A Comprehensive Model of a Miniature-Scale Linear Compressor for Electronics Cooling

TL;DR: In this paper, a model of a miniature-scale linear compressor for electronics cooling is presented, which incorporates all of the major components of the linear compressor including dynamics associated with the piston motion.

Abstract: A comprehensive model of a miniature-scale linear compressor for electronics cooling is presented. Linear compressors are appealing for refrigeration applications in electronics cooling. A small number of moving components translate to less theoretical frictional losses and the possibility that this technology could scale to smaller physical sizes better than conventional compressors. The model developed here incorporates all of the major components of the linear compressor including dynamics associated with the piston motion. The results of the compressor model were validated using experimental data from a prototype linear compressor. The prototype compressor has an overall displacement of approximately 3 cm 3 , an average stroke of 0.6 cm. The prototype compressor was custom built for this work and utilizes custom parts with the exception of the mechanical springs and the linear motor. The model results showed good agreement when validated against the experimental results. The piston stroke is predicted within 1.3% MAE. The volumetric and overall isentropic efficiencies are predicted within 24% and 31%, MAE respectively.

Summary

1. Introduction

• Miniature refrigeration systems offer distinct advantages for use in electronics cooling relative to other technologies.
• Trutassanawin et al. (2006) reviewed the available technologies for vapor compression systems in electronics cooling.
• It has been reported that the overall performance and size of these systems is still not at a level that is desired for desktop and portable electronic systems (Cremaschi et al., 2007).
• Early investigations of a linear compressor were conducted by Cadman and Cohen (1969a,b) for traditional refrigeration systems.
• Pollak et al. (1979) investigated one-dimensional, nonlinear dynamics of the piston and electrical systems and confirmed such confounding effects.

2. Model Formulation

• The major components of the linear compressor model are described in this section.
• The comprehensive compressor model consists of a solution to two compression process equations that provide the temperature and density, and thus fix the state within each control volume.
• These equations require inputs from the five sub-models representing the valve flows, leakage flows, motor losses, heat transfer from the cylinder, and piston dynamics.
• The compression cycle is discretized and an initial guess of temperature and density is made within each control volume.
• The compression process equations are then used to step through a compression cycle using the numerical techniques described in Section 2.7.

2.1. Compression Process Equations

• The compression process is modeled using mass and energy conservation over a control volume.
• In order to determine the state of the working fluid in the compression chamber at any point during the compression process, it is necessary to determine two independent fluid properties to fix its state.
• The first is the compression chamber for the compressor, while the second consists of the remaining volume within the compressor.
• With a further assumption that the changes in kinetic and potential energies are negligible, the left-hand side of Equation (1) can be expanded in terms of changes in internal energy and mass.
• Due to the nonlinear nature of these equations, a numerical solution approach is adopted.

2.2. Valve Model

• Reed valves are used in the present work as is typical for reciprocating compressors.
• The valve body is constructed to only allow valve motion in the direction of desired flow for each valve; this ensures that pumping occurs and can be seen in Figure 3.
• Early in its deflection the stagnation pressure driving the valve reed is assumed equal to the high side pressure.
• This valve lift is known as the transitional valve lift because below this value the pressure-dominated model is used and above the mass-flux dominated model is used.

2.3. Leakage Model

• The leakage model only focuses on leakage past the piston.
• The only other leakage paths, those past the reed valves, are ignored since they are negligibly small compared to the leakage past the piston (Kim and Groll, 2007).
• The piston leakage is modeled as an incompressible Couette-Poiseuille flow driven by the pressure difference across the piston and the movement of the piston.
• This assumption is valid because the flow Mach number in the simulations is found to be between 0.1 and 0.3 at the representative conditions in the experiment.

2.4. Heat Transfer Model

• The instantaneous heat transfer from each of the control volumes is calculated using the empirical approach of Fagotti and Prata (1998).
• By integrating these instantaneous heat transfer rates over the entire cycle, the total heat transfer from each of the two control volumes is calculated for use in the overall energy balance.

