Experimental verification of an analytical model for orifice pulse tube refrigeration

Experimental verification of an analytical model for orifice pulse tube refrigeration M.J.A. Baks, A. Hirschberg, B.J. van der Ceelen and H.M. Gijsman Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands

Received 31 May 1990 The orifice pulse tube refrigerator has the potential to become a small, reliable cryocooler. In linear approximation the refrigeration produced is due to two independent contributions: an enthalpy flow in the bulk of the gas and an enthalpy flow due to heat exchange with the wall. These effects are treated independently in the literature. For pulse tube operation they can be of the same order of magnitude and have to be combined. Experimental verification is obscured by the regenerator loss, which is difficult to determine accurately. By performing experiments at room temperature this loss can be made negligible. These experiments are in agreement with the theory.

Keywords: cryocoolers; refrigerators; pulse tube

For some time there has been increasing demand for small reliable cryocoolers, especially for space applications. The orifice pulse tube refrigerator, introduced in 19841 , has the potential to fulfil this need. Its principle of refrigeration is similar to the Stirling process, but unlike a Stirling machine there are no moving parts operating at low temperature, which improves reliability. In contrast with the compressor-driven pulse tubes of others 2-4, this pulse tube refrigerator is driven by a system of inlet and outlet valves connected to pressure vessels (see Figure 1). The expansion piston of the Stirling refrigerator is replaced by a tube, the pulse tube, with at one end an adjustable orifice connected to a reservoir. The inlet and the outlet valves create pressure fluctuations which cause an enthalpy flow in the pulse tube, and heat can be absorbed by the cold end heat exchanger. With this pulse tube a minimum temperature of 74 K has been reached with no thermal load. This modest performance in comparison with the 49 K reported by Liang et al. 2 is expected to be due mainly to the absence of a flow straightener and possibly a non-optimized

regenerator. This is, however, irrelevant for the experiments described here. So far, only Storch and Radebaugh 5 have presented an analytical model of orifice pulse tube refrigeration. Their theory, however, was not fully confirmed by their experiments. The research described in this paper concentrated on reviewing their theory and combining it with the theory of the surface heat pumping mechanism presented by Wheatley 6 •7 and Rott g,9' . Experimental tests have been made in which the negative influence of the regenerator was eliminated.

Enthalpy flow analysis After Radebaugh 5, an enthalpy flow analysis is given here to explain pulse tube refrigeration. This analysis is based on the first law of thermodynamics, taken over a control volume around the cold end heat exchanger for steady state operation of the pulse tube refrigerator, given by

<0> + = I

]

I

i I~I L ......................

I

i

I

I

1 Schematic diagram of the pulse tube refrigerator: A, regenerator; B, cold end heat exchanger; C, pulse tube; D, warm end heat exchanger; E, adjustable orifice; F, reservoir; G and H, inlet and outlet valve; I, vacuum vessel Figure

0011 - 2 2 7 5 / 9 0 / 1 1 0 9 4 7 - 05 (c) 1990 B u t t e r w o r t h - H e i n e m a n n

(2)

where Q is the heat flow to the heat exchanger, /"Ir the enthalpy flow through the regenerator and/:/the enthalpy flow in the pulse tube. The enthalpy flow is taken to be positive in the direction of the warm end of the pulse tube. The angle brackets denote an average value over one cycle. The heat transport due to the interaction with the wall is not taken into account here. The cooling power (Q) can be increased either by minimizing the regenerator loss (]-~r) or by maximizing the enthalpy flow ( H ) in the tube. At the lowest temperature which can be reached, the enthalpy flow in the tube and

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Experimental verification of an analytical model: M.J.A. Baks et al.

the regenerator loss are equal, other losses due to radiation and conduction being negligible. Assuming that an ideal gas is used as working fluid, the average enthalpy flow is given by

(I21) = cp(pC T)

(2)

tuation in a closed tube with no heat exchange between the gas and the tube wall.

Calculation of enthalpy flow in the pulse tube

where Cp is the heat capacity at constant pressure, which is assumed constant, p is the density, cb the volume flow, which is positive in the direction of the warm end of the pulse tube, and Tthe temperature. Using the ideal gas law

A fixed amount of gas between the cold and warm end heat exchangers undergoes adiabatic expansions and compressions. The average volume of this gas is the volume of the pulse tube, lit. Due to the volume flows ~1 and O2 at, respectively, the cold and warm ends of the pulse tube, the volume of gas changes in a period dt by

p=pRT

dV = -cb~dt + ~2dt

(3)

where p is the pressure and R the specific gas constant, Equation (2) becomes

( [21) = (cp/R) (pcb)

