Device Having Enhanced Heat Transfer in Natural Convection by Means of Liquid Metals and Partitioned Domains

Abstract

A heat transfer device comprises: an outer wall, a first end wall connected to a first end of the outer wall, and a second end wall connected to a second end of the outer wall, wherein the outer wall, first end wall, and second end wall define a cavity; at least one partition wall located in the cavity, each partition wall being spaced inward from the first and second end walls; and a working fluid in the cavity, the working fluid being selected from liquid metals and liquid metal alloys, wherein the fluid has a final melting point at or below an operating temperature of the device, wherein the heat transfer device transfers heat from a heat source adjacent the first end wall to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source toward the second end wall.

Description

CROSS-REFERENCES TO RELATED APPLICATIONS

[0001] This application is based on, claims benefit of, and claims priority to U.S. Application No. 63 / 643,025 filed on May 6, 2024, which is hereby incorporated by reference herein in its entirety for all purposes.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] Not Applicable.FIELD OF THE INVENTION

[0003] This invention relates to a device having enhanced heat transfer in natural convection by means of liquid metals or metal alloys and partitioned domains.BACKGROUND

[0004] Rayleigh-Benard convection (RBC) can be applied for cooling in systems with intense heat generation, such as high-performance computer chips, power electronics, or large electric batteries. In comparison to heat exchangers relying solely on conduction and forced convection, RBC offers the advantage of significant convection heat transfer not requiring an energy input.

[0005] What is needed is improved devices and methods wherein an RBC flow can be manipulated to increase the rate of heat transfer without changing the fluid volume and the temperature difference between the hot and cold walls.SUMMARY

[0006] The foregoing needs are met by a device according to the present disclosure having enhanced heat transfer in natural convection by means of liquid metals or liquid metal alloys and partitioned domains.

[0007] In one aspect, the present disclosure provides a heat transfer device comprising: an outer wall, a first end wall connected to a first end of the outer wall, and a second end wall connected to an opposite second end of the outer wall, wherein the outer wall, the first end wall, and the second end wall define a cavity; at least one partition wall located in the cavity, each partition wall being spaced inward from the first end wall and the second end wall; and a working fluid contained in the cavity, the working fluid being selected from liquid metals and liquid metal alloys, wherein the working fluid has a final melting point at or below an operating temperature of the heat transfer device. The heat transfer device transfers heat from a heat source adjacent the first end wall to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source toward the second end wall.

[0008] In one embodiment of the device, the working fluid is selected from the group consisting of gallium, mercury, sodium, and eutectic alloys. In one embodiment of the device, the working fluid is selected from the group consisting of gallium, mercury, sodium, a eutectic alloy of gallium, indium, and tin, and a eutectic alloy of bismuth, lead, tin, and cadmium. In one embodiment of the device, the working fluid comprises gallium. In one embodiment of the device, the working fluid has a final melting point of 100° C. or below. In one embodiment of the device, the working fluid has a final melting point of 80° C. or below. In one embodiment of the device, the working fluid has a final melting point of 40° C. or below.

[0009] In one embodiment of the device, the at least one partition wall extends laterally between a first inner surface of the outer wall and a second inner surface of the outer wall. In one embodiment of the device, the outer wall has a cylindrical shape such that the cavity has a height and a diameter. In one embodiment of the device, a height-to-diameter ratio of the cavity is 3 or greater. In one embodiment of the device, the at least one partition wall extends laterally along the diameter of the cavity between a first inner surface of the outer wall and a second inner surface of the outer wall. In one embodiment of the device, each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, and a gap distance-to-height ratio (δ) is in a range of 0.001 to 0.3. In one embodiment of the device, each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, and a gap distance-to-height ratio (δ) is in a range of 0.02 to 0.6.

[0010] In one embodiment of the device, a diameter-to-height ratio (AR) of the cavity is in a range of 0.1 to 10. In one embodiment of the device, a diameter-to-height ratio (AR) of the cavity is in a range of 4 to 6. In one embodiment of the device, a Rayleigh number (Ra) of the working fluid is in a range of 105-108. In one embodiment of the device, a Rayleigh number (Ra) of the working fluid is in a range of 105-107.

[0011] In one embodiment of the device, the at least one partition wall is spaced inward from the first end wall by a first gap distance, the at least one partition wall is spaced inward from the second end wall by a second gap distance, and a gap ratio (a) of the second gap distance to the first gap distance is not 1. In one embodiment of the device, the gap ratio (a) in a range of 0.5 to 1.5. In one embodiment of the device, the gap ratio (a) in a range of 0.7 to 0.8.

[0012] In one embodiment, the device further comprises: the heat source, wherein the heat source is positioned adjacent the first end wall such that the first end wall has a first temperature higher than a second temperature of the second end wall, and the first end wall is located at a lower level than the second end wall. In one embodiment of the device, the heat source is selected from engines, electrochemical devices, power electronics, computer components, and heating, ventilation, and air conditioning systems. In one embodiment of the device, the outer wall has a polygonal shape.

[0013] In another aspect, the present disclosure provides a method for cooling a heat source. The method comprises: (a) providing a heat transfer device comprising: (i) an outer wall, a first end wall connected to a first end of the outer wall, and a second end wall connected to an opposite second end of the outer wall, wherein the outer wall, the first end wall, and the second end wall define a cavity, (ii) at least one partition wall located in the cavity, each partition wall being spaced inward from the first end wall and the second end wall; and (iii) a working fluid contained in the cavity, the working fluid being selected from liquid metals and liquid metal alloys, wherein the working fluid has a final melting point at or below an operating temperature of the heat transfer device; (b) thermally coupling the first end wall of the heat transfer device with a heat source; and (c) cooling the heat source by transferring heat from the heat source to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source toward the second end wall.

[0014] In one embodiment of the method, the working fluid is selected from the group consisting of gallium, mercury, sodium, and eutectic alloys. In one embodiment of the method, the working fluid is selected from the group consisting of gallium, mercury, sodium, a eutectic alloy of gallium, indium, and tin, and a eutectic alloy of bismuth, lead, tin, and cadmium. In one embodiment of the method, the working fluid comprises gallium. In one embodiment of the method, the working fluid has a final melting point of 100° C. or below. In one embodiment of the method, the working fluid has a final melting point of 80° C. or below. In one embodiment of the method, the working fluid has a final melting point of 40° C. or below.

[0015] In one embodiment of the method, the at least one partition wall extends laterally between a first inner surface of the outer wall and a second inner surface of the outer wall. In one embodiment of the method, the outer wall has a cylindrical shape such that the cavity has a height and a diameter. In one embodiment of the method, a height-to-diameter ratio of the cavity is 3 or greater. In one embodiment of the method, the at least one partition wall extends laterally along the diameter of the cavity between a first inner surface of the outer wall and a second inner surface of the outer wall. In one embodiment of the method, each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, and a gap distance-to-height ratio (δ) is in a range of 0.001 to 0.3. In one embodiment of the method, each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, and a gap distance-to-height ratio (δ) is in a range of 0.02 to 0.6. In one embodiment of the method, a diameter-to-height ratio (AR) of the cavity is in a range of 0.1 to 10. In one embodiment of the method, a diameter-to-height ratio (AR) of the cavity is in a range of 4 to 6. In one embodiment of the method, a Rayleigh number (Ra) of the working fluid is in a range of 105-108. In one embodiment of the method, a Rayleigh number (Ra) of the working fluid is in a range of 105-107.

