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Rip currents: A spontaneous heat transfer enhancement mechanism in a wickless heat pipe
International Journal of Heat and Mass Transfer 149 (2020) 119170
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Rip currents: A spontaneous heat transfer enhancement mechanism in a wickless heat pipe Thao T.T. Nguyen a, Jiaheng Yu a, Peter C. Wayner Jr a, Joel L. Plawsky a,∗, Akshay Kundan a, David F. Chao b, Ronald J. Sicker b a b
The Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute Troy, NY 12180, United States NASA Glenn Research Center, Cleveland, OH 44135, United States
a r t i c l e
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Article history: Received 3 August 2019 Revised 3 November 2019 Accepted 4 December 2019 Available online 13 January 2020 Keywords: Heat pipe Two Phase Heat Transfer Microgravity Rip current Marangoni flow Capillary flow
a b s t r a c t The liquid-vapor distribution and its effects on the fluid dynamics and heat transfer occurring within a wickless heat pipe are little understood, especially in a microgravity environment. Such information is vital to the design of thermal management systems for deep space robotic and manned exploration missions, especially if unexpected behaviors arise. We observed an unusual analog of a terrestrial rip current during the operation of a wickless heat pipe on the International Space Station. The current arose spontaneously as the heat input increased, flowing along the flat surfaces of the device toward the heater end, and was driven by a pair of counterrotating vortices that formed from the interaction of opposing surface tension and capillary driven corner flows. The current served as a natural way of increasing the total contact line length within the device and this enabled higher evaporation rates than would have been possible based on the engineered geometry of the device alone.
1. Introduction A heat pipe is a light, efficient, reliable, and widely used heat transfer device that combines thermal conduction and phase change to transfer energy between a heat source and a heat sink. Most contain wicks, but wickless heat pipes have an ideal environment to study the interaction between interfacial forces, fluid dynamics, and two-phase heat transfer. Due to their robustness, heat pipes continue to be studied extensively in academia and industry [1-13]. In collaboration with NASA, we studied a wickless heat pipe in a microgravity environment where interfacial forces overwhelm gravitational forces. Several new phenomena were revealed. Among them was the consequence of Marangoni forces, driven by the large surface tension gradient near the heater end, flooding the heater region and limiting device performance [14-16]. The same forces also created an instability in the thin liquid film on the wall surface near the heater end that led to condensation on highly superheated surfaces [17]. Herein, we report on another unexpected feature, the analog of a terrestrial rip current that developed spontaneously as the heat load to the device increased. Rip currents are defined as narrow,
∗
Corresponding author. E-mail addresses: [email protected] (P.C. Wayner Jr), [email protected] (J.L. Plawsky), [email protected] (D.F. Chao), [email protected] (R.J. Sicker). https://doi.org/10.1016/j.ijheatmasstransfer.2019.119170 0017-9310/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
intense, off-shore directed jets of water that originate within the surf zone and dissipate outside the breaking zone [18-25]. They flow at, and near, the surface with velocities up to one meter per second. Rip currents are driven by energy contained in ocean waves and have been studied over the last several decades both experimentally and numerically. Castelle et al. [21] created a laboratory system to investigate their characteristics over different representative beach morphologies and were able to find a correlation between the rip current intensity and beach shoreline nonuniformity. Numerical studies of rip current evolution have provided insights into rip current behavior, especially the vorticity mechanism generating the current [22-25]. The consensus is that there are two main components to a rip current; a pair of counterrotating vortices at the neck from which issues a narrow, intense, off-shore directed current, and an onshore flow that feeds water back into the neck region to sustain the current. In our wickless heat pipe, thermal energy drives the process and a rip current was established by a pair of counterrotating vortices that were formed by a Marangoni flow originating at the heater end and an opposing capillary flow originating at the cooler end. The capillary and Marangoni flows feed the vortices at the neck and the walls act as the shoreline inhomogeneity, that confines the flow and leads to the formation of the ejected current. Fluid vortices within a narrow tube, driven by Marangoni forces, have been previously reported [26]. However, those vortices were small and widely separated so no currents of the kind we
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The heat input was varied from 0.1 W to 3 W, the latter being the safety limits of the device. The cooler end was kept constant at 19 °C. To study the fluid mechanics within the heat pipe, we used optical interferometry. The technique allowed us to calculate the liquid film thickness and the shape of the vapor-liquid interface. This method is fast and doesn’t interfere with the physics of the heat pipe. A detailed description of how the interferometry technique works can be found in [27]. To obtain high resolution data, we used a microscope objective with 10X and 50X magnifications. The data in this paper are from the 10X magnification images. For presentation purposes, we included only compressed images. The original, high resolution images were used for the data analysis process and can be obtained from NASA’s Physical Sciences Informatics Database. Fig. 1. Experimental setup.
2.2. Data analysis see evolved. This is the first time a complete, rip current-type flow, driven solely by interfacial forces within a micro-channel, was observed. The current served to increase the contact line length available for evaporation, thereby promoting phase change processes inside the pipe. Based on the results of our experiments, we believe that for the rip current phenomenon to occur, we first need an appropriate geometry with broad, flat walls. The square geometry we used or a triangular cross section should be able to support the flows. The vortices generated by the structure must (i) span nearly the entire width of the wall and leave a small gap as in an automatic pitching machine and (ii) be strong. Many more experiments would have to be done to understand the maximum size of the heat pipe that could support a rip current and under what conditions such a current could occur. Our experiments were performed in the microgravity environment of the International Space Station where the effect of gravity was eliminated. This environment and the materials of our system promoted the formation of strong, opposing, capillary and Marangoni flows that generated strong vortices. For the rip current phenomena to occur under gravity, the heat pipe will likely have to be small enough and driven hard enough for capillary and Marangoni flows to overwhelm the effect of gravity.
From the thickness profile obtained using interferometry [27], we could calculate the curvature of the vapor-liquid interface and hence, the pressure profile in the liquid. Along the liquid-vapor interface, the pressure jump is given by the augmented Young– Laplace equation:
Pv − Pl = σ (K1 + K2 ) + where: •
• •
•
•
2. Materials and methods 2.1. Experimental setup The wickless heat pipe experiment was run on the International Space Station. The apparatus consisted of a fused silica cuvette with sharp corners that was partially filled with pure pentane as the working fluid. The cuvette was 3 mm x 3 mm in the internal cross-section and 5.5 mm x 5.5 mm in the external cross-section. A constant heat input was supplied to the heater end by a resistance heater and the cooler end was kept at a constant temperature by a cold finger connected to a set of thermoelectric coolers. Both the heat input and the cooler end temperature could be precisely varied to achieve different experimental conditions. Type E thermocouples were drilled into the walls of the cuvette at 1.5 mm intervals along the heat pipe axis to measure the axial temperature profile between the heater and condenser. Several thermocouples were also placed in the transverse direction to estimate the temperature change going from the center of a wall to the corner. Prior to launch of the experiment, the thermocouples were calibrated against known standards. The accuracy and resolution of the temperature measurements were ±0.5 °C. The overall pressure of the heat pipe was measured by a pressure transducer installed downstream from the cold finger. The pressure was measured to within an accuracy of ±350 Pa. The full view of the heat pipe setup was recorded by a surveillance camera (Fig. 1).
