일정한 열선속을 가진 경사판 위를 흐르는

증발액막의 안정성 해석

이 재 성, 김 효, 김 덕 찬

서울시립대 화학공학과

The Stability of Evaporating Liquid Film down an Inclined Plane

with Constant Heat Flux

Jae Sung Lee , Hyo Kim, and Dok Chan Kim

Dept. of Chemical Engineering, The University of Seoul

INTRODUCTION

The stability of thin film with free surface is of importance to many processing arising in the fields of chemical and mechanical engineering. Nonisothermal films have been focused for a number of years because thin evaporating liquid film provides excellent removal of heat on the hot solid surfaces due to its high transfer surface of heat and mass in comparison with the volume of the thorough-flow film. The present study is devoted to the mechanisms of the film flow down an inclined plate with constant heat flux. Long wave approximation is applied by deriving an evolution equation for two dimensional disturbances. Numerical analysis for the nonlinear stability is compared by the Crank-Nicolson scheme using Newton-Raphson iteration  and Fourier-Spectral method.

FORMULATION

An incompressible Newtonian fluid of constant density[eqn: rho], viscosity[eqn: mu], thermal conductivity [eqn: k_1], and thermal diffusivity[eqn: kappa] is flowing down, under the action of gravity, on an inclined plate at a constant heat flux [eqn: q]. The layer is bounded above by its vapor of temperature[eqn: T_S] and has a mean thickness of [eqn: d_o]. As the volatile liquid flows, the liquid vaporizes into vapor particles at the interface and the film height is decreasing by evaporation in time. Evaporation is occured only in a direction normal to the interface. Cartesian coordinates are used to describe the system. In two dimensional case the normal coordinate[eqn: z] is taken to be zero at the solid surface and the interface is located at [eqn: z=h(x,t)], where [eqn: h] is a function of the lateral coordinate [eqn: x] and time[eqn: t] (Fig 1.). The unit vectors [eqn: n] and[eqn: t] are the outward normal and tangent vectors, respectively.

The Reynolds number based on the maximum velocity for an undisturbed layer becomes [eqn: G ~sin beta], where

(1)	[eqn: G~~=~~{g d_o ^3 } over nu]

is a parameter measuring the film thickness, [eqn: beta] is the anlge of inclination, [eqn: g] is the gravitational acceleration, and [eqn: nu ~=mu/rho].

The characteristic length [eqn: l] of the interfacial disturbance is much larger than the thickness [eqn: d_o] of the liquid layer. So we consider long-wave disturbance periodic in [eqn: x] and define as a small parameter [eqn: epsilon~= d_o / l] . The long wave asymptotics ([eqn: xi= epsilon x],[eqn: zeta=z],[eqn: tau= epsilon t] ) proposed by Benney[1] allows description of the flow development. The full system of the Navier-Stokes and energy equations then can be reduced into a single evolution equation for the local layer thickness [eqn: h(xi,tau)], which is derived by the following form :

(2)  [eqn: h_tau + bar E + &G sin beta h^2 h_xi + epsilon ( bar S h^3 `h_{xi xi xi} + 2 over 15 G^2 sin^2 beta h^6 h_xi - 1 over 3 G cos beta h^3 h_xi + A over h h_xi )_xi ## & + epsilon `G sin beta bar E h^3 h_xi ( 1 over 2 Pr + 5 over 6 ) + epsilon {barE}`^2 h Pr ``=``0 ~,]

where [eqn: Pr ~(=~ nu/ kappa~)] is the Prandtl number, [eqn: S~(=~ {sigma_o d_o }/{3 rho mu^2 } )] mesures the surface tension, and [eqn: A~(=~ A' /{6 pi d_o `rho `nu`^2 } )] describes the van der Waals attractions.

And the evaporation number, [eqn: E], describing the volatility of the liquid becomes

(3)	[eqn: E ~=~ {q `d_o } over {rho nu L }] ,

where [eqn: L] is the latent heat. Here, the parameters are [eqn: O(1)] except [eqn: E= epsilon barE] and[eqn: S = epsilon ^-2 ``BARS].

The terms proportional to [eqn: E] in Eq (2) means the mass loss due to evaporation, while the third term describes wave propagation and steepening. The fourth, fifth, sixth, and seventh terms mean surface tension, mean shear flow, hydrostatic effects, and van der Waals attractions, respectively.