[0016] In one embodiment of the method, the at least one partition wall is spaced inward from the first end wall by a first gap distance, the at least one partition wall is spaced inward from the second end wall by a second gap distance, and a gap ratio (a) of the second gap distance to the first gap distance is not 1. In one embodiment of the method, the gap ratio (a) in a range of 0.5 to 1.5. In one embodiment of the method, the gap ratio (a) in a range of 0.7 to 0.8. In one embodiment of the method, the heat source is positioned adjacent the first end wall such that the first end wall has a first temperature higher than a second temperature of the second end wall, and the first end wall is located at a lower level than the second end wall. In one embodiment of the method, the heat source is selected from engines, electrochemical devices, power electronics, computer components, and heating, ventilation, and air conditioning systems. In one embodiment of the method, the outer wall has a polygonal shape.

[0017] The foregoing and other aspects and advantages of the invention will appear from the following description. In the description, reference is made to the accompanying drawings which form a part hereof, and in which there is shown by way of illustration example embodiments of the invention. Such embodiments do not necessarily represent the full scope of the invention, however, and reference is made therefore to the claims and herein for interpreting the scope of the invention.BRIEF DESCRIPTION OF DRAWINGS

[0018] The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0019] FIG. 1 shows the results of the simulations of the flow at Ra=106, AR=1, δ=0.16 which are discussed in detail in section 3 of the Example. Instantaneous distributions of temperature in the vertical cross-section are shown. Left: RBC in a convective cell without a partition showing the typical flow organization. Right: RBC in a partitioned convective cell demonstrating stronger and more coherent large-scale circulation.

[0020] FIG. 2 shows the geometry of a convective cell according to one example embodiment of the present disclosure. Left: Top-down cross section of the center of the convective cell. Right: Front facing perspective of the convective cell.

[0021] FIG. 3 shows the Nusselt number Nu as a function of number of elements N. Results are shown for the non-partitioned domain at Ra=106 AR=1 (red triangle) and for the partitioned domain at Ra=107 AR=3 δ=0.016 (green square).

[0022] FIG. 4 shows a graph for the total rate of heat transfer at the top and bottom of the cell during the entire simulation for Ra=106 AR=1. The trend over time shows convergence of the heat transfer rate at both top and bottom walls to approximately 275 W.

[0023] FIG. 5 shows a zoomed in region of the mesh at the partition gap showing increased mesh resolution within the thermal boundary layers near the top and bottom surfaces.

[0024] FIG. 6 shows graphical representations of normalized values for the Nusselt Number.

[0025] FIG. 7 shows Left: Graphical representation of the inverse relationship between δmax and combinations of Ra and AR. Right: Normalized values of Nu at δmax as a function of Ra and AR showing a direct relationship that becomes more extreme as Ra decreases.

[0026] FIG. 8 shows in panels a-h, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=106, AR=1 are shown.

[0027] FIG. 9 shows in panels a-e, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=106, AR=2 are shown.

[0028] FIG. 10 shows in panels a-e, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=106, AR=1 are shown.

[0029] FIG. 11 shows in panels a-f, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=106, AR=5 are shown.

[0030] FIG. 12 shows in panels a-e, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=107, AR=1 are shown.

[0031] FIG. 13 shows in panels a-e, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=107, AR=2 are shown.

[0032] FIG. 14 shows in panels a-f, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=107, AR=3 are shown.

[0033] FIG. 15 shows in panels a-g, a typical distribution of temperature in the vertical axial cross-section and perpendicular partition. Flows with Ra=107, AR=5 are shown.

[0034] FIG. 16 shows temperature and velocity contours of x-velocity showing evidence of horizontal jets in the gaps between the top / bottom walls and the partitions. Instantaneous distributions for x-velocity contours Left: Ra=106, AR=1, δ=0.14; Right: Ra=107, AR=3, δ=0.04.

[0035] FIG. 17 shows horizontal cross sections of fluid temperature for Ra=106, AR=1, and δ=0.14. Left: Cross sections above and below the partition. Right: Cross section through cylinder center.

[0036] FIG. 18 shows: Top Left: Normalized values of the Nusselt number as a function of the single parameter δ / δBL. Top Right: δ / δBL as a function of AR (dependence on the geometry of the cylinder). Bottom: Average boundary layer thickness as a function of δmax.

[0037] FIG. 19 shows axial profiles of time-averaged temperature in flows without partitions at Ra=106.

[0038] FIG. 20 shows axial profiles of time-averaged temperature in flows without partitions at Ra=107.

[0039] FIG. 21 shows results of modifying α for Ra=106, AR=1, and δ=0.06.DETAILED DESCRIPTION

[0040] Before any embodiments of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of components set forth in the following description or illustrated in the following drawings. The invention is capable of other embodiments and of being practiced or of being carried out in various ways. Also, it is to be understood that the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,”“comprising,” or “having” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

[0041] The following discussion is presented to enable a person skilled in the art to make and use embodiments of the invention. Various modifications to the illustrated embodiments will be readily apparent to those skilled in the art, and the generic principles herein can be applied to other embodiments and applications without departing from embodiments of the invention. Thus, embodiments of the invention are not intended to be limited to embodiments shown but are to be accorded the widest scope consistent with the principles and features disclosed herein. Skilled artisans will recognize the examples provided herein have many useful alternatives and fall within the scope of embodiments of the invention.

[0042] FIG. 2 shows a heat transfer device 30 according to one non-limiting example embodiment of the present disclosure. The heat transfer device 30 comprises an outer vertical cylindrical wall 32 of height H, a first end wall 34 connected to a first end of the outer wall 32, and a second end wall 36 connected to an opposite second end of the outer wall 32, wherein the outer wall 32, the first end wall 34, and the second end wall 36 define a cavity 38. A vertical partition wall 40 is located in the cavity 38. The partition wall 40 passes through the cylinder's axis and extends wall-to-wall horizontally while leaving gaps of size Hg between the partition wall 40 and the first end wall 34 and the second end wall 36. The width of the partition wall 40 is 1 / 10 of the cylinder diameter D. A working fluid is contained in the cavity 38. The working fluid can be selected from liquid metals and liquid metal alloys, wherein the working fluid has a final melting point at or below an operating temperature of the heat transfer device 30. The heat transfer device 30 transfers heat from a heat source 50 adjacent the first end wall 34 to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source 50 toward the second end wall 36.

[0043] For the heat transfer device 30 to function efficiently, it is preferred that there be a temperature difference between the first end wall 34 and the second end wall 36. Specifically, the second end wall 36 must have a lower temperature than the first end wall 34. In one embodiment, this means that the second end wall 36 has to have some cooling applied to it. Any kind of heat sink suitable to the specific application environment can be used. This can be an active cooling (e.g., by fan) or passive (simply exposing the outer surface of the second end wall 36 to a cold environment). In one embodiment, fins are attached to the outer surface of the second end wall 36 for cooling.

[0044] In one embodiment, the first end wall 34 has a first temperature and the second end wall 36 has a second temperature wherein the second temperature is at least 20° K lower temperature than the first temperature, or the second temperature is at least 30° K lower temperature than the first temperature, or the second temperature is at least 40° K lower temperature than the first temperature, or the second temperature is at least 50° K lower temperature than the first temperature, or the second temperature is at least 60° K lower temperature than the first temperature, or the second temperature is at least 70° K lower temperature than the first temperature, or the second temperature is at least 80° K lower temperature than the first temperature. Greater differences between the first temperature and the second temperature lead to increased heat transfer in the heat transfer device 30. The heat transfer device 30 will also work for smaller or larger temperature differences. Slight adjustments of the height of the cavity can be implemented to produce the same heat flux.

[0045] In one embodiment of the heat transfer device 30, the first end wall 34 comprises a first material having a first thermal conductivity, the second end wall 36 comprises a second material having a second thermal conductivity, the outer vertical cylindrical wall 32 comprises a third material having a third thermal conductivity, and the partition wall 40 comprises a fourth material having a fourth thermal conductivity. In one embodiment, the first thermal conductivity of the first material is greater than the third thermal conductivity of the third material, the first thermal conductivity of the first material is greater than the fourth thermal conductivity of the fourth material, the second thermal conductivity of the second material is greater than the third thermal conductivity of the third material, and the second thermal conductivity of the second material is greater than the fourth thermal conductivity of the fourth material. In one embodiment, the first material and the second material are independently selected from metallic materials. In one embodiment, the third material and the fourth material are independently selected from polymeric materials.

