Design Methodology for the Micronozzle-Based Electrospray Evaporative Cooling Devices

Thermal management of microelectronics demands higher heat flux removal capabilities due to the rapid increase in component and heat flux densities generated by integrated circuits (ICs). Electrospray evaporative cooling (ESEC) is a potential package-level thermal management solution for the next generation of microelectronics. In this paper, a design methodology is presented using numerical electrostatic field modeling to indirectly design proof-of-concept, micronozzle-based ESEC chambers. The results of the numerical modeling and heat transfer experiments indicate that the potential distribution near the micronozzle tip of the ESEC chamber dominates the heat transfer performance of ESEC cooling devices. The surface charge density at the micronozzle tips has a minor impact on the heat transfer performance. The maximum enhancement ratio of 1.87 was achieved by the 8-nozzle ESEC chamber at the lowest heat flux investigated, indicating that the heat transfer capability of ESEC chambers declines as the heat source density increases. The study demonstrates that increasing the number of micronozzles and decreasing the flow rate per nozzle may not effectively improve the heat transfer performance of ESEC devices.


Introduction
The thermal management of microelectronics faces a critical challenge because of rapid technology advancements, which have led to enormous component densities and heat flux generation, leaving very little physical room for thermal engineers to work.Although conventional rotary fan cooling technology has been widely adopted for industrial users and consumers due to its simplicity for large form factor electronics, it is no longer considered a viable thermal management solution for advanced microelectronics.
Electrospray evaporative cooling (ESEC), which relies on Coulomb forces for energy-efficient fluid atomization, has great potential to precisely control the formation of droplet sizes and droplet distribution, and hence, can be adapted to create a uniform temperature over the surfaces of electronic devices.In the past, the electrospray technology was primarily used for applications in areas, such as: mass spectroscopy [22][23][24], microthrusters [23], and nanofibers [25].The application of electrospray in heat transfer thermal management of electronics is currently limited.Feng and Bryan [26] investigated the heat transfer performance enhancement through the application of electrohydrodynamics (EHD) on traditional impinging liquid jets over a thermal exchange surface by using cooling chambers of different numbers of capillary tubes in an enclosed cooling loop system.Their experimental results show that the application of the potential on the traditional impinging liquid jets can enhance the heat transfer rate of the cooling chamber by approximately 1.7.Better enhancement of the heat transfer rate is also achieved by the chamber with multiple capillary tubes at a lower heat flux condition.Wang and Mamishev [27] have achieved a maximum enhancement ratio (defined in Equation ( 8)) and heat removal ratio of 1.61% and 61%, respectively, with an ESEC device utilizing four micronozzles.The corresponding calculated heat dissipation and convection heat transfer coefficients were 123.19 W/cm 2 and 3.99 W/cm 2 •K, respectively.The investigation also shows that increasing the number of micronozzles and decreasing the flow rate per nozzle is an effective way of improving the heat transfer performance of ESEC chambers.
Deng and Gomez [28] have achieved a heat dissipation of 96 W/cm 2 with a cooling efficiency of 97% by operating a microfabricated multiplexed electrospray system (MES) in cone-jet mode.The MES uses 19 and 37 nozzles with a packing density of 253 nozzles/cm 2 .The results suggest that increasing the number of electrospray nozzles per unit area is feasible for cooling microelectronics in a broad range of applications.
Although the cooling capability of ESEC chambers over a CPU-sized thermal exchange surface has been demonstrated experimentally, a potential methodology to estimate the thermal management characteristics of these devices has not yet been presented.One way to design ESEC chambers is to use multiphysics modeling; however, there are still many aspects of the electrospray processes that have not been thoroughly investigated, and therefore, are not well understood.For example, the physics behind a mechanism triggering transitions between different electrospray modes [29] are not understood as a whole.Therefore, we believe that the evaluation of the thermal management characteristic of ESEC devices through numerical electrostatic field modeling is an appropriate and potential tool for design assistance.
The purpose of this paper is to present a methodology to design the proof-of-concept micronozzle-based ESEC microfluidic chambers.This design methodology uses numerical electrostatic field modeling to indirectly estimate the heat transfer performance of three different ESEC microfluidic chambers.The heat transfer performance achieved by the designed ESEC microfluidic chambers, associated with their numerical electrostatic field strength near the tips of the ESECs' micronozzles, is discussed as well.