(1)
Pv is the vapor pressure. For a specific running condition of the heat pipe, the vapor pressure stays almost constant. Pl is the liquid pressure. σ is the surface tension of the liquid. σ varies slightly with small changes in temperature across the field of view of the microscope. For the several regions of interest in this paper, the change in curvature is significantly larger than the change in surface tension. In those cases, we assume a constant surface tension. K1 , K1 are the radii of curvature of the meniscus. In general, only one radius of curvature dominates the expression due to the high magnification and small field of view of our interferometric images. is the disjoining pressure. is negligible in the thick meniscus region which is the focus of this paper.
Under the near-constant vapor pressure in the heat pipe, the liquid pressure difference between two points, in close proximity to one another, follows:
−
dPl dK =σ dx dx
Or in more discrete terms:
Pl,1 − Pl,2 ≈ σ (K2 − K1 ) = σ
(2)
1 r2
−
1 r1
(3)
where: •
r = 1/K is the radius of curvature.
Eq. (3) shows that for near constant Pv , liquid flows from regions of high pressure (Pl,1 ) to regions of low pressure (Pl,2 ), or from regions with small curvature to large curvature, or from a large radius of curvature to a small radius of curvature. For liquid films exhibiting a positive radius of curvature, liquid flows from a high radius of curvature to a low radius of curvature; if the film has a negative radius of curvature (e.g. along the axial direction of the rip current), liquid flows from a low absolute radius of curvature to a high absolute radius of curvature. In this paper, to make it easy to follow, we keep the sign of the obtained curvature values in the plots so that fluid flows from a high radius of curvature to a low radius of curvature regardless of the film is concave or convex.
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Fig. 2. (a) 10X interferometry image of the CVB heat pipe at 3 W heat input when the rip current is the strongest. This composite image was stitched together from many small frames. (b) A magnified image of the heat pipe highlighting the rip current and the three main sub-regions of interest (A, B, and C).
The curvature of a meniscus can be calculated as the second derivative of the thickness profile with respect to position [27]. However, for 10X resolution images, there is generally not enough data available from the fringe images right at the contact line and so taking the second derivative of that experimental data generally introduces large errors. To overcome this, we used an older method developed by Sujanani and Wayner [28] √ that relates the change in the square root of the film thickness, δ , with respect to position to the local curvature. This method assumes that, outside of the immediate contact line region, the meniscus can be represented by a parabola. If so, then the curvature can be described by:
K=2
√ 2 d δ dx
(4)
If the meniscus is isothermal, the curvature should be constant √ and a plot of δ vs. x will be a straight line. A convex plot indicates evaporation along the contact line (Figs. 4(a) and 5(a)). 3. Results and discussion In microgravity, Marangoni and capillary forces control the macroscopic liquid flow within the heat pipe [16, 17]. A capillary force drives the liquid from the condenser end, where we have the liquid pool, to the heater end along a curvature gradient in the liquid-vapor interface (Fig. 2(a)). A Marangoni force drives liquid from the heater end to the condenser end along large temperature and surface tension gradients near the heater. When these two
opposing flows meet, a thick liquid drop, the “central drop”, containing two counterrotating vortices is formed on the flat surface of the heat pipe (region A in Fig. 2(b)) [16]. The strong Marangoni flow also creates a thick liquid film region in the corners near the heater end, called the “flooded” or interfacial flow region [16]. We developed a thermal model based on the measured wall temperature profile to calculate the heat flux to and from the inside wall of the heat pipe [27]. The model was useful to help understand the organization of the heat pipe. Based on the sign of the heat flow to the interior wall of the heat pipe, q , the heat pipe can be divided into three primary regions: an interfacial flow region, a classic evaporation region, and a classic condensation region as shown in Fig. 3. The interfacial flow region extends from the heater end to the drop and includes regions of condensation and evaporation [17]. It is within this region that the rip current appears. The model shows that evaporation is maximized at the central drop where q reaches a minimum. “Classic” heat pipe performance behavior occurs in the region from the central drop to the condenser. In this paper, we will focus on region (B) of Fig. 2 where the current appears. 3.1. Liquid flow patterns on the heat pipe surfaces We used interferometry to measure the shape of the liquidvapor interface and thereby infer flow patterns and regions of evaporation on the flat surfaces and in the corners of the heat pipe. Three regions within the interfacial flow region were of
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toward the first dark fringe at the drop’s edge (0 m) to replace liquid evaporating at the contact line. Interestingly, the curvature gradient at point A4, near 0, is noticeably larger than the other four positions measured. This means there is more fluid flow and evaporation along the axis of the drop that is colinear with the centerline axis of the cuvette. The A4 location is also where the two counterrotating vortices reach their closest approach to one another.
Fig. 3. Net heat flux to and from the inside wall of the heat pipe at 3 W heat input. The figure shows the different regions of the heat pipe corresponding with the visual division in Fig. 2(a) and (b).
interest and are shown in Figs. 2 and 3. The first is the source of the current at the central drop (A); the second is the "rip" current (B); and the third is the condensation region (C). In the rip current region, the liquid stream emanating from the drop is well-defined. At the boundary between regions (B) and (C), where q crosses 0, the heat transfer process switches from net evaporation (B) to net condensation (C). The large Marangoni forces in this region help drive film instability there and act to dissipate the rip current.
3.1.1. The central drop region The central drop was formed by opposing capillary and Marangoni flows. Both flows fed the drop which quickly became thick enough to refract light as the bright center of the drop in Fig. 2(b) shows. The central drop region exhibits the highest evaporation rate, and that rate should be strongest at the contact line running along the edge of the drop. We checked the film thickness profiles at five locations (A1-A5 in Fig. 2(b)) to verify that those regions were actively evaporating liquid and transporting liquid from the interior of the drop toward the contact line. In all regions shown in Fig. 4(a), a plot of the square root of film thickness vs. position was convex, indicating evaporation in the contact line region [29]. As the increasing radii of curvature with position in Fig. 4(b) show, liquid flows from the drop’s interior (3 × 10−5 m)
3.1.2. The rip current region Although there was a large amount of evaporation along the outer rim of the central drop, it could not keep pace with all the liquid flowing into the drop. The excess liquid was ejected as a stream flowing back toward the heater end (Fig. 2(b)). We investigated three representative slices at the beginning (B1), the middle (B2), and near the end (B3) of the stream. The square root of thickness profiles shown in Fig. 5(a) are also convex indicating that evaporation is occurring. This additional contact line length increases the amount of evaporation the device can sustain. The radii of curvature, shown in Fig. 5(b), are increasingly negative indicating the current has an overall convex shape and that liquid flows from the central drop (B1) toward the heater end (B3). This flow direction is confirmed by the increase in fringe spacing along the current’s centerline going from (B1) to (B3). Looking carefully at Fig. 2(b) between B2 and B3, we found a region of denser fringes, highlighted by the oval, that indicates a shallow, secondary drop. Liquid accumulates there because the rip current flow toward the heater is opposed by a Marangoni flow from the heater due to the large temperature gradients in the region. Beyond this local maximum, above point B3, the rip current gets torn apart by Marangoni forces, film instabilities, and condensation near the heater end [17]. Looking at the rip current at different heat inputs, three distinct behavioral regimes arise from the 10X images (Fig. 6(a)) and pressure data (Fig. 6(b)). From 0.1 W to 0.7 W, the heat supplied serves primarily to increase evaporation and with it, the pressure inside the cell. The vapor bubble, which is attached to the cold finger at 0 power input, moves toward the heater wall and the cell pressure increases sharply. From 0.7 W to 2.0 W, the added heat goes into expanding the interfacial flow region. This regime is associated with a slower rise in pressure as the location of maximum evaporation moves away from the heater end and film thicknesses increase in the corners. There is a subtle sign of the rip current in the images starting at a heat input of about 1.0 W. A distinct current signature occurs at heat inputs beyond 2.0 W, where the interfacial flow region no longer expands and the pressure begins
Fig. 4. (a) The convex square root of thickness profiles showing evaporation from the meniscus at the edge of the central drop. (b) Radii of curvature for regions along the edges of the central drop showing the liquid flows from the drop’s interior (3 × 10−5 m) toward the drop’s edge (0 m).