[0046] In one embodiment of the heat transfer device 30, the working fluid is selected from the group consisting of gallium, mercury, sodium, and eutectic alloys. In one embodiment of the heat transfer device 30, the working fluid is selected from the group consisting of gallium, mercury, sodium, a eutectic alloy of gallium, indium, and tin, and a eutectic alloy of bismuth, lead, tin, and cadmium. In one embodiment of the heat transfer device 30, the working fluid comprises gallium. One non-limiting example eutectic alloy comprises 68.5% Ga, 21.5% In, and 10.0% Sn (by weight) and is commercially available as Galinstan®. Another non-limiting example eutectic alloy comprises 50% bismuth, 26.7% lead, 13.3% tin, and 10% cadmium by mass and is commercially available as Wood's metal.

[0047] In one embodiment of the heat transfer device 30, the working fluid has a final melting point of 100° C. or below. In one embodiment of the heat transfer device 30, the working fluid has a final melting point of 80° C. or below. In one embodiment of the heat transfer device 30, the working fluid has a final melting point of 40° C. or below. One skilled in the art appreciates that metals and eutectic alloys have a single melting temperature and that non-eutectic alloys have melting temperature ranges wherein a final melting point is the temperature when the non-eutectic alloy is completely melted. As used herein, “a final melting point” also encompasses a single melting temperature of a metal or a eutectic alloy.

[0048] In one embodiment of the heat transfer device 30, the partition wall 40 extends laterally between a first inner surface of the outer wall 32 and a second inner surface of the outer wall 32. In one embodiment of the heat transfer device 30, the outer wall 32 has a cylindrical shape such that the cavity has a height H and a diameter D. In one embodiment of the heat transfer device 30, a height-to-diameter ratio of the cavity is 3 or greater. In one embodiment of the heat transfer device 30, the partition wall 40 extends laterally along the diameter of the cavity 38 between a first inner surface of the outer wall 32 and a second inner surface of the outer wall 32.

[0049] In one embodiment of the heat transfer device 30, the partition wall 40 is spaced inward from the first end wall 34 and the second end wall 36 by a gap distance Hg, and a gap distance-to-height ratio (δ) is in a range of 0.001 to 0.3. In one embodiment of the heat transfer device 30, the gap distance-to-height ratio (δ) is in a range of 0.02 to 0.6.

[0050] In one embodiment of the heat transfer device 30, a diameter-to-height ratio (AR) of the cavity is in a range of 0.1 to 10. In one embodiment of the heat transfer device 30, the diameter-to-height ratio (AR) of the cavity is in a range of 4 to 6.

[0051] In one embodiment of the heat transfer device 30, a Rayleigh number (Ra) of the working fluid is in a range of 105-108. In one embodiment of the heat transfer device 30, the Rayleigh number (Ra) of the working fluid is in a range of 105-107.

[0052] In one embodiment of the heat transfer device 30, the partition wall 40 is spaced inward from the first end wall 34 by a first gap distance Hg, the partition wall 40 is spaced inward from the second end wall 36 by a second gap distance Hg, and a gap ratio (a) of the second gap distance to the first gap distance is not 1. In one embodiment of the heat transfer device 30, the gap ratio (a) in a range of 0.5 to 1.5. In one embodiment of the heat transfer device 30, the gap ratio (a) in a range of 0.7 to 0.8.

[0053] In one embodiment of the heat transfer device 30, the heat source is positioned adjacent the first end wall 34 such that the first end wall 34 has a first temperature higher than a second temperature of the second end wall 36, and the first end wall 34 is located at a lower level than the second end wall 36.

[0054] In one embodiment of the heat transfer device 30, the heat source is selected from engines, electrochemical devices, power electronics, computer components, and heating, ventilation, and air conditioning systems.

[0055] The outer wall 32 is not limited to the cylindrical shape. For example, in one embodiment of the heat transfer device 30, the outer wall 32 has a polygonal shape (e.g., a cuboid).

[0056] The present invention also provides a method for cooling a heat source. The method can use any of the embodiments of the heat transfer device described herein. In a non-limiting example of the method, the first end wall 34 of the heat transfer device 30 is thermally coupled to the heat source 50; and the heat source 50 is cooled by transferring heat from the heat source 50 to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source 50 toward the second end wall 36. One example role of the heat transfer device 30 is to transfer the heat from the heat source to a place, where the heat can be easily dissipated into a colder environment. An example is the heat transfer from the heating elements within a computer box (a source) toward the room air. This can be achieved by a heat transfer device with the first end wall 34 is attached to the source and the second end wall 36 is exposed to the atmosphere.EXAMPLE

[0057] The following Example has been presented in order to further illustrate the invention and is not intended to limit the invention in any way. The statements provided in the Example are presented without being bound by theory.1.1 Overview of Example

[0058] Heat generation by commonly used systems and components, such as the large batteries used for energy storage, powerful instrumentation in computing, and advanced HVAC and climate control systems, has continued to increase and is further augmented by technological advancement. Assuming progress continues, the research of heat transfer efficiency remains a meaningful and worthwhile endeavor. This Example explores possible ways to increase the effectiveness of heat transfer based on natural convection for systems at relatively low temperatures, which increases the range of applications for which it can be applied. It is hypothesized that the high energy density and high thermal conductivity of liquid metals and the effects of vertical partitions on flow organization in a fluid cavity can positively impact the heat transfer rate of a convective cell. The hypothesis is explored for a geometry of a cylindrical cavity with a single partition using Ansys Fluent CFD simulations. The aspect ratio of the cylinder, the Rayleigh number of the convective fluid flow, and the gap height between the top and bottom cylinder surfaces and a partition are considered as factors of a parametric optimization study. The results of this Example show manyfold enhancement of the heat transfer rate by a partition and indicate a strong potential in heat transfer applications.1.2 Introduction to Example

[0059] Understanding the effects of partitioned Rayleigh-Benard convection (RBC) for a liquid metal flow requires comprehension of the factors affecting the heat transfer in RBC. Formation of a thermal boundary layer is expected for flows with a temperature difference between a surface and fluid flowing over it. This effect should be notable for fluids with a low Prandtl number fluid which dissipate heat quickly, such as liquid metals [Ref. 11]. For simulation of a vertically mounted cylindrical convective cell, thermal boundary layers are predicted to form near the cold and hot regions of the top and bottom surfaces, respectively. Identifying these thermal boundary layers, which are a source of resistance to heat transfer, provides information on the effectiveness of RBC and how much of an effect the thermal boundary layer resistance has. There exists a plethora of research investigating methods for reducing this resistance, such as deformation of the boundary layers. In one such study, boundary layer deformation of the standing-wave type, that being a combination of two waves at the same amplitude and frequency, changes global responses to convection turbulence, given that the deforming amplitude of the standing waves is close to or larger than the boundary layer thickness of the flow in RBC [Ref. 5].

[0060] The large-scale circulation caused by RBC in a convective cell also has a prominent effect on the heat rate transfer [Ref. 8]. Modifying the large-scale circulation may, therefore, be used to increase the heat transfer rate. Due to a limited number of published studies on liquid metals, value can be found in works published on more common working fluids. Research from a study on large-scale circulation used empirical data produced through experimentation to study different types of convective domains for modifying RBC bulk flow [Ref 4]. For two different flow domain setups, type 1 with a square grid suspended in a convective cell and type 2 with the grid fully extending to the top and bottoms of the convective cell, heat transfer efficiency within the type 1 domain was enhanced by up to 14%. It was opined that increased plume coherency caused this effect. Furthermore, heat transfer efficiency in the type 2 domain, with longer segments or “sub-units” of flow, increased by as much as 30%. These results confirm that the organization of the flow is a component of heat transfer optimization.

