UNIVERSITY OF PISA
Civil and Industrial Engineering Department Master of Science in Space Engineering
TWO PHASE PASSIVE HEAT TRANSFER DEVICES FOR SPACE APPLICATION
SUPERVISORS
Prof. Sauro FILIPPESCHI Prof. Salvo MARCUCCIO Dr. Mauro MAMELI
CANDIDATE
Aditya Reddy RAMIREDDY
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Acknowledgements
Firstly, I would like to thank Prof. Sauro Filippeschi and Prof. Salvo Marcuccio for giving the opportunity to work with them. Every discussion we had, helped me boost my confidence. I would be indebted to them forever, as their suggestions not only improved my knowledge but also my thought process. I would like to specially mention the suppor given by Dr. Mauro Mameli, who mentored me in every phase of my thesis.
I would like to thank all my friends for their constant support. I would like to mention the organisation “Associazione Sante Malatesta Onlus” that became more like a family to me by supporting me in all my endeavours during my stay.
Finally, I would like to thank my mother, father, brother and sister in law, for their constant support and boosting my morale. A special mention to my late grandmother, who always believed in my abilities.
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Abstract
With an aim to reach farther in to space, research is going on to develop advanced propulsion systems and miniaturisation of electronics. This leads to the need to dissipate excess heat from the systems and high heat fluxes. Two phase devices fit in to the solution set due to their capability to transfer heat up to 10 m distance, and with an accuracy up to 0.1 K. They can be used over wide ranges of temperatures. Pulsating heat pipe is one such device which can cater to the needs of deep space mission’s thermal control requirements. It is light, reliable, passive, cheap and has satisfactory thermal performance. A hybrid heat pipe, also known as, loop thermosyphon has been proposed for space applications. To better understand the physical phenomenon of the loop thermosyphon, experimental and analytical study is required. Hence, a numerical model has been studied and is validated against the experimental results.
For this purpose, two models have been studied. First model considered, is based on conservation of momentum equation. Mass flow rate is calculated for different working fluids at various fill ratios. The mass flow rate is as a function of input heat power. The second model is based on the consideration that the two-phase flow is a homogeneous mixture which is in thermal equilibrium. The flow model is solved with a hyperbolic solver based on Godunov method. The model considers subcooled liquid and overheated vapor as well as phase transition. To carry out the numerical analysis, the equations are discretised in a one-dimensional space using finite volume approach. MATLAB code has been developed to give the initial steady state condition of the flow.
The first model is based on lumped capacitance model and it accurately depicts the changing trends in the mass flow rate with the variation of power. It is validated against the experimental results for the working fluid ethanol and water for different fill ratios. The second model is thermodynamic equilibrium model which can reproduce satisfactorily the steady state response of a classical loop and variation of temperature, density, vapor mass fraction with the change in gravity. As a future work, both the models can be merged to create a new model which can estimate the variation of operating parameters and can be used as an initial point for better understanding of the dynamics of loop thermosyphon in space.
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Contents
Acknowledgements ...ii
Abstract ... iii
List of figures ... vii
List of tables ... viii
Chapter 1 ... 1
Introduction ... 1
1.Thermal control in space ... 1
2.Technology Readiness Levels ... 1
3.State of the art of thermal control systems ... 3
3.1 Active thermal control system ... 3
3.1.1 Thermal straps ... 3
3.1.2 Heaters ... 3
3.1.3 Cryocoolers ... 4
3.2 Passive thermal control system ... 4
3.2.1 Multilayer insulation ... 4
3.2.2 Coatings ... 5
3.2.3 Thermal fillers ... 6
3.2.4 sun shields ... 6
3.2.5 Flexible thermal straps ... 7
3.2.6 Thermal Louvers ... 7
3.2.7 Radiators ... 8
3.2.8 Heat pipe capillary-based loops ... 9
4.Conclusion ... 10
Chapter 2 ... 11
Two phase passive systems ... 11
1. Working principle ... 11
2. Types of heat pipes ... 12
2.1 Flat heat pipes ... 12
2.2 Variable conductance heat pipes ... 13
2.3 Diode heat pipes ... 14
2.4 Pulsating heat pipes ... 15
2.5 Loop heat pipes ... 15
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2.7 Sorption heat pipes ... 17
Chapter 3 ... 18
Closed Loop Thermosyphon Numerical Simulation ... 18
1. Thermosyphon ... 18
1.1 Two-phase systems: Literature survey ... 19
1.2 Flow Regimes ... 20
1.3 Defining parameters for a two-phase flow ... 22
1.4 Definition of basic quantities for a two-phase flow ... 26
2. Modelling ... 28
2.1 Two-phase conservation equations ... 28
2.2 Treatment of Phase interface ... 28
2.3 Mass conservation ... 29
2.4 Energy conservation equation ... 29
2.5 Momentum conservation equation ... 30
2.5 Closure of the system of equations ... 31
3. Modelling Approaches ... 32
3.1 Lumped capacitance model ... 32
3.2 Thermodynamic equilibrium Model... 38
3.2.1 Design ... 39
3.2.2 Dimensions ... 40
Chapter 4 ... 41
Methodology ... 41
1. Lumped capacitance model ... 41
1.1 Flow solver ... 41
1.2 Simulation Strategy ... 42
2. Thermodynamic equilibrium model ... 42
2.1 Assumptions ... 42
2.2 Flow solver ... 48
2.3 Simulation strategy ... 49
2.4 Numerical analysis... 50
2.4.1 Initial steady state ... 50
Chapter 5 ... 52
Results and Conclusion... 52
1. Results of lumped capacitance model ... 52
2. Preliminary results of thermodynamic equilibrium model ... 61
vi 3.1 Conclusion ... 63 3.2 Future Prospects ... 64 Appendices ... 65 Appendix A ... 65 MATLAB codes ... 65
A.1 Lumped Capacitance Model: ... 65
A.2 Thermodynamic Equilibrium model ... 73
Nomenclature ... 75
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List of figures
Figure 1 TRL ladder from concept to flight validation (ESA, Technology readiness levels, 2015) ... 2
Figure 2 Cryocooler developed by Creare (Creare, 2015) ... 4
Figure 3 Multi-Layer Insulation blanket on payload module (Sylvain, 2012) ... 5
Figure 4 Sun shield on infrared space observatory (ESA, Spacecraft, 2005) ... 6
Figure 5 I shaped thermal strap (space, 2015) ... 7
Figure 6 Thermal louvres in Rosetta spacecraft (ESA, Space in images, 2015) ... 8
Figure 7 International Space Station photographed by an STS-134 crew member of the Space Shuttle Endeavour on May 29, 2011. (NASA, 2011) ... 9
Figure 8 Heat pipe experimental set-up used on ISS (Avino, 2017) ... 10
Figure 9 Representation of a heat pipe ... 12
Figure 10 A flat heat pipe on a telecommunication equipment (Meyer, 2016) ... 13
Figure 11 Variable conductance Heat pipe (engineering, 2018) ... 14
Figure 12 representation of a reverse mode liquid trap diode ... 14
Figure 13 Representation of a pulsating heat pipe (Reay D, 2006) ... 15
Figure 14 Representation of a loop heat pipe (Reay D, 2006) ... 16
Figure 15 Schematic of a micro heat pipe (Reay D, 2006) ... 17
Figure 16 Schematic of a sorption heat pipe (Reay D, 2006) ... 17
Figure 17 Schematic of working of a heat pipe (Mameli M. , 2012) ... 19
Figure 18 Flow pattern observed by Triplett (Triplett, 1999) ... 22
Figure 19 Comparison of selected criteria as a function of reduced pressure (Baldassari, 2013) .. 26
Figure 20 Control volume of a two-phase flow ... 28
Figure 21 Control volume for momentum equation ... 31
Figure 22 Schematic representation of a loop thermosyphon used for lumped capacitance analysis ... 34
Figure 23 variation of two-phase frictional multiplier with steam quality for water for different mass flow rates ... 38
Figure 24 Schematic representation of two phase loop thermosyphon ... 39
Figure 25 Sketch of one-dimensional finite volume cells and related cell faces ... 49
Figure 26 Variation of mass flow rate with power for ethanol at fill ratio of 0.2 ... 53
Figure 27 Variation of mass flow rate with power for ethanol at fill ratio of 0.25 ... 54
Figure 28 Variation of mass flow rate with power for ethanol at fill ratio of 0.32 ... 55
Figure 29 Comparison of mass flow rate regimes for ethanol at various fill ratios ... 55
Figure 30 Variation of mass flow rate with power for water at fill ratio of 0.2 ... 56
