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Modelling of efficiency of synthetic jet actuators
Accepted Manuscript Title: Modelling of efficiency of synthetic jet actuators Author: Michele Girfoglio Carlo Salvatore Greco Matteo Chiatto Luigi de Luca PII: DOI: Reference:
S0924-4247(15)30084-4 http://dx.doi.org/doi:10.1016/j.sna.2015.07.030 SNA 9262
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
13-11-2014 27-7-2015 27-7-2015
Please cite this article as: Michele Girfoglio, Carlo Salvatore Greco, Matteo Chiatto, Luigi de Luca, Modelling of efficiency of synthetic jet actuators, Sensors & Actuators: A. Physical (2015), http://dx.doi.org/10.1016/j.sna.2015.07.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Modelling of efficiency of synthetic jet actuators
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Michele Girfoglio, Carlo Salvatore Greco, Matteo Chiatto, Luigi de Lucaa,b,c,d a
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PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. b PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. c PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. d Professor, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy.
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Abstract
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A comprehensive and detailed modelling to evaluate the efficiency of energy conversion of piezo-electric actuators driving synthetic jets is developed. The contribution is original because the analysis is based on the energy equations of the two coupled oscillators, the membrane and the acoustic one, which are directly derived from the corresponding motion equations. The modelling is validated against numerical as well as experimental investigations carried out on home-made actuators having brass and aluminum shims on which the piezo-disk is bonded. A major result is that for aluminum shim the global efficiency (representing the conversion of input Joule power to kinetic power) decreases with increasing the applied voltage. Another finding is that the conversion process of mechanical power transferred, by means of the driving membrane, to Helmholtz oscillator kinetic power scales dramatically with the coupling degree of the oscillators. The coupling degree influences the efficiency of two cavities actuators sharing the same piezo-diaphragm as well. Considerations are reported to relate the theoretical orifice efficiency to the practical jet efficiency issuing in the external field. Keywords: efficiency, synthetic jet, piezo-electric
Preprint submitted to Sensors and Actuators A: Physical
June 4, 2015
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1. Introduction
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Continuous and pulsed jets are nowadays quite largely used as devices for flow control (e.g., Glezer and Amitay [1], Cattafesta and Sheplak [2]) and heat transfer applications (e.g., Chaudhari et al. [3], Valiorgue et al. [4], Greco et al. [5]). The continuous request for a size reduction of such systems has led researchers to focus their efforts on the design of new technology devices. Synthetic jet actuators well fit this need because they produce jets with zero-net mass flux synthesized directly from the fluid system in which the device itself is embedded (Smith and Glezer [6]), avoiding the need for an external input piping and making them ideal for reduced-space and lowweight applications. A synthetic jet is generated by a membrane oscillation (generally driven by a piezo-ceramic element) in a relatively small cavity, which produces a periodic cavity volume change and thus pressure variation. As the membrane oscillates, fluid is periodically entrained into- and expelled out from an orifice connecting the cavity with the external ambient (to be controlled). During the expulsion phase of the cycle, due to the flow separation, a vortex ring forms near the orifice exit section which, under favorable operating conditions (Holman et al. [7]), convects away towards the far field and breaks up due to the viscous dissipation eventually ”synthesizing” a turbulent jet always directed downstream (Smith and Glezer [6]). Since the jet formation depends on the ability of the vortex ring to escape to the subsequent ingestion phase, following Smith and Glezer [6], a basic ¯ parameter characterizing the jet strength is the so called stroke length L, namely the integral of the (spatially averaged) velocity at the orifice exit over the ejection phase only of the cycle: ¯= L
Z
T /2
U(t)dt
(1)
0
where T is the actuation period and U(t) is the fluid velocity at the exit section. The stroke length can also be expressed conveniently as the product ¯ = U¯ T , where U¯ is a proper reference velocity defined as: L R T /2
U(t)dt (2) T Therefore, it is natural to expect that the jet is formed or not according to whether the parameter L/do is greater or less than a certain critical value (e.g. 0.16π, as suggested by Holman et al. [7]), with do being the orifice ¯= U
0
2
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¯ o is generally referred to as the diameter. In the literature the parameter L/d reciprocal of the Strouhal number (Cater and Soria [8]). ¯ lies in The importance of the stroke length, or of the average velocity U, the fact that to compare the performance of a synthetic jet with that of a continuous jet, it is usual to refer to a Reynolds number based on the velocity ¯ i.e.: U,
¯ o ρUd (3) µ where ρ is air density, µ is air dynamic viscosity. Although such a topic has been so widely explored, only few studies on the synthetic jet efficiency have been carried out. One of the first works on the efficiency of synthetic jet was undertaken by Tesar and Zhong [9] who based their definition of efficiency on the capability of jet generation rather than on energy conversion considerations. They found experimentally a constant efficiency value equal to 7.2% for several values of Reynolds number (ranging from 4000 to 8000) and for two values of dimensionless stroke length L/do (equal to 223.9 and 527.8). They also compared their experimental evaluation with a numerical prediction made by using the commercial code FLUENT, and found that this last is twice the experimental one. Subsequently Crowther and Gomes [10] studied the system costs associated with the application of flow control system to civil transport aircraft based on the use of electrically powered synthetic jet actuators. They defined the efficiency of the actuator as fluid power (scaling with the cube of the exit velocity) divided by absorbed electrical power, and analyzed it as a function of the operating conditions and actuator geometry, which were chosen reasonably close to those expected for industrial applications. For this reason the experiments were carried out basically with a chamber depth to orifice diameter ratio equal to 0.56 and an orifice depth to diameter ratio equal to 2.1. By resorting to the energy balance principle, Crowther and Gomes [10] considered that the difference between the supplied electrical power and the gained fluid power was lost because of electrical impedance (ascribed to the piezo-electric actuator), mechanical impedance (related to the diaphragm dynamics) and acoustic impedance (due to the fluid-acoustic coupling of the flow within the cavity and through the orifice), however without accurately quantifying each term of the balance. They showed their experimental findings in terms of a map of the electric-fluidic conversion efficiency, where such an efficiency was reported as a function of excitation voltage amplitude (for
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peak-to-peak excitation voltage up to 250V) and actuation frequency (up to 4000Hz). They found that the efficiency attains a maximum equal to about 14% and noted that it did not correspond to the condition of maximum exit velocity because of the dielectric saturation effect affecting the commercial piezo-electric patch (i.e., the piezo-element bonded to the brass shim). A more recent work, focusing specifically on the energy conversion efficiency of synthetic jets devices, was presented by Li et al. [11] who tried to express on an analytical basis every term contributing to the energy rate balance. They argued that once the synthetic jet has received electric energy input, due to the capacitance of the piezo-electric actuator, a part of the energy is stored as electric potential energy while the rest of the energy is converted to mechanical energy accompanied by energy dissipation. The mechanical energy includes vibration of piezo-electric actuators and kinetic energy of air flow. For piezo-electric actuators two forms of energy, i.e. strain energy and kinetic energy, are temporarily stored by the vibrating structure. Energy dissipation (namely energy loss) occurs in piezo-electric actuators due to the deflection dynamics, as well as in the air flow motion (i.e., the head losses) when traversing the jet orifice. The synthetic jet device efficiency is defined as the ratio of the kinetic power of the air flow to the input electric energy. The authors carried out experiments on two slot synthetic jets having orifice length of 4mm and 15mm, for two values of voltage amplitude (80V and 100V) and actuation frequency ranging from 200Hz to 1100Hz. Li et al. [11] found that the efficiency of energy conversion is dependent on the orifice size and on the operating conditions (namely, voltage and frequency) showing a peak (of about 40% for the 15mm orifice) close to the mechanical resonance frequency of the actuator. They claim that the uncertainty of their experimental evaluation was of about 34%. The aim of the present paper is to assess a rigorous and comprehensive physical approach to the evaluation of the efficiency of piezo-electrically driven synthetic jet actuators, based on a rather detailed modelling of the actuator dynamics, already developed by two of the authors, de Luca et al. [12], in the past. The peculiarity of the present approach is that the energy balance equation, properly averaged over the actuation period, is derived directly from the equations governing the dynamics of the actuator. The advantage is that the presence of each term of the resulting energy balance draws its justification from the corresponding dynamics term and, in principle, there is no need to perform experimental measurements to evaluate the actuator efficiency at the orifice exit section. Since the physical model 4
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mentioned above has been already tested and calibrated against a systematic experimental campaign, the uncertainty of the efficiency estimation should be reduced. The analysis methodology and a part of the results will highlight the key role played by the coupling degree of the two oscillators introduced in the physical model, i.e. the oscillating membrane and the acoustic one. Roughly speaking, the coupling degree is the ratio of the cavity air stiffness to the membrane stiffness. It will be shown that the conversion efficiency of mechanical power, transferred by means of the driving membrane, to Helmholtz oscillator kinetic power increases with increasing the coupling degree of the oscillators. However, in the case of actuators with two cavities, sharing the same membrane, we will see that for high coupling degree the global efficiency remains almost the same, namely, in other terms, the efficiency of operation of the single cavity is nearly halved. It is worth noting here that, since the modelling is of ”lumped parameters” kind, it yields the evaluation of the energy conversion efficiency intrinsically at orifice; however, in various practical applications the efficiency in the external jet field is needed, which should be estimated approximately at the stagnation point (or saddle point) separating, during the suction phase, the ingestion near field from the ejection far field. The connection between the ”efficiency at orifice” and the ”efficiency in the jet” will be addressed in the final part of the paper, after discussing the results. The paper is organized as follows: section 2 reports the basic equations governing the dynamics of the actuator, whilst section 3 is devoted to the development of the energy balance, directly derived from the dynamics equations. The experimental validation and calibration of the proposed model are discussed in section 4, useful practical remarks are included in section 5 and, finally, conclusions are drawn in section 6. 2. Governing equations system In this section a short description of the motion equations governing the dynamics of a typical piezo-electrically driven synthetic jet actuator will be given. Details regarding the physical modelling can be found in the original paper of de Luca et al. [12]. It should be pointed out that the approach followed by the present authors is fluid-dynamics based, and was inspired by the previous original paper of Sharma [13].
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The membrane behavior is modelled as a one-degree of freedom forced mass-spring-damper system mwt x¨w + cwt x˙ w + kw xw = F − pi Aw
(4)
k w d A Va sin ωt = kw ∆xw sin ωt Aw
(5)
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F = Fo sin ωt =
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where mwt is the diaphragm total mass (including shim, piezo-element and air added mass), xw is the membrane displacement at a generic time instant t, cwt is the total damping coefficient (including the structural damping coefficient and a further damping coefficient due to the interaction with the external air), kw is the equivalent stiffness of the membrane (the clamped edge condition is assumed here), F is the electrodynamic force, pi is the cavity (internal) differential pressure, Aw is the membrane area. The electrodynamic force F is related to the piezo-electric effect due to the applied sine voltage and is given by
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where Fo is the forcing magnitude, dA is the effective acoustic piezo-electric coefficient, Va is the voltage amplitude, ω = 2πf is the operating angular frequency (with f being the natural frequency, f = 1/T ) and ∆xw is the (average) linear ”static” displacement of the membrane due to the piezoelectric effect. The application of the unsteady Bernoulli’s equation between a point inside the cavity where the flow velocity is practically null and a point, just outside the cavity, representing the location where the pressure matches the unperturbed external ambient value, yields Ma
dU 1 = pi Ao − Kρa Ao U|U| dt 2
(6)
in which Ao is the orifice area, K is the head loss coefficient, ρa is the air density, Ma = ρa le Ao is the mass of air oscillating inside the orifice (Helmholtz oscillator), with le being the effective length of the orifice (i.e., the distance between the two locations of application of the Bernoulli’s equation) and U being is the instantaneous orifice jet-flow velocity, as already defined in section 1. The equation of evolution of the third unknown, the cavity pressure pi , is obtained by enforcing the conservation of air mass in the cavity; by relating the density and pressure variations by means of an isentropic compression/expansion transformation, the continuity equation can be formulated in 6
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Vc dpi = Aw x˙ w − Ao U γpo dt
(7)
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the following way
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where Vc is the cavity volume, po is the ambient pressure, γ is the specific heat ratio.
