COMBINED OPTIMAL DESIGN AND CONTROL OF A NEAR …lixxx099/papers/DSCC_2015Combined_Opt.pdf · Proceedings of the ASME 2015 Dynamic Systems and Control Conference DSCC2015 October

Proceedings of the ASME 2015 Dynamic Systems and Control ConferenceDSCC2015

October 28-30, 2015, Columbus, Ohio, USA

DSCC2015-9957

COMBINED OPTIMAL DESIGN AND CONTROL OF A NEAR ISOTHERMAL LIQUIDPISTON AIR COMPRESSOR/EXPANDER FOR A COMPRESSED AIR ENERGY

STORAGE (CAES) SYSTEM FOR WIND TURBINES

Mohsen SaadatDept. of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

Email: [email protected]

Perry Y. Li(Corresponding author)

Dept. of Mechanical EngineeringUniversity of MinnesotaMinneapolis, MN 55455

Email: [email protected]

ABSTRACTThe key component of Compressed Air Energy Storage

(CAES) system is an air compressor/expander. The roundtrip ef-ficiency of this energy storage technology depends greatly on theefficiency of the air compressor/expander. There is a trade offbetween the thermal efficiency and power density of this compo-nent. Different ideas and approaches were introduced and stud-ied in the previous works to improve this trade off by enhancingthe heat transfer between air and its environment. In the presentwork, a combination of optimal compression/expansion rate, op-timal chamber shape and optimal heat exchanger material dis-tribution in the chamber is considered to maximize the powerdensity of a compression/expansion chamber for a given desiredefficiency. Results show that the power density can be improvedby more than 20 folds if the optimal combination of flow rate,shape and porosity are used together.

INTRODUCTIONThe availability of a cost-effective, scalable and dispatchable

energy storage system is the key to eliminating the most pressingintegration issue of renewable energy and the grid: integratingunpredictable and intermittent green energy, such as wind andsolar energy, into the electrical grid. In our previous work [1–4],we have proposed a novel Compressed Air Energy Storage sys-tem (Fig. 1) for wind turbines that can store energy prior to elec-tricity generation. With the use of a novel open accumulator sys-

tem architecture and a near isothermal liquid piston air compres-sor/expander, this system can be efficient, power dense and cost-effective. A critical component of this system is the high pres-sure (200 bar) air compressor/expander which is responsible forthe conversion between mechanical work and compressed air instorage. To be useful, the compressor/expander must be efficientand power dense. For a fixed compression/expansion ratio, in-creasing power density means that the compressor/expander willoperate in shorter cycle time. A smaller component can thereforebe used to achieve the same power capability, reducing capitalexpense (CAPEX).

Air

Oil

Water

Oil Water

Storage Vessel

(Accumulator)

Wind Turbine

Generator

Grid Air Compressor/Expander

Hydraulic Pump/Motor

FIGURE 1: Open accumulator architecture CAES system

1 Copyright c© 2015 by ASME

There is a natural tradeoff between efficiency and powerdensity as the thermodynamic efficiency of air compres-sion/expansion is highly dependent on heat transfer. When op-erating the compressor/expander slowly, there is more time forheat transfer to take place and efficiency increases. This is atthe expense of reducing power density so that a large compres-sor/expander will be needed. The reverse is true when operatingthe compressor/expander quickly. Power density increases at theexpense of efficiency since time for heat transfer becomes lim-ited. In our approach, a liquid piston air compressor/expanderwith porous media insert is used. Here a liquid column is usedto compress or expand the air above it. The interface betweenliquid and air column stays stable if the maximum accelerationof the liquid (upward or downward) remains lower than a certainvalue [5]. Since liquid can flow through tortuous path, addingporous media augments heat transfer through the increase in sur-face area and heat capacitance [6–9]. It has been shown throughCFD and experiments that introduction of porous media can in-crease the power density over the case with no porous media bymore than an order of magnitude without sacrificing efficiency.Yet another approach to improving the efficiency-power densitytradeoff is to optimize the compression and expansion trajecto-ries. It has been shown that optimized trajectories can also in-crease power density by an order of magnitude over ad-hoc lin-ear/sinusoidal trajectories [10–12].

