp HEAT-BALANCE INTEGRAL METHOD FOR HEAT TRANSFER IN SUPERFLUID HELIUM by Bertran BAUDOUY Orig i nal sci en tific pa per UDC: 56.48:5.1:55.:517.9 BIBLID: , 1 (9),, 11-1 DOI: 1.98/TSCI911B The heat-bal ance in te gral metho is use to solve the non-lin ear heat if fu sion eua tion in static tur bu lent superflui he lium (He II). Al though this is an ap prox i - mate metho, it has proven that it gives so lu tions with fairly goo ac cu racy in non-lin ear flui y nam ics an heat trans fer. Us ing this metho, it has been pos si ble to e velop pre ic tive so lu tions that re pro uce an a lyt i cal so lu tion an ex per i men tal ata. We pres ent the so lu tions of the clampe heat flux case an the clampe tem - per a ture case in a semi-in fi nite us ing in e pen ent vari able trans for ma tion to take ac count of tem per a ture e pen ency of the thermophysical prop er ties. Goo ac cu - racy is ob taine us ing the Kirchhoff trans form whereas the metho fails with the Goo man trans form for larger tem per a ture range. Keywor: heat-balance integral metho, superflui helium, heat transfer Introuction An a lyt i cal treat ment of tran sient heat trans fer in He II has re ceive not enough at ten - tion, con si er ing the sub stan tial in ter est as it re lates to the cool ing an sta bil ity of mag net sys - tems e sign. An a lyt i cal treat ment is use ful for pro vi ing a phys i cal e scrip tion of the phe nom e - non an scal ing laws for en gi neer ing to e sign cool ing sys tem of cryo gen ics e vice coole by superflui he lium. Only, Dresner, us ing sim i lar ity so lu tions metho, has e vel ope an a lyt i cal so lu tions for the clampe tem per a ture an heat flux cases an the pulse-source prob lem [1, ]. These cases eal with lin ear boun ary con i tions an tem per a ture in e pen ent prop er ties in a semi-in fi nite me ia. These so lu tions are not pre ic tive since an a e uate tem per a ture has to be cho sen to set the ther mal prop er ties in or er to fit the ex per i men tal ata [1, ]. For the clampe heat flux prob lem, the Dresner s so lu tion has even a free pa ram e ter that has to be a - juste to fit ex per i men tal ata [1]. The so lu tions for the clampe heat flux case [1] an for the pulse source prob lem [] re pro uce with a high ac cu racy ex per i men tal re sults an give goo phys i cal e scrip tion of tran sient heat trans fer in superflui he lium. An a e uate metho in the so lu tion of non-lin ear heat if fu sion prob lems is the heat- -bal ance in te gral metho (HBIM), e vel ope by Goo man [], be cause of its ca pa bil ity of solv - ing non-lin ear prob lems where the non-lin ear ity can be foun ei ther in the if fer en tial eua tion it self or in the boun ary con i tions. With ex act metho, the re sult ing so lu tion sat is fies lo cally the sys tem of eua tions over the en tire range of space an time. Such so lu tions are rather if fi - cult to ob tain when the if fer en tial eua tion is non-lin ear or if the boun ary con i tions in volve are non-lin ear. The HBIM in the so lu tion of time-e pen ent boun ary-value prob lems gives so -

2 1 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... lu tions, which sat isfy the if fer en tial sys tem only on the av er age over the re gion con si ere rather than con si er ing a lo cal so lu tion. It is of ten suf fi cient for en gi neer ing cal cu la tions in which many more ap prox i ma tions are use to moel com plex sys tems. Heat trans fer in superflui he lium is non-lin ear since the Fou rier law is re place by a non-lin ear law be tween the heat flux an the tem per a ture gra i ent as it will be e vel ope in the next para graph [4]. The heat if fu sion eua tion con structe is non-lin ear an is a per fect can i - ate to be solve by the HBIM. More over, the ther mal prop er ties of superflui he lium are strongly e pen ent on tem per a ture, es