Heat Transfer between a Plane Surface and Air Containing Suspended Water Droplets William C. Thomas Department of Mechanical Engineering, Virginia Polytechnic Institute, Blacksburg, Va. 24061
J. Edward Sunderland Department of Mechanical and Aerospace Engineering, North Carolina State University,Raleigh, N . C. 27607
An analytical and experimental investigation was made to determine heat transfer and liquid film thickness for a wedge-shaped body in a two-phase two-component stream. The rate of heat transfer was increased over that for the gas (air) phase alone as a result of evaporation and sensible heating of the continuous liquid film that formed on the solid surface. Typically, heat transfer rates were increased about 20 times by adding 5% liquid water to the air stream. The liquid film formed was 0 to 0.008 inch thick. The integral energy equation was solved to obtain local and average heat transfer coefficients. Solutions were obtained in closed form and compared with measured values. A closed-loop subsonic wind tunnel with a 1 -sq-foot test section was used in the experimental phase. This investigation considered the mechanisms involved for a plane surface. The liquid film thickness measurements showed good agreement with the analytical solution.
As
IXCREASISG EMPHASIS is currently being directed toward reducing the size and weight of heat transfer equipment. This paper is concerned with the increase in heat transfer from a plane surface caused by introducing liquid droplets (spray) into a gas stream. Liquid droplets entrained in a gas stream as shown in Figure 1 increase heat transfer from the body by sensible heating and evaporation of the liquid in the film. (The liquid and gas boundary layers shown in Figure 1 are disproportionately large.) This technique is closely related to modern cooling applications such as film, ablation, and transpiration cooling. The results of an analytical and experimental investigation are reported for a wedge-shaped profile exposed to a twophase two-component flow stream consisting of water droplets and air. A survey of the literature reveals that several extensive analyses (Acrivos et al., 1964; Goldstein et al., 1967; Hodgson, 1967; Hoelscher, 1965; Hodgson et al., 1968; Takahara, 1966) have been recently carried out for twophase two-component crossflow over a circular cylinder. All the experimental and analytical studies confirmed the high potential for increasing heat transfer by this method. A thin continuous liquid film was observed on all surfaces directly exposed to droplet-air flow. Accurate heat transfer solutions were obtained assuming straight-line trajectories for impinging droplets. Film thickness solutions were obtained for cylinders and subsequently used to obtain a heat transfer solution. Heat transfer solutions were generally verified experimentally, but not film thickness solutions. Tiff ord (1964) conducted an exploratory analytical study that is applicable to a flat surface. Goldstein (1965) considered a flat plate that is oscillating and oriented parallel to the two-phase stream direction. The current investigation presents a study of the heat transfer from a geometrically simple surface exposed to a two-phase two-component stream. A wedge-shaped body is analyzed, since it is subjected to a positive uniform droplet impingement and produces a symmetrical flow pattern. Approximate solutions are obtained by solving the integral
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Eng. Chem. Fundam., Vol. 9, No. 3, 1970
energy equation with different simplifications and are conipared with experimental findings. The analytical heat transfer solutions show the relative contribution to total heat transfer of sensible heating of the film and evaporativeconvection cooling a t the film-stream interface. Analysis
The following assumptions are made throughout the analysis: Flows in the film and the two-component stream are laminar, steady, incompressible, and involve Newtonian fluids with constant properties. Film surface tension, wall adhesion, wave, and turbulence effects are neglected. All impinging droplets are captured by the film with negligible mass loss Fy surface eruption. Time average values describe the impingement effect. The effect of evaporation is negligible in the hydrodynamic analysis (but significant in the heat transfer analysis). Far upstream, liquid droplets are a t the same temperature as the gas and travel with the same velocity in a direction parallel to the wedge center line. A droplet is assumed to move through the gas surrounding the wedge and strike the film with its upstream direction and speed. Thus, uu =
u, cosp
and -vu
=
C , sinP
Droplets are uniformly distributed in the gas. The local heat transfer coefficient is defined by
where 4” is determined from the energy equation. Neglecting x-direction conduction and viscous dissipation, an energy balance for the control volume in Figure 2 gives the integral form of the energy equation as
3
z
0
0 0
0 0
0
wDROPLETS 0 0
I
LIQUID FILM INTERFACE Figure 3. Film thickness correlation for all stream velocities and water-air flow rates
so t h a t Figure 1 .
/T0(DROPLET O
9 = ncr[-U,]X 2
Physical model
The change of momentum within the film, gravity force, and pressure force are negligible for the thin film when compared with the x-direction momentum flux of the impinging liquid, the shear stress a t the solid-liquid interface
6 GAS TEMPERATURE)
k
-\I\
(4)
IJ,
and the shear stress exerted by the gas on the liquid film, rg. The latter is assumed to be the same as that exerted on a stationary (dry) flat plate
Figure 2.
Energy balance for
a
liquid film
element
The solution of Equation 2 is based on assuming an appropriate velocity and temperature profile. A very thin (less than 0.010 inch) liquid film was observed for all flow conditions encountered, and a linear velocity profile
u
= ug
[+-I
d dx
nudy
= PI
d dx
-
~ g 6
[I]
is the flux of impinging droplets PLnU[-Uol
dx
u6 dx 6
-
--
+
76
dx
=
0
