Applied Energy xxx (2009) xxx–xxx

ARTICLE IN PRESS

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Applied Energy

journal homepage: www.elsevier .com/locate /apenergy

Liquid piston gas compression

James D. Van de Ven a,*, Perry Y. Li b,1

a Worcester Polytechnic Institute, Department of Mechanical Engineering, 100 Institute Rd., Worcester, MA 01609, USAb Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN 55455, USA

a r t i c l e i n f o

Article history:Received 19 March 2008Received in revised form 25 November 2008Accepted 2 December 2008Available online xxxx

Keywords:Liquid pistonGas compressionAir compressorCompression efficiency

0306-2619/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.apenergy.2008.12.001

* Corresponding author. Tel.: +1 508 831 6776; faxE-mail addresses: vandeven@wpi.edu (J.D. Van de

Li).1 Tel.: +1 612 624 4992; fax: +1 612 626 7165.

Please cite this article in press as: Vaj.apenergy.2008.12.001

a b s t r a c t

A liquid piston concept is proposed to improve the efficiency of gas compression and expansion. Becausea liquid can conform to an irregular chamber volume, the surface area to volume ratio in the gas chambercan be maximized using a liquid piston. This creates near-isothermal operation, which minimizes energylost to heat generation. A liquid piston eliminates gas leakage and replaces sliding seal friction with vis-cous friction. The liquid can also be used as a medium to carry heat into and out of the compressionchamber. A simulation is presented of the heat transfer and frictional forces for a reciprocating pistonand a liquid piston. In the application of an air compressor, with a pressure ratio of 9.5:1 and a cycle fre-quency of 20 Hz, the liquid piston decreased the energy consumption by 19% over the reciprocating pis-ton. The liquid piston and the reciprocating piston exhibited a total efficiency of 83% and 70%respectively. The liquid piston demonstrated significant improvements in the total compression effi-ciency in comparison to a conventional reciprocating piston. This gain in efficiency was accomplishedthrough increasing the heat transfer during the gas compression by increasing the surface area to volumeratio in the compression chamber.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction/background

When rapidly compressing a gas to a specific pressure, withminimal heat transfer, significant energy is converted into increas-ing the gas temperature. As the compressed gas cools at constantpressure in a storage reservoir, the potential energy of the gas de-creases, reducing the efficiency of the overall cycle. By increasingthe heat transfer during compression to near-isothermal condi-tions, the energy required to compress the same mass of gas tothe same final, cooled state decreases. The reduction in work doneon the gas improves the efficiency of the gas compression orexpansion process and enables efficient energy storage throughgas compression.

Current applications involving compression and expansion ofgases, such as air compressors, internal and external combustionengines, and air motors, typically utilize mechanical methods ofsealing the gas while changing the volume. These mechanical com-pression and expansion methods, such as the reciprocating piston,screw compressors, and vane motors, provide poor heat transferbetween the mechanical boundaries and the gas. The heat transferis poor in these machines due to the high frequency operation andthe practical geometry requirements that dictate a poor surfacearea to volume ratio.

ll rights reserved.

: +1 508 831 5680.Ven), pli@me.umn.edu (P.Y.

n de Ven JD, Li PY, Liqu

Heat transfer during reciprocating piston compression has beeninvestigated extensively. The heat transfer within a compressioncylinder is quite complex due to the varying flow regimes withinthe cylinder and the varying convection coefficient [1]. The analy-sis is further complicated by the fact that during compression andexpansion in a reciprocating cylinder, the heat transfer can be outof phase with the temperature difference between the bulk gas andthe wall [2,3]. A prime reason that the bulk gas temperature doesnot closely follow the wall temperature is that the surface areato volume ratio in reciprocating piston systems is generally quitelow. There have been efforts to improve this surface area to volumeratio by using a small diameter cylinder with a long stroke or alarge diameter cylinder with a short stroke. Attempts to improvethis ratio have also included placing protrusions on the movingpiston that slide into cavities in the cylinder endplate, increasingthe surface area of the compression space [4].

Beyond poor heat transfer, mechanical methods of compressingand expanding gases suffer from a design trade-off between gasleakage and high sealing friction. The field of seal design has seenmuch concentration on both modeling and experimental work.Work in this area has ranged from attempting to minimize frictionthrough seal geometry, analyzing seal friction and power loss asso-ciated with various lubricants, experimentally determining the lifeof seals before failure, and analyzing the leakage of various seal de-signs [5–9]. There has been a large amount of work done specifi-cally on internal combustion engine seals, including analyzingfriction and sealing for various piston ring designs [10]. Akalinand Newaz present a more extensive literature review on the

id piston gas compression, Appl Energy (2009), doi:10.1016/

Fig. 1. Liquid piston configuration where a hydraulic pump drives two liquid pistonchambers using a switching valve. In this setup, one chamber is always filling whilethe other is emptying.

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tribology between a piston ring and cylinder bore in an internalcombustion engine [11]. Balancing seal friction with leakage is animportant issue for other mechanical compression and expansionmethods as well, such as scroll compressors, screw compressors,and vane compressors [12–14].