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3. Derivation of the energy balance equation and definition of efficiency Multiplying the equation (4) by the membrane velocity x˙ w , yields:
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dEw = F x˙ w − pi Aw x˙ w − cw x˙ 2w dt
(8)
where Ew is the diaphragm energy defined as
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1 1 Ew = mw x˙ 2w + kw xw 2 2 2
(9)
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that takes into account both kinetic and elastic strain contributions. Similarly, the product between the equation (6) and the jet exit velocity U yields dEo 1 = pi Ao U − Kρa Ao |U|3 (10) dt 2 where Eo is given by 1 (11) Eo = Ma U 2 2 and represents the kinetic energy flow rate of the air mass through the orifice. The (instantaneous) energy balance equation of the actuator system can be obtained by summing the equations (8) and (10). Moreover, since we are interested in characterizing the actuator behavior over an operating cycle, it is convenient to apply the time average operator on the resulting equation, defined as Z 1 T ϕdt (12) ϕ¯ = T 0 where ϕ is the generic time-dependent variable.
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T
0
1 − T |
Z
1 d(Ew + Eo ) = T {z } | ∆E T
cwt x˙ 2w dt −
0
{z
D¯s
}
1 T |
Z
Z
T 0
T 0
1 F x˙ w dt + T {z } | P¯e
Z
T
0
pi (Ao U − Aw x˙ w )dt {z } P¯m
1 1 (K − 1)ρa Ao |U|3 dt − 2 T {z } |
Z
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Z
T
1 ρa Ao |U|3 dt (13) 2 {z }
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The energy balance equation averaged over an actuation cycle (i.e., over a time equal to the period T ) is given by
D¯f
P¯k
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An accurate description of the various contributions of equation (13) is reported hereafter:
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• ∆E is the total energy variation. Note that this term is null because, for each cycle, there is no change for Ew and Eo . • P¯e is the electrodynamic power provided to the membrane by the applied voltage.
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• P¯m is the mechanical power due to the work done by the differential pressure pi which acts on the wall surface Aw and on the orifice surface Ao . By using the equation (7), it can be shown that this term is proportional to 12 (pi 2 (T ) − pi 2 (0)) and, therefore, it does not give any contribution because pi assumes the same value at the beginning and end of each cycle. One can reach the same result by observing that the pressure work is conservative by definition. • D¯s is the power dissipation due to the structural damping effects of the membrane. • D¯f is the power dissipation due to the head loss of fluid dynamics type at the orifice.
• P¯k is the kinetic power of air flow at the orifice. Note explicitly that the kinetic power here refers by definition to the entire cycle, i.e. the suction phase included. We will continue this discussion in the last section 4.
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Then, by deleting ∆E and P¯m terms, the equation (13) becomes (14)
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P¯e − D¯s − D¯f − P¯k = 0
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and, once defined the kinetic efficiency ηk as the ratio of the kinetic power of the exit flow P¯k to the electrodynamic power P¯e , one obtains
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P¯k D¯s + D¯f ηk = ¯ = 1 − (15) Pe P¯e It is worthwhile to stress that in practice the global efficiency of an actuator has to quantify the amount of Joule power provided to the system P¯j that is actually converted in P¯e . This can be done by introducing the electrodynamic transduction efficiency ηe (16)
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P¯e ηe = ¯ Pj
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Hence, finally, one can define the (global) efficiency of the actuator η as the product of ηk by ηe P¯k η = ηk ηe = ¯ (17) Pj where the external Joule power supply P¯j provided to the actuator is calculated as Z 1 T ¯ V Idt (18) Pj = T 0 with V = Va sin ωt being the applied voltage and I the electric current flowing through the piezo-electric element. The rate of P¯j not turned into P¯e is converted into a variation of internal energy Q˙ of the air inside the cavity (heat generation per unit time), which is transferred in part to the external ambient through a natural convection mechanism, and, in part, into enthalpy flow rate of the air leaving the orifice P¯j − P¯e = Q˙ ≡ hAe ∆T + mc ˙ p ∆T
(19)
where h is the convective heat transfer coefficient by natural convection, Ae is the exchange surface (which depends on the relevant geometry of actuator), ∆T is the average-per-cycle temperature difference between the system and ¯ o and we remember that U¯ the colder surrounding external air, m ˙ = ρa UA is the average velocity associated to the stroke length (eq. 2), and cp is the 9
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specific heat coefficient at constant pressure. For the sake of simplicity we assumed that the air inside the cavity is isothermal with the device case. From equation (19) one obtains also Q˙ ηe = 1 − ¯ Pj
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(20)
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Moreover, it is important to note that the energy conversion process from electrodynamic energy to flow kinetic energy depends intrinsically on the coupling degree between the two oscillators modeling the device behavior, i.e. the membrane and the Helmholtz one. From the dynamical point of view the coupling effect is represented by the internal pressure term acting as a forcing term on the Helmholtz oscillator in equation (6); from the energy point of view, the coupling effect can be seen as a transfer of mechanical power (e.g, the work done by the internal pressure) to the Helmholtz oscillator, expressed by Z 1 T ¯ pi Aw x˙ w dt (21) PmH = T 0
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It is clearly evident that, although the net mechanical power due to the pressure work introduced in equation (13) is null, P¯mH defined in the above equation is not, and it is convenient to split ideally the conversion process from electrodynamic energy to flow kinetic energy into two steps: the conversion of electrodynamic energy P¯e to mechanical power due to the pressure work (within the membrane oscillator), namely P¯mH , and the conversion of mechanical power to air flow kinetic energy (within the Helmholtz oscillator). In equation terms, once introduced the internal efficiencies P¯mH P¯e
(22)
P¯k η ∗∗ = ¯ PmH
(23)
η∗ =
the kinetic efficiency of equation (15) can be interpreted as ηk = η ∗ η ∗∗
(24)
In discussing the results, we will see that, while η ∗∗ is generally very high, as perhaps expected η ∗ depends crucially on the coupling degree between the 10
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two oscillators. In the case of reduced coupling, η ∗ can be very low and it reduces dramatically also the kinetic efficiency of the device. Furthermore, note that a limited coupling degree does not imply a low air flow velocity; it means that a certain value of air jet velocity is obtained at the price of a very low energy efficiency. 4. Validation of the model and results
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For fixed values of the geometrical and electro-mechanical parameters of a typical synthetic jet actuator, the modeling developed before has been employed to estimate the various contributions of the energy budget equations (13, 14) by carrying out direct numerical simulations of the governing equations (4, 6, 7). For a proper operating frequency f , the time trends of the relevant quantities, in practice the membrane velocity and the air velocity at the orifice exit, have been used to evaluate, ultimately, the energy conversion efficiencies (15, 16, 17). Numerical integration has been performed by means of a standard 4th order Runge-Kutta method in MATLAB environment with ode45 routine. Initial conditions of xw = 0, x˙ w = 0, pi = 0, and U = 0 have been assumed for all the computations and it has been observed that the quasi-steady oscillatory solution is generally reached in about 20 cycles (de Luca et al. [12]). The time averages here performed refer typically to the 23th cycle. The uncertainty errors of the numerical data have been evaluated especially with respect to variations of the values of the head loss coefficient as well as of the effective orifice length (de Luca et al. [12]). For K varying in the range 1 < K < 1.46 the variations of the resonance frequencies are restricted within a maximum of 2%, while the corresponding variations of the air velocity peaks are generally of the order of 10%. The uncertainty of data of le is reflected on the peak frequencies which vary within a corresponding uncertainty band of 5% which may grow up to a maximum of 10% for relatively low frequencies, while the average width of such a band is of 10% for the peak exit velocities. To validate the modeling, numerical values of efficiency have been compared with measured data for three typical synthetic jet actuators, the characteristics of which are summarized in Table 1. The membrane was built inhouse by gluing a LZT piezo-ceramic disk (manufactured by PIEZO System inc.) on a thin aluminum foil, or it is constituted by a commercially available brass shim fabricated by Murata Manufactoring Co. The schematic of 11
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Synthetic Jet
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U
Vortex Train
Orifice
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Membrane
Figure 1: Modular structure of the actuator (de Luca et al. [12])
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the device is also shown in Fig. 1 highlighting its modular structure, which permits independent variations of cavity diameter and height, orifice diameter, and piezo-electric diaphragm (de Luca et al. [12]). Here we adopted the value K = 1.14 for the head loss coefficient, while the effective length of the orifice le , was basically evaluated according to the description of Sharma [13], namely le /do = lo /do + ∆le , with ∆le =0.71. For a given operating voltage, the evaluation of the Joule power supply (equation (18)) needs the knowledge of the current intensity. This has been done by means of the theoretical relationship It = 2πf CVa
(25)
where It denotes the theoretical current intensity peak, and C is the electric capacitance of the piezo-ceramic disk (furnished by the manufacturer). To verify this estimation, the current intensity peak has been also measured directly by means of the experimental apparatus sketched in Fig. 2. A waveform generator (Wavetek model 164) creates an electrical sinusoidal signal which is amplified (by EPA 104 PIEZO System inc. unit) and sent to the synthetic jet device. Voltage and current are acquired by a two channels oscilloscope (Tektronix tds 2024). In Table 2, experimental measurements of the electric current peak Ie 12
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Figure 2: Sketch of the experimental setup.