2.5. Vibration Model

• A linear compressor is a free-piston device for which the stroke is not fixed by a crank mechanism but is instead determined by chosen geometry, the linear motor, and the mechanical springs used.
• Both the desired linear motion of the piston as well as its undesirable rotation due to eccentricity in the mechanical springs are considered.
• The driving force, Fdrive, is the sinusoidal force applied from the motor.
• The stiffness associated with the gas, kgas, is determined by linearizing the force generated by the gas over an entire compression cycle.
• The effective damping term is made up of two components: a frictional term and the boundary work performed on the gas (Pollak et al., 1979).

2.6. Overall Energy Balance

• To ensure that all of the compressor components satisfy an energy balance, a thermal network is constructed to account for heat transfer from the compression chamber (Chen et al., 2002a, Kim and Groll, 2007, Mathison et al., 2008).
• The energy balance for the compressor assumes that the heat transfer between the two control volumes is negligible and that the heat only flows to the compressor shell.
• A lumped-mass thermal network can then be constructed consisting of a single lumped mass to represent the compressor shell with two heat inputs and one heat output to the ambient.
• The thermal network elements are also shown in Figure 2.
• This network adds the following relation which is solved simultaneously with the compression process equations.

2.7. Solution Approach

• The model developed above for the linear compressor consists of two non-linear firstorder differential equations for the compression process and one non-linear equation from the energy balance.
• An explicit closed-form solution is not available, and the equations are instead solved numerically.
• Once each sub-model has been called the overall compression process solver is called which solves Equations (5) to (7) and calculates the internal state in the compressor.
• The initial conditions in both control volumes were set to the inlet conditions for each operating condition that was tested.
• The number of iterations required for convergence varied between approximately 20 and 150 depending on operating conditions.

3. Experiments

• No linear compressors are commercially available in the capacity and pressure ranges desired for electronics cooling.
• The compressor was built using a moving-magnet type linear motor (H2W Tech).
• The prototype linear compressor was tested on a compressor load stand specifically built for testing miniature-scale compressors.
• The compressor operates between state points 1 and 2.
• This process was then repeated for subsequent compressor inlet conditions which are tabulated in Table 2.

3.1. Experimental Uncertainty

• A variety of measurements were obtained in the experiments including temperature and pressure at the suction and discharge ports, mass flow rate of refrigerant, piston stroke, frequency, and input power.
• The measurements of pressure, temperature, and stroke have absolute uncertainties of 4.6 kPa, 0.5 C, and 25.40 µm, respectively.
• The Coriolis mass flow meter has relative uncertainty values between 0.350 and 1.25 %, depending on the quantity of mass flow measured.
• The uncertainty in the reported efficiency values is calculated using an uncertainty propagation analysis (Fox et al., 2004).

4. Validation of Model Predictions

• The experimental results are compared to the predictions from the model.
• The model performance is quite sensitive to several parameters: the leakage gap between the piston and cylinder, g, the eccentricity of the piston, , the dry friction coefficient, f , and the linear motor efficiency, ηmotor.
• The predicted mass flow rates are shown in Figure 11 and agree with the measured values to within 20% MAE.
• The operating principles underlying the model and the prototype are identical, and therefore, the experiments serve the purpose of validating model predictions.
• The model developed in this work can be applied in the analysis of any linear compressor application.

5. Conclusions

• A comprehensive model of a miniature-scale linear compressor for electronics cooling applications is developed.
• The valve and leakage models are developed from first principles.
• Both the trends and quantitative values of the mass flow rate, volumetric, and overall isentropic efficiencies, respectively, are also predicted to within reasonable bounds.
• These parameters should be further investigated in an effort to determine an optimum design for electronics cooling.
• Starting with a desired operating condition, the model can be used to optimize the compressor stroke, cylinder diameter and leakage gap.

Figures

Table 2: Summary of experimental data.

Figure 11: Comparison of experimental and predicted mass flow rates.

Table 1: Linear compressor prototype parameters.

Figure 3: Photos of reed valve assembly used in compressor prototype.

Figure 2: Representation of the compression process in a linear compressor.

