how moisture content a ects the performance of a liquid...
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How Moisture Content A↵ects the Performance of a Liquid Piston AirCompressor/Expander ,II I
Anirudh Srivatsaa, Perry Y. Lia,⇤
aDepartment of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Abstract
For a compressed air energy storage (CAES) system to be competitive for the electrical grid, the air com-pressor/expander must be capable of high pressure, e�cient and power dense. However, there is a trade-o↵between e�ciency and power density mediated by heat transfer. This trade-o↵ can be mitigated in a liquid(water) piston air-compressor/expander with enhanced heat transfer. However, in the past, dry air hasbeen assumed in the design and analysis of the compression/expansion process. This paper investigates thee↵ect of moisture on the compression e�ciency and power. Evaporation and condensation of water playcontradictory roles - while evaporation absorbs latent heat enhancing cooling, the tiny water droplets thatform as water condenses also increase the apparent heat capacity. To investigate the e↵ect of moisture, a0-D numerical model that takes into account the water evaporation/condensation and water droplets hasbeen developed, assuming equilibrium phase change. The 0-D model is also extended to a 1-D model toinvestigate the spatial e↵ect. To increase computational e�ciency, a uniform pressure in the 1-D deformablemodel is assumed. Results show that inclusion of moisture improves the e�ciency-power trade-o↵ minimallyat lower flow rates, high e�ciency cases, and more significantly at higher flow rates, lower e�ciency cases.This e↵ect is the same regardless of whether air is assumed to be an ideal gas or a real gas. The improvementis primarily attributed to the increase in apparent heat capacity due to the increased propensity of waterto evaporate. While the 1-D model does capture the spatial e↵ect, the 0-D model is found to be su�cientlyaccurate in predicting the e�ciency and power density of the compressor.
Keywords: Compressed air energy storage (CAES), Isothermal compressor/expander, Liquid piston, Heattransfer, Evaporation/condensation, Wet compression, Overspray fogging
Hightlights
• Quantifies moisture e↵ect on isothermal com-pressor/expander (C/E) for CAES
• 0-D and 1-D models for liquid piston C/E in-clude evaporation, condensation
• Dominant moisture e↵ect due to propensity ofwater to evaporate
IA portion of the results in this paper was presented in[1] in abbreviated form. The current paper contains moreextensive results (such as derivation and results of the 1-Dmodel) and expanded exposition and discussion.
IIResearch supported by the National Science Foundationunder grant EFRI-1038294.
⇤Corresponding authorEmail addresses: [email protected] (Anirudh
Srivatsa), [email protected] (Perry Y. Li)
• Moisture e↵ect only significant for high flowrates, low e�ciency, high temp cases
• 0-D and 1-D models predict e�ciency andpower similarly
1. Introduction
A grid scale energy storage that is economical anddispatchable is key to meeting the challenge of inte-grating more and more renewable energy in the elec-trical grid. Since renewable energy such as wind orsolar are intermittent, variable and unpredictable,without energy storage, backup power plants areneeded to compensate for the mismatch betweenpower supply and demand. Currently, most of theseback up power plants, known as “peaker” plants arenatural gas turbine generators that use fossil fuels
Preprint submitted to Energy Storage April 23, 2018
and are expensive to construct, maintain and op-erate. Compressed air energy storage (CAES) iswidely believed to be a viable means for storinglarge amount of energy.In recent years, an Open Accumulator Isother-
mal Compressed Air Energy Storage (OAICAES)system has been proposed [2] as a cost e↵ective,scalable, fossil-fuel free, dispatchable approach forgrid scale energy storage. Whereas a traditionalCAES stores the compressed air in undergroundsalt caverns and reuses the energy by mixing thecompressed air with fuel in natural gas turbine,the OAICAES does not use any fossil fuel andstores and reuses energy by compressing and ex-panding the air using a near isothermal compres-sor/expander. This results in a much higher overalle�ciency. In addition, with the open accumulatorarchitecture, the energy storage density can be in-creased by 5-6 times1 so that compressed air canbe stored in pressure vessels without geographicalrestrictions on the availability of underground cav-erns. The system can also be directly integratedwith a wind turbine.A key element of the OAICAES is the near-
isothermal air compressor/expander that com-presses/expands air from/to atmospheric pressureto/from 200bar. The compressor/expander wouldbe most e�cient if the compression/expansion pro-cesses are isothermal. However, this is often atthe expense of power density as long cycle timesare needed to allow for heat transfer, so that large(and expensive) compressor/expanders are neededto satisfy the required power. To improve the trade-o↵ between e�ciency and power density, one ap-proach is to spray tiny water droplets into the sys-tem to absorb the compression heat [3]. This isespecially useful for low pressure stage of the com-pressor/expander. Another approach, better suitedfor high pressure stage, is a liquid (water) pistoncompressor/expander that uses the movement ofthe water column to compress and expand the airabove it (Fig. 1) [4]. This allows the use of porousmedia to dramatically increase the heat transferarea [5, 6, 7]. The liquid also forms an excellentseal for the compressed air. By optimizing the liq-uid piston’s trajectory, further improvement can beobtained [8, 9, 10, 11, 12, 13, 14]. Overall, two or-ders of magnitude increase in power density can be
1This is due to the possibility of maintaining air pressureto be constant even as compressed air is depleted and to theuse of higher pressure.
achieved without sacrificing e�ciency [12].
Figure 1: A gourd shaped liquid piston compressor/expanderwith porous media. Note that the current study assumes acylindrical compressor/expander.
