MODELING AND TRAJECTORY OPTIMIZATION OF WATER SPRAY COOLING IN ALIQUID PISTON AIR COMPRESSOR

Mohsen SaadatDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: saad002@umn.edu

Farzad A. ShiraziDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: fshirazi@umn.edu

Perry Y. LiDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: perry-li@umn.edu

ABSTRACTAn efficient and sufficiently power dense air compres-

sor/expander is the key element in a Compressed Air Energy Stor-age (CAES) approach. Efficiency can be increased by improvingthe heat transfer between air and its surrounding materials. Oneeffective and practical method to achieve this goal is to use wa-ter droplets spray inside the chamber when air is compressing orexpanding. In this paper, the air compression cycle is modeledby considering one-dimensional droplet properties in a lumpedair model. While it is possible to inject water droplets into thecompressing air at any time, optimal spray profile can result inmaximum efficiency improvement for a given water to air massratio. The corresponding optimization problem is then definedbased on the stored energy in the compressed air and the requiredinput works. Finally, optimal spray profile has been determinedfor various water to air mass ratio using a general numericalapproach to solve the optimization problem. Results show thepotential improvement by acquiring the optimal spray profile in-stead of conventional constant spray flow rate. For the specificcompression chamber geometry and desired pressure ratio andfinal time used in this work, the efficiency can be improved up to4%.

INTRODUCTIONGas compression and expansion has many applications in

pneumatic and hydraulic systems, including in the Compressed

Air Energy Storage (CAES) system for offshore wind turbine

that has recently been proposed in [1, 2]. Since the air compres-

sor/expander is responsible for the majority of the storage energy

conversion, it is critical that it is efficient and sufficiently pow-

erful. This is challenging because compressing/expanding air in

high compression ratios (200-300) heats/cools the air greatly, re-

sulting in poor efficiency, unless the process is sufficiently slow

which reduces power [3]. There is therefore a trade-off between

efficiency and power.

Most attempts to improve the efficiency or power of the air

compressor/expander aim at improving the heat transfer between

the air and its environment. One approach is to use multi-stage

processes with inter-cooling [4]. Efficiency increases as the num-

ber of stages increase. To improve the efficiency of the compres-

sor/expander with few stages, it is necessary to enhance the heat

transfer during the compression/expansion process. A liquid pis-

ton compression/expansion chamber with porous material inserts

has been studied in [5]. The porous material greatly increases

the heat transfer area and the liquid piston prevents air leak-

age. Numerical simulation studies of fluid flow and enhanced

heat transfer in round tubes filled with rolled copper mesh are

studied in [6]. Application of porous inserts for improving heat

transfer during air compression has also been investigated [7]. In

addition, the compression/expansion trajectory can be optimized

and controlled to increase the efficiency for a given power or to

increase power for a given efficiency [3, 7–9].

Another approach to increase the air compression efficiency

is to employ a water spray. The large number of small size

droplets with a high heat capacity can provide a high total surface

area for heat transfer [10–12]. However, the presence of signifi-

cant liquid volume in the piston chamber must also be accommo-

Proceedings of the ASME 2013 Heat Transfer Summer Conference HT2013

July 14-19, 2013, Minneapolis, MN, USA

HT2013-17611

1 Copyright © 2013 by ASME

dated. A simple theoretical analysis of a single droplet transport

phenomena in humid air and the prediction of the life time of a

freely-falling droplet is investigated in [13]. A descriptive mathe-

matical model for energy and exergy analysis is presented in [14]

for a co-current gas spray cooling system. One-dimensional sim-

ulations of liquid piston compression with droplet heat transfer

has been recently investigated in [15] to determine the conditions

required for significant improvement of compression efficiency.