Figure 31 Variation of mass flow rate with power for water at fill ratio of 0.25 ... 57
Figure 32 Variation of mass flow rate with power for water at fill ratio of 0.32 ... 58
Figure 33 Comparison of mass flow rate regimes for water at various fill ratios ... 58
Figure 34 Design of the CLTPT experimental system established (a) Schematic system design; and (b) a picture of the experimental apparatus (Alessandro Franco, 2013) ... 59
Figure 35 Mass flow rate as a function of the heat input for ethanol at different values of filling ratio represented by the parameter H/L at atmospheric pressure (Alessandro Franco, 2013) ... 60
Figure 36 Mass flow rate as a function of the heat input for water at different values of filling ratio represented by the parameter H/L at atmospheric pressure (Alessandro Franco, 2013) ... 60
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Figure 38 Variation of temperature in the loop thermosyphon in steady state ... 62
Figure 39 Variation of velocity in the loop thermosyphon in steady state ... 62
Figure 40 Variation of vapor mass fraction in the loop thermosyphon in steady state ... 63
List of tables
Table 1 Technology readiness levels (ESA, Technology readiness levels, 2015) ... 2Table 2 Useful non-dimensional numbers ... 23
Table 3 Critical diameter as proposed by different authors compared with equivalent Eotvos number ... 25
Table 4 Geometric data for the simulation ... 52
Table 5 Thermal conductivity of different components of Loop Heat pipe ... 52
Table 6 Condenser evaporator distance for fill ratio of 0.2 ... 53
Table 7 Fluid properties for Ethanol... 53
Table 8 Condenser evaporator distance for fill ratio of 0.25 ... 54
Table 9 Condenser evaporator distance for fill ratio of 0.32 ... 55
Table 10 Fluid properties of water ... 56
Table 11 Condenser evaporator distance for fill ratio of 0.2 ... 56
Table 12 Condenser evaporator distance for fill ratio of 0.25 ... 57
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Chapter 1
Introduction
1.Thermal control in space
In spacecraft design the function of thermal control system is to maintain thermal environment for all spacecraft components subject to environmental range of conditions, operating modes and spacecraft lifetime (Marcuccio, 2012-2013). The external environments can vary when the spacecraft is exposed to deep space or to solar or planetary flux and the heat, which is generated due to the operating of the spacecraft itself, ejected to the space. Thermal control is required to guarantee the optimum performance and the success of the mission, if the spacecraft is subject to temperatures too high or too low. Thermal control is also what keeps the specified temperature stability for delicate electronics or optical components to ensure that they perform as efficiently as possible. Thermal control for space applications covers a very wide temperature range, from the cryogenic level (down to -270 0C) to high-temperature thermal protection systems (more than 2000 0C).
Thermal control subsystem can be used to achieve two goals.
• To Protect the equipment from overheating, either by thermal insulation from external heat fluxes such or the sun or the planetary infrared and albedo flux or by proper heat removal from internal sources such as the heat emitted by the internal electronic equipment.
• To protect the equipment from temperature that are too cold, by thermal insulation from external sinks, by enhanced heat absorption from external sources, or by heat release form internal sources.
2.Technology Readiness Levels
It is a method to estimate the technological maturity of a critical technology elements, like, program concepts, technology requirements, and demonstrated technology capabilities of a program. TRL are based on a scale 1 to 9 and 9 being the most mature technology. Figure
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1 shows the technology readiness chart with the different stages of the usability of the technology.
TRL Description
1 Basic principles observed
2 Technology concept formulated
3 Experimental proof of concept
4 Technology validated in lab
5 Technology validated in relevant environment
6 Technology demonstrated in relevant environment
7 System prototype demonstration in operational environment
8 System complete and qualified
9 Actual system proven in operational environment
Table 1 Technology readiness levels (ESA, Technology readiness levels, 2015)
Figure 1 TRL ladder from concept to flight validation (ESA, Technology readiness levels, 2015)
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3.State of the art of thermal control systems
Thermal control subsystem can be composed of two types.
• Active thermal control system • Passive thermal control system
3.1 Active thermal control system
Active thermal control system includes components which depend on input power for operation. These are associated with higher precision and are known to be more effective. Active thermal control devices are electrical resistance heaters, coolers or the use of cryogenic materials.
3.1.1 Thermal straps
Active thermal straps are used to increase thermal performance that is associated with high concentrated heat fluxes on the electronics. The advanced thermally conductive path on the strap supplies a reliable mitigation method for reducing hot spots while also limiting integration overhead and space. Load Path Aerospace Structures currently have Flexible and Enhanced Active Thermal Straps that are capable of heat dissipation up to 50 Wcm -2 and cooling capacity of 35 W. The status of the technology is TRL7.
3.1.2 Heaters
Heaters are used to maintain battery temperature during cold cycles of the orbit and are controlled by a thermostat or temperature sensor. These are used to maintain thermal regulation for the entire spacecraft to maintain thermal regulation during eclipses (Gilmore, 2002). Actively-controlled resistance heaters for precise temperature maintenance for their biological payloads with close loop temperature feedback to maintain the biology temperatures. Usually manufacturers produce flexible strip heaters equipped with polyimide insulation. The status of the technology is TRL 9.
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3.1.3 Cryocoolers
Cryogenic coolers, or cryocoolers, are used on instruments or subsystems requiring cryogenic cooling, such as high precision IR sensors. The low temperature improves the dynamic range and extends the wavelength coverage. Further, the use of cryocoolers is associated with longer instrument lifetimes, low vibration, high thermodynamic efficiency, low mass and supply cooling temperatures less than 50 K. Figure 2 shows single-stage turbo-Brayton cryocooler, developed by creare (Creare, 2015), that operates between a cryogenic heat rejection temperature and the primary load temperature.
Figure 2 Cryocooler developed by Creare (Creare, 2015)
3.2 Passive thermal control system
Passive thermal control requires no input power for thermal regulation within a spacecraft. This can be achieved using several methods and is highly advantageous to spacecraft designers as passive thermal control systems are associated with low cost, volume, weight and risk, and have been shown to be reliable. The following are the different technologies used for the passive thermal control system.
3.2.1 Multilayer insulation
Multilayer insulation is the most common thermal control technique used on spacecrafts. It protects the spacecraft from excessive solar or planetary heating as well from excessive cooling when exposed to deep space. Spacecraft component such as propellant tanks,
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propellant lines, batteries and solid rocket motors are also covered in multilayer insulation blankets to maintain ideal operating temperature. It consists of an outer cover layer, interior layer and an inner cover layer. The outer cover layer needs to be opaque to sunlight, generate a low amount of particulate contaminates and be able to withstand the environment and temperature to which the spacecraft will be exposed. In Figure 3 a multi-layer insulation blanket can be seen wrapped around the payload of a metrological satellite. The insulation blanket is attached to the payload using a Velcro and some other pieces are punctured to insert a stud and closed by a clip. Some pieces can be taped to the structure. The status of this component is TRL 9.
