Test and analysis of a liquid 3He target for intermediate energy physics experiments

Nuclear Instruments and Methods in Physics Research A234 (1985) 421-425 North-Holland, Amsterdam

421

T E S T A N D A N A L Y S I S O F A L I Q U I D 3He T A R G E T F O R I N T E R M E D I A T E E N E R G Y P H Y S I C S EXPERIMENTS H.C. M E I J E R a n d H. P O S T M A Department of Applied Physics, University of Technology, Lorentzweg 1, 2628 CI Delft, The Netherlands W.B. B L O E M ECN, Netherlands Energy Research Foundation, P.O. Box 1, 1755 ZG Petten, The Netherlands Received 27 September 1984

Experiments with a liquid 3He target with convectional circulation and a liquid 4He cooling system are described. A theoretical treatment is given which agrees with the experimental results. If properly constructed, the target can take up more than 1 W. In that case the 3He leaving the reaction region must have a temperature near 3 K, which is higher than the temperatures measured directly. A small expansion vessel on top of the heat exchanger appeared to be important for the functioning of the target under conditions of high power dissipation. On the basis of our calculations, the variation of the experimental averaged target thickness with the beam power of the NIKHEF 3He target at Amsterdam can be largely explained.

1. Introduction

2. Experiment

Liquid 3He targets [1-6] are used when studying nuclear reactions with small cross sections of 3He for various kinds of projectiles. With this low-temperature technique, an areal target density of the order of 10 22 nuclei per cm2 and a target thickness of about 1 cm can be realized while relatively thin walls of about 20 # m can be used. Experience shows that not all targets are capable of taking up the same power. Most of the reported liquid 3He targets [1-4] are meant for rather low heat dissipation, say of the order of 0.1 W. In other cases the heat generated by the beam in the target may be as large as 1 W [5,6]. The problem of how to remove this heat can be solved relatively easy by cooling the circulating 3He in a heat exchanger with superfluid 4He. In the N I K H E F target system [6], which has a low flow impedance, the 3He liquid circulates due to a difference in mean density between the warm rising and the cold descending liquid, that is by using the buoyancy effect. In order to obtain some insight into the rather complicated system of circulating 3He, driven by an internal heat source and cooled by 4He in a heat exchanger, an experimental and theoretical study has been undertaken, which is presented below. Several results obtained in this research have been used during the construction of the N I K H E F 3He target at Amsterdam [6].

Fig. 1 shows the principle of the main part of an experimental set-up which has been constructed as a test facility during the design of the Amsterdam N I K H E F target [6]. Two copper tubes of 5 mm i.d. for the rising and returning 3He, respectively, are connected to the test cell C. The liquid 3He can be heated inside a region comparable to the beam size by means of a 77 ~2 heater of spiralized manganin wire inside the cell. The warm 3He flows upwards to a heat exchanger H, consisting of about 80 cm of copper tube, which is cooled by low-temperature liquid 4He. The temperature of the 4He varies between 1.0 and 1.8 K when the heat load is increased from 0 to 1 W. The 4He bath, which is cooled by reducing its vapour pressure with a 60 m3/h pump, is filled at intervals from a 4.2 K liquid He reservoir by means of a needle valve. Level indicators and pressure gauges give information about the behaviour of the system. In the later experiments a small expansion vessel E of 6 cm3 had been placed on top of the heat exchanger; this point will be discussed below. 3He is condensed into the cell and the heat exchanger through capillary A. Capillary S is added for safety in order to prevent overpressure in case of a plugged filling capillary. Four calibrated Allen-Bradley 56 ~ resistors, numbered 1 to 4 in fig. 1, were mounted at crucial places.

0168-9002/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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11.C. Meijer et a L / A fiquid 3He target

PUMP

Itl "///////~////A F/[ A '3He

~

80

70

- ~ - 2.79K 60

2

r [ K]---~

3

Fig. 2. The density of liquid 3He as a function of temperature. The calculated temperature region in which the system operates has been indicated. 4He

",/////"//

-1.7 7K~

~,///////

4 •11.77K)1 0.12W

!0.(

1

2.19KK~ (2.18K)

~= 1.6cm/s

~=12.8cm/s 2.98K

(~=lW (1.90K) 1.85K

1.85K)

