Numerical Simulation of Electrospray Thruster Extraction · 2019-12-19 · Numerical Simulation of Electrospray Thruster Extraction IEPC-2019-565 Presented at the 36th International

The 36th International Electric Propulsion Conference, University of Vienna, Austria September 15-20, 2019

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Numerical Simulation of Electrospray Thruster Extraction

IEPC-2019-565

Presented at the 36th International Electric Propulsion Conference University of Vienna • Vienna, Austria

September 15-20, 2019

Henry Huh1, Richard E. Wirz2 University of California, Los Angeles, CA, 90095 USA

The UCLA Plasma & Space Propulsion Laboratory developed an Electrohydrodynamics (EHD) OpenFOAM solver (PSPL-EHD) based on the electrowetting model to understand the physics of cone-jet and droplet formation of electrospray thrusters. The code requires no simplification of charge conservation laws and includes improvements to handle the uniquely challenging properties of highly conductive propellants. Gauss’s law and the charge conservation equation are solved alongside the Navier-Stokes equation for flow field and charge distribution. The full charge conservation equation is applied in order to consider the limits of both leaky dielectrics and perfect conductors in this work. The model captures unsteady and time-dependent behavior, which are crucial for investigating the lifetime and performance of electrospray thrusters. To verify and validate the model, simulations were performed for a heptane electrospray at an applied voltage of 5.5 kV between an emitter and extractor plate and a liquid flow rate at 10 cm3/s. The droplet diameters are calculated in the range of 20-27 µm, in reasonable agreement with the 10 µm average size reported in experimental results. Also, simulations where the applied voltage is increased steepen the shape of the cone-jet and shrink the droplet diameter, consistent with trends from electrospray theory and observations.

Nomenclature

a = Taylor semi-cone angle 𝛼 = volume fraction 𝛼"#$ = air volume fraction 𝛼%#& = liquid volume fraction 𝑑( = non-dimensional droplet diameter 𝐸 = electric field ε = electrical permittivity ε( = electrical permittivity in vacuum 𝐹, = electrostatic force 𝐹-. = surface tension force 𝛾 = surface tension coefficient 𝐽 = current density 𝜅 = interface curvature 𝑛 = unit normal vector

𝜇 = dynamic viscosity 𝑃 = pressure 𝑄6#7 = droplet diameter 𝜌 = density 𝜌: = volumetric charge density R = local droplet radius 𝑅<,> = principal radii of curvature 𝜎 = electrical conductivity 𝜎%#& = liquid electrical conductivity 𝜎"#$ = air electrical conductivity t = time 𝑢 = velocity

1 Ph.D. student, Department of Mechanical & Aerospace Engineering, [email protected] 2 Professor, Department of Mechanical & Aerospace Engineering, [email protected]


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I. Introduction HE UCLA Plasma & Space Propulsion Laboratory (PSPL) [1] is supporting the LISA Colloid Microthruster Technology (CMT) development plan via experiments and modeling [2]. To support this effort PSPL has

developed an Electrohydrodynamic (EHD) OpenFOAM solver (PSPL-EHD) based on the electrowetting model [3] to understand the physics of cone-jet and droplet formation of an electrospray within extraction region defined in Figure 1. The code requires no simplification of charge conservation laws and includes improvements to handle the uniquely challenging properties of highly-conductive propellants. Gauss’s law and the charge conservation equation are solved with the Navier-Stokes equation for flow-field and charge distribution. In response to a voltage applied between the emitter and extractor plate, charges begin to accumulate at the liquid-vacuum interface. Due to the concentration of charge, Coulombic forces result in a cone of liquid. A jet is generated from the meniscus when the electrostatic stress exceeds the surface tension. For an ionic liquid of higher conductivity, the onset voltage required for droplet emission is lower than that for lower conductivity fluids.

Figure 1: Computational domains of an Electrospray Thruster

Extraction modeling in electrospray devices has been widely investigated. Taylor, Melcher, and Saville first developed the idea of the leaky dielectric model, where tangential stress plays a more important role than in the perfect dielectric and perfect-conductive models [4] [5] [6]. Lastow et al. used commercial CFD software (ANSYS-CFX) to simulate the leaky dielectric model using Laplace’s equation [7]. Lopez-Herrera et al. developed an Electrohydrodynamic solver with the Volume of Fluid method in the software Gerris [8]. In the present study, the open source CFD tool, OpenFOAM, is used. There are several advantages of OpenFOAM over other commercial software for electrospray modeling. Due to the complexity of the two-phase fluid flow coupled with electrostatic effects, it is difficult to model the physics in commercial software. Access to the source code in OpenFOAM enables direct implementation of the governing equations best fit for EHD modeling of electrospray emitters.