Figure 3 Multi-Layer Insulation blanket on payload module (Sylvain, 2012)
3.2.2 Coatings
Coatings are the simplest and least expensive of the thermal control system techniques. A coating can be paint or a more sophisticate chemical applied to the surfaces of the spacecraft to lower or increase heat transfer (Jaworske, Optical and calorimetric evaluation of Z-93-P and other thermal control coatings, 1996). The characteristics of the type of coatings depends on their absorptivity, emissivity, transparency and reflectivity. These coatings change the thermos optical properties of external surfaces (Jaworske, 1998). The main disadvantage of coating is that it degrades quickly due to operating environment. The status of this technology is TRL 9
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3.2.3 Thermal fillers
Thermal fillers are materials used to fill the voids formed by the surface peaks and valleys appearing in the contact region when two materials are brought into contact. With these materials, the area of the heat transfer path is maximized, and therefore, heat conduction is enhanced. The thermal fillers are usually made of soft materials with high thermal conductivity, and in some cases also high electrical impedance so that they can provide electrical isolation. These thermal fillers are generally used to improve the heat transfer between units with high power dissipation, such as electronics units, and their mounting base-plates. The status of this technology is TRL 2
3.2.4 sun shields
A space sunshade or a sunshield is an umbrella like structure that diverts or sometimes reduces the radiation of a star preventing them from hitting a spacecraft and thereby reducing the solar irradiance. This helps in reducing the heating of a spacecraft (J Meseguer, 2012). This sun shield can also be used to generate solar power in space. In Figure 4 a sunshield with solar cells can be seen installed on an infrared observatory of a metrological satellite launched by European space agency. The status of this technology is TRL 8.
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3.2.5 Flexible thermal straps
Thermal straps are used to transport heat from source to thermal sink with reduced stiffness. These are simple to implement and depends on system considerations. System considerations include cost operating temperature, thermal conductance, mass, stiffness, geometry and contamination. Thermal straps available in thin aluminium or copper foil layers or a copper braid. Figure 5 shows an I shaped thermal strap developed by thermal space (space, 2015).The status of technology is TRL 9 for metal straps, TRL 8 for composite straps.
Figure 5 I shaped thermal strap (space, 2015)
3.2.6 Thermal Louvers
Louvers are usually placed over external radiators and can also be used to control heat transfer between internal spacecraft surfaces or be placed on openings on the spacecraft walls. A louver in its fully open state can reject six times as much heat as it does in its fully closed state, with no power required to operate it. The most commonly used louver is the bimetallic, spring-actuated, rectangular blade louver also known as venetian-blind louver. Louver radiator assemblies consist of five main elements: baseplate, blades, actuators, sensing elements, and structural elements. Figure 6 shows thermal louvres of Rosetta spacecraft launched by European space agency. The status of the technology is TRL 8.
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Figure 6 Thermal louvres in Rosetta spacecraft (ESA, Space in images, 2015)
3.2.7 Radiators
Excess waste heat created on the spacecraft is rejected to space using radiators. Radiators come in several different forms, such as spacecraft structural panels, flat-plate radiators mounted to the side of the spacecraft, and panels deployed after the spacecraft is on orbit. Whatever the configuration, all radiators reject heat by infrared (IR) radiation from their surfaces. The radiating power depends on the surface's emittance and temperature. The radiator must reject both the spacecraft waste heat and any radiant-heat loads from the environment. Most radiators are therefore given surface finishes with high IR emittance to maximize heat rejection and low solar absorptance to limit heat from the sun. Most spacecraft radiators reject between 100 and 350 W of internally generated electronics waste heat per square meter. Radiators weight typically varies from almost nothing, if an existing structural panel is used as a radiator, to around 12 kg/m2 for a heavy deployable radiator and its support structure. Figure 7 shows ten radiators installed on international space station. The status of the technology is TRL 9.
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Figure 7 International Space Station photographed by an STS-134 crew member of the Space Shuttle Endeavour on May 29, 2011. (NASA, 2011)
3.2.8 Heat pipe capillary-based loops
A capillary-based, two-phase heat transport technology involves the use of a loop where there are separate liquid and vapor lines and the wick is located only at the evaporator. Such loops are called loop heat pipes (LHP) or capillary pumped loops (CPL). The advantages of this technology include transport of heat over long distances with low temperature drop, inherently self-regulating, zero to no maintenance, doesn’t need mechanical pump and can last indefinitely. Figure 8 shows the experimental setup (Avino, 2017) of a heat pipe sent on international space station to study the microgravity effects on heat pipe. The status of the technology is TRL 6. Detailed review on heat pipes is discussed in next chapter.
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Figure 8 Heat pipe experimental set-up used on ISS (Avino, 2017)
4.Conclusion
Most of the heat transfer devices rely on single-phase processes, involving heat conduction and diffusive processes, with or without convection. However, systems exploiting phase change, and thus latent heat, promise to transfer larger amounts of heat at small temperature differences, which is a highly desirable achievement in practical applications. Such systems are typically more compact, also reducing the environmental impact on the utilisation of resources. They already emerged in numerous applications, e.g. heat pipes in space and on ground, power plants, air conditioners, industrial boilers, two-phase cooling loops. Previous space experiments demonstrated the large-scale behaviour of two-phase systems in weightlessness. Hence, in the following chapters, a model that can be used to accurately depict the behaviour of the two-phase phenomenon has been studied for a single loop.
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Chapter 2
Two phase passive systems
Two phase flows are a special case of multiphase flows in which the flow consists of liquid and gas. Two phase flows can occur in different forms like, pure liquid phase transitioning to pure vapour phase due to external heating, dispersed flows, droplets, bubble flows. The transport devices which use the two flows to transfer heat are called as two-phase passive systems. Heat pipe is one such heat transfer device to efficiently transport large quantities of heat over a large distance without any external electrical or heat input.
1. Working principle
Heat pipes uses a process of circulation where in the liquid changes to vapor at evaporator and the vapor changes to liquid at the condenser. In this way the heat is transported from evaporator to condenser with high thermal conductivity. This circulation is achieved by pressure difference and capillary effect, hence there is no need of any external pump. The heat pipe operates in a two-phase flow regime by exploiting latent heat of vaporisation and transferring heat over a large distance with small amount of temperature difference. At the evaporator the heat is transferred to the working fluid using conduction and then the vaporisation starts. Due to this vaporisation, there is a rise in the local vapor pressure at the evaporator which helps the vapor to move towards the condenser, thereby transporting latent heat of vaporisation. At the condenser, the energy is extracted and the vapor is condensed at the surface and thus releasing the latent heat. closed circulation is made possible by the capillary action and/or bulk forces. Figure 9 shows various sections of a basic heat pipe.
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Figure 9 Representation of a heat pipe
2. Types of heat pipes
There are different types of heat pipes including standard, constant conductance heat pipe. The following are the types of heat pipes from the state of the art.
2.1. Flat heat pipes
2.2. Variable conductance heat pipes 2.3. Diode heat pipes
2.4. Pulsating heat pipes 2.5. Loop heat pipes 2.6. Micro heat pipes 2.7. Sorption heat pipes
2.1 Flat heat pipes
A flat heat pipe is like normal heat pipe and has a hollow cylinder, a working fluid and capillary circulation system. Also, posts are used to avoid collapsing of flat top and bottom in the event of pressure drop to less than atmospheric pressure. These vapor chamber or flat heat pipes are used when there is a need for high power and high heat fluxes are applied to small evaporator. Most flat heap pipes do not depend on gravity and the position
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of condenser and evaporator can be interchanged. These are generally used to cool electronic chip or laser diodes. These flat heat pipes are used when there is a lot of space for fins, presence of high ambient or low air flow, power densities are high and low height is present (Meyer, 2016). Figure 10 shows a flat heat pipe installed on a telecommunication equipment (Meyer, 2016).