The liquid piston is certainly not a new concept. The earliestknown use, dating back to 1906, was in an internal combustion en-gine used for pumping water known as the Humphrey pump [15].The Humphrey pump ran on an Atkinson cycle and demonstratedefficiencies between 5% and 10% [16]. Another water pumping en-gine has utilized a liquid piston is the fluidyne Stirling cycle. Theprimary research related to this engine has focused on tuning theoscillating frequency of the liquid piston columns and design fordependable operation for remote environments [17–19]. FurtherStirling engine concepts [20,21] and Stirling heat pump concepts[22,23] can be found in patents. None of these works have dis-cussed exploiting the liquid piston to improve the heat transfer be-tween the gas and the working chamber.

One work that utilized a liquid piston to improve heat transferin a Stirling engine is by Gerstmann and Hill. Their work recog-nized the need to improve the surface area to volume ratio in theworking chamber. They proposed using an auxiliary pump toatomize the liquid and spray it into the compression and expansionchambers to improve heat transfer between the liquid and the gas.In the introduction of their concept, they mentioned modifying thechamber geometry to increase the surface area to volume ratio, butdismissed the idea based on the belief that the resultant viscousforces would be unacceptable [24].

Fig. 2. Multiple liquid pistons individually coupled to the pistons of a hydraulicmotor. In this figure, each piston of the radial piston controls the liquid pistoncolumn in a separate gas chamber.

2. Liquid piston gas compression/expansion concept

At the basic level, the liquid piston gas compression concept uti-lizes a column of liquid to directly compress a gas in a fixed volumechamber. The liquid piston eliminates the mechanical sliding sealsassociated with kinematic compressors. By replacing these sealswith a liquid column, gas leakage is eliminated and the sliding fric-tion is replaced with viscous fluid forces. While the majority of thispaper will focus on gas compression, the liquid piston can also beused for the expansion of gas, such as in internal and external com-bustion engines and air motors. Recognize that the gas tempera-ture of various applications will influence the selection of theoperating liquid.

The liquid piston can be driven through a variety of methods. Inthe simplest form, the liquid piston acts as a direct hydraulic topneumatic transformer. In the transformer arrangement, the liquidpiston is driven as part of a hydraulic system where the flow in andout of the liquid piston chamber is controlled with valves. To cou-ple the liquid piston to a rotating shaft, analogous to an electricmotor driving an air compressor, a hydraulic pump is used to gen-erate the flow of the liquid pistons. The liquid pistons can be con-trolled with a valve by switching the inlet and outlet of thehydraulic pump between two chambers, similar to previous workby Lemofouet and Rufer [25]. In this arrangement, as seen inFig. 1, one liquid piston is emptying while the other is filling. Inan alternative arrangement, a liquid piston can be coupled to eachcylinder of a hydraulic pump, such as a radial piston unit, wherethe cylinders are stationary. In this configuration, previously pro-posed by the author and colleagues, seen in Fig. 2, each hydraulicpiston directly drives the liquid column into and out of the gaschamber [26].

Because the liquid column can conform to an irregularly shapedgas chamber, the heat transfer between the gas and the gas cham-ber can be drastically improved by using a liquid piston. To realizethe possible heat transfer benefits of the liquid piston, the surfacearea to volume ratio of the chamber needs to be drastically in-

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creased. This can be accomplished using numerous small diametercylinders, fins, a mesh, or a variety of other geometries. The designof the chamber geometry must balance increasing the surface areato improve heat transfer with increasing the viscous forces createdby small passages. This balance of heat transfer to viscous forcesrequires a design optimization for each specific application.

To be successful, there are a few complexities of the liquid pis-ton gas compression system that need to be addressed. First, be-cause a gas and a liquid are in direct contact at high pressures, aportion of the gas will become entrained in the liquid. In a hydrau-lic system, this can create problems with decreasing the bulk mod-ulus of the liquid and possibly causing cavitation in low pressureareas of the system. This issue can be addressed through a varietyof manners. First, the liquid chamber needs to be designed to min-imize splashing of the liquid, especially for high frequency opera-tion. Second, a fluid can be selected with a low gas solubility, tominimize the gas entrainment. A final option it to use a bladder

id piston gas compression, Appl Energy (2009), doi:10.1016/

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or diaphragm in the liquid column that separates the liquid pistonfluid from the hydraulic fluid and allows transfer of pressure acrossthe diaphragm with minimal loss. Alternatively a steppedpiston with two different areas could be used to separate the liquidpiston fluid from the hydraulic fluid while creating pressureintensification.

Another possible issue that can arise from a liquid piston gascompression system is liquid leaving the gas chamber whenexhausting the gas through valves. The first approach to this issueto minimize splashing through the geometric design of the gaschamber, making the division between the gas and the liquid welldefined. A second approach to this issue is to provide an auxiliarymethod of separating the gas and the liquid with a liquid trap inthe gas passage outside of the chamber. These and other issues cer-tainly do need to be addressed for a successful liquid piston sys-tem, but they are not insurmountable.

Fig. 3. Reciprocating piston machine demonstrating the conceptual air reservoir.The reservoir is a constant pressure chamber, regulated by the weight acting againstthe moving boundary. Once the gas is pumped out of the working chamber, it coolsto ambient temperature in the reservoir, while maintaining a constant pressure.