(4)

Now consider the enthalpy flow in the pulse tube. If it is assumed that the pressure and the volume flow rate can be split into average components P0 and ~0, respectively, and fluctuations p' and ~', respectively, then

(I':I) = (cp/R)((p'cb') + po~bo)

(5)

because (p') = 0 and ( ¢ ' ) = 0. Equation (5) is an exact non-linearized expression for the average enthalpy flow in the pulse tube. ~0 is calculated using the fact that the net transfer of mass through a cross section of the tube is zero (pCI,) = 0

(6)

Similarly, if the density is written as the sum of the average density and the fluctuation p', then 0 0 = --(p'O')/po

(7)

If it is assumed that the fluctuations in pressure and density are small in comparison with the mean values, then the average volume flow ¢0 is thus small compared to its fluctuation ~'. However, because the enthalpy flow is quadratic in the fluctuations PoO0 cannot be neglected in Equation (5). If heat transfer between the gas and the tube wall is neglected, because the thermal boundary layer is small in comparison with the tube radius, then because the process is adiabatic P'/Po = P'/~Po

(8)

with 3' the specific heat ratio. Combining this expression with equation (7) gives, after substitution in Equation (5) (H) = (p'O')

(9)

which is equivalent to the result obtained by Storch and Radebaugh for harmonic fluctuations. This expression shows that the amount of heat transferred from the cold end to the hot end of the pulse tube per unit time is determined by the component of the volume flow which is in phase with the pressure fluctuation. It also shows the need for an orifice at the hot end of the pulse tube, since the volume flow is 7r/2 out of phase with the pressure fluc-

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Cryogenics 1990 Vol 30 November

(10)

Since the processes in the pulse tube are adiabatic the following expression is obtained, assuming that the displacements of the gas are small in comparison with the tube length and that the pressure fluctuations are small in comparison with the mean pressure

E

-¢bldt + ¢b2dt = - - dp 7P0

(11)

which for harmonic fluctuations with frequency o~results in t~ 1 = (I) 2 "I-

( ko Vt /"/po)p '

(12)

Using a position independent pressure p' in Equation (11) is equivalent to the assumption that the pulse tube length is small compared with the acoustic wave length. This is a reasonable assumption for operational frequencies of 3 Hz as used in the present set-up. This equation shows that gas which flows into the pulse tube causes a volume flow at the warm end of the pulse tube and a pressure rise in the pulse tube. The relation between the pressure fluctuation p' and the volume flow O2 through the orifice is determined by a combination of orifice impedance and volume of the reservoir, which should be designed such that the pressure fluctuations in the reservoir are negligible in comparison with the pressure fluctuations in the pulse tube. Therefore, the volume flow resistance R of the orifice and the volume Vr of the reservoir should be sufficiently large to make the following expression valid for operational frequencies

R~oVrl3,p ~> 1

(13)

Under this condition the volume flow through the orifice is in phase with the pressure fluctuation p' in the pulse tube, which is equal to the pressure drop over the orifice, provided that the mean pressures in the pulse tube and reservoir volume are equal. The volume flows which appear in Equation (12) may be illustrated by a vector diagram in which the amplitudes of cI,~ and p', denoted as ~z and p respectively, are represented on the real axis (see Figure 2). The time average enthalpy flow in the pulse tube can be calculated with equation (9). Taking the product of the component of the volume flow in phase with the pressure fluctuation, neglecting higher order terms, the following is obtained for the enthalpy flow in the pulse tube

( H ) = 1/2p~ 2

(14)

Experimental verification of an analytical model: M.J.A. Baks et al. 40

Im td A

30 o

.O"

