Proceedings of the 1st Thermal and Fluid Engineering Summer Conference, TFESC August 9-12, 2015, New York City, USA

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*Corresponding Author: huang386@umn.com

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ANALYSIS OF DISSOLVED GAS IN THE APPLICATION OF

LIQUID PISTON GAS COMPRESSION Longzhong Huang1*, Terrence W. Simon1, Perry Y. Li1, Edward J. Hummelt2

1 Mechanical Engineering Department, University of Minnesota,

111 Church St. S.E., Minneapolis MN, 55455 2 Eaton Corporation, W126 N7250 Flint Drive, Menomonee Falls, WI, 53051

ABSTRACT

Liquid piston compression technology is being explored as a way to achieve high-pressure gas compression in

Compressed Air Energy Storage (CAES) systems. When combined with a porous medium, a liquid piston

compressor is able to provide near-isothermal compression, and high efficiency. However, the gas being

compressed can dissolve into the liquid resulting in loss of compressed gas output. Also, the dissolved gas in the

liquid lines can cause aeration that causes damage to pumps and valves. To understand this gas transport

phenomenon in the liquid and to evaluate the total amount of gas dissolved, a numerical model was developed.

Dispersion due to the porous medium is found to be the dominant mode of mass transfer. The dispersion

coefficient is calculated based on an experimental correlation that is given in terms of flow velocity, porous

medium geometry and molecular diffusion coefficient. The numerical model is used to investigate a case study

in which a water liquid piston is used for air compression. The amount of dissolved gas under various conditions

is calculated, such as gas concentration of the inlet flow to the compressor and discharge pressure to the storage

tank at the end of compression. The effects of different types of liquids are also quantified by applying in the

model the different solubility values of gases in liquids.

KEY WORDS: Liquid piston, Dissolved gas, Porous media, Dispersion, Mass transfer

1. INTRODUCTION

Liquid piston compression has been proposed for various gas compression applications, one of which is the

compressor/expander in a high pressure compressed energy storage system [1, 2]. A liquid piston

compressor/expander utilizes a column of liquid to compress and subsequently expand the gas above it [3]

Compared to a traditional reciprocating solid piston, the liquid piston is able to eliminate gas leakage and

sliding friction. In addition, a porous medium can be put inside the compression chamber that the liquid can

also pass through. The solid porous medium, with its significant increase in surface area, is able to absorb

heat energy from the gas in real time improving the efficiency of the compression process. Reference [1]

found that the liquid piston compressor consumed 19% less energy than the reciprocating piston compressor

in the application of air compression with a pressure ratio of 9.5:1 and a cycle frequency of 20 Hz [3]. Using

this concept, an energy storage system for offshore wind turbines was designed [4-6]. The wind energy is

used to drive the liquid piston to compress air so that the energy is stored in the compressed air. This system

is more flexible and the efficiency is higher than a traditional wind power plant. Different types of porous media, such as metal foams and interrupted plates were investigated with liquid piston flow using CFD

methods [7-11] and experimentally [4-6]. It was found that the porous medium is effective in absorbing heat

from the compressed gas and makes the compression a nearly isothermal process. In all these studies, the

focus has been on compression efficiency and heat transfer. To our knowledge, no one has investigated the

mass transfer that also occurs within the compression chamber during the compression process.

Although conceptually simple, the interfaces between liquid and gas allows for gas sorption and

desorption to occur during the compression and intake stroke, respectively. The overall effect is to decrease

the total mass of gas that is delivered by the machine, thus decreasing compression efficiency. Residual

dissolved gas in the liquid can also cause aeration problems as it goes through the rest of the hydraulic

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system. Thus, it is important to understand mass transfer of dissolved gas in the liquid and to predict the gas

solution and dissolution in the liquid under liquid piston compression. However, the local flow is complex

due to the existence of a porous medium. The complex flow generates considerable mixing, which enhances

mass transfer. This mixing is called dispersion and is a common element in chemical engineering systems.