pe cially the euiv a lent ther mal con uc tiv ity [5]. To take ac count of these two char ac ter is tics, Bauouy use the Goo man s metho to tackle the non-lin ear heat if fu sion eua tion with tem per a ture e pen ent ther mal prop er ties in superflui he lium [6, 7]. The e vel op ment is base on a Kirchhoff or Goo man trans form where the HBIM metho is solve for a new vari able which is the in te gral of the main ther mal properties with re - spect to tem per a ture. This e vel op ment takes in ac count both the non-lin ear ity of the heat if fu - sion eua tion of He II an the tem per a ture e pen ency of the ther mal properties. Nev er the less, some ap prox i ma tions have been one, mainly to sim plify the e vel op ment to pro uce sim ple so lu tion forms to be use by en gi neers for the e sign of sys tem coole by superflui he lium. Even if these ap prox i ma tions lea to some in ac cu racy in the fi nal re sults, the main char ac ter is - tics of the tran sient heat trans fer in He II were ob taine such as the evo lu tion of the tem per a ture at the cool ing sur face sub jecte to a heat flux or the time to reach the crit i cal tem per a ture of phase-change be tween superflui he lium an nor mal he lium [6]. Ex per i men tal re sults for the pulse source problem is fitte with goo accuracy with no ajusting parameter [7]. We pro pose in this pa per to cover in e tails the treat ment of the heat trans fer in superflui he lium by the HBIM an to pres ent the clampe tem per a ture an heat flux cases with both trans forms (Kirchhoff s an Goo man s) with a slight im prove ment. Heat trans fer in superflui he lium Ac cor ing to the the ory of Lan au [8], be low a tem per a ture name T l (T l =.17 K), he lium un er goes a phase tran si tion from nor mal he lium (He I) to superflui he lium (He II). It is viewe as a mix ture of a nor mal com po nent, the nor mal flui, hav ing a en sity r n an a ve loc ity fiel v n an a superflui com po nent, the superflui, hav ing a en sity r s an a ve loc ity fiel v s. The superflui com po nent is as so ci ate with the en ergy groun state an the nor mal flui com - po nent is as so ci ate with en ergy ex ci ta tions such as phon ons an rotons. The en sity of the en - tire flui is e fine from the en si ties of nor mal flui an superflui as: r = r s + r n (1) For a given tem per a ture, there is a sin gle ra tio r s /r an this ra tio tens to unity when the tem per a ture ap proaches the ab so lute zero an goes to zero when to tem per a ture reaches the tran si tion tem per a ture T l. This tem per a ture sep a rates the two li ui phases that ex ist for he lium, the nor mal he lium (He I) above T l an the superflui he lium (He II) be low T l. In this two-flui moel, the nor mal flui is con si ere as a clas si cal flui in the New to - nian point of view, an there fore it car ies en tropy an is as so ci ate to a vis cos ity, m. Given that the superflui com po nent is as so ci ate with the en ergy groun state, the mo tion of superflui is ba si cally if fer ent from any clas si cal flui since its vis cos ity is null. The superflui com po nent oes not carry en tropy. One can thus con nect the flow of en tropy to the ve loc ity fiel of the nor - mal flui by s rsv n, an the heat flux is ex presse as: rst v n ()

