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Heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets
International Journal of Heat and Mass Transfer 149 (2020) 119207
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Heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets Onur Ozkan, Vaibhav Bahadur∗ Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA
a r t i c l e
i n f o
Article history: Received 6 July 2019 Revised 20 September 2019 Accepted 9 December 2019
Keywords: Leidenfrost state suppression Electrowetting Boiling Heat transfer Droplet
a b s t r a c t A liquid drop levitates on its own vapor on a sufficiently hot surface (the Leidenfrost effect). Application of an electrical potential difference across the vapor gap suppresses the Leidenfrost state via electrostatic attraction of liquid towards the surface. The resultant transient solid–liquid contacts significantly reduce the thermal resistance to heat transfer across the insulating vapor gap. This work is an indepth study on the heat transfer enhancement accompanying electrostatic suppression of the Leidenfrost state. A purely analytical model is developed to predict the heat transfer enhancement based on the transient wetting dynamics observed during suppression. This model is based on semi-infinite conduction heat transfer coupled with a physics-based prediction of the voltage-dependent wetted fraction. The model is strongly validated by comparison with inhouse experiments and related experimental data from literature. These experiments measure the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state for three fluids, at various applied voltages and superheats. The experimentally-validated model predicts that heat transfer enhancement scales as the square of the wetted fraction. Results show a 4X enhancement in heat transfer by electrostatic suppression of the Leidenfrost state of water droplets; higher enhancements are measured for organic solvents. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Nucleate boiling heat transfer is characterized by high heat flux dissipation capacity, resulting from the high latent heat of vaporization and bubble-related convection currents at the surface [1–3]. However, heat transfer reduces drastically above the critical heat flux (CHF) limit owing to the formation of continuous pockets of thermally insulating vapor on the surface [1–3]. At the Leidenfrost temperature, the surface is completely blanketed by a vapor film; this mode of boiling is known as film boiling. The Leidenfrost phenomena has been studied extensively using pool boiling and droplet-based configurations [4–6]. Many recent studies have explored surface engineering as a tool to elevate the Leidenfrost temperature and CHF [7–11]. This study deals with heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. The use of an externally applied electric field to suppress/eliminate the Leidenfrost state has been studied in pool boiling [12–15] and more recently, in droplet-based configurations [16–26]. These studies have analyzed electrostatic suppression of the Leidenfrost state for electrically insulating (oils, refrigerants) and electrically conduct∗
Corresponding author. E-mail address: [email protected] (V. Bahadur).
https://doi.org/10.1016/j.ijheatmasstransfer.2019.119207 0017-9310/© 2019 Elsevier Ltd. All rights reserved.
ing (water, organic solvents) fluids. For fluids which are perfect electrical insulators, an electric field exists inside the liquid volume (in addition to the vapor layer); the liquid is not equipotential [27]. Conversely, for fluids with a finite (but small) electrical conductivity, the applied electrical potential difference is entirely expressed across the vapor gap, and the liquid can be considered equipotential [28]. Moreover, the vapor layer is typically very thin (O (100 μm)) during film boiling. Therefore, the applied voltage results in a high electric field within the vapor layer for electrically conducting fluids as compared to insulating fluids. Consequently, the voltage requirement for electrostatic suppression is an order of magnitude smaller for electrically conducting fluids, as compared to insulating fluids [28]. The above discussion highlights the differences in the nature of electrostatic suppression of electrically conducting and insulating fluids. It is reasonable to expect that the heat transfer enhancement will also show differences. A literature review indicates that there is a significantly greater number of studies on electric fieldinduced heat transfer enhancement of electrically insulating [27– 33] fluids, as compared to conducting fluids. The most detailed studies with conducting fluids are in fact from the 1960 s [12–14]. These studies were conducted with water and isopropanol (IPA) and involved pool boiling measurements on a horizontal tube. The heat flux was seen to increase with the electrical conductivity of
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Nomenclature A Bi∗ C F g K k h0 h00 hfg
Footprint area of droplet Modified Biot number Correlation constant Force Gravitational acceleration Dimensionless interface temperature Thermal conductivity Vapor gap thickness Initial vapor gap thickness Latent heat of vaporization
qtr R T tc V X
Transient heat flux during contact Droplet radius Temperature Duration of contact Voltage Wetted fraction
Greek Symbols α Thermal diffusivity q s Heat flux enhancement ɛ Permittivity μ Dynamic viscosity ρ Density σ Surface tension τ Characteristic period of Taylor wave; period of oscillations Subscripts 0 Non-contact region (vapor gap thickness) 00 Initial (vapor gap thickness) avg Average (vapor gap thickness) el Electrostatic l Liquid s Solid or surface sat Saturation th Threshold up Upward v Vapor w Wall
the fluid for the same voltage and superheat values. Other pool boiling-based studies for conducting fluids have derived the most unstable wavelength and the Leidenfrost point heat flux based on linear stability analysis [28]. The minimum heat flux during film boiling was calculated with the knowledge of the most unstable wavelength and its frequency (from linear stability analysis) [34,35]. The minimum heat flux (Leidenfrost point) under electrostatic suppression was estimated by a simple modification of the most unstable wavelength and frequency under electrostatic forces for insulating and conducting liquids [28–31]. Most recent studies on electrostatic suppression of the Leidenfrost state involve droplet-based experiments [15–26]. Celestini and Kirstetter [20] conducted interferometric measurements to quantify variations in the vapor layer thickness below the threshold voltage (for electrostatic suppression). The present group has conducted extensive recent studies on various aspects of electrostatic suppression of the Leidenfrost state (for conducting fluids), which include: demonstration of suppression at temperatures exceeding 500 °C [22]; accelerated and tunable cooling during quenching of metals [15]; linear instability analysis associated with electrostatic suppression [23]; influence of AC frequency on electrostatic suppression [25]; electrostatic suppression on liquid substrates [24]; acoustic detection of electrostatic suppression [21];
and impedance measurements-based quantification of wetted fractions associated with electrostatic suppression [36]. A fundamental study on the heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets does not exist, and is the focus of this work. Most previous heat transfer studies have used pool boiling configurations. Pool boiling models cannot be used for droplets, since these models involve bubble-related parameters (size, distance between bubbles, detachment frequency) which are not applicable in a droplet configuration. The nature of any heat transfer model or correlation is thus expected to be fundamentally different for droplets as compared to pool boiling configurations. The only reported heat transfer measurements [16,19] associated with electrostatic suppression of Leidenfrost droplets involved ethanol and R113 droplets. In fact, data from Ref. [19] is used in this study to benchmark the model developed in this work. The present work is an indepth study (combining modeling and experiments) on the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. A fundamentalsbased analytical model is developed to predict the heat transfer enhancement. This model is based on semi-infinite conduction-based estimates of heat transfer associated with transient solid–liquid contacts coupled with another fundamentals-based prediction of the voltage-dependent wetted fraction. The model is validated via comparison with our own experiments as well as with experimental data provided in Ref. [19]. Overall, three fluids are analyzed (water, isopropanol and ethanol). It is noted that this model matches the experimental data very well, and involves only one empirical parameter. 2. Modeling heat transfer enhancement associated with electrostatic suppression An electric field in the vapor gap leads to periodic (both spatial and temporal) solid–liquid contact, with the wetted fraction dependent on the surface superheat and the applied voltage [36]. The applied voltage must be larger than a threshold value to initiate solid–liquid contact. The value of the threshold voltage increases with superheat due of faster evaporation and larger vapor layer thickness [22,25,36]. The wetted fraction increases with the applied voltage (for voltages higher than the threshold voltage) at a given superheat. Similarly, smaller wetted fractions result when the superheat is increased at a constant applied voltage [36]. It is noted that this mode of boiling is not directly analogous to classical film boiling, which involves a continuous vapor film as opposed to periodic solid–liquid contact. This boiling mode also cannot be considered analogous to nucleate boiling, since the small duration of transient contacts is not sufficiently long for bubble nucleation to occur. The boiling mode in this work most resembles transition boiling which is observed when the surface temperature is lowered below the Leidenfrost point and the heat transfer is not sufficient to maintain a vapor film. It must be cautioned that ‘film boiling under electrowetting’ is fundamentally different from classical transition boiling since the wetted fraction depends on the electric field and not solely on the surface superheat. With this background, the heat transfer model developed in this work is based on an approach utilized by Ramilison and Lienhard [37] to model transition boiling. Another analytical sub-model to estimate the voltage-dependent wetted fraction feeds into this heat transfer model. 2.1. Modeling heat transfer Fig. 1 shows a high magnification snapshot during electrostatic suppression (100 V) of a saturated deionized (DI) water Leidenfrost droplet on a 250 °C surface. The diameter of the droplet is approximately 6 mm. The vapor layer and two solid–liquid contact
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3
Fig. 1. Solid–liquid contact during electrostatic suppression of a Leidenfrost droplet on a 250 °C surface. The liquid is saturated deionized water.