In this paper, we develop a dynamic model of the water

droplets spray in a liquid piston air compressor. This model al-

lows us to investigate the effect of spray flow rate profile on the

air compression efficiency and optimize that profile for a given

set of desired parameters. The rest of the paper is organized as

follows: the dynamic model of the system describing the com-

pression cycle including water spray is derived based on an Eu-

lerian approach. Finite volume method is then used to transform

the partial differential equations (PDE) into a system of ordinary

differential equations (ODE) validated through a sample case

study. Next, the optimal problem is introduced by defining the

profit function as well as constraints. The resulted optimal con-

trol problem is then solved by discretization of the control input

over the time interval. Comparison between the optimal and non-

optimal spray profiles has been finally shown in the last section.

ModelingCompression Chamber: A liquid piston air compres-

sor consists of a vertical chamber in which the conventional solid

piston is replaced by a column of liquid. This liquid column is

driven into the chamber by a variable displacement pump con-

nected to the chamber inlet flow [5]. The chambers length and

diameter are shown by L and D, respectively. It is assumed that

initially, the chamber is filled with air which means the initial liq-

uid column height is zero. Since the heat capacity of the chamber

walls and the liquid column is much larger than the air, it is as-

sumed that the walls and liquid piston temperature maintain at

ambient temperature over the compression cycle. In addition,

due to good sealing property of the liquid column, no leakage is

considered for the air inside the chamber.

Water Droplets: Analysis of interaction between water

droplets and air inside a compression chamber is naturally a com-

plicated phenomena. While the droplets can collide and make

bigger droplets, they may also touch the chamber walls as well

as the liquid surface (piston) and get vanished. Moreover, droplet

size can change due to mass transfer between the liquid phase to

the gas phase. This interphase mass transfer is a complicated

function of several properties such as droplet temperature, air

temperature, pressure and humidity. Therefore, a precise dy-

namic model of such a process is difficult to be obtained. How-

ever, a simple model can be used to understand the basic behavior

of this system for further purposes. Here, a one-dimensional dis-

tribution for water droplet’s properties is considered in a lumped

air model. While all the air properties are assumed to be constant

over the spatial domain, a linear distribution is used to describe

air velocity in the chamber as:

U(x,t) =− Y(t)L−Y(t)

x (1)

where U is the air velocity and Y is the liquid piston height in-

side the chamber. Here, x shows the location inside the chamber

with respect to the coordinate system with origin located at the

top of the chamber and directed toward its bottom (liquid piston

surface). Thus, the air velocity is zero at the top (x = 0) while it

is maximum at the liquid surface (x∗ = L−Y(t)). From realistic

point of view, there is a mass transfer between liquid phase (water

droplets) and gas phase (air). However, no mass transfer (evapo-

ration) is considered between these phases due to the fact that the

overall temperature rise of droplets during the compression pro-

cess is less than saturated temperature for evaporation. By using

this assumption, no variation in droplet size and mass takes place

during the compression cycle. In summary, the droplets leave the

spray nozzle (at top of the chamber), move inside the air toward

the chamber’s bottom and collide into the liquid piston surface

and get accumulated into it (no droplet to droplet collision is con-

sidered). More details are shown in Fig. 1.

FIGURE 1. WATER SPRAY INSIDE LIQUID PISTON AIR COM-

PRESSOR

Defining r as the number of droplets per unit length of the cham-

ber (drop/m) and v as absolute droplet velocity and then applying

2 Copyright © 2013 by ASME

the conservation of mass principal, we will get

∂ r∂ t

+∂∂x

(rv) = 0. (2)

While a droplet is traveling in air, two different forces act on it

due to i) gravity and ii) drag. The gravity force is always constant

and directed toward the bottom of the chamber. However, the

drag force is a function of the relative speed between droplet and

air as well as the air density. Here, the drag force is modeled as:

fdrag(t) =1

2CdAρ(t)(v(x,t)−U(x,t))

2 (3)

where Cd is the drag coefficient and A is the reference area. For a

spherical droplet moving in air, Cd is about 0.47 and A is πd2

4 in

which d is the droplet diameter assumed to be constant over the

whole process. Now, by applying the conservation of momen-

tum principal, the second PDE describing the droplet’s velocity

dynamic is obtained as:

∂v∂ t

− ∂∂x

(v2

2)+g− fdrag

m= 0 (4)

where g is the acceleration of gravity and m is the droplet mass.