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Piezo-electric
Aluminum 1 42 0.24 31.8 0.191 42 3 2 2 7.31x1010 0.31 2780 6.6x1010 0.31 7800
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Shim
Brass 35 0.4 23 0.23 35 3 2 2 9.7x1010 0.36 8490 6.7x1010 0.31 8000
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Geometry
Property Shim diameter (mm) Shim thickness (mm) Piezo-electric diameter (mm) Piezo-electric thickness (mm) Cavity diameter (mm) Cavity height (mm) Orifice diameter (mm) Orifice length (mm) Young’s Module (Pa) Poisson’s Module Density (Kg/m3 ) Young’s Module (Pa) Poisson’s Module Density (Kg/m3 )
Aluminum 2 80 0.25 63.5 0.191 80 4 5 2 7.31x1010 0.31 2780 6.6x1010 0.31 7800
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Table 1: Features of the tested devices
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are compared to the corresponding theoretical values It for the aluminum 2 device, which has been first employed to obtain insights on the energy efficiency. The data spread is generally less than 9% except for Va = 50V , for which it appears to be about 13%. The efficiency values computed for this device at the modified Helmholtz resonance frequency (which is of about 900Hz, as reported by de Luca et al. [12]) are listed in Table 3, together with the two power dissipation terms. Here we limit to remember that the modified (circular) frequencies, ωm , can be estimated to a good approximation by the relationship (with the Helmholtz one corresponding to the sign +): q 2 2 2 2 2 ωh ωh ωh ωwc 2 2 −(1 + + ) ± (1 + ωωwc 2 2 2 + ω2 ) − 4 ω2 ωm ωw ωw w w w = (26) ωw2 2 where the (first mode) structural frequency of the membrane is given by r kw , (27) ωw = mwt the uncoupled Helmholtz frequency is
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It (A) Ie (A) 0.038 0.036 0.045 0.044 0.052 0.054 0.059 0.056 0.066 0.060 0.073 0.064 0.079 0.074 0.085 0.080 0.093 0.086 0.099 0.094 0.106 0.104
te
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Va (V ) 25 30 35 40 45 50 55 60 65 70 75
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Table 2: Comparison of theoretical current peak with measured data (Ampere)
Table 3: Efficiencies and dissipation terms at the modified Helmholtz frequency for aluminum 2 device
ηe (%) 90.7 85.4 79.6 74.8 69.6 66.8 64.2 59.1 58.8 54.6 51.5
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Va (V ) ηk (%) 25 79.2 30 79.5 35 79.7 40 80.7 45 80.3 50 80.2 55 82.3 60 81.7 65 81.7 70 80.1 75 81
¯ f /P¯e (%) η(%) D 71.8 11.1 67.9 11.1 63.4 11.2 60.3 11.3 55.9 11.2 53.6 11.2 52.8 11.1 48.3 11.4 48 11.1 43.7 11.2 41.7 11.3
¯ s /P¯e (%) D 9.3 9.3 8.7 8.2 8.4 8.5 7.2 7.2 7.8 8.7 7.8
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r
ka , Ma
and the coupling frequency is defined as s r γA2w po /Vc γAw po ωwc = = mw mw H
(28)
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γA2o po /Vc = ρa le Ao
cr
ωh =
s
(29)
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which, following Sharma [13], may be interpreted as the frequency of the pneumatic spring made of the air enclosed within the cavity of volume Vc and of the diaphragm mass mw . Note that the height of the cavity, H, is explicitly introduced. The coupling degree between the two oscillators can be quantified, as made by de Luca et al. [12], by means of the parameter CR: 2 ωwc CR = (30) ωw
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which can be interpreted as the ratio of the cavity air stiffness to the membrane stiffness. It is evident that if CR ≪ 1 both the modified resonance frequencies tend to the natural frequencies of the membrane and Helmholtz oscillators. As one can observe (Table 3), the influence of the driving voltage is rather ¯ f /P¯e and D ¯ s /P¯e ; conversely, ηe decreases with increasing Va , weak for ηk , D thus determining the same trend for the global efficiency η. For the sake of completeness, to give the reader the possibility to check the calculations, the following additional parameters have been considered, according to the complete description of the actuator under examination made by de Luca et al. [12]: ρa = 1.205 Kg/m3 ; kw = 9.48 · 104 N/m; dA = 4.01 · 10−9 m3 /V ; cwt = 2.75 Ns/m, evaluated at the frequency of 900Hz. Note that the energy balance equation is satisfied numerically to within a maximum error of 6 × 10−3 (for Va = 55V and Va = 65V). Hereafter we will show that the trends against voltage of the quantities shown in Table 3 are well fitted by simple power laws which, in turn, can be justified on the basis of scaling theoretical considerations. From the numerical integration of the governing equations we found that the variations of exit kinetic power and electrodynamic power follow the following trends 16
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0.35 0.3
cr
0.2 0.15
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0.25
0.1
30
40
50 Va (V )
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70
80
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Figure 3: Kinetic power at orifice exit versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (31)
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P¯k ≈ Va 1.8 P¯e ≈ Va 1.8
(31) (32)
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as is depicted in Figs. 3 and 4, respectively, where circle symbols refer to computed data e continuous lines refer to the analytical relationships. Therefore, being ηk given by the ratio between two quantities both proportional ¯ f has the same to Va 1.8 , it does not depend on Va . Since one expects that D ¯ f /P¯e is constant too. scaling law as P¯k , the fluid dynamic dissipation rate D ¯ f vary as Indeed, according to their definitions in equation (13), P¯k and D 3 ¯ U0 and Pe as F0 x˙ w0 (with U0 and x˙ w0 being the fluid and membrane peak velocities, respectively), and therefore one could expect that they scale as Va3 and Va2 , respectively. However, due to the complex coupling between the two oscillators, the membrane and the acoustic one, in the presence of significant non linear damping acting on the acoustic one, such theoretical laws are remarkably modified, as discussed in detail by de Luca et al. [12] and de Luca et al. [14]. Here we found that U0 ≈ Va 0.6 and x˙ w0 ≈ Va 0.8 . Since F0 scales with Va , the scaling relationships of equations (31) and (32) are justified.
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0.4 0.35
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Figure 4: Electrodynamic power versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (32)
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Fig. 5 shows that the Joule power, evaluated experimentally, increases with increasing the voltage amplitude following a power law of the kind (33)
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P¯j ≈ Va 2.3
Note that if the behavior of the piezo-ceramic element was governed perfectly by the Ohm’s law, i.e. if the capacitance was constant, P¯j would be directly proportional to Va 2 . In effect, the relationship (33) tells us that the piezo-element capacitance varies as a function of the voltage and, furthermore, that the output instantaneous current is a distorted sine wave. Finally, as depicted in Fig. 6, the behavior of the electrodynamic transduction efficiency ηe defined by equation (16) and relating the Joule power to the electrodynamic one, taking into account also the relationship (32), is well fitted by a power law of the following type ηe ≈ Va −0.5
(34)
A similar trend holds for the global efficiency η (equation (17)). It is interesting to observe that the scaling laws discussed above and referring to the aluminum 2 actuator cannot be extended in a straightforward 18
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0.8 0.7 0.6
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Figure 5: Measured Joule power versus voltage for aluminum 2 device. Circle symbols are experimental data, continuous line represents eq. (33)
0.8
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Figure 6: Electrodynamic efficiency versus voltage for aluminum 2 device. Circle symbols are measured data, continuous line represents eq. (34)
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η ∗ (%) 5.5 6.3 6.9 7.5 7.9 8.2 8.7 8.9 9.1 9.2 10.1
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η ∗∗ (%) 85.5 85.3 85 84.8 85.5 88.6 86.6 85.6 87.1 87.8 83.7
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Va (V ) ηk (%) 25 4.7 30 5.3 35 5.9 40 6.4 45 6.8 50 7.3 55 7.5 60 7.6 65 7.9 70 8.1 75 8.5
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Table 4: Efficiencies of brass actuator at modified resonance structural frequency. CR=0.06
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way to all the actuators since, on one hand the electrodynamic transduction depends on the electrical capacitance of the piezo-ceramic element, and, on the other one, the energy conversion towards the air jet kinetic energy is strongly influenced by complex resonance phenomena which, in turn, are significantly influenced by the non linear fluidic damping effects. Therefore, rather we prefer to raise a brief discussion about the influence of the coupling degree of the oscillators on the kinetic energy efficiency, by monitoring the internal efficiencies η ∗ and η ∗∗ defined by equations (22) and (23). This will be made with the aid of Tables 4, 5, 6, where the relevant efficiencies are reported as functions of the applied voltage, for brass, aluminum 1, aluminum 2 actuators, respectively. The coupling parameter of equation (30) is equal to 0.06, 0.3, 1.88, respectively. Note that for the first two devices data refers to the modified resonance structural frequency (about 2200Hz for brass, which coincides practically with the uncoupled value, and 1600Hz for aluminum 1), whereas for the third device data refers to the modified resonance Helmholtz frequency (about 900Hz). As anticipated in the previous section, it is evident that, while η ∗∗ is generally very high for all the devices, η ∗ depends crucially on the coupling degree between the two oscillators. In the case of brass shim device having CR=0.06 the amount of mechanical power transferred from the membrane oscillator to the Helmholtz one is remarkably reduced, hence η ∗ is very low. Of course this reduces dramatically also the kinetic efficiency of the device 20
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η ∗ (%) 41.5 41.9 43.6 41.8 42.8 42.6 42.6 42.1 42.2 41.6 40.6
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η ∗∗ (%) 87.6 88.5 86.8 89.5 88 87.3 86.9 88.6 87.2 88 89
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Va (V ) ηk (%) 25 36.4 30 37.1 35 37.8 40 37.4 45 37.6 50 37.2 55 37 60 37.3 65 36.8 70 36.6 75 36.2
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Table 5: Efficiencies of aluminum 1 actuator at modified resonance structural frequency. CR=0.3
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Table 6: Efficiencies of aluminum 2 actuator at modified resonance Helmholtz frequency. CR=1.88
Va (V ) ηk (%) 25 79.2 30 79.5 35 79.7 40 80.7 45 80.3 50 80.2 55 82.3 60 81.7 65 81.7 70 80.1 75 81
η ∗∗ (%) 86.6 87.6 88.2 86.4 87.7 87.9 88.9 87.5 87.9 87.3 87.9
η ∗ (%) 91.5 90.7 90.4 93.4 91.6 91.2 92.6 93.4 92.9 91.8 92.2
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(35)
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Aw x˙ w = Ao U
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and finally the global efficiency. For the other two actuators the coupling parameter progressively increases up to CR=1.88 for the aluminum 2, which exhibits significantly high values of η ∗ . Finally, it should be observed that a low value of kinetic efficiency does not corresponds necessarily to a low value of air jet velocity at exit orifice, but it means that a certain air jet is achieved by a low energy conversion efficiency. This is shown in Figs. 7, 8, and 9 where the frequency response of brass, alluminun 1 and alluminum 2 actuators, respectively, is depicted in terms of orifice jet peak velocity, Uo . The straight line in these plots refers to the so called incompressible solution:
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namely, the volume rate entering the cavity as a consequence of the membrane displacement equals the volume rate of air expelled through the orifice. Although the peak velocity of the brass device at resonance structural frequency of about 2200Hz, slightly greater than 25m/s, is almost the same as that of the aluminum 2 device at the resonance Helmholtz frequency of about 900Hz, Tables 4 and 6 clearly show that the kinetic efficiency of the brass device is dramatically less than that of the aluminum 2.
