MODELING AND DEVELOPMENT OF A ...

MODELING AND DEVELOPMENT OF A SUPERFLUID MAGNETIC PUMP WITH NO MOVING PARTS

A. E. Jahromi1, F. K. Miller1, and G. F. Nellis1 1

Solar Energy Laboratory, University of Wisconsin-Madison, WI, 53706, U.S.A.

ABSTRACT Current state of the art sub Kelvin Superfluid Stirling Refrigerators and Pulse tube Superfluid Refrigerators use multiple bellows pistons to execute the cycle. These types of displacers can be replaced by a newly introduced pump, a Superfluid Magnetic Pump, with no moving parts. Integration of this pump in the Pulse tube Superfluid Refrigeration system will make it a sub Kelvin Stirling refrigeration system free of any moving parts that is suitable for use in space cooling applications. The Superfluid Magnetic Pump consists of a canister that contains Gadolinium Gallium Garnet particles that is surrounded by a superconducting magnetic coil. The driving mechanism of this pump is the fountain effect in He II. A qualitative description of one cycle operation of the Superfluid Magnetic Pump is presented followed by a numerical model for each process of the cycle. KEYWORDS: Superfluid Magnetic Pump, Fountain effect pump, Helium II pump, Sub Kelvin refrigerator, Superfluid pump, Pump with no moving parts INTRODUCTION The current state of the art detectors for space science infrared and x-ray missions are cryogenic detectors; either microwave kinetic inductance detectors or micro-calorimeters with transition edge sensors (TES). Both detector types require operation at sub Kelvin temperatures for the highest sensitivity applications. The current sub Kelvin refrigeration systems that use helium as a working fluid have one common feature: they must either displace or compress and expand the helium working fluid. Examples of such sub Kelvin coolers are the cold cycle dilution refrigerator, Active Magnetic Regenerative Refrigerator, Superfluid Stirling Refrigerator (SSR), and Pulse tube Superfluid Refrigerator (PSR). We

will discuss the latter two refrigerators as an example for the application of the Superfluid Magnetic Pump (SMP). The SSR uses the Stirling cycle to attain sub Kelvin temperatures by compressing and expanding 3He in a 3He-4He mixture as a thermodynamic working fluid [1]. This refrigerator uses bellows pistons on the cold and warm ends of the system. A new type of SSR was later introduced that replaced the single acting bellows pistons with double acting at either ends of the system [2]. The performance of this type refrigerator was verified by Watanabe et. al. in which it cooled to 168 mK [3]. Later a new type of refrigerator, the PSR, was developed by replacing the double acting bellows pistons with two pulse tubes, two heat exchangers, and an orifice at the cold end of the system [4]. Despite its low efficiency compared to the SSR, the absence of moving parts from the cold end played a significant advantage that outweighed the lower efficiency for potential space applications. FIGURE 1 shows a schematic of the PSR, earlier proposed by Miller et.al. [5]. This refrigerator eliminates the remaining moving parts from the earlier versions of the PSR by replacing the bellows pistons with the SMP at the warm end of the system. The system is two refrigerators that operate 180 degrees apart from each other. A 3He-4He mixture is used as the thermodynamic working fluid. An SMP, a cold heat exchanger, a pulse tube and two warm heat exchangers are used in each half of the refrigerator. An orifice and a recuperative heat exchanger connect the two halves of the refrigerator at the cold and warm end of the system respectively. The SMP acts as a compressor in the Stirling cycle used to attain sub Kelvin temperatures in this newly designed PSR.

FIGURE 1. The newly proposed Pulse tube Superfluid Refrigerator (PSR), HX denotes heat exchanger; TH and TC denote the hot and cold temperature platform respectively.

FIGURE 2. A schematic of the SMP in the proof of concept experimental setup.