Fig. 1. The principle of the test facility, described in the text. C is the test cell, with a heater inside. E is an expansion vessel of 6 cm3, H the heat exchanger, consisting of a copper tube of 5 mm i.d. and 7 mm o.d. The 3He is condensed through capillary A. S is a safety capillary. Numbers 1 to 4 indicate calibrated Allen-Bradley 56 12resistors. The indicated temperatures, heat flows and velocities were calculated in the situation that 1 W was dissipatedby the heater. Measured quantities are given in brackets. The first experiments were performed without the expansion vessel. In this situation, the system became unstable at a power dissipation of 0.4 W. Probably, violent boiling of the 3He occurred in this case. When the expansion vessel was added to the system, up to 1.1 W could be dissipated without difficulties. The expander serves as an overflow for the 3He which expands strongly upon heating, especially above = 2 K, see fig. 2. At the same time it is a reservoir from which 3He gas is supplied to the capillaries and the room temperature 3He vessel when the temperature of the system rises with a correspondingly increasing 3He vapour pressure. When the power in the 3He cell was increased from 0 to 1.1 W, all temperatures and pressures increased

monotonically. At a dissipation of 1 W in a stationary situation the following temperatures were measured (see fig. 1): T1 = 2 . 1 8 K, T2=1.90 K, T3=1.85 K and T4 = 1.77 K. The temperature errors are typically 0.01-0.03 K. The pressure of the 3He gas above the liquid was 210 Torr, corresponding to a temperature of 2.19 K at the top of the liquid in the expander. The temperatures T2 and T3 must be very near the temperature of the incoming 3He. The value of T1, which seems to be in good agreement with the 2.19 K of the expander, however, must be considered with some caution. Since there is a copper tube between this thermometer and the 1.77 K flange, T1 is determined both by the temperature of the 4He vessel and that of the rising 3He liquid. In the next paragraph this point will be discussed further.

3. Theory It is supposed that the circulation of liquid 3He is caused by a difference Ap in the average density between the warm rising liquid and the colder descending liquid. This buoyancy effect depends on the difference in pressure Ap given by: A p = g h A p = 1.3Ap,

(1)

where g is the gravitational acceleration and h the height of the two columns of liquid. In SI units, g is given in m / s 2, h in m, Ap in k g / m 3 and Ap in Pa. The direction of the circulation is dictated by the geometry. Only steady state circulation at 1 W power dissipation is considered in this paper. The power Q dissipated in the cell equals h ( H h - H 1), where h is the n u m b e r of moles of 3He circulated per second, and H a and H l are the molar enthalpies at the highest and lowest temperature in the system, respectively. Since h is equal to o A v N , where p is the liquid

H.C Meijer et al. / A liquid3He target density in k g / m 3, A the cross sectional area (A = 2 × 10 -5 m2), o the average velocity in m / s , and N is the number of moles per kg ( N = 103/3), the relation (~ = 6.7 × lO-3pV(Hh -- H1)

(2)

should be obeyed. The estimation of H(T) (or the specific heat C(T)) provides some difficulties: Firstly, experimental results of the specific heat of liquid 3He at temperatures above 2.5 K are, to our knowledge, not available [7,8]. Secondly, the question arises whether the specific heat at constant pressure Cr or at the saturated vapour pressure Cs should be used. A preliminary calculation, based on the relations (1) and (2) and the geometry of the tubes learn us, that the temperature of the liquid must rise to about 3 K in order to explain how 1 W of power is absorbed and carried off by the liquid 3He. The vapour pressure of 210 Torr, measured above the liquid 3He in the expander, corresponds with an equilibrium temperature of the liquid of 2.19 K. This temperature is considerably closer to the temperature of the 4He bath than that of the heated 3He. We conclude, that the 3He in the expander is very efficiently cooled by the liquid 4 He. This means, that changing of the heat input in the 3He cell will affect the temperatures of the 4He bath and that of the 3He in the expander only to a small degree. This in turn suggests that in the calculations rather Cp than C~ should be used. As mentioned above, data of Cp of 3He around 3 K are not readily available. In order to obtain a practical estimated relation of Cp of 3He as a function of temperature, we assumed that at not too low temperatures 3He and 4He will show a similar thermal behaviour. With the aid of the available data for 4He [9] and the relation 0V

dp

Cp = Cs + ( - ~ ) p T - ~ ,

(3)