3. Analysis

The liquid piston gas compression concept will be demon-strated through the analysis of a typical application of gas com-pression, an air compressor for industrial pneumatic applications.To provide a contrast to existing technology, a single compressioncycle will be analyzed for both a conventional reciprocating pistonchamber and a liquid piston compression chamber. The analysiswill estimate the heat transfer from the gas to the compressionchamber and estimate the frictional forces due of the mechanicalsliding of the reciprocating piston and the viscous drag of the liquidpiston. Because the purpose of this analysis is to illustrate the li-quid piston concept, assumptions will be made to simplify theanalysis in the interest of developing understanding.

The application for this analysis is a single stage air compressorwith a single cylinder used for industrial pneumatic applications.The air compressor will intake air from atmosphere at 101 kPaand 20 �C and compress it with a pressure ratio of 9.5:1 to 960 kPa.When running at a frequency of 20 Hz (1200 RPM), the compressorintakes 0.025 m3/s of air at atmospheric temperature and pressure.To intake this volumetric flow rate, the displaced piston volume is1.25 l.

Once the gas in the compression chamber reaches the pressureratio, it is pumped out of the chamber at a constant pressure into astorage reservoir. In the physical system, the gas will maintain aconstant pressure in the reservoir as additional gas is added andother gas removed for various industrial processes. Due to the sizeof the tank, the compressed gas cools to near atmospheric temper-ature before exiting the tank. This physical system is modeled for asingle cycle by pumping the air out of the compression chamberinto a reservoir, which is maintained at a constant pressure by amoving boundary loaded by a weight, as seen in Fig. 3. Once thegas is pumped into the reservoir, the gas cools at a constant pres-sure while the volume decreases. The cool pressurized air is thenreleased from the chamber and expanded to do work. The maxi-mum potential energy of the compressed cool gas in the reservoiris quantified by assuming an isothermal expansion.

The following section develops the equations for the numericalmodel analyzing the heat transfer and the energy loss due to fric-tion during gas compression. The piston displacement will be sinu-soidal with a phase angle of interest between h = 0 and h = p. Thenumerical model will use 1000 evenly spaced steps across the dis-placement of interest.

3.1. Heat transfer analysis

The heat transfer analysis for the reciprocating piston and theliquid piston will be developed together by applying the same

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equations to different geometries. The reciprocating piston cham-ber dimensions will be ‘‘square,” meaning that the piston strokeis equal to the cylinder diameter. The maximum piston volume,Vmax, is described by

Vmax ¼ pr2l ð1Þ

where r is the piston radius and l is the piston stroke. Using l = 2r,allows solving for the piston radius:

r ¼ffiffiffiffiffiffiffiffiffiffiVmax

2p3

r: ð2Þ

The reciprocating piston is driven by a conventional crankshaftand connecting rod, creating a sinusoidal displacement. The dis-placement of the piston is described by

xðhÞ ¼ l2ð1þ cosðhÞÞ ð3Þ

where h is the angular displacement of the crankshaft. Recall thatthe analysis will take place across a phase angle of h = 0 to h = p.The volume of the gas chamber is similarly described as

VðhÞ ¼ pr2l2ð1þ cosðhÞÞ ð4Þ

By taking the derivative of the piston displacement functionwith respect to h and using the definition of angular velocity,x ¼ dh

dt, allows solving for the piston velocity:

_xðhÞ ¼ dxdt¼ �xl

2sinðhÞ ð5Þ

where _x is the piston velocity and x is the angular velocity of thecrankshaft.

The liquid piston gas chamber will consist of many small diam-eter cylinders that will be simultaneously filled by the liquid pis-ton. Because all of the individual cylinders are symmetric, theheat transfer and viscous force analysis for the liquid piston gaschamber will focus on a single cylinder. This simplification allowsthe equations developed for the larger diameter reciprocating pis-ton to be used for the liquid piston. To further align the two com-pression methods and allow direct comparison, the displacement

id piston gas compression, Appl Energy (2009), doi:10.1016/

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of the liquid column will also be sinusoidal. For this analysis, waterwill be used for the liquid, although many other options exist.

Calculating the convective heat transfer coefficient within acompression chamber is very complex due to the non-uniformfluid velocity, circulating flows, pressure waves within the gas,thermal gradient across the gas, and the changing volume and sur-face area of the chamber. For a rough estimate of the gas flow andheat transfer with the purpose of building understanding, the sys-tem is modeled using a fully-developed pipe flow analysis. The firststep in the thermal analysis is to determine the flow regime, de-fined by the Reynolds number:

Re ¼_xmdm

ð6Þ

where _xm is the mean gas velocity, estimated as ½ the piston veloc-ity, d is the characteristic length, which is the diameter of the piston,and m is the kinematic viscosity. For pipe flow, a Reynolds numberless than approximately 2300 indicates laminar flow, a Reynoldsnumber greater than 4000 indicates turbulent flow, and intermedi-ate values signify transitional flow [27].

The kinematic viscosity of gases increases with temperature asdescribed by Sutherland’s equation

m ¼ l0

qT0 þ CT þ C

TT0

� �3=2

ð7Þ

where l0 is the reference dynamic viscosity, q is the mass density ofthe gas, T0 is the reference temperature, T is the gas temperature,and C is the Sutherland constant. For air, the Sutherland constant,C = 120 K [28].