3 THERMAL SCIENCE: Vol. 1 (9), No., pp where the ther mal con uc tion in the li ui is ne glecte. In most prac ti cal cases, the clas si cal ther mal con uc tion is neg li gi ble com pare to the spe cific heat trans fer in He II. The mo men tum of He II is sim ply writ ten: rv r v r v () s s n n where v is the barycentric ve loc ity of the bulk he lium flow. The heat trans port pro cess in superflui he lium im plies that any if fer ence in tem per a ture or heat flux leas to a move ment of nor mal flui in the op po site i rec tion of the superflui. This in ter nal con vec tion is the or i gin of the re mark able prop er ties of the trans port of heat in He II. For large heat flux, what is calle the tur bu lent re gime of superflui he lium, a force ap pears be tween the two com po nents, calle the mu tual fric tion force. It is the phys i cal e scrip tion for the cre ation of uantize vor tex, which can be viewe as the cir cu la tion of the superflui com po nent aroun a core of nor mal he - lium an it is cor re late to the rel a tive ve loc ity of the two com po nents. This force has been ex - presse by Gorter et al. [4] as: F Ar r v v ( v v ) (4) ns s n n s n s This force om i nates the eua tions of mo tion of the two com po nents, which are writ - ten as: v s rs rs rsv s( v s ) rsst p t r rs r n v n v s Ar n rs v n v s v n v ( s ) (5) r an v n r n r n r n v n( v n ) rsst p mdv n t r r n rs v n v s s n v n v Ar r s ( v n v s ) (6) r In steay-state re gime an for static he lium, i. e. r v, by com bin ing es. (5) an (6) an us ing es. (1), (), an (), we ob tain the ex pres sion of the tem per a ture gra i ent: h Ar n T Dv n v n v s ( v n v s ) (7) r s s s In most of prac ti cal cases, i. e. for large heat flux, the first term is neg li gi ble com pare to the sec on term [9]. In that case, the tem per a ture gra i ent can be ex presse in e. (7) in term of the heat flux us ing es. (1), (), an (): r s s4t T f ( T ) T (8) Ar n where f(t) is ther mal con uc tiv ity func tion in the tur bu lent re gime of He II. So ac cor ing to e. (8), the heat flux is pro por tional to the cube root of the tem per a ture gra i ent, which is usu ally name the Gorter-Mellink law. In steay-state re gime, this law has been com pare suc cess - fully to ex per i men tal ata nu mer ous time (see ref er ence [9] for ex am ple) but to solve e. (8), in - te gra tion of f(t) is nec es sary since it var ies largely with tem per a ture as the fig. 1 shows. He II has a ther mal con uc tiv ity func tion about two or ers of mag ni tue larger than for high-pu rity met als at superflui he lium tem per a tures. To il lus trate the high heat trans fer rate in He II, the

4 14 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... Figure 1. Thermal conuctivity function of He II in sat u rate superflui he lium [1] cal cu la tion of the euiv a lent ther mal con uc tiv ity is 1 kw/mk at 1.8 K an for a heat flux en sity of 1 kw/m. In tran sient heat trans fer, sev eral en ergy in puts have to be con si ere to con struct a moel. Ex am - in ing es. (5) an (6), ki netic en ergy is the first to come to min, as so ci ate with the ac cel er a tion of the com po nent s ve loc ity but it is rather small. The sec on is the en ergy to cre ate the superflui tur bu - lence but for large space, in the or er of 1 m in length, this en ergy is small too. In fact, the prin ci pal en ergy om i nat ing tran sient heat trans fer in He II is re late the enthalpy of the he lium [9] an this tran - sient phe nom e non is con trolle by heat trans port an enthalpy vari a tion, there fore e. (8) is use in an en ergy con ser va tion eua tion an yiels to a non-lin ear heat if fu sion eua tion: rc p T t f ( T ) T (9) Ob vi ously, such par tial if fer en tial eua tion is if fi cult to solve be cause of the large vari a tion of the ther mal con uc tiv ity func tion with tem per a ture an its non lin ear ity. Heat trans fer in a semi-in fi nite me ia with tem per a ture e pen ent prop er ties us ing a Kirchhoff transformation For the tur bu lent re gime of He II, the heat flux is given by the Gorter-Mellink law (8), ne glect ing the is si pa tion ef fects in He II, the par tial if fer en tial eua tion mo el ing our sys tem for one space i men sion is: T T rc p f ( T ) in x an for t (1) t where r is the en sity, C p the spe cific heat at con stant pres sure, an f(t) the He II ther mal con uc tiv ity func tion. For a pre scribe tem per a ture the boun ary con i tion is: T = T at x = an for t > (11) an for the clampe heat flux case, the boun ary con i tion is writ ten as: T f at x an for t (1) where is the heat flux at x =. At the ini tial time, the en tire me ia is at con stant tem per a ture T b, so the ini tial con i - tion is: T = T b in x an at t = (1) As it is a semi-in fi nite me ia, the nec es sary sec on boun ary con i tion is a con stant tem per a ture when x or prac ti cally for large x, i. e. the tem per a ture fiel is not is turbe for large x. This con i tions is ex presse by:

5 THERMAL SCIENCE: Vol. 1 (9), No., pp T = T b for x an for t > (14) In the HBIM, we as sume that the so lu tion of the is turbe tem per a ture fiel is lim ite by a is tance (t), calle the ther mal layer, af ter which the tem per a ture fiel is not is turbe, i. e. T = T b for x (t). For a semi-in fi nite me ia, the ther mal layer is e fine as be ing al ways in fe rior to the length of the sys tem. From this ef i ni tion we can mo ify the boun ary con i tion (14) as: T = T b at x = (t) an for t > (15) To take ac count of the tem per a ture e pen ence of the ther mal con uc tiv ity func tion of He II, we use a Kirchhoff trans for ma tion: T f ( T )T an the es. (1)-(1) an are trans forme into: 1 in x an for t (a) a t at x an for t ( b -1) or at x an for t ( b - ) at x ( t) an for t ( c) = in x at t () Tb (16) (17) where a = f(t)/rc p. If (17a) is in te grate with re spect to space over the ther mal layer the re sult ing eua tion is calle the heat bal ance in te gral eua tion. No tic ing that in our sys tem / x is null be cause of the ef i ni tion of the ther mal boun ary since T / x is null, the en ergy eua tion is then trans forme into: x a a t x (18) where we con sier a con stant. In pre vi ous work, we use the value a at x = for sim pli fi ca tion. But the ther mal prop er ties are far for be ing in e pen ent of tem per a ture how ever to sim plify the an a lyt i cal treat ment of the HBIM, we e cie to use the arith me tic av er age value of a name a over the tem per a ture range to take ac count of its vari a tion. With the rule of if fer en ti a tion, the in te gral on the left han-sie of e. (18) is trans - forme into: x x a t xt (19) One can no tice that, ue to the boun ary con i tions, the sec on term of the left han-sie of e. (19) is null which re uces it to a sim pler for mu la tion: t x a Eua tion () is the heat-bal ance in te gral eua tion for our prob lem. The sec on term of the right han-sie of e. () will be eval u ate with the knowl ege of the so lu tion. ()

6 16 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... Let as sume that has a poly no mial form as = a (t) + a 1 (t)x + a (t)x + a (t)x where the co ef fi cients a i (t) are func tion of time an there fore of the ther mal layer (t). Ob vi ously, is an ap prox i mate so lu tion of the sys tem an to fin the if fer ent co ef fi cients, we nee to use if - fer ent boun ary con i tions: the nat u ral con i tions, which en sue from the prob lem, an e rive con i tions, which are con structe from ei ther the if fer en tial eua tion or the nat u ral boun ary con i tions. For this ex pres sion of the so lu tion, we nee two ex tra boun ary con i tions. The first one we choose is straight for war an co mes from the ef i ni tion of the ther mal layer: at x for t (1) One can no tice that this con i tion has been al reay use to con struct the heat bal ance in te gral eua tion. The sec on one co mes from the if fer en tial eua tion at x = where the e riv a - tive of the tem per a ture with re spect to space is null be cause of con i tion (17c). We have, what it is calle a e rive con i tion: at x for t () By the use of the nat u ral boun ary con i tions (17b s), (17c), an (1), an the e rive one (), we can for mu late a so lu tion for as a func tion of : x 1 () T where = T f ( T )T for the clampe tem