Modeling of pulsating heat pipe (PHP)
by V.S. Nikolayev
One of the contemporary technological challenges is a reduction of mass of transportation means in order to reduce their energy consumption and CO2 emission. This requires replacing of metals by lighter synthetic materials (composites, ceramics, etc.), which, however, are poor heat conductors and thus require special thermal management solutions for their cooling capability to handle heat loads up to several kW. On the other hand, with the increase of power levels related to the miniaturization of electronics progressing towards multi chip modules, conventional cooling technologies and thermal management are facing growing challenges including the cooling heat fluxes of 100 W/cm2, long term reliability, and very low costs for consumer market products, among others. This necessitates the development and use of more efficient, nontraditional cooling approaches. Special devices called heat pipes are used more and more widely to transfer the excessive heat to a colder environment.
A heat pipe is a container tube filled with the working fluid. One end of this tube (called evaporator section) is brought in thermal contact with a hot point to be cooled. The other end (called condenser section) is connected to the cold point where the heat can be dissipated. A portion of the tube between evaporator and condenser is called adiabatic section.
The working fluid and its pressure are chosen in such a way that the saturation temperature is between the evaporator temperature Te and condenser temperature Tc. The fluid is thus vaporized in the evaporator section. The created vapor is transported to the condenser section and condenses there. The liquid is transported back to the evaporator section. The heat is transferred mainly due to the latent heat absorption in the evaporator and its release in the condenser. Since the latent heat is large, the heat pipes are quite efficient. They are capable of evacuating up to 100-200 W/cm2. There are different kinds of heat pipes. They differ by their geometry and a mechanism of fluid transport inside the heat pipe.
Pulsating (or oscillating) heat pipe, invented in early 1990s present promising alternatives for the removal of high localized heat fluxes to provide a necessary level of temperature uniformity across the components that need to be cooled. PHP is a capillary tube (with no wick structure) bent into many turns and partially filled with a working fluid. Because the tube is thin, the liquid plugs and vapor bubbles are formed inside it.
|Two possible PHP geometries.|
When the temperature difference between evaporator and condenser exceeds a certain threshold, the gas bubbles and liquid plugs begin to oscillate spontaneously back and forth. The amplitude of oscillations is quite strong and the liquid plugs penetrate into both condenser and evaporator. The heat is thus transferred not only by the latent heat tansfer like in other types of heat pipes, but also by sweeping of the hot walls by the colder moving fluid and vice versa. This phenomenon is the reason of high efficiency of PHPs in comparison with other types of heat pipes. Compared to other cooling solutions, PHPs are simple and thus more reliable and cheap. They are capable to transfer several kW to distances of the order of 1 m even when their orientation with respect to gravity is unfavorable. For comparison, advanced heat pipes used for spatial applications have heat transfer capacity (measured in W∙m) order of value smaller. The heat transfer capacity of the conventional heat pipes used for cooling of microelectronic devices like laptop computers is 2 to 3 orders of value smaller than that of PHPs. These features make PHP extremely promising for the thermal management of the next generation of electronic and other systems. However, the functioning of PHPs is not completely understood. A complicated interplay of different hydrodynamic and phase-exchange phenomena needs to be accounted for in the modeling approaches. Unlike other heat transfer devices, the functioning of PHPs is non-stationary and thus difficult to model. A number of PHP studies have been carried out since 1990s. Researchers agree that the oscillations are driven by an instability that appears due to coupling of the adiabatic vapor compression and evaporation/condensation mass exchange. However this instability has not been studied until recently. Such important parameters as oscillation threshold, heat transfer coefficient, and maximum heat load cannot be predicted from calculations. It is not even clear whether the oscillations are persistent or not and at which regimes. For these reasons the PHP applications are very limited. The PHP parameters are adjusted empirically, often without any certainty. To our knowledge, only a couple of small companies in the world produce them. A comprehensive introduction to PHPs can be found in PhD theses in English or French.
There are two possible PHP geometries: open loop and closed loop PHP. In the open loop PHP, the ends may be open or closed. It is however generally recognized that the closed loop PHP is more efficient. For this reason, we target this type of the PHP in our numerical multibubble modeling presented below.
There is a children's toy that is called "click-click" or "putt-putt" or "pop-pop" boat. Its name comes from the sound made by this toy while it moves. The sound is made by the cover of the fluid tank made of the thin metal membrane. The latter deforms when the pressure inside the tank varies and makes the sound. Web sites in English and in both English and French present interesting features of the steam boats and how to do it yourself (the toy is sold in traditional toy shops in France). They also give references to other web pages and to a discussion group on these wonderful toys.
|Steam boat||Scheme of the steam boat|
One can mention that the boat engine is similar to the PHP. The tubes play the role of the condenser and adiabatic sections. The water reservoir (tank) works as the evaporator.
|Scheme of the engine of the steam boat|
This boat works according to the same principle as the PHP. The water plugs oscillate inside the open tubes and the water is alternatively expulsed or sucked up. During the expulsion, the water flow is directed backwards while the suction is nearly isotropic. The created differential momentum propels the boat forward.
