Second sound wave heat transfer, thermal boundary layer formation and boiling: highly transient heat transport phenomena in He II

Cryogenics 35 (1995) 645451 0 1995 Elsevier Science Limited Printed in Great Britain. All rights resewed OfIll-2275/95/$10.00

Second sound wave heat transfer, thermal boundary layer formation and boiling: highly transient heat transport phenomena in He II T. Shimazaki,

M. Murakami

Institute of Engineering Mechanics, Tsukuba, lbaraki 305, Japan Received

16 February

and T. lida University

of Tsukuba, Tennoudai

l-l-l,

1995

Highly transient heat transport phenomena in He II are experimentally investigated. High sensitivity temperature measurements are carried out by means of a superconductive temperature sensor. It is found that the waveform of a thermal pulse in the second sound wave mode loses its dependence on heating time in the case of strong heating. Only a limited, small amount of thermal energy is transported in this mode. In such a situtation, dense quantized vortices are generated and accumulate in the vicinity of the heating surface to form a thermal boundary layer which has a large temperature gradient. The time variation of the temperature distribution in the thermal boundary layer is measured. Boiling in the layer is usually observed in the case of strong heating and the heat transport phenomena seem to be governed by boiling. The rest of the energy which is not transported in the second sound wave mode is slowly transported in a diffusive process or is consumed upon phase transition of the He II. Keywords: strong pulsative

heating; second sound wave; thermal

It is well known that superhuid helium (He II) exhibits several unique properties, such as an extremely high apparent thermal conductivity and the ability to flow through even extremely fine channels without any appreciable pressure drop. He II is expected to be an efficient coolant for superconducting magnets and space-borne infrared telescopes. An in-depth understanding not only of steady He II heat transport but also of transient heat transfer is needed for further practical application of He II. For instance, an understanding of transient heat transport is vital for predicting the thermal behaviour of He II in the case of the quench of a He II cooled superconducting magnet. In this case a very high heat flux is applied for quite a short time and even boiling may be observed in the vicinity of the heating surface. Heat transport phenomena in He II have been investigated by many researchers using various approaches. The two-fluid equations’ can give both qualitatively and quantitatively satisfactory explanations of steady heat transport phenomena in He II where superfluidity is not the breakdown state. In the superfluid breakdown state, the Gorter-Mellink’ mutual friction formula is usually introduced to the two-fluid equation system to take account of the effects of quantized vortices. However, the validity of the formula is found to be limited to steady or very slow

boundary

layer

thermo-fluid dynamic phenomena. Further investigations are still needed for highly transient heat transport phenomena. When the time-scale of the bulk thermo-fluid dynamic behaviour of He II becomes comparable to or shorter than that of the development of quantized vortices, heat transport phenomena must be treated by taking account of the effects of the evolution and decay of quantized vortices, which were formulated by Vinen3-5 and Schwarz6. Heat transport in the second sound wave mode with a propagation speed of ~20 m s-’ may be a typical example of highly transient heat transport phenomena in He II. In such cases the thermal pulse develops into a thermal shock wave. Pioneering work on thermal shock waves and their interaction with quantized vortices has been carried out by Liepmann and Laguna7. Second sound waves can also be used as powerful tools for diagnosis of the state of quantized vortices in He II. For a sufficiently small heat flux, a second sound wave is generated whose waveform is similar to the time variation of the initial heat input. However, when the heat flux exceeds a critical value, the second sound wave is strongly affected by interaction with the dense quantized vortices generated as a result of the superfluid breakdown, and develops into a thermal shock wave. The effects of interaction with quantized vortices appear in both waveform deformation and decay during propagation*. Recently,

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et al.

extensive research has been carried out at the Max-PlanckInstitut9 and the Institute of Thermophysics”*“. At the Max-Planck-Institut, experimental and numerical investigations on thermal shock waves and the tangled mass of quantized vortices have been carried out successfully. The results of these investigations clearly indicate the significance of the effects of quantized vortices. However, some aspects of highly transient heat transport in He II still remain open to question. The multidimensional features and very slow decay of high density quantized vortices, and the formation of a thermal boundary layer adjacent to the heating surface as a result accumulation of vortices are some such areas open to question. In the present investigation, high resolution measurements in both time and space are carried out by means of a superconductive temperature sensor. Particular emphasis is placed on heat transport in the second sound wave mode, the influence of thermal boundary layer formation and the occurrence of boiling in response to a highly transient heat input with a very large peak heat flux.