The perfect dielectric and the conductive models have no tangential electric stress acting on the liquid interface, so cone-jet formation and droplet breakup were not observed. The leaky dielectric model was only able to capture cone-jet formation, but not droplet breakup. The presented EHD code uses the full charge conservation equation with no other simplifications, enabling capture of not only cone-jet formation but also droplet breakup. This is made possible due to the consideration of both the conduction and convection of charge. Furthermore, this unconstrained model has the capability to capture both instabilities and unsteady behavior during extraction.

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II. Governing Equations

1. Fluid flow The simulation of fluid flow is governed by the incompressible EHD continuity and momentum equations

𝛻 ∙ 𝑢 = 0 (4)

𝜌 EFEG+ 𝑢 ∙ 𝛻𝑢 = −𝛻𝑃 + 𝜇𝛻>𝑢 + 𝜌𝑔 + 𝐹: + 𝐹-. (5)

where the electrostatic force 𝐹:, surface tension force 𝐹-. is added to Navier-Stokes equation for density 𝜌, pressure 𝑃, dynamic viscosity 𝜇, and velocity 𝑢. The surface tension force 𝐹-. is determined to be a volumetric force by the continuum surface force (CSF) model [9]

𝐹-. = 𝛾𝜅𝑛 = 𝛾 −𝛻 ∙ 𝑛 𝑛 (6)

for surface tension coefficient γ, interface curvature κ, and unit normal vector 𝑛. EK

EG+ 𝛻 ∙ 𝑢𝛼 = 0 (7)

The model uses a Volume-Of-Fluid(VOF) method to capture the interface between liquid and vacuum by transport

equation in Eq. (7) where 𝛼 is volume fraction, and t is time.

2. Electrostatics The model is governed by electrohydrodynamics as magnetic phenomena are negligible. Therefore, Maxwell’s

equations are reduced to the electrostatic equation in Eq. (8) and Gauss’s law for Eq. (9) where 𝐸 is electric field, and 𝜌: is volumetric charge density.

𝛻×𝐸 = 0 (8)

𝛻 ∙ 𝜀𝐸 = 𝜌: (9)

Under electrostatic conditions, and assuming homogeneous conductivity σ and permittivity ε, ohmic charge

conduction and charge convection are summed to yield current density (𝐽 = 𝜎𝐸 + 𝜌:𝑢). The charge conservation equation is reduced to

ENO

EG+ 𝛻 ∙ 𝜌:𝑢 = −𝛻 ∙ 𝜎𝐸 (10)

Gauss’s law and the charge conservation equation under electrostatic assumptions calculate charge density 𝜌: and

the electric field 𝐸for the region. Electrostatic force 𝐹: is represented by the summation of Coulombic and polarization force by

𝐹: = 𝜌:𝐸 − P

Q𝐸>𝛻𝜀 (11)

The computational flow diagram for the Plasma & Space Propulsion Laboratory’s Electrohydrodynamic (PSPL-EHD) model and a phenomenological diagram of forces acting during extraction are depicted in Figure 2. Micro/nano-sized droplets are generated in the extraction region due to the forces of Figure 2. In order to capture the interface of a Taylor cone and the jet, the Volume-of-Fluid method is employed for two-phase flow simulations.


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Figure 2: (Left) Forces acting on the fluid interface during extraction (Right) Flow chart of governing equations in source code

III. Results The computational domain was defined as shown in Figure 3 and simulated with previously stated operating

conditions. The domain utilized a 2-D axisymmetric simulation to reduce total computational time. In Figure 3, the charge density field shows charge concentration at the fluid interface where the electrostatic force is applied due to the charge. The electric field concentration shown per Figure 3 on the droplet and meniscus surface explains the smaller size and faster speed of the droplets [10].

Figure 3: Computational domain with 61,850 mesh cells and representative results for heptane at the following conditions: 5.65×10-<( 𝑚X 𝑠 flow rate, 8 kV voltage (permittivity ε = 1.762×10-<< 𝐹 𝑚, density ρ = 684𝑘𝑔 𝑚X, surface tension coefficient γ = 0.021 N/m)


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Electrospray simulation results from the PSPL-EHD model are shown in Figure 4 for extraction voltages of 4 kV, 8 kV, and 12 kV. When the extraction voltage increases, the cone develops a sharper tip resulting in smaller droplets. At 4 kV, the applied voltage is not enough to create a steady cone-jet, but shows pulsation of the droplets (pulsating mode) [11]. At 8 kV and 12 kV, steady cone-jet emission with droplet generation was observed. Note that due to hydrodynamic effects, a satellite droplet was formed. The high charge-to-mass ratio of the satellite droplet in the transition region can cause off-axis movement due to the clustering effect in the Interaction Region, thus generating a plume [12]. At 12 kV, the electric field normal to the electrode made the cone angle steeper. As the voltage increases, droplet size and the length of the jet progressively decrease in a power law inversely relating volume charge density to cone-jet length [10].