Figure 10 A flat heat pipe on a telecommunication equipment (Meyer, 2016)
2.2 Variable conductance heat pipes
Standard heat pipes are constant heat conductance devices, where the operating temperature depends on the temperatures of source and sink i.e., the thermal resistances from the source to the heat pipe and thermal resistance from heat pipe to the sink. For space applications the electronics will be overcooled at low powers. Hence these heat pipes are used to maintain the operating temperature of the equipment with the change in power and sink conditions. Hence these heat pies have additionally a reservoir and non-condensable gas chamber. This heat pipe works by varying the active length of the condenser. The heat pipe vapor pressure and temperature increases with the increase in the power. Figure 11 shows a variable conductance heat pipe developed by CRS engineering
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Figure 11 Variable conductance Heat pipe (engineering, 2018)
2.3 Diode heat pipes
A Diode heat pipes are like standard heat pipes except the heat is transferred in only one direction. It can be used as a thermosyphon when placed in an appropriate direction to use gravity effects. Due to initial investment in the manufacturing of the diode heat pipes and complexity in fitting it into a system it is not as widely used as other heat pipes. This heat pipe is useful particularly if the requirement of heat transfer is only in one direction. (Hassam Nasarullah Chaudhrya, 2012). Figure 12 shows a schematic of a liquid tap diode in reverse mode (Reay D, 2006).
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2.4 Pulsating heat pipes
A pulsating heat pipe consists of circuit like channel evacuated and filled with working fluid. Heat is transported to the latent heat of vapor and the heat is transferred by liquid slugs (Hassam Nasarullah Chaudhrya, 2012). When the evaporator section of the heat pipe is heated the fluid evaporates and thus increasing the vapor pressure, which results in the formation of bubbles. This results in transferring the liquid towards the condenser section of the heat pipe. In condenser, the cooling results in the reduction of vapor pressure and hence condensation of bubbles. The increase and decrease of bubbles in respective sections of the heat pipe results in the oscillator or pulsating motion with in the capillary tube. Figure 13 shows a schematic of pulsating heat pipe.
Figure 13 Representation of a pulsating heat pipe (Reay D, 2006)
2.5 Loop heat pipes
Loop heat pipes are similar in functionality with respect to standard heat pipe. Loop heat pipes have an advantage in terms of its ability to transfer heat over a large space without any constraint on the path of the liquid or vapor lines. Also, it has a robust operation and
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greater heat flux potential (Reay D, 2006). Capillary force in the evaporator section is the reason for operating loop heat pipes and they doesn’t need any external pump as input. Figure 14 shows a schematic of loop heat pipe.
Figure 14 Representation of a loop heat pipe (Reay D, 2006)
2.6 Micro heat pipes
Micro heat pipes are used in applications where small to medium heat transfer rates are needed. The rates of cooling achieved from these heat pipes are small when compared to conventional heat pipes. The capability to control temperatures in variable heat load environments and its compact structure allows it to be utilized in various applications (Reay D, 2006). Figure 15 shows the representation of a heat pipe.
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Figure 15 Schematic of a micro heat pipe (Reay D, 2006)
2.7 Sorption heat pipes
Sorption heat pipe is similar to loop heat pipe which uses sorption phenomenon on the heat pipe. This heat pipe constitutes of a condenser, evaporator and working fluid. It is shown in the literature that the integrity of a sorption cooler with a loop heat pipe provides higher heat fluxes and evaporator thermal resistances. Figure 16 shows a representation of a sorption heat pipe.
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Chapter 3
Closed Loop Thermosyphon Numerical Simulation
1. Thermosyphon
Due to increasing spacecraft power level, higher amount of heat is generated. Also, power consumption is being concentrated, miniaturizing and integrating electronic components. Hence higher heat flux needs to be removed, transported and dissipated by thermal control systems. Hence there is need for higher performing heat transfer devices. European space agency, in the framework of Horizon 2020 research funding opportunities, has stated the development of advancement thermal control systems as urgent in particular two-phase systems. (European space agency, 2015).
The main requirements to best meet this challenge is encountered by three main requirements to be faced in near future of thermal management systems. They are dissipating higher heat fluxes, containing power requirements, balancing weight and costs. Various technologies available are discussed in the previous chapter. Among these technologies two-phase systems have demonstrated to be very effective in space applications. Also, they satisfy the three requirements listed above. Hence, for these reasons, two-phase systems are best candidates to meet near future thermal control system challenges.
Most of the heat pipes uses wick structure to facilitate the circulation without the dependence of orientation. However, the maximum elevation of evaporator over condenser is relatively small and in the order of 25 cm long for a typical water heat pipe. Thermosyphon uses gravity for the condensate from the downcomer to reach the evaporator. Evaporator is placed below the condenser and they can be separated by practically any length.
In the thermosyphon pipe high heat fluxes gives high vapor velocity from evaporator which interact with counter current condensate and cause counter-current liquid flow limitation to the evaporator. Due to this limitation the thermal capacity of the thermosyphon loop is higher than a thermosyphon pipe of the same diameter (Rahmatollah Khodabandeh, 2008). Hence loop thermosyphon is of paramount interest in the current research.
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Two phase loop thermosyphon is a passive thermal system used for transportation of energy. It has coexisting liquid and vapor phases of the working fluid. It consists of a condenser, evaporator, downcomer and riser. Evaporator is the locus of the heat supplied to the working fluid. Condenser is the point where the extraction of heat to the sink takes place. The riser is the vapor line which connects the outlet of the evaporator and the inlet of the condenser. The downcomer is the liquid line which connects the outlet of the condenser and inlet of the evaporator. Two phase loop thermosyphon can manage high power density and has low thermal resistance. Hence it is ideal for many applications including space applications. In the present work, the modelling of such a system in steady state conditions.
Figure 17 Schematic of working of a heat pipe (Mameli M. , 2012)
1.1 Two-phase systems: Literature survey
Two-Phase flows:
Two-Phase flows are special case of multiphase flows where two phases exist in the flow.
Gas-Liquid flows: This is the most important two-phase flow, which is seen in most of the industrial applications including condensers, boilers, evaporators, combustion systems, air conditioning plants.
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Gas-Solid flows: These are found in sand transport, aluminium transport, pneumatic conveying etc.
Liquid-solid flows: This kind of flows are commonly found in flowing mudslides, quicksand etc.
Two phase systems can be classified as wicked and wickless. Wicked two-phase systems include Heat pipes, capillary pumped loop and loop heat pipes. Wickless two-phase systems include thermosyphon, loop thermosyphons and pulsating heat pipes. (Landi, 2017)
Wicked systems are popular since decades and are already used in the space industry. Wickless systems are gaining popularity and active research is being conducted on these systems since recent years due to their capability in transporting high rates of heat over long distances, less to no maintenance, low cost, reliability and relatively simpler design and modelling process.
The questions about the flow regime with respect to diameter, gravity, mass flow rate and the transition criteria among them remained unanswered due to the delay in the invention of wickless systems. However, these systems predictably solve the future need of spacecraft thermal control.
1.2 Flow Regimes
The flow pattern in the wickless systems, is responsible for the way these systems operate. The main forces acting inside the device depends on the distribution of liquid and vapor either in a stratified or confined way. In a stratified flow, buoyancy is the dominating force and the heat supplied results in pool boiling condition. In the confined flow, surface tension is the dominant force. The transition between stratified and confined flow is not sharp. There is a transitional region represented by loop thermosyphon.
The flow confinement problem has been an ongoing research for some time. A flow is said to be confined, if the bubble grows in length rather than in diameter then the flow is said to be confined and is also known as elongated bubble regime. (Baldassari, 2013). The work was carried out as a geometric one. The flow regime was distinguished between microchannel (surface tension dominated), macro channel (buoyancy dominated) and meso channel (transition between surface tension and buoyancy). However, the
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geometrical characterisation is not sufficient to understand the nature of the problem. The problem concerns more about the relative importance of acting forces rather than the geometric correlation because diameter is a parameter which changes the ratio between them. An alternative classification was provided by Haricharan (Harirchian, 2010) was that a flow inside a tube can be confined or unconfined. This classification was later supported in a review by Marengo (Baldassari, 2013).