APPARATUS A qualitative description of a cycle that uses the SMP as a compressor was presented in an earlier work by Miller et. al. [5]. In this work we present a numerical model for the operation of the SMP during one thermodynamic cycle. Our SMP consist of a 4.5 cm diameter and 7.5 cm tall canister that is filled with Gadolinium Gallium Garnet (GGG) particles. A superconducting magnetic coil surrounds the canister. This magnetic coil will be able to apply magnetic flux densities as high as 3 Tesla during the pump’s operation. The canister has two ports; the inlet houses a vycor glass superleak (a porous piece of vycor glass) which will only allow superfluid helium to pass through it, and a normal opening that allows He II to enter or exit the canister. The proof of concept experiment will use only one of the two SMP’s shown in FIGURE 1. A schematic of the future proof of concept experimental setup is shown in FIGURE 2. The experiment consists of one SMP, a vycor glass superleak, a pre-cooler/heater, a Venturi flow meter and a capacitor type pressure gauge that will be used to measure the pressure difference between the inlet and the throat of the flow meter for mass flow metering purposes. We discuss the operation of only one SMP for the rest of this paper. PUMP OPERATION AND MODELING The operation of the SMP during one cycle consists of four processes: I. Adiabatic Magnetization, II. Isothermal Magnetization, III. Adiabatic Demagnetization, and IV. Isothermal Demagnetization. A description and model of each process is described: I.

Adiabatic Magnetization

The constituents of the SMP, the GGG particles and the helium surrounding them, are at state 1, (Tlow) and (Bmin=0 Tesla) at the beginning of this process. No heat is transferred to and from the canister therefore the constituents of the SMP are undergoing an adiabatic process. Electric current is sent through the superconducting magnetic coil, thus increasing the externally applied magnetic flux density on the canister. By increasing the magnetic flux, the temperature of the GGG particles must increase during this adiabatic process. Heat transfer between the GGG particles and the surrounding helium causes the helium temperature to rise along with the GGG. The magnetic flux is increased until the SMP’s

constituents reach a new equilibrium temperature at state 2, (Thigh). The magnetic flux density has increased to an intermediate value at (Bint1) by the end of this process. This value is calculated by applying an entropy balance on the constituent’s of the canister:

mHe ΔsHe,1 to 2 + mGGG ΔsGGG ,1 to 2 = 0

(1)

where mHe and mGGG is the mass of helium and GGG within the canister. Lounasmaa provides an equation for the specific entropy of GGG [6] and Miller et al. provides an equation for the specific entropy of helium [7]. Note the magnetization time during this process is short so we assume that no mass exits the canister during the two adiabatic processes. II.

Unknown Field Code Cha

Isothermal Magnetization

The electric current through the superconducting magnetic coil is increased, further increasing the externally applied magnetic flux density on the canister. This magnetization continues until state 3, (Bhigh). The additional magnetic field continues to heat the GGG particles. This heat is transferred from the GGG particles to the surrounding helium. Energy is conserved for the helium within the canister therefore in order to keep this an isothermal process the additional energy in the helium within the canister must be equal to the net amount of energy (enthalpy) flowing out. The normal component of helium carries energy out of the canister and only superfluid helium at ground state enters the canister through the vycor glass superleak and replenishes the helium that has left the canister. During this process the pre-cooler/heater exchanges heat with the helium inside the closed SMP loop once it exits the canister. An energy balance on the helium within the canister yields: 3

min, II hsup erfluid − mout , II h3 − mGGG ∫ T ds = 0

(2)

2

Unknown Field Code Cha

where hsuperfluid and h3 are the enthalpy of superfluid helium and helium at state 3 respectively. Conservation of mass requires the mass of helium entering (min,II) and leaving (mout,II) the canister to be equal to one another. The third term on the left hand side of Eq. (2) is the heat transfer from the GGG to the surrounding helium during this isothermal process. Note that superfluid flow limitation through the vycor glass superleak is discussed later in this paper. III. Adiabatic Demagnetization The electric current is decreased through the superconducting magnetic coil thus the externally applied magnetic flux density on the canister is decreased. No heat transfer occurs between the SMP’s constituents and its surroundings, therefore the process is adiabatic. By decreasing the magnetic flux, the temperature of the GGG decreases. Heat transfer between the GGG and the surrounding helium causes the helium temperature to fall. The magnetic field is decreased until the temperature of the SMP’s constituents’ returns to state 4, (Tlow). The magnetic flux has decreased to a second intermediate value (Bint2) by the end of this process. Similar to the adiabatic magnetization, this value is calculated by applying the following entropy balance on the constituents of the canister:

mHe ΔsHe,3 to 4 + mGGG ΔsGGG , 3 to 4 = 0

(3)