𝑑( = (𝛾𝜀(> 𝜌𝜎>)</X (12) Based on the EHD model and the low conductivity simulation results, UCLA has also conducted high conductivity

simulations with one-tenth of EMI-Im’s conductivity with the properties and operating conditions given in Figure 3 (with 𝜎< = 0.88×10-1 S/m, 𝜎abc-cd = 0.88S/m). Due to the higher conductivity, the simulation shows higher charge density and electric field strength generating smaller droplet diameters and a shorter jet length consistent with the power law [10]. The simulation uses 273,580 mesh cells, the height of each cell near the axis being 80 nm. The result does not show any droplet breakup since the cells around the jet are not small enough to capture droplets. According to Gañán-Calvo et al., non-dimensional droplet diameter validation with experimental results is derived as Eq. (12). Using the properties of the EMI-Im from Gamero-Castaño et al., the droplet diameter is around 8 nm and the jet diameter is twice that of the droplets [13].

Figure 4: (Left) Simulation results for cone-jet formation of heptane at extraction voltages of 4 kV, 8 kV and 12 kV. (Right) Simulation results for higher electrical conductivity (a) Volume fraction (b) Electric field (σ = 0.88×10-< S m⁄ , electrical permittivity ε = 8.85×10-<< F m⁄ , density ρ =1520 kg mX⁄ , surface tension coefficient γ = 0.0349 N/m)


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Figure 5: Recirculation flow for low conductivity liquid near the emitter inlet for different flow rates (a) Qa=3.6´10-10 m3/s, (b) Qb=7.2´10-10 m3/s, (c) Qc=14.4 ´10-10 m3/s

Investigating instability and unsteady behavior in electrospray thrusters is crucial, as unsteady modes are considered somewhat responsible for lifetime-reduction [14] [15]. For low conductivity fluids, the electrostatic tangential stress at the liquid interface means recirculation flow is present near the emitter inlet, affecting jet formation and droplet breakup. In Figure 5, recirculation flow is modeled across time at three different flow rates. With flow rate Qa=3.6´10-10 m3/s, the recirculation region increases in time where the hydrodynamic effects dominate the extraction behavior, and the recirculation propagates back into the capillary. With increased flow rate (Qb=7.2´10-10 m3/s and Qc=14.4 ´10-10 m3/s), both the size and growth rate of the recirculation region decreased. Thus, the recirculation flow region grows faster and larger with lower flow rates such as those found in the high conductivity, EMI-Im propellant thrusters.

According to Gañán-Calvo et al., the minimum flow rate stability limit is derived by dimensional analysis and is validated by experiments [16].

𝑄6#7 = 𝛾𝜀( 𝜌𝜎 (13)

With the properties of the low conductivity propellant in Figure 3, the minimum flow-rate stability limit is analytically calculated as Qmin=6.2´10-10 m3/s and is observed between cases (a) and (b) of Figure 5. It appears in Figure 5a that recirculation dominates the extraction region, inducing instabilities with flow rates lower than the minimum flow rate 𝑄6#7. Instabilities are expected to be amplified when using higher conductivity propellants [17]. Currently, the UCLA Plasma & Space Propulsion Laboratory is developing a PSPL-EHD OpenFOAM solver to observe instabilities in high conductivity regimes.


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Figure 6: Electric field distribution (absolute) in the extraction region under start-up conditions (a) ta=0.1 ms, (b) tb=0.2 ms, (c) tc=0.3 ms, (d) td=0.4 ms, (e) te=0.5 ms, (f) tf=0.6 ms, (g) tg=3.7 ms (steady)


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Figure 7: Electric field (absolute) plotted along 𝑦 = 8µ𝑚 (a) ta=0.1 ms, (b) tb=0.2 ms, (c) tc=0.3 ms, (d) td=0.4 ms, (e) te=0.5 ms, (f) tf=0.6 ms, (g) tg=3.7 ms (steady)