Background
In the ESEC system, the working fluid is atomized through an electrospray technique, shown in Figure 1.When voltage is applied between the nozzles and the collecting electrode, charges within the fluid are forced to the fluid meniscus surface of each nozzle.As the applied voltage increases, the electrostatic field strength and the charge density at the surface increases as well.The Coulomb forces acting on the charges in the fluid cause the fluid meniscus to deform into the shape of a Taylor cone [30][31][32].At the critical electrostatic field intensity, the forces on the charged fluid in the Taylor cone overcome the intra-molecular forces of the fluid, such as viscosity, surface tension, and liquid momentum,

Model Geometry
The geometry of the ESEC devices for electrostatic field simulations was designed following the existing microfluidic chamber prototype in our laboratory.The detailed geometry of the ESEC microfluidic chamber is shown in Figure 2. The flow rate per nozzle was relatively low compared to the rate of conventional spray cooling technologies; to achieve considerable heat transfer performance, 1-nozzle, 4-nozzle, and 8-nozzle microfluidic chambers, as shown in Figure 3, were designed.The microfluidic chambers were made of polycarbonate.The inner diameter, the height, and the thickness of the chamber was 20 mm, 20 mm, and 2 mm, respectively.One side of the microfluidic chamber was fitted with a circular copper plate with 1, 4, or 8 microchannels fabricated through each copper plate, in three chambers.Microchannels were connected by stainless steel micronozzles with an inner diameter (D i ) of 210 µm, an outer diameter (D o ) of 410 µm, and a length of 6.98 mm.
According to our previous experimental results [27], a high heat transfer enhancement was achieved at a distance of 7.5 mm between the tips of the micronozzles and the thermal exchange surface; therefore, this distance was adopted as a fixed distance for the numerical modeling and experimental investigation.

Governing Equations
The classic Poisson's equation is derived from the combination of the definition of potential from Gauss's law and the equation of continuity.Under static conditions, the electrical potential, V, is defined by the relationship (1) where E is the electrostatic field.The electrical dis-V   E placement field, D, can be expressed as where ε 0 and P are permittivity of the vacuum and the electrical polarization, respectively.By combining Equation (1) and Equation ( 2), Gauss's law can be modified as Poisson's equation by the constitutive relationship between E and D.
where ρ is space-charge density.For the axisymmetric electrostatics application mode, since the field and geometry are axially symmetric, the electrical potential is constant in the φ direction, implying that the electrostatic field is tangential to the rz-plane.In cylindrical coordinates, when multiplying Equation (3) by r to avoid singularities at r = 0, the equation becomes

Modeling Result Analysis
e surface charge dis- The geometry of the electrostat microfluidic chamber was established in the COMSOL AC/DC module with three different boundary conditions applied to the boundary of the model, including zero charge/symmetry, ground, and specific electrical potential.Figure 4 indicates where the boundary conditions were applied to the components of the model.The zero charge/symmetry boundary cond plied to the exterior boundaries of the model to eliminate the effect of the geometry of the model on the simulation results.In addition, since the positive potential was adopted, the ground boundary condition was applied to the copper plate and the electric potential boundary condition was applied to an ESEC microfluidic chamber.The general physical meaning of these boundary conditions can be found in the user manual for the AC/DC module.

Space Charge Distribution
Figures 5(a), (b), and (c), show th tribution over the tip of one of the micronozzles of the 1-nozzle, 4-nozzle, and the 8-nozzle ESEC chambers, respectively, at the applied potential of 7.0 kV.The results show that the surface charge distribution near the inner diameter of the micronozzle is more uniform, while it is highly non-uniform near the outer diameter of the micronozzle.e outer diameter of the micronozzle tip for the 1-nozzle microfluidic chamber is more uniform than those of the 4-nozzle and 8-nozzle ESEC chambers.The non-uniform distribution of the space charge over the micronozzle tips is mostly a result of the influence of the adjacent micronozzles.The relative geometric orientation among the micronozzles, the micronchannel-based copper plate, and the collection electrode directly under the micronozzles is also another factor influencing the distribution of the surface charge at the tip of the micronozzle of the ESEC chambers.
When an ulti-jet mode, the distribution of the droplet diameters are from submicron meters to several hundred micron meters [32].Observing all of the electrified jets through photography becomes complicate.Figure 6 shows the multi-jet EHD functioning mode achieved by the 4-nozzle chamber at the applied potential of 7.0 kV and total flow rate of 2 cm 3 /hr.Three observable electrified ethyl alcohol jets are formed at the tip of one of the micronozzles of the 4-nozzle ESEC chamber.The number of the electrified jets observed is very close to the number of regions where local surface charge density is relatively higher, as shown in Figure 5(b).
Although the formation of the local arge density regions from the simulation results could explain the formation of the electrified jets of the multi-jet EHD mode, the actual number of the electrified jets from the experiments might not exactly match the simulation results because the model is assumed to be an ideal model.The experimental observation highly depends on the fabrication and integration processes of the components of the ESEC chambers.In addition, a slight deviation from the length of the micronozzles and the inclination angle between the tip of the micronozzles and the collection electrode is also the major factor affecting the  EHD functioning mode near the tip of all the micron zles.To obtain more accurate modeling results regarding the formation of the electrified jets, a complete multiphysics model, including electrostatics, fluid mechanisms, and heat transfer modes, is necessary.
Figure 7 shows the average surface charge at the tip of the micronozzle of three microfluidic chambers.As the applied DC potential is increased, the average surface charge is raised linearly for all three ESEC chambers.Additionally, although the average surface charge of the 1-nozzle and 4-nozzle chambers is almost identical, while that of the 8-nozzle chamber is relatively lower than the others, the total surface charge of the 8-nozzle chamber is the highest, indicating that more charges can be distributed to each nozzle to efficiently electrify the fluid at the tip of the nozzle.Furthermore, higher total surface charge at the tip of the micronozzles means more charge can be carried by the electrified droplets.The charged droplets with higher amounts of charge can be accelerated toward the thermal exchange surface, which induces the convective heat transfer rate near the thermal exchange surface, although the majority of the heat transfer is due to the phase change of the fluid on the thermal exchange surface.