is noted that each contact area shown in Fig. 2 is transient. Contacts are modeled as oscillations of the vapor-liquid interface (associated with the Taylor wave) that result in wetting for a fraction of their oscillation periods. Therefore, the surface heat flux (qs ) will be proportional to qtr as per:
t qs ∼ qtr (X ) c τ
(3)
where X is the wetted fraction (fraction of footprint area of droplet which is in contact with liquid) and tc is the contact time. The period of oscillations in the absence of an electric field (τ ) is assumed to be equal to the characteristic period of the associated Taylor wave and can be estimated as [37]: Fig. 2. Modeling electrostatic suppression: Schematic view from bottom showing the solid–liquid contact areas. It is noted that the arrangement of contact points does not have to be regular.
areas are clearly seen. The heat transfer enhancement associated with suppression can be attributed to the low thermal resistance pathways enabled by these solid–liquid contacts. The actual distribution of these contact areas has not been characterized in any study to date. Fig. 2 shows a schematic visualization of a 2D array of periodic contact areas at a particular time instant. The location of these contact areas will change, since these contacts are transient. It is noted that the presently developed model does not need a regular arrangement of contact areas. The primary parameter in the model which captures these contacts is the wetted fraction, which is modeled in Section 2.2. It is noted that the present model is valid for any other arrangement of contact areas as long as the heat transfer associated with these areas is not influenced by the surrounding contact areas. The heat flux enhancement (qs ) can be calculated if 1) the transient heat flux qtr during contact, 2) the duration of the contact (tc ), and 3) the area of each contact are known. Since each contact area is small compared to the total surface area and tc is short (order of milliseconds), the heat transfer during each contact is modeled as the heat transfer between two semi-infinite media brought into sudden contact. The transient heat transfer at each contact equals the cooling of the semi-infinite solid [37] as:
qtr =
ks (Tsat − Tcontact )
(αs τ )1/2
=
ks (Tw − Tsat )
(αs τ )1/2
K
(1)
where α s is the thermal diffusivity of the solid, τ is the temporal period of each contact and ks is the thermal conductivity of the solid. (α s τ )1/2 is thus the characteristic length of the solid. K is the dimensionless interface temperature which depends on the thermal conductivity and diffusivity of both media as [37]: kf
K≡
αl1/2 kl
αl1/2
+
ks
αs1/2
=
Tw − Tcontact Tw − Tsat
(2)
The transient heat flux in Eq. (1) is the heat flux across the contact area. However, the wetted fraction is smaller than 1 and the contact time is smaller than the period of each oscillation. It
τ=
σ
1/4 (4)
g3 (ρl − ρg )
where σ is the surface tension; g is the gravitational acceleration and ρ l and ρ g are the densities of the liquid and the vapor, respectively. The period of oscillations with electric fields can be calculated using Eqn. 4 and Ref [29]. During each contact event, the solid–liquid contact starts from a point, and then increases up to the maximum wetted fraction. Subsequently, the wetted area in the contact region decreases due to vapor pressure buildup, with the liquid eventually detaching from the surface. In this study, it is assumed that the dimensionless contact time equals the wetted fraction since increasing the contact area would take longer. It is noted that a similar assumption was made in another study [37] which modeled the heat transfer in transition boiling. This implies:
t c
τ
=X
(5)
It is important to note that the heat flux term qs is the heat flux enhancement resulting from electrostatic suppression (transient solid–liquid contacts), and not the total heat flux from the surface. As an approximation, the total heat flux can be estimated by adding the heat flux associated with the Leidenfrost droplet, which can be estimated using existing correlations [38]. Eqs. (1)– (5) can be rearranged to estimate qs as:
q s ∼
ks (Tw − Tsat )
(αs τ )1/2
K (X )2
(6)
qs can be non-dimensionalized as:
Bi∗ =
qs (αs τ )1/2 = C (K )(X )2 ks (Tw − Tsat )
(7)
where C is a constant (value determined from experimental results). Bi∗ is the modified Biot number, defined as the product of the heat flux enhancement and the length-scale divided by the product of the thermal conductivity of the solid and the surface superheat [37]. It is seen that Bi∗ is strongly dependent on the wetted fraction (X), which is estimated next as a function of the surface superheat (T) and voltage (V).
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Fig. 3. Schematic (side view) of the sub-model used to estimate the wetted fraction. Upon the application of an electric field, part of the liquid wets the surface, while the remainder of the liquid is attracted closer to the surface.
2.2. Modeling the wetted fraction
which can be estimated as [20]:
Presently, the wetted fraction is determined by the interplay between the vapor pressure and the electrostatic pressure. The model developed ahead is based on the following assumptions:
Fel = −
• •
•
• •
The vapor layer thickness is uniform. van der Waals interactions between the liquid and the solid are neglected. Weight of the droplet and buoyancy is negligible compared to other forces involved. Solid–liquid contact angle (no boiling) is 90°. Shape of the vapor pocket remains constant with time (Refer to Fig. 3), i.e.
h0 = constant 1−X where h0 is the vapor gap thickness and X is the wetted fraction. Presently, the wetted fraction associated with electrostatic suppression is modeled based on the schematic depicted in Fig. 3, which shows the side view of the solid–liquid contact. The initial vapor layer thickness is h00 . As the wetted fraction X increases (due to the applied voltage), the vapor layer thickness decreases. The decrease in vapor layer thickness in non-contact regions at low voltages [25] can be observed via side view imaging. However, as the wetted area increases, the side view gets blocked by multiple liquid columns. In this study, it is assumed that the reduction in thickness in non-contact regions occurs in a way such that the aspect ratio of the vapor pocket remains unchanged. If the vapor layer thickness in non-contact regions did not reduce with an increase in the wetted fraction, the vapor pockets would assume an oblong shape with an aspect ratio that increases with wetted fraction. This would result in high variations in the interface curvature. The initial vapor layer thickness (h00 ) and the vapor pocket thickness (h0 ) at two different voltages (where V2 > V1 ) are shown in Fig. 3. It is noted that the portion of the liquid surface not contacting the solid will always be drawn towards the surface due to electrostatic attraction. It is also noted that all of the liquid–solid contact points are clubbed together; the size and distribution of the contacting points does not influence the determination of X in the present model. Next, the mathematical framework to estimate X is developed. To start with, the vapor flow in the vapor gap is assumed to be Poiseuille flow [20,22]. As per previous studies [20,22], the force Fup on the Leidenfrost droplet due to evaporation-induced pressure buildup can be estimated as:
Fup =
3π 2
μ k v T R 4 A h f g ρv h 4
(8)
where μ is the viscosity of the droplet, kv is the thermal conductivity of vapor, R is the droplet radius; A is the footprint area of the droplet, hfg is the latent heat of vaporization, ρ v is the vapor density and h is the vapor gap thickness. The upward force on the droplet is balanced by the electrostatic force Fel on the droplet,
V 2 εA h2
(9)
where V and ɛ are the applied voltage and the permittivity of the vapor, respectively. It is assumed that the vapor gap thickness h, is equal to the average thickness of the vapor gap:
h = hav g = h0 ( 1 − X )
(10)
The average thickness of the vapor gap in Eq. (10) is the hypothetical equivalent thickness of the vapor gap without any interfacial instabilities. The thickness of a perfectly flat vapor gap will decrease due to the electrostatic force; however the liquid cannot touch the surface without the interfacial instabilities. The average thickness (h) therefore approximates the no-contact equivalent thickness. Eq. (10) is essentially used to represent a partial wetting situation as an equivalent non-wetting situation. It is assumed that the shape of the vapor pocket remains the same as shown in Fig. 3. Therefore, the vapor gap thickness in the no-contact zone becomes:
h0 = h00 (1 − X )
(11)
The force balance between Eqs. (8) and (9) results in the following equation:
Fup = Fel
3π 2
μ k v T R 4 A V 2 ε A = h f g ρv h 4 h2
(13)
Substituting Eqs. (10) and (11) into (13), we get:
3π μ k v T R 4 V 2ε = 8 2 h f g ρv h 4 ( 1 − X ) h200 (1 − X )4 00
(14)
The wetted fraction X can be obtained by rearranging the terms as:
3 π μ k v R 4 T T X = 1 − CX 2 (1 − X ) = 2 h f g ρv h200 ε V 2 V 4
1 / 4 (16)
where CX is a constant. Wetting starts when the applied voltage is larger than the threshold voltage Vth . Therefore, CX can be estimated using the threshold voltage for the specified superheat. Since the wetted fraction is zero below threshold voltage, we obtain:
CX =
Vth2
T
(17)
Finally, the following equation is obtained after combining Eqs. (7), (16) and (17).