Experimental

set-up

Figure 1 shows the main assembly of the experimental apparatus immersed in a He II Dewar. It consists of three essential parts, a thin film heater, a cylindrical shock tube and a superconductive temperature sensor. The thin film heater, 27 mm x 27 mm, fixed at the bottom of the shock tube, is composed of a Ni/Cr thin film vacuum deposited on to a Pyrex glass plate. The typical Ni/Cr film thickness is 400 A* and the resistance 30 R. A thermal shock wave is generated by Joule heating in the heater. A trapezoidally varying current pulse from a programmable arbitrary waveform generator is fed to the heater via a high speed power amplifier. The rising and falling times of the heating pulse are arranged so as both to be 6 /.Ls.The shock tube is made of a cylindrical Teflon tube, having a height of 150 mm and an inner diameter of 25 mm. The tube prevents the thermal shock wave from diffracting and maintains the onedimensional character of a plane wave due to its adiabatic walls. The end of the tube in contact with the thin film heater is shaped into a knife edge to maintain a tight seal with the heater surface and also to minimize the heat capacity of the contacting portion. The propagating thermal

_--sensor /

,

(1) where p and cp are the density and the heat capacity of He II, C,, is the propagation speed of second sound, and qP is the peak value of applied heat flux. The calibration coefficient (Yis given as follows (Y= AVIAT,,,

(2)

where AV is the detected voltage increment with the superconductive temperature sensor for a temperature rise. The sensitivity attained is as large as 100 pV per mK, although it depends on the bath temperature and applied bias current.

stem

metal needles

_,shock

shock wave is measured with a superconductive temperature sensor inserted via the upper opening of the shock tube. Figure 2 shows a schematic drawing of the superconductive temperature sensor. This type of superconductive sensor for use in He II experiment is quite similar to the original design developed at the Max-Planck-Institut’2~‘3. The fine wire configuration of the sensor element, which resembles a conventional hot wire probe for flow velocity measurement in aerodynamic experiments, is selected to minimize undesirable reflection of thermal shock waves. The sensing element is a superconductive thin film consisting of 200 A of gold and 1000 8, of tin, which are vacuum deposited on to the outer surface of a fine quartz fibre 2 mm long and 40 pm in diameter. The temperature variation is measured through the abrupt resistance change of the sensing element due to a superconducting to normal state transition. The transition temperature can be shifted to the superfluid helium temperature range by carefully choosing the ratio of film thicknesses of gold and tin. Fine adjustment of the working temperature is achieved by varying the applied bias current to the sensor. The response time of the sensor is found to be very small, at most several microseconds, due to the small heat capacity of the sensing element. The sensors are calibrated dynamically in the following manner. A weak (
tube

sensing element

I Ni/Cr thin film _ heater Amp. Figure 1 Main assembly of experimental in He II Dewar

apparatus

immersed

1OflYfl Figure 2 sensor

* 1 A = lO-‘O m

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Schematic

drawing

of superconductive

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Transient

The measurements are started after the He II in the bath reaches a specified equilibrium state at a bath temperature TB. A thermal pulse is generated by pulsed heating from the heater. The detected signal is amplified 100 times with the aid of a low noise preamplifier and is then ‘captured’ with a digital ‘storagescope’. All signals are transmitted to a personal computer to be stored on a floppy disk. The stored experimental data are further analysed with a personal computer. The sensor can be moved parallel to the shock tube axis in the shock tube, and can be fixed at any distance from the heater from 0.5 to 150 mm with a 0.1 mm resolution. Each successive measurement is performed after a rest time tR of 120 s to minimize the uncertainty resulting from the initial vortex line density in the He II field. The strong dependence of the observed waveforms on the initial vortex line density has been emphasized in a number of reports”,‘5.

