가열면에 부착된 기포의 성장과 이탈에 미치는 전기장의 영향

조 혜정, 강 인석*, 김 무환**

첨단유체공학 연구센터, 포항공대 화학공학과*, 포항공대 기계공학과**

The Effect of an Electric Field on the Growth and

Departure of a Bubble Attached to a Heating Surface

Hye Jung Cho, In Seok Kang*, Moo Hwan Kim**

Advanced Fluid Engineering Research Center(AFERC), Chemical Engineering Department of Postech*, Mechanical Engineering Department of Postech**

Introduction

Recently, there has been a growing interest in application of an electric field to enhance the rate of boiling heat transfer in liquid in order to develop a heat exchanger of better performance. Several important effects of an electric field on boiling include the increase of maximum heat flux, the suppression of film boiling, and the decrease of bubble departure size and time. In order to provide a theoretical basis for the development of new type of heat exchanger enhanced by an electric field, it is natural that fundamental studies on the mechanisms of nucleate boiling are necessary. Specially, the analyses for the effects of an applied electric field on the basic mechanisms of bubble growth and departure are of greatest importance.

In the present study, numerical analyses are performed to investigate the effects of an electric field on the growth and departure of a bubble attached to a heating surface. The bubble growth process is considered to be controlled by the unsteady heat transfer between the liquid and gas bubble. During the bubble growth, unsteady solutions on the equilibrium bubble shape in an electric field are obtained by solving the free boundary problem without any approximation to the bubble shape. The free boundary problem consists of the governing equations for electric field and  temperature field and the normal stress condition at the bubble surface. The numerically generated orthogonal grid system is employed for a finite-difference solution of the free boundary problem. The bubble departure time and volume are  also determined from the force balance on the bubble under the electric field.

PROBLEM STATEMENT

We are concerned with the growth and departure of a bubble attached to a heating surface in an electric field as shown in figure 1. The bubble is assumed to sit on a heating surface with contact angle[eqn: theta_c] and contact radius[eqn: r_c]. The initial bubble volume is assumed to be [eqn: 2pi a_0^3 /3], where [eqn: a_0] is the radius of an equivalent hemispherical bubble attached to a heating surface. The bubble surface is assumed to have a uniform temperature at saturation conditions,[eqn: T_{sat}] and in turn the bubble surface is characterized by a uniform surface tension[eqn: gamma~]. The electric conductivity and permittivity of the gas inside the bubble are assumed to be negligible in comparison with those of the surrounding dielectric liquid. Also, it is assumed that the electric conductivity and permittivity of the liquid are the uniform in the whole of surrounding liquid. The surrounding liquid is assumed to be incompressible and it is also assumed that there is no bulk free charge in the surrounding liquid. The applied electric field far from the bubble is uniform in space and constant in time. The far field can be expressed in terms of the electric potential, which is defined by [eqn: E_{infty}=-nabla phi_{infty}] as

[eqn: phi_{infty} = - E_{infty} ~r ~cos theta]                                (1)

where [eqn: theta ~]is the angle measured from the symmetric axis.

The dimensionless governing equations are

[eqn: nabla^2 phi = 0]                                        (2)

for the electric potential,

[eqn: Ja {{partial T}over{partial t}} + Gr^{1over2} Pr (bold u cdot {bold nabla}T) = nabla^2 T ~~]                   (3)

for the temperature field of the liquid, and

[eqn: Gr^{1over2}({bold u} cdot nabla {bold u}) = -Gr^{1over2}{bold nabla}p_d + nabla^2 {bold u}+Gr^{1over2}T {bold e}_x. ~~]          (4)

for the quasi-steady state liquid motion. In the above, the dimensionless numbers are defined by

[eqn: Ja={{rho_l C_p T_c}over{rho_g h_{fg}}},~ Gr={{beta g T_c a_0^3}over{nu^2}},~ Pr={{nu}over{alpha}}.~]

Above dimensionless governing equations are solved by the following boundary conditions