Electrostatic Field S Figure 8(a)
shows the numerical mo distribution of the potential of all three ESEC microfluidic chambers.This distribution of the potential lies along the r axis (defined in Figure 8(b)) from the center of the tip of one of the micronozzles, to the distance 3 mm away from the tip center in the radial direction of the chambers, parallel to the collection electrode.The applied potential is 7.0 kV for all three ESEC microfluidic chambers.The result shows that at a certain distance away from the centerline of the micronozzle, the 8-nozzle ESEC chamber still retains higher potential than the other two chambers.This is attributed to the fact that the effect of the adjacent micronozzles of the 8-nozzle ESEC chambers is stronger than that of the 4-nozzle ESEC chamber.Additionally, although the geometric orient m EC chambers and the respective ESEC chambers differ, Figure 8 indicates that the potential distribution of the 1-nozzle and 4-nozzle ESEC chambers in the horizontal direction is almost identical when the distance is within 1.5 mm.This phenomenon implies that, although there are four micronozzles within the 4-nozzle ESEC chamber, the electrostatic field of the individual micronozzle is almost the same as that of the 1-nozzle chamber.Therefore, the electrostatic field of each of the micronozzles of the 4-nozzle ESEC chamber near the outer diameter of the tip of the micronozzle can be regarded as an isolated unit.
However, when the distance is beyond 1.5 mm, the declining slope of the potential for the 4-nozzle and 8no he m chambers, we modified the analy n is zzle chambers is almost the same, i.e., at a distance of 3.0 mm, shown in Figure 8(a).The decline slope is even larger than that of the 1-nozzle chamber.This is because of the geometric position of the micronozzle to the centerline of the ESEC chamber, as shown in Figure 3.
Therefore, the greater the amount of micronozzles used, the wider the equipotential distribution around t icronozzles.For example, the 5.0 kV potential for the 1-nozzle and 4-nozzle chamber is 0.5 mm away from the centerline of the micronozzle, while the 8-nozzle chamber extends to around 0.8 mm away from the centerline of the micronozzle.The broadened equipotential distribution results in the wider electrospray angle between the centerline of the micronozzle and the tangential direction of the electrified jet at the outer diameter of the tip of the micronozzle when the electrospray is operated in the multi-jet mode.The definition of the electrospray angle is shown in Figure 9.
To estimate the average electrostatic field strength of the ESEC microfluidic tical model describing the electrostatic field strength of a hyperboloid-to-plane EHD configuration [26,33] to fit our electrostatic field strength modeling results.Some physical parameters, such as the distance of 7.5 mm between the tips of the micronozzle and the thermal exchange surface, as well as the diameter of the micronozzles, are fixed values, while the only variable for modeling is the applied potential on three ESEC chambers.
The general form describing the electrostatic field strength of a hyperboloid-to-plane EHD configuratio where r 0 and H are the outer radius o object and the distance between the tip of the hyperbol-f the hyperboloid oid object and the thermal exchange surface, respectively.The constant, A, is determined from experimental results of different EHD configurations [34][35][36].For the ESEC chambers, r 0 is the outer radius of the micronozzle.There are two reasons why we chose r 0 as the key parameter.First, the operation of the ESEC device primary depends on the potential distribution between the tips of the micronozzles and the collecting electrode.The outer diameter of the micronozzle affects the potential distribution between the tips of the micronozzles and the collecting electrode.Second, although the electrostatic field modeling results shown in Figure 8 indicates that there is a potential distribution inside the micronozzle, for practical operating of the ESEC device, the working fluid fills the entire inner part of the micronozzle; therefore, the potential distribution inside the micronozzle should be more uniform in comparison to the potential distribution outside of the micronozzle.
For the hyperboloid-to-plane EHD configuration, A usually ranges from 2 to 2. However, the estimation of vi three ESEC microfluidic chambers at diffe mental Results s Figure 11 shows the experimental apparatus for the heat the electrostatic field strength of the 1-nozzle ESEC chamber using the pre ously determined value of A results in an electrostatic field strength 50 times higher than that generated by the numerical results in this paper because the analytical model is based on a tip radius of the hyperboloid needle much smaller than the outer radius of the micronozzle of the 1-nozzle ESEC chamber.Therefore, we suggest that the constant, A, of the simplified analytical model for the geometry of the micronozzle of the 1-nozzle ESEC chamber is modified as 2.89 × 10 −2 , two orders lower than that of the previously determined values.
Figure 10 also shows the average electrostatic field strength of the rent applied DC potentials.Figure 10 also shows the plot of the electrostatic field strength of the analytical model with the modified constant, A, of 2.89 × 10 −2 .As the applied DC potential is increased, the average electrostatic field strength of the 8-nozzle chamber is always higher than that of the 4-nozzle and the 1-nozzle chambers.The average electrostatic field strength of the 1nozzle and 4-nozzle chambers is almost the same, which can explain the difference of the experimental heat transfer performance of the three chambers discussed in Section 4.  To position the collecting electrode in place and minimize the heat loss from its peripheral surfaces, the electrode was clamped using four plastic screws inside the thermal insulation block, which was made of Teflon.Only the top surface of the electrode was exposed to the surrounding air.Assuming approximately moderate heat flux generated from the heater, the estimated heat loss from the peripheral surfaces and the bottom surface were calculated at less than 5% and 1%, respectively.Four holes were drilled into the sidewalls of both the thermal insulation block and the collecting electrode to position the electrically insulated thermocouples used to measure tempera th collecting electrode and the rest of them were positioned 7 mm deep inside the electrode from the remaining sidewalls.In addition, a ceramic-based thermal compound layer was placed between the collecting electrode and the electrical heater to ensure that no electrical conduction path exists in between, as well as to minimize the interfacial thermal resistance.

