Modeling and Trajectory Optimization of Water

Proceedings of the ASME 2013 Heat Transfer Summer Conference HT2013 July 14-19, 2013, Minneapolis, MN, USA

HT2013-17611

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

Mohsen Saadat Dept. of Mechanical Engineering University of Minnesota Minneapolis, MN 55455 Email: [email protected]

Farzad A. Shirazi Dept. of Mechanical Engineering University of Minnesota Minneapolis, MN 55455 Email: [email protected]

Perry Y. Li Dept. of Mechanical Engineering University of Minnesota Minneapolis, MN 55455 Email: [email protected]

ABSTRACT An efficient and sufficiently power dense air compressor/expander is the key element in a Compressed Air Energy Storage (CAES) approach. Efficiency can be increased by improving the heat transfer between air and its surrounding materials. One effective and practical method to achieve this goal is to use water droplets spray inside the chamber when air is compressing or expanding. In this paper, the air compression cycle is modeled by considering one-dimensional droplet properties in a lumped air model. While it is possible to inject water droplets into the compressing air at any time, optimal spray profile can result in maximum efficiency improvement for a given water to air mass ratio. The corresponding optimization problem is then defined based on the stored energy in the compressed air and the required input works. Finally, optimal spray profile has been determined for various water to air mass ratio using a general numerical approach to solve the optimization problem. Results show the potential improvement by acquiring the optimal spray profile instead of conventional constant spray flow rate. For the specific compression chamber geometry and desired pressure ratio and final time used in this work, the efficiency can be improved up to 4%.

sor/expander is responsible for the majority of the storage energy conversion, it is critical that it is efficient and sufficiently powerful. This is challenging because compressing/expanding air in high compression ratios (200-300) heats/cools the air greatly, resulting in poor efficiency, unless the process is sufficiently slow which reduces power [3]. There is therefore a trade-off between efficiency and power.

INTRODUCTION Gas 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-

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

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 number of stages increase. To improve the efficiency of the compressor/expander with few stages, it is necessary to enhance the heat transfer during the compression/expansion process. A liquid piston 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 leakage. 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].

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 mathematical model for energy and exergy analysis is presented in [14] for a co-current gas spray cooling system. One-dimensional simulations 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 allows 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 compression cycle including water spray is derived based on an Eulerian 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 control problem is then solved by discretization of the control input over the time interval. Comparison between the optimal and nonoptimal spray profiles has been finally shown in the last section.

of this system for further purposes. Here, a one-dimensional distribution 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) x L −Y(t)

(1)

where U is the air velocity and Y is the liquid piston height inside 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 (evaporation) is considered between these phases due to the fact that the overall temperature rise of droplets during the compression process 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 considered). More details are shown in Fig. 1.

Modeling Compression Chamber: A liquid piston air compressor 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 connected 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 liquid column height is zero. Since the heat capacity of the chamber walls and the liquid column is much larger than the air, it is assumed 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 complicated 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 dynamic model of such a process is difficult to be obtained. However, a simple model can be used to understand the basic behavior

FIGURE 1. PRESSOR

WATER SPRAY INSIDE LIQUID PISTON AIR COM-

Defining r as the number of droplets per unit length of the chamber (drop/m) and v as absolute droplet velocity and then applying

2


the conservation of mass principal, we will get

From Sutherland’s formula, the dynamic viscosity of air as a function of its temperature can be calculated as follows:

∂r ∂ + (rv) = 0. ∂t ∂ x

(2) μ(t) = μr

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: 1 fdrag (t) = Cd Aρ(t) (v(x,t) −U(x,t) )2 2

(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 water 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 volume is chosen to contain both liquid piston and water droplet (Fig. 2).