Cussler et al. [12] discussed the dispersion concept and the factors that affect it in fluid systems. Dispersion

rate is strongly dependent on the flow velocity. At low flow velocities, the magnitude of dispersion is similar

to diffusion. At high flow velocities, the dispersion coefficient is almost linear to flow velocity. The type of

porous medium is another important factor that affects the dispersion coefficient. Different types of porous

media will generate different flow fields and, thus, affect the mixing differently. Delgado [13] summarized

the topic for packed beds, which are widely used in chemical engineering. They developed empirical

correlations for dispersion coefficients based on experimental data. These correlations can be used to predict

dispersion for many applications. In this study, a computation model is established to understand the mass

transfer processes of gas solution in a liquid when the liquid is compressing the gas. The model results can

be used in the design of a degassing system, if one is needed.

2. NUMERICAL MODEL

In the application of a liquid piston for energy storage, the liquid is pumped into a compression chamber

containing a porous medium and a charge of gas. As the liquid enters, the gas is compressed heating the gas

and simultaneously transferring most of the compression heat into the porous medium. Thus, the energy

stored inside the gas mainly as mechanical pressure energy. At the end of the compression stroke, an exhaust

valve opens allowing the gas to be transferred from the compression chamber to a storage tank. After the gas

is expelled, exhaust and intake valves close and open, respectively, while the liquid is withdrawn from the

compression chamber. This allows a fresh charge of gas to enter the compression chamber readying it for the

next compression cycle.

A numerical model is developed to calculate the mass transfer that occurs during the compression

and intake process (Figure 1). The compression chamber is a cylinder with a length of one meter and

diameter of 0.33 meter that is filled with small spheres of 1.0 mm diameter stacked to a porosity of 0.4. The

porous medium is effective in absorbing the thermal energy of the compressed gas during compression but

also causes pressure drop as liquid passes through it. The large pressure gradient in the axial direction of the

compression chamber allows the flow through the porous media to be modeled as one-dimensional flow.

When the liquid is compressing the gas, the gas is inclined to dissolve into the liquid.

The gas concentration in the liquid (C) changes due to dispersion and convection. Dispersion

describes the mixing and transport of gas in a liquid as it flows through a porous medium which is

proportional to the dispersion coefficient and the gradient of gas concentration. The speed of the velocity

determines the rate of the convection term. The governing equation that describes the mass transfer process

is thus given by:

y

CU

y

CE

t

Cz

2

2

The dispersion coefficient Ez (PeL = U×d/ Ez) is calculated based on an experimental correlation which is a

function of flow velocity, ball diameter, diffusion coefficient, and Schmidt number [13] (equations (1~5)).

The dimensionless Peclet number (PeL) is used to describe the ratio of convection to dispersion. It is used to

obtain the correlation. The diffusion coefficient (DAB) of gas through a liquid is calculated based on the

Wilke and Chang equation [14]. For boundary conditions, the concentration of the interface between gas and

liquid is calculated from Henry’s law [12], which states that the concentration is proportional to the gas

pressure and a constant (Ci (y= interface) = P×kH). Henry’s constant, kH, which is dependent only on

temperature, describes the solubility of gas in the liquid. The source of the liquid is assumed to be at constant

concentration Cin. Thus, the boundary condition at the inlet concentration of the compression chamber is (C =

Cin). For initial conditions, the initial gas pressure and temperature are given. The pressure and temperature

of the gas is calculated by ideal gas law. The initial gas concentration in the liquid is also given. The gas

pressure used to calculate the gas concentration at the interface is the same as the gas pressure inside the

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compression chamber. It increases as the liquid compresses the gas. The gas temperature change is within

5°C. The velocity of the flow at the interface is determined by the flow rate.

