APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation in static turulent superflui helium (He II) is propose y mean of integral metho. Although this is an approximate metho, it has proven that it gives solutions with fairly goo accuracy in nonlinear flui ynamics an heat transfer, especially in ounary layer theory. his analytical metho is aequate for this class of equations ecause of its capaility of solving non-linear prolems an it proposes also a simpler alternative metho to numerical calculation. o present the metho an compare its accuracy, a simple case solution is compare with the exact solution an experimental ata. A more general solution, taking account of the temperature epenence of the thermoynamic properties is also propose. INRODUCION Analytical treatment of transient heat transfer in He II has receive not enough attention, consiering the sustantial interest as it relates to the cooling an staility of magnet systems. Dresner using similarity solutions metho has evelope three analytical solutions 1,. hese cases eal with linear ounary conitions an temperature inepenent properties for semi-infinite meia. Several solutions are still of interest of esigner to investigate the cooling performance an staility of magnet systems, such as solutions in a finite meia an temperature epenent properties. An aequate metho in the solution of heat iffusion prolems is the integral metho ecause of its capaility of solving nonlinear prolems where the non-linearity can e foun either in the ifferential equation itself or in the ounary conitions. his metho is analogous to the metho employe to solve thermal an momentum ounary layer in flui mechanics. With exact metho, the resulting solution satisfies locally the system over the entire range of space an time. Such solutions are rather ifficult to otain when the ifferential equation is non-linear or if the ounary conitions involve are non-linear. Integral metho, in the solution of timeepenent ounary-value prolems, gives solutions, which satisfy the ifferential system
only on the average over the region consiere rather than consiering a local solution. It is often sufficient for engineering calculations in which many more approximations are use to moel complex cryogenics systems. SOLUION IN A SEMI-INFINIE MEDIA WIH CONSAN PROPERIES Case for an Clampe Heat Flux In this paper we examine the solution of a system in which the ifferential equation is non-linear ut not the ounary conitions. he simplest case that can e stuie is where the thermoynamic properties are temperature inepenent an the meia is consiere semi-infinite. We etail, to present the metho, the case of a heat flux step where at t a constant heat flux q is applie at the ounary x. It has een alreay proven that the iffusion equation is ale to moel He II transient heat transfer 1,. he main reason is ecause for sufficient heat flux an length (~1 m) the heat transfer is ominate y the enthalpy variation of the He II. For a fully evelope turulent state, the heat flux is given y the Gorter-Mellink law 4, neglecting the issipation effects in He II, the partial ifferential equation moeling our system for one space imension is, t ƒ ρc 1 p x x 1 in x ÃDQGÃIRUÃW! (1) where ρ is the ensity, C p the specific heat at constant pressure an ƒ the He II turulent thermal conuctivity function. he first ounary conition is 1 ƒ q at x an for t>, () x where q is the heat flux at x. At the initial time, the entire meia is at constant temperature, so the initial conition is in x ÃDQGÃDWÃW () As it is a semi-infinite meia, the necessary secon ounary conitions is a constant temperature when x ÃRUÃSUDFWLFDOO\ÃIRUÃODUJHÃ[ÃLHÃWKHÃWHPSHUDWXUHÃILHOGÃLVÃQRWÃGLVWXUEHG for large x. his conitions is expresse y for x ÃDQGÃIRUÃW! (4) We are only intereste y the solution of the isture temperature fiel which is limite y a istance δ(t), calle the thermal layer, after which the temperature fiel is not isture. For a semi-infinite meia, the thermal layer is efine as eing always inferior to the length of the system. From this efinition we can moify the ounary conition Eq. (4), at xδ(t) an for t>. (5) Introucing a set of non-imensional variales as θ, x χ an L τ ρc p L 4 ƒ 1 ( ) t, (6) where is the temperature corresponing to the lama transition, L the length of the omain which is suppose to e thermally semi-infinite in this case, θ, χ an τ respectively the non-imensional temperature, space imension an time, the system is transforme into
