Limit of transient heat absorption by superfluid helium for very large heat pulses

Limit of transient heat absorption by superfluid helium for very large heat pulses B.A. Danilchenko*, M.O. Lutset** and V.N. Poroshin* *Institute of Physics of the Ukraine, SSR Academy of Sciences, Kiev 252650, Prospect Nauki 144, USSR **Institute of Thermophysics of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk 630090, Prospect Lavrentyeva 1, USSR

Received 9 December 1987; revised 13 July 1988 Large heat pulse transfer from film heater into helium at bath temperature 1.7-3 K is considered. Experiments were carried out at vaporization onset and at the film heater temperature in He I and He II. The presence of nonequilibrium layer accounts for no difference in the mechanisms of heating up to the vaporization onset of He I and He II in the measurements. The critical processes localized near the heater are noted to be developed after the vaporization onset of helium.

Keywords: superfluid; helium; transient; transfer

The propagation of heat pulses in superfluid helium has been investigated in numerous works 1-13. It has been revealed 5'6 that the pulses of amplitude W and duration At propagate in accordance with Khalatnikov's theory 14, if W< Wet(At)with Wcr depending on the pulse duration At. With small frequencies of the pulse sequence (smaller than inverse times of the heater cooling down, which depend on W and At) no frequency influence has been observed at W < Wcr6'13. When W is in excess of W,, the waves of the first sound occur 6 which are associated with the onset of helium vaporization, and the pulse shape begins to change according to its sequential number 13, i.e., the helium layers adjacent to the heater acquire memory and the deviation from the theory is recorded 14'15. Different hypotheses have been put forward with respect to the nature of Wet. In Reference 7 W¢, is related to the attainment of fundamental critical velocity for relative motion of normal and superfluid components; in Reference 16 the reason is the generation of quantum vortices by the front of shock wave of the second sound; in Reference 17 Wet is associated with the superheating of the helium layers adjacent to the heater, its vaporization onset and subsequent turbulization. What is the quantity of attainable superheating for superfluid helium? The answer to this question can be found in Reference 18. For the temperature range 1.5 KTa in the isobaric process of heating the obtained superheating limit of a liquid is in good agreement with the theory of homogeneous nucleation (see Figure 1). On the basis of these data, the temperature of vaporization onset for the helium adjacent to the heater in the process of rather powerful heat pulse generation can be considered to be equal to the temperature of limiting superheating obtained in Reference 18. The reason for it is a slight time of superheated state existence under the pulse heating, which defines the fluctuating character of vaporization onset. The present paper reports the results concerning the 0011-2275/89/040444-04 $03.00

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measurements of heating surface temperature at the moment of vaporization onset which we compare with the helium temperature, and obtain the estimate of 'heater-helium' thermal resistance at this moment. We

He 1 /

~-J==~

X-line

102

2 N

"IE E

!o

He II

2 ._

I"

J

101

1 i 1

2

3 To(K)

4

5

Figure 1 Phase diagram of helium 18. Measurements of Tm are by the dashed line, The intervals of uncertainty for Tm determination are also plotted

shown

Transient heat absorption by superfluid helium: B.A. Danilchenko et al. obtain information on the mechanism of He II superheating, i.e., of Wcr nature, proceeding from the measurements of the vaporization onset time versus the bath temperature.

Experiment Copper film (of 9 mm 2 area and 1500 ~ thickness) was deposited on the polished surface of sapphire C-cut. The film was heated by current pulses of duration less than 1 0 - 6 S and power 1-100W cm -2. The energy radiated by the heater H to the substrate propagates in it as ballistic phonon pulses and is recorded by superconducting indium bolometer 1 (see Figure 2) located at the crystal side opposite with respect to the heater. The crystal is retained against the lower flange of the measuring cell so that the heater can be in contact with vacuum (10 -4 mm Hg) or with the liquid helium. The heater temperature TN and the quantity of the heat flux QHc to helium was determined as follows. First, the amplitude of L and T phonons detected by bolometer 1 versus electrical power in the heater at its contact with vacuum was recorded. The heater temperature was calculated by the formula 19' 17 with an error not more than 15 %. W = c~(T~ - T~), ct = n2~k4/40 h3s 2

(1)

where Z,=0.126 is the mean coefficient of the phonon transmission through the 'copper-sapphire' interface, = 2.54 x l05 cm s - 1 is the mean sound velocity in the heater and T O is the temperature of substrate (bath). Then the analogous dependence at 'heater-liquid helium' contact was recorded. The same amplitude of bolometer 1 signals in contact with vacuum and helium occurs at the same temperatures of the heater, and the difference between the electric powers of both cases defines the heat flux to helium. The signals propagating in helium are recorded by bolometer 2 with a 3 mm 2 cross-section which is located at a distance of 1.7 mm from the heater. During the pulse duration of less than 1 0 - r s the perturbed region in helium did not exceed 5 x l0 -2 mm. That was considerably less than the distance between bolometer and heater. The reflected waves had time to attenuate during the time interval between the pulses.