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5. Additional remarks
Hereafter we will end the paper with some practical remarks. Remember that the kinetic power defined by equation (13), and the related kinetic efficiency of equation (15), refer by definition to the entire cycle, i.e. the suction phase included. In effect, in some specific applications where, for instance, the final goal is to obtain a proper trust linked to the downstream momentum flux, or in heat transfer situations where the goal is to achieve a proper jet Reynolds number based on the stroke length velocity of equation (2), the kinetic power should be referred to the ejection phase only and evaluated just downstream of the so called saddle point, namely the stagnation point along the jet axis separating, during the suction phase, the ingestion near field and the ejection far field. Of course, an accurate evaluation of the time variation of the jet velocity at the saddle point could be made by proper measurements (e.g., by means of PIV techniques) or by detailed CFD 22
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Figure 7: Frequency response of exit flow peak velocity for the brass actuator at Va = 35V . The straight line refers to eq. (35)
U0 [m/s]
30 25 20 15 10 5
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Figure 8: Frequency response of exit flow peak velocity for the aluminum 1 actuator at Va = 35V . The straight line refers to eq. (35)
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Figure 9: Frequency response of exit flow peak velocity for the aluminum 2 actuator at Va = 35V . The straight line refers to eq. (35)
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computations. However, according to the experimental findings of Smith and Glezer [6], one can admit that the jet velocity peak is about the same as the air velocity peak at the orifice; thus, under the assumption of sine temporal variation of the jet velocity just downstream of the saddle point during the ejection phase, one could conclude that the kinetic power is roughly speaking the half of the quantity estimated by equation (13). In practice, due both to a temporal distortion of the velocity signal from the sine trend, and to addition fluid dynamic dissipations in the external air field, the kinetic power can lower down to about 1/3 of the quantity defined in equation (13). Thus, the values of efficiency reported in Table 3 should be reduced by a factor ranging from 1/2 to 1/3 accordingly. For instance, the global efficiency of 41.7% corresponding to the voltage of Va = 75V would be reduced down to about 14%. In the previous section it has been found that the conversion efficiency of mechanical power transferred from the driving membrane to Helmholtz oscillator kinetic power increases with increasing the coupling degree of the oscillators. However, it is very interesting to note that, in the case of actuators with two cavities sharing the same membrane (e.g., see Luo et al. [15]), for 24
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high coupling degree the global efficiency remains almost the same, namely, in other terms, the efficiency of operation of a single cavity is nearly halved. The efficiency performances of the aluminum 2 shim and brass shim actuators have been analysed by running the numerical code properly adapted to the two cavity version, where each cavity has the same geometrical characteristics as the basic single version, orifices included. Of course, due to the perfect geometrical symmetry of the two cavities, one has to expect a corresponding symmetry of the flow characteristics. This finding has been recovered in the computations, which show that the time periodic pressure trends inside the two cavities, as well as the air jets velocity trends at the two orifices, are perfectly in phase opposition. For the aluminum 2 actuator, operating at the voltage of 70V, a global kinetic efficiency of 78% has been computed, that is to say the kinetic efficiency of each single cavity is of 39%. This value has to be compared to the corresponding value of ηk =80.1% re¯ f /P¯e =11.2% and D ¯ s /P¯e =8.7%, ported in Table 3, where one can also read D while for the two cavities version the last two quantities result to be equal, respectively, to 5.5% (i.e., the global fluidic dissipation is 11%) and 11.1%. Thus, when comparing the single and two cavities configurations of the actuator exhibiting the cavity air stiffness greater than the membrane stiffness (CR=1.88), one can conclude that the global kinetic efficiency is almost the same. It is important to stress further that the halved value of the kinetic efficiency of a single cavity is not reflected remarkably on the values of the air jet peak velocities Uo , which are equal to 34.7m/s in the case of double cavity and 38.2m/s in the case of base single cavity. This finding is roughly explained by remembering that the kinetic power scales with the third power of the jet velocity. The efficiency performances of a two cavities actuator change entirely when one considers a low coupling degree between the membrane and Helmholtz oscillators, as it is the case of the brass actuator with CR=0.06. Since in this case the cavity air stiffness is practically negligible as compared to the membrane stiffness, the membrane dynamics is not affected by the compression/expansion phases within the cavities and the global kinetic efficiency of the device almost doubles. In fact, for the operating condition of voltage Va =35V, computations yielded for the single orifice of the two cavities configuration ηk =4.5% (thus the global kinetic efficiency is 9%), whereas Table 4 showed that at the same voltage the single cavity configuration is working with ηk =5.9%. Finally, when a numerical modelling is not available to estimate the elec25
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trodynamic power, the electric efficiency can be evaluated through the equivalent relationship (20), which requires essentially the measurement of the mass flow rate of air at the orifice, the Joule power and the average-per-cycle temperature difference between the system and the colder surrounding external air ∆T introduced in equation (19). Here equation (19) has been employed within a reverse procedure to estimate ∆T . If we assume h = 10W/(m2K), which is a typical value in natural convection, for the standard value of the specific heat coefficient at constant pressure at ambient temperature, ∆T ranges from 0.02◦ C (corresponding to the lower voltage) to 0.78◦ C. 6. Conclusions
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In this paper we presented a comprehensive and detailed physical modelling to evaluate the efficiency of synthetic jet actuators, driven by piezoelectric effect. To the knowledge of present authors this approach is original because is based on relevant energy equations written for both the membrane and acoustic oscillators, which are derived from the corresponding motion equations. It has been shown that the global efficiency, which is defined as the ratio of the useful kinetic power to the input Joule power, can be expressed as the product of the electrodynamic efficiency (defining the conversion of Joule power to electrodynamic power) and the kinetic efficiency (defining the conversion of electrodynamic power to kinetic energy). Both the efficiency can be also estimated in terms of the relevant dissipation terms. In particular, the electrodynamic efficiency is related to the variation of internal energy of air inside the cavity, or heat generation, which is dissipated towards the external ambient both by natural convection on the walls of the actuator case, and through the enthalpy flow rate traversing the orifice. On the contrary, the kinetic efficiency is produced by fluid dynamics and structural dissipations. The physical model has been validated against experimental measurements carried out on built-in-house devices having the piezo-ceramic disk glued on brass and aluminum shims. Combined numerical and experimental investigations allowed us to find that, for the aluminum actuators under examination, the kinetic efficiency does not vary with applied voltage, whereas the electrodynamic efficiency scales with the reciprocal of the voltage square root. These trends have been also justified according to theoretical scaling laws discussed for each relevant power term. Moreover, it has been found that the conversion process of mechanical power transferred, by means 26
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References
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References
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of the driving membrane, to the kinetic power of the Helmholtz oscillator scales dramatically with the coupling degree of the oscillators. However, for two cavities configurations which share the same membrane, computations showed that for high coupling degree (which is the case of aluminum 2 actuator discussed in the present paper) the global kinetic efficiency remains almost the same as compared to the base single cavity configuration, and therefore the kinetic efficiency of the single cavity is halved. A discussion has been made on the connection between the energy conversion efficiency at orifice (which is the base subject of the present paper) and the corresponding efficiency in the external jet field, estimated approximately at the stagnation point (or saddle point) separating, during the suction phase, the ingestion near field from the ejection far field. It has been argued that the saddle point efficiency is lower than the orifice efficiency by a factor that can range from 1/2 to 1/3. Finally, it has been shown that the electrodynamic efficiency can be conveniently evaluated by measuring, among other standard quantities, the average-per-cycle temperature difference between the air inside the cavity and the external ambient. In the present application a reverse procedure allowed us to evaluate such a temperature difference, which is limited to a maximum of almost one degree centigrade.
[1] Glezer, A., and Amitay, M., “Synthetic jets”, Annual Review of Fluid Mechanics, Vol. 34, 2002, pp. 503-529. doi: 10.1146/annurev.fluid.34.090501.094913.
[2] Cattafesta, L. N. III, and Sheplak, M., “Actuators for Active Flow Control”, Annual Review of Fluid Mechanics, Vol. 43, 2011, pp. 247-272. doi: 10.1146/annurev-fluid-122109-160634.
[3] Chaudhari, M., Puranik, B., and Agrawal, A., “Heat transfer characteristics of synthetic jet impingement cooling”, Int. J. Heat Mass Tran., Vol. 53, No. 5, 2010, pp. 1057-1069.
[4] Valiorgue, P., Persoons, T., McGuinn, A., and Murray, D.B., “Heat transfer mechanisms in an impinging synthetic jet for small jet-to-surface 27
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spacing”, Experimental Thermal and Fluid Science, Vol. 33, No. 4, 2009, pp. 597-603.
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[5] Greco, C.S., Ianiro, A. and Cardone, G., “Time and phase average heat transfer in single and twin circular synthetic impinging air jets”, International Journal of Heat and Mass Transfer, Vol. 73, 2014, pp. 776-788.
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[6] Smith, B.L., and Glezer, A., “The formation end evolution of synthetic jets”, Phys.Fluids, Vol. 10, No. 9, 1998, pp. 2281-2297.
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[7] Holman, R., Utturkar, Y., Mittal, R., Smith, B.L., and Cattafesta, L., “Formation criterion for synthetic jets”, AIAA J., Vol. 43, No. 10, 2005, pp. 2110-2116.
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[8] Cater, J.E., and Soria, J., “The evolution of round zero-net-mass-flux jets”, Journal of Fluid Mechanics, Vol. 472, 2002, pp. 167-200.
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[9] Tesar, V. and Zhong, S., “Efficiency of generating the synthetic jets”, Transactions of Aeronautical and Astronautical Society of the Republic of China, Vol. 35, No. 1, 2003, pp. 45-53.
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[10] Crowther, W. J., and L. T. Gomes. “An evaluation of the mass and power scaling of synthetic jet actuator flow control technology for civil transport aircraft applications”, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 222.5 (2008), pp. 357-372. [11] Li, R., Sharma, R., and Arik, M. “Energy conversion efficiency of synthetic jets”, ASME 2011 Pacific Rim Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Systems. American Society of Mechanical Engineers, 2011. [12] de Luca, L., Girfoglio, M., and Coppola, G., “Modeling and experimental validation of the frequency response of synthetic jet actuators”, AIAA J., Vol. 52, No. 8, 2014, pp. 1733-1748. doi: 10.2514/1.J052674 [13] Sharma, R., “Fluid Dynamics Based Analytical Model for Synthetic Jet Actuation”, AIAA Journal, Vol. 45, No. 8, 2007, pp. 1841-1847. [14] de Luca, L., Girfoglio, M., Chiatto, M., and Coppola, G., “Characterization of synthetic jet resonant cavities”, Springer International Publishing 28
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Switzerland 2015, E. Ciappi et al. (eds.), Flinovia - Flow Induced Noise and Vibration Issues and Aspects, pp. 101-118, Book ISBN: 978-3-31909712-1. doi: 10.1007/978-3-319-09713-8_6
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[15] Luo, Z.-b., Xia, Z.-x, and Liu, B., “New generation of synthetic jet actuators”, AIAA Journal, Vol. 44, No. 10, 2006, pp. 2418-2419. doi:10.2514/1.20747
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Supplementary Material
List of Figures
6
7 8
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. 18
. 19
. 20
. 20 . 24 . 24 . 25
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Modular structure of the actuator (de Luca et al. [12]) . . . Sketch of the experimental setup. . . . . . . . . . . . . . . . Kinetic power at orifice exit versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (31) . . . . . . . . . . . . . . . . . . . . . . . Electrodynamic power versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured Joule power versus voltage for aluminum 2 device. Circle symbols are experimental data, continuous line represents eq. (33) . . . . . . . . . . . . . . . . . . . . . . . . . . Electrodynamic efficiency versus voltage for aluminum 2 device. Circle symbols are measured data, continuous line represents eq. (34) . . . . . . . . . . . . . . . . . . . . . . . . . Frequency response of exit flow peak velocity for the brass actuator at Va = 35V . The straight line refers to eq. (35) . . Frequency response of exit flow peak velocity for the aluminum 1 actuator at Va = 35V . The straight line refers to eq. (35) . Frequency response of exit flow peak velocity for the aluminum 2 actuator at Va = 35V . The straight line refers to eq. (35) .
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*Highlights (for review)
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• The lumped parameters modelling is based on the relevant energy equations.
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• The kinetic efficiency depends on structural dissipations and fluidic dissipations at exit orifice.
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• The efficiency at orifice is correlated to the efficiency in the external jet field.
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• The coupling degree of membrane and acoustic oscillators strongly influences the kinetic efficiency of single and two cavities configurations.
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*Photographs and biographies of all authors
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Michele Girfoglio, born in 1985, graduated in Astronautic and Aerospace Engineering in 2011 summa cum laude. He is going to get the PhD degree at the University of Naples Federico II. Current fields of interest include the characterization of synthetic jet actuators and the analysis of the unsteady global dynamics of liquid sheet flows.
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Carlo Salvatore Greco achieved summa cum laude his degree in Aerospace and Astronautical Engineering at the University of Naples Federico II in 2011. He currently is a PhD student at the University of Naples Federico II, Experimental Thermo-Fluid Dynamics sector. His research mainly focuses on fluid-dynamics and heat transfer problems, with particular interest in synthetic jets characterization.
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Matteo Chiatto, born in 1988, is presently a PhD candidate of University of Naples Federico II. He got the master degree in 2013 summa cum laude and the Research Master in Fluid Dynamics at von Karman Institute in 2014. Current field of interest is the flow control by means of piezo-electric and plasma actuators.
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Luigi de Luca graduated in Aeronautical Engineering in 1977 summa cum laude. Presently he is full professor of thermo-fluid-dynamics and fluid dynamics stability at University of Naples Federico II. He is active in the fields of numerical and experimental thermo-fluid-dynamics.
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S0924-4247(15)30084-4 http://dx.doi.org/doi:10.1016/j.sna.2015.07.030 SNA 9262
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
13-11-2014 27-7-2015 27-7-2015
Please cite this article as: Michele Girfoglio, Carlo Salvatore Greco, Matteo Chiatto, Luigi de Luca, Modelling of efficiency of synthetic jet actuators, Sensors & Actuators: A. Physical (2015), http://dx.doi.org/10.1016/j.sna.2015.07.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Modelling of efficiency of synthetic jet actuators
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Michele Girfoglio, Carlo Salvatore Greco, Matteo Chiatto, Luigi de Lucaa,b,c,d a
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PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. b PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. c PhD student, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy. d Professor, Department of Industrial Engineering, Aerospace sect., Universit` a di Napoli Federico II, Italy.