Fig. 1 Numerical model to calculate mass transfer with governing equation and boundary conditions

z

LE

dUPe

(1)

AB

mD

dUPe

(2)

ABD

Sc

(3)

)75

exp()48.0

2

1(

48.015.015.0

mPe

Sc

ScScp

(4)

mm

mm

L PePepppp

Pep

Pe

Pe

11]

)1(

5exp[)1(

25)1(

5

1 32

2 (5)

3. Case Study: Water piston used for compressing air

The above numerical model is used to generate a solution for the particular case in which water is the liquid

and air is the gas. The combination of water and air is the most widely used in energy storage. In the present

application, the length of the compression chamber (L) is taken to be one meter with a length-to-diameter aspect ratio (L/D) of 3. The porosity of the porous medium is 0.4 and the ball diameter is 1 mm. All other

model parameters are as shown in table 1. Water is compressing the air at a constant flow rate (Q = 220

liters/min) giving a constant liquid piston velocity of 0.04 m/s. Note that this is for illustrative purpose only

as for optimal efficiency and power density, the flow rate would not be constant [4, 6]. The gas is

compressed from 5 bars to 250 bars in 10 seconds (Figure 2). The air concentration of inlet water is Cin = 0

kg/L. The Henry’s constant for air and water is assumed to be 20 mg/L/bar [12].

To describe the mass transfer process, the gas concentration profiles at different times are computed

(shown in Figure 3). During the compression process, the gas pressure continues increasing, causing the

concentration of dissolved gas in the liquid at the interface to increase, in accord with Henry’s law. Thus, the

gas concentration at the interface is always the highest within the liquid during compression. This high

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concentration drives gas into the liquid due to the concentration difference and the penetration rate is

determined by the dispersion coefficient. A steep concentration curve near the interface is clearly seen in the

figure due to the rapid increase of interface concentration associated with the rapidly rising gas pressure

(Figure 2). After the gas reaches its highest pressure (250 bars), the discharge valve opens and the gas is

expelled from the compression chamber to a storage tank with minimal pressure drop. At the end of the

discharge process, the exhaust valve closes and the intake valve opens to draw in a fresh charge of air for the

next compression cycle. During the intake process (t >10s), the gas pressure decreases rapidly as the liquid

recedes, which causes the gas concentration at the interface to drop to a low level (5 bar). However, some

distance into the liquid, the gas concentration remains high. Here, the liquid is supersaturated and the gas

may nucleate to form bubbles. In this simulation, all the nucleated gas is assumed to remain entrained in the

liquid (buoyancy forces ineffective). The bubble concentration (Figure 4) is calculated by subtracting the

saturation concentration (equal to the gas concentration at the gas to liquid interface) from the local gas

concentration. As the intake stroke progresses, it is observed that the gas disperses over the length of the

liquid column with a concentration distribution that is lognormal in appearance (Figure 3). In addition, gas

continues to exit the liquid at the gas to liquid interface decreasing the total amount of gas still contained

within the liquid.

The amount of gas sorbed by the liquid is important because it affects the volumetric efficiency of

the machine. The total dissolved gas is calculated by integrating the concentration profile through the liquid

volume. Figure 5 shows the increase of total mass of air in the water for one compression. At the end of the

compression cycle, 0.9 gram of air is dissolved into the water, which is about 0.45% of the total initial air

mass. During the intake process, about 0.65 gram of the air comes out of the liquid through the interface and

re-enters into the compression chamber. So 0.25 gram of the air is in the liquid at the end of the full stroke. It

is worth noting that this residual gas raises the concentration of the gas in the liquid in the next compression

cycle, and unless removed, can reduce volumetric compression efficiency and cause cavitation within the

machine.

Table 1 Air conditions and Parameters of water piston to compress air

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Fig. 2 Pressure of air inside the compression chamber

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Fig. 3 Air concentration in the water at different times.

Fig. 4 Bubble concentration in the water at different times.

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Fig. 5 Amount of Air in water at different times.