Since the drag force is always toward top of the chamber, a nega-

tive sign is used before drag force in Eqn. (4). The conservation

of energy is applied to derive the temperature dynamic of droplet.

After a few mathematical manipulations, we have:

∂E∂ t

+ v∂E∂x

+6

Csρwdh(x,t)(E(x,t)−T(t)) = 0 (5)

where E is the droplet temperature, T is the air temperature

and Cs is the specific heat capacity of water. While the heat

transfer area for each droplet is constant over time (due to fixed

droplet diameter), the convective heat transfer coefficient (h) is a

function of Reynolds number as well as air temperature. Based

on Ranz-Marshall correlation, the heat transfer coefficient of a

spherical droplet can be calculated as:

Nu(x,t) = 2+0.6Re12

(x,t)Pr13 (6)

where Re is the Reynolds number defined based on relative speed

between droplet and air as:

Re(x,t) =ρ(t)d|v(x,t)−U(x,t)|

μ(t)(7)

From Sutherland’s formula, the dynamic viscosity of air as a

function of its temperature can be calculated as follows:

μ(t) = μrTr +C

T(t) +C(

T(t)Tr

)32 (8)

In this equation, μr is the reference dynamic viscosity of air at

reference temperature Tr and C is the Sutherland’s constant for

air.

Air and Liquid Piston Dynamics: While the liquid

piston level is mainly governed by the liquid flow rate provided

by the hydraulic pump, the accumulation of water droplets into

the liquid column can also increase its level inside the chamber.

Such a consideration becomes more important when the liquid

piston is close to chamber’s top and the pressure ratio is large. In

this situation, even addition of a small amount of water as wa-

ter spray can cause a large change in air pressure due to its low

volume. To find the liquid piston height dynamics, consider a

control volume located at the piston surface. This control vol-

ume is chosen to contain both liquid piston and water droplet

(Fig. 2).

FIGURE 2. CONTROL VOLUME AT LIQUID PISTON SURFACE

By applying the conservation of mass principle for the total water

inside this control volume, we will have:

ddt(Apδw +V

∫δd

rdx) = F p + rvV (9)

where Ap is the cross sectional area of the chamber, V is droplet

volume and F p is the flow rate of liquid driven into the chamber

by the hydraulic pump. Now, if we let both δd and δw approach

to zero, Eqn. (9) will become:

Apδw +V r∗δd +V r∗δd = F p + r∗v∗V (10)

where * means the value of property at piston location (x∗). No-

tice that the third term on the left hand side of Eqn. (10) is zero

3 Copyright © 2013 by ASME

since δd approaches to zero. Considering the fact that δw = Y and

δd = −Y , the piston height dynamics can be finally determined

as:

Y(t) =F p(t) + r(x∗,t)v(x∗,t)V

AP − r(x∗,t)V(11)

The air temperature dynamics can be simply calculated

based on the ideal gas law and the total heat transfer of air. As

shown in Fig. 3, the air inside the chamber has heat transfer

to both water droplets and the surrounding materials. The heat

transfer coefficient between air and solid walls as well as liquid

piston surface is assumed to be constant (h). However, the heat

transfer coefficient between the air and droplets is a function of

local Reynolds number given by Eqn. (7). Combining these facts

and assumptions, air temperature dynamic is:

dTdt

= (1− γ)T(t)V(t)

V(t) +π

mairCv

⎛⎜⎜⎜⎝d2∫ x∗

0r(x,t)h(x,t)

(T(t)−E(x,t)

)dx︸ ︷︷ ︸

heat to droplets (H2)

+

(Dx∗+

D2

2

)h(T(t)−Twall

)︸ ︷︷ ︸

heat to walls (H1)

⎞⎟⎟⎟⎠ (12)

where γ is the heat capacity ratio of air, mair is the air mass inside

the chamber (fixed), Cv is the heat capacity of air and h is the

constant heat transfer coefficient between air and its surrounding

walls as well as liquid surface.