The presence of transition between confined and unconfined flow region is shown in the literature as explained above (intuitively it is comparable to the presence of transition region between laminar and turbulent flow). Approaching a single critical value simplifies the problem
Different flow patterns can eb observed in a confined flow regime. They are:
1. Slug flow: The gas bubbles almost fill the diameter of the pipe and separated from the wall surface by a thin liquid film. The liquid which is present in between the bubbles is in the form of plugs and even smaller bubbles are present.
2. Churn flow: This flow evolves from the slug flow when the bubbles start to breakdown and the flow is strongly stochastic
3. Annular flow: Along the channel, the gaseous phase pattern is distributed as a single bubble with a liquid film attached to the walls. Then the flow is called as annular flow. Figure 18 shows the flow pattern map as observed by Triplett (Triplett, 1999) for a 1.4mm diameter circular test section. The transition lines shown are indicative. Different flows discussed here are shown along with other patterns depending on the superficial velocity of liquid and gaseous phase.
22
Figure 18 Flow pattern observed by Triplett (Triplett, 1999)
1.3 Defining parameters for a two-phase flow
Parameters relating to flow confinement are shown in this section.
From the review of Baldassari (Baldassari, 2013) non-dimensional numbers that are used to understand flow confinement are shown in Table 2.
Number Formula Description
Reynolds number 𝑅𝑒 = 𝜌𝑢𝑑/𝜇 Ratio of inertia and viscous force
Prandtl number 𝑃𝑟 = 𝜈/𝛼 Ratio of kinematic viscosity and
thermal diffusivity
Boiling number 𝐵𝑙 = 𝑞̇/𝐺ℎ𝑙𝑣 Ratio of evaporation mass flux to
total mass flux flowing in a channel
Capillary Number 𝐶𝑎 = 𝜇1𝑢𝑙/𝜎 Ratio of viscous and surface
23 Confinement number 𝐶0 = [ 𝜎 𝑔(𝜌𝑙− 𝜌𝑣)𝑑2 ]
1/2 Ratio of surface tension forces and gravity
Eotvos number 𝐸0 = 𝑔(𝜌𝑙− 𝜌𝑣)𝐿2/𝜎 Ratio of gravity and surface
tension forces
Bond number 𝐵0 = 𝑔(𝜌𝑙− 𝜌𝑣)𝑑2/𝜎 Ratio of gravity and surface
tension forces
Garimella number 𝐺𝑎 = 𝐵𝑜0.5. 𝑅𝑒𝑙 Weighted ratio of gravity dot
inertia forces to the surface tension dot viscous forces
Weber number 𝑊𝑒 = 𝜌𝑙𝑢𝑙2𝑑/𝜎 Ratio of inertia to surface
tension forces
Table 2 Useful non-dimensional numbers
The difference between Eotvos and Bond number is that in Eotvos number L is the characteristic length and d in the bond number is the hydraulic diameter.
Tube Diameter:
The internal diameter is an important geometric parameter and acts as a tuning factor among the other forces. The diameter affects the distribution of liquid and vapor in a stratified or confined flow. Also, transition or critical diameter identification is important to understand the heat transfer process occurring.
This problem is approached by considering the forces acting on the motion of bubble rising in a channel filled with liquid and defining the ratios of the forces as non-dimensional numbers.
Some transition criteria has been proposed in literature by Kew and cornwell (Kew, 1997) based on confinement number, Cheng and Wu (Cheng, 2006) based on Bond number and Ullman and Brauner (Ullmann, 2007) based on Eotvos number, Haricharan (Harirchian, 2010) based on Garimella number, Gu et al (Gu, 2004) based on Weber number. The first three are static criteria as they neglect inertia and viscosity and the remaining two are dynamic.
Confinement number has been described as a discriminating parameter by Kew performing flow boiling experiments in a single channel for a heat exchanger and set 𝐶𝑜 = 0.5 for the
24
transition to occur. Therefore, if 𝐶𝑜 is greater than 0.5 for confined flows and less in other cases.
The transition occurrence was redefined by Ong (Ong, 2011) by measuring film thickness and modified Kew’s coefficient. According to Ong if 𝐶𝑜 < 0.34 the flow is not confined and if 𝐶𝑜 > 1 it was confined and between these values the transition takes place from unconfined to confined.
Li and Wang (Li, 2003) recommended using capillary length which is given by
𝐿𝑐 = √ 𝜎
𝑔(𝜌𝑙− 𝜌𝑣)
And if 𝑑𝑐𝑟 < 2𝐿𝑐 the flow is confined and if 𝑑𝑐𝑟 < 19𝐿𝑐 the flow is not confined while if the critical diameter is between these two values transition occurs from confined to unconfined.
Cheng defined the flow confinement basing on the Bond number. If 𝐵𝑜 ≤ 0.05 the flow is confined and if 𝐵𝑜 > 3 it is not confined. In between the values surface tension is still dominant but the effects of gravity increases. When 𝐵𝑜 ≤ 3 which means (𝐶𝑜 ≥ 0.58) gives more restrictive inner diameter than the one by Kew.
In the work presented by Ullmann, they defined a criterion based on Eotvos number. A criterion for confined flow has been proposed by them basing on the flow patter deviation for experiments in pipes which is 𝐸𝑜 ≤ 1.6 which is equivalent to 𝐶𝑜 ≥ 0.79.
When heat is given to the evaporator the motion in the flow is started and inertia and viscosity cannot be neglected. Hence, a dynamic criterion is needed. Harirchian proposed a non-dimensional number known as convective confinement number or Gariimella number which takes in to account both the inertia and viscosity.
𝐺𝑎 = 𝐵𝑜0.5. 𝑅𝑒 𝑙 = ( 𝑔(𝜌𝑙− 𝜌𝑣)𝑑2 𝜎 ) 0.5 .𝜌𝑙𝑢𝑙𝑑 𝜇𝑙 ≤ 160
In terms of Eotvos number it is 𝐸𝑜 = (160𝑅𝑒 𝑙)
25
Considering the criteria mentioned above, the critical diameter for each case compared with equivalent Eotvos number is shown in Table 3
Kew and Cornwell 𝐸𝑜𝑐𝑟 = 4
𝑑𝑐𝑟 = 2. √ 𝜎 𝑔(𝜌𝑙− 𝜌𝑣) Li and Wang 𝐸𝑜𝑐𝑟 = 3.06 𝑑𝑐𝑟 = 1.75. √ 𝜎 𝑔(𝜌𝑙− 𝜌𝑣) Cheng and Wu 𝐸𝑜𝑐𝑟 = 3 𝑑𝑐𝑟 = √3√ 𝜎 𝑔(𝜌𝑙− 𝜌𝑣)
Ullmann and Brauner 𝐸𝑜𝑐𝑟 = 1.6
𝑑𝑐𝑟 = √1.6√ 𝜎
𝑔(𝜌𝑙− 𝜌𝑣)
Harirchian and Garimella
𝐸𝑜𝑐𝑟 = (160 𝑅𝑒𝑙) 2 𝑑𝑐𝑟 =160 𝑅𝑒𝑙√ 𝜎 𝑔(𝜌𝑙− 𝜌𝑣) Gu et al. 𝑑𝑐𝑟 = 4. ( 𝜎 𝜌𝑙𝑢𝑙2)
Table 3 Critical diameter as proposed by different authors compared with equivalent Eotvos number
Since surface tension and fluid density are both functions of temperature. Difference in the selected criterion is given in Figure 19 at the same temperature and as a function of reduced pressure.
26
Figure 19 Comparison of selected criteria as a function of reduced pressure (Baldassari, 2013)
1.4 Definition of basic quantities for a two-phase flow
Volume flux: Volume flux (𝑈𝑖), also known as superficial velocity, is defined as
𝑈𝑖 = 𝑉̇𝑖
𝑆
Where 𝑉̇𝑖 is the volume flow rate of the 𝑖th phase ( 𝑚3
𝑠 ) and 𝑆 is the cross-sectional area (𝑚2). The total superficial velocity is given by
𝑈 = ∑ 𝑈𝑖
𝑖=𝑙,𝑔
Where 𝑙, 𝑔 are liquid and gas phases
Flow quality 𝑥𝑖 of 𝑖th phase is defined as
𝑥𝑖 =
𝑚̇𝑖 ∑𝑖=𝑙,𝑔𝑚̇𝑖
27
Where 𝑚̇𝑖 is the mass flux at 𝑖th phase, which is given by 𝑀̇𝑖
𝑆 where 𝑀̇𝑖 is the mass flow rate of the 𝑖th phase through the channel.
