Dr Jagannath K Professor, Dept of Mechanical and Mfg Engg, MIT Manipal, Manipal University, India

Transcription

1 Dr Jagannath K Professor, Dept of Mechanical and Mfg Engg, MIT Manipal, Manipal University, India 1

2 Nucleate Boiling Nucleate Boiling is an efficient mode of heat transfer. It has useful application in many areas such as refrigeration, power generation, chemical processing and nuclear reactors. Avoiding the Critical Heat Flux is an engineering problem in heat transfer applications, such as nuclear reactors, where fuel plates must not be allowed to overheat. 2

3 Boiling phenomena Boiling occurs at solid-liquid interface. Characterized by formation of bubbles which grow and subsequently detach from the surface. POOL BOILING Liquid is quiescent. It s motion is induced by free convection and mixing induced by bubble growth and detachment. CONVECTIVE BOILING Fluid motion is induced by external means. 3

4 Boiling Facts When the liquid is at a lower temperature than the bubble, heat will be transferred from the bubble into the liquid, causing some of the vapor inside the bubble to condense and the bubble to collapse eventually. When the liquid is at a higher temperature than the bubble, heat will be transferred from the liquid to the bubble, causing the bubble to grow and rise to the top under the influence of buoyancy. 4

5 Boiling Facts The boiling processes in practice do not occur under equilibrium conditions. The temperature and pressure of the vapor in a bubble are usually different than those of the liquid. The pressure difference between the liquid and the vapor is balanced by the surface tension at the interface. The temperature difference between the vapor in a bubble and the surrounding liquid is the driving force for heat transfer between the two phases 5

6 Boiling Curve Natural convection Heat transferred by single phase natural convection. Nucleate Boiling Bubble production commences on surface. Initially small number of nucleation sites are active. At higher flux number of nucleation sites increases Bubbles coalesce and form irregular columns of vapor leaving the surface. At departure from nucleate boiling occurs also known as Critical Heat Flux 6

7 Boiling Curve Transition BoilingVapor film begins to form. Thermal conductivity of vapor is much less than that of liquid. Flux decreases. Film Boiling Heated surface is covered by a continuous film of vapor. Heat is transferred mainly by radiation. Flux increases. International Conference on Mechanical, Aeronautical and 7

8 Pool Boiling Process 8

9 Bubble Nucleation For a bubble of radius r to grow, the internal pressure must overcome the collapsing effect of surface tension This excess pressure can be converted to the amount of superheat by Clausius-Clapeyron equation. Smaller nucleus than equilibrium size will collapse and a larger nucleus will grow 9

10 Bubble Growth Thermal Diffusion Controlled Growth -The temperature of the bubble interface must be lower than that of the surrounding liquid so that heat is supplied to cause evaporation. Inertia Controlled Growth -The pressure inside the bubble must exceed that some distance away in the liquid, both to do work to increase the kinetic energy of the surrounding liquid and to overcome the interfacial pressure difference caused by surface tension. 10

11 Introduction In the fast breeder reactor, sodium flows from the cold pool to the hot pool through the subassemblies where heat is convected from the subassembly to sodium. In the present work the mechanism involving the dynamics of bubble formation, its detachment from the surfaces, rise of the bubble and finally the collapse and the resulting hydrodynamic forces generated due to the collapse is studied. 11

12 Nuclear Reactor 12

13 Fast Breeder Reactor 1 Fuel (fissile material) / 2 Fuel (breeder material) / 3 Control rods / 4 Primary Na pump / 5 Primary Na coolant / 6 Reactor vessel / 7 Protective vessel / 8 Reactor cover / 9 Cover / 10 Na/Na heat exchanger / 11 Secondary Na / 12 Secondary Na pump / 13 Steam generator / 14 Fresh steam / 15 Feedwater pre-heater / 16 Feedwater pump / 17 Condenser / 18 Cooling water / 19 Cooling water pump / 20 High pressure turbine / 21 Low pressure turbine / 22 Generator 13 / 23 Reactor building

14 Core Flowering Core flowering: a phenomenon where the core of a typical nuclear reactor expands in radial direction. This could result in the increase in the gap between the subassemblies in the core leading to the decrease of reactivity of the core which is directly related to the thermal power. A sudden change in reactivity will lead to the automatic shutdown of the reactor. 14

15 Core Flowering 15

16 Core Flowering Why Core flowering happens? There are several possible scenarios which can lead to this phenomenon. Transients happening in a single subassembly can cause local flowering of the core and associated reactivity effects Collapse of vapor bubbles on a subassembly of the core is one of the major cause of core flowering. 16