[0061] Geometry and orientation of the convective cell cavity play a significant factor in RBC heat transfer. For a cylinder, the ratio of the height to diameter, known as the aspect ratio AR, influences RBC by defining the shape in which flow can occur. Research suggests an increase in heat transfer can be observed in convective cells with a lower AR, caused by modification of large-scale circulation. In a confined geometry, the amount and intensity of hot and cold plume clusters increases and are more energetic, which has a significant influence in reducing the thickness of the thermal boundary layers [Ref. 6]. However, the organization of the flow aided by the partitions may facilitate increased heat transfer even with an increase in the surface area of a taller convective cell of the same diameter. Additionally, the effect of a smaller convective cell must also be noted, which leads to a decrease in the volume of fluid and thus reduces the amount of heat transport that can be accommodated.

[0062] Looking at the effects of just the partitions on the convective cell geometry shows promise for improved heat transfer. Partitioned RBC may lead to reduction of heat exchange between hot ascending and cold descending jets which reduce heat transfer [Ref. 5,7]. Further research shows that adding vertical partitions in a convective cell with a high-Prandtl number liquid increases convective heat transfer in a liquid medium by increasing Nusselt number Nu when compared to non-partitioned cases [Ref. 2]. Investigation of the causes of this increase suggests that partitions in a large-scale circular fluid flow within a fluid cavity create a symmetry-breaking bifurcation, causing the fluid to organize into a unidirectional flow along the partition walls. This can create a disruption of the thermal boundary layer where the partitions extend close to the top and bottom walls [Ref. 2,8]. Furthermore, it has been observed that mean velocity and temperature fields are correlated due to the increased coherency of the flow as number of partitions increases, leading to a meaningful, albeit small, improvement in heat flux. It should be noted that as the number of partitions increases, the volume of fluid within the cell decreases and impedance from the no-slip condition of the partitioned walls increases. Adding volume to the convective cell to compensate for the loss of working fluid may prove to negate these effects.

[0063] Additional investigation into the effects of partitions on a convective cell could yield an optimized configuration for increased convective heat transfer. It has been observed that the Nusselt number increases monotonically as the number of partitions aligned perpendicularly in a convective cell increase [Ref. 2]. Aided by the partitions, the working fluid is forced through the gaps at the top and bottom of the partitions and leads to a pressure distribution at the gaps that sustain flow, creating horizontal jets, which will sweep the thermal boundary layer, disturbing these layers and further increase thermal efficiency [Ref. 2]. The size of the gap height also influences the fluid flow and heat transport and may influence the flow in relation to the cell height [Ref. 7].

[0064] There are several challenges when looking at generating valuable data for liquid metal flows. Liquid metals are opaque, and so measuring velocity using optical techniques, such as PIV, is impossible. What is more, systems using liquid metal tend to operate at much higher temperatures than conventional liquids such as air or water, creating difficult working conditions and energy requirements [Ref 1]. Regardless, an advantage of liquid metals that cannot be overlooked is their high thermal conductivity that facilitates more heat transfer.

[0065] It must be stressed that the effect of partitions on heat transfer in RBC is, while evident, not very strong in conventional fluids such as water. Increase of the Nusselt number by up to 30% is reported in [Ref 2, 7, 8]. As we will see in the discussion of the results in the Results and Discussion section below, the effect is much stronger in fluids with low Prandtl numbers, e.g., liquid metals.

[0066] Understanding the advantages of partitioned RBC provides evidence that an optimized AR for increased RBC, in combination with a specific number of partitions with a given gap height, can be used to design a fluid model for which heat transfer can be optimized. The high thermal conductivity and low viscosity of the liquid gallium could potentially add to optimized heat transport, outweighing the possibility of reduced heat transfer from fluid volume reduction and increased surface area the partition will add. Thus, this Example will report on simulations combining the effects of liquid metal and partitioned flow in a convective cell.2. Methods

[0067] This Section will detail methods used to set up and verify the accuracy of simulations. The results will be reported and discussed in Section 3.2.1. Governing Equations and Physical Parameters

[0068] An unsteady three dimensional flow in a partitioned cylindrical cavity acting as a convective cell is calculated. Several assumptions are made to simplify the model. The Oberbeck-Boussinesq approximation is assumed in which all physical properties of the fluid are assumed constant except density in the gravity force term. Density in this case is assumed to be a linear function of temperature, thus providing the buoyant force effect. The sidewalls and the wall of the partition are assumed adiabatic with heat transfer only occurring at the top and bottom walls of the cell. No-slip boundary conditions are assumed at the walls of the cavity and along the partition walls. Under these conditions, with the cylinder height H, free-fall velocity U, and ΔT as the typical scales, flow within the cavity can be represented by the governing equations:∇·u=0(1)∂u¯∂t+(u·∇)⁢u=-∇p+Pr12⁢Ra-12⁢∇2u+Tez,(2)∂T∂t+u·∇T=(RaPr)-1 / 2⁢∇2T.(3)

[0069] With boundary conditions

[0070] Top Surface: z=1, T=T2 Bottom Surface: z=0,

[0071] Cylinder Side Wall:r=12⁢AR,∂T∂r=0,Partition Walls:∂T∂r=0The control parameters for this model are the Rayleigh number Ra, the aspect ratio of the cylinder AR, and the ratio of the partition gap height to the cylinder height δ. A change in the rate of heat transfer is anticipated by modifying AR of the convective cell cylinder, a second parameter identified as the ratio of the gap height Hg between the partition fixed to the center of the cylinder δ, and the Ra. While AR and δ are directly correlated to the model geometry, Ra and the Prandtl number Pr (ratio of kinematic viscosity to thermal diffusivity) represent design parameters determined by the properties of the fluid and the temperature difference ΔT=T1−T2. Specifically, Ra is the relationship between the Grashof Number Gr (ratio of buoyant force to viscous force) and the Prandtl number Pr. A relationship between these properties and the design parameters is shown in (1) and (2). H is the characteristic length represented by the height of the cylinder, D is the cylinder diameter, g is the acceleration of gravity, Thot is the heat source temperature along the bottom surface, and Tcold is the fluid bulk temperature.TABLE 1Material properties of liquid gallium at 60° C. [Ref. 9].Properties listed are as follows: melting temperature Tmelt,thermal diffusivity χ, coefficient of thermal expansion β,thermal conductivity λ, density ρ, kinematicviscosity ν, and dynamic viscosity μ.Material PropertiesTmelt302.95Kχ1.2 × 10−5m2 / sβ1.2 × 10−4K−1λ30W / m*Kρ6040kg / m3ν3.0 × 10−7m2 / sμ1.81 × 10−3kg / m · sRa=g⁢β⁡(Thot-Tcold)⁢H3χυ,AR=DH,δ=HgH,Pr=vχ(4)Ra=Gr*Pr(5)Research from Section 1.1 is improved in the modeled environment for simulation of partitioned RBC with liquid gallium, the properties for which are listed in Table 1. The work presented in this Example is for the case of a single partition; however, it is contemplated that the use of multiple partitions can be beneficial. It is anticipated that the actual width of the partition will have minimal effects on the model assuming the partition width remains thin, and so the width of the partition will be kept at a constant value of 10% of H for each model.In a fully developed flow, the heat fluxes through the top and bottom walls fluctuate around the same constant mean Qconv. This value is produced by averaging instantaneous values of Qconv,inst in time over a long period of evolution of fully developed flow. Qconv is then calculated using (6) and is then used to calculate the Nusselt number Nu in (8). This value represents the ratio of convective over conductive heat transfer. Conductive heat transfer Qcond will be calculated for each simulation using (7) which is a function of the cylinder height, temperature difference, and fluid properties. Higher values for Nusselt number Nu indicate increased convection, indicating which models produce better results for convective efficiency.Qconv=Qconv,instNT(6)Qcond=λ⁢STB⁢Δ⁢TH(7)Nu=QconvQcond(8)2.2. Approach to SimulationsAs described below, the problem is solved computationally using Ansys Fluent, which requires the problem to be presented in dimensional units. The dimensional parameters are found for a given set of AR and Ra by fixing the temperature difference ΔT and using the physical properties of Ga reported in Table 1. Values for H, D, and Hg are then calculated, which provides the desired values of AR, Ra, and δ. Simulations are carried out for Ra=106, 107 and AR=1, 2, 3, 5 and the partition gaps for δ=0.04, 0.06, 0.08, 0.10. This creates a minimum of five simulations for each combination of Ra and AR, including the non-partitioned cases. The goal of the simulations is to determine for each combination of AR and Ra, which values of δ produces the greatest heat transfer, resulting in the highest value for Nusselt number Nu. Additional simulations for higher or lower δ are run in case a local maximum is not found within the initial simulation set.