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To investigate the evolution of the electric field strength during the emission process, results involving heptane emitted into vacuum were examined. Figure 6 and Figure 7 show the cone-jet formation (a-d) and droplet extraction (e-g), along with the evolution of electric field strength. To verify the accuracy of the results of figures, the electric field strength normal to the surface of a Taylor cone can be estimated by

𝐸 = 2[𝛾 cos 𝑎/(𝜀(𝑅)]</> (12)

with a surface tension coefficient of 𝛾, radius of R, a Taylor semi-cone angle of 𝑎(49.29°), and permittivity of free space, 𝜀( [18]. In Figure 6, the electric field E at the tip of the jet increases during cone-jet formation through times (a-d). The electric field strength at the tip of the jet is 7.5×10uV/m at 𝑡" = 0.1ms and 1×10xV/m at 𝑡y = 0.4ms. In Figure 7, the electric field strength at the meniscus at steady state (𝑡z = 3.7ms) is 6.8×10uV/mwhere the local radius R is relatively small and thus consistent with the result from Eq. (12) [18]. Due to accumulated charge at the tip of the jet and the strong electric field due to the small local radius, the electrostatic force in Eq. (11) exceeds the local surface tension at the tip and generates droplets.

Figure 8: Pressure distribution in the extraction region under start-up conditions (a) ta=0.1 ms, (b) tb=0.2 ms, (c) tc=0.3 ms, (d) td=0.4 ms, (e) te=0.5 ms, (f) tf=0.6 ms, (g) tg=3.7 ms (steady)


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Figure 8 depicts the pressure distribution during extraction under the start-up conditions. According to the Young-Laplace equation in Eq. (15),

∆𝑃 = 𝛾( <}P+ <

}Q), (15)

the pressure difference over the liquid interface is defined in terms of 𝛾 and the principal radii of curvature R1 and R2 [19]. As can be seen in Figure 9, pressure peaks around 7 kPa at the tip of the jet where jet breakup occurs. The pressure at the tip of the jet is larger than the pressure at the droplet. It is implied that the electrostatic force applied to the tip of the jet due to the accumulation of charge is high enough to overcome the surface tension at the tip, which can also be observed in Fig. 6.

Figure 9: Pressure plot along 𝑦 = 8µ𝑚 at 𝑡 = 3.7𝑚𝑠 (steady)

IV. Conclusion

Electrohydrodynamic governing equations without simplification have been implemented in the PSPL-EHD model. The model can capture steady, unsteady, and time-dependent behavior (e.g., start-up, shutdown, thrust changes) as well as details of both cone-jet formation and droplet breakup during the extraction of fluid in an electrospray thruster. Several simulations were conducted at different voltages with the low conductivity propellant, heptane. By increasing the applied voltage, the steeper slope of the cone was shown due to the high electrostatic force acting tangentially at the interface. This high electrostatic force also generates higher frequencies of droplet generation resulting in smaller droplets. An EMI-Im simulation was also conducted with one-tenth of the EMI-Im conductivity. In the extraction region, electric and pressure fields during startup present where their maxima change over time.

With the PSPL-EHD model, unsteady behaviors were observable, such as satellite droplets and growth of recirculation flow during extraction when the flow rate was below the minimum stability limit. Instabilities increase with higher conductivity and lower flow rate. Therefore, investigating unsteady behaviors and instabilities in electrospray thrusters is crucial [17]. The model is also capable of developing a new emitter design with better performance. PSPL has also developed a novel hybrid emitter design to achieve dual-mode propulsion, which was qualitatively validated with the PSPL-EHD model [20].

Acknowledgments This effort was supported by a grant from NASA’s Jet Propulsion Laboratory, California Institute of Technology

to support the LISA CMT development plan (NASA/JPL Award No. 1580267) and was additionally supported by the Air Force Research Lab at Edwards AFB, CA (AFRL 16-EPA-RQ-09). The author would like to thank Adam L. Collins and Jared Magnusson of UCLA and Sarah H. Lee of USC for useful conversations.


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References [1] R. E. Wirz, A. L. Collins, Z. Chen, C. Huerta, G. Z. Li, S. A. Samples, A. Thuppul, P. L. Wright, N. M.

Uchizono, H. Huh, M. J. Davis and A. Ottaviano, "Electric Propulsion Activities at the UCLA Plasma & Space Propulsion Laboratory," in The 36th International Electric Propulsion Conference, IEPC-2019-547, 2019.

[2] R. E. Wirz, A. L. Collins, A. Thuppul, P. L. Wright, N. M. Uchizono, H. Huh, M. J. Davis, J. K. Ziemer and N. R. Demmons, "Electrospray Thruster Performance and Lifetime Investigation for the LISA Mission," in 2019 Propulsion and Energy Forum, AIAA 2019-3816, Indianapolis, IN, 2019.