Experimental Apparatu
The working fluid, ethyl alcohol, was pumped into the chamber using a syringe pump.The inlet temperature of the working fluid is room temp t flow rates were set at 1, 2, 4, and 8 cm 3 /hr.A micropositioning xyz optical stage was used to control the distance of 7.5 mm from the nozzle tip to the thermal exchange surface.During the experiment, the electrical heater was connected in series with a resistor to an AC power supply.The copper plate in the microfluidic chamber was connected to a high voltage power supply with positive polarity and the collection electrode, which is also the thermal exchange surface, was connected in series with a resistor to ground.
Experiments were conducted in ambient conditions.All temperatures were measured at the steady state condition, which defined that the tem 0.15˚C with respect to an average temperature and lasted for at least 5 minutes.

Operated EHD Modes
According to the investiga al. [37] and Jaworek et al. [29] modes of a single-capillary-to general, can be classified as dripping modes and jet modes.The dripping mode is the mode that only fragments of liquid are electrified directly from the nozzle tip.The jet mode is the mode that liquid is electrified directly into a long fine jet, which can either be stable or move in any regular way.The cone-jet mode and the multi-jet mode are both referred to as the jet mode.
The stable cone-jet mode has the advantage of generating uniform submicron droplets [32,38,39], which has the ability to generate the appropriate droplet's size for op ts in hi cribing the average corresponding coefficient is [41] timal heat transfer.However, the operating range of the potential to achieve the cone-jet mode is narrow and highly influenced by the properties of the working fluid, the quantity of electrospraying nozzles, the arrangement of the nozzles, and so on [39,40].The critical voltage for a given flow rate to change from the cone-jet to the multi-jet mode has not been quantitatively investigated.
Although the droplet control ability in multi-jet mode is poor, the electrified liquid jets can cover larger portions of the thermal exchange surface, which resul gher heat transfer performance.Therefore, in this investigation, all ESEC chambers were operated in the multi-jet EHD mode.