FIGURE 3. HEAT TRANSFER BETWEEN AIR AND WATER

DROPLETS AS WELL AS AIR AND SURROUNDING WALLS

Finally, applying the conservation of mass principal for the com-

pression chamber, the air volume dynamic can be determined by:

dVdt

=−(

F p(t) +Fs

(t)

)(13)

where Fs(t) is the flow rate of water spray into the chamber.

Solution Method: Complete dynamics of this system is

determined by Eqn. (2), (4), (5), (11), (12) and (13). The first

three equations are PDE with respect to time and space. Finite

Volume Method (FVM) is used to transform PDE system into an

ODE system. Resulted ODE system in addition to Eqn. (11),

(12) and (13) describe the complete dynamic behavior of the

whole system. This ODE system (including 3n+ 3 differential

equations, n is the number of finite volumes used in FVM) is

then solved in MATLAB R© using available ODE solvers.

Sample Case StudyA numerical simulation has been performed for a sample

case to show how the system’s states vary over the compression

cycle. Here, a constant flow rate is assumed for the liquid piston

(F p). While initially there is no water droplet in the chamber, a

constant flow rate spray is injected into the chamber starting at

t = 0.4 sec and ends at t = 0.8 sec. The compression ends when

the desired compression ratio is achieved (rd = 50). The liquid

piston flow rate is chosen for a total compression time of about 1

sec. The rest of the constant parameters used in this simulation

are given in Table 1.

TABLE 1. CONSTANT PARAMETERS USED FOR NUMERICAL

SIMULATIONS

Property Value Unit Property Value Unit

L 30 cm T0 293 K

D 5 cm Twall 293 K

d 50 μm Tr 291.15 K

g 9.806 m/s2 P0 1.01 bar

ρw 998 Kg/m3 μr 1.83e-5 Pa.s

Cs 4200 J/Kg.K C 120 K

Cd 0.5 − h 10 W/m2.K

R 286.9 J/Kg.K γ 1.4 −Pr 0.7 − Knz 8e-9 −

Results of the simulation are shown in Fig. 4. Due to extra heat

transfer area provided by water droplets after injection, the air

is cooled down and its temperature drops for a while. However,

4 Copyright © 2013 by ASME

after the spray stops, the air temperature rises again until the final

desired pressure ratio is achieved.

0 0.2 0.4 0.6 0.80

0.3

0.6

0.9

1.2

1.5

Sp

ray

Flo

w R

ate

(cc/

s)

Time (s)0 0.2 0.4 0.6 0.8

250

350

450

550

650

750

Air

Tem

per

atu

re (

K)

0 100 200 300 400 500 6000

1

2

3

4

5

6x 10

6

Air Volume (cc)

Air

Pre

ssu

re (

Pa)

Adiabatic Compression Compression with Water Spray Isothermal Compression

FIGURE 4. AIR TEMPERATURE AND WATER SPRAY FLOW

RATE VS. TIME (TOP), AIR PRESSURE VS. AIR VOLUME (BOT-

TOM)

Droplet density, velocity and temperature during this sample case

study have been shown in Fig. 5. While droplets leave the spray

nozzle with a large velocity, they decelerate fast and traverse the

rest of their trip between the nozzle and liquid piston with a much

smaller velocity. Consequently, the droplets are accumulated in a

region between the nozzle and liquid piston surface. The temper-

ature of droplets is equal to the ambient temperature when they

leave the spray nozzle. This is while due to heat absorption from

the compressing air, they heated up and reach the liquid piston

surface with a larger temperature. Once the injection stops, this

temperature rise gets even larger due to vanishing number of wa-

ter droplets.