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Abstract
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A comprehensive and detailed modelling to evaluate the efficiency of energy conversion of piezo-electric actuators driving synthetic jets is developed. The contribution is original because the analysis is based on the energy equations of the two coupled oscillators, the membrane and the acoustic one, which are directly derived from the corresponding motion equations. The modelling is validated against numerical as well as experimental investigations carried out on home-made actuators having brass and aluminum shims on which the piezo-disk is bonded. A major result is that for aluminum shim the global efficiency (representing the conversion of input Joule power to kinetic power) decreases with increasing the applied voltage. Another finding is that the conversion process of mechanical power transferred, by means of the driving membrane, to Helmholtz oscillator kinetic power scales dramatically with the coupling degree of the oscillators. The coupling degree influences the efficiency of two cavities actuators sharing the same piezo-diaphragm as well. Considerations are reported to relate the theoretical orifice efficiency to the practical jet efficiency issuing in the external field. Keywords: efficiency, synthetic jet, piezo-electric
Preprint submitted to Sensors and Actuators A: Physical
June 4, 2015
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1. Introduction
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Continuous and pulsed jets are nowadays quite largely used as devices for flow control (e.g., Glezer and Amitay [1], Cattafesta and Sheplak [2]) and heat transfer applications (e.g., Chaudhari et al. [3], Valiorgue et al. [4], Greco et al. [5]). The continuous request for a size reduction of such systems has led researchers to focus their efforts on the design of new technology devices. Synthetic jet actuators well fit this need because they produce jets with zero-net mass flux synthesized directly from the fluid system in which the device itself is embedded (Smith and Glezer [6]), avoiding the need for an external input piping and making them ideal for reduced-space and lowweight applications. A synthetic jet is generated by a membrane oscillation (generally driven by a piezo-ceramic element) in a relatively small cavity, which produces a periodic cavity volume change and thus pressure variation. As the membrane oscillates, fluid is periodically entrained into- and expelled out from an orifice connecting the cavity with the external ambient (to be controlled). During the expulsion phase of the cycle, due to the flow separation, a vortex ring forms near the orifice exit section which, under favorable operating conditions (Holman et al. [7]), convects away towards the far field and breaks up due to the viscous dissipation eventually ”synthesizing” a turbulent jet always directed downstream (Smith and Glezer [6]). Since the jet formation depends on the ability of the vortex ring to escape to the subsequent ingestion phase, following Smith and Glezer [6], a basic ¯ parameter characterizing the jet strength is the so called stroke length L, namely the integral of the (spatially averaged) velocity at the orifice exit over the ejection phase only of the cycle: ¯= L
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where T is the actuation period and U(t) is the fluid velocity at the exit section. The stroke length can also be expressed conveniently as the product ¯ = U¯ T , where U¯ is a proper reference velocity defined as: L R T /2
U(t)dt (2) T Therefore, it is natural to expect that the jet is formed or not according to whether the parameter L/do is greater or less than a certain critical value (e.g. 0.16π, as suggested by Holman et al. [7]), with do being the orifice ¯= U
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¯ o is generally referred to as the diameter. In the literature the parameter L/d reciprocal of the Strouhal number (Cater and Soria [8]). ¯ lies in The importance of the stroke length, or of the average velocity U, the fact that to compare the performance of a synthetic jet with that of a continuous jet, it is usual to refer to a Reynolds number based on the velocity ¯ i.e.: U,
¯ o ρUd (3) µ where ρ is air density, µ is air dynamic viscosity. Although such a topic has been so widely explored, only few studies on the synthetic jet efficiency have been carried out. One of the first works on the efficiency of synthetic jet was undertaken by Tesar and Zhong [9] who based their definition of efficiency on the capability of jet generation rather than on energy conversion considerations. They found experimentally a constant efficiency value equal to 7.2% for several values of Reynolds number (ranging from 4000 to 8000) and for two values of dimensionless stroke length L/do (equal to 223.9 and 527.8). They also compared their experimental evaluation with a numerical prediction made by using the commercial code FLUENT, and found that this last is twice the experimental one. Subsequently Crowther and Gomes [10] studied the system costs associated with the application of flow control system to civil transport aircraft based on the use of electrically powered synthetic jet actuators. They defined the efficiency of the actuator as fluid power (scaling with the cube of the exit velocity) divided by absorbed electrical power, and analyzed it as a function of the operating conditions and actuator geometry, which were chosen reasonably close to those expected for industrial applications. For this reason the experiments were carried out basically with a chamber depth to orifice diameter ratio equal to 0.56 and an orifice depth to diameter ratio equal to 2.1. By resorting to the energy balance principle, Crowther and Gomes [10] considered that the difference between the supplied electrical power and the gained fluid power was lost because of electrical impedance (ascribed to the piezo-electric actuator), mechanical impedance (related to the diaphragm dynamics) and acoustic impedance (due to the fluid-acoustic coupling of the flow within the cavity and through the orifice), however without accurately quantifying each term of the balance. They showed their experimental findings in terms of a map of the electric-fluidic conversion efficiency, where such an efficiency was reported as a function of excitation voltage amplitude (for
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peak-to-peak excitation voltage up to 250V) and actuation frequency (up to 4000Hz). They found that the efficiency attains a maximum equal to about 14% and noted that it did not correspond to the condition of maximum exit velocity because of the dielectric saturation effect affecting the commercial piezo-electric patch (i.e., the piezo-element bonded to the brass shim). A more recent work, focusing specifically on the energy conversion efficiency of synthetic jets devices, was presented by Li et al. [11] who tried to express on an analytical basis every term contributing to the energy rate balance. They argued that once the synthetic jet has received electric energy input, due to the capacitance of the piezo-electric actuator, a part of the energy is stored as electric potential energy while the rest of the energy is converted to mechanical energy accompanied by energy dissipation. The mechanical energy includes vibration of piezo-electric actuators and kinetic energy of air flow. For piezo-electric actuators two forms of energy, i.e. strain energy and kinetic energy, are temporarily stored by the vibrating structure. Energy dissipation (namely energy loss) occurs in piezo-electric actuators due to the deflection dynamics, as well as in the air flow motion (i.e., the head losses) when traversing the jet orifice. The synthetic jet device efficiency is defined as the ratio of the kinetic power of the air flow to the input electric energy. The authors carried out experiments on two slot synthetic jets having orifice length of 4mm and 15mm, for two values of voltage amplitude (80V and 100V) and actuation frequency ranging from 200Hz to 1100Hz. Li et al. [11] found that the efficiency of energy conversion is dependent on the orifice size and on the operating conditions (namely, voltage and frequency) showing a peak (of about 40% for the 15mm orifice) close to the mechanical resonance frequency of the actuator. They claim that the uncertainty of their experimental evaluation was of about 34%. The aim of the present paper is to assess a rigorous and comprehensive physical approach to the evaluation of the efficiency of piezo-electrically driven synthetic jet actuators, based on a rather detailed modelling of the actuator dynamics, already developed by two of the authors, de Luca et al. [12], in the past. The peculiarity of the present approach is that the energy balance equation, properly averaged over the actuation period, is derived directly from the equations governing the dynamics of the actuator. The advantage is that the presence of each term of the resulting energy balance draws its justification from the corresponding dynamics term and, in principle, there is no need to perform experimental measurements to evaluate the actuator efficiency at the orifice exit section. Since the physical model 4
Page 4 of 32
Ac ce p
te
d
M
an
us
cr
ip t
mentioned above has been already tested and calibrated against a systematic experimental campaign, the uncertainty of the efficiency estimation should be reduced. The analysis methodology and a part of the results will highlight the key role played by the coupling degree of the two oscillators introduced in the physical model, i.e. the oscillating membrane and the acoustic one. Roughly speaking, the coupling degree is the ratio of the cavity air stiffness to the membrane stiffness. It will be shown that the conversion efficiency of mechanical power, transferred by means of the driving membrane, to Helmholtz oscillator kinetic power increases with increasing the coupling degree of the oscillators. However, in the case of actuators with two cavities, sharing the same membrane, we will see that for high coupling degree the global efficiency remains almost the same, namely, in other terms, the efficiency of operation of the single cavity is nearly halved. It is worth noting here that, since the modelling is of ”lumped parameters” kind, it yields the evaluation of the energy conversion efficiency intrinsically at orifice; however, in various practical applications the efficiency in the external jet field is needed, which should be estimated approximately at the stagnation point (or saddle point) separating, during the suction phase, the ingestion near field from the ejection far field. The connection between the ”efficiency at orifice” and the ”efficiency in the jet” will be addressed in the final part of the paper, after discussing the results. The paper is organized as follows: section 2 reports the basic equations governing the dynamics of the actuator, whilst section 3 is devoted to the development of the energy balance, directly derived from the dynamics equations. The experimental validation and calibration of the proposed model are discussed in section 4, useful practical remarks are included in section 5 and, finally, conclusions are drawn in section 6. 2. Governing equations system In this section a short description of the motion equations governing the dynamics of a typical piezo-electrically driven synthetic jet actuator will be given. Details regarding the physical modelling can be found in the original paper of de Luca et al. [12]. It should be pointed out that the approach followed by the present authors is fluid-dynamics based, and was inspired by the previous original paper of Sharma [13].
5
Page 5 of 32
ip t
The membrane behavior is modelled as a one-degree of freedom forced mass-spring-damper system mwt x¨w + cwt x˙ w + kw xw = F − pi Aw
(4)
k w d A Va sin ωt = kw ∆xw sin ωt Aw
(5)
M
F = Fo sin ωt =
an
us
cr
where mwt is the diaphragm total mass (including shim, piezo-element and air added mass), xw is the membrane displacement at a generic time instant t, cwt is the total damping coefficient (including the structural damping coefficient and a further damping coefficient due to the interaction with the external air), kw is the equivalent stiffness of the membrane (the clamped edge condition is assumed here), F is the electrodynamic force, pi is the cavity (internal) differential pressure, Aw is the membrane area. The electrodynamic force F is related to the piezo-electric effect due to the applied sine voltage and is given by
Ac ce p
te
d
where Fo is the forcing magnitude, dA is the effective acoustic piezo-electric coefficient, Va is the voltage amplitude, ω = 2πf is the operating angular frequency (with f being the natural frequency, f = 1/T ) and ∆xw is the (average) linear ”static” displacement of the membrane due to the piezoelectric effect. The application of the unsteady Bernoulli’s equation between a point inside the cavity where the flow velocity is practically null and a point, just outside the cavity, representing the location where the pressure matches the unperturbed external ambient value, yields Ma
dU 1 = pi Ao − Kρa Ao U|U| dt 2
(6)
in which Ao is the orifice area, K is the head loss coefficient, ρa is the air density, Ma = ρa le Ao is the mass of air oscillating inside the orifice (Helmholtz oscillator), with le being the effective length of the orifice (i.e., the distance between the two locations of application of the Bernoulli’s equation) and U being is the instantaneous orifice jet-flow velocity, as already defined in section 1. The equation of evolution of the third unknown, the cavity pressure pi , is obtained by enforcing the conservation of air mass in the cavity; by relating the density and pressure variations by means of an isentropic compression/expansion transformation, the continuity equation can be formulated in 6
Page 6 of 32
Vc dpi = Aw x˙ w − Ao U γpo dt
(7)
ip t
the following way
cr
where Vc is the cavity volume, po is the ambient pressure, γ is the specific heat ratio.
us
3. Derivation of the energy balance equation and definition of efficiency Multiplying the equation (4) by the membrane velocity x˙ w , yields:
an
dEw = F x˙ w − pi Aw x˙ w − cw x˙ 2w dt
(8)
where Ew is the diaphragm energy defined as
M
1 1 Ew = mw x˙ 2w + kw xw 2 2 2
(9)
Ac ce p
te
d
that takes into account both kinetic and elastic strain contributions. Similarly, the product between the equation (6) and the jet exit velocity U yields dEo 1 = pi Ao U − Kρa Ao |U|3 (10) dt 2 where Eo is given by 1 (11) Eo = Ma U 2 2 and represents the kinetic energy flow rate of the air mass through the orifice. The (instantaneous) energy balance equation of the actuator system can be obtained by summing the equations (8) and (10). Moreover, since we are interested in characterizing the actuator behavior over an operating cycle, it is convenient to apply the time average operator on the resulting equation, defined as Z 1 T ϕdt (12) ϕ¯ = T 0 where ϕ is the generic time-dependent variable.