17 Core Flowering Why Core flowering happens? Blockage of sodium flow inside the SA or if there is a sudden power transient in a SA, there is a possibility of sodium boiling occurring inside that SA. The boiling process is characterized by the rapid formation and growth of vapour bubbles at the solid liquid interface (nucleate boiling). 17

18 Core Flowering Result of core flowering: Increase in the gap between the subassemblies in the core leading to the decrease of reactivity of the core which is directly related to the thermal power. A sudden change in reactivity may lead to the automatic shutdown of the reactor. 18

19 Boiling Phenomena: Boiling The boiling process is characterized by the rapid formation and growth of vapour bubbles at the solid liquid interface (nucleate boiling). At some diameter of the bubble, the bubble detaches from the surface and rises up. The resulting condensation may cause the bubble to collapse which may induce hydrodynamic forces on the nearby structures 19

20 Importance Importance of present work: It is important to study the conditions at which the hydrodynamic forces generated by the collapse of the bubble could lead to the phenomenon of core flowering. Bubble detachment, growth, velocity and collapse are important parameters. 20

21 Importance It is found that when a bubble of 1 mm condenses, a pressure of app 200 Mpa is generated at the interface of the bubble. 19 bubbles have to collapse on one face of the subassembly to cause a movement of 1 mm. Situations can occur in the reactor where power transients can occur in parts of subassemblies leading to sodium boiling resulting in bubble formations. high pressure is generated during the collapse, which can lead to the movement of the subassemblies, hence could result in core flowering. 21

22 Status of work Large amount of literature are available on cavitation phenomenon which deals with the collapse of bubbles when it enters a region of higher pressure. Similarly, literature dealing with the process of condensation of vapor bubbles are available. Recent developments include the use of advanced CFD tools to study interaction between multiple fluid phases. 22

23 Objective: Content of work To simulate bubble growth, detachment and rising of a single bubble inside a cylinder using CFD software FLUENT. To determine the bubble detachment diameter obtained from the simulation and compare the result obtained with the established results. To determine the terminal velocity of the bubble, determine the shape of the bubble and compare the result obtained with established results. 23

24 Bubble formation and growth The boiling process is characterized by the rapid formation of vapour bubbles at the solid liquid interface (nucleate boiling) with pre-existing vapour or gas pockets. Once the vapour bubble on the heater surface is large enough at a given temperature that nucleation is assured, it will begin to grow in size. With the increase in temperature difference with the associated pressure difference causes the bubble to grow. 24

25 Bubble formation and growth At some diameter of the bubble, the buoyancy force overcomes the restraining effects of the surface tension attaching the bubble to the heater surface. The bubble detaches from the surface and rises up. As the bubble moves up, it translates from a high pressure regime (hydrostatic) to a lower one. Hence, the bubble internal pressure also reduces. This will be accompanied by increase in size of the bubble. 25

26 Bubble formation and growth In some cases, the influence of surface tension can cause instability effects resulting in the breaking of the bubble into smaller ones. Thermal conditions prevailing in the transit path of the bubble also influence the dynamic behavior of the bubble. As bubble rises towards the relatively colder region, the resulting condensation may cause the bubble to collapse 26

27 Forces acting on a Bubble The condensation as well as the breaking of bubbles may induce hydrodynamic forces on the nearby structures and result in core flowering. FB = Buoyancy force FC = Surface tension force FD = Dynamic force FCP = Contact pressure force 27

28 FB=ρl gv B Fc=2πr σ sinθ Forces acting on a growing Bubble 0 Fcp=πr 2 ( Pg Pl ) 0 Bubble Evolution in dynamic region is governed by interplay of inertial, viscous, surface tension and buoyancy forces. Attaching forces : Surface tension Drag force Inertia force Detaching forces: Buoyancy force Momentum influx of injected air through orifice 28

29 Important non-dimensional Reynolds number = ρ ld 0 U 0 μ l Eotvos number = (ρ l ρ g )gd 0 2 Morton number = gμ l 4 σ numbers ρ l σ 3 Measures the importance of surface tension force in characterizing the shape of the bubble (compared to Body force) The effect of the liquid properties on the dynamical bubble shape is complex. The bubbles moving through low-morton-number liquids are much deformable than those moving through high-morton-number liquids. A high value of the Eötvös or Bond number indicates that the system is relatively unaffected by surface tension effects; a low value (typically less than one) indicates that surface tension dominates. 29

