MODELING AND TRAJECTORY OPTIMIZATION OF WATER SPRAY COOLING IN ALIQUID PISTON AIR COMPRESSOR
Mohsen SaadatDept. of Mechanical Engineering
University of MinnesotaMinneapolis, MN 55455
Email: [email protected]
Farzad A. ShiraziDept. of Mechanical Engineering
University of MinnesotaMinneapolis, MN 55455
Email: [email protected]
Perry Y. LiDept. of Mechanical Engineering
University of MinnesotaMinneapolis, MN 55455
Email: [email protected]
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.
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