Bi∗ =
V 1/2 2 q (αs τ )1/2 th = CK 1 − ks (Tw − Tsat ) V
(18)
Eq. (18) is the correlation to predict the heat transfer enhancement resulting from electrostatic suppression of the Leidenfrost state. It clearly shows that increasing the voltage will enhance heat transfer, primarily by enhancing the wetted fraction. The heat
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Fig. 4. Schematic of the experimental setup to quantify the heat transfer enhancement.
transfer enhancement is predicted to be proportional to the square of the wetted fraction. The threshold voltage for the specified superheat value is an experimentally measured quantity [21–23]. K is the dimensionless interface temperature, which depends on properties of the fluid and the surface (Eq. (2)). The constant C is a fitting parameter which can be extracted from measurements. 3. Measurements of heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets The model described in the previous section was validated by comparison with experiments with three liquids (water, isopropanol and ethanol). Heat transfer experiments with water and isopropanol (IPA) were conducted in the present study. Experimental data in Ref. [19] on similar experiments with ethanol droplets was also used for validation. Fig. 4 shows the schematic of the experimental setup presently used to measure the heat flux to the Leidenfrost droplets. A syringe pump continuously fed the Leidenfrost droplet to maintain its shape and volume; the flow rate was adjusted to equal the evaporation rate. The dispensed liquid was preheated to a temperature 2 °C lower than the boiling point. Preheating was done by passing the tubing through a cylindrical thermal mass with a flexible heater wrapped around it. Water was degassed prior to the experiments to avoid gas formation and expansion inside the tubing, which led to uncontrolled dispensing. A stainless steel retaining ring (shown in Fig. 4) was used to keep the Leidenfrost droplets in place and to ensure a constant and well-defined heat transfer area. The presence of this ring is especially important for experiments with water, since water droplets tend to move around very vigorously upon the application of an electric field. This ‘violent’ motion of electrostatically suppressed Leidenfrost droplets of water has been previously reported [19,20,25,36]. In fact, Ref. [19] (which provides experimental data on ethanol) reported the non-feasibility of water-based experiments (compared to ethanol) due to vigorous droplet motion. The retaining ring is attached to the preheating section which is 2 °C below the boiling point; consequently, the heat loss from the liquid to the ring can be neglected. The bottom section of the setup consisted of a stainless steel rod with embedded cartridge heaters. Five thermocouples (K type with an error of ±1.1 °C or 0.4% of measured value, whichever is greater) were located along the axis of the rod, 10 mm apart. Stainless steel was used (instead of higher thermal conductivity metals
5
like copper or aluminum) to ensure that the temperature difference between the thermocouples was large compared to the uncertainty in measurements. The side wall of the heating rod was insulated radially and a hollow stainless steel cylinder was placed around the insulation as shown in Fig. 4. This hollow cylinder was heated from the bottom and was insulated circumferentially. The heated cylinder around the central heating rod minimized heat loss from the central rod and ensured a linear axial temperature profile under steady state conditions. The experimental procedure is briefly described ahead. Before starting the experiment, the heater and the preheater were turned on to reach steady state without any liquid. Next, the syringe pump was turned on and kept on until steady state conditions were achieved without any voltage. Thermocouple measurements can be used to quantify the heat flux in the Leidenfrost state. Next, a voltage was applied and the flow rate of the syringe pump was increased such that the droplet volume and the footprint area remained constant (as tracked by a sideview camera). The temperatures in the heating rod reduced when the voltage was applied; these measurements can be used to quantify the heat transfer enhancement. It is noted that the present experimental setup is largely similar to the one used by Takano et al. [19]. Ref. [19] also provides the experimental data for ethanol droplets which is used presently for model validation. The two notable differences in the present experiments as compared to Ref. [19] are the use of a retaining ring, and the fact that the applied voltage and the heater power are fixed. The heat flux was estimated via linear regression of the data from the five axial thermocouples. The temperature gradient was averaged over 20 s of measurements after steady state was achieved. The error in the slope was calculated by accounting for the uncertainty in temperature measurements and the standard deviation of the mean (average of 20 measurements over 20 s). The surface temperature was estimated by extrapolating the linear regression of the temperature profile. The diameter of the water and isopropanol droplets (and the retaining ring) was 6 mm and 4 mm, respectively. The droplet diameter was chosen so as to be relevant to similar experiments with ethanol by Takano et al. [17–19]. The surface tension and density of ethanol and IPA are very close to each other. Therefore, the diameter of the IPA droplets was chosen to be the same as the ethanol droplets used in Ref. [19]. The diameter of the water droplets was 6 mm such that the dimensionless diameter (diameter /capillary length) for water was the same as that for ethanol and IPA droplets. This approach is important for the consistency of the experimental results, noting that the wetting dynamics depends strongly on droplet size [36]. The threshold voltage values for all fluids are taken from literature [19,22,36] and interpolated for different temperatures if required. The threshold voltage values for ethanol [19] range from 33 V to 93 V for superheat values ranging from 120 °C to 500 °C. The threshold voltage values for IPA and water are taken from References [22,36], respectively. The threshold voltage for IPA ranges from 38 V to 68 V for superheats ranging from 150 °C to 350 °C. The threshold voltage for DI water ranges from 40 V to 65 V for superheats ranging from 120 °C to 280 °C. 4. Results and discussions The experimental setup described in the previous section was used to measure the heat flux enhancement associated with suppression of the Leidenfrost state of deionized (DI) water and isopropanol (IPA) droplets. Fig. 5 shows the heat flux versus superheat curves for water droplets with applied voltages of 0 V, 100 V, 200 V and 300 V. The superheats range from 117 °C to 280 °C. Clearly, electrostatic suppression of the Leidenfrost state leads to
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Fig. 5. Heat flux versus superheat at various voltages corresponding to electrostatic suppression of the Leidenfrost state of water droplets.
Fig. 6. Heat flux versus superheat at various voltages corresponding to electrostatic suppression of the Leidenfrost state of isopropanol droplets.
a significant enhancement in heat flux. At the lowest superheat (~ 120 °C), a 4X enhancement in heat transfer over the baseline (no-voltage case) was observed at the highest voltage (300 V). The corresponding heat transfer enhancement (with 300 V) for a superheat of ~ 170 °C is 3X, suggesting that the influence of the applied voltage is reduced with increasing superheats. This should be expected, since the extent of electrostatic suppression will reduce at higher temperatures, resulting in lower wetted fractions [36]. In fact, the heat fluxes at 0 and 100 V converge to the same value at high superheats, as seen in Fig. 5. This can be attributed to the threshold voltage increase with temperature; heat transfer enhancement is not observed below the threshold voltage [22,36]. It is noted that the 300 V experiments were not conducted beyond a superheat of 170 °C, due to practical issues resulting from vigorous motion of the Leidenfrost pool, and uncertainties in maintaining a constant droplet volume. Interestingly, such experiments (with water) were not possible in the study reported in Ref. [19]. Presently, the use of a retaining ring enabled successful experiments. Fig. 6 shows the heat flux versus superheat curves for IPA droplets with applied voltages of 0 V, 150 V and 300 V. The superheat values range from 160 °C to 372 °C. At the lowest superheat (~ 160 °C), a 2.5X and 3.5X enhancement in heat transfer over the baseline (no-voltage case) is observed for applied voltages of 150 V
Fig. 7. Heat flux versus voltage for various superheats for electrostatic suppression of the Leidenfrost state of ethanol droplets. Data is extracted from Reference 19.