The models used in our previous work for the de-sign and analysis of the liquid piston near isother-mal air compressor/expander have assumed thatthe air in the chamber is completely dry despite itbeing in contact with the water piston and the pos-sible presence of water films on the porous media.This paper aims to study the e↵ect of moisture inthe air on the compressor/expander performance.The presence of moisture has two contradictory ef-fects. On the one hand, evaporation absorbs latentheat that helps to keep the compressed air cool;on the other hand, condensation into tiny waterdroplets suspended in air increases the heat capac-itance and heat transfer surface area. Evaporationand condensation are enhanced respectively by theincrease in temperature and the increase in pres-sure. Which e↵ect is more important is not cleara-priori. This is compounded by the fact that sig-nificant temperature and pressure variations, whicha↵ect evaporation/condensation, occur during thecompression/expansion cycle.
To investigate the e↵ect of moisture, a 0-D nu-merical model that considers phase change and heattransfer to/from water droplets is developed in thispaper. The two phases of water are assumed tobe in quasi-equilibrium at equal temperature withsaturated vapor, and air is assumed to be either
2
an ideal gas or a real gas. The 0-D model, (withair assumed to be an ideal-gas), is also extendedto 1-D spatially distributed model to capture thespatial variation. In order to increase computa-tional e�ciency, the 1-D model assumes that thepressure is uniform within the compression cham-ber, thus avoiding the need to solve the 2nd or-der full Navier-Stokes equation. This assumptionwas previously verified in detailed 2D CFD simu-lation using COMSOL software. The 0-D and 1-D models are exercised in the compression mode,with and without moisture, and for di↵erent liq-uid piston speeds. Results show that the presenceof moisture improves the e�ciency-power trade-o↵minimally at lower flow rates, high e�ciency, lowertemperature cases, and more significantly at higherflow rates, lower e�ciency and higher temperaturescases. This e↵ect is the same for both the ideal gasand real gas assumptions for air.The beneficial e↵ect of injecting water into the
compression stage of a gas turbine (known as wetcompression) has been studied and documented inthe literature (e.g. [15, 16, 17, 18, 19, 20]). The sig-nificant reduction in compression work has been at-tributed to the evaporative cooling (overspray fog-ging) e↵ect. However, a key di↵erence between theliquid piston CAES and the gas turbine applicationsis that the former has, by design, additional heattransfer intended to operate at lower temperatureat near isothermal condition. In the latter, theseadditional heat transfer is often negligible and ig-nored in analysis [18].The rest of the paper is organized as follows. In
Section 2, the system description and the 0-D modelis developed. The extension to a 1-D system modelis presented in Section 3. Simulation results fordi↵erent compression rates are presented in Section4. Discussion and concluding remarks are given inSections 5 and 6.
2. System Description and the 0-D Model
We consider the 2nd stage liquid piston com-pressor/expander for compressing and expandingair between 7bar and 200bar. The compres-sion/expansion chamber is cylindrical with a diam-eter of 76mm and length of 483mm which corre-sponds to the experimental setup in [7]. The cham-ber is initially filled with air at 7bar pressure. Aswater is pumped into the chamber from the bot-tom, the air volume decreases and the air pressure
increases. The chamber can be empty or filled uni-formly with porous media. In the latter case, theporous media increases the surface area for heattransfer.
The air is assumed to be saturated with watervapor at all times such that the vapor pressure ofwater is the saturated vapor pressure. As cham-ber volume decreases, water is assumed to condensehomogeneously into tiny droplets suspended in theair. The diameters of the water droplets are as-sumed to be small such that they are in thermalequilibrium with the air surrounding them. Hence,the suspended water droplets serve to increase theheat capacitance of the air.
As temperature increases, the saturated vaporpressure increases and there is an increased ten-dency for the water to evaporate, either from thedroplets or from the liquid piston. Latent heat isabsorbed or released by the water during evapo-ration and condensation respectively. Evaporationtends to keep the air cool but condensation tendsto increase the air temperature.