Hel

I

2

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t8

Figure 2

Measuringscheme and typical bolometer 1 signal

The moment t a of vaporization onset was registered by two methods, i.e., by sharp increase of L, T pulse amplitude (see Figure 2) recorded by bolometer 1 and by the appearance of the first sound signals recorded by bolometer 2. Two methods yield data which are in good agreement.

Results and discussion Measurements of t B versus W and To are well described by the relation

(tB)½W = A(To)

(2)

The coefficient A as a function of temperature is shown in Figure 3 (left scale). Here the maximum temperatures TIn(To) of the helium superheating obtained in measurements is in the region T o < Tz and 2° in the region To > Tz (right scale) are also plotted. For comparison, the heater temperature TB at the moment tB obtained in our experiments is presented here. A surprising fact is the coincidence of TB and T= within the ranges of measurement accuracy. The extrapolation of Kapitza's resistance measurements at small heat fluxes to the region of large powers appears to be, by our estimation, of the order of tens of degrees Kelvin for realized heat fluxes. The coincidence between the temperature of the heat generated surface and that of helium adjacent to it testifies to the absence of acoustic resistance and the fulfilment of classical boundary condition with no temperature jump for large heat fluxes. The result obtained is consistent with the investigations by Kinder and co-authors 21. By the Kinder model 22, with the temperature increase the surface defects are excited, which absorb the energy of solid phonons and reradiate it predominantly into a liquid. An additional channel of energy transport to superfluid helium appears, which is generally called a direct phonon transmission. It is natural that this mechanism of heat transfer should approach the temperature of solid surfaces with that of helium. A strong nonequilibrium state of the adjacent helium layer (similar to Knudsen's nonequilibrium state in the boundary layer of rarified gas) appears, which relaxes at some distance from the heater sufficient for thermolization of phonons emitted from the heating surface. A similar but weak nonequilibrium state was recorded in the experiment 23 where the temperature profile was measured across the Knudsen layer in H e l I . As one would expect, the temperature profile varied along exponential curve with the linear scale of the order of average length for a free run of elementary excitation. At present the structure of highly nonequilibrium Knudsen layer is unknown for He II. It should be defined by the kinetics of highly nonequilibrium phonons, which is unknown to the authors. However, the following statement seems to be evident. The flow of nonequilibrium phonons which defines the heat flux from the heater should transport its energy to liquid, thus being converted in the second sound wave with thermalization. The temperature (in kinetic sense) in nonequilibrium layer varies from Tm to To, i.e., the difference Tin-To amounts to about 2 K. Note that in the wave of the second sound (when the normal component is close to an equilibrium) the temperature amplitude does not exceed 0.l K even at large heat fluxes up to 100 W cm -2. Hence, the transfer

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445

Transient heat absorption by superfluid helium." B.A. Danilchenko et al, 10 -1 10-3

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Figure 3 Measurements and calculation of A and Tm. Coefficient A (left ordinate): O, our measurements; z~, measurementsr; 4, 5, calculation by Equation (4) in To > Tx and TO< Ta. Superheating temperature (right ordinate): 0 , our measurements of Ts - To; 6, measurements of Tm - Tols'2°.

10-7 i 10

102

W(W cm -2)

mechanism in a nonequilibrium layer differs basically from that of an equilibrium one. The information on heat transfer near the surface is incorporated in the empirical relation (Equation (2)). In the region To > Ta for tB < 10-2s (in our experiments tB < 10 -6 s) the only mechanism of heat transfer is the heat conductivity, and the vaporization onset occurs when helium attains temperature Tm at which the probability of fluctuating appearance of critical vapour nucleation at the interval t B becomes close to unity. As TB = Tm within the ranges of measurement accuracy, Kapitza's jump may be neglected. It follows from our measurements x9 that the heat flux to helium can be considered to be constant up to the moment of vaporization onset and is equal to QH= = 7 Wwhere

7 = 0.77 7 = 0.82

for TO> T~ for To < T~

(3)