and 300 V, respectively. A comparison of the heat fluxes with those obtained by water (Fig. 5) shows higher heat flux values for water. This is expected since water is a better heat transfer fluid (has a larger latent heat of vaporization). As an illustration, the heat flux associated with suppression (300 V) at a superheat of ~ 163 °C is 51 W/cm2 and 34 W/cm2 for water and IPA, respectively. It is noted that measurements at higher superheats and voltages were not taken due to the stability-related issues mentioned earlier. In addition to water and isopropanol, this study uses the data from similar experiments with ethanol droplets in Ref. [19]. In that study, the experiments involved changing the applied voltage to keep the surface temperature constant, while ethanol droplets (3.5 mm diameter) were being constantly fed with preheated ethanol (to compensate for evaporation). The experimental setup used was similar to the present one. Fig. 7 shows the heat flux versus the applied voltage for five different superheat values for ethanol droplets (data extracted from [19]). It is noted that the measurements are presented as heat flux versus voltage at different superheats, since the study [19] achieved steady state by changing the applied voltage to bring the surface temperature to a desired value. This is different from the present experiments (Figs. 5 and 6), wherein the voltage and heater power are kept constant, and the surface temperature is allowed to reach a steady state value. Fig. 7 shows the heat flux increasing with the superheat and the applied voltage, similar to Figs. 5 and 6. The heat transfer was enhanced by 7.6X and 3.6X at superheats of 120 °C and 500 °C (at 250 V), respectively. This again illustrates the reduced influence of electric fields (manifested in lower wetted fractions) as the superheat increases. Also, the slopes of the heat flux curves at low superheat (eg. 120 °C) decrease as the voltage increases, whereas the heat fluxes at high superheats (eg. 500 °C) increase linearly with voltage. This can be attributed to the fact that wetted fractions do not increase indiscriminately with voltage, and will saturate. More insights on the combined influence of superheat and the applied voltage on wetted fractions is available in a recent study by the present authors [36]. All the experimental results (present experiments and the data from Ref. [19]) for the three fluids are captured together in Fig. 8. Fig. 8 shows the heat transfer enhancement coefficient (h+ = q /T ) versus applied voltage at different superheats. It is noted that h+ is not the heat transfer coefficient, but an equivalent parameter to quantify the enhancement. The presently developed model does not predict the total heat flux (or heat transfer coef-
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7
Fig. 8. Voltage-dependent heat transfer enhancement coefficient (q /T) associated with electrostatic suppression of Leidenfrost droplets of water, isopropanol and ethanol. The color map shows the values of the superheat.
ficient), but instead calculates the heat transfer due to transient contacts, which is essentially the heat transfer enhancement. The heat flux enhancement at a given superheat (q ) is calculated by subtracting the baseline (no-voltage) heat flux from the experimentally measured total heat flux in Figs. 5–7. Since the superheat values in Figs. 5 and 6 are randomly distributed, they are shown in the color map in Fig. 8. The heat transfer enhancement coefficient increases with voltage and decreases with superheat for all three liquids. This is a direct consequence of larger wetted fractions at higher voltages and lower superheats, also keeping in mind that the heat transfer enhancement scales as the square of the wetted fraction, as per Eq. (7). The heat transfer enhancement with water is larger than the heat transfer enhancement with IPA, because of the better heat transfer characteristics of water as compared to IPA. IPA and ethanol have similar thermal properties, which implies that the heat transfer enhancement numbers should be similar. However, the heat transfer enhancement with IPA is approximately half of the heat transfer enhancement with ethanol. This discrepancy can be explained by noting that the heat transfer due to transient contact depends on the thermal conductivity and diffusivity of the solid surface (in addition to the fluid), as per Eq. (1). Stainless steel and platinum surfaces were used for IPA and ethanol experiments, respectively. √ The thermophysical properties-depended parameter ( αs τ /K ks ) in Eq. (7), is two times larger for the IPA-stainless steel combination as compared to the ethanol-platinum combination. When the heat transfer enhancement values of both liquids are multiplied √ by the thermophysical properties-depended variable ( αs τ /K ks ), the dimensionless heat transfer enhancement becomes similar for the same superheat values and applied voltages. This aspect can also be understood via the subsequent discussion related to Fig. 9. Next, the predictions of presently developed heat transfer enhancement model are compared to the heat flux measurements reported in Figs. 5–8. Fig. 9 shows the experimental data reported in Fig. 8 in terms of the parameters used in the heat transfer model (Bi∗ , K, X). It is seen that the sparse data in Fig. 8 collapses on one line in Fig. 9. The slope of Eq. (18) (constant C) is found to be 25,435 as
Fig. 9. Correlation to predict heat transfer enhancement at different surface superheats and applied voltages.
per the linear regression. Therefore, Eq. (18) becomes:
V 1/2 2 q (αh τ )1/2 th Bi = = 25435K 1 − kh (Tw − Tsat ) V ∗
(19)
Eq. (19) is the experimentally validated correlation for predicting the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. Only 3 out of 50 data points have an absolute error larger than 10 0 0 in Bi∗ /K. The relative error (error divided by the prediction as per Eq. (19)) increases at lower values of X2 (which is proportional to heat transfer enhancement). The relative error is always less than 10% when X2 is larger than 0.25. The heat transfer enhancement should also be compared to the heat flux of the Leidenfrost droplet (at zero voltage) for a better understanding of the accuracy of the present model. The heat flux enhancement term in Bi∗ /K can be replaced by the heat flux q0 of the Leidenfrost droplet (no-voltage). This yields the no-voltage
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modified Biot number Bi∗0 as:
References
1/2
Bi∗0 q (αh τ ) = 0 K ks (Tw − Tsat )K
(20)
It is noted that the heat flux divided by superheat (q0 /T ) is simply the heat transfer coefficient, which reduces with increasing temperature in the film boiling region [3]. The mean values of Bi∗0 /K are 40 0 0, 550 0 and 1500 for water, IPA and ethanol, respectively. Therefore, the absolute errors in Bi∗ /K in Fig. 9 are smaller than the Bi∗0 /K value of the Leidenfrost droplets (no-voltage) for most cases. The mean absolute error in Bi∗ /K over all data points in Fig. 9 was 524. One of the key findings of this work is the heat transfer enhancement scaling as the square of the wetted fraction. This is suggested by the analytical model and the experimental results support this hypothesis very well. It should be noted that such a relation will not necessarily be valid for classical nucleate boiling or transient boiling situations. Presently, heat transfer occurs via transient solid–liquid contacts, as opposed to bubble-related phenomena that influence nucleate and transition boiling. It is also noted that the wetted fraction will not increase indiscriminately with voltage. Our recent study [36] suggests that the wetted fraction will saturate in the 70–80% range, which suggests a limit on the heat transfer enhancement. Overall, Eq. (19) can be used to predict the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state under strong electric fields. The heat transfer enhancement can then be added to the heat transfer rate in the Leidenfrost state (no voltage), which can be calculated using existing correlations [38]. It is noted that the present model has been experimentally validated only for droplet sizes of the order of the capillary length of the fluid. Wetting dynamics and the related heat transfer enhancement is expected to be different at smaller or larger length scales [31]. 5. Conclusions Overall, this work outlines a comprehensive study of heat transfer enhancement enabled by electrostatic suppression of the Leidenfrost state. A first principles-based analytical model is developed to predict the heat transfer enhancement as a function of the applied electric field, surface temperature and fluid and surface properties. This model is strongly validated by experimental measurements (inhouse as well as by another group) with three fluids. The strength and utility of the modeling approach is further underscored by the presence of only one empirical parameter in the model. The results suggest that electrostatic suppression of the Leidenfrost state can enhance heat transfer by 4X for water, and enable larger enhancements for organic solvents. Declaration of Competing Interest The authors report no conflict of interest Acknowledgments The authors acknowledge National Science Foundation grants CBET-1605789, CBET-1653412 for supporting this work. This work was also partly supported by the UT Austin Portugal program and Google Faculty Research Award. The assistance of Awan Bhati in conducting some experiments is acknowledged. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2019.119207.