Figure 12: Comparison of experimental and predicted volumetric efficiencies.

Figure 7: A section view of the prototype linear compressor.

Figure 6: Solution flowchart of linear compressor model (equations solved at each step given in blocks).

Figure 8: Schematic diagram of miniature-compressor hot gas bypass load stand.

Figure 1: Pressure over enthalpy diagram of typical R134a miniature-scale refrigeration cycle for electronics cooling.

Figure 13: Comparison of experimental and predicted overall isentropic efficiencies.

Figure 9: Pressure enthalpy plot of compressor load stand operation.

Figure 5: Free body diagram of linear compressor piston assembly.

Figure 10: Comparison of experimental and predicted piston displacements.

Figure 4: Free body diagrams for the valve reed.

Full text

Purdue University
Purdue e-Pubs
Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering
1-1-2011
A Comprehensive Model of a Miniature-Scale
Linear Compressor for Electronics Cooling
Craig R. Bradshaw
Purdue University - Main Campus, cbradsha@purdue.edu
Eckhard A. Groll
Purdue University - Main Campus, groll@purdue.edu
Suresh V. Garimella
Purdue University, sureshg@purdue.edu
Follow this and additional works at: h5p://docs.lib.purdue.edu/herrick
4is document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for
additional information.
Bradshaw, Craig R.; Groll, Eckhard A.; and Garimella, Suresh V., "A Comprehensive Model of a Miniature-Scale Linear Compressor
for Electronics Cooling" (2011). Publications of the Ray W. Herrick Laboratories. Paper 1.
h5p://docs.lib.purdue.edu/herrick/1

A Comprehensive Model of a Miniature-Scale Linear Compressor
for Electronics Cooling
Craig R. Bradshaw
a,b,1
, Eckhard A. Groll
a,b
, Suresh V. Garimella
b
a
Herrick Laboratories, Purdue University, West Lafayette, IN 47907
b
Cooling Technologies Research Center, Purdue University, West Lafayette, IN 47907
Abstract
A comprehensive model of a miniature-scale linear compressor for electronics cooling
is presented. Linear compressors are appealing for refrigeration applications in electronics
cooling. A small number of moving components translates to less theoretical frictional losses
and the possibility that this technology could scale to smaller physical sizes better than con-
ventional compressors. The model developed here incorporates all of the major components
of the linear compressor including dynamics associated with the piston motion. The results
of the compressor model were validated using experimental data from a prototype linear
compressor. The prototype compressor has an overall displacement of approximately 3cm
3
,
an average stroke of 0.6 cm. The prototype compressor was custom built for this work and
utilizes custom parts with the exception of the mechanical springs and the linear motor. The
model results showed good agreement when validated against the experimental results. The
piston stroke is predicted within 1.3% MAE. The volumetric and overall isentropic efficiencies
are predicted within 24% and 31%, MAE respectively.
Keywords: linear compressor, compressor model, electronics cooling, miniature compressor
1
Corresponding Author, cbradsha@purdue.edu
Preprint submitted to International Journal of Refrigeration April 12, 2011

Nomenclature
A area [m
2
]
C
D
drag coefficient for flow past reed valve [−]
C
v
specific heat capacity at constant volume [J kg
−1
K
−1
]
D diameter [m]
E total energy in the control volume [kW ]
F
drive
driving force from linear motor [N]
J
CG
rotational moment of inertia for the piston about the CG [kg m]
M moving mass [kg]
MAE Mean absolute error [−]
P pressure of gas [kP a]
P
high
higher pressure [kP a]
P
low
lower pressure [kP a]
R
amb
thermal resistance between the compressor shell and ambient [W K
−1
]
Re Reynolds number[−]
T temperature of gas[K]
V volume of control volume [m
3
]
W work [J]
˙
Q total heat transfer into the control volume [kW ]
˙
W total work on the control volume [kW ]
2