Data Reduction
The general form des convection heat transfer where q″ is the heat flux, and T s is the average temperature of the entire thermal exchan electrode).Since the copper is a ge surface (collecting thermally conductive material and it is thin (4 mm), we assumed that the entire copper (collecting electrode) temperature T s is uniform.T ∞ is the ambient temperature.The heat flux (q″) was assumed uniform over the entire thermal exchange surface, and it is expressed as where I is the current from the heater, V is the applied voltage on the electric heater, and of the thermal exchange surface, which is ion heat transfer co A s is the surface area 900 mm 2 (30 mm × 30 mm), faces the micronozzles.
During the experiments, the heat flux from the heater was kept constant.Therefore, the enhancement ratio (ER) of the average corresponding convect efficient was where 0 h and 1 h er coe are average corresponding convection heat transf fficients calculated at n pray and electrospray conditions, respectively.T 0,s and th tion (Q on-electros-T 0,∞ are e average surface and environmental temperatures during non-electrospray condition, respectively.T 1,s and T 1,∞ are the average surface and environmental temperatures during electrospray condition, respectively. At the steady state condition, the total heat (Q t ) is transferred through the thermal conduction to the thermal exchange surface, and is then transferred by convec conv ), vaporization (Q v ) and thermal radiation (Q rad ) at the thermal exchange surface.Therefore, heat removed by the ESEC device, Q ESEC , was calculated and expressed as where Q l is the heat loss of the thermal stand.In the analysis, according to the design data of stand, the maximum heat loss was assume ws the relationship between the enhancethe mass flow rate of the 8-nozzle ESEC erent applied potentials at a increases.At th the thermal d to be approximately 5% of the total heat flux.Furthermore, according to the experiments, the heat loss due to thermal radiation is less than 0.1%; therefore, Q rad can be neglected in Equation (9).The uncertainty of this investigation mainly comes from the measurement of the surface temperature of the thermal exchange surface and the measurement of the current from the electric heater.