Results and discussion Non-linear

waveform

deformation

A second sound wave is free from the effect of quantized vortices, as long as the applied heat flux is below a certain level and the heating time is short, so that generation of quantized vortices is negligibly small. Under this condition, a second sound wave with a finite amplitude evolves nonlinearly into a thermal shock wave exhibiting a discontinuity in temperature. The waveform then deforms according to the non-linear features of thermo-fluid dynamics. One of the interesting features of thermal shock waves is the variation in the location of discontinuity in the waveform. The local propagation speed at a point with temperature amplitude AT within a second sound waveform is given, up to the second order of smallness, by

where subscript 0 denotes a quantity in an equilibrium state and AT is the local temperature amplitude within a second sound waveform. The sign of the coefficient b, which is a thermodynamic function, specifies the location of a discontunity, according to the form b = T :T In

The coefficient b can be both positive and negative, depending on the temperature. In the temperature range where b is positive, points in the peak region within a waveform travel faster than those in the trough region, so that the peak region tends to catch up with the trough region. Consequently, a temperature discontinuity is formed at the front of the wave. This type of shock wave, termed a frontal shock wave, is quite familiar in classical fluids. On the other hand, in the temperature range where b is negative, the trough travels faster than the peak and hence a discontinuity is formed at the back of the wave. This is termed a back shock wave, the counterpart of which is not seen in classical fluids in the normal state. If the temperature is just below 1.884 K where b vanishes, a sufficiently strong heat pulse will develop into a double shock wave in which discontinuities appear both at the front and back of a wave.

heat transport

et al.

in He II: T. Shimazaki

During propagation, a waveform is further deformed according to the non-linear features as follows. A trapezoidal waveform turns into a right-angled triangular form as a result of the formation of a shock wave. This, in turn, reduces the wave height and increases the length of the wave (being subject to the equal area rule, which is equivalent to the energy conservation law). A detailed discussion on this subject is found in reference 16.

Waveform deformation due to interaction with quantized vortices Waveforms generated under various initial conditions for a heat flux qP and heating time tn are presented in Figure 3. These forms are measured with a temperture sensor fixed at a distance z = 30 mm from the heater. ‘Ihis distance is selected so as to be long enough to form a shock front but not long enough to make other non-linear features noticeable. The form generated by a small heat flux is free from the influence of interactions with vortices (forms 1 and 2), as mentioned in the last section. The wave height and length are in proportion to those of an initial trapezoidal heating pulse. The deformation of a waveform becomes noticeable when the heat flux or the heating time becomes sufficiently large or long. In the case of waveform 3, heating time rH is rather long (1000 /.s), though the heat flux is not so large. It is seen that the wave height at the front is almost equal to the theoretical prediction [see equation ( 1)], but gradually diminishes towards the trailing edge, and that the length becomes longer than the initial pulse, and moreover a diffusive tail is formed almost continuously following the main body of a propagating thermal pulse. Long heating times allow vortices to develop with sufficient density to diminish the wave height in the rear portion, even though the initial vortex development is slow because qP is not so large. Quantized vortices at a location far from the heater do not start to develop until a pulse front reaches that point. They start to develop on the arrival of a sufficiently large heat pulse. This is superfluid breakdown. When the vortex density becomes high enough, wave deformation resulting from interaction with the vortices becomes apparent. It takes a finite time for vortices to become dense enough to interact appreciably with the flow field. The state in which a flow field is filled with quantized vortices is called the superfluid turbulent state. The characteristic time of the vortex development and the final density

50 @ 40 g

30

20 % ‘” 10

EO rS Time (0.2ms/div) Figure 3

Superposed waveforms measured with superconducsensor fixed at a distance of z= 30 mm under various initial conditions (as indicated in the figure). Ts = 1.70 K

tive temperature

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are functions of the initial vortex density and the strength of the thermal pulse. Similar deformation is observed in the case of large heat flux (form 4). The interaction with vortices results in a reduction of the wave height in a more significant manner, even very close to the front. From numerous experimental results of this series of measurements an order estimation of the characteristic time for vortex development can be drawn at bath temperature of TB= 1.70 K. This ranges from several milliseconds for qP= 5 W cm-* to less than 200 /LS for qp of more than 30 W cm-*.