7
Page 7 of 32
T
0
1 − T |
Z
1 d(Ew + Eo ) = T {z } | ∆E T
cwt x˙ 2w dt −
0
{z
D¯s
}
1 T |
Z
Z
T 0
T 0
1 F x˙ w dt + T {z } | P¯e
Z
T
0
pi (Ao U − Aw x˙ w )dt {z } P¯m
1 1 (K − 1)ρa Ao |U|3 dt − 2 T {z } |
Z
0
cr
Z
T
1 ρa Ao |U|3 dt (13) 2 {z }
us
1 T |
ip t
The energy balance equation averaged over an actuation cycle (i.e., over a time equal to the period T ) is given by
D¯f
P¯k
an
An accurate description of the various contributions of equation (13) is reported hereafter:
M
• ∆E is the total energy variation. Note that this term is null because, for each cycle, there is no change for Ew and Eo . • P¯e is the electrodynamic power provided to the membrane by the applied voltage.
Ac ce p
te
d
• P¯m is the mechanical power due to the work done by the differential pressure pi which acts on the wall surface Aw and on the orifice surface Ao . By using the equation (7), it can be shown that this term is proportional to 12 (pi 2 (T ) − pi 2 (0)) and, therefore, it does not give any contribution because pi assumes the same value at the beginning and end of each cycle. One can reach the same result by observing that the pressure work is conservative by definition. • D¯s is the power dissipation due to the structural damping effects of the membrane. • D¯f is the power dissipation due to the head loss of fluid dynamics type at the orifice.
• P¯k is the kinetic power of air flow at the orifice. Note explicitly that the kinetic power here refers by definition to the entire cycle, i.e. the suction phase included. We will continue this discussion in the last section 4.
8
Page 8 of 32
Then, by deleting ∆E and P¯m terms, the equation (13) becomes (14)
ip t
P¯e − D¯s − D¯f − P¯k = 0
cr
and, once defined the kinetic efficiency ηk as the ratio of the kinetic power of the exit flow P¯k to the electrodynamic power P¯e , one obtains
an
us
P¯k D¯s + D¯f ηk = ¯ = 1 − (15) Pe P¯e It is worthwhile to stress that in practice the global efficiency of an actuator has to quantify the amount of Joule power provided to the system P¯j that is actually converted in P¯e . This can be done by introducing the electrodynamic transduction efficiency ηe (16)
M
P¯e ηe = ¯ Pj
Ac ce p
te
d
Hence, finally, one can define the (global) efficiency of the actuator η as the product of ηk by ηe P¯k η = ηk ηe = ¯ (17) Pj where the external Joule power supply P¯j provided to the actuator is calculated as Z 1 T ¯ V Idt (18) Pj = T 0 with V = Va sin ωt being the applied voltage and I the electric current flowing through the piezo-electric element. The rate of P¯j not turned into P¯e is converted into a variation of internal energy Q˙ of the air inside the cavity (heat generation per unit time), which is transferred in part to the external ambient through a natural convection mechanism, and, in part, into enthalpy flow rate of the air leaving the orifice P¯j − P¯e = Q˙ ≡ hAe ∆T + mc ˙ p ∆T
(19)
where h is the convective heat transfer coefficient by natural convection, Ae is the exchange surface (which depends on the relevant geometry of actuator), ∆T is the average-per-cycle temperature difference between the system and ¯ o and we remember that U¯ the colder surrounding external air, m ˙ = ρa UA is the average velocity associated to the stroke length (eq. 2), and cp is the 9
Page 9 of 32
ip t
specific heat coefficient at constant pressure. For the sake of simplicity we assumed that the air inside the cavity is isothermal with the device case. From equation (19) one obtains also Q˙ ηe = 1 − ¯ Pj
cr
(20)
M
an
us
Moreover, it is important to note that the energy conversion process from electrodynamic energy to flow kinetic energy depends intrinsically on the coupling degree between the two oscillators modeling the device behavior, i.e. the membrane and the Helmholtz one. From the dynamical point of view the coupling effect is represented by the internal pressure term acting as a forcing term on the Helmholtz oscillator in equation (6); from the energy point of view, the coupling effect can be seen as a transfer of mechanical power (e.g, the work done by the internal pressure) to the Helmholtz oscillator, expressed by Z 1 T ¯ pi Aw x˙ w dt (21) PmH = T 0
Ac ce p
te
d
It is clearly evident that, although the net mechanical power due to the pressure work introduced in equation (13) is null, P¯mH defined in the above equation is not, and it is convenient to split ideally the conversion process from electrodynamic energy to flow kinetic energy into two steps: the conversion of electrodynamic energy P¯e to mechanical power due to the pressure work (within the membrane oscillator), namely P¯mH , and the conversion of mechanical power to air flow kinetic energy (within the Helmholtz oscillator). In equation terms, once introduced the internal efficiencies P¯mH P¯e
(22)
P¯k η ∗∗ = ¯ PmH
(23)
η∗ =
the kinetic efficiency of equation (15) can be interpreted as ηk = η ∗ η ∗∗
(24)
In discussing the results, we will see that, while η ∗∗ is generally very high, as perhaps expected η ∗ depends crucially on the coupling degree between the 10
Page 10 of 32
cr
ip t
two oscillators. In the case of reduced coupling, η ∗ can be very low and it reduces dramatically also the kinetic efficiency of the device. Furthermore, note that a limited coupling degree does not imply a low air flow velocity; it means that a certain value of air jet velocity is obtained at the price of a very low energy efficiency. 4. Validation of the model and results
Ac ce p
te
d
M
an
us
For fixed values of the geometrical and electro-mechanical parameters of a typical synthetic jet actuator, the modeling developed before has been employed to estimate the various contributions of the energy budget equations (13, 14) by carrying out direct numerical simulations of the governing equations (4, 6, 7). For a proper operating frequency f , the time trends of the relevant quantities, in practice the membrane velocity and the air velocity at the orifice exit, have been used to evaluate, ultimately, the energy conversion efficiencies (15, 16, 17). Numerical integration has been performed by means of a standard 4th order Runge-Kutta method in MATLAB environment with ode45 routine. Initial conditions of xw = 0, x˙ w = 0, pi = 0, and U = 0 have been assumed for all the computations and it has been observed that the quasi-steady oscillatory solution is generally reached in about 20 cycles (de Luca et al. [12]). The time averages here performed refer typically to the 23th cycle. The uncertainty errors of the numerical data have been evaluated especially with respect to variations of the values of the head loss coefficient as well as of the effective orifice length (de Luca et al. [12]). For K varying in the range 1 < K < 1.46 the variations of the resonance frequencies are restricted within a maximum of 2%, while the corresponding variations of the air velocity peaks are generally of the order of 10%. The uncertainty of data of le is reflected on the peak frequencies which vary within a corresponding uncertainty band of 5% which may grow up to a maximum of 10% for relatively low frequencies, while the average width of such a band is of 10% for the peak exit velocities. To validate the modeling, numerical values of efficiency have been compared with measured data for three typical synthetic jet actuators, the characteristics of which are summarized in Table 1. The membrane was built inhouse by gluing a LZT piezo-ceramic disk (manufactured by PIEZO System inc.) on a thin aluminum foil, or it is constituted by a commercially available brass shim fabricated by Murata Manufactoring Co. The schematic of 11
Page 11 of 32
Synthetic Jet
ip t
U
Vortex Train
Orifice
cr
l0
le d0
us
Cavity
H
an
dw
Piezo
M
Membrane
Figure 1: Modular structure of the actuator (de Luca et al. [12])
Ac ce p
te
d
the device is also shown in Fig. 1 highlighting its modular structure, which permits independent variations of cavity diameter and height, orifice diameter, and piezo-electric diaphragm (de Luca et al. [12]). Here we adopted the value K = 1.14 for the head loss coefficient, while the effective length of the orifice le , was basically evaluated according to the description of Sharma [13], namely le /do = lo /do + ∆le , with ∆le =0.71. For a given operating voltage, the evaluation of the Joule power supply (equation (18)) needs the knowledge of the current intensity. This has been done by means of the theoretical relationship It = 2πf CVa
(25)
where It denotes the theoretical current intensity peak, and C is the electric capacitance of the piezo-ceramic disk (furnished by the manufacturer). To verify this estimation, the current intensity peak has been also measured directly by means of the experimental apparatus sketched in Fig. 2. A waveform generator (Wavetek model 164) creates an electrical sinusoidal signal which is amplified (by EPA 104 PIEZO System inc. unit) and sent to the synthetic jet device. Voltage and current are acquired by a two channels oscilloscope (Tektronix tds 2024). In Table 2, experimental measurements of the electric current peak Ie 12
Page 12 of 32
ip t cr us an M d te Ac ce p
Figure 2: Sketch of the experimental setup.