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Fig. 5. (a) The convex square root of thickness profiles showing evaporation along the edges. (b) The radius of curvature along the center of the rip current showing the liquid flows from the central drop toward the heater end (B1 to B3).
Fig. 6. (a) 10X interferometry images of the heat pipe at different heat inputs showing the development of the interfacial flow region, central drop, and the rip current. (b) Internal pressure at different heat inputs showing three distinct behavioral regimes. (c) Total heat of evaporation in the heat pipe and heat of evaporation in the rip current region alone for different heat inputs. The overall heat of evaporation increases with heat input, but the heat of evaporation within the rip current region stays almost constant up to 2 W. From 2 W heat input, the rip current starts growing quickly and this leads to increase of heat evaporation within the rip current region (square-marker line).
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increasing quickly due to added evaporation from the edges of the rip current and the central drop (Fig. 6(c)). As the heat input increases, capillary pumping from the condenser end and Marangoni flow from the heater end both increase, leading to a greater flow of liquid into the drop and an increase in vorticity. The vorticity generates a higher pressure and a larger force to pump the liquid toward the hot end. The total amount of heat absorbed by liquid evaporating in the heat pipe in Fig. 6(c) was calculated by integrating the net heat flux between the two locations where q crosses the zero-axis and taking the absolute value of the obtained value. The first location where q = 0 is the boundary between the condensation region and the rip current region; and the second location where q = 0 is the boundary between the classic evaporation region and the classic condensation region. To calculate the total amount of heat absorbed by evaporation in the region of the rip current, we integrated the net heat flux within the rip current region only, or from the first location where q = 0 to the location where q reaches its minimum value or the location of central drop, and plot its magnitude as well in Fig. 6(c). The figure shows a jump of about 25% in the evaporation rate at a heat input beyond 2 W as the rip current becomes fully established and the interfacial region stops expanding toward the condenser. We attribute the jump to the evaporation from the edges of the current though much more detailed temperature profile data would be needed to quantify exactly how much of that increase is due solely to evaporation from the rip current itself. To see how large the pressure gradient of the rip current was relative to other driving forces inside the heat pipe, we formed a rough estimate of the gradient of the capillary pressure driving the liquid from the cold end toward the drop:
P 2σ = x r∗x
•
•
In Fig. 8(a), we connect the flow in the three regions to generate a full picture of the processes leading to the rip current pattern. In this case, the capillary flow (blue arrows) and the Marangoni flow (two large black arrows along the side) act as the rip current feeder. When the two flows meet, a large liquid drop containing the two vortices develops and leads to the formation of a rip current shooting out toward the heater direction (the large red arrow). The current forms an additional channel that increases the overall length of the contact line between the heater wall and the drop and so promotes additional evaporation. 3.2. Modeling the fluid flow in the interfacial flow region
(5)
where: •
Fig. 7. Pressure gradient along the axial direction of the rip current at different heat inputs.
σ is the surface tension of pentane. For estimation purposes, σ ≈ 0.016 (N/m). r is the radius of curvature in the liquid pool at the cold end of the heat pipe. Given the internal dimension of the cuvette is 3 mm by 3 mm, r ≈ 1.5 (mm). x is the length of the vapor bubble from the cold end to the central drop and x ≈ 25 (mm).
Therefore, the capillary pressure gradient was about 850 Pa/m. The pressure gradient driving the rip current was estimated using the radii of curvature differences at B1 and B3 for each power input. These are shown in Fig. 7. The pressure gradient driving the rip current is the same order of magnitude as the capillary pressure gradient driving liquid from the cold end to the central drop. 3.1.3. The condensation region Near the heater end at the top of Fig. 8(a), the temperature gradient becomes very large, and the Marangoni force drives liquid from the hot end to the cold end (three large black arrows in Fig. 8(a)). On the flat surface, this Marangoni flow directly opposes the rip current and helps to dissipate and stop the current entirely in region (C). In the transverse direction perpendicular to the heater/condenser axis, the temperature difference between the centerline and the corners near the heater (Fig. 9) drives the fluid from the center of the wall to the corners of the heat pipe. With increasing heat input, this temperature difference increases, there is increasing condensation, a thicker liquid film on the wall surface [17], and stronger Marangoni flow from the wall center to the corners of the heat pipe. Thus, flow from the center toward the corners helps feed the Marangoni flow in the corners.
The liquid flows and heat transfer that give rise to the phenomena we observe are fully three-dimensional phenomena and arise from a communication between fluid-thermal processes occurring throughout the heat pipe. We hypothesize that the current occurs as a result of two counter-rotating vortices contained within the central drop region. As these vortices spin, they take the fluid feeding the central drop from the Marangoni and capillary flows in the corners and from the thin film on the flat wall downstream of the drop and then eject that fluid in the form of a surface current. This mechanism is similar to how rip currents are generated at a shoreline [18-26]. While a full simulation of the process is well beyond the scope of this paper, we were interested if this minimalist approach, proven useful for describing the features of the rip current neck, would give us a flow, resembling what we observe. Here, the hypothesis was that as we increase the heat input, the Marangoni and capillary return flows get stronger, the vortices within the central drop spin faster and so we have a more well-defined and larger ejected current and wake. We set up a simple model similar with the model used to reproduce the rip current neck in [24]. We used a two-dimensional geometry of the same size and shape as the heat pipe, 3 mm wide and 40 mm long. We then placed two counter-rotating cylinders at a location that is a similar distance from the heater as the central drop appeared in the experiments when heat input is above 2 W. The cylinders also mimicked the size of the central drop, spanning nearly the entire width of the cell. We applied no-slip conditions on the boundaries of the simulation domain and on the surface of the cylinders. We then spun the cylinders, one clockwise and the other counterclockwise, at high and low linear velocities to mimic high and low heat inputs. Fig. 8(b) and (c) show the results of the simulation for both magnitudes of rotation rate. At a relatively low rotation rate (Fig. 8(b)), the current flowing toward the heater is weak, and at
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Fig. 8. Full picture of the fluid flow in the interfacial flow region. (a) Experimental result. Black arrows show the Marangoni flow. Red arrows show the flow due to pressure gradient. Blue arrows show the capillary flow from the cold end. (b) Simulation result for slow rotation rate. (c) Simulation result for high rotation rate (an order of magnitude faster than (b)).
pointing toward the condenser end that also exhibits a kind of nascent current. So, the model, though crude, seems to support some of the experimental observations including the odd wake pattern that exists in the thin film region downstream of the drop. Much more 3-D simulation work will be required to fully understand and simulate all aspects of what we observed in this system.