~Po'~

@~ 20

///1/ ///

10

//t 1/// 0 Figure 2

Vector diagram of volume flows in the pulse tube

Experimental results To verify the theory the cooling power of the pulse tube refrigerator was measured as a function of the enthalpy flow in the tube, as given by Equation (14). The pressure fluctuations in the pulse tube were measured at the warm end of the pulse tube by means of a pressure transducer (Micro Switch 242PC150G). To determine the volume flow at the warm end of the pulse tube the orifice was replaced by a capillary, with which similar cooling powers could be reached. Having calibrated the capillary it is possible to determine the volume flow 4,2 through it from the measured pressure fluctuations p in the pulse tube to an accuracy of 5 % of the total volume flow. A thermal load is applied to the cold end heat exchanger by means of an electrical heater. To be able to compare this cooling power with the enthalpy flow in the pulse tube, losses must be made negligibly small. Losses due to radiation and conduction are minimized by surrounding the pulse tube and regenerator with multi-layer radiation shields (superinsulation) and placing the assembly in a vacuum vessel. The regenerator loss, which is linearly dependent on the temperature difference along the regenerator, cannot be ignored at low temperatures. The result is a decreasing cooling power of the pulse tube refrigerator. To avoid the negative influence of this loss, which is difficult to predict analytically, the cold end heat exchanger is kept at ambient temperature (294 K) by supplying an appropriate amount of heat. As the temperature difference over the regenerator is kept zero, the regenerator loss vanishes. The enthalpy flow in the pulse tube results in a temperature rise of the warm end heat exchanger up to 15 K above ambient temperature. The pulse tube on which the measurements were made had a length of 400 mm and an inner diameter of = 18 mm. The working fluid was helium gas. The mean pressure during the experiments was between 1.7 and 4.3 bar* and the frequency was 3 Hz (co = 19 rad s-t). The pressure fluctuations p varied up to 1.7 bar and the volume flows cI,2 at the outer end of the tube were of order of magnitude 150 cm 3 s -I. The results of the experiment are shown in Figure 3. The dependence of cooling power on enthalpy flow l/2pO2 is linear, but the slope of "1 bar = 10 5 Pa

0

I 10

-~½~2[watt]

20

Figure 3 Cooling power of the orifice pulse tube: O, experimental values; - - , enthalpy flow calculated according to Equations (14) and (1'5), respectively

the curve is about 50% larger than that predicted by Equation (14) (dashed line in Figure 3).

Influence of heat transfer to the wall of the pulse tube The results of this experiment show qualitatively good agreement with the values predicted by Equation (14). They are, however, in contradiction with the results reported by Storch and Radebaugh, who measured a cooling power a factor three to five times lower than predicted by their theory, although it has to be taken into account that at lower temperatures there is a considerable infuence of the regenerator losses. However, there is still a deviation of 50% in the present results which should be explained. To find out if this might be due to the thermal contact between the gas in the thermal boundary layer near the wall and the wall of the pulse tube, the mechanism of heat transfer between the gas and the wall has been studied. First there is the heat pumping mechanism, which has been treated by Wheatley 67, which results in a heat flow from or to the cold end heat exchanger, depending on the temperature gradient along the tube wall. If there is no temperature gradient, which is the case in the experiment described here, gas which is moved in the direction of the orifice during compression will have a higher temperature than the adjacent wall at the moment that maximum pressure is reached. This results in the transfer of heat from the gas to the wall. During the expansion the gas temperature falls while gas moves back in the direction of the cold end heat exchanger. While at a lower temperature during the low pressure period the gas will absorb heat from the wall which is to be transported in the direction of the warm end heat exchanger. The amount of heat which is pumped by this mechanism is determined by the amount of gas in the thermal boundary layer of which the displacements are in phase with the pressure fluctuations. Hence, the velocity of this gas, or the volume flow in the

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Experimental verification of an analytical model: M.J.A. Baks et al.

boundary layer, must be out of phase with the pressure fluctuations. Due to simplifications in the theory of Wheatley 6"7 a second effect has been neglected which is present in the more accurate theory of Rott 8'9. Richardson l° also presents a detailed model for this effect. This second effect, which causes an additional heat flow away from the cold end heat exchanger, is caused by an enthalpy flow in the pulse tube in the direction of the wall. During the high pressure period the gas in the boundary layer will shrink due to the heat transfer to the wall, causing a volume flow outside the boundary layer in the direction of the wall. The component of this volume flow which is in phase with the pressure fluctuation causes a non-zero average enthalpy flow perpendicular to the wall. The enthalpy flow at the warm end of the pulse tube given by Equation (14) must, therefore, be augmented with the enthalpy flow perpendicular to the wall to give the enthalpy flow at the cold end heat exchanger. This effect appears to be similar to the heat pumping mechanism because it is dependent on the volume flow in the axial direction in the pulse tube which is out of phase with the pressure fluctuation. It does not, however, depend on the temperature gradient. This effect is described by Landau and Lifshitz 11 as the absorption of acoustic waves by a wall due to thermal effects. Both effects caused by thermal contact between the gas and the pulse tube wall are dependent on the volume flow out of phase with the pressure fluctuation p'. They also depend on the temperature fluctuations in the boundary layer, which are directly linked to the pressure fluctuation. In Figure 2 it is shown that the volume flow at the cold end heat exchanger, which has a phase difference of ~'/2 with p, is given by pwVt/TPo. The additional heat which is extracted from the cold end heat exchanger should be linearly dependent on p2wVtlTpo and is within a linear approximation independent of the enthalpy flow at the outer end of the pulse tube, ½P~2 (Equation (14)). If the results derived by Wheatley 6 and by Landau and Lifshitz 11 are written such that they are suitable for comparison with the present results, the following expression is obtained for the cooling power

(Q) = 2

a\w/

[2r +

Vg-]

pzo~v,17po (15)

where ~ is the thermal diffusivity of the gas, a the pulse tube radius and G a dimensionless measure for the temperature gradient along the wall

G = 1 - V T,~,~uc/AT

.i.i

5

O

A

• 0' V

4

1 3 2

0 0

1

AT/Ax = (7 -- 1)To/L

950

4

5

.