In this paper, we consider the following questions:

1. If a total amount of porous media is to be introduced, howshould it be distributed within the compression/expansionchamber?

2. How does the shape of the compression/expansion chamberaffect the efficiency/power density tradeoff? What will bean optimal shape for optimizing the power density withoutsacrificing power density?

3. How should the compression/expansion trajectory be opti-mized in combination with the optimization of the porositydistribution and shape?

The rest of this paper is structured as follows: First, a zero-dimensional thermodynamic dynamic model is derived for theair being compressed/expanded in the chamber. A heat transfercorrelation is used in this derivation that is found for the spe-cific porous material geometry considered in this work. In thenext step, the trajectory optimization problem is defined and for-mulated as an optimal control problem in order to maximize thepower density of the chamber. Shape and porosity distributionoptimizations are then defined as the high level optimization andsolved iteratively to find the combined optimal shape, porositydistribution and compression trajectory for the chamber. Finally,a step-by-step procedure is presented to demonstrate how the de-veloped optimization algorithm can improve the power densityof a compression chamber.

Thermodynamic ModelA schematic of the air compression/expansion chamber is

shown in Fig. 2. Air is compressed/expanded by a liquid (water)piston that is driven by a hydraulic pump/motor. The cross sec-tion area (θ ) of the compression/expansion chamber varies alongits length. Heat exchanger material (porous media) in the formof parallel plates is inserted into the chamber. While the platethickness is fixed (0.8mm), the plate spacing can vary along thechamber axis (Fig. 3). Therefore, the full geometry of the cham-ber can be realized by specifying the cross section area and platespacing at different air volumes corresponding to different liquidpiston levels (i.e. θ(v) and d(v) where v ∈ [0,Vc], and Vc is themaximum chamber volume). Here, we assume that the air tem-perature is uniform in the chamber (i.e. zero-dimensional modelfor air temperature). Considering real gas properties for air [13],a zero-dimensional thermodynamic dynamic model for air undercompression/expansion can be derived as:

m∂e∂P

P = (P− ∂e∂ρ

ρ2)Q(t)−hA(P,ρ,Q)(T(P,ρ)−Tw) (1)

ρ =ρ2

mQ(t) (2)

where P, ρ , m and e are the pressure, density, mass and spe-cific internal energy of the air, respectively. Note that the airmass remains constant during the process (i.e. no air leakage isassumed). Additionally, Q is the liquid piston flow rate into thechamber which can be controlled by varying the pump/motor dis-placement attached to the chamber. The specific internal energyand temperature of air depend on its state, given by air pressureand density:

e = e(P,ρ)⇒ e =∂e∂P

P+∂e∂ρ

ρ

T = T(P,ρ)⇒ T =∂T∂P

P+∂T∂ρ

ρ

Due to the high thermal capacity of the porous media (rel-ative to air), the heat exchanger temperature (Tw) is assumed toremain constant. hA is the volume averaged product of the con-vective heat transfer coefficient (h) and the effective heat transferarea (A) of the air inside the chamber. A specific correlation forNusselt number has been found for the heat exchanger geometryused in this work, given by [7]:

Nu =hs(v)K(T )

= a+b.Rem.Prn (3)

where Nu is the Nusselt number, Re is the Reynolds number andPr is the Prandtl number for air, while a, b, m and n are constants(a=9.7, b=0.0876, m=0.792 and n=0.33). Note that s is the local


to/from Air Reservoir (low pressure @ 7bar)

to/from Storage Vessel (high pressure @ 200bar)

Porous Material (interrupted plates)

Liquid (Water) Piston

Air V

olume

Porosity (%) Φ(V)

Chamber Diameter

D(V)