In the isothermal limit ([eqn: E~ -> ~0] ), Eq (2) without the effect of van der Waals attractions reduces to the evolution equation for falling films obtained by Benney[1] and Atherton & Homsy[2].

BASIC STATE

The basic state is represented by a flat layer draining downward and disappears in a finite time. The layer thickness is independent of [eqn: xi], and can be obtained by solving the evolution equation with [eqn: PARTIAL_xi ~=~0]. The velocity component in the [eqn: x]- direction has a parabolic profile and that in the [eqn: z]-direction is zero.

The basic state solution for the film height and temperature are obtained by

(4)	[eqn: h_o (tau)```=``1~- ~bar E` ~tau],

(5)	[eqn: T_o (zeta) ``=~`-~zeta~ +~K~+~h] ,

where [eqn: K] measures the degree of nonequilibrium at the evaporating interface.

LINEAR STABILITY

To take into consideration of the stability of the liquid film, first we employ  linear stability theory. Write the solution of Eq (2) as follows :

(6)	[eqn: h~=~h_o (tau) ~+~delta``[H(tau) exp^{ik xi } ~+c.c.]],

where [eqn: delta] is the initial amplitude of the disturbance, [eqn: H] is the time-dependent normal mode amplitude having [eqn: H(0)=1/2], [eqn: k] is the scaled disturbance wavenumber, and [eqn: c.c] denotes the complex conjugate. We substitute Eq (6) into Eq (2) and linearize in [eqn: delta] to obtain

(7)	[eqn: DOT H /H ~=~ GAMMA ~+~ GAMMA_ev ~-~ i` k` c(tau)],

where

(8)	[eqn: GAMMA_ev ~=~- epsilon {barE}`^2 Pr] ,

(9)	[eqn: GAMMA ~=~ epsilon k^2 ``[- k^2 ``bar S `` h_o ^3 + 2 over 15 G^2 sin^2 beta h_o ^6 - 1 over 3 G cos beta h_o ^3 + A over h_o ]] ,

(10)	[eqn: c(tau)~=~G `sin beta`` h_o ^2 `` (1+ 5 over 6 epsilon`` h_o + 1 over 2 epsilon` barE`` Pr `h_o )].

Here,[eqn: GAMMA] is defined as an effective growth rate, [eqn: c] is the linearized phase speed, and [eqn: GAMMA_ev] is real, but does not affect the effective growth. The phase speed [eqn: c] decreases with time as the layer thins, and so indicates that the wave propagation can be affected by evaporation. The effective growth rate shows the destabilizing effects of mean flow and van der Waals attractions and the stabilizing effects of hydrostatic pressure and surface tension.

NONLINEAR STABILITY

The nonlinear development of the instability can be studied in the two methods by posing an initial small disturbance on a spatially-periodic domain. As an initial condition, we set

[eqn: h`(xi,0)``=``1- delta ` cos(`k `xi`)],	(17)

where[eqn: delta] is 0.1. First, the evolution equation is solved by a Fourier Spectral method. For the another method the difference scheme in space are employed with the Crank- Nicolson scheme used in time and then the difference equations are solved by Newton-Raphson iteration.

CONCLUSIONS

We studied the evolution equation of the system which the bounding plate has a fixed heat flux. Unlikely the result from a fixed wall temperature condition studied by Joo et al[3] and Lee et al[4], vapor recoil and thermocapillarity do not exist because the mass flux and temperature along the interface are constant. Fig. 2 indicates that the maximum growth rate increases with time while the range of unstable wavenumbers gets wider. Fig. 3 and Fig. 4 show the surface-wave instability with initial wavenumber [eqn: k] = 0.7 located in the unstable zone obtained from the linear stability anaylsis. Fig. 3 is the case of isothermal film flow and Fig. 4 shows the evaporating liquid film flow. In both cases the disturbances increases with time.

REFERENCES

1. Benny, D.J.: J. Math & Phys., 45, 150(1966)

2. Atherton, R.W. and Homsy, G.M.: Chem. Engng Commun., 2, 57(1976)

3. Joo, S.W., Davis, S.H., and Bankoff, S.G.: J. Fluid Mech., 230, 117(1991)

4. Lee, J.S., Kim, H., and Kim, D.K., Proceedings of '97 KICHE Spring Meeting, 3, 1161(1997)

Fig 4. Free surface shape

of evaporating liquid film flow

Fig 3.  Free surface shape

of isothermal liquid film flow

Fig 1. The physical configuration

Fig 2. Growth rate vs. wavenumber