per a ture case an b = / for the clampe heat flux case. For the pre scribe tem per a ture case, if we sub sti tute e. () in the heat in te gral e. (), a first or er if fer en tial eua tion for ther mal layer is ob taine: an the so lu tion of e. (4) with the ini tial con i tion (17b-1) is: 4a (4) t a 8 4 t (5) The so lu tion of our prob lem is then com pose of the es. () an (5) for the pre - T scribe tem per a ture prob lem where T f ( T ) T. b For the clampe heat flux case, the heat bal ance in te gral e. () is trans forme with the boun ary con i tion (17b-): t 4 x a Sim i larly, the first or er or i nary if fer en tial eua tion for the ther mal layer is: (6) t ( ) 1 a (7) The so lu tion of e. (7) sub jecte to the ini tial con i tion (17b-) an ex clu ing the neg a tive so lu tion is: at (8)

7 THERMAL SCIENCE: Vol. 1 (9), No., pp For the clampe heat flux prob lem, the so lu tion is given by es. () an (8). Since the boun ary sur face tem per a ture is not yet known, e. (8) can not be i rectly use to eval u - ate but we can elim i nate the ther mal layer in the ex pres sion of an have a tran scen en tal eua tion for since a is also a func tion of : a t (9) The evo lu tion of the ther mal prop er ties of superflui he lium with tem per a ture use to solve e. (9) with a sim ple rou tine are taken from the spe cific ata base for he lium [1]. Heat trans fer in a semi-in fi nite me ia with tem per a ture e pen ent prop er ties us ing the Goo man trans form To take ac count of the tem per a ture e pen ency of the ther mal prop er ties of the he lium in the tran sient heat bal ance e. (1), an other trans for ma tion can be use, the Goo man trans for - ma tion []: T Q rc ( T ) T () The sys tem of es. (1)-(1) an (15) are trans forme into: Tb Q Q a t in x an for t (a) Q = Q at x an for t (b -1) a Q o at x an for t (b - ) Q = at x = ( t) an for t (c) Q in x at t () where a is also f(t)/rc p. Fol low ing the same pro ce ure than in the pre vi ous para graph, the heat bal ance in te - gral eua tion is writ ten as: Q Qx a () t As be fore, we use the tem per a ture pro file Q = b (t) + b 1 (t) x + b (t)x + b (t)x an two a i tional boun ary con i tions ien ti cal to the ones use ear lier: an by: p (1) Q at x for t () Q at x for t (4) Us ing the boun ary con i tion (1b s), (1c), (), an (4) the so lu tion for Q is given x Q Q 1 where for the pre scribe tem per a ture case Q T rc p ( T ) T an for the clampe heat flux case b Q ( /a ). Sim i larly to the pre vi ous trans form, if we in tro uce e. (5) into the heat bal - ance e. (), then the ther mal layer for the pre scribe tem per a ture case is: T (5)

8 18 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... an for the pre scribe heat flux: 8 a 4 4 Q t (6) a t (7) For the pre scribe tem per a ture prob lem, the so lu tion of our prob lem is then com pose T of the es. (5) an (6) where Q T rc p T. b For the clampe heat flux prob lem, the so lu tion is given by es. (5) an (7). As be - fore, the boun ary sur face tem per a ture Q is not yet known, there fore e. (7) can not be i rectly use to eval u ate but we can elim i nate the ther mal layer in the ex pres sion of Q an have a tran - scen en tal eua tion: Q a t (8) Fi nally, one have to note that the use of the Goo man trans form oes not lea to any ap prox i ma tion on the con trary of the Kirchhoff trans form. But the vari a tion of the ther mal prop - erty, tak ing in ac count with this metho, is re late to the enthalpy vari a tion whereas the Kirchhoff trans form takes ac count of the ther mal prop er ties re late to the heat trans fer. One ex - pects to have some if fer ence in the com par i son with an a lyt i cal so lu tions or ex per i men tal ata since the ther mal con uc tiv ity func tion of He II var ies by sev eral or er of mag ni tue in the prac - ti cal tem per a ture range of He II (1.6 K;.17 K). Comparison