Variable Displacement Pump/Motor

FIGURE 2: Schematic of the liquid piston air compressor/expanderwith inserted heat exchanger materials (parallel plates)

hydraulic diameter of the heat exchanger (s = 2×d) and K is theheat conductivity of air considered to be a function of air temper-ature. The range of Reynolds number for which this correlationis valid can be found in [7]. The local Reynolds number of airinside the chamber is calculated based on the local air speed as:

Re =ρs(v)|u(v)|

µ(T )(4)

In this equation, µ is the dynamic viscosity of air (depends on airtemperature) and u is the local Darcian velocity of air given by:

u =v

V θ(v)Q (5)

Note that Eqn. (5) is obtained assuming that the air flow rate at across section of the chamber is linear to the volume of air abovethe section (i.e. the air flow rate is equal to water flow rate at thewater surface, while it is zero at the top cap). V(t) is the total airvolume inside the chamber at a given time; v is a cumulative air

Parallel Plates

Open Area

Plate Thickness

Plate Spacing d(v)

Diameter D(v)

FIGURE 3: Cross section area of the chamber. Inner diameter of thechamber D(v) and plate spacing d(v) are functions of chamber cumula-tive volume v

volume (above a cross-section) that can vary between 0 and V(t).Thus, each v defines a level and cross-section of the chamber.Using Eqns. (3), (4) and (5), the volume average of hA can becalculated as:

hA =∫ V(t)

0h(P,ρ,Q)β(v)dv (6)

where β is the local heat transfer area density (m2/m3) and h isthe local convective heat transfer coefficient calculated based onEqn. (3). After some mathematical manipulations, the final formof hA can be found as:

hA = aK(T )M(V )+bK(T )

(ρ

µ(T )V

)m

PrnQmN(V ) (7)

where M and N are given by:

M(V ) =∫ V

0

β(v)

s(v)dv

N(V ) =∫ V

0

β(v)

s1−m(v)

(u

θ(v)

)m

dv(8)

By integrating Eqns. (1) and (2) with respects to time, the historyof air states (pressure and density) can be evaluated for a givencompression/expansion flow rate trajectory and initial conditioninside the chamber (P0 and ρ0).

Compression Trajectory Optimization for a GivenChamber Geometry

For a given chamber shape θ(v) and plate spacing d(v) distri-butions, an optimal control problem can be formulated to maxi-mize the power density by minimizing the required compressiontime while achieving the desired thermal efficiency. By manip-ulating Eqns. (1) and (2), it would be possible to omit the time


differentiation and combine the equations as:

mepdPdV

=hAQ

(T(P,ρ)−Tw

)−(P−ρ

2eρ

)(9)

where ep and eρ denote the partial derivative of the specific inter-nal energy of air with respect to pressure and density. In addition,time can be considered as a dependent variable and calculated as:

t =−∫ V (t)

V0

dVQ

(10)

where V0 is the air volume at initial time (V(t=0) =V0). The costfunction of the optimization problem is defined as the total timerequired for compressing air from initial pressure (P0) to the finalpressure (Pf ):

t f =−∫ V f

V0

dVQ

(11)

where V0 and Vf are the total air volume in the chamber at pres-sure P0 and Pf , respectively. Due to the limited heat transferbetween air and its environment, there is a trade off between thecompression time and compression efficiency. Since compres-sion time is considered to be the cost, compression efficiencyneeds to be an equality constraint in this optimal control prob-lem formulation. This equality constraint can be realized as:

−∫ V f

V0

PdV +Pf .Vf −P0.V0︸︷︷︸Total Compression Work

=Es

η∗(12)

where η∗ is the desired efficiency and Es is the energy stored inair after compression (i.e. if the input work is equal to Es/η∗ thenthe compression efficiency would be η∗) (Fig. 4). Note that thetotal compression work is the summation of the required work tocompress air from initial pressure to the final pressure, isobariccooling work (to the ambient temperature) and work required topush the compressed air out of the chamber [4]. Finally, an in-equality constraint is required to reflect the limitation on flowrate that can be provided by the hydraulic pump:

0≤ Q≤ Qmax (13)

The optimal control problem is then defined as:

Q∗(P) = arg.minQ(.)