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Nishio, Enhancement of evaporation of a droplet using ehd effect (Onset of instability of gas–liquid interface under electric field applied in a stepwise manner), JSME Int. J. 38 (2) (1995) 288–294. [19] K. Takano, I. Tanasawa, Enhancement of evaporation of a droplet using EHD effect (Measurement of steady-state heat flux during evaporation of a single droplet), JSME Int. J. 39 (1996) 583–589. [20] F. Celestini, G. Kirstetter, Effect of an electric field on a Leidenfrost droplet, Soft Matter 8 (22) (2012) 5992. [21] A. Shahriari, P.S. Wilson, V. Bahadur, Acoustic detection of electrostatic suppression of the Leidenfrost state, Phys. Rev. E 98 (May) (2018) 13103. [22] A. Shahriari, J. Wurz, V. Bahadur, Heat transfer enhancement accompanying Leidenfrost state suppression at ultrahigh temperatures, Langmuir 30 (40) (2014) 12074–12081. [23] A. Shahriari, S. Das, V. Bahadur, R.T. Bonnecaze, Analysis of the instability underlying electrostatic suppression of the Leidenfrost state, Phys. Rev. Fluids 2 (2017). [24] A. Shahriari, O. Ozkan, V. Bahadur, Electrostatic suppression of the Leidenfrost state on liquid substrates, Langmuir 33 (46) (2017) 13207–13213. [25] O. Ozkan, A. Shahriari, V. Bahadur, Electrostatic suppression of the Leidenfrost state using AC electric fields, Appl. Phys. Lett. 111 (14) (2017) 141608. [26] A. Shahriari, P. Birbarah, J. Oh, N. Miljkovic, V. Bahadur, Electric field-based control and enhancement of boiling and condensation, Nanoscale Microscale Thermophys. Eng. 21 (2) (2017) 102–121. [27] S. Laohalertdecha, P. Naphon, S. Wongwises, A review of electrohydrodynamic enhancement of heat transfer, Renew. Sustain. Energy Rev. 11 (5) (2007) 858–876. [28] T.B. Jones, K.R. Hallock, Surface wave model of electrohydrodynamically coupled minimum film boiling, J. Electrostat. 5 (C) (1978) 273–284. [29] T.B. jones, Electrohydrodynamically enhanced heat transfer in liquids—a review, Adv. Heat Transf. 14 (1979) 107–148 vol. Volume. [30] T.B. Jones, R.C. Schaeffer, Electrohydrodynamically coupled minimum film boiling in dielectric liquids, AIAA 10th Thermophysics Conference, 1975. [31] J. Berghams, Prediction of the effect of electric fields on the most unstable wavelength during film boiling on small wires, J. Electrostat. 5 (1978) 265–272. [32] T.G. Karayiannis, Electrohydrodynamic enhancement of nucleate boiling heat transfer in heat exchangers, Chem. Eng. Commun. 81 (1) (1989) 15–24. [33] M. Zaghdoudi, Electric field effects on pool boiling, J. Enhanc. Heat Transf. 9 (5–6) (2002) 187–208. [34] N. Zuber, and M. Tribus, “Further remarks on the stability of boiling heat transfer,” 1958.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets Onur Ozkan, Vaibhav Bahadur∗ Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA
a r t i c l e
i n f o
Article history: Received 6 July 2019 Revised 20 September 2019 Accepted 9 December 2019
Keywords: Leidenfrost state suppression Electrowetting Boiling Heat transfer Droplet
a b s t r a c t A liquid drop levitates on its own vapor on a sufficiently hot surface (the Leidenfrost effect). Application of an electrical potential difference across the vapor gap suppresses the Leidenfrost state via electrostatic attraction of liquid towards the surface. The resultant transient solid–liquid contacts significantly reduce the thermal resistance to heat transfer across the insulating vapor gap. This work is an indepth study on the heat transfer enhancement accompanying electrostatic suppression of the Leidenfrost state. A purely analytical model is developed to predict the heat transfer enhancement based on the transient wetting dynamics observed during suppression. This model is based on semi-infinite conduction heat transfer coupled with a physics-based prediction of the voltage-dependent wetted fraction. The model is strongly validated by comparison with inhouse experiments and related experimental data from literature. These experiments measure the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state for three fluids, at various applied voltages and superheats. The experimentally-validated model predicts that heat transfer enhancement scales as the square of the wetted fraction. Results show a 4X enhancement in heat transfer by electrostatic suppression of the Leidenfrost state of water droplets; higher enhancements are measured for organic solvents. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Nucleate boiling heat transfer is characterized by high heat flux dissipation capacity, resulting from the high latent heat of vaporization and bubble-related convection currents at the surface [1–3]. However, heat transfer reduces drastically above the critical heat flux (CHF) limit owing to the formation of continuous pockets of thermally insulating vapor on the surface [1–3]. At the Leidenfrost temperature, the surface is completely blanketed by a vapor film; this mode of boiling is known as film boiling. The Leidenfrost phenomena has been studied extensively using pool boiling and droplet-based configurations [4–6]. Many recent studies have explored surface engineering as a tool to elevate the Leidenfrost temperature and CHF [7–11]. This study deals with heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. The use of an externally applied electric field to suppress/eliminate the Leidenfrost state has been studied in pool boiling [12–15] and more recently, in droplet-based configurations [16–26]. These studies have analyzed electrostatic suppression of the Leidenfrost state for electrically insulating (oils, refrigerants) and electrically conduct∗
Corresponding author. E-mail address: [email protected] (V. Bahadur).
https://doi.org/10.1016/j.ijheatmasstransfer.2019.119207 0017-9310/© 2019 Elsevier Ltd. All rights reserved.
ing (water, organic solvents) fluids. For fluids which are perfect electrical insulators, an electric field exists inside the liquid volume (in addition to the vapor layer); the liquid is not equipotential [27]. Conversely, for fluids with a finite (but small) electrical conductivity, the applied electrical potential difference is entirely expressed across the vapor gap, and the liquid can be considered equipotential [28]. Moreover, the vapor layer is typically very thin (O (100 μm)) during film boiling. Therefore, the applied voltage results in a high electric field within the vapor layer for electrically conducting fluids as compared to insulating fluids. Consequently, the voltage requirement for electrostatic suppression is an order of magnitude smaller for electrically conducting fluids, as compared to insulating fluids [28]. The above discussion highlights the differences in the nature of electrostatic suppression of electrically conducting and insulating fluids. It is reasonable to expect that the heat transfer enhancement will also show differences. A literature review indicates that there is a significantly greater number of studies on electric fieldinduced heat transfer enhancement of electrically insulating [27– 33] fluids, as compared to conducting fluids. The most detailed studies with conducting fluids are in fact from the 1960 s [12–14]. These studies were conducted with water and isopropanol (IPA) and involved pool boiling measurements on a horizontal tube. The heat flux was seen to increase with the electrical conductivity of
2
O. Ozkan and V. Bahadur / International Journal of Heat and Mass Transfer 149 (2020) 119207
Nomenclature A Bi∗ C F g K k h0 h00 hfg
Footprint area of droplet Modified Biot number Correlation constant Force Gravitational acceleration Dimensionless interface temperature Thermal conductivity Vapor gap thickness Initial vapor gap thickness Latent heat of vaporization
qtr R T tc V X
Transient heat flux during contact Droplet radius Temperature Duration of contact Voltage Wetted fraction
Greek Symbols α Thermal diffusivity q s Heat flux enhancement ɛ Permittivity μ Dynamic viscosity ρ Density σ Surface tension τ Characteristic period of Taylor wave; period of oscillations Subscripts 0 Non-contact region (vapor gap thickness) 00 Initial (vapor gap thickness) avg Average (vapor gap thickness) el Electrostatic l Liquid s Solid or surface sat Saturation th Threshold up Upward v Vapor w Wall
the fluid for the same voltage and superheat values. Other pool boiling-based studies for conducting fluids have derived the most unstable wavelength and the Leidenfrost point heat flux based on linear stability analysis [28]. The minimum heat flux during film boiling was calculated with the knowledge of the most unstable wavelength and its frequency (from linear stability analysis) [34,35]. The minimum heat flux (Leidenfrost point) under electrostatic suppression was estimated by a simple modification of the most unstable wavelength and frequency under electrostatic forces for insulating and conducting liquids [28–31]. Most recent studies on electrostatic suppression of the Leidenfrost state involve droplet-based experiments [15–26]. Celestini and Kirstetter [20] conducted interferometric measurements to quantify variations in the vapor layer thickness below the threshold voltage (for electrostatic suppression). The present group has conducted extensive recent studies on various aspects of electrostatic suppression of the Leidenfrost state (for conducting fluids), which include: demonstration of suppression at temperatures exceeding 500 °C [22]; accelerated and tunable cooling during quenching of metals [15]; linear instability analysis associated with electrostatic suppression [23]; influence of AC frequency on electrostatic suppression [25]; electrostatic suppression on liquid substrates [24]; acoustic detection of electrostatic suppression [21];
and impedance measurements-based quantification of wetted fractions associated with electrostatic suppression [36]. A fundamental study on the heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets does not exist, and is the focus of this work. Most previous heat transfer studies have used pool boiling configurations. Pool boiling models cannot be used for droplets, since these models involve bubble-related parameters (size, distance between bubbles, detachment frequency) which are not applicable in a droplet configuration. The nature of any heat transfer model or correlation is thus expected to be fundamentally different for droplets as compared to pool boiling configurations. The only reported heat transfer measurements [16,19] associated with electrostatic suppression of Leidenfrost droplets involved ethanol and R113 droplets. In fact, data from Ref. [19] is used in this study to benchmark the model developed in this work. The present work is an indepth study (combining modeling and experiments) on the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. A fundamentalsbased analytical model is developed to predict the heat transfer enhancement. This model is based on semi-infinite conduction-based estimates of heat transfer associated with transient solid–liquid contacts coupled with another fundamentals-based prediction of the voltage-dependent wetted fraction. The model is validated via comparison with our own experiments as well as with experimental data provided in Ref. [19]. Overall, three fluids are analyzed (water, isopropanol and ethanol). It is noted that this model matches the experimental data very well, and involves only one empirical parameter. 2. Modeling heat transfer enhancement associated with electrostatic suppression An electric field in the vapor gap leads to periodic (both spatial and temporal) solid–liquid contact, with the wetted fraction dependent on the surface superheat and the applied voltage [36]. The applied voltage must be larger than a threshold value to initiate solid–liquid contact. The value of the threshold voltage increases with superheat due of faster evaporation and larger vapor layer thickness [22,25,36]. The wetted fraction increases with the applied voltage (for voltages higher than the threshold voltage) at a given superheat. Similarly, smaller wetted fractions result when the superheat is increased at a constant applied voltage [36]. It is noted that this mode of boiling is not directly analogous to classical film boiling, which involves a continuous vapor film as opposed to periodic solid–liquid contact. This boiling mode also cannot be considered analogous to nucleate boiling, since the small duration of transient contacts is not sufficiently long for bubble nucleation to occur. The boiling mode in this work most resembles transition boiling which is observed when the surface temperature is lowered below the Leidenfrost point and the heat transfer is not sufficient to maintain a vapor film. It must be cautioned that ‘film boiling under electrowetting’ is fundamentally different from classical transition boiling since the wetted fraction depends on the electric field and not solely on the surface superheat. With this background, the heat transfer model developed in this work is based on an approach utilized by Ramilison and Lienhard [37] to model transition boiling. Another analytical sub-model to estimate the voltage-dependent wetted fraction feeds into this heat transfer model. 2.1. Modeling heat transfer Fig. 1 shows a high magnification snapshot during electrostatic suppression (100 V) of a saturated deionized (DI) water Leidenfrost droplet on a 250 °C surface. The diameter of the droplet is approximately 6 mm. The vapor layer and two solid–liquid contact
O. Ozkan and V. Bahadur / International Journal of Heat and Mass Transfer 149 (2020) 119207
3
Fig. 1. Solid–liquid contact during electrostatic suppression of a Leidenfrost droplet on a 250 °C surface. The liquid is saturated deionized water.