Unknown Field Code Cha

IV. Isothermal Demagnetization The electric current through the superconducting magnetic coil is decreased, further decreasing the externally applied magnetic flux density on the canister until it returns to state 1, (Bmin=0). By decreasing the externally applied magnetic flux the GGG particles continue to cool throughout this process. Therefore heat is transferred from the helium to the GGG. During this isothermal process energy is conserved for the helium within the canister. Therefore the amount of energy reduction in the helium within the canister must be equal to the net energy flowing in. The normal component of helium carries energy into the canister and only superfluid helium at ground state flows out through the vycor glass superleak. The pre-cooler/heater exchanges heat with the superfluid helium that exits the canister. An energy balance on the constituents of the canister yields: 1

min, IV hp _ ch − mout , IV hsup erfluid − mGGG ∫ T ds = 0

(4)

4

Unknown Field Code Cha

where the subscript p_ch denotes the pre-cooler/heater. Note that similar to the adiabatic magnetization process, the demagnetization time for this process is significantly shorter than that for the isothermal process therefore mass displacement during this process is ignored. The cycle must be thermodynamically closed in order to determine the required temperature of the pre-cooler/heater. An energy balance on the helium within the SMP loop is: IV

∑Q

He ,canister ,i

i=I

+ (QHe, p _ ch , II + QHe , p _ ch , IV ) = 0

(5)

where the first term on the left hand side is the total heat transfer to the helium within the canister for the entire cycle and the second term is the total heat transfer to the helium in the SMP loop and in the pre-cooling/heating stage. FIGURE 3 shows a schematic of the SMP undergoing the four different processes described above. Flow Limitation through the Superleak The velocity of superfluid helium through the vycor glass superleak is limited to values below the critical velocity of superfluid helium otherwise it can no longer flow through the porous medium. According to investigations by Nakai et. al. a critical mass flow rate associated with the critical velocity through porous materials such as a vycor glass superleak in a thermomechanical pump is experimentally obtained [8]:

m! c = 2.4 × 10−3[m5/4 / s]

ρ ε Ac 4

d

(6)

where ρ is the fluid density, ε is the porosity, Ac the cross sectional area and d is the average pore diameter of the porous medium. Therefore for the vycor glass superleak with a diameter of 0.63 cm sized for this work, the critical mass flow rate will be 439 mg/s. In order for the pump to operate accordingly the mass flow rate produced must be kept below this value during the cycle for the prototype used in this work.

Unknown Field Code Cha

FIGURE 3. The four processes of the SMP during one cycle.

RESULTS The dimensions used for the SMP in the model matches with our prototype SMP. High porosities in the canister result in a lower mass flow rate produced by this pump and low porosities create a more resistive flow path for helium. Our calculations show optimal results for a porosity on the order of 30%. The porosity estimated for this model is 38%. Our model initially accounted for a maximum externally applied magnetic flux density of 3 Tesla. The maximum and minimum temperatures within the canister were set to 1.5 and 1.9 K respectively. Our model predicts a mass flow rate of 80 mg/s for a cycle time of 100s for the specified conditions. The pre-cooler/heater temperature must be set at 1.638 K with the specified conditions. FIGURE 4 shows a parametric study of the mass flow rate produced by the SMP. We varied the externally applied magnetic flux density between 1.5 and 6 Tesla for various 600

Thigh=1.9 Thigh=1.75 Thigh=1.6

500

mout [mg/s]

400 300 200 100 0 1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Externally applied magnetic flux density [Tesla]

FIGURE 4. Mass flow rate produced by our SMP as a function of externally applied magnetic flux density for various maximum temperatures. The low temperature is set to 1.5 K and the cycle time is 100s.