4. Various effects on dissolved gas in the liquid

To further characterize the dissolved gas in the liquid piston, effects such as solubility of gas in the liquid

and the compression chamber geometry are considered.

First, the solubility of gas in the liquid, as described by Henry’s law, was investigated. High

solubility indicates high saturated gas concentration in the liquid for a given pressure. The chemical makeup

of the gas and liquid and their temperature are major factors leading to different solubilities of the gas in the

liquid. Henry’s law provides the equation to describe solubility, C = P×kH, in which C is gas concentration in

the liquid, P is the gas partial pressure adjacent to the liquid interface and kH is Henry’s constant. Figure 6

shows the total dissolved gas in the system for different values of Henry’s constant. In this study, Henry’s

constant is the only parameter that is changed. It is clear that, in this case, since Cin = 0 [kg/L], the total

dissolved gas is linear with Henry’s constant. The partial pressure of gas in the liquid is another important

parameter in the application since it will determine the possibility of cavitation in pumps, lines and valves

that may be in the system. It is found that the partial pressure of gas in the liquid is constant, regardless of the

value of Henry’s constant. To confirm this result, the governing equations for gas concentration (C) in the

liquid as well as boundary conditions were rewritten as an explicit function of partial pressure (P) (Figure 7).

The result shows the governing equation to be independent of Henry’s constant. From this we conclude that

for the same compression pressure profile, the partial pressure of gas in the liquid will also be the same.

Another factor that may affect the amount of dissolved gas in the liquid is the geometry of the

compression chamber. Table 2 shows five different compression chambers with different length-to-diameter

aspect ratios, but with the same total volume. It was found that total amount of dissolved gas decreases with

an increase of length-to-diameter ratio (L/D). To explain this behavior, the original governing equation

(equation (6)) is normalized by length, time and concentration (y’ = y/L, C’ = C/C0, t’ = t/T). The final

normalized equation is expressed as equation (9) in which y’, C’, t, are normalized position, concentration,

time and L, C0, T are length of compression chamber, inlet concentration, and total compression time. The

coefficient on the convection term, U/L, can be transformed to the ratio of flow rate to total volume of

compression chamber (Q/Vc) in equation (10). The flow rate and total volume are kept the same in the five

cases, which indicates that the convection term is not affected by L/D. The coefficient for the dispersion term,

Ez/L2, is transformed to the product of d/PeL, Q/Vc, and 1/L in equation (11). The ball diameter (d) is kept the

same and the Peclet number (PeL) is also the same for the five cases, which indicates that the values of d/PeL

are the same. Thus, the coefficient of dispersion term decreases with increases of length of the compression

chamber. As the convection term doesn’t change, the total mass transfer rate decreases with an increase of

length of the compression chamber. We conclude that the total dissolved gas content decreases with

increases of length-to-diameter ratio (L/D) of the compression chamber.

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A third factor investigated was the effect of incoming fluid gas concentration. In this analysis, the

incoming fluid partial pressure was changed while other parameters were kept the same. The amount of

dissolved gas added to the cycle was found to decrease with an increase in beginning gas concentration

pressure (Figure 9). The higher starting gas concentration decreases the difference of gas concentration

between the interface and the bulk liquid which decreases the potential to drive the gas through the liquid. As

a result, less gas is dissolved into the liquid.

y

CU

y

CE

t

Cz

2

2

(6)

L

yy ' ,

0

'C

CC ,

T

tt ' (7)

'

'

'

'

'

' 0

2

2

2

00

y

CU

L

C

y

CE

L

C

t

C

T

Cz

(8)

'

'

'

'

'

'12

2

2 y

C

L

U

y

C

L

E

t

C

T

z

(9)

cV

Q

LD

Q

L

U

12

(10)

LV

Q

Pe

d

LD

Q

Pe

d

LPe

dU

L

E

cLLL

z 1112222

(11)

Finally, it is of interest to know how the variable of time affects gas uptake. Two completing factors

are at work; at higher flow velocities, the rate of dispersion is higher but the time for adsorption, dispersion

and diffusion is shorter. To understand which factor dominates, the effect of total compression time was

investigated. Fig. 10, shows that the mass of total dissolved gas increases with increased compression time.