Thus, to calculate tn at T O> Tx it is sufficient to solve the equation of heat conductivity for half-space with constant heat flux set at the boundary and initial temperature equal to T o. The moment when the helium temperature at the boundary attains the value Tm is assumed to be t B. There is a fundamental difficulty in solving this problem. Isobaric heat capacity cp is reduced to infinity on spinodal T,v and therefore in a small neighbourhood T,v depends highly on T,p-Tin, the value of which cannot be measured accurately at present for helium. In this situation, for the lower estimate of ta we shall use the solution of heat conductivity equation with constant coefficients. According to the choice of the lower estimate of A, Tm is taken as a characteristic temperature at which 2, Cp, p are calculated*. In this case the solution is in the form of

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Figure 4 Comparison of vapour formation onset and quantum turbulence appearance, ts(W): 1, our measurements; O, Reference 24; 0 , A2t(W) 6; A , At(W)S; A, A2t(W)13; 6, A3t(W)13, Equation (2) where

A( To) = ~ - (Tin- To)(2Cpp)½/y

(4)

The calculated values of A are smaller than the measured ones by about a factor 1/1.76. If the values 1.76 A are plotted in comparison to the measured values of A, good agreement within the accuracy of measurements can be found, as shown in Figure 3. The extrapolation (Equation (4)) to the region of superfluid helium agrees with our measurements and experiment 24 with the same accuracy. Hence, one can state that within measurement accuracy no difference is observed in the mechanisms of heating up the layers of He I and He II adjacent to the heater at pulse heat generation, though the part of heat flux to liquid from the pulse (Equation (3)) generated in superfluid helium turns out to be greater 19. The evaluation of the superheated layer thickness at t B can be obtained by the penetration depth of thermal wave equal to (2tB/cpp)~ which at T o = 1 . 8 K and W = 3 0 W cm - z amounts to 10 - s c m or approximately 20 lengths of free run of thermal phonons in helium. The fact that the predominant transfer mechanism in superfluid helium near the heater is the heat diffusion (heat conductivity) seems to be strange. However, as is known from solid physics xT, the process of phonon *The example of helium temperature variation before vaporization onset is shown in Figure 1

Transient heat absorption by superfluid helium: B.A. Danilchenko et al. thermalization under collision regime is well described by the e q u a t i o n of diffusion. In o u r o p i n i o n , the heat c o n d u c t i v i t y b e c o m e s the d e t e r m i n i n g m e c h a n i s m of heat transfer due to s t r o n g d e v i a t i o n from the e q u i l i b r i u m of n o r m a l c o m p o n e n t a n d the d e g e n e r a t i o n of the transfer m e c h a n i s m by the s e c o n d sound. The p r o b l e m o f h e a t c o n d u c t i v i t y d u e to the temp e r a t u r e g r a d i e n t related to a q u a n t u m turbulence has been discussed p r e v i o u s l y (see References 25, 26). W e discuss the s a m e p r o b l e m due to the t e m p e r a t u r e g r a d i e n t of n o n - e q u i l i b r i u m K n u d s e n layer.

Sequence of critical process evolution In the a b o v e e x p e r i m e n t s we used the densities of heat fluxes (more t h a n 30 W c m - z ) w h i c h u n d e r s t e a d y state c o n d i t i o n s c o r r e s p o n d e d to a film b o i l i n g of superfluid helium. The q u e s t i o n arises as to w h a t processes acc o m p a n y these large b u t s h o r t - t i m e heat p e r t u r b a t i o n s . First, when A t = tB(W) we o b s e r v e d the v a p o r i z a t i o n onset o f helium. Second, p r e v i o u s l y 5 the d e v i a t i o n s of the p r o p a g a t i o n velocity of h e a t pulses from K h a l a t n i k o v ' s t h e o r y were o b s e r v e d if t h e i r d u r a t i o n exceeded a certain value of Alt(W). T h i r d , w h e n At exceeded A2t(W ) then the s y m p t o m s o f q u a n t u m t u r b u l e n c e occurrence in the bulk of helium were r e c o r d e d 6'13 F o u r t h , the time interval A3t(W) necessary to a c t i v a t e the v a p o u r film was r e c o r d e d 13. All the m e a s u r e m e n t s of time intervals A~t, Azt , A3t, t a versus the pulse p o w e r are given in Figure 4. The c o m p a r i s o n shows t h a t for a n y value of W investig a t e d by the a u t h o r s the c o n d i t i o n At > tB(W) results in the e v o l u t i o n of all critical processes, which were discussed at the b e g i n n i n g of this paper.

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