To determine the convection coefficient, the average Nusseltnumber can be expressed in terms of the Reynolds and Prandtlnumbers by

Nu ¼ hdk¼ aRemPrn ð8Þ

where Nu is the Nusselt number, h is the convective heat transfercoefficient, k is the thermal conductivity, Pr is the Prandtl number,and a, m, and n are constants. Across a wide range of operating tem-peratures, the Prandtl number is approximately 0.7 for air. The ther-mal conductivity of air does vary with temperature, but thisanalysis will use a value for the average gas temperature duringthe compression process. The three constants in Eq. (8) have beendetermined empirically and experimentally and vary with the flowregime. For laminar flow in a pipe, a = 0.664, m = 1/2, and n = 1/3.For turbulent flow, a = 0.023, m = 0.8, and n = 0.3 [27]. Solving thisEq. (8) for h yields:

h ¼ kd

aRemPrn ð9Þ

Using the properties of the gas at the previous timestep, thesimulation approximates h, the convective heat transfer coeffi-cient. For each timestep the temperature increase of the gas dueto compression is calculated assuming adiabatic compression ofan ideal gas according to

T ¼TprevVk�1

prev

Vk�1 ð10Þ

where T and Tprev are the temperature of the gas for the current andprevious timesteps respectively, V and Vprev are the gas chambervolume for the current and previous timesteps respectively, and kis the specific heat ratio. Through sufficient external heat exchange,the walls of the compression chamber are assumed to maintain aconstant temperature during the compression cycle. The heat trans-fer rate from the gas to the chamber walls is calculated using New-ton’s law of cooling:

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_Q conv ¼ hAðTs � �TgÞ ð11Þ

where _Q conv is the heat transfer rate, A is the area of the gas cham-ber, including the piston and end cap, Ts is the surface temperatureof the compression chamber walls, and �Tg is the average gas tem-perature. Using the heat transfer rate, the average temperature ofthe gas after each timestep can be calculated using the followingequation:

_Q ¼ mCpDTg

Dtð12Þ

where m is the mass of the gas, Cp is the specific heat of the gas, DTg

is the change in the average gas temperature, and Dt is the time-step. Finally, knowing the mass, temperature, and volume of thegas, the pressure of the gas for each timestep is calculated usingthe ideal gas law:

PV ¼ mRT ð13Þ

where P is the pressure and R is the universal gas constant.Once the pressure in the chamber has reached the desired com-

pression ratio, in this case 9.5, the gas is pumped out of the cham-ber at constant pressure. The volume of compressed gas leaving thechamber during a single compression cycle is

VC ¼ ½VðhÞ�h¼hCRh¼p ¼ pR2h

2ð1þ cosðhÞÞ

" #h¼hCR

h¼p

ð14Þ

where hCR is the angular position of the crankshaft when the pres-sure in the chamber has reached the compression ratio. The com-pressed gas pumped out of the chamber will enter a storagereservoir where it will cool to ambient temperature before beingused. During this cooling, the pressure is kept constant and the vol-ume of the gas decreases according to the ideal gas law. Thus thefinal volume of the cooled gas pumped from the chamber duringa single compression cycle held at a constant pressure is

V final ¼VCTambient

TCð15Þ

where Vfinal is the final volume of the compressed cooled gas,Tambient is the ambient temperature, and TC is the compressed gastemperature when it exits the compression chamber. The energystored in this compressed gas volume is described by the area underthe pressure–volume curve, which is a maximum for isothermalexpansion. This expansion consists of two phases, a constant pres-sure pumping of the gas from the reservoir to the expansion cham-ber and the expansion. The stored energy, subtracting the workdone by atmosphere is expressed as

E ¼Z½PðVÞ � Patm�dV

¼Z V final

0ðPC � PatmÞdV þ

Z Vexpanded

V final

PCV final

V� Patm

� �dV

¼ ðPC � PatmÞV final þ PCV final lnVexpanded

V final

��������� PatmðVexpanded � V finalÞ

¼ PCV final lnPC

Patm

�������� ð16Þ

where PC is the gas pressure at the compression ratio, Patm is atmo-spheric pressure, and Vexpanded is the volume of the gas when it isexpanded to atmospheric pressure during isothermal expansion.

To allow calculation of the compression efficiency, the workdone on the gas is required. By performing a numerical integrationof the area under the pressure–volume curve during the compres-sion and subtracting the work done by atmosphere, the work doneon the gas is

W �XðPðhÞ � PatmÞDVðhÞ from h ¼ 0 to h ¼ p: ð17Þ

id piston gas compression, Appl Energy (2009), doi:10.1016/

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Finally, the compression efficiency, gcomp, neglecting viscousforces, friction, and leakage, is

gcomp ¼E

W� 100%: ð18Þ

3.2. Sliding and viscous forces analysis

The major source of friction in a kinematic reciprocating pistonis the sliding friction of the piston seals [29]. For most air compres-sor applications, the piston seals consist of split rings typicallymade of cast iron. The rings are preloaded against the cylinderwalls by spring force in the rings. The design of the piston ringgroove, seen in Fig 4, allows the gas in the chamber to apply out-ward radial pressure to the piston ring, further increasing the pres-sure with the wall.