30 Computational Fluid Dynamics CFD is the area which is concerned with applications of mathematical, numerical and computational techniques to the fluid flow simulation. The main goal of CFD is to obtain results comparable with wind tunnel experiments, to avoid (at least partially) expensive and time consuming measurements and to simulate process which cannot be realized experimentally.. It is the analysis of the system involving fluid flow, heat transfer and associated phenomena such as chemical reaction by mean of computer-based simulation. Because of its ability to simulate fluid flow digitally, CFD code has become a widely used tool in engineering, biomedical, and environmental research and development in the past few years. 30

31 Finite volume method The Finite Volume Method (FVM) is one of the most versatile discretization techniques used in CFD. Based on the control volume formulation of analytical fluid dynamics, the first step in the FVM is to divide the domain into a number of control volumes (aka cells, elements) where the variable of interest is located at the centroid of the control volume. The next step is to integrate the differential form of the governing equations (very similar to the control volume approach) over each control volume. Interpolation profiles are then assumed in order to describe the variation of the concerned variable between cell centroids. The resulting equation is called the discretized or discretization equation. In this manner, the discretization equation expresses the conservation principle for the variable inside the control volume. 31

32 Finite volume method The most compelling feature of the FVM is that the resulting solution satisfies the conservation of quantities such as mass, momentum, energy, and species. This is exactly satisfied for any control volume as well as for the whole computational domain and for any number of control volumes. Even a coarse grid solution exhibits exact integral balances. 32

33 Governing equations For all flow related problems conservation equations for mass and momentum has to be solved. For flows involving heat transfer or compressibility, an additional equation for energy conservation is to be solved. Additional transport equations are also solved when the flow is turbulent. 33

34 Governing Equation The Mass Conservation Equation: ρ t +. ρv = S m. The source S m is the mass added to the continuous phase from the dispersed second phase (e.g., due to vaporization of liquid droplets) and any user-defined sources. Momentum Conservation Equation: (ρv) t +. vv =. p +. μ v + ρg + F Where p is the static pressure, μ v is the stress tensor term, μ is the viscosity and ρg and F are the gravitational body International force and Conference external body on Mechanical, forces (e.g., Aeronautical centrifugal force), and 34 respectively.

35 Governing Equation Volume Fraction equation: The tracking of the interface(s) between the phases is accomplished by the solution of a continuity equation for the volume fraction of one (or more) of the phases. For the q th phase, this equation has the following form: 1 ρ q α t qρ q +. α q ρ q v q = S αq + p=1 (m pq m qp ) Where m qp is the mass transfer from phase q to phase p and International m pq is the mass Conference transfer from on Mechanical, phase p to phase Aeronautical q. and 35 n

36 Volume of Fluid (VOF) model The VOF model can model two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. Typical applications include the prediction of jet breakup, the motion of large bubbles in a liquid, motion of liquid after a dam break, and the steady or transient tracking of any liquid-gas interface. 36

37 Forces acting on vapour bubble The prime forces acting on a vapour bubble during the later phases of its growth are buoyancy and hydrodynamic drag forces attempting to detach it from the surface and surface tension and liquid inertia forces acting to prevent detachment. The liquid inertia force is a dynamic force resulting from the displacement of bubble during bubble growth. 37

38 Forces acting on vapour bubble The liquid inertia force is a dynamic force resulting from the displacement of bubble during bubble growth. The growth velocity of a bubble and hence the initial force is a strong function of the liquid superheat which, in turn, is inversely proportional to the size of the active cavity. A small cavity thus forms a bubble with a faster growth rate than from a large cavity. 38

39 Forces acting on vapour bubble For small cavity, bubble size at departure is dictated mainly by balance between buoyancy and liquid inertia forces. For larger cavity sizes, the growth rate decreases, the dynamic forces become small, and the bubble size at departure is set by a balance between buoyancy and surface tension forces. Equation for bubble detachment diameter; D d = θ σ g(ρ l ρ g ) 1 2 [Fritz and Ende], θ is the angle of contact. 39

40 Forces acting on vapour bubble Terminal velocity of the bubble: Terminal velocity is defined as the steady velocity that the bubble reaches when there is a balance between buoyancy and drag forces. ρ L ρ g gv B = πa 2 ρ 2 lu o f 2 The relative terminal rising velocity of a single gas bubble, moving into a liquid phase, is determined by its size, by the interfacial tension, by the density and viscosity of the surrounding liquid. Both shape and velocity are strongly interacting. 40