Two-phase density (𝜌𝑡𝑝) can be defined as the mass of the two-phase mixture per unit channel volume and is given by
𝜌𝑡𝑝= ∑ 𝜖𝑖𝜌𝑖 𝑖=𝑙,𝑔
Where 𝜖𝑖 is defined as the mean phase constant, which is the time-averaged of volume fraction (or area fraction) in cross-section of a channel. The mean phase constant for a gas phase is called as void fraction.
Design parameters in a two-phase flow:
The design parameters for a two-phase flow include the following:
Pressure drop: The pressure losses occur due to friction, acceleration and gravity effects. When a fixed flow is required, pressure drop is used to determine the total input power required. On the other hand, if the available pressure drop is fixed, the mass flow rate occurring in the circulation can be predicted.
Heat transfer coefficient: Heat transfer coefficient is an important parameter to determine the size of the heat exchanger in a two-phase system.
Mass transfer coefficient: This design parameter is used to estimate the heat and mass transfer in in condensation of vapor mixtures.
Mean phase content: It is the design parameter which represents the fraction by volume or by cross section area of a particular phase. It is called void fraction in liquid- gas phase flow. It governs the gravitational pressure gradient.
Flux limitations: Limitations in heat and mass fluxes play a vital role in two-phase systems. One of such flux limiter is critical flow, which occurs in low velocities in a two-phase flow. Heat flux limitation is necessary in boiling. If the critical heat flux is exceeded, the system performance is affected or can cause damage to the equipment due to increase in the wall temperature.
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2. Modelling
2.1 Two-phase conservation equations
2.1.1 Local Instantaneous equations
Local instantaneous equations are basis for the understanding of two-phase flow modelling procedures. These equations are used directly in the study of bubble dynamics, film flows etc. For the study of flow in pipes, the averaged form of the instantaneous equations is used. The formulation of local instantaneous equations involves deriving of appropriate conservative equations and closure of the set of equations depending on the applications. The conservative equations are similar to that of a single phase, however the behaviour of the interface is accounted for.
2.2 Treatment of Phase interface
A phase interface can be treated as a three-dimensional region which separates both phases and the constitutive equations may differ from the equations in the region where a single phase is present.
Consider an arbitrary two-phase control volume as shown in the figure.
Figure 20 Control volume of a two-phase flow
The surfaces considered for the derivation of single phase conservation equations are flow or convective surfaces and wall or fixed surfaces. The convective surface (at 𝑧 and 𝑧 + 𝑑𝑧)
29
account for the transport of the mass, momentum and energy through the normal convective flow of the fluid. The fixed surfaces account for the heat and momentum (drag forces). In two-phase flow, the interface between the liquid and gas phase provides an additional surface for convective and sources of mass, momentum and energy transfer. has to be considered. If we assume that the phase change occurs at heated surfaces and at the interface between the liquid and vapor (𝛿𝑖′), where 𝛿′ represents phase change rate per unit length. No assumptions for phases are made here, which means that the liquid can be subcooled and the vapor can be superheated.
2.3 Mass conservation
The two-phase conservation equations are obtained by performing mass, momentum and energy balances on the individual phases.
𝑑𝑀𝑙 𝑑𝑡 = ∑ 𝑚̇𝑙𝑖𝑛− ∑ 𝑚̇𝑙𝑒𝑥𝑖𝑡 𝑒𝑥𝑖𝑡 𝑖𝑛 Where, 𝑀𝑙 = 𝜌𝑙𝑉𝑙 = 𝜌𝑙𝛼𝑙𝑉 = 𝜌𝑙𝛼𝑙𝐴𝑥Δ𝑧 𝑚̇𝑙 = (𝜌𝑙𝑣𝑙𝐴𝑙) = (𝜌𝑙𝑣𝑙𝛼𝑙𝐴𝑥) ∴ 𝐴𝑥Δ𝑧ⅆ(𝛼𝑙𝜌𝑙) ⅆ𝑡 = (𝜌𝑙𝑣𝑙𝛼𝑙𝐴𝑥)|𝑧− (𝜌𝑙𝑣𝑙𝛼𝑙𝐴𝑥)|𝑧+Δ𝑧− 𝛿̇ ′Δ𝑧
We can write this equation in differential form for liquid phase as
𝐴𝑥𝜕(𝛼𝑙𝜌𝑙)
𝜕𝑡 +
𝜕
𝜕𝑡(𝛼𝑙𝜌𝑙𝑣𝑙𝐴𝑥) = 𝛿̇′
We can write this equation in differential form for vapor phase as
𝐴𝑥𝜕(𝛼𝑔𝜌𝑔)
𝜕𝑡 +
𝜕
𝜕𝑡(𝛼𝑔𝜌𝑔𝑣𝑔𝐴𝑥) = 𝛿̇′
2.4 Energy conservation equation
30 𝑄̇ + ∑ 𝑚̇𝑖𝑛(𝑒𝑖𝑛+ 𝑃𝑖𝑛𝜈𝑖𝑛) = 𝑑𝑀𝑒 𝑑𝑡 + ∑ 𝑚̇𝑒𝑥𝑖𝑡(𝑒𝑒𝑥𝑖𝑡+ 𝑃𝑒𝑥𝑖𝑡νexit) 𝑒𝑥𝑖𝑡 𝑖𝑛 Where 𝑒 ≡ 𝑢 +𝑣2 2 + 𝑔𝐻
And after neglecting the external work terms we have,
The liquid phase energy equation is
𝐴𝑥 𝜕(𝛼𝑙𝜌𝑙𝑒𝑙) 𝜕𝑡 + 𝜕 𝜕𝑧(𝛼𝑙𝜌𝑙𝑒𝑙𝑣𝑙𝐴𝑥) + 𝜕 𝜕𝑧(𝛼𝑙𝑃𝑙𝑣𝑙𝐴𝑥) = −𝛿̇ ′(𝑒 𝑙+ 𝑃𝑙𝑣𝑙)𝑖+ 𝑞̇𝑤,𝑙′ + 𝑞̇𝑖,𝑙′
The vapor phase energy equation is
𝐴𝑥𝜕(𝛼𝑔𝜌𝑔𝑒𝑔) 𝜕𝑡 + 𝜕 𝜕𝑧(𝛼𝑔𝜌𝑔𝑒𝑔𝑣𝑔𝐴𝑥) + 𝜕 𝜕𝑧(𝛼𝑔𝑃𝑔𝑣𝑔𝐴𝑥) = −𝛿̇′(𝑒𝑔+ 𝑃𝑔𝑣𝑔) 𝑖 + 𝑞̇𝑤,𝑔 ′ + 𝑞̇ 𝑖,𝑔′
2.5 Momentum conservation equation
Considering the arbitrary two-phase control volume as shown in the figure below, we have
One-Dimensional momentum equation for liquid phase is given by,
𝐴𝑥 𝑔𝑐 𝜕𝛼𝑙𝜌𝑙𝑣𝑙 𝜕𝑡 + 1 𝑔𝑐 𝜕 𝜕𝑧(𝛼𝑙𝜌𝑙𝑣𝑙𝜈𝑙𝐴𝑥) = − 1 𝑔𝑐𝛿̇ ′(𝑣 𝑙)𝑖− 𝛼𝑙𝐴𝑥 𝜕𝑃𝑙 𝜕𝑧 − (𝜏𝑤𝑃𝑤)𝑙+ (𝜏𝑖𝑃𝑖) − 𝛼𝑙𝜌𝑙𝐴𝑥 𝑔 𝑔𝑐𝑠𝑖𝑛𝜃 + Δ𝑃𝑝′ 𝑙𝛼𝑙𝐴𝑥
31
Figure 21 Control volume for momentum equation
One-Dimensional momentum equation for vapor phase is given by,
𝐴𝑥 𝑔𝑐 𝜕𝛼𝑔𝜌𝑔𝑣𝑔 𝜕𝑡 + 1 𝑔𝑐 𝜕 𝜕𝑧(𝛼𝑔𝜌𝑔𝑣𝑔𝜈𝑔𝐴𝑥) = − 1 𝑔𝑐𝛿̇ ′(𝑣 𝑔)𝑖 − 𝛼𝑔𝐴𝑥 𝜕𝑃𝑔 𝜕𝑧 − (𝜏𝑤𝑃𝑤)𝑔+ (𝜏𝑖𝑃𝑖) − 𝛼𝑔𝜌𝑔𝐴𝑥 𝑔 𝑔𝑐𝑠𝑖𝑛𝜃 + Δ𝑃𝑝′ 𝑔𝛼𝑔𝐴𝑥
2.5 Closure of the system of equations
There are total of ten variables for the above system of equations and we have six
equations. Hence considering the equation of state,
𝜌𝑘 = 𝜌𝑘(𝑢𝑘, 𝑃𝑘) where 𝑘 = 𝑙, 𝑔
In addition, we have volume constraint which is
∑ 𝛼𝑘
𝑘=𝑙,𝑔
32
We have nine equations, hence for closure we consider the equation relating the
pressures.