τ 1 in χ ÃDQGÃIRUÃτ>, (7-a) 1 φ at χ an for τ>, (7-) θ at χ an for τ>, (7-c) θ in χ ÃDQGÃDWÃτ, (7-) where we efine a non-imensional heat flux φ an thermal layer as φ ƒ q L ( ) an δ. (8) L If Eq. (7-a) is integrate with respect to space over the thermal layer the resulting equation is calle the Heat-Integral Equation. With this integration, terms in space graient can e remove from the energy equation. Following these irections, the energy equation is then transforme into 1 1 χ χ τ. (9) 1 With the use of the ounary conitions (7-) an noticing that in our system null ecause of the efinition of the thermal ounary, Eq. (9) is reuce to is 1 χ. (1) τ When the rule of ifferentiation is use on Eq. (1), the integral on the left han-sie is transforme into 1 χ θ χ θ τ. (11) τ One can notice that ue to the ounary conitions the secon term of the right hansie of Eq. (11) is null which reuces it to a simpler formulation, τ θχ 1 φ. (1) Eq. (1) is the Heat-Integral Equation for the clamp heat flux prolem; it coul e use to treat non-linear ounary conition too. Let assume that the temperature has a polynomial form as θa+χ+cχ +χ where the coefficients a,, c an are function of the thermal layer. Oviously, θ is an approximate solution of the system an to fin the ifferent coefficients, we nee to use ifferent ounary conitions: the natural conitions, which ensues from the prolem, an erive conitions, which are constructe from either the ifferential equation or the natural ounary conitions. For this expression of the solution we nee two extra ounary conitions. he first one we choose is straightforwar an comes from the efinition of the thermal layer,
at χ for τ>. (1) One can notice that this conition has een alreay use to construct the Heat-Integral (XDWLRQÃ7KHÃVHFRQGÃRQHÃFRPHVÃIURPÃWKHÃGLIIHUHQWLDOÃHXDWLRQÃDWÃ ÃZKHUHÃGHULYDWLYHÃRI the temperature with respect to space is null ecause of conition Eq. (7-c). We have, what it is calle a erive conition, θ at χ for τ>. (14) By the use of the natural ounary conitions Eq. (7-), (7-c) an (1) an the erive one, Eq. (14), we can formulate a solution of θ as a function of, φ χ θ 1. (15) By sustituting Eq. (15) into the Heat-Integral Equation, Eq. (1), we otain a first orer orinary ifferential equation for the thermal layer thickness, 1. (16) τ φ he solution of Eq. (16) sujecte to the initial conition Eq. (7-) gives φ τ. (17) he negative solution of Eq. (17), which has no physical meaning in our prolem, has een eliminate. he general solution is compose of Eq. (15) an Eq. (17). he time constant of the system is φ /1 t, when transforme into imensional variales it is consistent with a imensional analysis of Eq. (1), giving a time constant of ρc p L q /1ƒ. Comparison with Existing Solution an Experimental Data If we compare the solution of Eq. (15) at χ which is given y θ an can e expresse as a function of the imensional variales as φ ρc p q ƒ( τ, (18) ) t, (19) one can notice that this formulation is similar to Dresner s moel with the exception of the coefficient (1.15), which is.8 in his solution 1. We can formulate also the time where the temperature of the helium reaches the lama temperature at the ounary. Accoring to this, θ 1 an the time t is efine y ƒρcp ( ) t. () 4 q his formulation is also similar to Dresner s formulation with the exception of the coefficient /4 which is 1.4 in his moel ut it agrees on the quartic epenence on the 4
heat flux with experimental results reporte y Van Sciver 5. Dresner s coefficients are foun y ientification with experimental results reporte y Van Sciver which means that these coefficients are only vali for the thermoynamic conitions of Van Sciver s experiment. A comparison with experimental ata is encouraging, when we look at the proportional function etween the time t an q 4. he experimental results of Van Sciver give a value of 11 W 4 s cm -8 for 1.8 K whereas Eq. () gives a value comprise etween 5 an 141 W 4 s cm -8 for a ath temperature comprises etween 1.8 K an. K. he approximate solution Eq. (15) is plotte on Figure 1 with the ata otaine y Van Sciver. As we are assuming that the thermoynamic properties are constant, we nee to efine an average temperature to evaluate ρ, C p an ƒ. For this plot, the est match have een foun to e for an average temperature of 1.99 K. here is a goo agreement for small temperature variation an small x whatever the time ut for large x the solution reaches a null temperature variation too soon. It is the limitation of the moel an in fact the null temperature variation space location is a function of time an correspons to the thermal layer. he moel unerestimates the length of the thermal layer which comes from the profile of the approximate solution an also the associate ounary conitions taken to calculate the solution. Other profile an ounary conitions have een investigate in the following paragraph. Other Solutions For heat conuction prolems, it has een shown that taking a polynomial form with a egree higher than three oes not improve necessarily the accuracy of the solution. he reason is that for each of the polynomial coefficient, which are time epenent i.e. function of the thermal layer, a ounary conition has to e provie. For four coefficients, we have to provie two supplementary conitions, one natural an another erive, whereas for a fourth egree polynomial, another erive ounary conition has to e use for the fifth coefficients an the choice of the extra conition can reuce the accuracy..15.1 1.8 K q o. 1 4 W/m Van Sciver s ata Eq. (15) Eq. (4) Eq. () /t 1/ (ms -1/ ).9.6...4.6.8 1 1. x/t 1/ (ms -1/ ) Figure 1. Comparison etween ifferent solutions an Van Sciver s experiment 5.