13
Page 13 of 32
M
Piezo-electric
Aluminum 1 42 0.24 31.8 0.191 42 3 2 2 7.31x1010 0.31 2780 6.6x1010 0.31 7800
cr
us
Shim
Brass 35 0.4 23 0.23 35 3 2 2 9.7x1010 0.36 8490 6.7x1010 0.31 8000
an
Geometry
Property Shim diameter (mm) Shim thickness (mm) Piezo-electric diameter (mm) Piezo-electric thickness (mm) Cavity diameter (mm) Cavity height (mm) Orifice diameter (mm) Orifice length (mm) Young’s Module (Pa) Poisson’s Module Density (Kg/m3 ) Young’s Module (Pa) Poisson’s Module Density (Kg/m3 )
Aluminum 2 80 0.25 63.5 0.191 80 4 5 2 7.31x1010 0.31 2780 6.6x1010 0.31 7800
ip t
Table 1: Features of the tested devices
Ac ce p
te
d
are compared to the corresponding theoretical values It for the aluminum 2 device, which has been first employed to obtain insights on the energy efficiency. The data spread is generally less than 9% except for Va = 50V , for which it appears to be about 13%. The efficiency values computed for this device at the modified Helmholtz resonance frequency (which is of about 900Hz, as reported by de Luca et al. [12]) are listed in Table 3, together with the two power dissipation terms. Here we limit to remember that the modified (circular) frequencies, ωm , can be estimated to a good approximation by the relationship (with the Helmholtz one corresponding to the sign +): q 2 2 2 2 2 ωh ωh ωh ωwc 2 2 −(1 + + ) ± (1 + ωωwc 2 2 2 + ω2 ) − 4 ω2 ωm ωw ωw w w w = (26) ωw2 2 where the (first mode) structural frequency of the membrane is given by r kw , (27) ωw = mwt the uncoupled Helmholtz frequency is
14
Page 14 of 32
ip t
M
an
us
It (A) Ie (A) 0.038 0.036 0.045 0.044 0.052 0.054 0.059 0.056 0.066 0.060 0.073 0.064 0.079 0.074 0.085 0.080 0.093 0.086 0.099 0.094 0.106 0.104
te
d
Va (V ) 25 30 35 40 45 50 55 60 65 70 75
cr
Table 2: Comparison of theoretical current peak with measured data (Ampere)
Table 3: Efficiencies and dissipation terms at the modified Helmholtz frequency for aluminum 2 device
ηe (%) 90.7 85.4 79.6 74.8 69.6 66.8 64.2 59.1 58.8 54.6 51.5
Ac ce p
Va (V ) ηk (%) 25 79.2 30 79.5 35 79.7 40 80.7 45 80.3 50 80.2 55 82.3 60 81.7 65 81.7 70 80.1 75 81
¯ f /P¯e (%) η(%) D 71.8 11.1 67.9 11.1 63.4 11.2 60.3 11.3 55.9 11.2 53.6 11.2 52.8 11.1 48.3 11.4 48 11.1 43.7 11.2 41.7 11.3
¯ s /P¯e (%) D 9.3 9.3 8.7 8.2 8.4 8.5 7.2 7.2 7.8 8.7 7.8
15
Page 15 of 32
r
ka , Ma
and the coupling frequency is defined as s r γA2w po /Vc γAw po ωwc = = mw mw H
(28)
ip t
γA2o po /Vc = ρa le Ao
cr
ωh =
s
(29)
M
an
us
which, following Sharma [13], may be interpreted as the frequency of the pneumatic spring made of the air enclosed within the cavity of volume Vc and of the diaphragm mass mw . Note that the height of the cavity, H, is explicitly introduced. The coupling degree between the two oscillators can be quantified, as made by de Luca et al. [12], by means of the parameter CR: 2 ωwc CR = (30) ωw
Ac ce p
te
d
which can be interpreted as the ratio of the cavity air stiffness to the membrane stiffness. It is evident that if CR ≪ 1 both the modified resonance frequencies tend to the natural frequencies of the membrane and Helmholtz oscillators. As one can observe (Table 3), the influence of the driving voltage is rather ¯ f /P¯e and D ¯ s /P¯e ; conversely, ηe decreases with increasing Va , weak for ηk , D thus determining the same trend for the global efficiency η. For the sake of completeness, to give the reader the possibility to check the calculations, the following additional parameters have been considered, according to the complete description of the actuator under examination made by de Luca et al. [12]: ρa = 1.205 Kg/m3 ; kw = 9.48 · 104 N/m; dA = 4.01 · 10−9 m3 /V ; cwt = 2.75 Ns/m, evaluated at the frequency of 900Hz. Note that the energy balance equation is satisfied numerically to within a maximum error of 6 × 10−3 (for Va = 55V and Va = 65V). Hereafter we will show that the trends against voltage of the quantities shown in Table 3 are well fitted by simple power laws which, in turn, can be justified on the basis of scaling theoretical considerations. From the numerical integration of the governing equations we found that the variations of exit kinetic power and electrodynamic power follow the following trends 16
Page 16 of 32
ip t
0.35 0.3
cr
0.2 0.15
us
P¯k (W )
0.25
0.1
30
40
50 Va (V )
60
70
80
M
0 20
an
0.05
d
Figure 3: Kinetic power at orifice exit versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (31)
te
P¯k ≈ Va 1.8 P¯e ≈ Va 1.8
(31) (32)
Ac ce p
as is depicted in Figs. 3 and 4, respectively, where circle symbols refer to computed data e continuous lines refer to the analytical relationships. Therefore, being ηk given by the ratio between two quantities both proportional ¯ f has the same to Va 1.8 , it does not depend on Va . Since one expects that D ¯ f /P¯e is constant too. scaling law as P¯k , the fluid dynamic dissipation rate D ¯ f vary as Indeed, according to their definitions in equation (13), P¯k and D 3 ¯ U0 and Pe as F0 x˙ w0 (with U0 and x˙ w0 being the fluid and membrane peak velocities, respectively), and therefore one could expect that they scale as Va3 and Va2 , respectively. However, due to the complex coupling between the two oscillators, the membrane and the acoustic one, in the presence of significant non linear damping acting on the acoustic one, such theoretical laws are remarkably modified, as discussed in detail by de Luca et al. [12] and de Luca et al. [14]. Here we found that U0 ≈ Va 0.6 and x˙ w0 ≈ Va 0.8 . Since F0 scales with Va , the scaling relationships of equations (31) and (32) are justified.
17
Page 17 of 32
ip t
0.4 0.35
cr
0.25 0.2
us
P¯e (W )
0.3
0.15
30
40
50 Va (V )
60
70
80
M
0.05 20
an
0.1
d
Figure 4: Electrodynamic power versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (32)
te
Fig. 5 shows that the Joule power, evaluated experimentally, increases with increasing the voltage amplitude following a power law of the kind (33)
Ac ce p
P¯j ≈ Va 2.3
Note that if the behavior of the piezo-ceramic element was governed perfectly by the Ohm’s law, i.e. if the capacitance was constant, P¯j would be directly proportional to Va 2 . In effect, the relationship (33) tells us that the piezo-element capacitance varies as a function of the voltage and, furthermore, that the output instantaneous current is a distorted sine wave. Finally, as depicted in Fig. 6, the behavior of the electrodynamic transduction efficiency ηe defined by equation (16) and relating the Joule power to the electrodynamic one, taking into account also the relationship (32), is well fitted by a power law of the following type ηe ≈ Va −0.5
(34)
A similar trend holds for the global efficiency η (equation (17)). It is interesting to observe that the scaling laws discussed above and referring to the aluminum 2 actuator cannot be extended in a straightforward 18
Page 18 of 32
ip t
0.8 0.7 0.6
cr
0.4
us
P¯j (W )
0.5
0.3
an
0.2 0.1
30
40
50 Va (V )
60
70
80
M
0 20
0.9
Ac ce p
0.85
te
0.95
d
Figure 5: Measured Joule power versus voltage for aluminum 2 device. Circle symbols are experimental data, continuous line represents eq. (33)
0.8
ηe
0.75 0.7
0.65 0.6
0.55
0.5 20
30
40
50 Va (V )
60
70
80
Figure 6: Electrodynamic efficiency versus voltage for aluminum 2 device. Circle symbols are measured data, continuous line represents eq. (34)
19
Page 19 of 32
cr
η ∗ (%) 5.5 6.3 6.9 7.5 7.9 8.2 8.7 8.9 9.1 9.2 10.1
us
η ∗∗ (%) 85.5 85.3 85 84.8 85.5 88.6 86.6 85.6 87.1 87.8 83.7
M
an
Va (V ) ηk (%) 25 4.7 30 5.3 35 5.9 40 6.4 45 6.8 50 7.3 55 7.5 60 7.6 65 7.9 70 8.1 75 8.5
ip t
Table 4: Efficiencies of brass actuator at modified resonance structural frequency. CR=0.06
Ac ce p
te
d
way to all the actuators since, on one hand the electrodynamic transduction depends on the electrical capacitance of the piezo-ceramic element, and, on the other one, the energy conversion towards the air jet kinetic energy is strongly influenced by complex resonance phenomena which, in turn, are significantly influenced by the non linear fluidic damping effects. Therefore, rather we prefer to raise a brief discussion about the influence of the coupling degree of the oscillators on the kinetic energy efficiency, by monitoring the internal efficiencies η ∗ and η ∗∗ defined by equations (22) and (23). This will be made with the aid of Tables 4, 5, 6, where the relevant efficiencies are reported as functions of the applied voltage, for brass, aluminum 1, aluminum 2 actuators, respectively. The coupling parameter of equation (30) is equal to 0.06, 0.3, 1.88, respectively. Note that for the first two devices data refers to the modified resonance structural frequency (about 2200Hz for brass, which coincides practically with the uncoupled value, and 1600Hz for aluminum 1), whereas for the third device data refers to the modified resonance Helmholtz frequency (about 900Hz). As anticipated in the previous section, it is evident that, while η ∗∗ is generally very high for all the devices, η ∗ depends crucially on the coupling degree between the two oscillators. In the case of brass shim device having CR=0.06 the amount of mechanical power transferred from the membrane oscillator to the Helmholtz one is remarkably reduced, hence η ∗ is very low. Of course this reduces dramatically also the kinetic efficiency of the device 20
Page 20 of 32
ip t
η ∗ (%) 41.5 41.9 43.6 41.8 42.8 42.6 42.6 42.1 42.2 41.6 40.6
us
η ∗∗ (%) 87.6 88.5 86.8 89.5 88 87.3 86.9 88.6 87.2 88 89
te
d
M
an
Va (V ) ηk (%) 25 36.4 30 37.1 35 37.8 40 37.4 45 37.6 50 37.2 55 37 60 37.3 65 36.8 70 36.6 75 36.2
cr
Table 5: Efficiencies of aluminum 1 actuator at modified resonance structural frequency. CR=0.3
Ac ce p
Table 6: Efficiencies of aluminum 2 actuator at modified resonance Helmholtz frequency. CR=1.88
Va (V ) ηk (%) 25 79.2 30 79.5 35 79.7 40 80.7 45 80.3 50 80.2 55 82.3 60 81.7 65 81.7 70 80.1 75 81
η ∗∗ (%) 86.6 87.6 88.2 86.4 87.7 87.9 88.9 87.5 87.9 87.3 87.9
η ∗ (%) 91.5 90.7 90.4 93.4 91.6 91.2 92.6 93.4 92.9 91.8 92.2
21
Page 21 of 32
(35)
an
Aw x˙ w = Ao U
us
cr
ip t
and finally the global efficiency. For the other two actuators the coupling parameter progressively increases up to CR=1.88 for the aluminum 2, which exhibits significantly high values of η ∗ . Finally, it should be observed that a low value of kinetic efficiency does not corresponds necessarily to a low value of air jet velocity at exit orifice, but it means that a certain air jet is achieved by a low energy conversion efficiency. This is shown in Figs. 7, 8, and 9 where the frequency response of brass, alluminun 1 and alluminum 2 actuators, respectively, is depicted in terms of orifice jet peak velocity, Uo . The straight line in these plots refers to the so called incompressible solution:
te
d
M
namely, the volume rate entering the cavity as a consequence of the membrane displacement equals the volume rate of air expelled through the orifice. Although the peak velocity of the brass device at resonance structural frequency of about 2200Hz, slightly greater than 25m/s, is almost the same as that of the aluminum 2 device at the resonance Helmholtz frequency of about 900Hz, Tables 4 and 6 clearly show that the kinetic efficiency of the brass device is dramatically less than that of the aluminum 2.