Enhancement Ratio
Figure 13 sho ment ratio and microfluidic chamber at diff heat flux of 4384.40W/m 2 .At the fixed mass flow rate, a higher applied potential results in a higher enhancement ratio.At the fixed potential, increasing the mass flow rate increases the enhancement ratio.The 4-nozzle ESEC microfluidic chamber and the 1-nozzle ESEC microfluidic chamber show the same relationship between the enhancement ratio and the mass flow rate.
Figure 14 shows that at a constant heat flux (4384.40W/m 2 ) and potential of 7.0 kV, the enhancement ratio increases as the mass flow rate of the fluid e same mass flow rate, the highest enhancement ratio was achieved by the 8-nozzle ESEC chamber, which Mass Flow Rate (kg/s) 0.0 500.0x10 - 1.0x10 -6 1.5x10 -6 2.0x10 -6   Enhancement Ratio indicates that the higher the number of micronozzles used, the more e cover larger surface areas of the thermal exchange surface for heat transfer.
At a heat flux of 43843.40W/m 2 , Figure 15 shows the relationship between the enhancement ratio and the average electrostatic field of all ESEC microfluidic chambers at a total flow rate of 8 cm 3 /hr.In general, at the same average electrostatic field, the 8-nozzle ESEC chamber has the highest enhancement ratio, followed by the 1-nozzle ESEC chamber and the 4-nozzle ESEC chamber.
To compare the enhancement ratio achieved by three ESEC chambers at different average electrostatic fields, we use linear curve regression to discuss the relationship between the average electrostatic field and the enhancement ratio of the ESEC chambers.
For the 8-nozzle ESEC chamber, the regression curve is For the 4-nozzle ESEC chamber, the regression curve is 7 1.4625 6.7105 10 For the 1-nozzle ESEC chamber, the regression curve is where E is the average electrostatic field and ER is the enhancement ratio.The regression curve slope for the 1-nozzle chamber and 8-nozzle chamber are of the same order, which indicates that the increase of the average electrostatic field in these two chambers results in the same increasing rate of the enhancement ratio.Furthermore, Figure 15   the same average electrostatic field intensity cannot achieve the same magnitu is cronozzle tip, Figure 8(a), of the ESEC chamber that influences the achievable enhancement ratio.The 4-nozzle ESEC chamber's electrostatic field intensity near the micronozzle tip (within 3 mm) is not highly influenced by adjacent micronozzles and each micronozzle on this ESEC chamber can be considered as an isolated micronozzle.Weak local electrostatic field intensity results in a smaller electrospray angle (Figure 9).Therefore, the electrified liquid jets must cover a small surface area, which reduces the heat transfer performance.In addition, our experimental enhancement ratios are within ± 5.0% of the linearly regressed curve for the ESEC chambers, which indicates that choosing linear curve regression should provide us with enough information regarding the relationship between the average electrostatic field intensity and the corresponding enhancement ratio for the ESEC design.
Figure 16(a) shows enhancement ratios achieved by each ESEC chamber at a potential of 7.0 kV, flow rate of 8 cm 3 /hr, and f e 8-nozzle microfluidic ESEC chamber always achieves the highest enhancement ratio, followed by the 1-nozzle ESEC chamber and the 4-nozzle ESEC chamber.The reason why the highest enhancement ratio was achieved by the 8-nozzle ESEC chamber is because the potential distribution near one of the tips of the micronozzle is higher than that of the other two ESEC chambers (Figure 8(a)).In this operating condition, the electrified jets are forced to eject in a higher electrospray angle, allowing the electrified jets to cover a larger portion of the thermal exchange surface, yielding greater heat transfer performance and leading to a significantly lowered initial surface temperature for the thermal exchange surface.
In addition, for the same ESEC chamber, increasing the heat flux results in a decreased enhancement ratio.Enhancement ratio differences among the three ESEC ch al, highest flow rate of 8 cm 3 /hr, and four di lowest heat flux of 7.8 kV and 4384.40W/m , re ac the ESEC chambers investigated.A ambers become noticeable when the ESEC chambers are operated in lower heat flux conditions.At the lower heat flux, the temperature difference between the ethyl alcohol and the thermal exchange surface is small, resulting in a lower heat transfer rate by conduction from the thermal exchange surface to the ethyl alcohol film accumulated on the thermal exchange surface.The ethyl alcohol film does not absorb enough heat to vaporize on the thermal exchange surface.Furthermore, the amount of coming ethyl alcohol from the ESEC chamber is larger than that of the vaporized ethyl alcohol on the thermal exchange surface.Therefore, the surface temperature of the thermal exchange surface is largely lowered by the net increased amount of ethyl alcohol.The accumulation of the electrified ethyl alcohol decreases as the heat flux becomes higher.This phenomenon occurs at all heat flux conditions.
Figure 16(b) shows the highest enhancement ratio achieved by each ESEC chamber at the highest achievable potenti fferent heat fluxes investigated in this paper.The highest achievable potential occurs before the breakdown between the ESEC chamber and the collection electrode and is therefore different for all three ESEC chambers.The highest achievable potential for the 1-nozzle chamber, 4-nozzle chamber, and 8-nozzle chamber is 7.0 kV, 7.7 kV, and 7.8 kV, respectively.The corresponding average electrostatic field strength for the 1-nozzle chamber, 4-nozzle chamber, and 8-nozzle chamber is 2.74 × 10 5 (V/m), 3.14 × 10 5 (V/m), and 3.48 × 10 5 (V/m), respectively.
The maximum enhancement ratio of 1.87 was achieved by the 8-nozzle chamber at the highest achievable potential and 2 spectively.The same tendency that the maximum enhancement ratio occurs at the lowest heat flux and the highest potential was also presented by Feng et al. [26].
Figure 16(b) shows that for heat flux higher than 5000 W/m 2 , the highest enhancement ratio achieved by the 1-nozzle ESEC chamber is slightly higher than that hieved by the 4-nozzle ESEC chamber.Although the average electrostatic field of the 4-nozzle chamber is higher than that of the 1-nozzle chamber, the potential distribution close to the tip of the micronozzle dominates the enhancement ratio.
Therefore, we conclude that the potential distribution close to the tip of the micronozzles dominates the enhancement ratios of all lthough the universal criterion that correlates the average electrostatic field and the enhancement of the ESEC chambers with different geometries is still not available, the modeling results regarding the potential distribution near the tip of the micronozzles of the ESEC chambers c ment ratio among the ESEC chambers.
To understand the effect that the increasing potential and total mass flow rate have on the enhancement ratio, the potential, total mass flow rate, and en e all normalized according to the smallest values for analysis.Figure 17 shows the effect of the potential increasing rate on the enhancement ratio increasing rate achieved by the three different ESEC microfluidic chambers at a heat flux of 4384.40W/m 2 .The slope of the regression curves among the three ESEC chambers indicates that, at the same potential increasing rate, the 8-nozzle ESEC chamber achieves the fastest increasing rate in the enhancement ratio, followed by the 1-nozzle ESEC chamber and the 4-nozzle ESEC chamber.The same performance behavior among the three ESEC chambers regarding the total mass flow rate increasing in Figure 18.
Additionally, for the same ESEC microfluidic chamber and heat flux, Figure 17 and Figure 18 also indicate that to achieve a higher enhancement ratio increasing rate, increasing the applied potential is more efficient than in f heat energy flows through it in unit time.From Newton's law of cooling, ection heat transfer is ex-creasing the total mass flow rate.

Thermal Resistance
Thermal resistance (TR) is the temperature difference across a structure when a unit o thermal resistance for conv pressed as where h and A are the convective heat transfer coefficient and the surface area of the thermal exchange surface, respectively.Lower thermal resi stance represents a higher  convective heat transfer coefficient in the same thermal exchange area.
At the same heat flux and applied potential, Figure 19 shows that increasing the mass flow rate decreases the average thermal resistance.At the same mass flow rate and heat flux, increasing the applied potential reduces the average thermal resistance.
At a heat flux of 43843.40W/m 2 , Figure 20 shows the relationship between the average thermal resistance and the average electrostatic field of all the ESEC microfluidic chambers at the total flow rate of 8 cm 3 /hr.At the same average electrostatic field, the 8-nozzle ESEC chamber has the lowest average thermal resistance, followed by the 1-nozzle and the 4-nozzle ESEC chambers , respectively.The regression curves regarding the relationship between the average electrostatic field and the average thermal resistance for all three chambers are described next.