Optimization of Spray Flow Rate for a Given MassLoading

In general, increasing the relative amount of water droplets

compared to air increases the compression efficiency by improv-

ing heat transfer. Compression efficiency defined as the ratio

between the stored energy in air (after compression) and the re-

quired input work. The stored energy in the air at pressure rP0

and ambient temperature is defined as the maximum work ob-

Time (s)

Ch

amb

er L

oca

tio

n (

m)

0 0.2 0.4 0.6 0.8

0.05

0.1

0.15

0.2

0.25

Den

sity

(d

rop

/m)

0.5

1

1.5

2

2.5x 10

7

Liquid Piston

Time (s)

Ch

amb

er L

oca

tio

n (

m)

0 0.2 0.4 0.6 0.8

0.05

0.1

0.15

0.2

0.25

Vel

oci

ty (

m/s

)

1

2

3

4

Liquid Piston

Time (s)

Ch

amb

er L

oca

tio

n (

m)

0 0.2 0.4 0.6 0.8

0.05

0.1

0.15

0.2

0.25

Tem

per

atu

re (

K)

300

320

340

360

380

400

420

Liquid Piston

FIGURE 5. DISTRIBUTION OF DROPLET DENSITY (TOP), VE-

LOCITY (MIDDLE) AND TEMPERATURE (BOTTOM) OVER THE

SPACE AND TIME DURING THE COMPRESSION CYCLE

tainable via an isothermal expansion as [3, 8]:

Wstored = rP0V ln(r) (14)

The input work is the summation of liquid piston work and the

water spray work (to inject water droplets into the high pressure

air). In addition, the energy loss due to pressure drop across

the spray nozzle is also a part of the required input work. This

pressure drop can be expressed as a function of spray flow rate:

ΔPnz(t) =

(Fs(t)

Knz

)2

(15)

5 Copyright © 2013 by ASME

where Knz is the discharge coefficient of the spray nozzle. Thus,

the input work can be calculated as:

Winput =−∫ Vf

V0

(P(t)−P0

)dV +P0(r−1)Vf +

∫ t f

0Fs(t)ΔPnz

(t)dt

(16)

where Vf is the final air volume at the end of compression (t =t f ). The compression efficiency is then defined as:

ηc = 100Wstored

Winput% (17)

The baseline compression efficiency is determined according to

the adiabatic compression. For a compression ratio of r = 50,

the adiabatic compression efficiency is about 54.4%. Consider-

ing the heat transfer from the surrounding walls (with the same

boundary conditions and constant parameters used for the pre-

vious case study) the compression efficiency increases to 57%.

However, by injecting the water droplets into the compressing air

as shown in Fig. 4, the compression efficiency increases to about

70.8% which is much higher than the case without spray. To

quantify how much water is added to the air (as droplet) during

the compression cycle, the spray mass loading (ML) is defined as

follows:

ML =mw

mair=

ρw∫ t f

0 Fs(t)dt

mair(18)

For the sample case study that resulted in 70.8% efficiency, MLis obtained to be about 0.5. It seems that increasing mass load-

ing always improves the compression efficiency by increasing

the heat transfer area. However, a quick look at Eqn. (16) reveals

the fact that increasing mass loading can have negative effect on

efficiency due to energy loss across the spray nozzle. Moreover,

due to dynamic behavior of droplets inside the air, the timing of

water spray is also important in improving the efficiency. For

example, spraying water into the air very early or late in time

can be useless. Therefore, it is important to find the best spray

profile (over time) for a given mass loading and liquid piston

profile. This problem is in fact an optimal control problem for

which the profit function is given by Eqn. (17) while the dy-

namic constraint is given by the air compression model includ-

ing water spray. Note that the algebraic constraints are the given

desired parameters such as compression ratio, compression time

and mass loading. Moreover, it is assumed that the liquid pis-

ton flow profile F p is also specified before hand. Based on these

definitions and assumptions, the optimal spray profile is:

Fs(t) = arg.max{ηc} (19)

In this paper, the continuous optimal control problem is parame-

terized as a finite dimensional problem and then solved numeri-

cally by standard algorithms for constrained parameter optimiza-

tion. The control input can be parameterized as:

Fs(t) =

T

∑i=1

fi.Ui(t) 0 ≤ t ≤ t f (20)

where fi’s are some constant parameters and Ui’s can be any

function. Here, we used linear function and Gaussian function

for Ui in different case studies. Once the control input defined

over the time interval, the dynamic states (i.e. droplet and air

properties) can be calculated over the time and space.