The force of the piston ring on the wall due to spring preload ofthe split piston ring is typically a fraction of the force created bythe maximum gas pressure applied to the inner diameter of thepiston ring [29]. The total pressure applied to the cylinder wallby the piston ring can be expressed as

PrðhÞ ¼ xpreloadPgas;max þ PgasðhÞ ð19Þ

where Pr is the ring pressure applied to the cylinder wall, xpreload isthe fraction of the maximum gas pressure preload on the ring,Pgas,max is the maximum gas pressure in the chamber, and Pgas isthe gas pressure in the chamber. The sliding friction can be calcu-lated from the normal force on the chamber wall and the coefficientof friction as follows:

Ff ðhÞ ¼ PrðhÞAl ¼ ðxpreloadPgas;max þ PgasðhÞÞðtpdÞl ð20Þ

where Ff is the sliding friction, d is the inside diameter of the pistonring, t is the vertical height of the piston ring, and l is the slidingfriction coefficient.

The friction of an oil lubricated reciprocating piston ring is com-plicated because of the changing piston velocity and radial force,which result in varying lubrication regimes. When the piston isat low velocity, as occurs when dwelling at the top and bottomof the stroke, boundary lubrication exists between the piston ringand the cylinder wall. When at a high linear velocity, in the pres-ence of sufficient oil, thick film lubrication exists. Typical valuesfor the coefficient of friction for boundary lubrication range from0.05 to 0.15 [30], while values for thick film lubrication are around0.001 [31]. For this analysis a constant frictional coefficient of 0.1 isassumed for the entire piston travel. A more complex sliding fric-tion model that includes transition between lubrication regimesis highly dependent on geometry and lubrication conditions, caus-ing it to be beyond the scope of this paper.

The viscous forces of the liquid piston will be estimated using afully-developed pipe flow analysis. The Darcy–Weisbach equation

Fig. 4. Piston with piston ring. Note that the gas in the chambe

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describes the pressure drop for fully developed, steady, incom-pressible flow as

DP ¼ fLD

q _x2

2ð21Þ

where DP is the pressure drop along the pipe, f is the friction factor,L is the length of the pipe, D is the diameter of the pipe, q is themass density of the liquid, and _x is the velocity of the liquid column[28]. For laminar flow, the friction factor f is f ¼ 64

Re, and is indepen-dent of the surface roughness. For turbulent flow, the friction factorcan be estimated using the Moody chart. Alternatively, the frictionfactor for turbulent flow can be approximated using the Swamee–Jain equation:

f ¼ 1:325

ln e3:7Dþ 5:74

Re0:9

� �h i2 ð22Þ

where e is the surface roughness of the pipe interior [28]. Becausethe density of air is three orders of magnitude smaller than the li-quid, in this case water, the pressure drop due to the friction ofthe gas is neglected.

Using the sliding friction force from the reciprocating piston orthe pressure drop from the viscous friction analysis allows thework due to friction to be calculated. The work due to friction inthe reciprocating piston system is calculated by numerically inte-grating the product of the force and the change in position. Thework due to viscous forces in the liquid piston system is calculatedby numerically integrating the product of the pressure drop andthe change in volume.

From the compression work, the work due to friction, and theenergy content of the compressed gas, the efficiency can be calcu-lated as follows:

g ¼ EW þW friction

� 100% ð23Þ

where g is the efficiency and Wfriction is the energy loss due to fric-tion. Note that this efficiency value neglects gas leakage and otherunmodeled sources of friction and loss.

3.3. Model inputs

Many inputs to the model are required to estimate the gas com-pression efficiency for the reciprocating piston and liquid pistonsystems using the above equations. These input variables are com-piled in Table 1.

4. Results

While the displacement of the reciprocating piston and the li-quid piston are both sinusoidal, the flow of the gas in the chambersis quite different. This difference is most readily apparent by the

r applies pressure to the top and inside of the piston ring.

id piston gas compression, Appl Energy (2009), doi:10.1016/

Table 1Input variables to the numerical simulation of the heat transfer and frictional forces.

Constant Symbol Reciprocatingpiston

Liquidpiston

Units

Maximum chamber volume Vmax 0.00125 m3

Number of cylinders N 1 50,000Bore diameter D = 2r 0.1168 0.0009 mPiston stroke l 0.1168 0.0393 mMaximum cylinder volume V 0.00125 2.5E�08 m3

Pressure compression ratio CR 9.5Cycle frequency f 20 1/sAverage thermal conductivity

of airk 0.0275 W/m/K

Specific heat of air Cp 1006 J/kg/KChamber wall surface

temperatureTsurf 303 K

Initial temperature Tinitial 293 KInitial pressure Pinitial 101,000 PaSutherland reference dynamic

viscosityl0 1.827E�05 kg/m/s

Sutherland referencetemperature

T0 291.15 K

Sutherland constant C 120 KPrandtl number Pr 0.7Angular step size Dh p/1000 radiansPiston ring sliding friction

coefficientl 0.1

Piston ring vertical height t 0.002 mPiston ring preload fraction xpreload 0.5Liquid density qliquid 998.2 kg/m3

Liquid kinematic viscosity mliquid 1.00E�06 m2/sChamber wall roughness e 1.50E�06 m