(3)

where Cd is the drag coefficient and A is the reference area. For a 2

spherical droplet moving in air, Cd is about 0.47 and A is πd4 in which d is the droplet diameter assumed to be constant over the whole process. Now, by applying the conservation of momentum principal, the second PDE describing the droplet’s velocity dynamic is obtained as: fdrag ∂ v2 ∂v − ( )+g− =0 ∂t ∂ x 2 m

Tr +C T(t) 3 ( )2 T(t) +C Tr

(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 negative 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 6 ∂E +v + − T(t) ) = 0 h (E ∂t ∂ x Cs ρw d (x,t) (x,t)

(5)

FIGURE 2.

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: 1 2

Nu(x,t) = 2 + 0.6Re(x,t) Pr

1 3

By applying the conservation of mass principle for the total water inside this control volume, we will have: d (A p δw +V dt

ρ(t) d|v(x,t) −U(x,t) | μ(t)

δd

rdx) = F p + rvV

(9)

where A p 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:

(6)

where Re is the Reynolds number defined based on relative speed between droplet and air as: Re(x,t) =

CONTROL VOLUME AT LIQUID PISTON SURFACE

A p δ˙w +V r∗ δ˙d +V r˙∗ δd = F p + r∗ v∗V

(10)

where * means the value of property at piston location (x∗ ). Notice that the third term on the left hand side of Eqn. (10) is zero

(7)

3


since δd approaches to zero. Considering the fact that δ˙w = Y˙ and δ˙d = −Y˙ , the piston height dynamics can be finally determined as:

Y˙(t) =

p F(t) + r(x∗ ,t) v(x∗ ,t)V

AP − r(x∗ ,t)V

pression chamber, the air volume dynamic can be determined by: dV p s = − F(t) + F(t) dt

(11)

(13)

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

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:

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 R using available ODE solvers. then solved in MATLAB

⎛ Sample Case Study A 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.

∗ T(t) π ⎜ dT ⎜ 2 x ˙ r(x,t) h(x,t) T(t) − E(x,t) dx V + = (1 − γ) ⎜d dt V(t) (t) mairCv ⎝ 0

⎞

⎟ D2 ⎟ + Dx∗ + h T(t) − Twall ⎟ 2 ⎠

heat to droplets (H2 )

(12)

heat to walls (H1 )

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.

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

−

FIGURE 3. HEAT TRANSFER BETWEEN AIR AND WATER DROPLETS AS WELL AS AIR AND SURROUNDING WALLS

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,

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

4


0.9

550

0.6

450

0.3

350

0 0

0.2

0.4 0.6 Time (s)

6

6

x 10

Adiabatic Compression Compression with Water Spray Isothermal Compression

1.5 0.15

4

Liquid Piston

0.1

1 0.5

0.05 0.2

0.4 0.6 Time (s)

0.8

0.25

4

0.2

3

0.15

Liquid Piston

0.1

2 1

0.05

3 0

0.2

2

0.4 0.6 Time (s)

0.8

1

420

100

200

300 400 Air Volume (cc)

500

Chamber Location (m)

0 0

600

FIGURE 4. AIR TEMPERATURE AND WATER SPRAY FLOW RATE VS. TIME (TOP), AIR PRESSURE VS. AIR VOLUME (BOTTOM)

0.25

400

0.2

380

0.15

360

Liquid Piston

0.1

340

Temperature (K)

Air Pressure (Pa)

5

2 0.2

0

250

0.8

0.25 Density (drop/m)

650

7

Velocity (m/s)

1.2


750


1.5

x 10 2.5

Air Temperature (K)

Spray Flow Rate (cc/s)

after the spray stops, the air temperature rises again until the final desired pressure ratio is achieved.

320

0.05

300 0

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 temperature 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 water droplets.