Temperature (K)
250 300 350 400 450 500
Sa
tura
tion
Va
po
r P
ress
ure
(P
a)
×106
0
0.5
1
1.5
2
2.5
3
Figure 2: Saturated Vapor Pressure (Ps(T )) of water as afunction of temperature
In the 0-D model, the entire air and moisture inthe compression/expansion chamber is assumed tobe uniform. Let the decreasing chamber volume beV (t), temperature be T (t) and the air density be⇢a
(t). The total pressure P is:
P = Pa
(T, ⇢a
) + Ps
(T ) (1)
where Pa
(T, ⇢a
) is the partial pressure of air satis-fying the ideal gas or the real gas assumption,
Pa
(T, ⇢a
) = ⇢a
Ra
T (2)
and Ps
(T ) is the saturated vapor pressure for steam
3
computed using the Antoine equation (with Ps
ex-pressed in mmHg and T expressed in degC) [21]:
Ps
(T ) = 10As� BsCs+T mmHg (3)
with As
= 8.07131, Bs
= 1730.63 degC, andC
s
= 233.426 degC for temperature below 100degC and A
s
= 8.14079, Bs
= 1810.97 degC,and C
s
= 244.485 degC for temperature above 100degC. Note that P
s
(T ) increases with temperature(Fig. 2) so that water needs to evaporate to keepthe air saturated as temperature increases.Let u
a
(T, ⇢a
) be the specific internal energy ofair so that C
va
= @/@T ua
(T, ⇢a
) is the constantvolume specific heat capacity of air. If treated asideal gas, C
va
= 718J/kg/K and if treated as areal gas, C
va
is a function of T and P . Similarly,let C
w
= 4200J/kg/K and Cvs
= 1410.8J/kg/Kbe the constant volume specific heat capacities ofliquid water and steam respectively.Applying the first law of thermodynamics and
neglecting the potential and kinetic energies of thegases, the energy equation can be written as
Utot
= �W �HT = �PV �HT (4)
where Utot
is the internal energy of the air, waterand steam within the air volume above the liquidchamber, �PV is the boundary work by the liquidpiston, with �V = Q, the liquid piston flow rate,P is the total pressure, and HT is the heat transfer(rate) out of the system. Changes in internal en-ergy of the air, water and steam due to temperaturechange and phase change is:
Utot
=ma
ua
(T (t), ⇢a
(t)) + (mw
(t)Cw
+ms
(t)Cvs
)T
� mw
ufg
=ma
✓@u
a
@TT +
@ua
@⇢a
⇢a
◆+ (m
w
Cw
+ms
Cvs
)T
� mw
ufg
=ma
✓@u
a
@TT � @u
a
@⇢a
⇢a
VV
◆+ (m
w
Cw
+ms
Cvs
)T
� mw
ufg
(5)
where ufg
= 2260kJ/kg is the specific internal heatof evaporation for water. It is assumed that as wa-ter evaporates, the aerosols are turned into steam,and as steam condenses, they condense back intoaerosols uniformly dispersed in the air-space, andthere is no condensation onto the surfaces. The
aerosol of condensed water is assumed to be at ther-mal equilibrium with the air in the chamber. Hence,
mw
= �ms
(6)
Steam is assumed to behave as an ideal gas so that:
ms
(t) =Ps
(T (t)) · V (t)
Rs
T (t)(7)
where Rs
is the gas constant for steam, we have
mw
= �ms
= � d
dT
✓Ps
(T )
T
◆V
Rs
T� Ps
(T )
Rs
· T V (8)
Substituting this into (5) results in:
Utot
= ma
@ua
@TT �m
a
@ua
@⇢a
⇢a
VV
+(mw
Cw
+ms
Cvs
)T
+d
dT
✓Ps
(T )
T
◆V u
fg
Rs
T +Ps
(T ) · ufg
Rs
TV
(9)
Further substituting into (4) and rearranging,
✓m
a
@ua
@T+ (m
w
Cw
+ms
Cvs
)+
d
dT
✓Ps
(T )
T
◆V (t)
Rs
ufg
◆T
+
✓P � @u
a
@⇢a
⇢2a
+Ps
(T )V
Rs
· T ufg
◆V +HT = 0
(10)
The equation can be further simplified using
d
dT
✓Ps
(T )
T
◆V (t)
Rs
ufg
= ma
✓dP
s
dT� P
s
T
◆ufg
⇢a
Rs
T(11)
so that Eq.(10) becomes:
A(T, ⇢a
,mw
,ms
)T = �B(P, T )V �HT (12)
where
A(T, ⇢a
,mw
,ms
) = ma
@ua
@T+ m
w
Cw
+ms
Cvs
+ma
✓dP
s
dT� P
s
T
◆ufg
⇢a
Rs
T| {z }✏(t)
(13)
B(P, T ) =
✓P � @u
a
@⇢a
⇢2a
+Ps
(T )
Rs
Tufg
◆
(14)
4
The expression @ua@T
in A is simply Cva
, the constantvolume specific heat capacity, and the expression@ua@⇢a
⇢2a
is in fact the internal pressure [22]:
⇡T
= �@ua
@⇢a
⇢2a
= T
✓@P
a
@T
◆
V
� Pa
(15)
which vanishes for an ideal gas.The terms A(T, ⇢
a
,mw
,ms
) and B(P, T ) are re-ferred to as the apparent heat capacitance and theapparent pressure.From Eqs.(12)-(14), we see that moisture
has e↵ect on both the apparent heat capaci-tance A(T, ⇢
a
,mw
,ms
) and on the apparent pres-sure B(P, T ). In the absence of moisture,A(T, ⇢
a
,mw
,ms
) is just the heat capacitance of airand B(P, T ) is just the pressure if air is an ideal gas.Moisture has two e↵ects on the apparent heat ca-pacitance A(T, ⇢
a
,mw
,ms
): 1) the water dropletsand steam o↵er additional heat capacitance; 2) itchanges the apparent heat capacitance of air due tothe fact that variation in temperature causes wa-ter to evaporate or condense. E↵ect on B(P, T )changes the apparent boundary work as changes involume induce water to evaporate or condense ab-sorbing and releasing latent heat.In the 0-D model as defined in Eqs.(12)-(14), air
can either be an ideal gas or a real gas. In thispaper, results are obtained with both the ideal gasand real gas assumptions.The heat transfer term HT in (12) is obtained
using a convective heat transfer model from the airvolume to the wall and to the porous media whichare assumed to be at ambient temperature. Thisassumption is an idealization of the actual situationin which the temperature of the porous media andchamber walls is repeatedly refreshed by the liquidpiston filling the chamber every cycle. Thus, heatis ultimately transferred out by the liquid piston tothe ambient environment. The actual temperaturerise in the porous media and walls is governed bythe heat capacitance and conductivity.Two heat transfer cases are considered:
1. an empty chamber in which air exchanges heatwith the chamber wall, top cap and the liquidpiston surfac
2. the chamber is filled with a porous medium.The porous medium is assumed to be the par-allel plate heat exchanger studied in [5, 7] witha porosity of 70%, a surface area per volumeof 655 m�1, and a hydraulic diameter D
h
of0.0025 m.