[0077] Beginning with the DesignModeler in Ansys, the geometry is built according to calculated values for H and D. The geometry is then meshed using the verified refined meshes discussed in Section 2.3. Setup for the simulations is done by setting a double precision and assigning CPUs. The number of CPUs per simulation is set based on the number of mesh elements, using an approximation of 1 CPU per 100,000 elements needed. Ansys Fluent is capable of running both pressure and density based solvers. These models will be run with a pressure-based transient solver due to the incompressible nature of liquid metal fluid flow and account for the change in flow over time from turbulent to fully developed flows. Model parameters are then set within the simulation by inputting the materials properties for gallium listed in Table 1, setting the flow to laminar, and having the energy equation turned on. The convective cell boundaries are set at temperatures of Tcold=323.15° K and Thot=363.15° K for the top and bottom surfaces, respectively, giving a value for ΔT=40K. Tcold and Thot are nondimensionalized by the initial bulk temperature of the fluid Tb=343.15K, giving a nondimensional temperature range of 0.94≤T≤1.06 The initial conditions are the distribution of temperature and vertical velocity represented by (9) and (10), respectively.uz(x,y)=-(x2+y2)+D24(9)T⁡(z)=Thot+(-Δ⁢TH)*z(10)

[0078] The flow evolution is computed with the time step Δt=0.1 s and up to 200 iterations performed per Δt. The simulations fully developed flows used for final data acquisition run for a minimum of 100 s. This period is increased as needed to ensure convergence of the computed statistical means to a steady value (see FIG. 4). The data collected are graphed for Qconv at the top and bottom surfaces and values recorded before Δt=40 s ignored when calculating Nusselt number Nu to account for the transient nature of the initial flow profile.2.3. Model Verification

[0079] Before partitioned models are simulated, an initial non-partitioned model is run, and the resulting data is compared to verified data of RBC at high-resolution simulations of RBC in an inclined cylinder with low Pr [Ref. 3]. The non-dimensional parameters are set at Ra=106 and Pr=0.1, matching the non-dimensional values in the referenced study. The geometry is built using calculated dimensions H and D and is then meshed with number of elements NE=130,000 using hexahedrons as the element type. Quality of the mesh is maintained by keeping the element aspect ratio below 6. These simulations are run at ΔT=10K and at varying angles between applied gravity and the z axis of the cylinder φ between 0 and 0.37π for comparison to the referenced study [Ref. 3]. These verification simulations are run at only 40 iterations for time steps Δt of 0.1 s. Collected data are averaged over a minimum time of 3 s after steady state convergence is reached. The resulting Qconv is used to calculate Nusselt number Nu, which is then compared to results of [Ref. 3]. These calculations show excellent qualitative agreement with Qconv increasing with φ as predicted in [Ref 3]. The quantitative agreement is within 5% shown in Table 2.TABLE 2Percent errors for Nu values in an incline cylinder betweensimulated results and the data of high resolutions [Ref. 3].φ00.1π0.2π0.3πNu[Ref. 3]7.258.108.508.70Nu[This Example]7.158.408.919.06Percent Error1.3%3.5%4.6%3.9%2.4. Grid Sensitivity Study

[0080] Additional testing of this model is performed to determine mesh independence, ensuring that values for heat transfer are only a function of the flow parameters and not of the mesh parameters. This sensitivity study will ultimately determine the element size, number of elements NE, and element type which will provide accurate results. First, the simulation of a flow in a non-partition cavity is run at Ra=105 and AR=1 for meshes with hexahedron elements at NE=133,104 and tetrahedral element meshes at NE=133,326. Increased mesh density spanning a distance from the top and bottom surfaces at approximately 5 mm is included for both models to account for the increased activity near the thermal boundary layers. Nusselt number Nu is then calculated as mentioned in Section 2.1, using the time average values of the instantaneous temperatures where both models reach a fully developed flow, this being after 600 time steps, and using (6), (7), and (8). Values of Nu for the hexahedron and tetrahedral models are found to be Nu=3.25 and Nu=3.23, respectively. This gives a percentage error of less than 1% and validates that the tetrahedral elements can be used in place of hexahedron elements. Although hexahedron elements can provide improved accuracy with less elements, tetrahedral elements are better for complex geometry which will be a factor once adding the partition to the model. Therefore, further meshing will be done with the tetrahedral elements.

[0081] Finding the minimum mesh resolution that provides a reasonably accurate solution is determined by running multiple tests of a non-partitioned models at Ra=106, AR=1 and a partitioned model at Ra=107, AR=3, δ=0.016 both with increasing NE. The accuracy is determined by finding the point at which the value of Nusselt number Nu change minimally as NE changes. As can be seen in FIG. 3, Nu decreases as the resolution increases, with Nu starting to level out at approximately NE=1.4 million elements. This number is taken as acceptable in providing sufficient accuracy for the purposes of this Example. Anticipated increased activity in and around the thermal boundary layers near the top and bottom surfaces requires a higher mesh resolution, which can be seen in FIG. 5. This increased resolution is especially important when adding the partition which falls within the thermal boundary for each partitioned model.

[0082] The accuracy of the numerical model used in the Example is further analyzed by using the computed maximum velocity found to calculate the Courant Coefficient c. The coefficient is used to determine the stability of schemes for hyperbolic equations and establishes the distance to which information is transported by velocity over a time step in relation to the mesh step [Ref. 10]. Explicit schemes for purely hyperbolic equations are generally stable if the Courant-Friedrichs-Lewy stability condition [Ref. 11] is met. Ansys simulations use an implicit scheme which can be considered stable even when the condition is not met, the values for which are listed in Table 4.TABLE 3Values of Nu as a function of NE whichis represented graphically in FIG. 3.Number ofNu forNu forElementsRa = 106Ra = 10729000006.25—22300006.276.992000000—7.011350000—7.0512000006.28—10600007.177500006.37—690000—7.37345000—7.5880000—8.09