This indicates that the compression time is more important than the flow velocity in affecting the dissolved

gas uptake.

Fig. 6 Total mass of dissolved gas in the liquid with different values of Henry’s constant.

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Fig. 7 Governing equations and boundary conditions in the form of partial pressure (P) of gas in liquid

Table 2 Compression chamber with different geometries

Fig. 8 Total mass of dissolved gas in the liquid with different values of L/D.

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Fig. 9 Additional dissolved gas in the liquid with different values of inlet partial pressure.

Fig. 10 Total mass of dissolved gas in the liquid with different values of total compression time.

6. CONCLUSIONS

The uptake and transport of dissolved gas in the application of a liquid piston compression machine was

investigated. A numerical model was developed that describes the transient mass transfer of the gas over a

full compression cycle. The change in gas concentration was found to be governed by dispersion and

convection effects with the dispersion coefficient calculated using published empirical correlations. The

numerical model successfully calculates the transport of gas within the liquid and was used to obtain a

solution for the specific case of water compressing air. For the model problem, the rate of rise in gas

concentration was found to be steep due to the rapid increase of gas pressure inside compression chamber

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and the total gas uptake was calculated to be less than 0.45% of the starting air mass. Using the numerical

model, the effects of solubility of gas in the liquid and the effects of the compression chamber geometry on

dissolved gas content were investigated. The results can be used to guide a choice of fluid and compression

chamber. The total dissolved gas was found to be linear to the solubility of the gas in liquid as described by

Henry’s law. Moreover, the gases partial pressure profile was determined to be independent of the gases

solubility, a result that was validated by analysing the governing equations. The effects of compression

chamber geometry were also evaluated. An increase in the length-to-diameter ratio (L/D) was found to

decrease the gas dissolved in the liquid and the normalized governing equations were used to analyse this

L/D effect. It is found that increasing L/D decreases dispersion and, thus, decreases dissolved gas content in

the liquid. Finally, the rate of gas uptake was found to decrease with higher gas concentration in the

incoming liquid and gas uptake was found to increase with increasing compression time. This result suggests

that the time for dispersion and diffusion dominates over the effect flow velocity has on the dispersion

coefficient. Overall, the developed model was found to be a useful tool for analysing the uptake and transfer

of absorbed gas within a liquid piston compression machine.

Acknowledgment: The information, data, or work presented herein was funded in part by the Advanced

Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-

AR0000258.

Disclaimer: The information, data, or work presented herein was funded in part by an agency of the United

States Government. Neither the United States Government nor any agency thereof, nor any of their

employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the

accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or

represents that its use would not infringe privately owned rights. Reference herein to any specific

commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not

necessarily constitute or imply its endorsement, recommendation, or favoring by the United States

Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily

state or reflect those of the United States Government or any agency thereof.

NOMENCLATURE

PeL: Peclet number for dispersion

Pem: Peclet number for diffusion

Sc: Schmidt number

DAB

: diffusion coefficient (m2/s)

Ez: dispersion coefficient (m2/s)

U: fluid velocity (m/s)

Vc: total volume of compression chamber (L)

d: ball diameter (m)

τ: tortuosity factor (0.7 for sphere)

L: Length of compression chamber (m)

C: Gas concentration (kg/L)

D: Diameter of compression chamber (m)

kH: Henry’s constant (kg/L/bar)

P: Pressure (Pa)

Q: liquid flow rate (m3/s)

ρ: density (kg/m3) μ: viscosity (Pa·S)

Ci: Saturated gas concentration (kg/L)

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[14] Wilke C. R. and Chang P., “Correlation of diffusion coefficients in dilute solutions,” A.I.Ch.E. Journal, pp. 264-270, (1955).