𝑃𝑙= 𝑃𝑙(𝑃𝑔)
Which can be written as 𝑃𝑙 = 𝑃𝑔 = 𝑃
3. Modelling Approaches
Modelling methods for two-phase flow includes the following
1. Homogeneous model: In a homogenous model, the two phases are considered to be flowing at the same velocity similar to that of a single-phase flow.
2. Separated flow model: In a separated flow model, two fluids are considered to be moving at different velocities and overall conservation equations are modified accordingly.
3. Multi-fluid model: In a Multi-fluid model, conservation equations are written for each phase by considering the interaction between the phases.
4. Drift flux model: In a drift flux model, the flow is described in terms of a distribution parameter and difference of averaged local velocity between the phases
5. Computational fluid dynamics model: CFD models usually involve tow or three dimensions to describe the full flow field.
3.1 Lumped capacitance model
A lumped capacitance model reduces the thermal system in to number of discrete lumps and assumes that the temperature difference inside each lump is negligible. This approximation is used to simplify the complex differential heat equations. It was developed as a mathematical modelling tool for electrical capacitance and later improved to include thermal analogy of electrical resistance. The lumped capacitance model is developed initially as an approximation in transient heat conduction and can be applied to systems where heat conduction with in an object is faster than the heat transfer across the boundary. Later research showed that this method can be applied to two-phase systems.
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Lumped parameter model simplifies the behaviour of complex spatially distributed systems by considering the dependant variables concentrated at specific points in space. Hence the dependant variables are function of time alone. It implies that a set of ordinary differential equations are solved. (Landi, 2017)
Holley and Faghri (Holley, 2005) developed a one-dimensional model for analysis of closed loop pulsating heat pipes with capillary wick. They calculation the variation of diameter with working fluid as water. The main assumption made in their work is that the working fluid is always in the form of liquid slugs and vapor plugs alternatively. This model does not depict the real pulsating heat pipe as it considers the presence of wick structure, uses water as the working fluid and neglected the local pressure drop due to the presence of turns. Also, it was only validated qualitatively.
Beginning from the model by Holley and Faghri, Mameli et al. (Mameli M. M., 2012), Manzoni, Mameli, Falco, Araneo, Filippeschi and Marengo (Manzoni M. M., 2014); Manzoni, Mameli, Flaco, Araneo, Filippeschi and Marengo (Manzoni M. M., 2016) have done an extensive research to develop a realistic Pulsating heat pipe lumped model in order to understand the device performance and the effects of gravity. They added different types of working fluids, local pressure drops due to meanderings, sub-cooled and over-heated conditions for liquid and vapor as a compressible van der Waals gas and homogenous as well as heterogeneous phase change model. The model holds some strong assumptions such as presence of slug and plug flow, which is valid only for a narrow ranges of flow), not considering the liquid film and the effects of contact angle.
A similar work was done by Senjaya and Inoue (Senjaya, 2013) by considering the presence of liquid film and bubble generation. But the hypothesis limits its usage to a special case and the model has not been improved further yet.
A model by Gursel et al (Gürsel, 2015) is a novel idea. They proposed that a pulsating heat pipe can also be seen as a quasi-one-dimensional mass -spring-damper model where the liquid slugs are represented by masses, the vapor plugs by non-linear springs and the friction forces and the capillary forces by non-linear dampers. The comparison with both numerical and experimental results shows good agreement in the slug-plug regime.
34
This current model used is based on the scheme as shown in the Figure 22. The evaporator is placed on the vertical section of the thermosyphon. Condenser is placed on the horizontal section of
Figure 22 Schematic representation of a loop thermosyphon used for lumped capacitance analysis
the loop. The height of the liquid in the downcomer is fixed. The condenser and evaporator sections are considered non-adiabatic sections. All other sections of the loop are considered to be adiabatic (Cifelli, 2012).
The analysis is carried out using the conservation of momentum equation. The steady state form of the equation is considered and is simplified to compute pressure gradient. The pressure gradient can be divided in to three components. The pressure drop due to gravity, friction and acceleration and are denoted by subscripts 𝑔, 𝑓 𝑎𝑛𝑑 𝑎 as shown in the equation below.
35 𝑑𝑝 𝑑𝑧= ( 𝑑𝑝 𝑑𝑧)𝑓+ ( 𝑑𝑝 𝑑𝑧)𝑎+ ( 𝑑𝑝 𝑑𝑧)𝑔
The gravitational component depends on the liquid-vapor mixture density 𝜌𝑚 , the gravity vector and the inclination angle to the horizontal 𝜃.
Hence it can be written as − (ⅆ𝑝ⅆ𝑧)
𝑔 = 𝜌𝑚. 𝑔. 𝑠𝑖𝑛𝜃
According to the separated flow model by Davis (Davis, 1987), the mixture density can be written as
𝜌𝑚 = 𝛼𝜌𝑣 + (1 − 𝛼)𝜌𝑙
Where 𝛼 is the void fraction and 𝜌𝑙, 𝜌𝑣 are the liquid density and vapor density respectively.
Considering a infinitesimally small differential element and using the momentum balance equation we have − (𝑑𝑝 𝑑𝑧)𝑎 = ( 𝑚̇ 𝐴) 2 𝑑 𝑑𝑧[ 𝑥2 𝜌𝑣𝛼+ (1 − 𝑥2) 𝜌𝑙(1 − 𝛼)]
This term can be neglected when compared to gravitational pressure drop and frictional pressure drop.
The frictional component is expressed in terms of single-phase pressure gradient when the total flow is considered as liquid and is given by
− (𝑑𝑝 𝑑𝑧)𝑓 = − (𝑑𝑝 𝑑𝑧)𝑙0 𝜙𝑙20 = [ 𝑓𝑙 2. 𝐷. 𝜌𝑙. ( 𝑚̇ 𝐴) 2 ] 𝜙𝑙20 Where 𝜙𝑙0
2 is known as two-phase frictional multiplier.
The mass flow rate is calculated as the flow rate at which there is no net change in the total system pressure,Δ𝑃𝑡𝑜𝑡.
Δ𝑃𝑡𝑜𝑡 is nothing but the total pressure drop in the total control volume which is given by
Δ𝑃𝑡𝑜𝑡 = ∮ (𝑑𝑝
𝑑𝑧) 𝑑𝑧 = 0
The model is based on the correlations used for void fraction 𝛼 and the two-phase multiplier𝜙𝑙20.