Limiting

profile

A further increase in the heat flux leads to further deformation of the waveform in the initial stage. Quantized vortex development occurs to a very high density in quite a short time, as seen in waveforms 5 and 6 in Figure 3. As a result of the quite rapid development of vortices, the waveform is subject to interaction from the wavefront, and thus the whole waveform becomes markedly shorter than that of the initial heating pulse. Another important experimental discovery from these waveforms is that, for very large qy the forms almost coincide, with a unique single profile, irrespective of heating time. This tendency was also observed in another experiment”. The unique profile is referred to as the limiting profile. The reduction to a limiting profile comes about upon passage through a layer with dense quantized vortices formed very quickly adjacent to the heater. This region may be considered to be a thermal boundary layer characterized by a large temperature gradient through which heat transport is subject to strong dissipation. The thermal energy generated by the heater is mostly consumed upon boiling and also strongly diffuses in the thermal boundary layer. Figure 4 shows typical examples of limiting profiles at the two temperatures T, of 1.70 K (frontal shock wave region) and 1.90 K (back shock wave region), respectively. There are appreciable differences in the propagation speed at these two temperatures. A slight difference in the profiles can be attributed to the location of discontinuity, i.e. the front or back of a profile. The general features of the limiting profile can be summarized as follows: the limiting profile is of nearly triangular shape and the half-value width 50

t’ 1(ms)

Time (0.2msIdiv)

Figure4 Two examples of limiting profile at TB= 1.70 and 1.90 K.The transient data records are drawn from 1 ms after the heater is switched on, i.e. &, = 1 ms. The difference in propagating speed of the second sound wave, depending on TB, is seen from the time difference in the figure and the position of each waveform. 4 = 40 W cm-*, tH = 1000 ps, z= 30 mm

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is =50-100 ps (i.e. the width is -1-2 mm), almost irrespective of tH. The limiting profile is capable of transferring only a limited amount of thermal energy, imposed only in the initial stage of heating when the vortices have not yet reached their fully developed value. Consequently the features of the limiting profile are determined by the very first pulse of heating, and the profile is not affected further by subsequent heating. These experimental facts suggest that, typically, a limiting profile is formed for heat fluxes over 30 W cm-* and for a heating duration longer than 200 ps. It is also noted that the limiting profile is accompanied by a very long tail with a slightly higher temperature, because a strong thermal pulse induces dense vortex lines which decay very slowly after the passage of a pulse.

Transient energy transport wave mode

in second sound

It is seen from Figure 4 that the amount of thermal energy transported by the second sound wave mode becomes less than the total amount of heat generated by the heater in the case of very large qP: The amount of energy passing through per unit area 1s calculated from the area of the waveform of a thermal pulse obtained experimentally as follows ‘w

Ew

=

C2Opcp

I

AT( t)dt

0

(5)

The integration is carried out from the arrival of a pulse to t,. The pulse duration t, is defined as double the half-value width of a measured waveform. The data are shown in Figures 5a and b at the bath temperatures TB of 1.70 K and 1.95 K respectively, where the abscissa and the ordinate represent the applied heat flux qP and the amount of energy transported by the second sound wave mode, respectively. The solid lines represents the total amount of energy applied from the heater. This is given by the following formula in the case of a trapezoidal heat pulse with rise and fall times of 6 us t= 0

E, = qp(t, + 2t2 + Q/2

(6)

where t, and t, are the rise and fall times, and t2 is the period of constant heat flux. The total heating time tH is equal to t, + t2 + t3. When qP is small, that is below a subcritical heating condition, applied energy is transported wholly in the second sound wave mode. As the heat flux increases, the experimental data deviate downwards from the solid line which indicates ideal heat transfer in the second sound wave mode, because of superfluid breakdown. The critical value qcTfor the onset of deviation is smaller for longer tH, With a further increase in qP, E, reaches a maximum value and then begins to decrease, in spite of qP still increasing. Such a drastic decrease may be caused by the onset of boiling. Finally at very large values of qP, E, almost loses its dependence on the heating time tH. In this case, the waveform reaches a limiting profile and only a limited amount of energy (only several per cent of the input energy) can be transported in the second sound wave mode. The experimental results observed at different bath temperatures are qualitatively the same. Figure 6 shows a com-

Transient heat transport in He II: T. Shimazaki et al.