4.4.
heat from the thermal exchange surface, the average cooling rate (C R ) was defined and expressed n the average electrostatic field and the enhancement ratio, the slope of the regression curve of all three ESEC chambers investigated in this paper are of the same order, indicating that increase of the ave sults in the same rate of increase of the average thermal res

Average Cooling Rate
To explain how fast different ESEC chambers can transfer as where ΔT is the temperature difference between the starting temperature and the steady state tem ling rate of 7.56 × 10 ˚C/s as the applied potential is 5 kV, while the minimum average cooling ved when the applied po-.8 kV. Figure 22 shows the transient re average electrostatic field and th followed by the 4-nozzle and 8-nozzle chambers.The perature, and Δt is the time required for the chambers to reach the steady state temperature from a defined starting temperature.Figure 21 shows that the 8-nozzle ESEC chamber might not be able to achieve a better average cooling rate at a higher applied potential or higher mass flow rate.For example, at a mass flow rate of 8.77 × 10 −7 kg/s, the 8-nozzle ESEC chamber achieves the maximum average coo −3 rate of 2.00 × 10 −3 ˚C/s is achie tential is 7.8 kV.In addition, the maximum average cooling rate of 8.75 × 10 −3 ˚C/s is achieved by the chamber at the applied potential of 7.0 kV instead of 7 lationship between the e average cooling rate for all three ESEC chambers at a heat flux of 4384.40W/m 2 and the total flow rate of 8 cm 3 /hr for the first five minutes after the ESEC cooling device is started.In general, increase the average electrostatic field results in an increase in the average cooling rate.For all three ESEC chambers, the slopes between the average electrostatic field and the average cooling rate are approximately 5.0 × 10 −8 K/s per average electrostatic field strength.
At the same average electrostatic field, the average cooling rate of the 1-nozzle ESEC chamber is the highest,   At steady state conditions, Figure 23 shows the relationship between the highest cooling rate and the heat fluxes among all three ESEC microfluidic chambers.In general, the highest cooling rate of the three ESEC chambers at different heat fluxes ranges from 8 × 10 −3 K e of the 4-nozzle and the 1-nozzle ESEC ch the pr nozzle of the chambers has an indirect effect on the heat transfer performance of rs, while the potential distribution micronozzle of the ESEC chambers /s to 1.4 × 10 −3 K/s.Furthermore, the results show that the 4-nozzle chamber could deal with a wider range of heat flux while still maintain the highest average cooling rate over the 1-nozzle and 8-nozzle ESEC chambers.
Additionally, although the actual highest cooling rates of the 4-nozzle and the 1-nozzle ESEC chambers are not the same, the average difference of the highest cooling rate between the 4-nozzle and the 1-nozzle ESEC chambers at different heat fluxes is approximately 3 × 10 −3 K/s.The potential distribution close to the tip of the micronozzle of the 1-nozzle and 4-nozzle ESEC chambers is also applicable to point out the tendency of the highest cooling rat ambers at different heat fluxes.

Conclusions
The design of ESEC chambers through direct full multiphysics modeling at the molecular level has not been developed to date and is consequently unavailable for practical design, especially when the size of the devices is at the micro-scale.Therefore, this paper focuses on oof-of-concept design tool of ESEC microfluidic chambers for the thermal management of microelectronics.We have adopted the direct simulation results of the electrostatic field strength to design the ESEC microfluidic chambers and have discussed the impact of the electrostatic field strength difference close to the tips of the designed ESEC chambers' micronozzles on the chambers' heat transfer performance.
The numerical results show that the surface charge distribution at the tip of the micro the ESEC chambe close to the tip of the has great influence on the thermal characteristics of the ESEC cooling devices.The latter indicates that the 8nozzle chamber would have the highest thermal characteristic, followed by the 1-nozzle and 4-nozzle chambers.
According to the experimental results, at the same heat flux and average electrostatic field, the 8-nozzle chamber has the highest enhancement ratio and the lowest average thermal resistance among all ESEC chambers.This is attributed to the effect of the potential distribution close to the tip of the micronozzles of the ESEC chamber.The maximum enhancement ratio of 1.87 was achieved by the 8-nozzle ESEC microfluidic chamber at the maximum  flow rate investigated and the heat flux of 4384.40W/m 2 .Finally, the results indicate that increasing the number of micronozzles on ESEC chambers may not be an effective way of improving the heat transfer performance of an ESEC cooling device.The enhancement ratio and the average thermal resistance of the 1-nozzle chamber are even better than those of the 4-nozzle chambers.The potential distribution near the tip of the micronozzle of ESEC chambers dominates the difference of the heat transfer performance of ESEC chambers.
Although the experimental results of the highest enhancement ratio and the lowest thermal resistance shows the model-predicted performance difference, the average and the highest cooling rate is different.However, the difference of the highest cooling rate between the 4-noz erical electrostatic field modeling result is still applicall.
zle and the 1-nozzle ESEC chambers at different heat fluxes is almost the same, demonstrating that the num ble to pre-estimate the heat transfer performance difference among different ESEC chambers.Further investigation and analysis to explore the achievable cooling rate among the ESEC devices will be necessary.
Future work will focus on applications of electrostaticfield-distribution-optimized ESEC chambers on the thermal management of microelectronics, where high heat flux thermal management solutions are necessary.Power consumption and the coefficient of performance of the ESEC chambers will be investigated as we