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5. Additional remarks
Hereafter we will end the paper with some practical remarks. Remember that the kinetic power defined by equation (13), and the related kinetic efficiency of equation (15), refer by definition to the entire cycle, i.e. the suction phase included. In effect, in some specific applications where, for instance, the final goal is to obtain a proper trust linked to the downstream momentum flux, or in heat transfer situations where the goal is to achieve a proper jet Reynolds number based on the stroke length velocity of equation (2), the kinetic power should be referred to the ejection phase only and evaluated just downstream of the so called saddle point, namely the stagnation point along the jet axis separating, during the suction phase, the ingestion near field and the ejection far field. Of course, an accurate evaluation of the time variation of the jet velocity at the saddle point could be made by proper measurements (e.g., by means of PIV techniques) or by detailed CFD 22
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Figure 7: Frequency response of exit flow peak velocity for the brass actuator at Va = 35V . The straight line refers to eq. (35)
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Figure 9: Frequency response of exit flow peak velocity for the aluminum 2 actuator at Va = 35V . The straight line refers to eq. (35)
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computations. However, according to the experimental findings of Smith and Glezer [6], one can admit that the jet velocity peak is about the same as the air velocity peak at the orifice; thus, under the assumption of sine temporal variation of the jet velocity just downstream of the saddle point during the ejection phase, one could conclude that the kinetic power is roughly speaking the half of the quantity estimated by equation (13). In practice, due both to a temporal distortion of the velocity signal from the sine trend, and to addition fluid dynamic dissipations in the external air field, the kinetic power can lower down to about 1/3 of the quantity defined in equation (13). Thus, the values of efficiency reported in Table 3 should be reduced by a factor ranging from 1/2 to 1/3 accordingly. For instance, the global efficiency of 41.7% corresponding to the voltage of Va = 75V would be reduced down to about 14%. In the previous section it has been found that the conversion efficiency of mechanical power transferred from the driving membrane to Helmholtz oscillator kinetic power increases with increasing the coupling degree of the oscillators. However, it is very interesting to note that, in the case of actuators with two cavities sharing the same membrane (e.g., see Luo et al. [15]), for 24
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high coupling degree the global efficiency remains almost the same, namely, in other terms, the efficiency of operation of a single cavity is nearly halved. The efficiency performances of the aluminum 2 shim and brass shim actuators have been analysed by running the numerical code properly adapted to the two cavity version, where each cavity has the same geometrical characteristics as the basic single version, orifices included. Of course, due to the perfect geometrical symmetry of the two cavities, one has to expect a corresponding symmetry of the flow characteristics. This finding has been recovered in the computations, which show that the time periodic pressure trends inside the two cavities, as well as the air jets velocity trends at the two orifices, are perfectly in phase opposition. For the aluminum 2 actuator, operating at the voltage of 70V, a global kinetic efficiency of 78% has been computed, that is to say the kinetic efficiency of each single cavity is of 39%. This value has to be compared to the corresponding value of ηk =80.1% re¯ f /P¯e =11.2% and D ¯ s /P¯e =8.7%, ported in Table 3, where one can also read D while for the two cavities version the last two quantities result to be equal, respectively, to 5.5% (i.e., the global fluidic dissipation is 11%) and 11.1%. Thus, when comparing the single and two cavities configurations of the actuator exhibiting the cavity air stiffness greater than the membrane stiffness (CR=1.88), one can conclude that the global kinetic efficiency is almost the same. It is important to stress further that the halved value of the kinetic efficiency of a single cavity is not reflected remarkably on the values of the air jet peak velocities Uo , which are equal to 34.7m/s in the case of double cavity and 38.2m/s in the case of base single cavity. This finding is roughly explained by remembering that the kinetic power scales with the third power of the jet velocity. The efficiency performances of a two cavities actuator change entirely when one considers a low coupling degree between the membrane and Helmholtz oscillators, as it is the case of the brass actuator with CR=0.06. Since in this case the cavity air stiffness is practically negligible as compared to the membrane stiffness, the membrane dynamics is not affected by the compression/expansion phases within the cavities and the global kinetic efficiency of the device almost doubles. In fact, for the operating condition of voltage Va =35V, computations yielded for the single orifice of the two cavities configuration ηk =4.5% (thus the global kinetic efficiency is 9%), whereas Table 4 showed that at the same voltage the single cavity configuration is working with ηk =5.9%. Finally, when a numerical modelling is not available to estimate the elec25
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trodynamic power, the electric efficiency can be evaluated through the equivalent relationship (20), which requires essentially the measurement of the mass flow rate of air at the orifice, the Joule power and the average-per-cycle temperature difference between the system and the colder surrounding external air ∆T introduced in equation (19). Here equation (19) has been employed within a reverse procedure to estimate ∆T . If we assume h = 10W/(m2K), which is a typical value in natural convection, for the standard value of the specific heat coefficient at constant pressure at ambient temperature, ∆T ranges from 0.02◦ C (corresponding to the lower voltage) to 0.78◦ C. 6. Conclusions
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In this paper we presented a comprehensive and detailed physical modelling to evaluate the efficiency of synthetic jet actuators, driven by piezoelectric effect. To the knowledge of present authors this approach is original because is based on relevant energy equations written for both the membrane and acoustic oscillators, which are derived from the corresponding motion equations. It has been shown that the global efficiency, which is defined as the ratio of the useful kinetic power to the input Joule power, can be expressed as the product of the electrodynamic efficiency (defining the conversion of Joule power to electrodynamic power) and the kinetic efficiency (defining the conversion of electrodynamic power to kinetic energy). Both the efficiency can be also estimated in terms of the relevant dissipation terms. In particular, the electrodynamic efficiency is related to the variation of internal energy of air inside the cavity, or heat generation, which is dissipated towards the external ambient both by natural convection on the walls of the actuator case, and through the enthalpy flow rate traversing the orifice. On the contrary, the kinetic efficiency is produced by fluid dynamics and structural dissipations. The physical model has been validated against experimental measurements carried out on built-in-house devices having the piezo-ceramic disk glued on brass and aluminum shims. Combined numerical and experimental investigations allowed us to find that, for the aluminum actuators under examination, the kinetic efficiency does not vary with applied voltage, whereas the electrodynamic efficiency scales with the reciprocal of the voltage square root. These trends have been also justified according to theoretical scaling laws discussed for each relevant power term. Moreover, it has been found that the conversion process of mechanical power transferred, by means 26
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References
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References
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of the driving membrane, to the kinetic power of the Helmholtz oscillator scales dramatically with the coupling degree of the oscillators. However, for two cavities configurations which share the same membrane, computations showed that for high coupling degree (which is the case of aluminum 2 actuator discussed in the present paper) the global kinetic efficiency remains almost the same as compared to the base single cavity configuration, and therefore the kinetic efficiency of the single cavity is halved. A discussion has been made on the connection between the energy conversion efficiency at orifice (which is the base subject of the present paper) and the corresponding efficiency in the external jet field, estimated approximately at the stagnation point (or saddle point) separating, during the suction phase, the ingestion near field from the ejection far field. It has been argued that the saddle point efficiency is lower than the orifice efficiency by a factor that can range from 1/2 to 1/3. Finally, it has been shown that the electrodynamic efficiency can be conveniently evaluated by measuring, among other standard quantities, the average-per-cycle temperature difference between the air inside the cavity and the external ambient. In the present application a reverse procedure allowed us to evaluate such a temperature difference, which is limited to a maximum of almost one degree centigrade.
[1] Glezer, A., and Amitay, M., “Synthetic jets”, Annual Review of Fluid Mechanics, Vol. 34, 2002, pp. 503-529. doi: 10.1146/annurev.fluid.34.090501.094913.
[2] Cattafesta, L. N. III, and Sheplak, M., “Actuators for Active Flow Control”, Annual Review of Fluid Mechanics, Vol. 43, 2011, pp. 247-272. doi: 10.1146/annurev-fluid-122109-160634.
[3] Chaudhari, M., Puranik, B., and Agrawal, A., “Heat transfer characteristics of synthetic jet impingement cooling”, Int. J. Heat Mass Tran., Vol. 53, No. 5, 2010, pp. 1057-1069.
[4] Valiorgue, P., Persoons, T., McGuinn, A., and Murray, D.B., “Heat transfer mechanisms in an impinging synthetic jet for small jet-to-surface 27
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spacing”, Experimental Thermal and Fluid Science, Vol. 33, No. 4, 2009, pp. 597-603.
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[5] Greco, C.S., Ianiro, A. and Cardone, G., “Time and phase average heat transfer in single and twin circular synthetic impinging air jets”, International Journal of Heat and Mass Transfer, Vol. 73, 2014, pp. 776-788.
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[6] Smith, B.L., and Glezer, A., “The formation end evolution of synthetic jets”, Phys.Fluids, Vol. 10, No. 9, 1998, pp. 2281-2297.
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[7] Holman, R., Utturkar, Y., Mittal, R., Smith, B.L., and Cattafesta, L., “Formation criterion for synthetic jets”, AIAA J., Vol. 43, No. 10, 2005, pp. 2110-2116.
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[8] Cater, J.E., and Soria, J., “The evolution of round zero-net-mass-flux jets”, Journal of Fluid Mechanics, Vol. 472, 2002, pp. 167-200.
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[9] Tesar, V. and Zhong, S., “Efficiency of generating the synthetic jets”, Transactions of Aeronautical and Astronautical Society of the Republic of China, Vol. 35, No. 1, 2003, pp. 45-53.
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[10] Crowther, W. J., and L. T. Gomes. “An evaluation of the mass and power scaling of synthetic jet actuator flow control technology for civil transport aircraft applications”, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 222.5 (2008), pp. 357-372. [11] Li, R., Sharma, R., and Arik, M. “Energy conversion efficiency of synthetic jets”, ASME 2011 Pacific Rim Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Systems. American Society of Mechanical Engineers, 2011. [12] de Luca, L., Girfoglio, M., and Coppola, G., “Modeling and experimental validation of the frequency response of synthetic jet actuators”, AIAA J., Vol. 52, No. 8, 2014, pp. 1733-1748. doi: 10.2514/1.J052674 [13] Sharma, R., “Fluid Dynamics Based Analytical Model for Synthetic Jet Actuation”, AIAA Journal, Vol. 45, No. 8, 2007, pp. 1841-1847. [14] de Luca, L., Girfoglio, M., Chiatto, M., and Coppola, G., “Characterization of synthetic jet resonant cavities”, Springer International Publishing 28
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Switzerland 2015, E. Ciappi et al. (eds.), Flinovia - Flow Induced Noise and Vibration Issues and Aspects, pp. 101-118, Book ISBN: 978-3-31909712-1. doi: 10.1007/978-3-319-09713-8_6
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[15] Luo, Z.-b., Xia, Z.-x, and Liu, B., “New generation of synthetic jet actuators”, AIAA Journal, Vol. 44, No. 10, 2006, pp. 2418-2419. doi:10.2514/1.20747
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Supplementary Material
List of Figures
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Modular structure of the actuator (de Luca et al. [12]) . . . Sketch of the experimental setup. . . . . . . . . . . . . . . . Kinetic power at orifice exit versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (31) . . . . . . . . . . . . . . . . . . . . . . . Electrodynamic power versus voltage for aluminum 2 device. Circle symbols are computed data, continuous line represents eq. (32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured Joule power versus voltage for aluminum 2 device. Circle symbols are experimental data, continuous line represents eq. (33) . . . . . . . . . . . . . . . . . . . . . . . . . . Electrodynamic efficiency versus voltage for aluminum 2 device. Circle symbols are measured data, continuous line represents eq. (34) . . . . . . . . . . . . . . . . . . . . . . . . . Frequency response of exit flow peak velocity for the brass actuator at Va = 35V . The straight line refers to eq. (35) . . Frequency response of exit flow peak velocity for the aluminum 1 actuator at Va = 35V . The straight line refers to eq. (35) . Frequency response of exit flow peak velocity for the aluminum 2 actuator at Va = 35V . The straight line refers to eq. (35) .
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*Highlights (for review)
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• The lumped parameters modelling is based on the relevant energy equations.
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• The kinetic efficiency depends on structural dissipations and fluidic dissipations at exit orifice.
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• The efficiency at orifice is correlated to the efficiency in the external jet field.
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• The coupling degree of membrane and acoustic oscillators strongly influences the kinetic efficiency of single and two cavities configurations.
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*Photographs and biographies of all authors
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Michele Girfoglio, born in 1985, graduated in Astronautic and Aerospace Engineering in 2011 summa cum laude. He is going to get the PhD degree at the University of Naples Federico II. Current fields of interest include the characterization of synthetic jet actuators and the analysis of the unsteady global dynamics of liquid sheet flows.
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Carlo Salvatore Greco achieved summa cum laude his degree in Aerospace and Astronautical Engineering at the University of Naples Federico II in 2011. He currently is a PhD student at the University of Naples Federico II, Experimental Thermo-Fluid Dynamics sector. His research mainly focuses on fluid-dynamics and heat transfer problems, with particular interest in synthetic jets characterization.
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Matteo Chiatto, born in 1988, is presently a PhD candidate of University of Naples Federico II. He got the master degree in 2013 summa cum laude and the Research Master in Fluid Dynamics at von Karman Institute in 2014. Current field of interest is the flow control by means of piezo-electric and plasma actuators.
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Luigi de Luca graduated in Aeronautical Engineering in 1977 summa cum laude. Presently he is full professor of thermo-fluid-dynamics and fluid dynamics stability at University of Naples Federico II. He is active in the fields of numerical and experimental thermo-fluid-dynamics.
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