3.3. Rip current dependence on condenser temperature
Fig. 9. The temperature difference between the center and the side of the wall of the heat pipe at the heater end showing that fluid flows from the center of the wall to the corners of the heat pipe due to a Marangoni driving force.
the high rate (Fig. 8(c)), the current is stronger, well-defined, and penetrates about 50% further toward the heater. The wake region downstream of the drop in Fig. 8(c), is also more well-defined, than in Fig. 8(b), and there is a faint region along the central axis
So far, there is evidence that the rip current adds an additional channel to help dissipate heat within the heat pipe, and that the strength of the current increases with increasing heat input. To test whether the heat input alone governs the size of the interfacial region and the appearance of a rip current, we held the heat input constant at 2 W and varied the condenser temperature from 25°C to −5°C. Given that the condenser could be nearly 40 mm from the interfacial region, the condenser temperature still significantly affected the extent of the interfacial region, the overall evaporation rate, and whether a rip current appeared (Fig. 10). When the cooler temperature decreased from 25°C to −5°C, the length of the interfacial region doubled. The rip current was evident at 25 °C but became weaker at 19°C and disappeared entirely at 10°C and −5°C.
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Fig. 10. Rip current and interfacial flow region at different condenser temperatures. (a) 10X images showing large changes in the rip current (black arrows) and interfacial flow region (red arrows). (b) A magnified version showing the rip current region.
Fig. 11. (a) Strong correlation between the length of the interfacial flow region and the central drop location and the surface tension gradient. (b) Surface tension gradient from the heater to the drop and from the drop to the cold end showing less fluid flow into the drop from both ends.
We can attribute the change in the interfacial length to the large increase in the overall surface tension gradient between the hot end and the cold end and the decrease in condenser temperature. The strong correlation between the length of the interfacial flow region and the central drop location and the total surface tension gradient between the heater end and the cooler end is evident in Fig. 11(a). The surface tension gradient in this figure and Fig. 11(b) is calculated as followed.
dσ = dx
|σ1 − σ2 | x
(6)
Where σ 1 and σ 2 are the surface tension at the two ends of the region of interest based on our measured temperatures
at those locations. For example, to calculate the total surface tension gradient in Fig. 11(a), we use the temperature at the first thermocouple at the heater wall and the thermocouple at the liquid pool. x is the distance between those two points. We can attribute the change in rip current to two effects. As the condenser temperature decreases, the cell pressure decreases and the point where wholesale condensation first appears occurs at a lower wall temperature. The overall evaporation and condensation rates both decrease and this leads to a decrease in total fluid flow within the heat pipe. At the same time, at low cooler temperature, the changes in central drop location and condenser end temperature leads to weaker Marangoni flow into the drop from the heater end and weaker capillary flow into the drop
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Fig. 12. (a) Temperature profiles with solid dots showing the location of central drop. (b) Total heat absorbed by evaporation in the heat pipe (classic evaporation region and rip current region) and heat absorbed by evaporation in the rip current region alone for different cooler temperatures. At −5 °C and 10 °C, there is no rip current, evaporation occurs along the four corners and central drop and therefore, there is no change in heat absorbed by the rip current region (square-marker line). At 19 °C and 25 °C, the rip current appears and grows stronger, and the amount of heat absorbed by evaporation in the rip current region increases.
from the condenser due to a stronger opposing Marangoni flow from the drop to the condenser end (Fig. 11(b)). In Fig. 10, the fainter appearance of the central drop with decreasing condenser end temperature indicates a thinner drop and this means there is weaker flow into the drop. A weaker flow into the drop also leads to slower vortex rotation and a weaker rip current. This explains the relatively strong rip current at 25°C and the absence of one at –5°C. The dissipation of the rip current at low condenser temperatures results in lower heat transfer efficiency. This is evidenced by the slower decrease in temperature near the heater end at low cooler temperatures in Fig. 12(a) and lower amounts of heat absorbed via evaporation (Fig. 12(b)). The lower heat transfer efficiency at these conditions is offset by the larger temperature difference between the heater and cooler end. The result from this section again shows the role of the rip current in increasing overall evaporation and the heat transfer efficiency. 4. Conclusions Using both experimental and simulation techniques, we attempted to explain some of the remaining odd flow behaviors we observed inside a wickless heat pipe in microgravity where the effect of the capillary and Marangoni forces dominate. We found that the two forces oppose one another to create two counterrotating vortices contained within the central drop region of the heat pipe. As these vortices spin, they take the fluid feeding the central drop from the corners and eject it in the form of a surface current. The whole fluid flow pattern follows the mechanism of a rip current along the shore line. The stronger the Marangoni and capillary flows, the stronger the rip current. It was interesting to find how extensive information transfer is within the heat pipe. Although the rip current is associated mainly with the heater end, its shape and size change significantly with changes in both heat input and condenser end temperature. The rip current plays a critical role in maintaining heat transfer effectiveness by providing an additional channel for evaporation in the region near the heater end. It also provides an outlet for the excess liquid entering the central drops that cannot be evaporated from the drop. This prevents the drops on each flat surface from growing, merging together, and completely choking off vapor flow between the heater and the cooler ends. Declaration of Competing Interest We have no conflicts of interest to disclose.
Acknowledgement This material is based on the work supported by the National Aeronautics and Space Administration under Grant number NNX13AQ78G and the National Science Foundation under Grant No. CBET-1603318. Special thanks to Dr. Tyson DiLorenzo for proofreading. References [1] A. Faghri, Heat Pipe Science and Technology, D.C. Washington (Ed.), Taylor and Francis, 1995. [2] G.P. Peterson, An Introduction to Heat Pipes: Modeling, Testing, and Applications, Wiley Series in Thermal Management of Microelectronic and Electronic Systems, New York, 1994. [3] T.P. Cotter, Principles and prospects of micro heat pipes, in: Proc. 5th Int. Heat Pipe Conf., Tsukuba, Japan, 1984, pp. 328–335. [4] G.P. Peterson, Overview of micro heat pipe research and development, Appl. Mech. Rev. 45 (5) (1992) 175–189. [5] A. Faghri, Review and advances in heat pipe science and technology, J. Heat Transfer 134 (2012) 123001. [6] I. Catton, G.R. Stroes, A semi- analytical model to predict the capillary limit of heated inclined triangular capillary grooves, J. Heat Transfer 124 (2002) 162–168. [7] H.B. Ma, G.P. Peterson, Experimental investigation of the maximum heat transport in triangular grooves, J. Heat Transfer 118 (1996) 740–746. [8] C.P. Migliaccio, K. Dhavaleswarapu, S.V. Garimella, Temperature measurements near the contact line of an evaporating meniscus V-groove, Int. J. Heat Mass Transfer 54 (2011) 1520–1526. [9] B.R. Babin, G.P. Peterson, D. Wu, Steady-state modelling and testing of a micro heat pipe, J. Heat Transfer 112 (1990) 595–601. [10] D. Khrustalev, A. Faghri, Thermal analysis of a micro heat pipe, J. Heat Transfer 116 (1994) 189–198. [11] L.W. Swanson, G.P. Peterson, The interfacial thermodynamics of micro heat pipes, J. Heat Transfer 117 (1995) 195–201. [12] W.J. Bowman, J.K. Storey, K.I. Svensson, Analytical comparison of constant area, adiabatic tip, standard fins and heat pipe fins, J. Thermophys. Heat Transfer 13 (2) (1999) 269–272. [13] W.J. Bowman, T.W. Moss, D. Maynes, K.A. Paulson, Efficiency of a constant-area, adiabatic tip, heat pipe fin, J. Thermophys. Heat Transfer 14 (1) (20 0 0) 112–115. [14] L. Yang, G.M. Homsy, Steady three-dimensional thermocapillary flows and dry-out inside a V-shaped wedge, Phys. Fluids 18 (2006) 042107. [15] R. Savino, D. Paterna, Marangoni effect and heat pipe dry-out, Phys. Fluids 18 (2006) 118103. [16] A. Kundan, J.L. Plawsky, P.C. Wayner Jr., D.F. Chao, R.J. Sicker, B.J. Motil, T. Lorik, L. Chestney, J. Eustace, J. Zoldak, Thermocapillary phenomena and performance limitations of a wickless heat pipe in microgravity, Phys. Rev. Lett. 114 (2015) 146105. [17] A. Kundan, T. Nguyen, J. Plawsky, P. Wayner, D. Chao, R. Sicker, Condensation on highly superheated surfaces: unstable thin films in a wickless heat pipe, Phys. Rev. Lett. 118 (2017) 094501. [18] F.P. Shepard, K.O. Emery, E.C. La Fond, Rip currents: a process of geological importance, J. Geol. 49 (1941) 337–369. [19] P. McKenzie, Rip current systems, J. Geol. 66 (1958) 103–113. [20] C.J. Sonu, Field observation of nearshore circulation and meandering currents, J. Geophys. Res. 77 (1972) 3232–3247.