with existing solution an experimental ata Clampe temperature case The HBIM so lu tions, given by es. () an (5) for the Kirchhoff trans form an by the es. (5) an (6) for the Goo man trans form are com pare with the ex act so lu tion e. (9), e vel ope by Dresner in fig. for the tem per a ture range [1.8;. K]: 4 T T z ( rc b p ) x 1 with z T Tb (9) T T 4 4 b 8 f t z Since the Dresner so lu tion oes not take ac count of the tem per a ture e pen ency of the ther mal prop er ties, e. (9) is plot te as an area with the up per limit cor re spon ing to the so lu - tion with the ther mal prop er ties at T b = 1.8 K an the lower limit cor re spon ing to the so lu tion with the ther mal prop er ties at T =. K. In this tem per a ture range, the HBIM so lu tions gives sim i lar re sults since the vari a tion of the ther mal prop er ties are not to im por tant, f(t) var ies by 5 % an a(t) by 5%. The is - crep ancy be tween the HBIM so lu tions an the ex act so lu tion can be as high as 5% for x/t /4 < 1 an i verges at higher x/t /4. This is crep ancy is in trin sic to the HBIM ue to the fact that it only sat is fies the orig i nal par tial if fer en tial eua tion av er age over a fi nite is tance. An since the ther mal layer is un er es ti mate by the HBIM as fig. shows, the is crep ancy is foun for large x or small t. Sev eral tem per a ture pro files an boun ary con i tions has been trie but none of them so far pres ents a better ac cu racy. For small x, the ac cu racy is foun to be within a few per cent an it is em on strate in com par ing the heat flux at the axis or i gin. The heat fluxes at the x-or i gin are e fine, re spec - tively, with the Kirchhoff an Goo man trans forms as:

9 THERMAL SCIENCE: Vol. 1 (9), No., pp an at (4) a 4 Q t At the wall we can con sier small vari a tion of ther mal prop er ties, hence an Q can be re place by av er age value as f /(T T b ) an rc p Q /(T T b ) an e. (4) is sim pli fie to: 1 4 rc f T T 4 p b (41) t which has a ien ti cal form than the heat flux given by the Dresner s so lu tion ex cept that the 1/ / (.87) co ef fi cient in the HBIM so lu tion is 1/4 / 1/ (.9) in Dresner s so lu tion. For a larger tem per a ture range such as [1.8 K;.17 K] as shown in fig., the two HBIM so lu tions i verge es sen tially be cause the large vari a tion of the euiv a lent ther mal con - uc tiv ity of He II is not taken in ac count by the Goo man trans form. The HBIM so lu tion with the Goo man trans form is not even lo cate in the area of pos si ble so lu tions given by the Dresner s eua tion. Ob vi ously, this so lu tion can be only use for small vari a tion of tem per a ture. As ex pecte the HBIM so lu tion with the Kirchhoff trans form, which takes ac count of the vari a - tion of the euiv a lent ther mal con uc tiv ity of He II, gives ac cept able re sults when com pare to the an a lyt i cal pos si ble so lu tions area. This con sti tutes for that prob lem the only an a lyt i cal pre ic tive so lu tion known since the Dresner s moel oes not take ac count of the tem per a ture e pen ency of the ther mal prop er - ties. Fig ure. Com par i son be tween the analytical moel of Dresner [] an the HBIM solutions. The Dresner's so lu tion is pre sente as a area lim ite by the so lu tions with the ther mal prop er ties at T b (up per limit) an T (lower limit). The HBIM so lu tion with the Kirchhoff trans form es. () an (5), is rep re sente with a soli black line an the HBIM so lu tion with the Goo man trans form es. (5) an 6), is rep - re sente with a ashe black line