is noted that each contact area shown in Fig. 2 is transient. Contacts are modeled as oscillations of the vapor-liquid interface (associated with the Taylor wave) that result in wetting for a fraction of their oscillation periods. Therefore, the surface heat flux (qs ) will be proportional to qtr as per:
t qs ∼ qtr (X ) c τ
(3)
where X is the wetted fraction (fraction of footprint area of droplet which is in contact with liquid) and tc is the contact time. The period of oscillations in the absence of an electric field (τ ) is assumed to be equal to the characteristic period of the associated Taylor wave and can be estimated as [37]: Fig. 2. Modeling electrostatic suppression: Schematic view from bottom showing the solid–liquid contact areas. It is noted that the arrangement of contact points does not have to be regular.
areas are clearly seen. The heat transfer enhancement associated with suppression can be attributed to the low thermal resistance pathways enabled by these solid–liquid contacts. The actual distribution of these contact areas has not been characterized in any study to date. Fig. 2 shows a schematic visualization of a 2D array of periodic contact areas at a particular time instant. The location of these contact areas will change, since these contacts are transient. It is noted that the presently developed model does not need a regular arrangement of contact areas. The primary parameter in the model which captures these contacts is the wetted fraction, which is modeled in Section 2.2. It is noted that the present model is valid for any other arrangement of contact areas as long as the heat transfer associated with these areas is not influenced by the surrounding contact areas. The heat flux enhancement (qs ) can be calculated if 1) the transient heat flux qtr during contact, 2) the duration of the contact (tc ), and 3) the area of each contact are known. Since each contact area is small compared to the total surface area and tc is short (order of milliseconds), the heat transfer during each contact is modeled as the heat transfer between two semi-infinite media brought into sudden contact. The transient heat transfer at each contact equals the cooling of the semi-infinite solid [37] as:
qtr =
ks (Tsat − Tcontact )
(αs τ )1/2
=
ks (Tw − Tsat )
(αs τ )1/2
K
(1)
where α s is the thermal diffusivity of the solid, τ is the temporal period of each contact and ks is the thermal conductivity of the solid. (α s τ )1/2 is thus the characteristic length of the solid. K is the dimensionless interface temperature which depends on the thermal conductivity and diffusivity of both media as [37]: kf
K≡
αl1/2 kl
αl1/2
+
ks
αs1/2
=
Tw − Tcontact Tw − Tsat
(2)
The transient heat flux in Eq. (1) is the heat flux across the contact area. However, the wetted fraction is smaller than 1 and the contact time is smaller than the period of each oscillation. It
τ=
σ
1/4 (4)
g3 (ρl − ρg )
where σ is the surface tension; g is the gravitational acceleration and ρ l and ρ g are the densities of the liquid and the vapor, respectively. The period of oscillations with electric fields can be calculated using Eqn. 4 and Ref [29]. During each contact event, the solid–liquid contact starts from a point, and then increases up to the maximum wetted fraction. Subsequently, the wetted area in the contact region decreases due to vapor pressure buildup, with the liquid eventually detaching from the surface. In this study, it is assumed that the dimensionless contact time equals the wetted fraction since increasing the contact area would take longer. It is noted that a similar assumption was made in another study [37] which modeled the heat transfer in transition boiling. This implies:
t c
τ
=X
(5)
It is important to note that the heat flux term qs is the heat flux enhancement resulting from electrostatic suppression (transient solid–liquid contacts), and not the total heat flux from the surface. As an approximation, the total heat flux can be estimated by adding the heat flux associated with the Leidenfrost droplet, which can be estimated using existing correlations [38]. Eqs. (1)– (5) can be rearranged to estimate qs as:
q s ∼
ks (Tw − Tsat )
(αs τ )1/2
K (X )2
(6)
qs can be non-dimensionalized as:
Bi∗ =
qs (αs τ )1/2 = C (K )(X )2 ks (Tw − Tsat )
(7)
where C is a constant (value determined from experimental results). Bi∗ is the modified Biot number, defined as the product of the heat flux enhancement and the length-scale divided by the product of the thermal conductivity of the solid and the surface superheat [37]. It is seen that Bi∗ is strongly dependent on the wetted fraction (X), which is estimated next as a function of the surface superheat (T) and voltage (V).
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O. Ozkan and V. Bahadur / International Journal of Heat and Mass Transfer 149 (2020) 119207
Fig. 3. Schematic (side view) of the sub-model used to estimate the wetted fraction. Upon the application of an electric field, part of the liquid wets the surface, while the remainder of the liquid is attracted closer to the surface.
2.2. Modeling the wetted fraction
which can be estimated as [20]:
Presently, the wetted fraction is determined by the interplay between the vapor pressure and the electrostatic pressure. The model developed ahead is based on the following assumptions:
Fel = −
• •
•
• •
The vapor layer thickness is uniform. van der Waals interactions between the liquid and the solid are neglected. Weight of the droplet and buoyancy is negligible compared to other forces involved. Solid–liquid contact angle (no boiling) is 90°. Shape of the vapor pocket remains constant with time (Refer to Fig. 3), i.e.
h0 = constant 1−X where h0 is the vapor gap thickness and X is the wetted fraction. Presently, the wetted fraction associated with electrostatic suppression is modeled based on the schematic depicted in Fig. 3, which shows the side view of the solid–liquid contact. The initial vapor layer thickness is h00 . As the wetted fraction X increases (due to the applied voltage), the vapor layer thickness decreases. The decrease in vapor layer thickness in non-contact regions at low voltages [25] can be observed via side view imaging. However, as the wetted area increases, the side view gets blocked by multiple liquid columns. In this study, it is assumed that the reduction in thickness in non-contact regions occurs in a way such that the aspect ratio of the vapor pocket remains unchanged. If the vapor layer thickness in non-contact regions did not reduce with an increase in the wetted fraction, the vapor pockets would assume an oblong shape with an aspect ratio that increases with wetted fraction. This would result in high variations in the interface curvature. The initial vapor layer thickness (h00 ) and the vapor pocket thickness (h0 ) at two different voltages (where V2 > V1 ) are shown in Fig. 3. It is noted that the portion of the liquid surface not contacting the solid will always be drawn towards the surface due to electrostatic attraction. It is also noted that all of the liquid–solid contact points are clubbed together; the size and distribution of the contacting points does not influence the determination of X in the present model. Next, the mathematical framework to estimate X is developed. To start with, the vapor flow in the vapor gap is assumed to be Poiseuille flow [20,22]. As per previous studies [20,22], the force Fup on the Leidenfrost droplet due to evaporation-induced pressure buildup can be estimated as:
Fup =
3π 2
μ k v T R 4 A h f g ρv h 4
(8)
where μ is the viscosity of the droplet, kv is the thermal conductivity of vapor, R is the droplet radius; A is the footprint area of the droplet, hfg is the latent heat of vaporization, ρ v is the vapor density and h is the vapor gap thickness. The upward force on the droplet is balanced by the electrostatic force Fel on the droplet,
V 2 εA h2
(9)
where V and ɛ are the applied voltage and the permittivity of the vapor, respectively. It is assumed that the vapor gap thickness h, is equal to the average thickness of the vapor gap:
h = hav g = h0 ( 1 − X )
(10)
The average thickness of the vapor gap in Eq. (10) is the hypothetical equivalent thickness of the vapor gap without any interfacial instabilities. The thickness of a perfectly flat vapor gap will decrease due to the electrostatic force; however the liquid cannot touch the surface without the interfacial instabilities. The average thickness (h) therefore approximates the no-contact equivalent thickness. Eq. (10) is essentially used to represent a partial wetting situation as an equivalent non-wetting situation. It is assumed that the shape of the vapor pocket remains the same as shown in Fig. 3. Therefore, the vapor gap thickness in the no-contact zone becomes:
h0 = h00 (1 − X )
(11)
The force balance between Eqs. (8) and (9) results in the following equation:
Fup = Fel
3π 2
μ k v T R 4 A V 2 ε A = h f g ρv h 4 h2
(13)
Substituting Eqs. (10) and (11) into (13), we get:
3π μ k v T R 4 V 2ε = 8 2 h f g ρv h 4 ( 1 − X ) h200 (1 − X )4 00
(14)
The wetted fraction X can be obtained by rearranging the terms as:
3 π μ k v R 4 T T X = 1 − CX 2 (1 − X ) = 2 h f g ρv h200 ε V 2 V 4
1 / 4 (16)
where CX is a constant. Wetting starts when the applied voltage is larger than the threshold voltage Vth . Therefore, CX can be estimated using the threshold voltage for the specified superheat. Since the wetted fraction is zero below threshold voltage, we obtain:
CX =
Vth2
T
(17)
Finally, the following equation is obtained after combining Eqs. (7), (16) and (17).