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[21] B. Castelle, H. Michallet, V. Marieu, F. Leckler, B. Dubardier, A. Lambert, C. Berni, P. Bonneton, E. Barthélemy, F. Bouchette, Laboratory experiment on rip current circulations over a moveable bed: drifter measurements, J. Geophys. Res. 115 (2010) C12008. [22] B.A. Ebersole, R.A. Dalrymple, Numerical modelling of nearshore circulation, in: B.L. Edge (Ed.), 17th Int. Conf. Coastal Engineering Sydney, ASCE, 1980, pp. 2710–2725. [23] K.A. Haas, I.A. Svendsen, M.C. Haller, Numerical modelling of nearshore circulation on a barred beach with a rip current, in: Proc. 26th Intl Conf. on Coast. Engineering – ASCE vol. 1 (ed. B. L. Edge), 1998, pp. 801–814. [24] A. Kennedy, A circulation description of a rip current neck, J. Fluid Mech. 497 (2003) 225–234. [25] B. Philippe, N. Bruneau, B. Castelle, F. Marche, Large-scale vorticity generation due to dissipating waves in the surf zone, Discr. Cont. Dyn. Syst. - B 13 (2010) 729–738.
[26] C. Buffone, K. Sefiane, C. Minetti, The effect of wall thickness and material on Marangoni driven convection in capillaries, Colloid. Surf. A 481 (2015) 384–392. [27] T. Nguyen, A. Kundan, P. Wayner, J. Plawsky, D. Chao, R. Sicker, Experimental study of the heated contact line region for a pure fluid and binary fluid mixture in microgravity, J. Colloid Interface Sci. 488 (2017) 48–60. [28] M. Sujanani, P.C. Wayner, Microcomputer-enhanced optical investigation of transport processes with phase change in near-equilibrium thin liquid films, J. Colloid Interface Sci. 143 (1991) 472–488. [29] S. DasGupta, I. Kim, P. Wayner, Use of the Kelvin-Clapeyron equation to model an evaporating curved microfilm, J. Heat Transfer 116 (4) (1994) 1007–1015.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Rip currents: A spontaneous heat transfer enhancement mechanism in a wickless heat pipe Thao T.T. Nguyen a, Jiaheng Yu a, Peter C. Wayner Jr a, Joel L. Plawsky a,∗, Akshay Kundan a, David F. Chao b, Ronald J. Sicker b a b
The Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute Troy, NY 12180, United States NASA Glenn Research Center, Cleveland, OH 44135, United States
a r t i c l e
i n f o
Article history: Received 3 August 2019 Revised 3 November 2019 Accepted 4 December 2019 Available online 13 January 2020 Keywords: Heat pipe Two Phase Heat Transfer Microgravity Rip current Marangoni flow Capillary flow
a b s t r a c t The liquid-vapor distribution and its effects on the fluid dynamics and heat transfer occurring within a wickless heat pipe are little understood, especially in a microgravity environment. Such information is vital to the design of thermal management systems for deep space robotic and manned exploration missions, especially if unexpected behaviors arise. We observed an unusual analog of a terrestrial rip current during the operation of a wickless heat pipe on the International Space Station. The current arose spontaneously as the heat input increased, flowing along the flat surfaces of the device toward the heater end, and was driven by a pair of counterrotating vortices that formed from the interaction of opposing surface tension and capillary driven corner flows. The current served as a natural way of increasing the total contact line length within the device and this enabled higher evaporation rates than would have been possible based on the engineered geometry of the device alone.
1. Introduction A heat pipe is a light, efficient, reliable, and widely used heat transfer device that combines thermal conduction and phase change to transfer energy between a heat source and a heat sink. Most contain wicks, but wickless heat pipes have an ideal environment to study the interaction between interfacial forces, fluid dynamics, and two-phase heat transfer. Due to their robustness, heat pipes continue to be studied extensively in academia and industry [1-13]. In collaboration with NASA, we studied a wickless heat pipe in a microgravity environment where interfacial forces overwhelm gravitational forces. Several new phenomena were revealed. Among them was the consequence of Marangoni forces, driven by the large surface tension gradient near the heater end, flooding the heater region and limiting device performance [14-16]. The same forces also created an instability in the thin liquid film on the wall surface near the heater end that led to condensation on highly superheated surfaces [17]. Herein, we report on another unexpected feature, the analog of a terrestrial rip current that developed spontaneously as the heat load to the device increased. Rip currents are defined as narrow,
∗
Corresponding author. E-mail addresses: [email protected] (P.C. Wayner Jr), [email protected] (J.L. Plawsky), [email protected] (D.F. Chao), [email protected] (R.J. Sicker). https://doi.org/10.1016/j.ijheatmasstransfer.2019.119170 0017-9310/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
intense, off-shore directed jets of water that originate within the surf zone and dissipate outside the breaking zone [18-25]. They flow at, and near, the surface with velocities up to one meter per second. Rip currents are driven by energy contained in ocean waves and have been studied over the last several decades both experimentally and numerically. Castelle et al. [21] created a laboratory system to investigate their characteristics over different representative beach morphologies and were able to find a correlation between the rip current intensity and beach shoreline nonuniformity. Numerical studies of rip current evolution have provided insights into rip current behavior, especially the vorticity mechanism generating the current [22-25]. The consensus is that there are two main components to a rip current; a pair of counterrotating vortices at the neck from which issues a narrow, intense, off-shore directed current, and an onshore flow that feeds water back into the neck region to sustain the current. In our wickless heat pipe, thermal energy drives the process and a rip current was established by a pair of counterrotating vortices that were formed by a Marangoni flow originating at the heater end and an opposing capillary flow originating at the cooler end. The capillary and Marangoni flows feed the vortices at the neck and the walls act as the shoreline inhomogeneity, that confines the flow and leads to the formation of the ejected current. Fluid vortices within a narrow tube, driven by Marangoni forces, have been previously reported [26]. However, those vortices were small and widely separated so no currents of the kind we
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The heat input was varied from 0.1 W to 3 W, the latter being the safety limits of the device. The cooler end was kept constant at 19 °C. To study the fluid mechanics within the heat pipe, we used optical interferometry. The technique allowed us to calculate the liquid film thickness and the shape of the vapor-liquid interface. This method is fast and doesn’t interfere with the physics of the heat pipe. A detailed description of how the interferometry technique works can be found in [27]. To obtain high resolution data, we used a microscope objective with 10X and 50X magnifications. The data in this paper are from the 10X magnification images. For presentation purposes, we included only compressed images. The original, high resolution images were used for the data analysis process and can be obtained from NASA’s Physical Sciences Informatics Database. Fig. 1. Experimental setup.