10 1 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... Clampe heat flux case The clampe heat flux prob lem is the most prac ti cal case en coun tere in cool ing with static superflui he lium. It has been stu ie an a lyt i cally again by Dresner [1] an ex per i men - tally by Van Sciver [11] for ex am ple. An im por tant cri te rion in e sign ing with superflui he lium is to make sure that the tem per a ture of the e vice to cool oes not go over the phase change tem - per a ture, name the lamba tem per a ture T l, be tween superflui he lium an nor mal he lium. This in for ma tion is crit i cal to op er a tion of super con uct ing mag nets, for ex am ple, since the heat trans fer in nor mal he lium is much lower than in superflui he lium. The Dresner s an a lyt i cal so lu tion for the tem per a ture at the heate sie (at x = ) is: T T T T a b l r C f ( T T ) b p l b where a was foun by ien ti fi ca tion with ex per i men tal ata of Van Sciver.The so lu tions given by the HBIM meth os are, re spec tively: a t an (4) t Q a for the Kirchhoff an Goo man trans form. Once more, to com pare the form of these so lu tions with the an a lyt i cal so lu tion, one can take the as sump tion that f /(T T b ) an rc p Q /(T T b ) are av er age val ues an there fore eua tions are sim pli fie to: T Tb t rc f The HBIM so lu tions have the same form than the an a lyt i cal so lu tion. Dresner ien ti - fie the co ef fi cient a to be.8 by fit ting the ex per i men tal ata of Van Sciver whereas the HBIM gives straightfully 1/ (1.15). An other im por tant in for ma tion to give is the time to reach the crit i cal tem per a ture T l. This time is given by re plac ing T by T l, an for the HBIM so lu tion it is: t l rc p ( Tl Tb ) 4 This for mu la tion is also sim i lar to Dresner s for mu la tion with the ex cep tion of the co - ef fi cient /4 which is 1.4 in his moel but it agrees on the ua ric e pen ence on the heat flux with ex per i men tal re sults re porte by Van Sciver [11]. Dresner s co ef fi cients are foun by ien ti fi ca tion with ex per i men tal re sults re porte which means that these co ef fi cients are only vali for the ther mo y namic con i tions of the ex - per i men tal work they were ex tracte from. A com par i son with ex per i men tal ata is en cour ag - ing, when we look at the pro por tional func tion be tween the time t l an 4. The ex per i men tal work of Van Sciver gives a value of 11 W 4 s/cm 8 for bath tem per a ture 1.8 K whereas e. (45) gives a value com prise be tween 5 an 141 W 4 s/cm 8 for tem per a ture be tween 1.8 K an. K. The HBIM so lu tions are com pare with the ex per i men tal ata of Van Sciver on the fig.. Only the so lu tions ob taine with the Kirchhoff trans form is pre sente since the so lu tion with the Goo man trans form oes not fit the ata. The Goo man trans form HBIM so lu tion oes not al low com put ing a so lu tion for time higher than.95 s. This time cor re spons to a ther mal layer 4 p t (4) (44) (45)

11 THERMAL SCIENCE: Vol. 1 (9), No., pp of.6 m, which is the cause of the fail ure of this moel, i. e. the ther mal layer is un er es ti mate. The HBIM so lu tion with the Kirchhoff trans - form is uite ac cu rate; the if fer ence be tween the mea sure ment an the HBIM so lu tion is lower than 5%. But it still un er es ti mates the ther mal layer. For t =.75 s, the ther mal layer is 1.19 m an at 1.5 s the ther mal layer is.91 m. The un er pre ic tion of the ther mal layer is prob a bly ue to the ap prox i ma tion of con stant a, fur ther work coul be con ucte to im prove this part of the moel. Fi nally, one in ter est ing re sult that can be e uce from the HBIM mo - el ing is that the tran sient heat trans fer in He II is more in flu ence by trans - port prop er ties f(t), than in trin sic prop er ties of he lium rc p, for large tem per a ture vari a tion, i. e. from 1.8 to.17 K an for heat flux in the range of kw/m, since the HBIM with the Goo man trans form fails to re pro uce ex per i - men tal ata or ex act so lu tion. This re mark is not vali for lower tem per a ture range or smaller heat flux, as it can be no tice for the clampe tem per a ture case. Con clu sions Figure. Comparison