[0083] It should be noted that implicit schemes are unconditionally stable but have a truncation error heavily influenced by numerical dissipation; this can result in the amplification of a rounding error. To mitigate this effect, further simulations continue with a small time step size at 0.1 s. These verifications lead to the final design parameters for data collection simulations.c=<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[LeftBracketingBar]"< / annotation>< / semantics>umax<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[RightBracketingBar]"< / annotation>< / semantics>⁢Δ⁢tΔ⁢xmin≤1(11)TABLE 4Courant coefficients c calculated for further mesh verification.InstantaneousMesh Length atCNon-Dimensional ParameterMax VelocityMax Velocity(Δt =Combinations at N ~1.4 Million(m / s)(m)0.1 s)Ra = 106 AR = 1 No Partition3.72E−21.50E−424.8Ra = 107 AR = 3 δ = 0.0161.59E−14.15E−438.32.5. Simulation ProcedureEach model geometry is built using values for H based on set combinations of the non-dimensional parameters Ra and AR. A non-partitioned case is first run to find a baseline value for the heat transfer at the top and bottom of the cells and calculate Nuo. Instantaneous velocities along the z-axis in each non-partitioned simulation are collected and averaged over time to establish upper and lower thermal boundary layer thicknesses δBL,upper and δBL,upperδBL,lower≡(Tb-Thot)⁢<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[LeftBracketingBar]"< / annotation>< / semantics>dTdz<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[RightBracketingBar]"< / annotation>< / semantics>z=0-1(12)δBL,upper≡(Tb-Tcold)⁢<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[LeftBracketingBar]"< / annotation>< / semantics>dTdz<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[RightBracketingBar]"< / annotation>< / semantics>z=H-1(13)where Tb is the bulk temperature of the fluid outside of the boundary layer. The models for partitioned domains are then built for δ=0.04, 0.06, 0.08, and 0.10. If a local maximum of Qconv is not found in these simulations, models for additional values for δ are built and simulated.3. Results and Discussion3.1. Effect of Partition on Heat TransferThe main results of the Example are summarized in Table 5, showing all but two simulations presenting a notable increase in Nu over the non-partitioned counterparts, the exceptions being δ=0.04 and δ=0.06 for Ra=106 AR=1. A value of δ was found for each combination of Ra and AR which produced the highest Nusselt number δmax, resulting in the greatest convective heat transfer amplification.The data in Table 5 present a trend in which δmax decreases as AR increases, which is further decreased with an increase in Ra as shown in FIG. 7. This demonstrates an inverse relationship between δ and combinations of AR and Ra. This trend stays consistent with the data for the Nusselt number of partitioned models normalized with the Nusselt number for non-partitioned models Nu / Nuo, shown in FIG. 6. Additionally, there is an observable point in which the decrease in δ no longer increases Nu, showing a local maximum δ in each instance.TABLE 5Time-averaged values of Nu computed in all completed simulation.Ra106107AR12351235No Partition6.287.843.061.0010.59.287.924.65δ = 0.01———————17.45δ = 0.02———10.91——18.6925.65δ = 0.045.777.8611.0713.0512.4017.0821.3325.25δ = 0.066.209.3211.4911.4012.5017.6619.3220.60δ = 0.086.409.8610.569.6512.8016.9817.9119.15δ = 0.106.809.569.578.3511.8016.3816.3817.70δ = 0.126.83———————δ = 0.146.90———————δ = 0.166.78———————TABLE 6Normalized values of the Nusselt Number (Nu / Nu0).Ra106107AR12351235δ = 0.01———————3.75δ = 0.02———10.90——2.365.51δ = 0.040.921.003.6213.101.181.842.695.43δ = 0.060.981.193.7511.051.191.902.444.43δ = 0.081.021.263.459.651.211.832.264.12δ = 0.101.081.223.138.351.121.772.073.81δ = 0.121.09———————δ = 0.141.10———————δ = 0.161.08———————3.2. Effect of Partition on Flow StructureFIG. 8 panel a through FIG. 15 panel g present the flow structure and discuss the results based on temperature profiles with a nondimensional range of T=0.94 (blue) to T=1.06 (red) The partition created changes in the flow structure in two significant ways. First, it can be observed in FIG. 8 panel a through FIG. 15 panel g that adding the partition creates the large-scale circulation of the flow, which was noted to influence heat transfer when modified [Ref. 8]. Second, having δ smaller than the thermal boundary layer leads to an increased heat flux by intensifying the heat transfer across the thermal boundary layer. However, a decrease of δ also has a detrimental effect on heat transfer, because a small gap creates obstruction to the flow which reduces its kinetic energy. The relation between δ and the thickness of the thermal boundary layer is further discussed in Section 3.3.Organization of the flow due to the addition of the partition is evident in the simulation of the temperature and velocity contours shown in FIGS. 16 and 17. Horizontal jets occur at the top and bottom gaps with the addition of the partition and large-scale circulation is produced. These jets flow through and disrupt the top and bottom thermal boundary layers, contributing to thermal efficiency. Given the results in Table 3, it can be assumed that organization of the fluid flow facilitates increased convective heat transfer. The one instance in which convective heat transfer experienced a decrease in Nusselt number Nu compared to its non-partitioned case may speak to the theory of increased heat transfer due to a more compressed model as this specific instance was set at AR=1 [Ref. 6]. Table 6 shows the results for the analysis of the thermal boundary layers and can be observed in FIGS. 18 and 19.

[0089] The case of Ra=106 AR=5 with no partition is an outlier of the non-partitioned cases having no velocity with no flow having been developed. This can be observed in FIG. 11 panel a where the temperature gradient remains perfectly stratified due to the lack of fluid flow. It has been shown through analysis [Ref. 13] and demonstrated experimentally [Ref. 14] that flows within cylindrical containers of AR=5 for a fluid at Ra=106 have a stability that falls near the Critical Rayleigh Number, Racr, the threshold for which convective flow begins to develop. Due to the strong numerical dissipation produced by the finite-volume scheme used by Fluent, it is possible that a weak convective flow just above Racr are not captured in simulation. The comparison of analyzed and experimental data presented by Muller, Neumann, and Weber in their work on natural convection in cylindrical cavities [Ref. 14] shows that a flow with Ra=106 AR=5 is weak and non-turbulent. From this, it can be assumed that flow produced at these conditions does not manifest in Fluent simulation, where flow computed in a non-partitioned domain shows zero velocity and only a purely conduction profile of temperature. This is not surprising, considering that the sidewall closeness increases Racr above the point at which convection first occurs, to about Racr=7×105 [Ref. 14]. Therefore, convection not occurring in the simulations at a slightly higher Ra=106 is attributable to the known sensitivity of instability to numerical dissipation of the numerical method. The dissipation is relatively high in the finite-volume solution of the models. Results presented in FIG. 11 show that a partition may generate convection flow in geometries where convection would not otherwise happen.3.3. Relation Between the Optimal Gap Size and Thermal Boundary Layer Thickness

[0090] The relationship between the size of the gaps between the partition and the top and bottom walls and the thickness of the thermal boundary layer is further explored in this Section. For calculating the boundary layer thickness, flows computed for non-partitioned domains are used to find the vertical profiles of time-averaged temperature along the cylinder axis. These results are presented in FIGS. 19 and 20 and show that, except for the case Ra=106, AR=5, in which no flow and, thus, no boundary layer exists, the boundary layer thicknesses can be determined using (12) and (13) and are shown in Table 7.

[0091] The thicknesses obtained in each case for the upper and lower walls are averaged to provide the final estimates for δBL and presented in Table 7, showing that the boundary layer thickness drops about twofold as Ra increases from 106 to 107. The effect of AR is weaker. At both values of Ra, δBL decreases as AR decreases from 1 to 2 and increases with AR at AR≥2. Analysis of the results in terms of the ratio between the partition gap size and the thickness of the thermal boundary layer δ / δBL is presented in FIGS. 20 and 21. Unfortunately, a hypothesis that the amplification of heat transfer can be approximated as a function of the single parameter δ / δBL is not supported by the data. The curves of Nu / Nuo obtained for various Ra and AR do not collapse into one curve if plotted with δ / δBL (see FIG. 18). Even more disappointing is that the ratio δmax / δBL varies strongly with Ra and AR. The scatter plot of δmax vs δBL shown in FIG. 21 also does not show any clear dependency. However, while the hypothesis that δmax / δBL being a defining parameter is disproved by the results, it can be concluded that the optimal gap size δmax is always smaller than the boundary layer thickness. This means that the partition must penetrate the thermal boundary layer to facilitate a strong heat transfer enhancement.TABLE 7Thicknesses of the upper and lower thermal boundary layers δBL (in mm) computed withdata from time-averaged temperature fields in domains without partitions (see text forexplanations). Note that the mesh refinement zones set in these simulations as 20% of thecylinder height completely cover the thermal boundary layer in all simulations.Ra106107AR12351235Tb343.08343.19343.13343.15343.40343.16342.90343.63<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[LeftBracketingBar]"< / annotation>< / semantics>dTdz<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[RightBracketingBar]"< / annotation>< / semantics>⁢lower298626411658N / A2390301222721094δBL,lower6.706.527.74N / A8.086.206.669.07δ⁢BL,lowerH0.1580.1540.183N / A0.0880.0680.0730.099<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[LeftBracketingBar]"< / annotation>< / semantics>dTdz<semantics definitionURL="">❘<annotation encoding="Mathematica">"\[RightBracketingBar]"< / annotation>< / semantics>⁢upper266025491686N / A2368293823031070δBL,lower7.516.797.70N / A8.746.436.528.85δ⁢BL,upperH0.1770.1600.181N / A0.0960.0700.0710.097δ⁢BL,averageH0.1670.1570.182N / A0.0920.0690.0720.0983.4. Further Heat Transfer Optimization