[eqn: (i)~ as ~r -> infty~:~phi -> -E_infty r cos theta, ~T -> 0,~ {bold u}-> 0~~~~~~~~~~~~~~~~~~~~~~# (ii) ~at ~theta = 0,~r ~> R(theta,t)~:~ {{partial phi}over{partial theta}}=0,~ {{partial T}over{partial theta}}=0,~ {{partial {bold u}}over{partial theta}}=0~~~~~~~~# (iii)~ at~ theta={{pi}over{2}},~r>R(theta,t)~:~ phi=0,~T=1,~{bold u}=0~~~~~~~~~~~~~~~~~# (iv) ~at ~r=R(theta,t)~:~ {bold n} cdot {bold nabla}phi = 0,~ T=0,~ {bold u} cdot{bold n} =-{1over{|{bold nabla}F|}} {{partial F}over{partial t}}.#]

The rate of change in the bubble volume is obtained from the energy  balance on the bubble surface as follows

[eqn: {{dV}over{dt}}=int_A - ( {{partial T}over{partial r}}) dA.~~]

When an electric field is applied, the dimensional force balance on a bubble at departure may be written as

[eqn: Delta rho g V_d = 2pi gamma r_c sin theta_c + int_A (-{bold e}_x )cdot ({bold n} cdot {bold T}^e ) dA ~~]       (5)

where [eqn: V_d~] and [eqn: A] denote the bubble volume and the bubble surface area at departure time. The Maxwell stress tensor,[eqn: bold T^e~], is given by

[eqn: {bold T}^e = epsilon {bold E E}- {1over2} E^2 {bold I}.~]

The LHS terms in (5) is the[eqn: {bold e}_x~]directional (upward) component of the buoyancy force, and the RHS terms in (5) are the[eqn: -{bold e}_x~]directional (downward) components of the surface tension and electric forces, respectively. During the bubble growth process, the deformed bubble shape with contact radius and contact angle at t=t is determined to compute the surface tension force (the first term of RHS of (5)) and the electric force (the second term of RHS of (5)). At a certain time, if the resultant force (surface tension force plus electric force) is just balanced with the buoyancy force, the departure time ([eqn: t_d~]) and departure volume ([eqn: V_d~]) are determined exactly.

Result and Discussion

Figure 2 shows the effects of an electric field on the equilibrium bubble shape at departure for several values of Ne. As observed in the figure, the bubble shape is a spherical one in the absence of an electric field (case of Ne=0.0). However, when the electric field is applied, the bubble is extended in the direction parallel to the applied electric field. It is also observed that the bubble has a narrow-necked shape. Due to the deformed bubble shape, the contact angle between the bubble and the heating surface also increases under the given electric field.

In addition to the bubble deformation, applied electric field also affects the dynamic conditions on the bubble growth and departure. In figure 3, the variation of the bubble volume during the growth process for several values of Ja is presented. As we can see in the figure, for both cases without (solid lines) and with (closed markers) electric field, the growth rate of a bubble is larger for a large Ja and thus the departure time decreases with the increase of Ja. When an electric field is applied, the bubble volume and time at departure increase with the increase of an applied electric field strength. However, it is found that the growth rate does not affected by an applied electric field.

Figure 4 shows the effects of an electric field on the variation of the forces exerted on the bubble during the growth period. When the electric field is applied, the surface tension and electric force slowly decrease with the increase of time. Thus, the resultant force (surface tension force plus electric force) also decreases. For the case without electric field, it is natural that the buoyancy force does not exceed the surface tension force throughout the bubble growth. On the other hand, under the electric field, the buoyancy (upward) force exceed the surface tension force beyond a certain time. However, after this time, the electric force plays a significant role for holding the bubble on the heating surface and thus the bubble departure does not occur. Finally, at departure time, the retaining force due to surface tension and electric field are just balanced with the buoyancy force due to bubble expansion.

Fig.1 A bubble attached to a           Fig. 2 Numerical solutions on

heating surface in an electric field      the bubble shape for several values

of Ne for Ja=10

Fig. 3 Numerical results on the bubble  Fig. 4 Numerical results on the forces

volume during the growth time for     exerted on the growing bubble surface

Ja=1.0,10,100, Ne=1.5                for Ja=1.0, Ne=1.5