Bi∗ =
V 1/2 2 q (αs τ )1/2 th = CK 1 − ks (Tw − Tsat ) V
(18)
Eq. (18) is the correlation to predict the heat transfer enhancement resulting from electrostatic suppression of the Leidenfrost state. It clearly shows that increasing the voltage will enhance heat transfer, primarily by enhancing the wetted fraction. The heat
O. Ozkan and V. Bahadur / International Journal of Heat and Mass Transfer 149 (2020) 119207
Fig. 4. Schematic of the experimental setup to quantify the heat transfer enhancement.
transfer enhancement is predicted to be proportional to the square of the wetted fraction. The threshold voltage for the specified superheat value is an experimentally measured quantity [21–23]. K is the dimensionless interface temperature, which depends on properties of the fluid and the surface (Eq. (2)). The constant C is a fitting parameter which can be extracted from measurements. 3. Measurements of heat transfer enhancement associated with electrostatic suppression of Leidenfrost droplets The model described in the previous section was validated by comparison with experiments with three liquids (water, isopropanol and ethanol). Heat transfer experiments with water and isopropanol (IPA) were conducted in the present study. Experimental data in Ref. [19] on similar experiments with ethanol droplets was also used for validation. Fig. 4 shows the schematic of the experimental setup presently used to measure the heat flux to the Leidenfrost droplets. A syringe pump continuously fed the Leidenfrost droplet to maintain its shape and volume; the flow rate was adjusted to equal the evaporation rate. The dispensed liquid was preheated to a temperature 2 °C lower than the boiling point. Preheating was done by passing the tubing through a cylindrical thermal mass with a flexible heater wrapped around it. Water was degassed prior to the experiments to avoid gas formation and expansion inside the tubing, which led to uncontrolled dispensing. A stainless steel retaining ring (shown in Fig. 4) was used to keep the Leidenfrost droplets in place and to ensure a constant and well-defined heat transfer area. The presence of this ring is especially important for experiments with water, since water droplets tend to move around very vigorously upon the application of an electric field. This ‘violent’ motion of electrostatically suppressed Leidenfrost droplets of water has been previously reported [19,20,25,36]. In fact, Ref. [19] (which provides experimental data on ethanol) reported the non-feasibility of water-based experiments (compared to ethanol) due to vigorous droplet motion. The retaining ring is attached to the preheating section which is 2 °C below the boiling point; consequently, the heat loss from the liquid to the ring can be neglected. The bottom section of the setup consisted of a stainless steel rod with embedded cartridge heaters. Five thermocouples (K type with an error of ±1.1 °C or 0.4% of measured value, whichever is greater) were located along the axis of the rod, 10 mm apart. Stainless steel was used (instead of higher thermal conductivity metals
5
like copper or aluminum) to ensure that the temperature difference between the thermocouples was large compared to the uncertainty in measurements. The side wall of the heating rod was insulated radially and a hollow stainless steel cylinder was placed around the insulation as shown in Fig. 4. This hollow cylinder was heated from the bottom and was insulated circumferentially. The heated cylinder around the central heating rod minimized heat loss from the central rod and ensured a linear axial temperature profile under steady state conditions. The experimental procedure is briefly described ahead. Before starting the experiment, the heater and the preheater were turned on to reach steady state without any liquid. Next, the syringe pump was turned on and kept on until steady state conditions were achieved without any voltage. Thermocouple measurements can be used to quantify the heat flux in the Leidenfrost state. Next, a voltage was applied and the flow rate of the syringe pump was increased such that the droplet volume and the footprint area remained constant (as tracked by a sideview camera). The temperatures in the heating rod reduced when the voltage was applied; these measurements can be used to quantify the heat transfer enhancement. It is noted that the present experimental setup is largely similar to the one used by Takano et al. [19]. Ref. [19] also provides the experimental data for ethanol droplets which is used presently for model validation. The two notable differences in the present experiments as compared to Ref. [19] are the use of a retaining ring, and the fact that the applied voltage and the heater power are fixed. The heat flux was estimated via linear regression of the data from the five axial thermocouples. The temperature gradient was averaged over 20 s of measurements after steady state was achieved. The error in the slope was calculated by accounting for the uncertainty in temperature measurements and the standard deviation of the mean (average of 20 measurements over 20 s). The surface temperature was estimated by extrapolating the linear regression of the temperature profile. The diameter of the water and isopropanol droplets (and the retaining ring) was 6 mm and 4 mm, respectively. The droplet diameter was chosen so as to be relevant to similar experiments with ethanol by Takano et al. [17–19]. The surface tension and density of ethanol and IPA are very close to each other. Therefore, the diameter of the IPA droplets was chosen to be the same as the ethanol droplets used in Ref. [19]. The diameter of the water droplets was 6 mm such that the dimensionless diameter (diameter /capillary length) for water was the same as that for ethanol and IPA droplets. This approach is important for the consistency of the experimental results, noting that the wetting dynamics depends strongly on droplet size [36]. The threshold voltage values for all fluids are taken from literature [19,22,36] and interpolated for different temperatures if required. The threshold voltage values for ethanol [19] range from 33 V to 93 V for superheat values ranging from 120 °C to 500 °C. The threshold voltage values for IPA and water are taken from References [22,36], respectively. The threshold voltage for IPA ranges from 38 V to 68 V for superheats ranging from 150 °C to 350 °C. The threshold voltage for DI water ranges from 40 V to 65 V for superheats ranging from 120 °C to 280 °C. 4. Results and discussions The experimental setup described in the previous section was used to measure the heat flux enhancement associated with suppression of the Leidenfrost state of deionized (DI) water and isopropanol (IPA) droplets. Fig. 5 shows the heat flux versus superheat curves for water droplets with applied voltages of 0 V, 100 V, 200 V and 300 V. The superheats range from 117 °C to 280 °C. Clearly, electrostatic suppression of the Leidenfrost state leads to
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Fig. 5. Heat flux versus superheat at various voltages corresponding to electrostatic suppression of the Leidenfrost state of water droplets.
