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In this paper, the growth of vapour bubble in a viscous and non-viscous fluid in superheated liquid is proposed with the new definition of parameter to find the analytical solution of vapour bubble radius. The mathematical model is formulated of mass, momentum, and heat equations. It's found that the growing of vapour bubble radius is proportional to the thermal diffusivity, while it is inversely proportional to the initial superheating, initial surface tension, and initial viscosity.
In this study, the growth of vapour bubble in a superheated liquid in non-viscous takes higher values than that in a viscous fluid, and the obtained results exhibit a reasonable agreement with the theoretical previous works.
A better agreement with the current model and some previous theoretical works is performed.
Bubble growth in liquids is a principle phenomena in many problems, such as in subcooled boiling, condensation, cavitations, bubble sonoluminescence and sonfusion. The procedures are complicated because the heat transfer occurs at a moving interface where the bubble radius is changing with time.
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Although a large amount theoretical modeling and experimental research to study bubble development was conducted in latest years, only a few experimental studies have been conducted for the collapse of bubble.
Moreover, little dynamical modeling work has been published on bubble collapse because of numerical instabilities. It is recognized that bubble growth dynamics can be described in terms of three different mechanisms: inertia-controlled, thermal diffusion controlled and mass diffusion controlled mechanisms.
Approximate analytical solutions for bubble radius as a function of time for bubble growth based on different assumptions have been derived.
However, these analytical solutions show significant deviations from reality under certain conditions, such as in the very early stage of bubble growth when the superheat is small or when the operating pressure is extremely low. Previous studies have been shown that numerical methods have been used for the growth of bubble, such as Meietal., and Heoetal.. Meietal. have performed a numerical analysis to study the bubble growth under the condition of saturated heterogeneous boiling.
Continuous energy transfer between the vapour bubble and liquid micro-layer or heating wall was regarded as influencing the bubble growth. The moving unstructured grid was considered by using the interface tracking method with the combination of the finite volume method. The control volume continuity, momentum, and energy equations were modified to include surface tension and discontinuous pressure and velocity.
The flow-directional local grid (MPS-MAFL) method has been used by Heoetal. to do the numerical study about the growth of bubble in transient pool boiling through moving particle semi-implicit with meshlessadvection. The growth process of a bubble with different initial radii was calculated under the condition of high heat flux and high sub cooling condition. Besides, some new models were used to study the bubble growth.
A new understanding was provided by Li etal. to study the bubble which was about the interfacial transport characteristics of in viscid spherical bubble with different geometric parameters, rising in a stagnant hot or bisolution liquid. The flow and temperature fields around bubbles and similarly sized rigid spheroids were computed numerically while the development of the physical model for vapor bubble growth in the condition of saturated boiling was provided by Liaoetal., including heat transfer through the micro-layer and the surrounding foam body superheated liquid and bubble growth.
Both asymptotic and numerical methods were carried out to study the liquid temperature field surrounding a hemispherical bubble and it indicated that there was a thin unsteady thermal boundary layer existing adjacent to the bubble dome. During the early stages of bubble growth, heat transfer to the bubble dome through the unsteady thermal boundary layer constituted a substantial contribution to vapor bubble growth.
The region around a single growing bubble was considered to be subdivided into three sub regions by Genske and Stephan, which were micro-region, bubble area, and surrounding liquid. Their results showed that the flow pattern in the liquid around a growing vapor bubble was determined by not only the movement of the bubble surface, but also the vapor flow that flied inside the bubble.
Heat conduction would be the dominant factor on the bubble growth in the regions away from the bubble. There are three stages for bubble growth: inertial, thermal, and diffusion. In the inertial stage, the bubble nucleus depends strongly on the interfacial mechanical interactions, pressure forces, and surface tension forces. This stage lasts a few milliseconds and thermal phenomena are negligible. Therefore, this stage is called isothermal. In the thermal stage, the radius of the nucleus increases and the growth becomes mostly dependent on the supply of heat that is consumed to vaporize the liquid on the bubble surface.
The rate of expansion of the bubble is much lower than during the isothermal stage. This stage of bubble growth is called isobaric. It is worth noting that the duration of the isothermal stage is very short compared with the isobaric stage. Most of the bubble growth occurs in a stage that is characterized by an essentially constant bubble pressure. This feature was first noted by Plesset and Zwick, in which the growth of the bubble is described by mass and momentum equations. The mixture of vapour and superheated liquid is considered incompressible and non-viscous. The momentum equation is described by Rayleigh–Plesset and solved under external forces.