The heat transfer is given by:
HT = h(t)a(t)(T (t)� T0) (16)
where h(t)a(t) is the combined product of the vary-ing heat transfer coe�cient and heat transfer sur-face area of the cylinder walls and porous mediasurfaces (if present) as a function of liquid pistonposition, T0 is the temperature of the surfaces, as-sumed to be the initial and constant ambient tem-perature. The heat transfer coe�cients h(t) for thechamber walls are derived from an empirical cor-relation which is obtained in turn from fitting thedata from multiple COMSOL CFD simulation runsunder various conditions (including variation is ge-ometry, speed and physical constants) (see [23] fordetails). The correlation is aggregated for all thewall surfaces as shown in Fig.3. h(t) for the porousmedia surfaces, when present, is given by the Nus-selt number correlation from [5]:
Nu =h(t)D
h
k= 2 + 0.0876Re0.792Pr
13 (17)
obtained using detailed CFD studies on the Repre-sentative Elementary Volume (REV) of the porousmedia, taking both natural and forced convectioninto account. In (17), k is the conductivity of air,Re is the Reynold’s number and Pr is the Prandtlnumber, D
h
is the hydraulic diameter of the porousmedia.
3. Extension to 1-D Model
The 0-D model is simple and computationally ef-ficient. However, it ignores spatial variation. In thissection, a 1-D model is constructed to enhance theresults. The chamber volume occupied by the airand moisture is partitioned into multiple sectionswith its own temperature T
i
, density ⇢ai
and waterdroplet and steam contents, m
wi
and msi
(see Fig.4). The elevations of the boundaries of partitioni (from top to bottom) are X
i�1(t) and Xi
(t). Adeformable mesh approach is used in which the airmass and water mass contained in each partitionare constant. i.e. the partitions move with the airand water masses in it. The movement of the par-tition boundaries X
i
(t) are determined so that thetotal pressure is uniform for all partitions. Withthis method, the inertia e↵ects of the air can be ne-glected. Integration of the full Navier-Stokes equa-tion can be avoided, thus improving computationale�ciency.
5
x = (ρ μ)*v^0.5*(μ k)^1.13
y=0.0026x^2+10.6 x + 1439
y=
(h/k
)*(L
/D)^
0.5
Figure 3: Empirical heat transfer coe�cient correlation forthe compression/expansion chamber walls. Here µ is thekinematic viscosity, k is the thermal conductivity, ⇢a is thedensity, v is the piston speed, L and D are the length anddiameter of a compression chamber. Solid line is the empir-ical correlation and dots are results from the CFD experi-ments. Given physical and geometric properties, the wallsheat transfer coe�cient hwall can be obtained from the cor-relation.
The individual partition is modeled similarly asthe 0-D model in Eq.(12) such that air and wa-ter masses remain constant within each partition.There are heat transfer to the walls and, whenpresent, to the porous media within that partition,similar to (16) except that h(t)a(t) for the cham-ber walls and porous medium are computed for thatpartition. For the bottom and top partitions, thereare also heat transfer to the liquid piston and tothe top cap. Di↵erent correlations of h(t), similarto Fig.3, are used for the sides and the top andbottom caps of the chamber; the correlation in (17)is used for the porous media. Conduction betweenpartitions is also included with the conduction dis-tance being the distance between the center pointsof two successive partitions:
HTi,conduct
=2A
p
· k(X
i
�Xi�2)
(Ti
� Ti�1)
+2A
p
· k(X
i+1 �Xi�1)
(Ti
� Ti+1)
where k is the thermal conductivity of air and Ap
is the cross sectional area.For the i�th partition, the pressure of the air
volume is the sum of the pressure of the dry air
Figure 4: 1-D Model with Heat Transfer Directions
and the pressure of the water/steam mixture,
Pi
= Pa
(Ti
, ⇢ai
) + Ps
(Ti
) (18)
Pi
can be calculated to be
Pi
= Pa
(Ti
(t), ⇢ai
(t)) +@
@TPs
(Ti
)Ti
=
✓@P
a
@T+
@Ps
@T
◆Ti
� @Pa
@⇢a
⇢a
VVi
(19)
Since Ti
is already known from (12), Pi
can be writ-ten as
Pi
= �✓✓
@Pa
@T+
@Ps
@T
◆B
i
Ai
+@P
a
@⇢a
⇢a
V
◆
| {z }Ci
Vi
�✓@P
a
@T+
@Ps
@T
◆1
Ai
HTi
| {z }↵i
(20)
where HTi
signifies the total heat transfer for par-tition i.
The deformation of the partitions, Vi
, will be de-fined such that P1(t) = P2(t) = . . . = P
n
(t). Toensure that this happens, we implement stable dy-namics:
�Pi
= ���Pi
(21)
where � > 0 is a small number and �Pi
:= Pi+1 �
Pi
. With this desired dynamics, the pressure dy-namics equations can be expressed in matrix form
6
as2
666664
↵2 � ↵1
↵3 � ↵2...
↵n
� ↵n�1
Q
3
777775+ �
2
666664
�P1
�P2...
�Pn�1
0
3
777775
=
2
666664
C1 �C2 0 . . . 00 C2 �C3 0 . . .
0 0. . .