[0092] In this Section, we consider the possible effects of asymmetry between the top and bottom gaps of the partition. The asymmetry is defined by the non-dimensional parameter α, or the ratio of Hg,upper and Hg,lower. A sampling of data was taken for the flow at Ra=106, AR=1, and δ=0.06, with a adjusted from 0.5 up to 1.25. A local maximum of the Nusselt number was found at α=0.75.

[0093] It was decided to test α=0.75 with the combinations of AR and δ that produced the highest values for Nu at Ra=106 and Ra=107. The results are presented in Table 8. While there is a similar increase in Nu for optimized conditions at Ra=107, there is a slight decrease for Ra=106. These findings suggest that there is a potential for further increase of heat transfer rate.TABLE 8Comparison of Nu as a function of α forset combinations of non-dimensional variables.Raδα = 0.75α = 11060.067.006.201060.146.856.901070.0814.212.84. Conclusion

[0094] This Example presented the results of numerical simulations of the Rayleigh-Benard convection in cylindrical cells with a vertical partition and liquid gallium as a working fluid. The results show that the use of a partition leads to very strong (more than tenfold in one case) amplification of the rate of heat transfer. The amplification is much stronger than what was observed in earlier studies with water [Ref. 2, 6, 7], which we attribute to the effect of the low Pr of gallium. Moreover, we found that with a configuration close to convection stability limit (Ra=106, AR=5) the presence of a partition may cause a convection flow, even while such a flow is not observed in the non-partitioned case. Unfortunately, our hypothesis that the effect of heat transfer amplification is largely determined by the ratio of the gap width between the partition and the cylinder walls and the thickness of the thermal boundary layer was not supported by our data. Nevertheless, it can be concluded that the optimal gap size δmax is always smaller than the boundary layer thickness, meaning the partition penetrating the thermal boundary layer does assist in heat transfer enhancement.

[0095] Supplementary exploration of the combined effects studied in this Example includes extending the analysis to higher Ra, higher and lower AR, and various values of a. Additionally, it is anticipated that the effect of heat transfer enhancement is not limited to the geometry of a cylinder with a single partition. Other cavity shapes (e.g., a cuboid) and the use of multiple partitions may prove beneficial. Lastly, an experiment confirming the effect can be performed and would further validate the simulated data.

[0096] It is contemplated that further research may include more accurate numerical simulations, which would be free from numerical dissipation and other accuracy-detrimental features of a commercial CFD model. Furthermore, research into the use of an asymmetrically positioned partition indicates the possibility of additional heat transfer amplification.

[0097] The simulated data from this Example may prove useful for thermal management of systems in which a significant amount of convective heat transfer is required. An example of this would be a horizontally mounted CPU with a hypothetical cold sink resting above it. A convective cell using a low temperature liquid metal could be used in between to facilitate a large amount of heat transfer from the CPU for cooling [Ref. 12]. Other promising applications are stationary battery energy storage or cooling for high-rate power electronics equipment.

[0098] Thus, the present invention provides: (i) a device having enhanced heat transfer in natural convection by means of liquid metals or liquid metal alloys and partitioned domains, and (ii) improved methods for cooling a heat source.LIST OF VARIABLESList of VariablesAR—Aspect Ratioα—Ratio of Top and Bottom Hgc—Courant Coefficientβ—Coefficient of Thermal ExpansionCp—Specific HeatΔT—Temperature DifferenceD—Cylinder DiameterΔt—Time Step Sizeg—Acceleration of GravityΔx—Mesh Element LengthGr—Grashof Numberδ—Ratio Between Partition Gap and CylinderH—Cylinder HeightHeightHg—Gap Height Above and Below PartitionsδB—Average Lower Thermal Layer BoundaryNE—Number of Elements of Computational GridThicknessp—PressureδBL, lower—Lower Thermal Layer Boundarypb—Fluid Bulk PressureThicknessPr—Prandtl NumberδBL, upper—Upper Thermal Layer BoundaryRa—Rayleigh NumberThicknessRacr—Critical Rayleigh Numberϕ—Angle of Applied GravitySTB—Top / Bottom Cylinder Surface Areaχ—Thermal DiffusivityT—Temperatureλ—Thermal ConductivityTb—Fluid Bulk Temperatureρ—DensityTcold—Cold Sink Surface Temperatureμ—Dynamic ViscosityThot—Heat Source Surface TemperatureΘ—Angle Between x and y MagnitudesTmelt—Melting Temperatureν—Kinematic ViscosityU—Free-Fall VelocityΩ—FrequencyREFERENCES1. J. Scheel and J. Schumacher, “Global and local statistics in turbulent convection at low Prandtl numbers,”Journal of Fluid Mechanics, vol. 802, pp. 147-173, August 2016

[0100] 2. Y. Bao, J. Chen, B.-F. Liu, Z. She, J. Zhang, and Q. Zhou, “Enhanced heat transport in partitioned thermal convection,”Journal of Fluid Mechanics, vol. 784, November 2015

[0101] 3. O. Shishkina and S. Horn, “Thermal convection in inclined cylindrical containers,”Journal of Fluid Mechanics, vol. 790, February 2016

[0102] 4. L. Zhang and K. Xia, “Achieving heat transfer enhancement via manipulation of bulk flow structures in turbulent thermal convection,”Physical Review Fluids, vol. 8, no. 2, February 2023

[0103] 5. L. Yuan, S. Zou, Y. Yang, and S. Chen, “Boundary-Layer disruption and Heat-Transfer enhancement in convection turbulence by oscillating deformations of boundary,”Physical Review Letters, vol. 130, no. 20, May 2023

[0104] 6. S. Di Huang, M. Kaczorowski, R. Ni, and K. Xia, “Confinement-Induced Heat-Transport enhancement in turbulent thermal convection,”Physical Review Letters, vol. 111, no. 10, September 2013

[0105] 7. J. Chen, Y. Bao, Z. Yin, and Z. She, “Theoretical and numerical study of enhanced heat transfer in partitioned thermal convection,”International Journal of Heat and Mass Transfer, vol. 115, pp. 556-569, December 2017

[0106] 8. P. K. Kar, U. Chetan, J. Mahato, T. L. Sahu, P. K. Das, and R. Lakkaraju, “Heat flux enhancement by regular surface protrusion in partitioned thermal convection,”Physics of Fluids, vol. 34, no. 12, December 2022

[0107] 9. O. Zikanov, I. A. Belyaev, Y. Listratov, P. Frick, N. G. Razuvanov, and V. G. Sviridov, “Mixed convection in pipe and duct flows with strong magnetic fields,”Applied Mechanics Reviews, vol. 73, no. 1, January 2021

[0108] 10. O. Zikanov, Essential computational fluid dynamics, 2nd Edition. John Wiley & Sons, 2019.