Thermal boundary layer

I

100 Heat fhk”(W/cm2) 100

0

10

A

1 1 0.1 1

I

10 Heat flux (W/cm2)

Figure5 Amount of energy per unit area transported in the second sound wave mode. Solid lines represent the total amount of energy applied from the heater. (a) Experimental results obtained with different heating times and for TB= 1.70 K, z=30mm. (b) Three cases of heating time for TB= 1.95K. 2=30mm

=;

m 0 3

x

B 5

d O.ll

I

10 Heat flux (W/cm2)

cl

100

Figure 6 Comparison of amount of energy transported in the second sound wave mode at different bath temperatures under the same conditions of heating time. tH = 1000 hs, z= 30 mm

Figure 7 shows several transient traces of temperature measured near the heater from z = 0.5 to 30 mm. Each of the traces exhibits a propagating thermal pulse, detected as an initial spiky response and a secondary gradual temperature rise in the later stage. It should be mentioned that the heating duration is 1 ms, corresponding to one division starting from the arrow marked by ‘Heater On’. However, the temperature rise continues beyond 20 ms. This fact implies that heat transport in the region is not only governed by the second sound wave mode. The secondary temperature rise appearing after the passage of a second sound thermal pulse indicates that a thermal boundary layer reaches the location of the temperature sensor. The peak of the secondary temperature rise moves away from the heater at a speed of the order of 1 m s-l. Even a third temperature rise can be observed at = 18 ms after the onset of heating on the trace for z = 0.5 mm in this figure. It is observed up to =Z = 10 mm under the present condition, but its appearance is rather random in terms of time. This third temperature rise is caused by film boiling, and the appearance of the boiling is a type of random phenomenon, the detection of which is not quite reproducible. The time variations of the temperature distribution in the vicinity of a heater including a thermal boundary layer are shown in Figures 8u and b, where the time after the onset of heating tD is taken as a parameter. Figure 8u shows the results at T, = 1.90 K. In the early stage (to = 2 and 3 ms), formation of a thermal boundary layer with a large temperature gradient is clearly observed adjacent to the heater. The thickness of the layer is found to be =5 mm. The layer does not disappear quickly after the end of the heating, but rather gradually expands diffusively and fades away with the passing of time, since the vortex lines decay very slowly after the end of heating. The results at the lower temperature, 1.70 K, are qualitatively similar to those at T, = 1.90 K but quantitatively are not quite the same. The maximum temperature is higher than that at TB= 1.90 K at every instant due to less active boiling at the lower temperature. The time variation of the temperature becomes slower and the thermal boundary layer becomes thicker compared with the values at 1.90 K. The slower recovery of the temperature profile to the initial quiescent state in the thermal boundary layer may be caused by the lower decay rate of the quantized vortex at the lower temperature. When the quantized vortex line density becomes low, heat transport by thermal counterflow again works efficiently to recover

120, , , , , , , , , , , , , / , , 1 ( ( , I

2 100 parison of E,,, at different bath temperatures under the same condition for tH. It is seen that the critical value qcris large at lower temperatures. It is also found that the amount of energy transported by the second sound wave mode abruptly decreases when the bath temperature is higher than 1.95 K in the range of moderately large heat flux values (approximately from 10 to 30 W cmw2). This may come from the strong dependence of vortex development on temperature, in particular above TB= 1.95 K. A similar tendency was also observed in another experiment17. Vortex development becomes quicker as bath temperature rises and heat transport in the second sound wave mode is disturbed by the interaction with dense quantized vortices.

g

80

a

60

.;

40

f

Heater ON

Time (lms/div)

Figure 7 Several transient traces of temperature near heater. The sensor location ranges from z= 0.5 to 30 mm. TB= 1.70 K, t, = 1000 ps, qP = 40 W crne2

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30

2

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10 0 Distance z (mm)

Oo

5

10

15

20

25

k,,..