between experimental ata [11] an the HBIM so lu tions with the Kirchhoff trans form, e. () an (8), for the clampe heat flux case. The ex per i men tal ata were ob taine at T b = 1.8 K an =. kw/m. The black suare cor re spon to the mea sure ment at.75 s an the black cir cle for 1.5 s af ter en er giz ing the heater. The HBIM so lu tion (Kirchhoff trans form) is rep re sente with soli black line for t =.75 s an t = 1.5 s The HBIM metho has been ap plie to solve the non-lin ear heat if fu sion eua tion for superflui he lium with tem per a ture e pen ent prop er ties, for the clampe tem per a ture an clampe heat flux. One of the main con tri bu tions of the pres ent work is that the HBIM leas to pre ic tive so lu tions an o not nee any pa ram e ter a just ment to fit ex per i men tal ata. Ac tu ally, to the best of our knowl ege, these so lu tions are the only an a lyt i cal pre ic tive so lu tions. More over, com pare to nu mer i cal tech niues, the pres ent anal y sis is much sim pler an pro vies an a lyt i cal forms that can be han le with any sprea sheet pro gram us ing a he lium prop erty ata base. The ac cu racy ob taine with the HBIM us ing the Kirchhoff trans form pro vies a goo ac cu racy (within few percents) for re pro uc ing ex per i men tal re sults or ex act so lu tion. Nomenclature A Gorter-Mellink coefficient, [mskg 1 ] a i polynomial coefficients of the Kirchhoff solution b i polynomial coefficients of the Gooman solution C p specific heat at constant pressure, [Jkg 1 K 1 ] F ns mutual friction force per unit volume, [Nm ] f(t) thermal conuctivity function in the turbulent regime of He II, [W m 5 K 1 ] p pressure, [Pa] heat flux ensity, [Wm ] s entropy, [Jkg 1 K 1 ]

12 1 Bauouy, B.: Heat Balance Integral Metho for Heat Transfer in... T temperature, [K] t time, [s] v velocity, [ms 1 ] x space, [m] Greek let ters a = f/rc p, [J m s 1 ] a average value of a over the temperature range, [J m s 1 ] thermal layer, [m] Kirchhoff temperature transform, [W m 5 ] Q Gooman temperature transform, [Jm ] m viscosity, [Pa s] r ensity, [kgm ] Subscripts b relate to the bulk temperature or bath temperature relate to the thermal layer s relate to the superflui component n relate to the normal flui component l relate to the lamba transition relate to the space origin (x = ) Su per script vector value mark Ref er ences [1] Dresner, L., Tran sient Heat Trans fer in Superflui He lium, A vances in Cryo genic En gi neer ing 7, Ple - num Press, New York, USA, 1981, pp [] Dresner, L., Tran sient Heat Trans fer in Superflui He lium. Part II, A vances in Cryo gen ics En gi neer ing 9, Ple num Press, New York, USA, 198, pp. - [] Goo man, T. R., Ap pli ca tion of In te gral Meth os to Tran sient Non lin ear Heat Trans fer, A vances in Heat Trans fer 1, Ac a emic press, Lon on, 1964, pp [4] Gorter, C. J., Mellink, J. H., On the Ir re vers ible Pro cess in Li ui He lium II, Physica, XV (1949), May, pp [5] Arp, V., Heat Trans port through He lium II, Cryogenics, 1 (197),, pp [6] Bauouy, B., Ap prox i mate So lu tion for Tran sient Heat Trans fer in Static Tur bu lent He II, A vances in Cryo genic En gi neer ing 45, Ple num Press, New York, USA, 1999, pp [7] Bauouy, B., In te gral Metho for Tran sient He II Heat Trans fer in Semi-In fi nite Do main, A vances in Cryo genic Egineering 47 B, AIP (1), pp [8] Lan au, L., The The ory of Superfluiity of He lium II, Jour nal of Phys ics, V (1941), 1, pp [9] Van Sciver, S. W., He lium Cryo gen ics, Ple num press, New York, USA, 1986 [1] ***, Cryoata, Hepak,.4 e. Flor ence SC, USA 951: Cryoata Inc. [11] Van Sciver, S. W., Tran sient Heat Trans port in He II, Cryogenics, 19 (1979), 7, pp Author's affiliation: B. Bauouy CEA Saclay DSM/Irfu/SACM Gif-sur-Yvette France bertran.bauouy@cea.fr Paper submitte: September 9, 8 Paper revise: September 7, 8 Paper accepte: October 1, 8