0.2

0.4 0.6 Time (s)

0.8

FIGURE 5. DISTRIBUTION OF DROPLET DENSITY (TOP), VELOCITY (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:

Optimization of Spray Flow Rate for a Given Mass Loading In general, increasing the relative amount of water droplets compared to air increases the compression efficiency by improving heat transfer. Compression efficiency defined as the ratio between the stored energy in air (after compression) and the required input work. The stored energy in the air at pressure rP0 and ambient temperature is defined as the maximum work ob-

nz ΔP(t)

5

=

s F(t)

Knz

2 (15)


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)V f +

0

tf

In this paper, the continuous optimal control problem is parameterized as a finite dimensional problem and then solved numerically by standard algorithms for constrained parameter optimization. The control input can be parameterized as:

s nz F(t) ΔP(t) dt

T

(16) where V f is the final air volume at the end of compression (t = t f ). The compression efficiency is then defined as: ηc = 100

Wstored % Winput

s = ∑ fi .Ui (t) F(t)

Optimal Spray Profile for Constant Piston Flow Rate Optimal 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 F s . 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.

t

(18)

The trend of these optimal spray profiles (Fig. 6-top) are expectable considering the fact that at the first half of the compression, there is enough heat transfer area provided by the surrounding walls while the air temperature is still not high. Thus, additional cooling with water droplet is not necessary in this phase. On the other hand, injecting droplets into the air when the liquid 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 optimal spray profile does a better job and reduces the air temperature more in the second half (since some droplets are saved from the first half). Hence, as expected, the overall compression efficiency 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

For the sample case study that resulted in 70.8% efficiency, ML is obtained to be about 0.5. It seems that increasing mass loading 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 dynamic constraint is given by the air compression model including 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 piston flow profile F p is also specified before hand. Based on these definitions and assumptions, the optimal spray profile is: s = arg.max{ηc } F(t)

(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.

(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%. Considering the heat transfer from the surrounding walls (with the same boundary conditions and constant parameters used for the previous 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: s dt ρw 0 f F(t) mw ML = = mair mair

0 ≤ t ≤ tf

i=1

(19)

6


Flow Rate (Normalized with ML)

loading of 0.5. Note that the compression ratio and compression time are the same for all cases. In particular, note that the compression efficiency decreases for very large mass loadings since the energy loss across the spray nozzle becomes a dominant term in the input work. 2.5

CAES system. For such a compressor, a minimum thermal efficiency of 90% is required to achieve a reasonable round-trip efficiency for the storage system. As discussed earlier, one effective 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%, optimization 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 efficiency 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 functions is used to parameterize the spray profile over the compression 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 compression 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.

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

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

Time (s) 900 ML=0.02 (Optimal) ML=0.02 (Constant) ML=0.22 (Optimal) ML=0.22 (Constant) ML=1.75 (Optimal) ML=1.75 (Constant)

Air Temperature (K)

800 700 600 500 400 300 200 0

100

Compression Efficiency %

85

200


500

600

Optimal Flow Rate Constant Flow Spray

80 75 70 65 60 55

−2

10

−1

10

0

10 Mass Loading

1

10

Conclusions Equipping a liquid piston air compressor with a water droplet spray can improve the compression efficiency significantly. 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 profile 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 optimizing the spray profile. Similarly, the spray profile was optimized for the optimal liquid piston profile in a compression chamber

FIGURE 6. COMPARISON BETWEEN OPTIMAL SPRAY FLOW RATE AND CONSTANT SPRAY FLOW RATE. NORMALIZED OPTIMAL SPRAY FLOW RATE (TOP), TEMPERATURE VS. VOLUME (MIDDLE), AND EFFICIENCY VS. MASS LOADING (BOTTOM)

Design of an Efficient and Power-Dense Air Compressor Although the optimal spray profile improves the compression efficiency, it is not still satisfactory for the application of