Fig. 6. Heat flux versus superheat at various voltages corresponding to electrostatic suppression of the Leidenfrost state of isopropanol droplets.
a significant enhancement in heat flux. At the lowest superheat (~ 120 °C), a 4X enhancement in heat transfer over the baseline (no-voltage case) was observed at the highest voltage (300 V). The corresponding heat transfer enhancement (with 300 V) for a superheat of ~ 170 °C is 3X, suggesting that the influence of the applied voltage is reduced with increasing superheats. This should be expected, since the extent of electrostatic suppression will reduce at higher temperatures, resulting in lower wetted fractions [36]. In fact, the heat fluxes at 0 and 100 V converge to the same value at high superheats, as seen in Fig. 5. This can be attributed to the threshold voltage increase with temperature; heat transfer enhancement is not observed below the threshold voltage [22,36]. It is noted that the 300 V experiments were not conducted beyond a superheat of 170 °C, due to practical issues resulting from vigorous motion of the Leidenfrost pool, and uncertainties in maintaining a constant droplet volume. Interestingly, such experiments (with water) were not possible in the study reported in Ref. [19]. Presently, the use of a retaining ring enabled successful experiments. Fig. 6 shows the heat flux versus superheat curves for IPA droplets with applied voltages of 0 V, 150 V and 300 V. The superheat values range from 160 °C to 372 °C. At the lowest superheat (~ 160 °C), a 2.5X and 3.5X enhancement in heat transfer over the baseline (no-voltage case) is observed for applied voltages of 150 V
Fig. 7. Heat flux versus voltage for various superheats for electrostatic suppression of the Leidenfrost state of ethanol droplets. Data is extracted from Reference 19.
and 300 V, respectively. A comparison of the heat fluxes with those obtained by water (Fig. 5) shows higher heat flux values for water. This is expected since water is a better heat transfer fluid (has a larger latent heat of vaporization). As an illustration, the heat flux associated with suppression (300 V) at a superheat of ~ 163 °C is 51 W/cm2 and 34 W/cm2 for water and IPA, respectively. It is noted that measurements at higher superheats and voltages were not taken due to the stability-related issues mentioned earlier. In addition to water and isopropanol, this study uses the data from similar experiments with ethanol droplets in Ref. [19]. In that study, the experiments involved changing the applied voltage to keep the surface temperature constant, while ethanol droplets (3.5 mm diameter) were being constantly fed with preheated ethanol (to compensate for evaporation). The experimental setup used was similar to the present one. Fig. 7 shows the heat flux versus the applied voltage for five different superheat values for ethanol droplets (data extracted from [19]). It is noted that the measurements are presented as heat flux versus voltage at different superheats, since the study [19] achieved steady state by changing the applied voltage to bring the surface temperature to a desired value. This is different from the present experiments (Figs. 5 and 6), wherein the voltage and heater power are kept constant, and the surface temperature is allowed to reach a steady state value. Fig. 7 shows the heat flux increasing with the superheat and the applied voltage, similar to Figs. 5 and 6. The heat transfer was enhanced by 7.6X and 3.6X at superheats of 120 °C and 500 °C (at 250 V), respectively. This again illustrates the reduced influence of electric fields (manifested in lower wetted fractions) as the superheat increases. Also, the slopes of the heat flux curves at low superheat (eg. 120 °C) decrease as the voltage increases, whereas the heat fluxes at high superheats (eg. 500 °C) increase linearly with voltage. This can be attributed to the fact that wetted fractions do not increase indiscriminately with voltage, and will saturate. More insights on the combined influence of superheat and the applied voltage on wetted fractions is available in a recent study by the present authors [36]. All the experimental results (present experiments and the data from Ref. [19]) for the three fluids are captured together in Fig. 8. Fig. 8 shows the heat transfer enhancement coefficient (h+ = q /T ) versus applied voltage at different superheats. It is noted that h+ is not the heat transfer coefficient, but an equivalent parameter to quantify the enhancement. The presently developed model does not predict the total heat flux (or heat transfer coef-
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Fig. 8. Voltage-dependent heat transfer enhancement coefficient (q /T) associated with electrostatic suppression of Leidenfrost droplets of water, isopropanol and ethanol. The color map shows the values of the superheat.
ficient), but instead calculates the heat transfer due to transient contacts, which is essentially the heat transfer enhancement. The heat flux enhancement at a given superheat (q ) is calculated by subtracting the baseline (no-voltage) heat flux from the experimentally measured total heat flux in Figs. 5–7. Since the superheat values in Figs. 5 and 6 are randomly distributed, they are shown in the color map in Fig. 8. The heat transfer enhancement coefficient increases with voltage and decreases with superheat for all three liquids. This is a direct consequence of larger wetted fractions at higher voltages and lower superheats, also keeping in mind that the heat transfer enhancement scales as the square of the wetted fraction, as per Eq. (7). The heat transfer enhancement with water is larger than the heat transfer enhancement with IPA, because of the better heat transfer characteristics of water as compared to IPA. IPA and ethanol have similar thermal properties, which implies that the heat transfer enhancement numbers should be similar. However, the heat transfer enhancement with IPA is approximately half of the heat transfer enhancement with ethanol. This discrepancy can be explained by noting that the heat transfer due to transient contact depends on the thermal conductivity and diffusivity of the solid surface (in addition to the fluid), as per Eq. (1). Stainless steel and platinum surfaces were used for IPA and ethanol experiments, respectively. √ The thermophysical properties-depended parameter ( αs τ /K ks ) in Eq. (7), is two times larger for the IPA-stainless steel combination as compared to the ethanol-platinum combination. When the heat transfer enhancement values of both liquids are multiplied √ by the thermophysical properties-depended variable ( αs τ /K ks ), the dimensionless heat transfer enhancement becomes similar for the same superheat values and applied voltages. This aspect can also be understood via the subsequent discussion related to Fig. 9. Next, the predictions of presently developed heat transfer enhancement model are compared to the heat flux measurements reported in Figs. 5–8. Fig. 9 shows the experimental data reported in Fig. 8 in terms of the parameters used in the heat transfer model (Bi∗ , K, X). It is seen that the sparse data in Fig. 8 collapses on one line in Fig. 9. The slope of Eq. (18) (constant C) is found to be 25,435 as
Fig. 9. Correlation to predict heat transfer enhancement at different surface superheats and applied voltages.
per the linear regression. Therefore, Eq. (18) becomes:
V 1/2 2 q (αh τ )1/2 th Bi = = 25435K 1 − kh (Tw − Tsat ) V ∗
(19)
Eq. (19) is the experimentally validated correlation for predicting the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state. Only 3 out of 50 data points have an absolute error larger than 10 0 0 in Bi∗ /K. The relative error (error divided by the prediction as per Eq. (19)) increases at lower values of X2 (which is proportional to heat transfer enhancement). The relative error is always less than 10% when X2 is larger than 0.25. The heat transfer enhancement should also be compared to the heat flux of the Leidenfrost droplet (at zero voltage) for a better understanding of the accuracy of the present model. The heat flux enhancement term in Bi∗ /K can be replaced by the heat flux q0 of the Leidenfrost droplet (no-voltage). This yields the no-voltage
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modified Biot number Bi∗0 as:
References
1/2
Bi∗0 q (αh τ ) = 0 K ks (Tw − Tsat )K
(20)
It is noted that the heat flux divided by superheat (q0 /T ) is simply the heat transfer coefficient, which reduces with increasing temperature in the film boiling region [3]. The mean values of Bi∗0 /K are 40 0 0, 550 0 and 1500 for water, IPA and ethanol, respectively. Therefore, the absolute errors in Bi∗ /K in Fig. 9 are smaller than the Bi∗0 /K value of the Leidenfrost droplets (no-voltage) for most cases. The mean absolute error in Bi∗ /K over all data points in Fig. 9 was 524. One of the key findings of this work is the heat transfer enhancement scaling as the square of the wetted fraction. This is suggested by the analytical model and the experimental results support this hypothesis very well. It should be noted that such a relation will not necessarily be valid for classical nucleate boiling or transient boiling situations. Presently, heat transfer occurs via transient solid–liquid contacts, as opposed to bubble-related phenomena that influence nucleate and transition boiling. It is also noted that the wetted fraction will not increase indiscriminately with voltage. Our recent study [36] suggests that the wetted fraction will saturate in the 70–80% range, which suggests a limit on the heat transfer enhancement. Overall, Eq. (19) can be used to predict the heat transfer enhancement associated with electrostatic suppression of the Leidenfrost state under strong electric fields. The heat transfer enhancement can then be added to the heat transfer rate in the Leidenfrost state (no voltage), which can be calculated using existing correlations [38]. It is noted that the present model has been experimentally validated only for droplet sizes of the order of the capillary length of the fluid. Wetting dynamics and the related heat transfer enhancement is expected to be different at smaller or larger length scales [31]. 5. Conclusions Overall, this work outlines a comprehensive study of heat transfer enhancement enabled by electrostatic suppression of the Leidenfrost state. A first principles-based analytical model is developed to predict the heat transfer enhancement as a function of the applied electric field, surface temperature and fluid and surface properties. This model is strongly validated by experimental measurements (inhouse as well as by another group) with three fluids. The strength and utility of the modeling approach is further underscored by the presence of only one empirical parameter in the model. The results suggest that electrostatic suppression of the Leidenfrost state can enhance heat transfer by 4X for water, and enable larger enhancements for organic solvents. Declaration of Competing Interest The authors report no conflict of interest Acknowledgments The authors acknowledge National Science Foundation grants CBET-1605789, CBET-1653412 for supporting this work. This work was also partly supported by the UT Austin Portugal program and Google Faculty Research Award. The assistance of Awan Bhati in conducting some experiments is acknowledged. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2019.119207.
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