[0109] 11. Y. Cengel, R. Turner, and J. Cimbala, Fundamentals of thermal-fluid sciences, 4th Edition. McGraw-Hill, 2012.

[0110] 12. J. Liu, Advanced liquid metal cooling for chip, device and system. 2022.

[0111] 13. G. S. Charlson and R. L. Sani, “On Thermoconvective instability in a bounded cylindrical fluid layer,”International Journal of Heat admass Transfer, vol. 13, no. 9, pp. 1479-1496, September 1970

[0112] 14. G. Müller, G. Neumann, and W. Weber, “Natural convection in vertical Bridgman configurations,”Journal of Crystal Growth, vol. 70, no. 1-2, pp. 78-93, December 1984The citation of any document is not to be construed as an admission that it is prior art with respect to the present invention.

[0113] In light of the principles and example embodiments described and illustrated herein, it will be recognized that the example embodiments can be modified in arrangement and detail without departing from such principles. Also, the foregoing discussion has focused on particular embodiments, but other configurations are also contemplated. In particular, even though expressions such as “in one embodiment”, “in another embodiment”, “in certain embodiments”, or the like are used herein, these phrases are meant to generally reference embodiment possibilities, and are not intended to limit the invention to particular embodiment configurations. As used herein, these terms may reference the same or different embodiments that are combinable into other embodiments. As a rule, any embodiment referenced herein is freely combinable with any one or more of the other embodiments referenced herein, and any number of features of different embodiments are combinable with one another, unless indicated otherwise.

[0114] Although the invention has been described in considerable detail with reference to certain embodiments, one skilled in the art will appreciate that the present invention can be practiced by other than the described embodiments, which have been presented for purposes of illustration and not of limitation. Therefore, the scope of the appended claims should not be limited to the description of the embodiments contained herein. Various features and advantages of the invention are set forth in the following claims.

Claims

1. A heat transfer device comprising:an outer wall, a first end wall connected to a first end of the outer wall, and a second end wall connected to an opposite second end of the outer wall, wherein the outer wall, the first end wall, and the second end wall define a cavity;at least one partition wall located in the cavity, each partition wall being spaced inward from the first end wall and the second end wall; anda working fluid contained in the cavity, the working fluid being selected from liquid metals and liquid metal alloys, wherein the working fluid has a final melting point at or below an operating temperature of the heat transfer device,wherein the heat transfer device transfers heat from a heat source adjacent the first end wall to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source toward the second end wall.

2. The device of claim 1 wherein:the working fluid is selected from the group consisting of gallium, mercury, sodium, and eutectic alloys.

3. The device of claim 1 wherein:the working fluid is selected from the group consisting of gallium, mercury, sodium, a eutectic alloy of gallium, indium, and tin, and a eutectic alloy of bismuth, lead, tin, and cadmium.

4. The device of claim 1 wherein:the working fluid comprises gallium.

5. The device of claim 1 wherein:the working fluid has a final melting point of 100° C. or below.

6. The device of claim 1 wherein:the working fluid has a final melting point of 80° C. or below.

7. The device of claim 1 wherein:the working fluid has a final melting point of 40° C. or below.

8. The device of claim 1 wherein:the at least one partition wall extends laterally between a first inner surface of the outer wall and a second inner surface of the outer wall.

9. The device of claim 1 wherein:the outer wall has a cylindrical shape such that the cavity has a height and a diameter.

10. The device of claim 9 wherein:a height-to-diameter ratio of the cavity is 3 or greater.

11. The device of claim 9 wherein:the at least one partition wall extends laterally along the diameter of the cavity between a first inner surface of the outer wall and a second inner surface of the outer wall.

12. The device of claim 9 wherein:each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, anda gap distance-to-height ratio (δ) is in a range of 0.001 to 0.3.

13. The device of claim 9 wherein:each partition wall is spaced inward from the first end wall and the second end wall by a gap distance, anda gap distance-to-height ratio (δ) is in a range of 0.02 to 0.6.

14. The device of claim 9 wherein:a diameter-to-height ratio (AR) of the cavity is in a range of 0.1 to 10.

15. The device of claim 9 wherein:a diameter-to-height ratio (AR) of the cavity is in a range of 4 to 6.

16. The device of claim 1 wherein:a Rayleigh number (Ra) of the working fluid is in a range of 105-108.

17. The device of claim 1 wherein:a Rayleigh number (Ra) of the working fluid is in a range of 105-107.

18. The device of claim 1 wherein:the at least one partition wall is spaced inward from the first end wall by a first gap distance,the at least one partition wall is spaced inward from the second end wall by a second gap distance, anda gap ratio (a) of the second gap distance to the first gap distance is not 1.

19. The device of claim 18 wherein:the gap ratio (a) in a range of 0.5 to 1.5.

20. The device of claim 18 wherein:the gap ratio (a) in a range of 0.7 to 0.8.

21. The device of claim 1 further comprising:the heat source,wherein the heat source is positioned adjacent the first end wall such that the first end wall has a first temperature higher than a second temperature of the second end wall, and the first end wall is located at a lower level than the second end wall.

22. The device of claim 21 wherein:the heat source is selected from engines, electrochemical devices, power electronics, computer components, and heating, ventilation, and air conditioning systems.

23. The device of claim 1 wherein:the outer wall has a polygonal shape.

24. A method for cooling a heat source, the method comprising:(a) providing a heat transfer device comprising: (i) an outer wall, a first end wall connected to a first end of the outer wall, and a second end wall connected to an opposite second end of the outer wall, wherein the outer wall, the first end wall, and the second end wall define a cavity, (ii) at least one partition wall located in the cavity, each partition wall being spaced inward from the first end wall and the second end wall; and (iii) a working fluid contained in the cavity, the working fluid being selected from liquid metals and liquid metal alloys, wherein the working fluid has a final melting point at or below an operating temperature of the heat transfer device;(b) thermally coupling the first end wall of the heat transfer device with a heat source; and(c) cooling the heat source by transferring heat from the heat source to the working fluid such that convection is induced within the working fluid that increases heat transfer from the heat source toward the second end wall.

25. The method of claim 24 wherein:the working fluid is selected from the group consisting of gallium, mercury, sodium, and eutectic alloys.

26. The method of claim 24 wherein:the working fluid is selected from the group consisting of gallium, mercury, sodium, a eutectic alloy of gallium, indium, and tin, and a eutectic alloy of bismuth, lead, tin, and cadmium.

27. The method of claim 24 wherein:the working fluid comprises gallium.

28. The method of claim 24 wherein:the working fluid has a final melting point of 100° C. or below.

29. The method of claim 24 wherein:the at least one partition wall extends laterally between a first inner surface of the outer wall and a second inner surface of the outer wall.

30. The method of claim 24 wherein:the outer wall has a cylindrical shape such that the cavity has a height and a diameter,a height-to-diameter ratio of the cavity is 3 or greater,each partition wall is spaced inward from the first end wall and the second end wall by a gap distance,a gap distance-to-height ratio (δ) is in a range of 0.001 to 0.3, anda diameter-to-height ratio (AR) of the cavity is in a range of 0.1 to 10.

31. The method of claim 24 wherein:a Rayleigh number (Ra) of the working fluid is in a range of 105-108.

32. The method of claim 24 wherein:the at least one partition wall is spaced inward from the first end wall by a first gap distance,the at least one partition wall is spaced inward from the second end wall by a second gap distance,a gap ratio (α) of the second gap distance to the first gap distance is not 1, andthe gap ratio (α) in a range of 0.5 to 1.5.

33. The method of claim 24 wherein:the heat source is positioned adjacent the first end wall such that the first end wall has a first temperature higher than a second temperature of the second end wall, and the first end wall is located at a lower level than the second end wall.

34. The method of claim 24 wherein:the outer wall has a polygonal shape.

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