The effect of gravity and vapour pressure change are neglected. The theory of the growth of a single vapour bubble in a superheated liquid has been considered by several authors. The inertia controlled growth was presented by Rayleigh, who determined the first equation of motion for a spherical bubble growth or collapse. The equation of motion of a spherical bubble growth (or collapse) was first presented by Rayleigh under the effect of the stage of inertia controlled.
The theoretical studies were followed to explain the other growth-controlled stages as the asymptotic solution that presented by Plesset and Zwick; which was under consideration of the thermal diffusion controlled growth, neglecting liquid inertia and thin thermal boundary layer. The approximated solution for the bubble wall temperature has good agreement with the experimental data of Dergarabedian for moderate superheats up to 6◦C.
The problem was presented in a new treatment by Scriven by solving the heat equation without assuming a thin thermal boundary layer. The solution for moderate superheat was identical to that of Plesset and Zwick. The inertia and thermal diffusion controlled growth were combined by Mikicetal. using the Clausius-Clapeyron equation. The result was a generalized expression which was valid over the entire growth range, and has good agreement with the previous works.
Besides the analytical and experimental treatments of the problem, a number of numerical computations were carried out by coupling various forms of equation of motion with various special forms of the energy equation, as in refs. In this study, under the influence of heat transfer growth, the growth of a vapour bubble in a viscous, superheated liquid is solved analytically by a new technique with the a new define of parameter getting the radius of vapour bubble in viscous and non-viscous fluid, which takes into account the effects of several physical parameters, including surface tension, dynamic viscosity, density ratio, and other parameters.
After obtaining the solution expression, a numerical calculation is performed, and some graphs are presented to discuss the effect of these parameters on the growth of the vapour bubble. Comparisons are then made between this work and some of the previous theoretical and experimental work. The resultant formula gives good agreement with these previous works in some initial superheated conditions. The structure of paper consists of formulation of physical and mathematical problem in section2. Growth of vapour bubble in a viscous and non-viscous flow are presented in sections 3, and 4. In section 5, the results and discussion is discussed. Finally, remarked concluded is given in section 6.
A single vapour bubble is considered to grow inside a superheated, incompressible and viscous liquid between two finite radius boundaries and (Fig. 1). The growth is affected by the pressure difference P between the bubble pressure , and surface tension of the liquid at the bubble boundary, liquid viscosity, and other physical parameters. We take into account the following assumptions.
Surface tension was kept constant. Vapour density distribution inside the bubble is assumed to be uniform except for a thin boundary layer near the bubble wall. Based on these assumptions, an ordinary differential equation for the variation of bubble radius R as a function of time for bubble growth can be derived from the momentum equation in a spherical coordinate for the liquid phase as follows:
The mathematical model describing this problem consists of three equations: mass, momentum, and heat equations.
The mass equation for an incompressible fluid takes the form. For the spherical symmetry of a vapour bubble, the solution of above equation gives the liquid velocity inside the mixture.
The momentum equation for the growth of vapour bubble in an incompressible and viscid fluid derived by Rayleigh–Plesset in this case is given by.
The growth of vapour bubble under the physical condition, the heat equation of vapour inside the bubble is increasing and the heat outside bubbles is decreasing, see Fig.1. Then, the heat equation of Newtonian fluid surrounded a growing of a vapour bubble is described by the following equation.
In this study, the problem of the growth of vapour bubbles is presented under the influences of vapoursuperheated liquid into the bubble and the effect of viscosity and non-viscosity. We solve the equation of motion analytically using with a new definition formula of parameter under the proposed initial conditions.
We suppose that , are constants, and are a real numbers. The equation of motion (3) that describing the growth of vapour bubble with viscosity is becomes.
Thus, is introduced by Mohammadein and etal where, The rate of increasing vapour inside the bubble equals to the rate flow inward across the bubble wall, thus then, =. (9) Substituting by equation (9) into equation (6), then, (10) and whereas, , then. (11) Then we have the final formula for the growth of vapour bubble radius in a viscous flow with a new defination of parameter , in terms of time on the form. (12) where, A, B and ξ is defined as follows.
The extended Rayleigh equation of vapour bubble for a non-viscous fluid and an incompressible flow has the form, where, the surface tension is defined as the form , and the pressure difference is modified to take the form, , are constants and are a real numbers. The equation of motion in this case will take the form. To find the constant , we apply the initial conditions (5) into equation (14), then. (15). (16). (17) The growth of vapour bubble in a non-viscous superheated liquid is defined as follows (18) where, , and is defined as follows,
Table 1: Initial values of physical parameters used in the simulations and calculations of bubble.