.

Cryogenics 1990 Vol 30 November

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Figure 4

Measured cooling power ( 0 ) of a closed pulse tube compared with values calculated according to Equation (15) ( )

a contact surface separating the cold and the hot gas will be found, across which there is a large temperature gradient. Therefore, linear theory will fail 12. To verify this theory the cooling power of the pulse tube refrigerator was measured as a function ofp:wVt/TPo in a closed pulse tube (~2 = 0). The conditions were the same as in the other experiments. The results are shown in Figure 4 and confirm the dependence of the cooling power o n p2¢oVt/',/po for a closed pulse tube. To verify the results for the cooling power for the orifice pulse tube described earlier, the total cooling power of the orifice pulse tube was calculated according to Equation (15) for each experimental point. The result is shown in Figure 3, where the solid line represents the values calculated according to Equation (15). The result is striking and proves that a difference of about 50% from the enthalpy flow according to Equation (14) can be explained. Note that the influence of the deviation of the pressure fluctuations p' from a pure harmonic fluctuation was not taken into account. The amplitude p was simply taken as the difference between pressure maxima and minima. A

rO .D Q.

I

I/2

3

(17)

where L is the length of the flow tube. Equation (15) is based on a boundary layer approximation which is valid only for a (d/co) < 1. Also, this theory Is not vahd for high amplitudes of gas displacements. For high amplitudes ]

2

(16)

V Tw is the temperature gradient along the pulse tube wall, Ax the amplitude of the displacement of the gas in the tube and ATthe amplitude of the temperature rise due to adiabatic compression of the gas. AT/Ax is in linear approximation given by

--

O

Q

0O

.

Figure 5

I

0,2

I

I

0,4

I

I

I

I

0,6 0,8 = time [s]

Pressure fluctuations in the pulse tube

Experimental verification of an analytical model: M.J.A. Baks et al. typical example of the time dependence of the pressure in the experiments is given in Figure 5.

Conclusions The cooling power of a pulse tube refrigerator can be expressed in terms of regenerator loss and average enthalpy flow through the pulse tube. Neglecting in a first approximation the heat exchange with the wall of the pulse tube, the enthalpy flow through the pulse tube is only dependent on the amplitudes of the pressure fluctuation in the pulse tube and the volume flow through the orifice, which is similar to the result Radebaugh presented. In contrast with his experimental results, which did not confirm the theory, a cooling power has been measured which is significantly larger than predicted by this calculated enthalpy flow. The deviation of the present experimental results from the simple theory is shown to be due to the thermal contact between the gas in the thermal boundary layer of the pulse tube and the wall of the pulse tube and can be explained by combining the theory of the heat pumping mechanism of Wheatley with an additional heat transfer to the wall corresponding to the absorption of acoustic waves on normal reflection at a hard wall.

The interpretation of the experiments is simplified by elimination of the influence of regenerator loss by keeping the cold end heat exchanger at ambient temperature. Further experiments will have to be done to take into account the influence of regenerator loss and to understand the discrepancy observed by Radebaugh.

Acknowledgement The authors thank Mr A.K. de Jonge for his support of this project.

References 1 2 3 4 5 6 7 8 9 10 11

Mikulin, E.L. Adv Cryog Eng (1984) 29 629 Liang, J. et al. Cryogenics (1990) 30 49 Radebaugh, R. et al. Adv Cryog Eng (1986) 31 779 Radebaugh, R. Jap J Appl Phys (1987) 26 2076 suppl 26-3 Storch, P.J. and Radebaugh, R. Adv Cryog Eng (1988) 33 851 Wheatley, J. et al. J Acoust Soc Am (1983) 74 153 Wheatley, J. et al. Am J Phys (1985) 53 147 Rott, N. J Appl Math Phys (1969) 20 230 Rott, N. J Appl Math Phys (1975) 26 43 Richardson, R.N. Cryogenics (1986) 26 331 Landau, L. and Lifshitz, E. M~canique des fluides l~ditions Mir, Moscow (1971) 382 12 Richardson, R.N. personal communication(1990)

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