˙m mass flow rate [kg s
−1
]
V gas velocity [m s
−1
]
f dry friction coefficient between piston and cylinder [−]
k thermal conductivity[W m
−1
K
−1
]
c
eff
effective damping of piston [N s m
−1
]
f resonant frequency of piston operation [Hz]
g leakage gap width [m]
h specific enthalpy [kJ kg
−1
]
k stiffness or spring rate [N m
−1
]
q heat flux[W m
−2
]
t time [sec]
u internal energy [kJ kg
−1
]
v specific volume of gas [m
3
kg
−1
]
x displacement [m]
Greek letter
α thermal diffusivity[m
2
s
−1
]
 eccentricity of spring force [m]
η efficiency [−]
γ heat capacity ratio[−]
ω
d
damped natural frequency[rad sec
−1
]
3

ρ density of gas [kg m
−3
]
θ rotational angle of piston [rad]
Subscripts
amb ambient
cv control volume
discharge gas discharged from compression chamber
exp experimental value
gas quantity resulting from gas
in into control volume
leak leakage
mech mechanical
o, is overall isentropic
out out of control volume
p piston
port valve port opening
shell compressor shell
suction suction gas
tr transitional
v constant volume process
valve reed valve
vol volumetric
4

Frequently asked questions

Q1. What are the contributions in "A comprehensive model of a miniature-scale linear compressor for electronics cooling" ?

A comprehensive model of a miniature-scale linear compressor for electronics cooling is presented. The prototype compressor was custom built for this work and utilizes custom parts with the exception of the mechanical springs and the linear motor. 

Q2. Why is vapor compression a viable solution for electronics cooling?

Because vapor compression utilizes two-phase heat transfer in the evaporator, it is possible to maintain spatially uniform chip temperatures. 

Q3. What are the key parameters that may explain errors in the model?

the sensitivity to small changes in the values of the key parameters identified in this work (i.e. leakage gap, eccentricity, motor efficiency, and dry friction coefficient) may also explain errors. 

Q4. What is the atypical challenge for refrigeration systems?

In electronics cooling applications, an atypical challenge for refrigeration systems is the relatively small temperature lift for cycle operation. 

Q5. What are the major concerns involving refrigeration systems?

The major concerns involving refrigeration systems are their cost and reliability, as well as miniaturization of the different components. 

Q6. Why was the prototype built in this work?

the purpose of the prototype built in this work was merely to validate the model and not to obtain optimized performance. 

Q7. What is the definition of a linear compressor?

A linear compressor is a free-piston device for which the stroke is not fixed by a crank mechanism but is instead determined by chosen geometry, the linear motor, and the mechanical springs used. 

Q8. What is the equation of motion for the pressure-dominated and mass flux-dominated modes?

Summing the forces in the direction of displacement, the equation of motion for the pressure-dominated and mass flux-dominated modes may be expressed as:Mvalveẍvalve + 12 CDρAvalveẋ2 valve + kvalvexvalve = (Phigh − Plow)Avalve +1 2 CDρV 2Avalve (8)Mvalveẍvalve +( 12 CDρAport + ρAvalve) ẋ2valve + kvalvexvalve = 12 CDρV2Avalve + ρVAport (9)These second-order, nonlinear equations can be solved for the position of the reeds at each time step throughout the compression process. 

Q9. What is the advantage of a linear compressor?

A linear compressor is appealing for electronics cooling applications because it offers several potential advantages over traditional compressor technology. 

Q10. What is the valve geometry used to calculate the leakage mass flow rate?

The leakage mass flow rate is calculated using the average leakage velocity obtained byintegrating the velocity profile as:ṁ = ρV̄leakAleak = ρAleak 1g ∫ g 0 Vleak(y)dy = ẋp 2 + g2 4µ ( −dP dx ) + g3 6µ ( dP dx ) (11)The instantaneous heat transfer from each of the control volumes is calculated using theempirical approach of Fagotti and Prata (1998). 

Q11. What is the work done on the gas?

Wfriction ωdxpπ(17)The work done on the gas is calculated by integrating the boundary work expression over the entire compression process (i.e. integrating pressure over volume ). 

Q12. What is the overall compression process solver?

Once each sub-model has been called the overall compression process solver is called which solves Equations (5) to (7) and calculates the internal state in the compressor.