Parameter
Value
Unit
Parameter
Value
Unit
1003 [25]
[25]
0.068 [23]
[19]
2.5 [19]
[23]
0.1
[23]
)
2
-1
The physical problem is described by mass, momentum and heat equations (1), (8) and (9) respectively. The growth of vapour bubble under the effect of viscous and non-viscous flow between two phase-flow in terms of time with the different physical parameters such as the initial viscosity , superheating , thermal diffusivity and initial void fraction are investigated.
It is observed that the growth of vapour bubble in a viscous flow is taken the lower values comparison with non-viscous flow. It is illustrated, in Fig. 2, and this physical reality because the viscosity force of the fluid inhibits and resists the growth of vapour bubble in the fluid, in addition this result is an agreement with Mohammadein and Mohamed in ref. [25]. The radius of vapour the bubble in terms of time increases with thermal diffusivity and initial surface tension as shown in Fig.3 and Fig. 4 respectively. It is observed that the behaviour of growth bubble is directly proportional with the thermal diffusivity and initial surface tension .
It is observed that in Figs. 5, and 6, the growth of vapour bubble radius proportional with superheating , however, in the case of the effect of the viscosity we find that the radius of vapour bubble takes less values than if it does not exist. It is illustrated in Fig. 7, the behaviour of vapour bubble radius as a function of time decreasing with higher value of the initial viscosity , this means that the growth of vapour bubble radius is faster in the case of non-viscous flow than viscous flow.
In Fig.8, explains the comparison between the current work and some of previous theoretical works [21, 23, and 24]. The results of growth bubble in a viscous fluid give us, the lower values when we compare with non-viscous fluid. These results are agreement with the physical situation, and Mohammadein and Mohammed in ref. [25].
Figure 2: Growth of vapour bubble radius in terms of time with the viscous and non-viscous flow.
Figure 3: Growth of vapour bubble radius in terms of time with the different values of thermal diffusivity in a viscous flow.
Figure 4: Growth of vapour bubble radius in terms of time with the different values of initial surface tension in a viscous flow.
Figure 5: Growth of vapour bubble radius in terms of time with the different values of initial superheating in a non- viscous flow.
Figure 6: Growth of vapour bubble radius in terms of time with the different values of initial superheating in a viscous flow.
Figure 7: Growth of vapour bubble radius in terms of time with the different values of initial viscosity in a viscous flow.
Figure: 8 Growth of vapour bubble radius in terms of time with the present work, Foster & Zuber theory [24], Mohammadeinetal. model [23], and Scriven theory [21].
The Growth of vapour bubble is obtained analytically with a new define of parameter . The influence of viscous and non-viscous fluid its clearly on the growth of vapour bubble process, in addition the variable surface tension and other physical parameters such as thermal diffusivity , initial superheating , initial surface tension , and initial viscosity . The values of physical parameters are given by
Table (1). The results and discussion of figures are concluded as following remarks:
The above concluded remarks prove the validity of the proposed model, and how to extend the present model in more properties of fluid and flow.
Nomenclature= , liquid thermal diffusivity [ Liquid specific heat at constant pressure Specific heat of liquid at constant pressure [ Specific heat of vapour at constant pressure [ Latent heat of vaporization [J/kg]; = Pressure difference [kg ]; Saturated vapour pressure [kg ]; Liquid pressure [kg ]; Distance to the bubble radius; Instantaneous bubble radius; Initial bubble radius ; Maximum value of bubble radius ; , instantaneous radial velocity of bubble boundary ; , instantaneous radial acceleration of bubble boundary ; Time of bubble growth ; Absolute temperature of liquid ; The velocity of bubble boundary; , constant taking effect of radial convection on bubble growth into account; Initial uniform liquid superheating above saturation temperature, or liquid superheating at great distance from bubble, or superheating of bulk liquid ; ; Density of vapour; Density of liquid ; Surface tension ; viscosity parameters ;
Influence of Viscosity and Non-viscosity on a Growing Vapour Bubble in a Superheated Liquid Between Two-Phase Flow. (2024, Feb 07). Retrieved from https://studymoose.com/document/influence-of-viscosity-and-non-viscosity-on-a-growing-vapour-bubble-in-a-superheated-liquid-between-two-phase-flow
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