2.2. Data analysis see evolved. This is the first time a complete, rip current-type flow, driven solely by interfacial forces within a micro-channel, was observed. The current served to increase the contact line length available for evaporation, thereby promoting phase change processes inside the pipe. Based on the results of our experiments, we believe that for the rip current phenomenon to occur, we first need an appropriate geometry with broad, flat walls. The square geometry we used or a triangular cross section should be able to support the flows. The vortices generated by the structure must (i) span nearly the entire width of the wall and leave a small gap as in an automatic pitching machine and (ii) be strong. Many more experiments would have to be done to understand the maximum size of the heat pipe that could support a rip current and under what conditions such a current could occur. Our experiments were performed in the microgravity environment of the International Space Station where the effect of gravity was eliminated. This environment and the materials of our system promoted the formation of strong, opposing, capillary and Marangoni flows that generated strong vortices. For the rip current phenomena to occur under gravity, the heat pipe will likely have to be small enough and driven hard enough for capillary and Marangoni flows to overwhelm the effect of gravity.
From the thickness profile obtained using interferometry [27], we could calculate the curvature of the vapor-liquid interface and hence, the pressure profile in the liquid. Along the liquid-vapor interface, the pressure jump is given by the augmented Young– Laplace equation:
Pv − Pl = σ (K1 + K2 ) + where: •
• •
•
•
2. Materials and methods 2.1. Experimental setup The wickless heat pipe experiment was run on the International Space Station. The apparatus consisted of a fused silica cuvette with sharp corners that was partially filled with pure pentane as the working fluid. The cuvette was 3 mm x 3 mm in the internal cross-section and 5.5 mm x 5.5 mm in the external cross-section. A constant heat input was supplied to the heater end by a resistance heater and the cooler end was kept at a constant temperature by a cold finger connected to a set of thermoelectric coolers. Both the heat input and the cooler end temperature could be precisely varied to achieve different experimental conditions. Type E thermocouples were drilled into the walls of the cuvette at 1.5 mm intervals along the heat pipe axis to measure the axial temperature profile between the heater and condenser. Several thermocouples were also placed in the transverse direction to estimate the temperature change going from the center of a wall to the corner. Prior to launch of the experiment, the thermocouples were calibrated against known standards. The accuracy and resolution of the temperature measurements were ±0.5 °C. The overall pressure of the heat pipe was measured by a pressure transducer installed downstream from the cold finger. The pressure was measured to within an accuracy of ±350 Pa. The full view of the heat pipe setup was recorded by a surveillance camera (Fig. 1).
(1)
Pv is the vapor pressure. For a specific running condition of the heat pipe, the vapor pressure stays almost constant. Pl is the liquid pressure. σ is the surface tension of the liquid. σ varies slightly with small changes in temperature across the field of view of the microscope. For the several regions of interest in this paper, the change in curvature is significantly larger than the change in surface tension. In those cases, we assume a constant surface tension. K1 , K1 are the radii of curvature of the meniscus. In general, only one radius of curvature dominates the expression due to the high magnification and small field of view of our interferometric images. is the disjoining pressure. is negligible in the thick meniscus region which is the focus of this paper.
Under the near-constant vapor pressure in the heat pipe, the liquid pressure difference between two points, in close proximity to one another, follows:
−
dPl dK =σ dx dx
Or in more discrete terms:
Pl,1 − Pl,2 ≈ σ (K2 − K1 ) = σ
(2)
1 r2
−
1 r1
(3)
where: •
r = 1/K is the radius of curvature.
Eq. (3) shows that for near constant Pv , liquid flows from regions of high pressure (Pl,1 ) to regions of low pressure (Pl,2 ), or from regions with small curvature to large curvature, or from a large radius of curvature to a small radius of curvature. For liquid films exhibiting a positive radius of curvature, liquid flows from a high radius of curvature to a low radius of curvature; if the film has a negative radius of curvature (e.g. along the axial direction of the rip current), liquid flows from a low absolute radius of curvature to a high absolute radius of curvature. In this paper, to make it easy to follow, we keep the sign of the obtained curvature values in the plots so that fluid flows from a high radius of curvature to a low radius of curvature regardless of the film is concave or convex.
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Fig. 2. (a) 10X interferometry image of the CVB heat pipe at 3 W heat input when the rip current is the strongest. This composite image was stitched together from many small frames. (b) A magnified image of the heat pipe highlighting the rip current and the three main sub-regions of interest (A, B, and C).
The curvature of a meniscus can be calculated as the second derivative of the thickness profile with respect to position [27]. However, for 10X resolution images, there is generally not enough data available from the fringe images right at the contact line and so taking the second derivative of that experimental data generally introduces large errors. To overcome this, we used an older method developed by Sujanani and Wayner [28] √ that relates the change in the square root of the film thickness, δ , with respect to position to the local curvature. This method assumes that, outside of the immediate contact line region, the meniscus can be represented by a parabola. If so, then the curvature can be described by:
K=2
√ 2 d δ dx
(4)
If the meniscus is isothermal, the curvature should be constant √ and a plot of δ vs. x will be a straight line. A convex plot indicates evaporation along the contact line (Figs. 4(a) and 5(a)). 3. Results and discussion In microgravity, Marangoni and capillary forces control the macroscopic liquid flow within the heat pipe [16, 17]. A capillary force drives the liquid from the condenser end, where we have the liquid pool, to the heater end along a curvature gradient in the liquid-vapor interface (Fig. 2(a)). A Marangoni force drives liquid from the heater end to the condenser end along large temperature and surface tension gradients near the heater. When these two
opposing flows meet, a thick liquid drop, the “central drop”, containing two counterrotating vortices is formed on the flat surface of the heat pipe (region A in Fig. 2(b)) [16]. The strong Marangoni flow also creates a thick liquid film region in the corners near the heater end, called the “flooded” or interfacial flow region [16]. We developed a thermal model based on the measured wall temperature profile to calculate the heat flux to and from the inside wall of the heat pipe [27]. The model was useful to help understand the organization of the heat pipe. Based on the sign of the heat flow to the interior wall of the heat pipe, q , the heat pipe can be divided into three primary regions: an interfacial flow region, a classic evaporation region, and a classic condensation region as shown in Fig. 3. The interfacial flow region extends from the heater end to the drop and includes regions of condensation and evaporation [17]. It is within this region that the rip current appears. The model shows that evaporation is maximized at the central drop where q reaches a minimum. “Classic” heat pipe performance behavior occurs in the region from the central drop to the condenser. In this paper, we will focus on region (B) of Fig. 2 where the current appears. 3.1. Liquid flow patterns on the heat pipe surfaces We used interferometry to measure the shape of the liquidvapor interface and thereby infer flow patterns and regions of evaporation on the flat surfaces and in the corners of the heat pipe. Three regions within the interfacial flow region were of
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toward the first dark fringe at the drop’s edge (0 m) to replace liquid evaporating at the contact line. Interestingly, the curvature gradient at point A4, near 0, is noticeably larger than the other four positions measured. This means there is more fluid flow and evaporation along the axis of the drop that is colinear with the centerline axis of the cuvette. The A4 location is also where the two counterrotating vortices reach their closest approach to one another.