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ARTICLE IN PRESS

contrasting average Reynolds number for the two compressionchambers, presented in Fig. 5. The flow in the liquid piston cham-ber is dominated by viscous forces due to the small diameter of thecylinders, resulting in a low Reynolds number and laminar flow.Conversely, the dominance of inertial forces in the reciprocatingpiston chamber result in a higher Reynolds number and turbulentflow for the majority of the piston stroke. As seen in Fig. 5, the Rey-nolds number for both the reciprocating piston and the liquid pis-ton reaches a sharp peak value around 80% stroke before beginningto decrease. This peak occurs when the pressure in the chamberhas reached the desired compression ratio and gas is beingpumped from the chamber. Because this final pumping occurs at

Reciprocating Piston - Reynolds Number vs. Piston Travel

0

20000

40000

60000

80000

Piston Travel (%)

Rey

nold

s N

umbe

r

Liquid Piston - Reynolds Number vs. Piston Travel

0

50100

150

200

250

300350

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Piston Travel (%)

Rey

nold

s N

umbe

r

a

b

Fig. 5. The Reynolds number for the reciprocating piston, shown in plot (a),indicates turbulent flow for the majority of the piston travel. The significantly lowerReynolds number of the liquid piston, plot (b), indicates laminar flow for the entirepiston travel.

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a constant pressure, the kinematic viscosity remains constantand only the gas velocity is changing.

From the Reynolds and Prandtl numbers the average Nusseltnumber is calculated. As seen in Fig. 6, the Nusselt number of thereciprocating piston is more than an order of magnitude greaterthan the liquid piston. This indicates a stronger influence of con-vection versus conductive heat transfer, as is expected for turbu-lent flow. The small discontinuity in the Nusselt number for thereciprocating piston around 2% piston travel is due to the changingcoefficients in the relationship between the Nusselt, Reynolds, andPrandtl numbers when transitioning from laminar to turbulentflow.

From the Nusselt number, the convective heat transfer coeffi-cient can be estimated. Although the Nusselt number for the liquidpiston is significantly lower than the reciprocating piston, the con-vection coefficient is much higher for the liquid piston than thereciprocating piston, as seen in Fig. 7. The convection coefficientfor the liquid piston is higher due to the smaller diameter cylindersin the compression chamber.

A prime concern of this work is the average temperature of thegas during compression. By calculating the temperature rise thatwould result from adiabatic compression and implementing theconvective heat transfer coefficient, the average temperature ofthe gas can be estimated, as found in Fig. 8. For this simulationthe wall temperature is set to 303 K. Because the wall temperatureis higher than the inlet gas temperature, the gas is initially heatedby the wall, as exhibited by the convex curve of the liquid pistontemperature around 8% piston travel. Note the different behaviorof the gas temperature of the two piston types once the compres-sion ratio has been reached near 80% piston travel. After this point,the mass of the gas in the chamber is decreasing as it is pumpedfrom the chamber. Due to the high convection coefficient, the tem-perature of the gas in the liquid piston decreases significantly dur-ing this pumping stage while the gas temperature in thereciprocating piston remains nearly constant. Due to the increasein pressure, Fig. 9, simultaneous to the increase in temperature,the water in the liquid piston chamber does remain below the sat-uration point. For certain applications the need to avoid vaporizingthe liquid will drive the choice of liquid.

From the temperature, volume, and mass of the gas, the pres-sure in the compression chamber can be calculated. The gas pres-sure in the chambers, shown in Fig. 9, increases from atmosphericpressure to the compression ratio, at which point the gas ispumped at a constant pressure out of the chamber. Notethat due to the higher gas temperature, the pressure in the

Nusselt Number vs. Piston Travel

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100Piston Travel (%)

Nus

selt

Num

ber

Liquid Piston

Reciprocating Piston

Fig. 6. Nusselt number calculated using the Reynolds and Prandtl numbers. Notethe discontinuity in the reciprocating piston Nusselt number occurs during thetransition from laminar to turbulent flow.

id piston gas compression, Appl Energy (2009), doi:10.1016/

Compression Work vs. Piston Travel

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60 70 80 90 100

Piston Travel (%)

Wor

k (J

)

Liquid Piston

Reciprocating Piston

Fig. 10. Compression work done on the gas as a function of piston travel. This plotdoes not include work to overcome frictional forces.

Convection Coefficient vs. Piston Travel

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60 70 80 90 100

Piston Travel (%)

Con

vect

ion

Coe

ffic

ient

(W

/m^

2/K

)

Liquid Piston

Reciprocating Piston

Fig. 7. Convective heat transfer coefficient for the two compression chambers. Thesignificantly higher value for the liquid piston is due to the small diameter of thecylinders in the compression chamber.

Temperature vs. Piston Travel

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90 100

Piston Travel (%)

Tem

pera

ture

(C

) .

Liquid Piston

Reciprocating Piston

Fig. 8. Temperature of the gas in the two chambers as a function of piston travel.Note the significantly lower temperature of the gas in the liquid piston chambercompared to the reciprocating piston chamber.

Pressure vs. Piston Travel

0

100.000

200.000

300.000

400.000

500.000

600.000

700.000

800.000

900.000

1.000.000

0 10 20 30 40 50 60 70 80 90 100

Piston Travel (%)

Pre

ssur

e (P

a)

Liquid Piston

Reciprocating Piston

Fig. 9. Gas pressure in the chambers during the compression process. Once thepressure reaches the compression ratio, the gas is pumped from the chamber at aconstant pressure.