(−∫ V f

V0

dVQ(P)

)(14)

such that Eqns. (12) and (13) are satisfied. Note that the controlvariable is the flow rate (Q) which is assumed to be a function of

air pressure (instead of time). By introducing a new variable λ

(Lagrange multiplier), the Lagrangian can be defined as [14]:

L =−∫ V f

V0

(1

Q(P)+λP

)dV +λ

(PfVf −P0V0−

Es

η∗

)(15)

Hence, the optimal control problem can be summarized as:

λ∗,Q∗(P)

= arg

(max

λ

(minQ(.)

(L(λ ,Q)))

)(16)

such that:

0≤ Q≤ Qmax

P

VPamb

P0

Pf

V0VfVfiso

Stored Energy Compression Work Isothermal Compression

P

VPamb

Pf

P0

VfisoVfV0

Available Energy Expansion Work Isothermal Expansion

FIGURE 4: Compression (left) and expansion (right) process shownon P-V diagrams. In the compression case, efficiency is defined as thestored energy divided by the total compression work whereas in the ex-pansion case, efficiency is defined as the extracted work from expansiondivided by the available energy in the air (that could be achieved throughan ideal isothermal expansion) [4].

Dynamic Programming (DP) method is used to solve theminimization problem and to find the optimal compression tra-jectory (Q∗(P)) for a given λ , while a golden search method isused to solve the maximization problem (for λ ∗). In discretevolume-pressure domain (see Fig. 5), the compression trajectorycan be realized by a V-P sequence. Using the forward differencemethod, dV/dP can be evaluated as:

(dVdP

)i→i+1

'V j

i+1−V ki

Pi+1−Pi(17)

So, if (P, V) at step i and (P, V) at step i+ 1 are known, Qi→i+1can be calculated from Eqn. (9). Once Qi→i+1 and ∆Vi→i+1 arefound, the compression time from step i to step i+ 1 (∆ti→i+1)can be easily calculated from Eqn. (10). Therefore, the total


Pn=Pf Pn-1 Pn-2 P0 P1 Pi

P

V V0

V1m

Vim

Vi1

Vn1

Vnm Vi+1

k

Vi j

V11

. . . . . .

Adiabatic Compression Isothermal Compression Optimal Compression

FIGURE 5: Discretization in volume-pressure domain. Optimal com-pression trajectory can be realized as a sequence of volumes over thepressure range ([P0,P1, ...,Pn−1,Pn])

.

Lagrangian (L0→n) can be evaluated through backward inductionusing Bellman equation.

A sample case study is used to illustrate how compressiontrajectory is optimized for maximizing the power density of agiven chamber geometry and desired compression efficiency. Acylindrical chamber with a uniform cross section area (θ ) of50cm2 and volume of 1875cc is chosen. A total volume of 375ccof heat exchanger material in the form of parallel plates is usedinside the chamber. The plate spacing is assumed to be 3mm(uniform along the chamber) with a thickness of 0.8mm. Notethat the chamber length is equal to 37.5cm based on these val-ues. The initial air pressure is 7bar, final pressure is 200bar andthe desired compression efficiency is 85%. The optimal com-pression trajectory resulted for these given parameters is shownin Fig. 6. The minimum compression time required to achievethe final pressure as well as the desired compression efficiencyis found to be 3.66 seconds (λ ∗=13.5). Considering size of thechamber, the maximum power density is 510kW/m3. It shouldbe mentioned that in the case of a non-optimal compression tra-jectory, a longer compression process is required to achieve thesame efficiency. For example, a constant flow rate of 0.2 lit/secwill result the same efficiency for the given chamber geometryand final pressure. However, the required time to achieve thefinal pressure with this constant flow rate compression is about7 seconds, which gives almost half power density (250kW/m3)compared to the optimal case.