Cp of 4He was calculated for temperatures between about 2.9 and 5.2 K. From these data Cp of 3He was estimated by scaling the 4He temperatures with a factor of 0.638 which is the ratio of the critical temperatures of the two liquids. It should be kept in mind, however, that Cp is larger than both Cs and Co, so in eq. (2) the largest possible differences of enthalpy have been used. In the full mathematical treatment a circulation scheme must be derived in which velocities and temperatures have such values that not only eqs. (1) and (2) are satisfied, but that also the difference in pressure Ap of eq. (1) produces a circulation rate h which is in agreement with the geometry of the system. Finally, by using the laws of heat transfer it must appear that, with the chosen temperatures and velocities, 1 W is transferred from the 3He through the walls of the copper tubes to the 4He. To facilitate the model calculations some simplifying

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assumptions were made: It is supposed that theliquid flow is turbulent: the critical velocity for turbulent flow is estimated to be about 1.4 c m / s , which is surpassed already at very low heating rates. For the heat transfer between the liquids and the copper tube, two mechanisms must be taken into account: Firstly, the classical heat transfer, which is determined by Nu = otsDi/)k [10] where Nu is the Nusselt number, a s the heat transfer coefficient, Di the inner tube diameter and )~ the heat conductivity of the liquid. Secondly, the heat transfer between He liquid and copper is opposed by the Kapitza resistance [11], which is caused by an acoustical mismatch between the phonon spectra of the liquid and the copper, respectively. After inspection of the corresponding heat resistances of the two processes, it appears that for the 3He-Cu transition the Kapitza resistance is negligible compared to the thermal resistance of the liquid, but that for the 4 H e - C u transition only the Kapitza resistance counts because of the very large heat conductivity of bulk 4He in the superfluid state [12]. No dependence on temperature of the heat conductivity h and the viscosity ~1 was taken into account. At the temperatures considered here, 77 is almost constant at 25 × 10 -7 Ns//m 2 [12]; for )~ an average value of 14 × 10 -3 W / m K was used [12]. It was supposed that the two copper tubes in the 4 He liquid between the expander and the bottom of the 4He vessel have both their own constant temperature. The temperatures calculated for these tubes, 1.88 and 1.83 K, respectively, indicate that probably no large errors are caused by this assumption (the heat conductivity of the copper tube was about 200 T W / m K ) . For the parts of the copper tubes in the vacuum space this condition is, of course, not valid. The temperature gradient along these copper tubes is essential. We have assumed that turbulent flow of liquid 3He in straight and spiralized circular tubes can be described by the well known eq. [10]: -

-

-

-

L pOE

Ap = ~x Di

2 '

(4)

where L is the length of the tube, and ~t is a flow resistance factor, given by: ~1 = ~(1 + 3.75x(Di/Dk)).

(5)

For a spiralized tube, such as the descending tube, D k is the outer diameter of the spiral. For a straight tube, D k = oo, so ~1 = ~. The empirical factor X depends on the Reynolds number Re = PoDi/rl. We estimate X to be about 8 [10]. For ~ the relation: ~ = 0.316 Re -°25 = 0.316

(6)

has been found [10]. We suppose that all other flow

H. C. Meijer et al. / A liquid SHe target

424

resistances are very small. Iterative calculations were necessary before the final result was achieved. The maximum temperature Th was used as the main variable in this process. When Re >/104 and Pr >/0.7, the Nusselt number Nu may be approximated by: Nu = 0.027 Re°'Spr °'33,

(7)

or other expressions which are not much different from this one [10]. Pr = Cprl/~ is the Prandtl number, with Cp being the specific heat per kilogram at constant pressure. The final results (v -~ 12 cm/s) give Re ~ 1.8 × 10 4, and Pr ~ 1.05, so the conditions given above are valid for eq. (7). Combining eq. (7) with N u - asDi/h gives the heat transfer coefficient as. It appears that as = 11 CO'33. The heat transferred to a small length 8x of the copper heat exchanger with a surface area 8A = ~rDiSx is given by: 8 0 ~sSA(ra - rb) =

1DrDi8 x ( Ce (Ta,) } 0.33( T . - r b ) ,

(8)

where Ta and Tb are the temperatures of the 3He and the tube, respectively, and Tat is very well approximated by (Ta + Tb)/2 [13]. The corresponding decrease 8Ta of T~ is given by:

8 0 = h38H(Ta) = h3Cp(Ta)STa,

(9)

where 8H(Ta) is the change of enthalpy of the 3He at an averaged temperature Ta along the small length 8x, mentioned in eq. (8), and 8Ta the change of Ta along 8x. Computer calculations along the length of the tube were carried out by choosing a fixed small value for 8Ta, e.g. 10 -2 K, then calculating 8 0 by means of eq. (9), and next determining 8x from eq. (8) for an estimated (fixed) value of Tb. In this way, the variation of temperature of the 3He in liquid 4He can be calculated in K steps for any desired length L ~ Eff=18xi, once the starting temperature is known. The temperature at which 3He enters the 4He vessel was calculated in a related, but slightly different way. In this calculation, the estimated highest temperature of the 3He, leaving the heating region, was taken as the starting temperature, but now also heat flow and thus differences in temperature along the copper of the riser tube in the vacuum space were taken into account. Next, the value of Tb in the 1 K-vessel must be determined, and thus the real amount of heat transferred. This was done separately for the two parts of the tube between the entrance and the expander, and between the expander and the exit, respectively. When the Kapitza resistance R K is given by R K = B T - " , the heat transferred from a copper tube of length L, outer diameter D l and temperature Tb, to the liquid 4He of temperature TOis given by: 0

~rD,L

B ( m + 1)

(T~,+I - Td.+l)"

(10)

When a correct value of Tb is chosen, the heat transfer between the tube and the 4He given by eq. (10) is equal to the power, transferred between the 3He and the tube, the latter amount of power being the sum of the values of the 8 0, obtained by combining eqs. (8) and (9). Since many different values for B and m have been reported (see e.g. refs. [12] and [14] and references therein), three different sets of values for B and m were used in our calculations. The first set (B = 5 × 10 -3, m = 3) is typical for T < 0.2 K [14], the second (B = 1.2 × 10 -3, m = 4.2 [15]) and third (B = 4.5 × 10 -3, m = 2 [16]) describe measurements between about 0.2 and 0.9 K. Experimental results for T < 1 K were used rather than those for T > 1 K, because the former measurements have a better accuracy. Besides, it appears that almost all of the Kapitza resistances measured between 1 and 2.17 K show values which lie between the extrapolated curves, determined by the last two sets of B and m, given above. The differences in the calculated temperatures of the 3He at the outlet of the heat exchanger, caused by these different choices of B and m, are at most 0.08 K. So, an averaged value for the lowest temperature could be used. The total amount of heat, transferred to the 1 K vessel should be 1 W. The main contribution comes from the tubes in direct contact with the liquid 4he, but the heat flowing along the "warm" copper tube in the vacuum cannot be neglected. The amount of heat, transferred through the "cold" copper tube in the vacuum is very small. The calculations described so far also provide the thermal and density profiles along the two parts of the heat exchanger. The latter gives AP.

4. Results After some iterations, a consistent picture appeared as follows: In the case of a 1 W load the 3He enters the cell at a temperature of 1.85 K, and leaves it with T = 2.98 K. Between the cell and the 4He vessel, 0.12 W is transferred to the copper tube and consequently to the 4He vessel. At the position of the expansion vessel, the temperature in the heat exchanger has decreased to 2.79 K. Up to that point another 0.16 W has been transferred to the 4He bath. In the downward spiral 0.7 W flows to the 4He bath. The 3He leaves the 4He vessel at a temperature of 1.97 K. On its way back to the cell the 3He finally transfers a small amount of power to the copper tube. Since, according to eq. (1), #v is a constant, the upward and downward speed vary continuously. Averaged velocities for the rising and descending flows are 0.128 and 0.116 m/s, respectively. This corresponds to a molar circulation rate h 3 of about 6 × 10 -2 mol/s.

H.C. Meijer et al. / A fiquid 3He target

Fig. 1 shows the most important data. It is essential that the temperature at position 1 was calculated to be 2.186 K, despite of the high temperature of the rising 3He. Also the fact that the pressure in the expander is 210 Torr, corresponding with T--- 2.19 K, can be understood; the liquid 3He in the expansion vessel hardly takes part in the circulation, so it may cool in its surroundings of 1.77 K to a temperature far below 2.79 K. 5. Remarks