Frictional Work vs. Piston Travel

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100Piston Travel (%)

Wor

k (J

)

Liquid Piston Viscous Friction

Reciprocating Piston Sliding

Fig. 11. The work due to sliding friction in the reciprocating piston chamber andthe viscous forces in the liquid piston chamber.

J.D. Van de Ven, P.Y. Li / Applied Energy xxx (2009) xxx–xxx 7

ARTICLE IN PRESS

reciprocating piston chamber rises more quickly than the liquidpiston chamber.

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The work required to compress the gas is computed by numer-ically integrating the product of the change in volume times thedifference of the gas pressure minus the atmospheric pressure.The compression work, as seen in Fig. 10, is higher for the recipro-cating piston than the liquid piston due to the more rapid increasein chamber pressure. Fig. 10 only includes the compression workdone on the gas, not the total work done on the system.

The final work terms accounted for in this simulation are thework due to sliding friction in the reciprocating piston and thework due to viscous forces in the liquid piston. A plot of thesetwo forms of frictional work is presented in Fig. 11. Note that themechanisms behind these two frictional losses are quite different,yet their magnitudes are similar for this set of operatingconditions.

After compression, the gas in the storage tank cools to ambienttemperature while maintaining constant pressure. Once the gas iscool, the energy content of the gas is the same for both the recip-rocating piston system and the liquid piston system. By dividingthe maximum potential energy content of the gas by the totalwork, which includes the compression work and the frictionalwork, the compression efficiency is computed. The energy, work,and efficiency can be found in Table 2.

5. Discussion

The simulation illustrates a significant improvement in gascompression efficiency by using a liquid piston in place of a recip-

id piston gas compression, Appl Energy (2009), doi:10.1016/

Table 2The simulated work, energy, and total efficiency for compressing the gas using thetwo different compression chambers.

Reciprocating piston Liquid piston Units

Compression work on the gas 399.4 332.6 JFrictional work 6.6 8.4 JTotal work 406.0 341.0 JEnergy of the gas 284.2 JTotal efficiency 70.0% 83.3%

Efficiency vs. Number of Cylinders

65

70

75

80

85

90

95

0 200.000 400.000 600.000 800.000 1.000.000Number of Cylinders

Eff

icie

ncy

(%)

Total Efficiency

Compression Efficiency

Fig. 12. The compression and total efficiency of the liquid piston chamber as afunction of the number of individual cylinders. At each data point, the cylinderdiameter and length were optimized for the maximum total efficiency whilemaintaining a constant total chamber volume.

Cylinder Diameter vs. Number of Cylinders

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 200.000 400.000 600.000 800.000 1.000.000

Number of Cylinders

Cyl

inde

r D

iam

eter

(m

)

Fig. 13. The optimized cylinder diameter in the liquid piston chamber that resultsin maximum total efficiency for a given number of cylinders.

8 J.D. Van de Ven, P.Y. Li / Applied Energy xxx (2009) xxx–xxx

ARTICLE IN PRESS

rocating piston. From the simulation, the liquid piston chamber re-quires 19% less work than the reciprocating piston chamber,improving the total compression efficiency from 70% to 83%. Theefficiency improvement is the result of removing more heat duringthe compression process, thus lowering the work required to com-press the gas. In the application of an air compressor, the com-pressed air is stored in a tank before use, allowing time for thegas to cool to near ambient temperature. Because the gas cools be-fore it is converted into another form of work through expansion,any heat energy added to the gas is lost. Thus by conducting anear-isothermal compression with a liquid piston, less work is re-quired to compress the gas to the same final pressure than a near-adiabatic compression such as in the reciprocating piston chamber.

Near-isothermal compression is accomplished in the liquid pis-ton chamber by maximizing the surface area to volume ratio. Bydecreasing the diameter of the individual cylinders in the liquidpiston chamber, the heat transfer coefficient was greatly improved.Unfortunately, decreasing the cylinder diameter also decreases theReynolds number, resulting in laminar flow, which creates a lowerNusselt number. Furthermore, small diameter cylinders also in-crease the viscous pressure drop of the liquid piston.

The design of a liquid piston chamber requires a careful balancebetween the convective heat transfer coefficient and the viscousfluid forces. As previously discussed, decreasing the diameter ofthe cylinders improves the heat transfer while negatively impact-ing the viscous pressure drop. One way to positively influence boththe heat transfer and the viscous forces is to decrease the operatingfrequency, yet this decreases the power of the machine. Anotherway to decrease the viscous forces is to decrease the fluid velocity.The viscous pressure drop is proportional to the square of thevelocity and the convective heat transfer coefficient is proportionalto the square-root of the velocity, for laminar flow. Another factorthat decreases the viscous pressure drop is decreasing the length ofthe cylinder.

By combining a decrease in the flow velocity and a decrease inthe cylinder length, the viscous pressure drop can be decreasedwhile positively impacting the heat transfer. This can be accom-plished by increasing the number of individual cylinders in the li-quid piston chamber and decreasing their length. The increase incompression and total efficiency versus the number of cylinders,for an optimized cylinder diameter, can be observed in Fig. 12.Note that both the compression efficiency and the total efficiency,which includes the compression work and the viscous work, in-crease with an increasing number of cylinders. As the number ofcylinders increases, the cylinder diameter resulting in maximumtotal efficiency decreases, as seen in Fig. 13.