36
Void fraction is calculated using the relation given by Winkler (J. Winkler, 2012). Here the assumption made is that there exists a no slip condition between liquid and vapor phase. The slip ratio is defined as the ratio of vapor velocity and liquid velocity. The slip ratio,𝑆, can be written from two-phase flow relations as
𝑆 = 𝑤𝑣 𝑤𝑙 = 𝑗𝑣 𝑗𝑙 .1 − 𝛼 𝛼
Where the ratio 𝑗𝑣/𝑙 is the superficial velocity or the vapor/liquid flux density.
𝑗𝑙 =𝑉̇𝑙 𝐴 = 𝑚̇ 𝐴 . 1 − 𝑥 𝜌𝑙 𝑗𝑣 =𝑉̇𝑣 𝐴 = 𝑚̇ 𝐴. 1 − 𝑥 𝜌𝑣
Re-arranging the equations we get the value of void fraction as
𝛼 = [1 + 𝑆 (1 − 𝑥 𝑥 ) .
𝜌𝑣 𝜌𝑙]
−1
The relation for slip ratio given by Zivi (Zivi, 1964) is
𝑆 = (𝜌𝑙 𝜌𝑣 )
1 3
The two-phase multiplier that is derived from the homogeneous model (Thome, 1994) is given by 𝜙𝑙20 = [1 + 𝑥. (𝜌𝑙− 𝜌𝑣 𝜌𝑣 )] [1 + 𝑥. ( 𝜇𝑙− 𝜇𝑣 𝜇𝑣 )] −14
According to friedel’s empirical correlation (Friedel, 1979) the two-phase multiplier can be written as 𝜙𝑙20 = (1 − 𝑥)2+ 𝑥2(𝜌𝑙 𝜌𝑣. 𝑓𝑣 𝑓𝑙) + (3.24. 𝑥0.78. (1 − 𝑥)0.224. (𝜌𝑙 𝜌𝑣) 0.91 . (𝜇𝜇𝑣 𝑙) 0.19 . (1 −𝜇𝜇𝑣 𝑙) 0.7 ) 𝐹𝑟0.045𝑊𝑒0.035
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The Froude number and the Weber number are calculated using the average density of the liquid-vapor mixture, which is given by
𝜌 = (𝑥 𝜌𝑣 +1 − 𝑥 𝜌𝑙 ) −1
This formulation takes in to account the effects of inertia, gravity and capillary forces in Froude and Weber numbers. Figure 23 shows the correlation between two-phase frictional multiplier and steam quality for water for different mass flow rates. It can be observed that Friedel’s relation gives higher values of pressure drop than the relationship derived from homogenous model. For the same conditions considered, the mass flow rate is lower for the relation developed by using homogenous model.
Hence the friction factor is a function of Reynolds number and is given by
𝑓𝑙(𝑣)= { 64 𝑅𝑒ⅆ,𝑙(𝑣) 𝑖𝑓 𝑅𝑒ⅆ,𝑙(𝑣)< 1187 0.316 𝑅𝑒ⅆ,𝑙(𝑣)0.25 𝑖𝑓 𝑅𝑒ⅆ,𝑙(𝑣)≥ 1187
In the model the heat exchange with environment has been neglected. Hence the vapor quality is given by
𝑥 =𝑄 − 𝑚̇. 𝑐𝑝. Δ𝑇𝑠𝑢𝑏 𝑚̇. ℎ𝑙𝑣
The liquid height in the downcomer ℎ has been considered to be a known quantity and is the measure of initial fill of the loop.
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Figure 23 variation of two-phase frictional multiplier with steam quality for water for different mass flow rates
3.2 Thermodynamic equilibrium Model
Vincent (Charles C.J.Vincent, 1992) presents a numerical model for two-phase co-current thermosyphon and studied the transient behaviour at the near steady-state conditions using control volume approach. They founded that the control volume approach limits the usage of number of variables and still depict the represents the two-phase flow in a closed loop thermosyphon. Some of their results are verified in experiment (Leidenfrost, 1987) and concluded that the predictability of the model can be improved by accurate representation of two-phase flow at evaporator and condenser.
In this study, a two-phase flow in thermodynamic equilibrium is considered. In each part of the loop same model is utilised in the computation. It can describe thermodynamic equilibrium of saturation or mixture as well as the non-equilibrium which is pure liquid or pure vapor phases. Also, phase transitions which are liquid-mixture-vapor transitions can be estimated with this current model.
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3.2.1 Design
Figure 24 Schematic representation of two phase loop thermosyphon
In Figure 24 a two-phase loop thermosyphon is depicted. It has an evaporator, a liquid line, condenser, a liquid line. The operation of the two-phase loop thermosyphon lies mainly on phase change of the working fluid and gravity, which is the driving force of the fluid flow. At the evaporator B-C, the working fluid is heated by the heat at the evaporator. Then there is a phase change and working fluid changes in to vapor and moves in the vapor line C-E towards the condenser E-F. The condenser is at lower temperature when compared to evaporator. Hence the vapor cools and with the help of the gravity flows in the liquid line F-A and then moves towards the evaporator and it is heated again. Thus, the closed loop circulation is possible. If x is coordinate along the loop, g is the gravitational constant, is the density of the mixture and depends on the direction of gravity, then
40 𝜖 = { −1 𝑖𝑓 𝑥 ∈ [𝐵, 𝐷] 0 𝑖𝑓 𝑥 ∈ [𝐴, 𝐵]⋃[𝐷, 𝐸] 1 𝑖𝑓 𝑥 ∈ [𝐸, 𝐴] ( 1 )
The integral
g dx
.
is not zero because the density of working fluid in the descending section E-A is greater than in the ascending section B-D in average. This is responsible for the circulation in the loop of the two-phase loop thermosyphon shown in the schematic representation.3.2.2 Dimensions
The dimension of the loop, in rectilinear coordinate system, are as follows.
I. The length of the section A-B, which is horizontal liquid line, is 0.25 metres (0 to 0.25)
II. The length of the section B-C, which is the evaporator section, is 0.25 metres (0.25 to 0.50)
III. The length of the section C-D, which is vertical vapor line, is 0.25 metres (0.50 to 0.75)
IV. The length of the section D-E, which is horizontal vapor line is 0.25 metres (0.75 to 1.00)
V. The length of the section E-F, which is the condenser section, is 0.25m (1.00 to 1.25) VI. The length of the section F-A, which is the horizontal liquid line, is 0.25 metres (1.25
to 1.50)
VII. The diameter of the pipe,𝐷ℎ, is 7 millimetres. VIII. The exchange surface area, 𝐴𝑒𝑥𝑡 , is 1.32 × 10−2 𝑚2
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Chapter 4
Methodology
1. Lumped capacitance model
Assumptions:
1. The liquid height in the downcomer is a know value and is a measure of initial fill of the loop.
2.The fluid properties have been calculated using a fixed saturation temperature.
3. Local pressure drop due to measuring instruments have been neglected.
4. The vapor quality in the evaporator and condenser is a linear function of the axial position.
1.1 Flow solver
Predictor corrector method:
Predictor-corrector method is a powerful tool to integrate ordinary differential equations to find an unknown function that satisfies the given differential equation. This usually follows two steps. In the first step, which is predictor, the process starts from a function values and derivative values at a previous set of point to extrapolate the function value at the next new point. In second step, which is corrector step, refinement of the initial approximation is done by using the predicted value of the function. When finding the numerical solution of ordinary differential equations, predictor-corrector method uses an explicit method for the predictor step and an implicit method for the corrector step. There are different variants of predictor-corrector method, depending on the number of times the corrector method is applied. They are Predict- Evaluate-Correct-Evaluate mode and Predict-Correct-Evaluate mode. The main difference in both the methods is after correcting the predicted values an additional evaluate step can be performed to obtain the convergence faster. Even though this method is more efficient, the later is used to save the computing power and time.