Fig. 3. Net heat flux to and from the inside wall of the heat pipe at 3 W heat input. The figure shows the different regions of the heat pipe corresponding with the visual division in Fig. 2(a) and (b).
interest and are shown in Figs. 2 and 3. The first is the source of the current at the central drop (A); the second is the "rip" current (B); and the third is the condensation region (C). In the rip current region, the liquid stream emanating from the drop is well-defined. At the boundary between regions (B) and (C), where q crosses 0, the heat transfer process switches from net evaporation (B) to net condensation (C). The large Marangoni forces in this region help drive film instability there and act to dissipate the rip current.
3.1.1. The central drop region The central drop was formed by opposing capillary and Marangoni flows. Both flows fed the drop which quickly became thick enough to refract light as the bright center of the drop in Fig. 2(b) shows. The central drop region exhibits the highest evaporation rate, and that rate should be strongest at the contact line running along the edge of the drop. We checked the film thickness profiles at five locations (A1-A5 in Fig. 2(b)) to verify that those regions were actively evaporating liquid and transporting liquid from the interior of the drop toward the contact line. In all regions shown in Fig. 4(a), a plot of the square root of film thickness vs. position was convex, indicating evaporation in the contact line region [29]. As the increasing radii of curvature with position in Fig. 4(b) show, liquid flows from the drop’s interior (3 × 10−5 m)
3.1.2. The rip current region Although there was a large amount of evaporation along the outer rim of the central drop, it could not keep pace with all the liquid flowing into the drop. The excess liquid was ejected as a stream flowing back toward the heater end (Fig. 2(b)). We investigated three representative slices at the beginning (B1), the middle (B2), and near the end (B3) of the stream. The square root of thickness profiles shown in Fig. 5(a) are also convex indicating that evaporation is occurring. This additional contact line length increases the amount of evaporation the device can sustain. The radii of curvature, shown in Fig. 5(b), are increasingly negative indicating the current has an overall convex shape and that liquid flows from the central drop (B1) toward the heater end (B3). This flow direction is confirmed by the increase in fringe spacing along the current’s centerline going from (B1) to (B3). Looking carefully at Fig. 2(b) between B2 and B3, we found a region of denser fringes, highlighted by the oval, that indicates a shallow, secondary drop. Liquid accumulates there because the rip current flow toward the heater is opposed by a Marangoni flow from the heater due to the large temperature gradients in the region. Beyond this local maximum, above point B3, the rip current gets torn apart by Marangoni forces, film instabilities, and condensation near the heater end [17]. Looking at the rip current at different heat inputs, three distinct behavioral regimes arise from the 10X images (Fig. 6(a)) and pressure data (Fig. 6(b)). From 0.1 W to 0.7 W, the heat supplied serves primarily to increase evaporation and with it, the pressure inside the cell. The vapor bubble, which is attached to the cold finger at 0 power input, moves toward the heater wall and the cell pressure increases sharply. From 0.7 W to 2.0 W, the added heat goes into expanding the interfacial flow region. This regime is associated with a slower rise in pressure as the location of maximum evaporation moves away from the heater end and film thicknesses increase in the corners. There is a subtle sign of the rip current in the images starting at a heat input of about 1.0 W. A distinct current signature occurs at heat inputs beyond 2.0 W, where the interfacial flow region no longer expands and the pressure begins
Fig. 4. (a) The convex square root of thickness profiles showing evaporation from the meniscus at the edge of the central drop. (b) Radii of curvature for regions along the edges of the central drop showing the liquid flows from the drop’s interior (3 × 10−5 m) toward the drop’s edge (0 m).
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Fig. 5. (a) The convex square root of thickness profiles showing evaporation along the edges. (b) The radius of curvature along the center of the rip current showing the liquid flows from the central drop toward the heater end (B1 to B3).
Fig. 6. (a) 10X interferometry images of the heat pipe at different heat inputs showing the development of the interfacial flow region, central drop, and the rip current. (b) Internal pressure at different heat inputs showing three distinct behavioral regimes. (c) Total heat of evaporation in the heat pipe and heat of evaporation in the rip current region alone for different heat inputs. The overall heat of evaporation increases with heat input, but the heat of evaporation within the rip current region stays almost constant up to 2 W. From 2 W heat input, the rip current starts growing quickly and this leads to increase of heat evaporation within the rip current region (square-marker line).
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increasing quickly due to added evaporation from the edges of the rip current and the central drop (Fig. 6(c)). As the heat input increases, capillary pumping from the condenser end and Marangoni flow from the heater end both increase, leading to a greater flow of liquid into the drop and an increase in vorticity. The vorticity generates a higher pressure and a larger force to pump the liquid toward the hot end. The total amount of heat absorbed by liquid evaporating in the heat pipe in Fig. 6(c) was calculated by integrating the net heat flux between the two locations where q crosses the zero-axis and taking the absolute value of the obtained value. The first location where q = 0 is the boundary between the condensation region and the rip current region; and the second location where q = 0 is the boundary between the classic evaporation region and the classic condensation region. To calculate the total amount of heat absorbed by evaporation in the region of the rip current, we integrated the net heat flux within the rip current region only, or from the first location where q = 0 to the location where q reaches its minimum value or the location of central drop, and plot its magnitude as well in Fig. 6(c). The figure shows a jump of about 25% in the evaporation rate at a heat input beyond 2 W as the rip current becomes fully established and the interfacial region stops expanding toward the condenser. We attribute the jump to the evaporation from the edges of the current though much more detailed temperature profile data would be needed to quantify exactly how much of that increase is due solely to evaporation from the rip current itself. To see how large the pressure gradient of the rip current was relative to other driving forces inside the heat pipe, we formed a rough estimate of the gradient of the capillary pressure driving the liquid from the cold end toward the drop:
P 2σ = x r∗x
•
•
In Fig. 8(a), we connect the flow in the three regions to generate a full picture of the processes leading to the rip current pattern. In this case, the capillary flow (blue arrows) and the Marangoni flow (two large black arrows along the side) act as the rip current feeder. When the two flows meet, a large liquid drop containing the two vortices develops and leads to the formation of a rip current shooting out toward the heater direction (the large red arrow). The current forms an additional channel that increases the overall length of the contact line between the heater wall and the drop and so promotes additional evaporation. 3.2. Modeling the fluid flow in the interfacial flow region
(5)
where: •
Fig. 7. Pressure gradient along the axial direction of the rip current at different heat inputs.
σ is the surface tension of pentane. For estimation purposes, σ ≈ 0.016 (N/m). r is the radius of curvature in the liquid pool at the cold end of the heat pipe. Given the internal dimension of the cuvette is 3 mm by 3 mm, r ≈ 1.5 (mm). x is the length of the vapor bubble from the cold end to the central drop and x ≈ 25 (mm).
Therefore, the capillary pressure gradient was about 850 Pa/m. The pressure gradient driving the rip current was estimated using the radii of curvature differences at B1 and B3 for each power input. These are shown in Fig. 7. The pressure gradient driving the rip current is the same order of magnitude as the capillary pressure gradient driving liquid from the cold end to the central drop. 3.1.3. The condensation region Near the heater end at the top of Fig. 8(a), the temperature gradient becomes very large, and the Marangoni force drives liquid from the hot end to the cold end (three large black arrows in Fig. 8(a)). On the flat surface, this Marangoni flow directly opposes the rip current and helps to dissipate and stop the current entirely in region (C). In the transverse direction perpendicular to the heater/condenser axis, the temperature difference between the centerline and the corners near the heater (Fig. 9) drives the fluid from the center of the wall to the corners of the heat pipe. With increasing heat input, this temperature difference increases, there is increasing condensation, a thicker liquid film on the wall surface [17], and stronger Marangoni flow from the wall center to the corners of the heat pipe. Thus, flow from the center toward the corners helps feed the Marangoni flow in the corners.