Optimal Spray Profile for Constant Piston Flow RateOptimal spray flow rate for different mass loadings are

found while the liquid piston flow rate is chosen to be constant.

Desired final pressure ratio r is 50 and the compression time

t f is about 1 sec. Other constant parameters describing the

compression chamber geometry, spray nozzle as well as initial

and boundary conditions are given in Table 1. Nine equally

spaced points over the time range are used to discretize the

control input Fs. The optimal spray flow rate for different mass

loadings are shown in Fig. 6. Note that each flow profile is

normalized based on its own mass loading. The thick blue

curve represents the time average of all optimal spray flow rates

resulted for different mass loadings.

The trend of these optimal spray profiles (Fig. 6-top) are ex-

pectable considering the fact that at the first half of the compres-

sion, there is enough heat transfer area provided by the surround-

ing walls while the air temperature is still not high. Thus, addi-

tional cooling with water droplet is not necessary in this phase.

On the other hand, injecting droplets into the air when the liq-

uid surface is close to the chamber’s top cannot be very effective

due to rapid transition of droplets from the spray nozzle into the

liquid piston. In this situation, injected droplets will not have

enough time to capture heat from air before touching the liquid

piston surface. As shown in Fig. 6-middle, the air temperature

of the optimal spray profile is higher than the constant flow spray

in the first half of the compression process. However, the opti-

mal spray profile does a better job and reduces the air tempera-

ture more in the second half (since some droplets are saved from

the first half). Hence, as expected, the overall compression ef-

ficiency of optimal profile is higher than the constant flow rate

case. Such an improvement is shown in Fig. 6-bottom where

the compression efficiency for different mass loadings is shown

for both optimal and constant spray flow rate. While for small

and large mass loadings the optimal and constant spray result in

similar efficiencies, their difference can get up to 2% for a mass

6 Copyright © 2013 by ASME

loading of 0.5. Note that the compression ratio and compression

time are the same for all cases. In particular, note that the com-

pression efficiency decreases for very large mass loadings since

the energy loss across the spray nozzle becomes a dominant term

in the input work.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time (s)

Flo

w R

ate

(No

rmal

ized

wit

h M

L)

ML= 0.11 ML= 0.22 ML= 0.44 ML= 0.88 ML= 1.76 ML= 3.48 ML= 6.92 Average

0 100 200 300 400 500 600200

300

400

500

600

700

800

900

Air Volume (cc)

Air

Tem

per

atu

re (

K)

ML=0.02 (Optimal) ML=0.02 (Constant) ML=0.22 (Optimal) ML=0.22 (Constant) ML=1.75 (Optimal) ML=1.75 (Constant)

10−2

10−1

100

10155

60

65

70

75

80

85

Mass Loading

Co

mp

ress

ion

Eff

icie

ncy

%

Optimal Flow Rate Constant Flow Spray

FIGURE 6. COMPARISON BETWEEN OPTIMAL SPRAY FLOW

RATE AND CONSTANT SPRAY FLOW RATE. NORMALIZED OP-

TIMAL SPRAY FLOW RATE (TOP), TEMPERATURE VS. VOL-

UME (MIDDLE), AND EFFICIENCY VS. MASS LOADING (BOT-

TOM)