7


3

[2] M. Saadat and P. Y. Li, “Modeling and Control of a Novel Compressed Air Energy Storage System for Offshore Wind Turbine,” in Proc. American Control Conference, pp. 30323037, Montreal, Canada, 2012. [3] C. J. Sancken and P. Y. Li,“Optimal Efficiency-Power Relationship for an Air Motor/Compressor in an Energy Storage and Regeneration System,” in Proc. ASME Dynamic Systems and Control Conference, pp. 1315-1322, Hollywood, USA, 2009. [4] J. D. Lewins, “Optimising and intercooled compressor for an ideal gas model,” Int. J. Mech. Engr. Educ., vol. 31, Issue 3, pp. 189-200, 2003. [5] J. D. Van de Ven and P. Y. Li, “Liquid Piston Gas Compression,” Applied Energy, vol. 86, Issue 10, pp. 2183-2191, 2009. [6] M. Sozen and T. M. Kuzay, “Enhanced Heat Transfer in Round Tubes with Porous Inserts,” Int. J. Heat and Fluid Flow, vol. 17, Issue 2, pp. 124-129, 1996. [7] A. T. Rice and P. Y. Li, “Optimal Efficiency-Power Tradeoff for an Air Motor/Compressor with Volume Varying Heat Transfer Capability,” in Proc. ASME Dynamic Systems and Control Conference, Arlington, USA, 2011. [8] M. Saadat, P. Y. Li and T. W. Simon, “Optimal Trajectories for a Liquid Piston Compressor/Expander in a Compressed Air Energy Storage System with Consideration of Heat Transfer and Friction,” in Proc. American Control Conference, pp. 1800-1805, Montreal, Canada, 2012. [9] F. A. Shirazi, M. Saadat, B. Yan, P. Y. Li and T. W. Simon,“Iterative Optimal Control of a Near Isothermal Liquid Piston Air Compressor in a Compressed Air Energy Storage System,” to appear in American Control Conference, Washington, DC, 2013. [10] S. Kachhwaha, P. Dhar, S. Kale, “Experimental Studies and Numerical Simulation of Evaporative Cooling of Air with a Water Spray- I. Horizontal Parallel Flow,” Int. J. Heat Transfer, vol. 41, pp. 447-464, 1998. [11] S. Sureshkumar, S. Kale, P. Dhar, “Heat and Mass Transfer Processes Between a Water Spray and Ambient Air - I. Experimental Data,” Applied Thermal Engineering, vol. 28, pp. 349-360, 2008. [12] W. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays, Cambridge University Press, 1999. [13] H. Barrow, C.W. Pope, “Droplet Evaporation with Reference to the Effectiveness of Water-Mist Cooling,” Applied Energy, vol. 84, pp. 404-412, 2007. [14] A. Niksiar, A. Rahimi, “Energy and Exergy Analysis for Cocurrent Gas Spray Cooling Systems Based on the Results of Mathematical Modeling and Simulation,” Energy, vol. 34, pp. 14-21, 2009. [15] C. Qin and E. Loth, “Liquid Piston Compression with Droplet Heat Transfer,” in Proc. 51st AIAA Aerospace Sciences Meeting, Grapevine, TX, 2013.

150 125

Optimal Spray Flow Rate (cc/sec)

2

100

1.5

75

1

50

0.5

25

0 0

0.1

0.2

0.3

900

0.4

0.5 0.6 Time (s)

0.7

0.8

0.9

Spray Flow Rate (cc/sec)

Piston Flow Rate (lit/sec)

Optimal Piston Flow Rate (lit/sec) 2.5

0 1

Adiabatic

800

Constant Piston Flow

61.2%

Temperature (K)

Optimal Piston Flow 700

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

74.4%

600

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

500 77.2% 400

90.7%

300 94.5% 200 0

100

200


500

600

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 VERSUS VOLUME FOR FIVE DIFFERENT CASES (BOTTOM)

with porous inserts. Combination of these heat transfer enhancement 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 increased 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 transition time of droplets between the nozzle and the liquid piston surface, it would be better to change the direction and/or location of spray nozzles. For example, spraying from the sides of the compression chamber (and close to the top) in a radial direction can be more effective as a result of longer lifetime of water droplets.

REFERENCES [1] P. Y. Li, E. Loth, T. W. Simon, J. D. Van de Ven and S. E. Crane, “Compressed Air Energy Storage for Offshore Wind Turbines,” in Proc. International Fluid Power Exhibition (IFPE), Las Vegas, USA, 2011.

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