1. It was mentioned above that only the flow resistances of the tubes were taken into account. Even so, the temperature in the target rises to about 3 K. If serious extra obstructions for the flow are present, much less power can be carried off at the same temperature, as has been experienced in other designs. 2. Although the construction of the Amsterdam target [6] is somewhat different from the target described in this paper, the general features are the same. In this target, the temperature of the returning (cold) 3He is measured by means of a carbon resistor, and the pressure of the 3He gas in the expansion vessel is measured as well. Exactly as in our system, both measurements are not fully representative of the temperature to which the 3He is heated. It is interesting to note that for this target the experimental densities, even after a temperature correction, were low by about 8 ± 2%/W. If the temperature of the 3He in the target beam changes from 1.85 to 2.98 K, the average temperature is calculated to be 2.52 K. The ratio of the 3He density at this temperature and at 2.20 K, used to analyze the Amsterdam target, is 1.046, so that 4.6%/W of the 8% mentioned above has been explained. 3. It was argued that the Cp, used in our analysis, is the maximum possible value of the specific heat. In reality, the 3He vapour pressure rises slightly upon heating of the 3He cell, so the true specific heat will be somewhat smaller. This means that, with the same power dissipation, the maximum temperature will nse above 3 K. In that case, also the average temperature in the target beam is higher, giving an even better agreement with measurements. Local boiling in the 3He might also play a role. 4. Despite of the rather extensive calculations the results can only be considered to be approximate, in view of the simplifications, used. Nevertheless, we think that the essentials of the behaviour of this system are well understood. 6. Condusions

1. In these experiments and calculations it has been shown that a target in which liquid 3He is circulated by the buoyancy effect and with a simple 4He heat ex-

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changer may take up more than 1 W, provided it has been well designed. 2. If 1 W is dissipated, the temperature of the liquid helium leaving the reaction region must be about 3 K, although this temperature has not yet been measured directly. 3. In order to maximize the power that can be dissipated in the liquid, as little flow resistance as possible should be present. 4. The mounting of a small expansion vessel on top of the heat exchanger is profitable. 5. The variation of the experimental averaged target thickness with the beam power of the Amsterdam target has been explained for the greater part. Thanks are due to Dr. M. Durieux of the Kamerhngh Onnes Laboratory, Leiden, The Netherlands, for advice and discussions about the specific heat problem of 3He, to Ir. T.H. van der Meer for very interesting and useful discussions about flow phenomena, and to Messrs. A. Kollen and W. Schot for their expert technical assistance.

References

[1] S. BtLlaler, Proc. 3rd Int. Cryogenic Engineering Conf., BErlin (1970) p. 306. [2] J.S. Vincent and W.R. Smith, Nucl. Instr. and Meth. 116 (1974) 551. [3] A. Christ, H.J. Gassen, G. Ntldeke, T. Reichelt and H. Stanek, Nucl. Instr. and Meth. 152 (1978) 367. [4] D.K. Hasell, R. Abegg, B.T. Murdoch, W.T.H. van Oers, H. Postma and J. Soukup, Nucl. Instr. and Meth. 189 (1981) 341. [5] E. Jans, private communication concerning the Saclay 3HE target. [6] H. Postma, J.P. Boogaard, P.H.M. Keizer, L. Prins and P.K.A. de Witt Huberts, Nucl. Instr. and Meth. 219 (1984) 292. [7] E.F. Hammel, Progress in Low Temperature Physics I, ed., C.J. Gorter (North-Holland Publ. Comp., Amsterdam, 1955) p. 78. [8] D.S. Greywall, Phys. REV. B27 (5) (1983) 2747. [9] R.D. McCarty, J. Phys. Chem. Ref. Data, voi. 2 (4) (1973). [10] Handbook of Heat Transfer, eds., W.M. Roshenow and J.P. Harnett (McGraw-Hill, New York, Maidenhead, 1973). [11] J.P. Harrison, J. Low Temp. Phys. 37 (1979) 467. [12] J. Wilks, Liquid and Solid Helium (Clarendon Press, Oxford, 1967). [13] E.R.G. Eckert and R.M. Drake, Jr., Analysis of Heat and Mass Transfer (McGraw-Hill, New York, Maidenhead, 1972). [14] O.V. Lounasmaa, Experimental PrinciplEs and Methods below 1 K, (Academic Press, New York, London, 1974). [15] A.C. Anderson, J. Connolly and J.C. Wheatley, Phys. Rev. 135 (1964) A910. [16] H.A. Fairbank and J. Wilks, Proc. Roy. Soc. A231 (1955) 545.