Design of an Efficient and Power-Dense Air Compres-sor

Although the optimal spray profile improves the compres-

sion efficiency, it is not still satisfactory for the application of

CAES system. For such a compressor, a minimum thermal ef-

ficiency of 90% is required to achieve a reasonable round-trip

efficiency for the storage system. As discussed earlier, one ef-

fective way to improve compression efficiency is to increase heat

transfer area inside the compression chamber by inserting some

porous materials into the chamber. This will also increase the

convective heat transfer coefficient between air and solid wall

due to reduction of hydraulic diameter. Additionally, the piston

flow rate can be optimized to improve the efficiency through a

better use of available heat transfer capacity. Let’s consider the

design of an air compressor for the second stage compression in

a CAES system, where the inlet pressure is 5bar and the desired

compression ratio is 40 (in the first stage, air is compressed from

the ambient pressure to final pressure of 5bar). Due to required

power density for this compressor, the total compression time

must be 1 sec. Considering the chamber geometry given in Table

1 with some porous inserts, the total heat transfer product (h.A)

can be increased by a factor of 50 [8]. While a constant piston

flow rate results in a compression efficiency of 74.4%, optimiza-

tion of the piston flow rate allows us to increase the efficiency

up to 77.2% (Fig. 7-top). By introducing water droplets to the

air during compression (for the optimal piston flow rate), the ef-

ficiency can rise to 90.7% (for a mass loading of 5). However,

the efficiency can be improved even more if the constant spray

flow is replaced by the optimal one. Here, in order to have a

smoother optimal spray profile, a combination of Gaussian func-

tions is used to parameterize the spray profile over the compres-

sion time. In this way, the optimization will be summarized as

finding the optimal set of amplitudes for these functions. For the

same mass loading (ML=5), the optimal spray flow rate is found

as shown in Fig. 7-top. Applying this spray profile, the compres-

sion efficiency can be increased to 94.5% which has a noticeable

difference compared to the constant spray flow. Fig. 7-bottom

shows the air temperature versus volume for these five different

compression cases. As shown, by reducing the air temperature

rise over the compression process, the compression efficiency

will be improved.

ConclusionsEquipping a liquid piston air compressor with a water

droplet spray can improve the compression efficiency signifi-

cantly. However, for a given compression chamber geometry and

liquid piston flow profile, the optimal spray profile can improve

the compression efficiency even more than constant flow spray

with the same mass loading. In this work, a general numerical

optimization approach was proposed to optimize the spray pro-

file for different mass loadings and liquid piston profiles. For

a constant liquid piston flow rate and compression ratio of 50,

up to 2% improvement in efficiency was obtained by optimiz-

ing the spray profile. Similarly, the spray profile was optimized

for the optimal liquid piston profile in a compression chamber

7 Copyright © 2013 by ASME

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

Pis

ton

Flo

w R

ate

(lit

/sec

)

Time (s)

0 0.2 0.4 0.6 0.8 10

25

50

75

100

125

150

Sp

ray

Flo

w R

ate

(cc/

sec)

Optimal Piston Flow Rate (lit/sec)

Optimal Spray Flow Rate (cc/sec)

0 100 200 300 400 500 600200

300

400

500

600

700

800

900

61.2%

74.4%

77.2%

90.7%

94.5%

Air Volume (cc)

Tem

per

atu

re (

K)

Adiabatic

Constant Piston Flow

Optimal Piston Flow

Optimal Piston Flow & Constant Spray Flow (ML=5)

Optimal Piston Flow & Optimal Spray Flow (ML=5)

FIGURE 7. OPTIMAL COMPRESSION PISTON PROFILE FOR

THE GIVEN COMPRESSION RATIO AND COMPRESSION TIME

WITH THE CORRESPONDING OPTIMAL SPRAY PROFILE FOR

THE GIVEN MASS LOADING OF 5 (TOP); TEMPERATURE VER-

SUS VOLUME FOR FIVE DIFFERENT CASES (BOTTOM)

with porous inserts. Combination of these heat transfer enhance-

ment methods allows us to design an efficient and power dense

air compressor where the compression efficiency is boosted up

from 74.2% to 94.5%. Potentially, this improvement can be in-

creased by simultaneous optimization of liquid piston and spray

profiles instead of individual optimizations that is the topic of

future studies in this field. In addition, it is observed that the

water spray is more needed at the end of compression process

where the air temperature is high. However, due to small tran-

sition time of droplets between the nozzle and the liquid piston

surface, it would be better to change the direction and/or loca-

tion of spray nozzles. For example, spraying from the sides of

the compression chamber (and close to the top) in a radial direc-

tion can be more effective as a result of longer lifetime of water

droplets.