. . . 00 . . . 0 C
n�1 �Cn
�1 �1 . . . �1 �1
3
777775
2
666664
V1
V2...Vi
Vn
3
777775
(22)
where Q is the liquid piston flow rate, and ↵i
andC
i
are defined in (20). The rate of change of parti-tion volumes V
i
can then be solved from (22). Thevelocity of the partition boundaries X
i
can be ob-tained from:
Xi
=1
Ap
iX
k=1
Vi
(23)
Note if porous media inserts were included in themodel, that A
p
is a function of the porous medium’sporosity.The velocity used to calculate heat transfer is ob-
tained as the mean partition velocity (Xi
+Xi�1)/2
from the previous time step. The velocity generallyincreases from the top cap towards the liquid pistonat the bottom of the chamber.
4. Results
This section presents the results for simulating aircompression from 7bar to 200bar, with and withoutmoisture. Section 4.1 considers the 0-D model withair assumed to be an ideal gas. Section 4.2 considersthe same model but with air assumed to be a realgas. Section 3 considers the 1-D model.
4.1. 0-D Model with air as an ideal gas
The 0-dimensional model is simulated with andwithout porous media for the compression processfrom 7bar to 200bar. Air is initially assumed tobe an ideal gas. Case studies with five di↵erentconstant flow rates are performed: 5 cc/s, 20 cc/s,100 cc/s, 400 cc/s, and 800 cc/s. The slowest flowrate case is close to an isothermal compression whilethe fastest flow rate case is close to an adiabaticcompression.Figs. 5-6 show the volume, temperature, droplet
mass and vapor mass for the di↵erent flow rates
for the cases without porous media. The time tra-jectories were viewed to intuitively determine thevalidity of the model but pressure was used as theindependent variable in the plots to allow di↵erentflow rates to be compared. In these figures, theprocesses proceeded from left to right.
From Fig. 5 (bottom), the temperature is greaterwith a higher flow rate, because the heat transferhas less time to take e↵ect. As a result, the volumein Fig. 5(top) for the empty chamber is greater forthe same pressure than for the porous media filledcase, according to the ideal gas law. Fig. 6 showsthe mass trajectories of the water droplets and va-por. Condensation into droplets only occur for theslower flow rates. The reverse is true for water va-por whose contents are higher for the high flow ratecases and decrease for low flow rate cases. The gen-eral trends correlate with temperature being higherfor the high flow rate cases and as compression pro-gresses, which inhibit condensation but encourageevaporation. The 20cc flow rate case is interest-ing in that droplets are formed initially but theyevaporate later on. This illustrates the oppositee↵ects of increasing pressure which encourages con-densation and of increasing temperature that en-courages evaporation. Notice also that even whenwater droplet mass remain at 0, evaporation canstill take place as the program uses the water fromthe liquid piston or films on the heat transfer sur-faces in such circumstances.
Figs. 7-8 show the results for the porous mediafilled cases. Compared to the empty chamber cases,for the same flow rate, the temperature and volumeare lower at the same pressure. This is expected andconsistent with computational and experimental re-sults in [12, 6, 7], demonstrating the porous media’se↵ectiveness to improve heat transfer. As for theempty chamber case, higher flow rates tend to in-crease evaporation and prohibit droplet formation.The trend that water condenses in the beginningand evaporates later on is also observed. However,for the same flow rate, less water vapor and morewater droplets occur in the porous media case. Thereason for this is that porous medium reduces tem-perature which favors condensation.
To evaluate the e↵ect of moisture on the com-pressor performance, the e�ciency of compressionis plotted against the power in Fig. 9. E�ciencyis defined to be the ratio between the isothermalwork to compress the same amount of air (whichis also the stored energy) and the actual work in-put [2]. This includes the isobaric ejection work
7
Figure 5: Empty chamber volume vs Pressure (top) and tem-perature vs pressure (bottom) relations for various flow rates.
but excludes the work by the ambient pressure of7 bar which is taken to be the pressure of the 1ststage compressor and of the case pressure of the liq-uid piston pump. The isothermal work is used as areference because it represents the minimum workinput (and the maximum work output of an ex-pander) if the heat sink/source is at the initial tem-perature of T0. Power is defined as the isothermalwork (or stored energy) divided by the compres-sion time. Fig. 9 shows that there is an e�ciency-power trade-o↵ with and without steam, for boththe empty chamber and porous media filled cases.For the empty chamber case, moisture increasesboth e�ciency (from 62% to 72% at 3000kW) andpower (from 500W to 2000W at 73% e�ciency) forthe high flow rate, high power density, low e�ciencycases, whereas for the low flow rates, high e�ciency,low power cases, the improvement is minimal. Forthe porous media filled chamber case, the e�ciencyis higher than the empty chamber case for the same
Figure 6: Empty chamber water droplet mass vs pressure(top) and water vapor mass vs pressure (bottom) for variousflow rates.
power. The presence of steam also increases e�-ciency at the same power, and increases power atthe same e�ciency. However, the improvements aremuch less. For example at the highest power, theimprovement in e�ciency is less than 1%.
To determine what contributes to the improve-ment in the steam case and why a larger improve-ment is observed at higher flow rates, the apparentheat capacity term A(T, ⇢
a
,mw
,ms
) and the appar-ent pressure term B(P, T ) in (13)-(14) are plottedin Figs. 10 for the empty chamber case. Increas-ing B(P, T ) will have a tendency to increase tem-perature and reduce e�ciency; whereas increase inA(T, ⇢
a
,mw
,ms
) has the tendency to reduce tem-perature. From Figs. 10, we see that B increasesmarginally at low flow rates but is doubled at highflow rates. On the other hand, A(T, ⇢
a
,mw
,ms
) in-creases by less than 50% for low flow rates, but it isincreased by more than 5 times at high flow rates.This suggests that at low rates, the e↵ect of B andA more or less cancel out; but at high flow rates,
8
Figure 7: Porous media filled chamber volume vs pressure(top), temperature vs pressure (bottom) for various flowrates.
the e↵ect of A increasing dominates, contributingto lower temperature and higher e�ciency.