Chamber Shape OptimizationOptimization of the compression trajectory for a given ge-

ometry (described in the previous section) is the low-level opti-

0 0.3 0.6 0.9 1.2 1.50

25

50

75

100

125

Cha

mbe

r Cro

ss S

ectio

n A

rea

(cm

2 )

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.550

60

70

80

90

100

Poro

sity

(%)

50 100 150 2000

0.5

1

1.5

Air Pressure (bar)

Flow

Rat

e (li

t/sec

)

Optimal Flow Rate Maximum Available Flow Rate

0 0.3 0.6 0.9 1.2 1.50

50

100

150

200

250

Air

Pres

sure

(bar

)

Air Volume (lit)

0 0.3 0.6 0.9 1.2 1.5250

300

350

400

450

Air

Tem

pera

ture

(K)

Pressure Temperature

FIGURE 6: Results of compression trajectory optimization for a givenchamber geometry. Chamber geometry (top); optimal flow rate (mid-dle); air temperature and pressure vs. volume (bottom)

mization of the combined shape and flow rate optimization prob-lem. Therefore, a high-level optimization problem needs to bedefined and solved to find the optimal geometry of the chamberin order to maximize its power density (Fig. 7). In other words,we want to find the optimal cross section area (θ ∗) and poros-ity distribution (φ ∗) that maximize the power density for a givenchamber and porous material total volume (while achieving thedesired compression efficiency). Note that the porosity at eachlocation is a function of chamber cross section area and the platespacing. Define Ω as the open area (i.e. total cross section areaminus the area occupied by heat transfer material):

Ω(v) = θ(v)×φ(v) (18)

where porosity φ ∈ [0,1]. If the open area and porosity areknown, the chamber cross section area θ can be found from Eqn.(18). Therefore, we can optimize the open area and porosity dis-tributions in order to calculate the optimal chamber geometry.While the total chamber volume is a fixed (given) parameter, the


total volume of the heat exchanger material that is allowed to beused must be also fixed in order to have a meaningful optimiza-tion problem1. Moreover, from manufacturing point of view, theaspect ratio of the chamber must remain in a reasonable range.This constraint can be included by adding an equality constrainton chamber length. Therefore, the optimization of the open areaand porosity distribution can be summarized as:

Ω∗(v),φ

∗(v)

= arg.min

Ω(.),φ(.)

(−∫ V f

V0

dVQ(P)

)(19)

such that: ∫ V0

0

dVφ(V )− V0

Φ= 0 (20)

lmin ≤∫ V0

0

dVΩ(V )

≤ lmax (21)

where Φ is the allowable total porosity of the chamber and lminand lmax are the minimum and maximum possible length for it.

Optimal Flow Rate (solving the optimal control problem

defined by Eqns. 12, 13 and 14)

Optimization for Shape (solving the optimization problem

defined by Eqns. 19, 20 and 21)

Compression Time

Q*

t f

Ω*& φ* t f

Low Level Optimization

High Level Optimization

FIGURE 7: Iterative optimization approach to find the optimal chambershape (high-level optimization) as well as optimal compression trajec-tory (low-level optimization)

Interior point method is used here to solve the corresponding op-timization problem and find the optimal chamber shape to min-imize the required compression time (i.e. maximize the powerdensity). This procedure will be repeated in an iterative manneruntil the open area and porosity converge to their optimal distri-butions.

Case Studies and Simulation ResultsA systematic approach is introduced and developed in the

previous sections to maximize the power density of a compres-sion chamber by combined optimization of its geometry and flow

1without this constraint, the optimal porosity distribution problem will havea trivial solution which is a chamber where all its volume is occupied by heatexchanger material, with zero space for air.

rate. In what follows this method is utilized to improve the powerdensity of a sample chamber step-by-step. The total chambervolume is assumed to be 1875cc where 375cc of this volume isdevoted to the heat exchanger material (parallel plates with thick-ness of 0.8mm). In this case, the total porosity would be 80%. Athermal efficiency of 92% is desired for the compression process.Initial air temperature and pressure are assumed to be 293K and7bar, while final pressure of 200bar is expected. Additionally, inorder to prevent water trapping between the parallel plates, theminimum local porosity in the chamber is set to be 70% (whilethe maximum is 100%, which means no porous material).

