numerical simulation and experimental verification of ...

Numerical simulation and experimental verification of a G-M type double inlet pulse tube refrigerator G. Q. Lu and P. Cheng Citation: AIP Conference Proceedings 613, 760 (2002); doi: 10.1063/1.1472092 View online: http://dx.doi.org/10.1063/1.1472092 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/613?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis of Loss Mechanisms in G‐M Type Pulse Tube Refrigerators AIP Conf. Proc. 710, 1451 (2004); 10.1063/1.1774838 An Experimental Set‐Up for Large‐Scale Pulse Tube Refrigeration AIP Conf. Proc. 710, 1301 (2004); 10.1063/1.1774818 Experimental study on GM/PT hybrid refrigerator AIP Conf. Proc. 613, 782 (2002); 10.1063/1.1472095 A five-watt G-M/J-T refrigerator for LHe target at BNL AIP Conf. Proc. 613, 776 (2002); 10.1063/1.1472094 4 K pulse tube refrigerator and excess cooling power AIP Conf. Proc. 613, 633 (2002); 10.1063/1.1472075

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NUMERICAL SIMULATION AND EXPERIMENTAL VERIFICATION OF A G-M TYPE DOUBLE INLET PULSE TUBE REFRIGERATOR

G.Q. Lu and P. Cheng Department of Mechanical Engineering Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong

ABSTRACT A 1-D transient numerical model has been developed to predict the performance and reveal the nonlinear dynamical characteristics of a G-M type double-inlet pulse tube refrigerator, where the oscillating amplitudes of the physical quantities are large. In this numerical simulator, governing equations consisting of the state equation, the conservation of mass and momentum in the fluid phase, as well as the energy equations for the fluid and the solid, are spatially and temporally conjugated. The boundary conditions and the initial conditions for these governing equations are discussed. The methods for the numerical discretizations of these governing equations are given. The assumption, that the refrigeration temperature at the cold-end heat exchanger is kept at a constant and known value during a cycle in the existing simulations, is released in our simulator. Instead, the refrigeration capacity is prescribed while the refrigeration temperature is determined from the numerical solution. Numerical results, such as cycle-averaged temperature distribution and fluctuations of the physical quantities in a single-stage G-M type pulse tube refrigerator, are analyzed. These numerical results are shown in good agreement with experimental data. This numerical simulator can be used not only to predict dynamical performance of a pulse tube refrigerator, but it can also be used to design a pulse tube refrigerator for optimal performance.

1.

INTRODUCTION The pulse tube refrigerator/cryocooler has many advantages over other cryocoolers due to its simplicity in construction with no moving part in the cold end, and therefore is more reliable for operation. For this reason, more and more attention has been paid on the design of these refrigerators/cryocoolers in recent years. One of the most important considerations in the design of these devices is to predict its performance. Because of the prohibitive computer time required for the numerical simulation of a two-dimensional transient viscous compressible flow model, 1-D transient numerical models have often been used for this purpose (see for example, Zhu 1994, Wang 1992, 1997 and Ju 1998). In these existing numerical models, the refrigeration temperatures at the cold-end heat exchanger were assumed to be at fixed and known values. In practice, however, there are two kinds of loadings at the cold-end heat exchanger of a pulse tube refrigerator. The first kind is those of constant heating power (such as an CP613, Advances in Cryogenic Engineering: Proceedings of the Cryogenic Engineering Conference, Vol. 47, edited by Susan Breon et al. © 2002 American Institute of Physics 0-7354-0059-8/027$ 19.00 760


electrical heater with constant power) which is mostly used in the experimental setup. The second is heat exchange between a specified loading and the cold-end heat exchanger with a finite heat transfer coefficient, which is commonly applied for cooling of an object in an industry application. In the above two cases, the solid temperature at the cold-end heat exchanger usually fluctuates with the fluid temperature, and will reach a cycle-steady temperature as the result of the oscillatory motion of the fluid. These equilibrium temperatures of the cold-end heat exchanger at the cycle-steady state, as well as the equilibrium temperatures in the regenerator and the pulse tube of a pulse tube refrigerator, can only be determined according to the cycle balances of mass, momentum and energy. Thus, in our numerical model, we have relaxed the assumption that the solid temperatures at the cold-end heat exchanger being at fixed and known values as in the previous models. In addition, the effect of the connecting tube between the cold ends of the regenerator and the pulse tube is also taken into consideration in our numerical model. 2.

GOVERNING EQUATIONS Fig.l shows a schematic diagram of a single-stage double-inlet pulse tube refrigerator. The geometrical sizes of the system are the same as those given in our experimental setup (Lu, 2001). In Fig.l, XH,R = 0, XQR = 150mm, XL,C =158.5mm, XR,C = 268.5mm, XC,P = 279mm, XH,p = 459mm and XR,H = 479mm. Solenoid Gas tanks operating

Double-inlet valve ———XI————

Straightener Cold-end heat exchanger Pulse tube

Hot.end

heat exchanger Reservoir Compressor

XH,R=O

FIGURE 1.