We solve the same prolem with a quaric polynomial form. In orer to o so, we have to use an extra ounary conition which is, θ at χ for τ>, (1) which comes from the erivation of the ounary conition at χ Eq. (7-). he solution for that case is φ θ 1 χ + χ χ 4 with 5 τ. () φ Another solution can e calculate with a cuic polynomial form to illustrate the effect of the ounary conition on the solution. We use a ifferent ounary conition then Eq. (14) erive from the ounary conition Eq. (7-), that is to say, he solution foun is θ φ 1 θ at χ for τ>. () χ 1 + χ with τ φ. (4) hese two solutions are also plotte on Figure 1 an it is interesting to note that the solutions o not iffer y a lot even if we can note that the quartic polynomial form is less accurate that the others. Solutions given y Eq. () has a lower accuracy than solution given y Eq. (15). Accuracy of these solutions epens on the ounary conitions an the profile of the approximate solution an is har to preict unless y comparison with the exact solutions. SOLUION IN A SEMI-INFINIE MEDIA WIH EMPERAURE DEPENDEN PROPERIES he system to solve is similar to the one efine y the system of Eq. (1), Eq. (), Eq. () an Eq. (5), such as the ifferential equation, 1 ρ C p ƒ in x ÃDQGÃIRUÃW! (5) t x x where in this case we consier the thermoynamic properties ρ, C p an ƒ temperature epenent. he ounary conitions are ientical to the previous case. By applying the Kirchhoff transformation Θ ƒ ( ), (6) the system Eq. (5), Eq. (), Eq. () an Eq. (5) is transforme into, 1 Θ α t Θ x x 1 in x ÃDQGÃIRUÃW! (7-a)
1 Θ q at x an for t>, (7-) x Θ in x ÃDQGÃIRUÃW (7-c) Θ at xδ(t) an for t>, (7-) where αƒ/ρc p. One can remark that αα(θ). he Heat-Integral Equation is now written t δ Θ x αq, (8) where in a first approximation we consier α(θ) constant. If we use the same polynomial form Θa+x+cx +x for the expression of the temperature, the solution is expresse as δ x Θ q 1, (9) δ when the following natural an erive ounary conitions use are, Θ x δ Θ an. () x One can oserve that the solution is again a function of the ounary conition at x, which is efine as Θ q δ/. he expression of δ is otaine y inserting Eq. (9) in Eq. (8) an using Eq. (7-) is δ δ αt. (1) q Since the ounary surface temperature Θ is not yet know, Eq. (1) cannot e irectly use to evaluate δ ut we can eliminate the thermal layer in the expression of Θ an have a transcenental equation for Θ when α is given as a function of Θ, Θ α( Θ ) q t. () It is possile to have the expression of α ƒ /ρ C p as a function of Θ y using analytical expressions of the He II turulent thermal conuctivity function an the specific heat as a function of the temperature 6, 4 ρ s ƒ A 4 5.7 1 5.7 an C p K 5.6. (),(4) where A ÃPVNJÃDQGÃ. Ã-NJ. 6.6. he solution is easy to fin when the ounary temperature ifference (at x) is set, we can evaluate α an Θ an calculate the time neee to reach this temperature ifference. As the solution is efine, y evaluating the thermal layer efine from the initial temperature, it is also easy to fin the value of the temperature for location ifferent than the ounary, i.e. x ÃZLWK 1 q Θ x Θ Θ (5)
.15.1 1.8 K q o. 1 4 W/m Van Sciver s Eq. (15) Eq. () /t 1/ (ms -1/ ).9.6...4.6.8 1 1. x/t 1/ (ms -1/ ) Figure. Comparison etween Eq. (15), Eq. () an Van Sciver s experiment 5. he solution is plotte on Figure an compare with Eq. (15) an ata otaine y Van Sciver. Not only the solution is more accurate than the solution given y Eq. (15), this solution gives a irect result, even with the constant α approximation, without the nee to evaluate an average temperature to fin the est fit of the experimental ata. For the same reason than the others solutions, this one preicts with less accuracy the evolution of the temperature for large x. It comes from the profile of the approximate solution which gives a thermal layer shorter, as a function of time, than the experimental ata s one. CONCLUSION Integral metho are suite to solve the non-linear heat iffusion equation for superflui helium with acceptale accuracy, ut further work is neee to improve it, especially in the choice of the approximate solution profile an the associate ounary conitions. aking account of the temperature epenence of the thermoynamics properties gives etter accuracy an a irect result without the nee of the evaluation of an average temperature to efine the thermoynamic properties. Further work shoul involve other ounary conitions such as clampe temperature or pulse-source prolem. REFERENCES 1. L. Dresner, ransient heat transfer in superflui helium, Av. Cryo. Eng. 7:411-419 (198).. L. Dresner, ransient heat transfer in superflui helium. Part II, Av. Cryo. Eng. 9:- (1984).. H. Schlichting. "Bounary-Layer heory", 7th E. McGraw-Hill Inc., New-York (1976). 4. C. J. Gorter an J. H. Mellink, On the irreversile process in liqui helium II, Physica. XV:851 (1949). 5. S. W. Van Sciver, ransient heat transport in He II, Cryogenics. 19:85-9 (1979). 6. S. W. Van Sciver. Helium Cryogenics, E. Plenum Press, New York (1986).