Step 1:As the base case, uniform porosity (80%) and uniform open

area (21.4cm2) are used for the chamber. In this case, the cham-ber length will be 70cm. Without flow rate optimization, a con-stant flow rate of 43cc/s is required to achieve the desired effi-ciency (92%). The total compression process takes about 33 sec-onds in order to reach the desired final pressure while satisfyingthe thermal efficiency requirement (Fig. 8).

Step 2:The compression (flow) rate is optimized for the chamber

geometry given in step 1. As it can be seen in Fig. 8, the optimalcompression trajectory starts with a large flow rate, followed bya slow compression rate that takes the main portion of the wholeprocess. At the end, a second fast compression results in the de-sired final pressure. Optimizing the compression trajectory with-out changing the chamber geometry reduces the required com-pression time by more than 3 times (from 33 seconds (step 1) toabout 10 seconds) which enhances the power density by the sameratio.

Step 3:To demonstrate the effect of heat exchanger material loca-

tion in the chamber, the porosity distribution is optimized overthe entire chamber volume, while the open area is considered tobe uniform (i.e. no optimization on shape yet). Without flowrate optimization, a constant flow rate of 149cc/s is required,which results in final pressure after 9.6 seconds. As shown inFig. 8, according to the optimal porosity distribution, all the heatexchanger material must be located in the upper region of thechamber (close to the top cap). In other words, the upper portionhas the minimum allowable porosity (70%) while the lower por-tion is empty (100% porosity). This result is not surprising andis consistent with the physics of the problem2.

2Maximum temperature of air under compression happens at the end of pro-cess where the compressed air is very close to the top cap. Therefore, more heatexchanger material is required close to the top cap in order to prevent air to gethot which reduces thermal efficiency.


Step 1

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90

100 Cross Section Area (3) Opening Area (+ )

Air Pressure (bar)50 100 150 200

Flow

Rat

e (li

t/sec

)

0

0.5

1

1.5 Flow Rate Maximum Flow Rate

Step 2

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90



Flow

Rat

e (li

t/sec

)

0

0.5

1


Step 3

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90



Flow

Rat

e (li

t/sec

)

0

0.5

1


Step 4

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90



Flow

Rat

e (li

t/sec

)

0

0.5

1


Step 5

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90

100

Cross Section Area (3) Opening Area (+ )


Flow

Rat

e (li

t/sec

)

0

0.5

1


Narrow Chamber Case

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Are

a (c

m2 )

0

25

50

75

100

Poro

sity

(%)

60

70

80

90



Flow

Rat

e (li

t/sec

)

0

0.5

1


FIGURE 8: Chamber geometry (cross section area, open area andporosity distribution) and compression flow rate shown for differentsteps during the power density optimization process.

Step 4:Now, by optimizing the compression rate for the optimal

porosity found in the previous step (3), the required compressiontime is reduced to 3.5 seconds. As a result, the power density ofthe chamber will be 669kW/m3. Note that the open area distri-bution (i.e. chamber shape) is not optimized yet.

Step 5:In the final step, chamber geometry is optimized in addition

to its porosity distribution. Combined geometry and flow rate

optimization results in power density of 1.5MW/m2 which is 20times larger than the non-optimal condition (base case). For theshape optimization, it has been assumed that the maximum al-lowable chamber length is 70cm (i.e. lmax=70cm). As shown inFig. 9, the optimal shape has a larger area at the bottom (wherewater enters the chamber), while the rest of the chamber has asmaller area.