The increase in apparent heat capacity A for theempty chamber case can be further broken downinto the ✏ term in (13) (Fig. 11) and the heat ca-pacities of water droplets and of steam (Fig. 12).The contribution due to additional heat capacitiesof water and steam is small compared to that of✏, especially at high flow rates. This is despite thefact that the heat capacities due to steam increasesup to 15% of the air at high flow rates. The ef-fect due to water droplets is minimal (< 2%). Thecontribution due to ✏ term is up to 400%. It con-tributes predominantly to the e↵ect of increasingthe e�ciency at higher flow rates. Recall that thephysical meaning of ✏ is the propensity of water toevaporate at higher temperature. Notice the formof ✏ in (13) and that P
s
increases super-linearly withtemperature (Fig. 2) cause ✏ to increase with tem-perature.
The same figures were plotted in Figs. 13-15 for
Figure 8: Porous media filled chamber water droplet mass vspressure (top) and water vapor mass vs pressure (bottom)for various flow rates.
the porous media filled case. From Fig. 13, thesame trends are apparent as with the empty cham-ber case. We see that B increases marginally at lowflow rates, but is increased by up to 35% at higherflow rates. On the other hand, A(·) increases byless than 10% for low flow rates, but is increased by2.5 times at higher flow rates. As with the emptychamber case, this suggests that at lower flow rates,the e↵ect of B and A more or less cancel out; but athigher flow rates, the e↵ect of A increasingly domi-nates contributing to lowering temperature and in-creasing e�ciency. However, the e↵ect is less pro-nounced for the porous media case than with anempty chamber.
The various contributors to the apparent heat ca-pacity A for the porous media filled case are shownin Figs. 14-15. The contribution due to additionalheat capacities of water and steam is small com-pared to that of ✏, especially at high flow rates.Heat capacities due to steam increase up to 3% ofdry air at high flow rates, and heat capacities due
9
Figure 9: E�ciency versus power with and without steam,and with 0-D and 1-D models. Empty chamber case (top),porous media filled case (bottom).
to water droplets are 1%. Therefore, the ✏ term isagain the majority contributor (140% of dry air) tothe e↵ect of increasing the e�ciency at higher flowrates. However, the relative contributions between✏, droplets and water vapor are closer for the porousmedia filled case.
4.2. 0-D model with air as a real gas
The 0-D results in section 4.1 has assumed airand steam to be ideal gases. The model developedin section 2 however are also appropriate if the airis a real gas. This section presents the 0-D resultswith air assumed to be a real gas. The real gasmodel used here is based upon experimental datapresented in [24]. The data is used to parameter-ize the equation of state as T (P
a
, ⇢a
), the constantvolume specific heat capacity as C
va
(T, Pa
) and theinternal pressure as ⇡
T
(Pa
, ⇢a
). The parameterizedmodels fit the data within 2% if temperature isgreater than 200K and pressure is between 1barand 500bar.
Figure 10: Empty chamber case: Normalized apparent heatcapacity A(T, ⇢a,mw,ms)/(maCva) versus pressure (top);apparent pressure normalized by pressure B(P, T )/P versuspressure (bottom).
Figure 16 shows the e↵ect of steam on thee�ciency-power trade-o↵ relationship with the realgas assumption. Compared to Fig. 9, using the realgas model decreases e�ciency by 0.5-1% over usingan ideal gas model, for both the empty and porousmedia filled cases with and without steam. Similarto using the ideal gas model, steam has negligible ef-fect, except for cases with low e�ciencies. At lowere�ciencies, steam has the e↵ect of increasing e�-ciency at the same the power, or increasing powerat the same e�ciency. As with the ideal gas model,the moisture e↵ect is much more prominent in theempty chamber case than in the porous media filledcase.
10
Figure 11: Empty chamber case: ✏(t)/Cva versus pressure.
Figure 12: Empty chamber case heat capacities ofwater droplets [mwCw/(maCva)] (top) and of steam[msCvs/(maCva)] (bottom) normalized by heat capacity ofair.
Figure 13: Porous media case: normalized apparent heatcapacity A(T, ⇢a,mw,ms)/(maCva) versus Pressure (top),apparent pressure normalized by pressure B(P, T )/P versuspressure (bottom).
Figure 14: Porous media case: ✏(t)/Cva versus pressure.
11
Figure 15: Porous media filled case heat capacitiesof water droplets [mwCw/(maCva)] (top) and of steam[msCvs/(maCva)] (bottom) normalized by heat capacity ofair
101 102 103 104
Power [W]
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Eff
icie
ncy
Empty chamber (real gas)
No steam
With steam
101 102 103 104
Power [W]
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
Eff
icie
ncy
Porous media filled chamber (real gas)
No steam
With steam
Figure 16: E�ciency versus power with and without steam,using a 0-D model with air assumed to be a real gas. Emptychamber case (top), porous media filled case (bottom).
12
4.3. 1-D Model with air as an ideal-gas
The 1-dimensional model ensures that pressureis spatially uniform throughout the air volume, butallows temperature to vary along the length of theair volume. Air is again assumed to be an ideal-gas.This is done by appropriately adjusting the volumeof each partition. In the 1-D simulations, 100 par-titions are used. Fig. 17 (20cc/s, empty cham-ber case) illustrates that pressure is indeed spatiallyuniform, confirming the validity of the method toenforce uniform pressure. The 1-D model is thenexercised for the same cases as for the 0-D model.Illustrative results for two cases representative ofother cases are described below.