maximum temperatures within the canister while holding the rest of the parameters constant. This plot shows two effects: a higher mass flow rate is produced by lowering the maximum temperature, and the mass flow rate reaches an asymptote for higher externally applied magnetic flux densities. The SMP produces higher mass flow rates for lower maximum temperatures at constant externally applied magnetic flux densities because less energy is spent for heating or cooling the GGG and consequently the surrounding helium during the two adiabatic processes and more energy is spent to drive helium in and out during the two isothermal processes. Regardless of the maximum temperature within the canister, an externally applied magnetic flux density in the range of 2 to 4 Tesla makes most use of the mass flow rate production for our SMP. A parametric study yields FIGURE 5 which shows the mass flow rate produced by our SMP as a function of the pressure difference calculated according to the fountain pressure of helium presented by Miller et. al. [7]. This pressure difference corresponds to the pressure of helium inside and outside of the canister as a function of temperature at various externally applied magnetic flux densities while holding the rest of the parameters constant (Tlow is kept constant at 1.5 K and Thigh varies). In order to produce higher mass flow rates and create a higher pressure difference, the externally applied magnetic flux density must be higher. Note that higher pressure difference corresponds to higher maximum temperature within the canister. The pressure difference becomes important for applications where compression is desired. Another parametric study yields FIGURE 6 which shows the compression power, volume of helium displaced multiplied by the pressure difference of helium due to the fountain effect inside and outside of the canister, that is produced by our SMP as a function of maximum temperature of the constituents of the canister at various values of externally applied magnetic flux densities while keeping other parameters constant (Tlow is kept constant at 1.5 K and Thigh varies). At lower maximum temperatures, the mass flow rate produced by the pump is higher while the pressure difference between the helium inside and outside of the canister is low (the volume displaced outweighs the pressure difference). At higher maximum temperatures, the mass flow rate produced by the pump is lower while the pressure difference is higher (the pressure difference outweighs the volume displaced).

FIGURE 5. Mass flow rate produced by our SMP as a function of pressure difference between the maximum and minimum temperatures for the back to back SMP’s at various externally applied magnetic flux densities.

FIGURE 6. Compression power produced by our SMP as a function of maximum temperature at various externally applied magnetic flux densities. The maximum (peak) power is also shown for flux densities between 1.5 and 4 Tesla.

The compression power, the displaced volume multiplied by the pressure difference, produces a maximum value between these two behaviors. The light dashed line shows the peak compression power for intermediate flux densities between 1.5 and 4 Tesla as a function of the maximum temperature. CONCLUSIONS The results from our model show that our SMP is capable of producing a mass flow rate as well as a compression work to drive the helium in the sub Kelvin refrigeration systems including the Stirling and superfluid pulse tube. Depending on the application, the results can be used to optimize the pump performance for either circulation or compression purposes. Our model will be used to verify the pump performance in a future experiment to be conducted at the UW-Madison and will enable us to modify the pump based on the experimental results if necessary. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

Kotsubo, V. and Swift, G.W. Journal of low temperature physics 83, pp. 217-224 (1991). Brisson, J.G., Kotsubo, V. and Swift, G.W. Physica B 194-196, pp. 45-46 (1994). Watanabe, A., Swift, G.W. and Brisson J.G., “Measurements with a recuperative superfluid Stirling refrigerator,” in Advances in Cryogenic Engineering 41 B, edited by P. Kittel, Plenum, New York, 1996, pp. 1527-1533. Watanabe, A., Swift, G.W. and Brisson J.G., “Superfluid orifice pulse tube refrigeration below 1 Kelvin,” in Advances in Cryogenic Engineering 41 B, edited by P. Kittel, Plenum, New York, 1996, pp. 1519-1526. Miller, F.K. and Brisson J.G., “A Superfluid Pulse Tube Driven by a Thermodynamically Reversible Magnetic Pump,” in cryocoolers 15, edited by S.D. Miller and R.G. Ross, ICC Press, Boulder, 2009, pp. 519-523. Lounasmaa, O.V., Experimental principles and methods below 1K, Academic Press, London, 1974, pp. 87-90. Miller, F.K. and Brisson J.G., Cryogenics 41, pp. 311-318 (2001). Nakai, H., Kimura, N., Murakami, M., Haruyama, T. and Yamamoto, A., Cryogenics 36, pp. 667-673 (1996).