REFERENCES[1] P. Y. Li, E. Loth, T. W. Simon, J. D. Van de Ven and

S. E. Crane, “Compressed Air Energy Storage for Offshore

Wind Turbines,” in Proc. International Fluid Power Exhibi-tion (IFPE), Las Vegas, USA, 2011.

[2] M. Saadat and P. Y. Li, “Modeling and Control of a Novel

Compressed Air Energy Storage System for Offshore Wind

Turbine,” in Proc. American Control Conference, pp. 3032-

3037, Montreal, Canada, 2012.

[3] C. J. Sancken and P. Y. Li,“Optimal Efficiency-Power Rela-

tionship for an Air Motor/Compressor in an Energy Storage

and Regeneration System,” in Proc. ASME Dynamic Systemsand Control Conference, pp. 1315-1322, Hollywood, USA,

2009.

[4] J. D. Lewins, “Optimising and intercooled compressor for an

ideal gas model,” Int. J. Mech. Engr. Educ., vol. 31, Issue 3,

pp. 189-200, 2003.

[5] J. D. Van de Ven and P. Y. Li, “Liquid Piston Gas Compres-

sion,” Applied Energy, vol. 86, Issue 10, pp. 2183-2191, 2009.

[6] M. Sozen and T. M. Kuzay, “Enhanced Heat Transfer in

Round Tubes with Porous Inserts,” Int. J. Heat and FluidFlow, vol. 17, Issue 2, pp. 124-129, 1996.

[7] A. T. Rice and P. Y. Li, “Optimal Efficiency-Power Trade-

off for an Air Motor/Compressor with Volume Varying Heat

Transfer Capability,” in Proc. ASME Dynamic Systems andControl Conference, Arlington, USA, 2011.

[8] M. Saadat, P. Y. Li and T. W. Simon, “Optimal Trajectories

for a Liquid Piston Compressor/Expander in a Compressed

Air Energy Storage System with Consideration of Heat Trans-

fer and Friction,” in Proc. American Control Conference, pp.

1800-1805, Montreal, Canada, 2012.

[9] F. A. Shirazi, M. Saadat, B. Yan, P. Y. Li and T. W. Si-

mon,“Iterative Optimal Control of a Near Isothermal Liquid

Piston Air Compressor in a Compressed Air Energy Storage

System,” to appear in American Control Conference, Wash-

ington, DC, 2013.

[10] S. Kachhwaha, P. Dhar, S. Kale, “Experimental Studies and

Numerical Simulation of Evaporative Cooling of Air with a

Water Spray- I. Horizontal Parallel Flow,” Int. J. Heat Trans-fer, vol. 41, pp. 447-464, 1998.

[11] S. Sureshkumar, S. Kale, P. Dhar, “Heat and Mass Transfer

Processes Between a Water Spray and Ambient Air - I. Ex-

perimental Data,” Applied Thermal Engineering, vol. 28, pp.

349-360, 2008.

[12] W. Sirignano, Fluid Dynamics and Transport of Dropletsand Sprays, Cambridge University Press, 1999.

[13] H. Barrow, C.W. Pope, “Droplet Evaporation with Refer-

ence to the Effectiveness of Water-Mist Cooling,” Applied En-ergy, vol. 84, pp. 404-412, 2007.

[14] A. Niksiar, A. Rahimi, “Energy and Exergy Analysis for

Cocurrent Gas Spray Cooling Systems Based on the Results

of Mathematical Modeling and Simulation,” Energy, vol. 34,

pp. 14-21, 2009.

[15] C. Qin and E. Loth, “Liquid Piston Compression with

Droplet Heat Transfer,” in Proc. 51st AIAA Aerospace Sci-

ences Meeting, Grapevine, TX, 2013.

8 Copyright © 2013 by ASME