XC,R

XC,P

XR,H=479 x(mm) XH,P

Schematic diagram of a double-inlet pulse tube refrigerator

We will make the following assumptions in our numerical model: (a). The oscillating flow in a pulse tube refrigerator is 1-D flow. (b). Working fluid is an ideal gas; this assumption is a reasonable simplification when the refrigeration temperature is higher than 50°K for helium gas (Scott, 1988). (c). The physical quantities in the reservoir, in the highpressure tank and in the low-pressure tank can be taken as uniform. Furthermore, pressure fluctuations in the two gas tanks are small due to the large volumes of the tanks; consequently the pressures in these two tanks can be considered as constants at known values, which can be measured experimentally. Governing equations in different parts (such as the pulse tube, the regenerator, the heat exchangers) in a pulse tube refrigerator are given as follows. (1)

761


dt du dt

du dx

0

dx -

(2)

uu

dp dx

(3)

2dh

„ far dT} dp dp , u\u2 d (1 dT} 1 1 /_ _x pC A— + u—\ = — + u —+ (T-T) JHfp—— +— \k—\ + h — 7 r pp,f ^ I x* ^ 1J 3V 3V r V 5 ^ 'f(~dt'3, ~dx~)~~dt ~dx ~2d^ ~dx( ~dx~ J L,

"\

ox

I

~

T

\

~ 5 / - * S

(4)

V

'

where p is pressure, T is temperature, u is velocity, p is density, R is gas constant, x and t are axial and temporal coordinates, respectively. In addition, k is thermal conductivity, Cp is specific heat, dh is hydraulic diameter, while h is the heat transfer coefficient and f is the friction factor. Subscripts "f' and "s" indicate the quantity associated with the solid phase (wall) and the fluid phase, respectively. LI and L2 in Eqs.(4) and (5) are the characteristic lengths of heat transfer, defined as: Control volume, F(ra 3 ) L = ——————————————————— Heat transfer area, F(m )

(6)

If the porous medium in the regenerator is made of wire screens, and if dp denotes (p-d dD wire diameter and cp denotes porosity, then L, - ———— and L2 - —-. In Eq.(5), qs 4(1 -(p) 4 indicates the heat source term which has different values for different parts of the system. The heat source term indicates heat transfer between solid and cooling water in the hot-end heat exchanger. This term is zero in the pulse tube or in the regenerator or in the connecting tube due to thermal insulation. The heat source term in the cold-end heat exchanger denotes refrigeration capacity per volume of the solid. In our experiments, electrical heater was used to balance refrigeration capacity; hence the refrigeration capacity was a known value by measuring the power of the electrical heater. In Eqs.(l)-(5), thermal properties such as thermal conductivities and specific capacities for both fluid and solid, and viscous coefficients of the fluid are functions of temperature, which have been correlated according to the data given in the literature (Scott, 1988). In addition, we have used the empirical results of the interfacial heat transfer and friction coefficient, given by Kays and London (1980), for the flow through the porous medium in our model. This is based on the fact that the flow behaved as if it were at a quasi-steady flow in the G-M type pulse tube refrigerator where the oscillating frequency is relatively low. Furthermore, we used the transient velocity in calculating the instantaneous Reynolds number at different time levels to reflect the transient flow characteristics. The heat transfer coefficient and the friction factor in the void tube (such as in the pulse tube and in the connecting tube), and the heat transfer coefficient for natural convection occurring between the surface of the reservoir and the ambient are all from the literature (Barron, 1999).

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3. BOUNDARY CONDITIONS, INTIAL CONDITIONS AND DISCRETIZATIONS

Boundary Conditions Eqs.(l)-(5) can be applied to different parts of a pulse tube refrigerator, by taking into consideration different values of friction factor, heat transfer coefficient, energy source term, and characteristic lengths of heat transfer. Because different parts in the pulse tube refrigerator may have different cross-section areas, the following continiuity relations will be imposed at any interface beteween two different parts:

p+ = p~,

P+ - P~',

j+=j-^

j,+ _j.- ^

p+u+A+=p~u~A~

(7)

where the superscripts "+" and "-" indicate right-hand side and left-hand side of the interface between sub-components of the system, respectively.

We now prescribe the boundary conditions at the hot end of the regenerator (X=XH,R=O). The fluid temperature boundary condition can be determined by the upwind method (Wang 1992; Ju 1998). For the solid temperature at this boundary, we can assume that there is an imaginary point outside of the regenerator whose temperature is at room temperature. Using energy equation (5), we can determine the boundary value of the solid temperature. The pressure at this boundary can be obtained from the Darcy law; we have —Y = 0 which is similar to those of a fully developed flow of an incompressible fluid at the exit of a tube. The boundary condition of the density can be calculated according to Eq.(l) after the boundary conditions of the temperature and the pressure are known. The mass flow rate at the boundary x=0 can be obtained: •

•

•

/o\

m = m S —wid •

(o) ^ '

•

where ms and md denote the mass flow rates through the solenoid valve and the doubleinlet valve, respectively, which can be formulated according to the flow characteristic through the valve (Lu and Cheng, 2000). For example: /2

m. =

RT

PH > Pi (9)

Pi > PL

where PH and PL are the pressures in the high-pressure tank and the low-pressure tank, respectively, which can be measured experimentally as known quantities. In Eq.(9), As is the cross-section area of the solenoid valve, and Cd is the coefficient which can be determined experimentally. Because of the inertia, the area As of the solenoid valve depends on time during the opening and closing of these valves (Lu, 2001). The boundary conditions for the solid temperature, the fluid temperature and the pressure on the right-hand side of the hot-end heat exchanger (x = XR,H) can be specified by the same way as those boundary conditions at the hot end of the regenerator. The mass flow rate at x = XR,H is:

763


m-mo-md

(10)

where m0 is the mass flow rate through the orifice valve, which depends on the temperature and pressure in the reservoir. In order to close the problem mathematically, we now consider the governing equations for the reservoir: PrVr=RmrTr

(11)

dm^ dt

(12)

,

;o