Figure 1 .
Figure 1.Schematic diagram of the fluid atomization and heat transfer mechanism of the ESEC device using an ESEC microfluidic chamber.and a jet of charged liquid is then sprayed from the tip of the cone.The break-up of the charged fluid expelled from the tip of the fluid cone repel each other, generating fine aerosol droplets.Those charged droplets are then accelerated by the electrostatic force toward the collecting electrode surface.The collecting electrode, as shown in Figure 1, doubles as the thermal exchange surface.As a result, two-phase heat transfer occurs at the thermal exchange surface, dissipating large amounts of heat because of the droplet's phase change from liquid to vapor.

Figure 2 .
Figure 2. Schematic diagram showing the geometry and the components of the ESEC microfluidic chamber.

Figure 3 .
Figure 3.The bottom view and arrangement of the micronozzles of the three different kinds of ESEC microfluidic chambers.The spacing of the 4-nozzle chamber and 8-nozzle chamber is 5 mm between the centerline of the chamber and the centerline of one of micronozzles.

Figure 5 .
Figure 5. Simulation results he surface charge distribu-of t tion at the tip of one of the micronozzles of the 1-nozzle, 4-nozzle, and 8-nozzle microfluidic chambers at the potential of 7.0 kV.(a) 1-nozzle microfluidic chamber; (b) 4-nozzle microfluidic chamber; (c) 8-nozzle microfluidic chamber.

Figure 6 .
Figure 6.Multi-jet EHD mode of the 4-noz ESEC microfluidic chamber operated at the potential of 7.0 kV and the

Figure 7 .Figure 8 .
Figure 7. Simulation results showing the amount of the average surface charge at the tip of the micronozzles of each ESEC microfluidic chamber at different DC potential.

Figure 9 .Figure 10 .
Figure 9. Schematic diagram showing the definition of the electrospray angle in the multi-jet EHD mode.

Figure 11 .
Figure 11.The experimental apparatus for the heat transfer enhancement measurement of the ESEC devices different ESEC microfluidic chambers.

Figure 12 .
Figure 12.Schematic diagram of the testing platform for heat transfer enhancement measurement of ESEC devices.The platform consists of an electrical and thermal i ula-

Figure 13 . 2 .Figure 14 .
Figure 13.The relationship between the mass flow rate and the enhancement ratio of the 8-nozzle ESEC chamber at different potentials.The heat flux is 4384.40W/m 2 .

Figure 15 .
Figure 15.The relationship between the average electrostatic field and the enhancement ratio among three different ESEC chambers at the flow rate of 8 cm 3 /hr and th

Figure 16 .
Figure 16.Enhancement ratios achieved by the three chambers at different heat fluxes.The highest enhancemen ratios occur at the flow rate cm 3 /hr.

Figure 18 .
Figure 18.The enhancement ratio increasing rate due to the increased total mass flow rate among the three ESEC microfluidic chambers.The heat flux is 4384.40W/m 2 .

Figure 19 .Figure 20 . 6 10
Figure 19.The relationship between the mass flow rate and the average thermal resistance of the 4-nozzle ESEC chamer at different potentials.The heat flux is 4384.40W/m 2 .b

Figure 21 .
Figure 21.The relationship between the mass flow rate and the average cooling rate of the 8-nozzle ESEC chamber at different potentials.The heat flux is 4384.40W/m 2 .

Figure 22 .
Figure 22.The relationship between the average electro

Figure 23 .
Figure 23.Highest cooling rate achieved by the three ESEC chambers at different heat fluxes.
also indicates that ESEC chambers with