Researchers agree that the oscillations are driven by an instability that appears due to coupling of the adiabatic vapor compression and evaporation/condensation mass exchange. This instability is not yet understood. What is its principal positive feedback that makes the system unstable? What provides the instability threshold? Is the stopping threshold (measured by lowering the heating/cooling power) differs from the oscillation start threshold? These questions need to be answered. A parametric study of the instability needs to be carried out (different fluids, temperatures, pressures). The behavior of the physical parameters in the vapor phase remains to be controversial. In some modeling approaches, the vapor is allowed to be strongly overheated due to its compression. It is assumed in the others to be at saturation temperature corresponding to its pressure, which is a behavior analogous to that observed in bubbles at boiling or in conventional heat pipes. However, the vapor compression is a moving force of the oscillations and its behavior needs to be clearly understood. We proposed recently a model that describes the simplest PHP that contains one bubble and one liquid plug. We have discussed the factors that define its frequency and the threshold of oscillations. The model shows the importance of the liquid films left on the internal wall of the tube by the receding liquid plugs. The model agrees with the experimental results obtained in collaboration with CETHIL.
To our knowledge, there is only one approach to the bubble-level modeling of the PHP. We developed a new approach based on the developed recently single bubble model. It is capable to describe the variable bubble number so that events like bubble coalescence can be accounted for. The computer code is object oriented and is written with C++. The volume of its output data might be (and usually is) very large and difficult to process. A special application, called PHP Viewer, has been developed. It visualizes data files created with the simulation program. The PHP Viewer allows visualization of the dynamics of gas-liquid interfaces and of the wetting (liquid) films that envelope the gas. The film dynamics is very important because their evaporation/condensation is the main moving force of the oscillations. The wetting films cover the internal tube walls completely when the gas exists in the condenser and adiabatic sections. The evaporator section may or may not be covered by the films. The film length in the evaporator changes because the films vaporize. The films are left on the walls by the receding gas-liquid menisci or "eaten up" by them when they advance. The next figure shows how the liquid plugs, the vapor and the films are represented by PHP Viewer. The evaporator area is shown with rose and the condenser with light blue color. Their temperatures and the simulated time moment are presented at the top.
An example of the PHP modeling (the working fluid: water) is presented on the video. It shows also some functionality of PHP Viewer 1.5 like animation speed control. Double click to open the video fullscreen, press Escape to exit.
This simulation is one dimensional. The only space variable x runs along the tube of the PHP. The modeling is performed by breaking the loop, "unbending" it, and imposing periodic boundary conditions at its ends as is shown in the figure below. The evaporator, adiabatic and condenser sections are indicated with the same colors as above and with the letters E,A,C, respectively.
The time evolution of positions of the gas-liquid interfaces is plotted below. Only a part of the whole x extension is shown. Several stages of the evolution can be distinguished. First, some bubbles disappear because the liquid plugs coalesce between them. The coalescence corresponds to the point where two interfaces meet each other. The instability develops next and the amplitude of oscillations grow with time. The last stage is that of the developed oscillations.
The PHP Viewer 1.6 can display the liquid temperature distribution shown by colors. The correspondence colors-temperatures are shown with a bar at the top of the screen. Red (blue) corresponds to the highest (lowest) displayed temperature. An example of the temperature variation in the liquid is presented in the video below where the gravity is directed to the right. It shows that the ends of the liquid plugs that enter the evaporator become warmer than the rest of the plugs. The temperature of the liquid-vapor interfaces is the saturation temperature that may quickly vary in time (following the pressure of the vapor). An example of the temperature variation in the regime of developed oscillations is shown in the video below. Double click to open the video fullscreen, press Escape to exit.
The corresponding heat transfer evolution is shown in the image below. The heat exchange with evaporator and condenser are shown. In the developed regime, a dynamic equilibrium is established: the time average of the power taken from evaporator is equal to that average power given to the condenser.
In this particular case, about 60% of the power given to evaporator is transferred due to the latent heat during film evaporation. The other part of the heat is taken due to the heating of the liquid plugs when they are situated inside the evaporator part. Since at any time moment there is one or several bubbles inside the condenser, the major part (99%) of the condenser heat exchange is performed via condensation on the films that cover the internal walls of the tube.
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