The liquid flows and heat transfer that give rise to the phenomena we observe are fully three-dimensional phenomena and arise from a communication between fluid-thermal processes occurring throughout the heat pipe. We hypothesize that the current occurs as a result of two counter-rotating vortices contained within the central drop region. As these vortices spin, they take the fluid feeding the central drop from the Marangoni and capillary flows in the corners and from the thin film on the flat wall downstream of the drop and then eject that fluid in the form of a surface current. This mechanism is similar to how rip currents are generated at a shoreline [18-26]. While a full simulation of the process is well beyond the scope of this paper, we were interested if this minimalist approach, proven useful for describing the features of the rip current neck, would give us a flow, resembling what we observe. Here, the hypothesis was that as we increase the heat input, the Marangoni and capillary return flows get stronger, the vortices within the central drop spin faster and so we have a more well-defined and larger ejected current and wake. We set up a simple model similar with the model used to reproduce the rip current neck in [24]. We used a two-dimensional geometry of the same size and shape as the heat pipe, 3 mm wide and 40 mm long. We then placed two counter-rotating cylinders at a location that is a similar distance from the heater as the central drop appeared in the experiments when heat input is above 2 W. The cylinders also mimicked the size of the central drop, spanning nearly the entire width of the cell. We applied no-slip conditions on the boundaries of the simulation domain and on the surface of the cylinders. We then spun the cylinders, one clockwise and the other counterclockwise, at high and low linear velocities to mimic high and low heat inputs. Fig. 8(b) and (c) show the results of the simulation for both magnitudes of rotation rate. At a relatively low rotation rate (Fig. 8(b)), the current flowing toward the heater is weak, and at
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Fig. 8. Full picture of the fluid flow in the interfacial flow region. (a) Experimental result. Black arrows show the Marangoni flow. Red arrows show the flow due to pressure gradient. Blue arrows show the capillary flow from the cold end. (b) Simulation result for slow rotation rate. (c) Simulation result for high rotation rate (an order of magnitude faster than (b)).
pointing toward the condenser end that also exhibits a kind of nascent current. So, the model, though crude, seems to support some of the experimental observations including the odd wake pattern that exists in the thin film region downstream of the drop. Much more 3-D simulation work will be required to fully understand and simulate all aspects of what we observed in this system.
3.3. Rip current dependence on condenser temperature
Fig. 9. The temperature difference between the center and the side of the wall of the heat pipe at the heater end showing that fluid flows from the center of the wall to the corners of the heat pipe due to a Marangoni driving force.
the high rate (Fig. 8(c)), the current is stronger, well-defined, and penetrates about 50% further toward the heater. The wake region downstream of the drop in Fig. 8(c), is also more well-defined, than in Fig. 8(b), and there is a faint region along the central axis
So far, there is evidence that the rip current adds an additional channel to help dissipate heat within the heat pipe, and that the strength of the current increases with increasing heat input. To test whether the heat input alone governs the size of the interfacial region and the appearance of a rip current, we held the heat input constant at 2 W and varied the condenser temperature from 25°C to −5°C. Given that the condenser could be nearly 40 mm from the interfacial region, the condenser temperature still significantly affected the extent of the interfacial region, the overall evaporation rate, and whether a rip current appeared (Fig. 10). When the cooler temperature decreased from 25°C to −5°C, the length of the interfacial region doubled. The rip current was evident at 25 °C but became weaker at 19°C and disappeared entirely at 10°C and −5°C.
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Fig. 10. Rip current and interfacial flow region at different condenser temperatures. (a) 10X images showing large changes in the rip current (black arrows) and interfacial flow region (red arrows). (b) A magnified version showing the rip current region.
Fig. 11. (a) Strong correlation between the length of the interfacial flow region and the central drop location and the surface tension gradient. (b) Surface tension gradient from the heater to the drop and from the drop to the cold end showing less fluid flow into the drop from both ends.
We can attribute the change in the interfacial length to the large increase in the overall surface tension gradient between the hot end and the cold end and the decrease in condenser temperature. The strong correlation between the length of the interfacial flow region and the central drop location and the total surface tension gradient between the heater end and the cooler end is evident in Fig. 11(a). The surface tension gradient in this figure and Fig. 11(b) is calculated as followed.
dσ = dx
|σ1 − σ2 | x
(6)
Where σ 1 and σ 2 are the surface tension at the two ends of the region of interest based on our measured temperatures
at those locations. For example, to calculate the total surface tension gradient in Fig. 11(a), we use the temperature at the first thermocouple at the heater wall and the thermocouple at the liquid pool. x is the distance between those two points. We can attribute the change in rip current to two effects. As the condenser temperature decreases, the cell pressure decreases and the point where wholesale condensation first appears occurs at a lower wall temperature. The overall evaporation and condensation rates both decrease and this leads to a decrease in total fluid flow within the heat pipe. At the same time, at low cooler temperature, the changes in central drop location and condenser end temperature leads to weaker Marangoni flow into the drop from the heater end and weaker capillary flow into the drop
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Fig. 12. (a) Temperature profiles with solid dots showing the location of central drop. (b) Total heat absorbed by evaporation in the heat pipe (classic evaporation region and rip current region) and heat absorbed by evaporation in the rip current region alone for different cooler temperatures. At −5 °C and 10 °C, there is no rip current, evaporation occurs along the four corners and central drop and therefore, there is no change in heat absorbed by the rip current region (square-marker line). At 19 °C and 25 °C, the rip current appears and grows stronger, and the amount of heat absorbed by evaporation in the rip current region increases.
from the condenser due to a stronger opposing Marangoni flow from the drop to the condenser end (Fig. 11(b)). In Fig. 10, the fainter appearance of the central drop with decreasing condenser end temperature indicates a thinner drop and this means there is weaker flow into the drop. A weaker flow into the drop also leads to slower vortex rotation and a weaker rip current. This explains the relatively strong rip current at 25°C and the absence of one at –5°C. The dissipation of the rip current at low condenser temperatures results in lower heat transfer efficiency. This is evidenced by the slower decrease in temperature near the heater end at low cooler temperatures in Fig. 12(a) and lower amounts of heat absorbed via evaporation (Fig. 12(b)). The lower heat transfer efficiency at these conditions is offset by the larger temperature difference between the heater and cooler end. The result from this section again shows the role of the rip current in increasing overall evaporation and the heat transfer efficiency. 4. Conclusions Using both experimental and simulation techniques, we attempted to explain some of the remaining odd flow behaviors we observed inside a wickless heat pipe in microgravity where the effect of the capillary and Marangoni forces dominate. We found that the two forces oppose one another to create two counterrotating vortices contained within the central drop region of the heat pipe. As these vortices spin, they take the fluid feeding the central drop from the corners and eject it in the form of a surface current. The whole fluid flow pattern follows the mechanism of a rip current along the shore line. The stronger the Marangoni and capillary flows, the stronger the rip current. It was interesting to find how extensive information transfer is within the heat pipe. Although the rip current is associated mainly with the heater end, its shape and size change significantly with changes in both heat input and condenser end temperature. The rip current plays a critical role in maintaining heat transfer effectiveness by providing an additional channel for evaporation in the region near the heater end. It also provides an outlet for the excess liquid entering the central drops that cannot be evaporated from the drop. This prevents the drops on each flat surface from growing, merging together, and completely choking off vapor flow between the heater and the cooler ends. Declaration of Competing Interest We have no conflicts of interest to disclose.
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