41 Terminal Velocity Bozzano and Dente have obtained equations to determine the terminal rising velocity of a single bubble. They have represented the bubble shape by means of the superposition of two oblate semi-spheroids having a common larger semi-axis. Asymptotically this shape can degenerate towards something resembling a spherical-cap or towards a spherical bubble. The validity of the proposed expression is restricted to Morton numbers less than International 1E-8. Conference on Mechanical, Aeronautical and 41

42 Terminal velocity By neglecting the gas density in comparison with that of the liquid, above equation gives the terminal rising velocity in infinite environment: U 0 2 = 4 3 gd 0 C D where C D is the drag coefficient and C D = f a R 0 2 where f is the friction factor which covers a wide range of Reynolds, Morton and Eotvos numbers 42

43 Experiment A cylinder of certain radius filled with water which is open to atmosphere. A small inlet is provided at the bottom of the cylinder through which water-vapor is injected at a constant mass flow rate. The jet rises inside the cylinder forming a vertical column due to the jet effects and the momentum of the incoming fluid. At some time, the buoyancy force overcomes the restraining effects of the surface tension attaching the bubble to the surface. Thereafter, the bubble detaches from the surface and rises up. Once the bubble has detached from the surface, injection of vapor into the cylinder is stopped in order to capture the effects of only one bubble. 43

44 Experiment Because of the surface tension, initially the bubble tends to a circular shape. During its upward motion, the bubble will be accelerated due to the net balance between buoyancy force and viscous drag. Finally when these forces exactly balance each other, it attains a constant terminal velocity. The terminal velocity of the bubble is captured by the monitor function provided in FLUENT. 44

45 Modeling and Meshing The above mentioned phenomenon is modelled by using a 2-D axi-symmetric slab model of a cylindrical container of radius 5 mm and height 60 mm as shown in figure The model is meshed with simple hexahedral meshes as shown 45

46 Boundary Conditions The inlet mass flow rate of water vapour is 1e-7 kg/s. The walls of the cylinder are assigned the wall boundary condition with slip condition. The angle of contact between water and water vapour is 90 Pressure outlet is provided at the top. Gravity o acceleration is taken equal to g = 9.81 m/s 2, the surface tension is N/m. 46

47 Material Properties In this problem two fluids are considered, namely water-vapour (light fluid) and water (heavy fluid).the physical parameters of the injected fluid i.e., vapour and the surrounding water as follows. Fluid Density(kg/m 3 ) C p (J/kg-K) Thermal conductivity (W/m-K) Water- liquid Water- vapour E Viscosity(kg/ms)

48 Mesh Independence study It is seen that the simulation results and the theoretical results match fairly. The difference between the diameters in case A1 and A2 is less than 3.5% and the difference between A2 and A3 is less than 2% i.e. when the mesh is made finer the value of the bubble detachment diameter trends towards a constant value. However mesh A3 being very fine consumes significant higher amount of time compared to mesh A2; hence mesh A2 is used for further simulations. Case Grid Size Bubble detachment Diameter, D d (mm) A1 0.20mm x 0.20mm 5.11 A2 0.15mm x 0.15mm 5.31 A3 0.10mm x 0.10mm 5.41 Theoretical (from equation 1)

49 Results and Discussion The sequence of bubble formation, departure and the change in bubble shapes during the rise of the bubble in the container at different time intervals is shown 49

50 Results and Discussion After the bubble has detached from the surface, the flow of vapour is stopped. The bubble is allowed to rise in the quiescent fluid. The sequence of bubble rise in the cylinder at different time intervals is shown 50

51 Results and Discussion The bubble rising velocity is plotted against time as shown in fig. It is seen that after initial oscillations, the bubble stabilizes and reaches a constant terminal velocity. 51

52 Results and Discussion The results obtained from simulation are compared with the theoretical results Cases Terminal velocity (m/s) Present simulation G.Bozzano, M. Dente [6] (solving equation 4.3) Clift R, Grace JR, Weber M [17], (solving equation 4.8)

53 Conclusion The sequence of bubble formation, departure and the change in bubble shapes during the rise of the bubble in the container at different time intervals is analysed using CFD There is some difference in the values between the theoretical and simulation result. This is because the equation available for detachment diameter is for small cavity sizes. However further decreasing the inlet diameter in the simulation model will lead to convergence issues 53

54 Conclusion As the mesh size decreases, the size of the bubble increases, however the value of the detachment converges. Once the bubble has detached from the surface, the velocity of the rising bubble is oscillating in the beginning. This is because the fluid inside the bubble keeps rotating inside. The values obtained for terminal velocity matches with the theoretical results. 54

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