Figure 17: Pressure profile for 1-D model 20 cc/s flow rate,empty chamber case.
Fig. 18 shows the spatial temperature profiles atvarious times during the compression process with20 cc/s flow rate for the empty chamber and with400 cc/s for the porous media filled cases. Forthe empty chamber case, the temperature profileis fairly uniform initially, but as compression pro-gresses, temperature at the top and bottom becomelower than in the middle (due to additional heattransfer areas). For the porous media case, the tem-perature tends to decrease from the top cap to thebottom. This shape is due to greater forced convec-tion heat transfer to the porous media towards thebottom of chamber where the air speed is highest.This e↵ect is less significant for the empty cham-ber case because heat transfer without the porousmedia is much smaller.The spatial distributions of the water droplet are
examined next in Fig. 19. At 20cc/s flow rate,water droplets are formed initially in the emptychamber and then disappear, consistent with the
Figure 18: Sample temperature profiles for 1-D Model.Empty chamber, 20 cc/s flow rate case (top); Porous me-dia filled, 400 cc/s flow Rate. case (bottom).
0-D model result in Fig. 6 (top). More droplets areformed at the bottom and top where the temper-atures are lower. For the porous media case, theamount of water droplet increases from the bottomto the top of the chamber. The temporal variationis also consistent with the 0-D model result in Fig.8 (top) that the water droplet mass increases ini-tially and later decreases to zero (as temperatureincreases).
The spatial distributions of steam for the samecases are shown in Figs. 20. Steam is initiallyuniformly distributed in the empty chamber anda bulged shape emerges as compression progresses.This shape is opposite to that of the water-dropletsin Fig. 19(top). The amount of steam decreasesand then increases consistent with Fig. 6 (bottom).For the porous media case, the general trend is thatthe total mass of steam increases over time as themean temperature increases, consistent with the re-
13
Figure 19: Sample 1-D model water mass spatial profiles:Empty chamber, 20cc/s flow rate (top); Porous media filled,400cc/s flow rate (bottom).
sults in Fig. 8(bottom). However, there is signif-icant spatial variation. Whereas the steam massnear the top of the chamber (with the highest tem-perature) increases successively in time; the amountof steam decreases over time near the bottom ofthe chamber (with the lowest temperature) in fa-vor of condensation into droplets. In fact, the top0.05m of the chamber only has steam and no waterdroplets.
The 0-D temperature, droplet mass and steammass results are also superimposed on Figs. 18-20for comparison. In general, the 0-D results and thespatial means of the 1-D results are close and followthe same trends. The results are very close for theporous media filled cases, but some deviations existfor the 0-D case. In the empty chamber case, the0-D temperatures are slightly lower than the 1-Dspatial averages, causing the 0-D droplet mass andsteam to be respectively higher and lower than their
1-D counterparts.
Figure 20: Sample 1-D model steam mass spatial profiles.Empty chamber, 20cc/s flow rate case (top); Porous mediafilled, 400cc/s flow rate case (bottom).
In regard to the e�ciency-power trade-o↵, the 1-D model results are similar to the 0-D results asshown in Fig. 9 although the e�ciency with the1-D model is slightly lower than the 0-D results.
5. Discussion
The results above indicate that the dominantcontributor to improved performance is the ✏ termin (13), while the added heat capacitances due tothe water droplets or to the steam are much lesssignificant. To understand further why the ✏ e↵ectis more prominent with higher flow rates than withlower flow rates, and in an empty chamber than ina porous media filled chamber, contour plots of ✏as a function of pressure and temperature for thevarious flow rates are shown in Fig. 21. The curve
14
Figure 21: Contour plots of ✏/Cva in (13) as a function oftemperature and pressure overlayed with ✏(t)/Cva for var-ious flow rates. Empty chamber case (top); Porous mediafilled case (bottom).
✏/Cva
= 1 indicates where ✏ would double the ap-parent heat capacity of dry air. Notice that ✏/C
va
increases with increasing temperature and decreas-ing pressure. For all cases, ✏(t) > 0 so that it al-ways contributes to increasing the apparent heatcapacity (13). Higher flow rates increase temper-ature and hence also ✏; whereas for the same flowrate, the porous media filled cases have lower tem-perature, leading to smaller ✏. From Fig. 21, inorder to increase ✏, one must increase temperature,which unfortunately leads to low e�ciency. This isfor this reason that the positive e↵ect of moisture ismost pronounced at high flow rate, lower e�ciencysituations.
In the literature on wet compression for gas tur-bines, significant reduction in compression has alsobeen reported [16, 17, 15, 18, 19, 20]. Analysis hasshown that evaporative cooling is a key contribu-tor, consistent with our findings that ✏ is the key.
As gas turbines tend to run at higher temperatures,this reduction (in the order of 15% [19]) is more sim-ilar to the empty chamber, low e�ciencies, higherpower cases than the high e�ciency (lower temper-atures), porous media filled cases of interests to theisothermal CAES application.