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shown in Figure 9~2, where the filled symbols indicate the boundary between the second sound wave mode and the diffusive process, and the open symbols indicate that between the diffusive process and evaporation. These results were obtained under the condition that TB = 1.70 K and tH = 1000 ps. If the heat flux is less than 5 W cm-‘, the entire input energy is transported in the second sound wave mode. When the heat flux increases beyond 5 W cm-‘, superfluid breakdown occurs and some portion of the input energy accumulates in a thermal boundary layer and is then diffusively transported outwards from the heater region. Above 15 W cm-*, the onset of boiling is observed in the thermal boundary layer and input energy begins to be consumed in the phase transition. The contribution of the second sound wave mode to heat transfer decreases and those of the diffusive process and evaporation increase as the heat flux rises. In this figure, the experimental results obtained at TB = 1.90 K and q,, = 40 W cm-* are also plotted for comparison. At the higher bath temperature, the superfluid breakdown occurs at a lower heat flux and the contributions of the diffusive process and evaporation become more significant even at lower heat flux values, i.e. the boundary between the three contributions in Figure 9u shifts leftwards. Figure 9b shows the absolute amounts of energy trans-

Distance z (mm) Figure8 Time variation of temperature distribution in the vicinity of heater including a thermal boundary layer, taking to as a parameter. (a) Ts = 1.90 K, qP = 40 W cm-‘, tH = 1000 ps. (b) For the case of small but still supercritical heat flux and T,=1.70K, qP=15Wcm-2, t,,=lOOO~s

the temperature uniformity. When the heat flux is not so large, a major part of the heat can be transported in the second sound mode. The density of the vortex lines does not evolve enough to form a thick thermal boundary layer for small qp. Figure 8b shows the experimental results for qp = 15 W cm-‘, which is slightly larger than the critical heat flux for superfluid breakdown. It should be noted in this figure that the thermal shock wave front reaches the positions of z = 15 mm and z = 30 mm at tD = 0.75 ms and rD = 1.49 ms, respectively. It is seen that a shock front, a discontinuity in temperature, is already formed at r, = 0.75 ms. At tD = 1.49 ms, the thermal boundary layer develops to a thickness of ==3 mm. The thin thermal boundary layer quickly disappears because of the low vortex line density due to small qp.

Heat transport layer

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It is seen from the above discussion that only a limited amount of the input thermal energy is transported in the form of a thermal wave in the case of strong heating. It is seen from Figures 8a and b that some portion of the energy accumulates in a thermal boundary layer and is then transported by a diffusion-like process and also by thermal counterflow. In addition, if the applied heat flux is large enough to induce boiling, a major part of the energy must be consumed upon phase transition, i.e. boiling. The ratio of the amount of energy transported by each process is roughly estimated from the present experiments. The results are

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Figure9 (a) Ratio of amount of energy transported by each process. Filled symbols indicate the boundary between the contribution of the second sound wave mode and that of the diffusive process. Open symbols indicate the boundary between the contribution of the diffusive process and that of evaporation. Ts= 1.70 K, tH = 1000 ~LS. A pair of data points obtained at TB= 1.90 K (denoted by squares) is also included for comparison. (b) Absolute amounts of energy transported by the three processes. Most of the data plotted are those obtained at Ts= 1.70 K, t,.,= 1000 ps, with a pair of data points for tH = 2000 ps (denoted by squares) added for reference. Broken lines represent the total energy transported per unit area for two cases, tH = 1000 and 2000 ps

Transient heat transport in He II: T. Shimazaki et al. ported by the three processes. Here, most of the data plotted are those obtained at TB = 1.70 K, fn = 1000 us, with two data points for rH= 2000 ps (denoted by squares) added for reference. The total energy transported per unit area is indicated by broken lines for two cases, tn = 1000 and 2000 ps. It is clearly seen from this plot that above qP= 15 Wcmm2 the contribution of the second sound wave mode decreases abruptly. On the other hand, the contributions of the diffusive process and evaporation increase with increase in qp. This results from the onset of film boiling. The amount of energy transported by the second sound wave mode in the case of tn = 2000 ps, qp = 20 W cmm2is almost the same as that in the case of tH = 1000 ps, q,=20Wcm-2 because propagating waveforms lose their dependence on heating time for large qp (the limiting profile).