Minimum and Maximum Length Cases:To clarify the effect of shape optimization on power den-

sity, two additional cases have been studied here. If we relaxthe length constraint in shape optimization, the optimal shape forthe chamber would be a narrow tube with an open area equalto 7cm2 with a total length of 214cm. On average, a narrowerchamber results in higher air velocity during compression. Thiswill improve the heat transfer between air and its environmentby increasing the convective heat transfer coefficient (see Eqns.(3), (4) and (5)). If the porosity distribution as well as the com-pression trajectory are both optimized for this narrow chamber,power density of 1.6MW/m3 will be achieved. On the other hand,a fat chamber with a uniform open area of 78cm2 and total lengthof 19cm results in a poor power density (423kW/m3) even thoughthe porosity distribution and compression trajectory are both op-timized for it. In reality, fabricating a long and narrow com-pression chamber is challenging. Difference in power densitybetween the optimal geometry and the narrowest shape is small,while manufacturing the former is more cost effective than thelatter geometry. Fig. 10 shows the air temperature versus air vol-ume during the compression process for various situations thathave been studied here. It should be mentioned that the powerdensity optimization for the expansion mode can be done in thesame manner, where the initial pressure P0 is 200 bar while thefinal pressure Pf is 7bar.

ConclusionsA systematic approach was introduced and used to maxi-

mize power density of a liquid piston air compression/expansionchamber by optimizing its geometry as well as its compres-sion/expansion rate. The combined shape and flow rate opti-mization problem is divided into two levels: In the low leveloptimization, the compression trajectory is optimized for a givenchamber geometry to maximize the power density by minimiz-ing the required compression time while achieving the desiredefficiency. Dynamic Programming technique is used for solvingthe optimal control problem associated with this level. Then, inthe high level optimization, open area and porosity distributionsof the chamber are optimized to maximize the power density. Aniterative method is used to find the combined optimal geome-try and flow rate for the chamber. According to the results, theoptimal chamber shape in addition to its corresponding optimal


Step Porosity Flow Rate Shape Chamber Length Efficiency Compression Time Power Density

1 uniform constant (43cc/s) uniform area (21.5cm2) 70cm 92% 33s 71.2 kW/m3

2 uniform optimal uniform area (21.5cm2) 70cm 92% 10.8s 217.3 kW/m3

3 optimal constant (149cc/s) uniform area (21.5cm2) 70cm 92% 9.6s 245.6 kW/m3

4 optimal optimal uniform area (21.5cm2) 70cm 92% 3.5s 669.3 kW/m3

5 optimal optimal optimal 70cm 92% 1.6s 1470 kW/m3

N optimal optimal narrowest (7cm2) 214cm 92% 1.47s 1600 kW/m3

F optimal optimal fattest (78.5cm2) 19cm 92% 5.5s 423.6 kW/m3

TABLE 1: Effect of optimization of porosity distribution, chamber shape and compression trajectory on compression time and power density for the given thermalefficiency (92%) and total porosity (80%). For all the cases, total chamber volume is 1875cc where 375cc is occupied by the heat exchanger material. The initial and finalpressure are 7bar and 200bar, respectively.

Optimal Geometry Uniform Geometry

FIGURE 9: Uniform geometry (right) and optimal geometry (left) re-sulted by combined optimization algorithm to maximize the chamberpower density for a given thermal efficiency.

flow rate have the potential to increase the power density from71kW/m3 to 1.6MW/m3 which is more than 20 times improve-ment (see Table 1). The optimal geometry shows that the bestplace for locating the heat exchanger material is at the top ofthe chamber where most of the hot air accumulates at the end ofcompression cycle. Results also show that the chamber is betterto have a larger diameter at the bottom (where water enters thechamber) while a smaller diameter for the rest.

Acknowledgements

This work is supported by the National Science Foundationunder grant ENG/EFRI-1038294.

Air Volume (lit)0 0.3 0.6 0.9 1.2 1.5

Air

Te

mp

era

ture

(K

)

280

300

320

340

360

380

400

Step 1 Step 2 Step 3 Step 4 Step 5 Narrow Tube Fat Tube

FIGURE 10: Temperature versus volume for air under compression fordifferent scenarios (steps) studied in this work.

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