The cases tested in this paper are with constantflow rates. However, it has been shown analyt-ically, numerically and experimentally [8, 14, 10,11, 12, 13] that the compression trajectories canbe optimized to maximize power for a given e�-ciency or to maximize e�ciency for a given power.Analysis using dry air model has shown that powercan be 2-3 times that of constant speed trajecto-ries. The optimized trajectories typically consist offast (nearly adiabatic) segments at the beginningand end, sandwiching a slow segment. A questionarises as to the e↵ect of moisture on the compres-sion performance when optimized trajectories areused. Based on the results in the paper, it is ex-pected that there will be minimal e↵ect for highe�ciencies, lower power trajectories; whereas someimprovement will be present for lower e�ciencies,higher power trajectories. It is because of the corre-lation between temperature and e�ciency, and be-tween temperature and ✏ as discussed above. Yet,it will be interesting in future work to answer thisquestion definitively and to optimize trajectories totake advantage of the moisture e↵ect.
The results have also shown that despite the sim-plicity, and the inability to provide spatial informa-tion, the 0-D model provides close estimates to themean values of properties such as temperature, wa-ter droplet and steam masses, e�ciency and power,as predicted by the 1-D model, especially for theporous media filled cases. The temperature estima-tion tends to be slightly lower and e�ciency slightlyhigher (by < 1%) for the 0-D model than for the 1-D model. This slight discrepancy is most likely dueto the 0-D model over-estimating the heat transferwhich is a function of which heat transfer coe�-cient h(t) in (16) is used. The di↵erence is mostlikely for the heat transfer to the chamber wallssince the 0-D model uses an aggregate correlationfor all walls whereas the 1-D model uses di↵erentcorrelations for the side and the top and bottomsurfaces. In terms of predicting the e↵ect of mois-ture on compression performance, both the 0-D and1-D results are very consistent. Given the simplicityof the model and computation e�ciency, the use ofthe 0-D model is recommended except when spatialinformation is needed.
15
At very high pressures and temperatures, thereal gas model, the compressibility factor of air isgreater than 1 [25] and a real gas model is moreappropriate than an ideal gas model. The e↵ect ofusing an a real gas model is that e�ciency tendsto be slightly lower (0.5-1%) for the same power.However, the qualitative e↵ect of moisture on thee�ciency-power trade-o↵ is unchanged whether anideal gas or a real gas assumption is used.This paper has only studied compression. The ef-
fect of moisture on expansion performance requiresfurther study. It is suspected that the e↵ect of wa-ter vapor and of ✏ will be smaller than in the com-pression case, because temperature will be low dur-ing expansion. The e↵ect due to droplet conden-sation may increase. There can also be a positivee↵ect due to water droplet freezing, releasing latentheat. On the other hand icing within the porousmedium or in the liquid piston can be catastrophiccausing the expander to lockup.Finally, the implications of some of the simplify-
ing assumptions made in the model are discussed:
• It is assumed that all the water is evaporatedfrom and condensed into droplets suspendedin the air space, with the exception that whenall the droplets have been depleted, water inthe liquid piston will evaporate. In reality, wa-ter can also condense as liquid films on sur-faces. This leads to a smaller amount of ac-tive droplets. Another possibility is that wa-ter evaporates from liquid films instead of fromdroplets, leaving a greater amount of dropletmass remaining. However, this latter case isless likely due to the small size and large sur-face areas of the droplets and the use of su-perhydrophobic surface on the porous mediawhich allow the condensed water to drain o↵rapidly. In either case, the amount of steamwould not be a↵ected significantly.
• The evaporation/condensation processes arealso assumed to happen instantly. If dynam-ics are involved and a time lag does exist, theair space may become un-saturated at sometime. The expected result would be a decreasein the benefit of the moisture because the fulle↵ect of the increase in apparent heat capac-itance will not come into full e↵ect. On bal-ance, with consideration of possible evapora-tion/condensation dynamics and of condensa-tion onto films, it is expected that the perfor-
mance improvement predicted due to moisturein this paper would be slightly optimistic.
• It is assumed that the porous media and thechamber walls are isothermal. In reality, ascompression progresses, their temperature willincrease. This will decrease the temperaturedi↵erence in (16) and heat transfer. This leadsto lower e�ciency for the same flow rates, orlower power for the same e�ciency. However,since the e↵ect will be common to both thecases with and without moisture, the improve-ment due to moisture itself may only increaseslightly (due to increased temperature) but notsignificantly.
6. Conclusion
The objective of this paper is to determine thee↵ect of moisture in the air volume during the com-pression phase for a CAES system. A 0-D and a 1-D model were developed to compare the e�ciency-power density trade-o↵ with or without moisture.By simulating the models with di↵erent constantflow rates, it was found that there is a thresh-old compression e�ciency (between 85-90%) abovewhich the di↵erences between the steam and nosteam cases are minimal. And at e�ciencies belowthat threshold, the steam case resulted in a highere�ciency than the no steam case. This meant thedi↵erences were greater for the empty chamber casesince the heat transfer from the air volume to theambient conditions were less than if porous mediawas included in the chamber.
The mechanism for increasing e�ciency is foundto be mainly the increase in the apparent specificheat capacity of air due to the increased propensityof water to evaporate (✏ term). Since this e↵ectis increased at increased temperature, the improve-ment in e�ciency due to moisture is also significantat high flow rate, low e�ciency, high temperaturescenarios. On the other hand, the contribution dueto additional heat capacities of water droplets andof steam is much smaller.
A comparison of the 0-D model and a more de-tailed 1-D model indicates that the 0-D model withno moisture is adequate for computing the e�ciencypower-density trade-o↵s when e�ciency is expectedto be high, such as when porous media and rela-tively long compression times are used. Moreover,the use of either the ideal gas or the real gas as-sumption for air does not change the qualitative
16
e↵ect of moisture on the e�ciency power-densitytrade-o↵.
Acknowledgment
This research is supported by the U.S.A. Na-tional Science Foundation under grant ENG/EFRI-1038294.
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