4.

diffusive process of heat transfer, usually leading to boiling. The contributions of the diffusive process and evaporation to energy transfer increase as the heat flux rises. The amount of energy transported in the second sound wave mode strongly depends on qP and the temperature. Under a subcritical heating condition it is = 100% of the applied heat, but decreases with increase in qP; in particular, it drops drastically when a waveform reaches the limiting profile and boiling starts. At lower temperatures, i.e. lower than 1.9 K, energy transport in the second sound mode becomes more efficient.

Acknowledgement This research was partly supported by the Japan Society for the Promotion of Science. References

Conclusions

1

Highly transient heat transport in He II is investigated experimentally with emphasis on the transient thermal behaviour in superfluid breakdown and the superfluid turbulent state. The following conclusions can be drawn. The effect of high density vortices, even on highly transient heat transfer phenomena, is significant. The order of the characteristic time for vortex development ranges from several milliseconds for the peak heat flux qp = 5 W cmm2 to less than 200 ps for values over 30 W cm-* at 1.70 K, although it depends on qp and the temperature. The shape of the thermal wave generated by pulsed heating loses its dependence on heating time and reduces to a unique profile termed the limiting profile for a sufficiently large and long heat flux, typically larger than 30 W crnw2and longer than 200 ps. A thermal boundary layer is formed in the vicinity of the heater after superfluid breakdown. The thickness of this layer is of the order of several millimetres, e.g. 5 mm in the case of tH = 1 ms and qP= 40 W cme2. It takes as long as several milliseconds to diminish after the onset of heating. Locally this layer brings about a

8 9 10 11 12 13 14

Landau, L.D. and Lifshitz, E.M. Fluid Mechanics Pergamon Press, Oxford, UK (1959) Ch 16 Gorter, CJ. and Mellink, J.H. Physica (1949) 15 285-305 Vinen, W.F. Proc Roy Sot A (1957) 240 114-127 Vinen, W.F. Proc Roy Sot A (1957) 240 128-143 Vinen, W.F. Proc Roy Sot A (1957) 242 493-515 Schwarz, K.W. Phys Rev B (1988) 38 2398-2417 Liepmann, H.W. and Laguna, G.A. Ann Rev Fluid Mech (1984) 16 139-177 Turner, T.N. Phys Fluids (1983) 26 3127-3241 Stamm, G., Olszok, T., Schwerdther, M. and Schmidt, D.W. Cryogenies (1992) 32 598-600 Nemirovskii, S.K. and Lebedev, V.V. Sov Phys JETP (1983) 57 1009-1016 Nemirovskii, S.K. and Tsoi, A.N. Cryogenics (1989) 29 985-994 Borner, H., Schmeling, T. and Schmidt, D.W. .I Low Temp Phys (1983) 50 405-426 Schwerdtner, M.v., Poppe, W. and Schmidt, D.W. Cryogenics (1989) 29 132-134 Fiszdon, W., Nemirovskii, S.K. and Schwerdtner, M.v. Physicu B (1991) 168 93-103

Fisrdon, W., Schwerdtner, M.v., Stamm, G. and Poppe, W. .I Fluid Mech (1990) 212 663-684 16 Shimazaki, T., Iida, T. and Murakami, M. Adv Cryog Eng (1994) 39B 1859-1864 17 Katsuki, Y., Murakami, M., Iida, T. and Shimazaki, T. Cryogen15

ics in press 18

Donnelly, RJ. Experimental Superjuidity University

of Chicago IL, USA (1967) Putterman, SJ. Superfluid Hydrodynamcis North-Holland Publishing Co., Amsterdam, The Netherlands (1974)

press, Chicago, 19

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