As can be seen from Figs. 12 and 13, the optimal liquid pistonchamber geometry consists of a very large number of small diam-eter cylinders. While this paper has only explored cylindricalgeometry in the liquid piston chamber, many other geometriesare possible. Some options include fins, layered wire mesh, or abundle of wire, of similar consistency to steel wool. A key to anyof these geometries is to maximize the surface area to volume ra-tio, to improve the heat transfer, while minimizing the viscous flowforces. A likely driver in the selection of the internal geometry is

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the manufacturability of the various options. The cylindrical geom-etry could be manufactured as a small diameter honeycomb styleextrusion or by tightly packing many small tubes, similar to hypo-dermic needles, in a chamber. Adding a fine wire mesh to a cham-ber or a porous metal matrix structure are more feasiblemanufacturing alternatives.

It does need to be clearly stated that the model makes assump-tions to simplify the analysis and illustrate the liquid piston con-cept. The equations used in model are for fully-developed steady-state flow, which does not exist in a sinusoidal compression cycle.Furthermore, an average temperature and velocity were used forthe gas, not accounting for the variations that have been discussedby previous researchers [2,3]. Despite these shortcomings, the sim-plified equations are valuable as they provide further insight intothe influence of the various parameters on the system efficiency.

Beyond the basic heat transfer model, there are additional ef-fects that were not modeled. First, the energy loss due to gas leak-age past the reciprocating piston seal was not modeled. This can bea significant source of energy loss, which would further decreasethe efficiency of the reciprocating piston system. Gas leakage inthe liquid piston system would be negligible, but of the form ofgas entrainment in the liquid. Second, the piston ring friction mod-el only incorporated a single ring and did not analyze the lubrica-tion regime of the piston. Typical reciprocating compressors use

id piston gas compression, Appl Energy (2009), doi:10.1016/

J.D. Van de Ven, P.Y. Li / Applied Energy xxx (2009) xxx–xxx 9

ARTICLE IN PRESS

two or three piston rings, including an oil control ring. By simplify-ing the analysis to a single ring the estimated friction is likely low-er than actual applications. Finally, the small scale effects ofsurface roughness, wetting of the liquid piston chamber walls,and surface tension were not modeled. As the size of the flow pas-sages decreases, the influence of these effects become more pro-nounced, making these effects an important addition to futuremodels.

The liquid piston concept not only enables excellent heat trans-fer from the gas to the compression chamber during compressionor expansion of a gas, it also provides a secondary path to removeor add heat to the system. In a typical reciprocating piston ma-chine, heat is added or removed only through the cylinder walls.With the liquid piston system, the compressing liquid enters thesame compression space previously occupied by the gas, allowingheat to be transferred from the gas chamber to the liquid. By circu-lating the liquid used for the liquid piston through an external heatexchanger, heat can be effectively added or removed from thesystem.

The combination of a near-isothermal compression or expan-sion with the ability to add or remove heat from the gas chambermakes the liquid piston concept advantageous for many applica-tions. While this paper has focused on the single application ofan air compressor, this concept can also be used for expanding agas, such as an air motor. An emerging application requiring effi-cient compression and expansion of air is the open accumulatorenergy storage concept, which greatly surpasses previous technol-ogy in energy density by compressing air from atmospheric pres-sure to pressures as high as 35 MPa [26]. Another application forthis technology is a liquid piston engine. One specific engine thatwould benefit from this concept is the Stirling engine, where heatis conventionally added and removed in heat exchangers that areexternal from the compression and expansion chambers. By inte-grating the heat exchanger and the compression space, the deadvolume can be decreased, improving the power density.

6. Conclusion

Liquid piston compression is an exciting concept that can signif-icantly improve the efficiency of compressing or expanding a gas.Through a simplified model, this concept was demonstrated to im-prove the efficiency of compression from 70% to over 83%. It wasfurther demonstrated that greater improvements in efficiencycould be gained by increasing the number of individual cylindersin the gas chamber and decreasing their diameter. Estimates ofthe frictional forces demonstrated that the magnitude of the slid-ing frictional work in a reciprocating piston and the viscous fric-tional work are similar.

The liquid piston concept relies on maximizing the surface areato volume ratio within the gas chamber, allowing high frequenciescompression or expansion to approach isothermal conditions. Thisconcept not only improves the heat transfer during compression,but also eliminates leakage past conventional reciprocating pistonrings. Furthermore, because the liquid piston does not involve slid-ing contact, it promises to be a low maintenance alternative.

While the model presented in this work allows a basic under-standing of the interaction of the design parameters, it makesassumptions that oversimplify the situation. To fully understandthe liquid piston system, a more advanced model is required that

Please cite this article in press as: Van de Ven JD, Li PY, Liquj.apenergy.2008.12.001

removes the oversimplify assumptions. Further future work alsoincludes exploring the application of liquid piston compressionand expansion to other applications including compressors, mo-tors, and internal and external combustion engines.

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id piston gas compression, Appl Energy (2009), doi:10.1016/