[3] I. Roghair, M. Musterd, D. Ende, C. Kleijin, M. Kruetzer and F. Mugele, "A Numerical Technique to Simulate Display Pixels Based on Electrowetting.," Microfluidics and Nanofluidics , vol. 19, pp. 465-482, 2015.

[4] G. I. Taylor and J. R. Melcher, "Electrohydrodynamics: Disintegration of water drops in an electric field," Royal Society, vol. 280, 1964.

[5] J. R. Melcher and G. I. Taylor, "Electrohydrodynamics: A review of the role of interfacial shear stresses.," Annu. Rev. Fluid Mech. 1, pp. 111-146, 1969.

[6] D. A. Saville, "Electrohydrodynamics: The Taylor-Melcher Leaky Dielectric Model," Annu. Rev. Fluid Mech. 29, pp. 27-64, 1997.

[7] O. Lastow and W. Balachandran, "Numerical simulation of electrohydrodynamic(EHD) atomization," Journal of Electrostatics, vol. 64, no. 12, pp. 850-859, 2006.

[8] J. M. Lopez-Herrera, S. Popinet and M. A. Herrada, "A charge-conservative approach for simulating electrohydrodynamic two-phase flows using Volume-Of-Fluid.," Journal of Computational Physics, vol. 230, pp. 1939-1955, 2011.

[9] J. U. Brackbill, D. B. Kothe and C. Zemach, "A Continuum Method for Modeling Surface Tension," Journal of Computational Physics, vol. 100, pp. 335-354, 1992.

[10] A. Suhendi, M. M. Munir, A. B. Suryamas, A. B. Nandiyanto, T. Ogi and K. Okuyama, "Control of cone-jet geometry during electrospray by an electric current," Advanced Powder Technology, pp. 532-536, 2013.

[11] P. L. Wright, A. Thuppul and R. E. Wirz, "Life-Limiting Emission Modes for Electrospray Thrusters," in 54th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA 2018-4726, Cincinnati, OH, 2018.

[12] M. J. Davis, A. L. Collins and R. E. Wirz, "Electrospray Plume Evolution Via Discrete Simulations," in The 36th International Electric Propulsion Conference, IEPC-2019-590, 2019.

[13] M. Gamero-Castaño, "Characterization of the electrosprays of 1-ethyl-3-methylimidazoliumbis (trifluoromethyl sulfonyl) imide in vacuum," Physics of Fluids, 20, p. DOI: 10.1063/1.2899658, 2008.

[14] A. L. Collins, A. Thuppul, P. L. Wright, N. M. Uchizono, H. Huh, M. J. Davis, J. K. Ziemer, N. R. Demmons and R. E. Wirz, "Assessment of grid impingement for electrospray thruster lifetime," in The 36th International Electric Propulsion Conference, IEPC-2019-213, University of Vienna, Austria, 2019.

[15] N. M. Uchizono, A. L. Collins, A. Thuppul, P. L. Wright, D. Q. Eckhardt, J. K. Ziemer and R. E. Wirz, "Electrospray Transient Emission Behavior and Spatial Stability," in The 36th International Electric Propulsion Conference, IEPC-2019-368, University of Vienna, Austria, 2019.

[16] A. M. Gañán-Calvo, J. M. López-Herrera, M. A. Herrada, A. Ramos and J. M. Montanero, "Review on the physics of electrospray: From electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray," Journal of Aerosol Science, pp. 32-56, 2018.

[17] A. M. Gañán-Calvo, "Electro-Flow Focusing: The High-Conductivity Low-Viscosity Limit," Physical Review Letters, p. DOI: 10.1103/PhysRevLett.98.134503, 2007.

[18] M. Gamero-Castaño and J. Fernández De La Mora, "Direct Measurement of Ion Evaporation Kinetics from Electrified Liquid Surfaces," The Journal of Chemical Physics, vol. 113, no. 2, pp. 815-832, 2000.

[19] T. Young, "An essay on the cohesion of fluids," Philosophical Transactions, 1805. [20] P. L. Wright, H. Huh, N. M. Uchizono, A. Thuppul and R. E. Wirz, "A Novel Variable Mode Emitter for

Electrospray Thrusters," in The 36th International Electric Propulsion Conference, IEPC-2019-650, 2019.

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Numerical Simulation of Electrospray Thruster Extraction · 2019-12-19 · Numerical Simulation of Electrospray Thruster Extraction IEPC-2019-565 Presented at the 36th International